Clark Hess1

COMMUNICATION CIRCUITS: ANALYSIS AND DESIGN

KENNETH K. CLARKE D O N A L D T. H E SS

C larke-H ess C om m unications R esearch C orpora tion F o rm erly : Polytechnic In s titu te o f Brooklyn

AADD ISO N -W ESLEY PUBLISH ING CO M PAN Y

R eading, M assachusetts • M enlo P ark , C alifornia • L ondon • D o n M ills, O n tario

This book is in theADDISON-WESLEY SERIES IN ELECTRICAL ENG INEERING

C onsulting E ditors DAVID K. CHENG LEONARD A. GOULD FRED K. MANASSE

Copyright © 1971 by Addison-Wesley Publishing Company, Inc.Philippines copyright 1971 by Addison-Wesley Publishing Company, Inc.All rights reserved. N o part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States o f America. Published simultaneously in Canada. Library of Congress Catalog Card No. 78-125610.

To our wives, Nona and Carole

PREFACE

This book has been w ritten on the basis o f a combined experience o f more than thirty years o f teaching about and working with electronic circuits o f the type used in present-day comm unications and control systems. In this book we deal neither with sem iconductor or vacuum tube m anufacture nor with overall system design, but with the understanding and use o f devices and configurations o f devices that bridge the gap between these two disciplines. Although we do not deal particularly with the problem s o f integrated circuits, many o f our results are indeed directly applicable to circuits in integrated form.

Chapter 1 offers a preview o f things to come. Chapters 2 and 3 may be considered as a review o f linear system concepts. Although the m aterial stressed in these chapters ought to be presented in linear systems courses or textbooks, it has been our experience tha t the viewpoints that we find useful are often somewhat slighted there.

Chapters 4 and 5 provide the foundation for the rest o f the book. Essentially they provide a reasonably rigorous but (we hope) intelligible account o f both the small- signal and large-signal operation o f both the single devices and the basic multiple device configurations that serve as the building blocks for all later circuits. These devices and configurations include the bipolar and field effect transistor, the differential pair, and the com bination o f resistance and reactance with these devices.

The approach taken allows one to make both large-signal and small-signal calculations w ithout any ambiguity as to the resultant distortions or nonlinear byproducts. While we did not invent all the results here, we have been using them and teaching them for some years. To our knowledge this is the first time that they have been coordinated and m ade available in one place.

Chapter 6 uses the vehicle of the sinusoidal oscillator to tie together all of the previous material. The techniques presented allow one to calculate the actual am plitude frequency and distortion of real oscillators rather than just to catalog a number o f circuits. The squegging phenom enon in oscillators is treated in a unique and readily usable manner.

C hapter 7 considers the deliberate use of the device non-linearity to produce mixers and frequency converters. It explores the amplitude lim itations upon “ linear” mixing and the effect of deliberate or accidental series resistance upon the mixing process. This chapter also examines the feedthrough and the feedback problems involved in small signal R F amplifiers and AG C systems.

C hapter 8 is concerned with multipliers and amplitude m odulators. It presents a step-by-step analysis o f the popular G ilbert integrated four-quadrant multiplier as

vii

viii PREFACE

well as a num ber o f other useful circuits. C hapter 9 discusses all types o f power amplifiers from linear broadband Class A types through both tuned and broadband Class D types. C hapter 10 explores the am plitude dem odulation problem in detail. I t presents useful design results for the com m on narrow band peak envelope detector, which is usually used in circuits bu t rarely discussed in textbooks. Chapters 11 and 12 present a large am ount o f new m aterial in their complete coverage o f FM generation and detection.

Because the general principles o f the first five chapters are applicable in some form to m ost o f the circuits in the rest o f the book, a unity is achieved tha t has often no t been apparent in past books in this field. Thus instead o f considering a seemingly endless variety o f apparently different oscillators o r detectors, one is able to group circuits into rather broad classes and show straightforward design or analysis p ro cedures applicable to all o f them.

Some o f the early versions o f this m aterial were originally pu t into riote form in 1962. All o f it, except our last-m inute revisions, has been used in various graduate and senior year courses a t the Polytechnic Institute o f Brooklyn. I t is no t reasonable to try to cover all this m aterial in a one-semester course. W ell-grounded students who can handle Chapters 2 and 3 by themselves, and who can absorb Chapters 4 and 5 in say three weeks, should be able to cover selected m aterial from the remaining chapters w ithout undue difficulty in a semester. A num ber o f selections o f coherent groups o f m aterial are possible. M ost instructors should have no problem picking ou t a set tha t is both interesting to them and instructive to their students.

Hom ework problem s are included a t the end o f each chapter. Illustrative examples are worked out in m ost chapters.

O ur form er colleagues and students a t the Polytechnic Institute o f Brooklyn deserve our thanks for their m any stim ulating criticisms and observations. Professors G erald Weiss, R onald Juels, and M arvin Panzer were particularly helpful in pointing out errors or areas in need o f clarification. A special debt o f gratitude is due to the various people who struggled with the typing and the drawings for the m anuscript and the various sets o f notes tha t preceded it.

As D epartm ent Head through m uch o f the period that the book was in preparation, Professor Edward J. Smith and the rest o f the adm inistration o f the Polytechnic were m ost kind in extending the use o f various typing and reproduction facilities.

W hile book writing is never really a pleasure, it is exciting to find a simple way to solve a heretofore difficult problem . We have had m any such exciting m om ents in preparing this book and we hope tha t the reader will be able to share some o f our excitement as he uses it.

New York M ay 1971

K . K . C. D. T. H.

C O N T E N T S

C hapter 1 Preview

1.1 Basic circu it b i a s i n g ............................................................................................. 11.2 W ideband am plifier lim its on “ sm all-signal” o p e r a t i o n .......................... 31.3 N arrow band am plifiers and l i m i t e r s .................................................................. 61.4 Frequency m u l t i p l i e r s .............................................................................................. 81.5 M i x e r s ........................................................................................................................ 81.6 Sine-wave o s c i l l a t o r s ..................................................................................................... 101.7 C o n c lu s io n s ..........................................................................................................................13

C hapter 2 B roadband and N arrow band Transform erlike Coupling N etworks

2.1 B roadband transfo rm er c o u p l in g ................................................................................. 162.2 Parallel RLC c i r c u i t ..........................................................................................252.3 Parallel L C c ircu it w ith series l o s s .......................................................................... 322.4 Parallel resonan t “ transform erlike” n e t w o r k s ......................................................382.5 Parallel resonan t t r a n s f o r m e r s ......................................................................... 482.6 Three-w inding parallel resonan t t r a n s f o r m e r ......................................................54

A ppendix to C hap ter 2: T ransfo rm er E quivalent C irc u i ts ...................................62

C hapter 3 Transm ission o f Signals Through N arrow band Filters

3.1 Low -pass equivalent netw orks fo r sym m etrical bandpass netw orks . 653.2 Im pulse and step re s p o n se .............................................................................................. 703.3 N arrow band netw orks w ith m odu la ted i n p u t s ......................................................723.4 N arrow band netw orks w ith periodic i n p u t s .............................................................783.5 T o ta l harm onic d i s to r t io n .............................................................................................. 82

A ppendix to C hap ter 3 : H igh-Q F ilter M easu rem en ts ................................. . 88

Chapter 4 Nonlinear Controlled Sources

4.1 G eneral c o m m e n ts ............................................................................................................ 9C4.2 Piecewise-linear source, single d isc o n tin u ity .............................................................914.3 M ultiple-segm ent piecewise-linear s o u r c e s .............................................................944.4 Square-law c h a ra c te r is t ic s ...............................................................................................984.5 T he exponential c h a r a c t e r i s t i c ............................................................................... 1044.6 T he differential c h a ra c te r is t ic ...................................................................................... 1144.7 O ther gradual nonlinearities— p e n t o d e s ............................................................ 1204.8 Effect o f series resistance on the exponential characteristic . . . . 1234.9 C lam p-biased square-law d e v i c e ................................................................................131

A ppendix to C hap ter 4: F ourier E x p a n s i o n s .....................................................144ix

X CONTENTS

Chapter 5 Reactive Element and Nonlinear Element Combinations

5.1 Capacitive coupling to nonlinear l o a d ...........................................................1495.2 Transient build-up to steady s t a t e ................................................................ 1595.3 Capacitively coupled transistor amplifier—constant current bias . . . 1625.4 Capacitively coupled transistor amplifier—resistor b i a s ..............................1695.5 Nonlinear loading of tuned c i r c u i t s ................................................................ 1815.6 Transfer function for low-index AM i n p u t .................................................... 195

Chapter 6 Sinusoidal Oscillators

6.1 Operating frequency and minimum gain conditions for linear-feedbacko s c i l l a to r s .......................................................................................................... 205

6.2 Amplitude-limiting m ech an ism s...................................................................... 2126.3 Frequency stab ility .............................................................................................. 2166.4 Self-limiting single-transistor o s c i l la to r .............................. 2226.5 Self-limiting differential-pair o s c i l l a to r ........................................................... 2366.6 Self-limiting junction field effect transistor o sc illa to rs ................................... 2416.7 Crystal o s c illa to rs .............................................................................................. 2436.8 S q u e g g in g .......................................................................................................... 2556.9 Bridge o s c i l la to r s .............................................................................................. 261

6.10 The one-port approach to o s c i l l a t o r s .......................................................... 2686.11 The phase plane approach.................................................................................. 2736.12 The distortion-operating frequency r e l a t i o n s h i p ........................................ 279

Chapter 7 Mixers; RF and IF Amplifiers

7.1 The superheterodyne co n c e p t............................................................................ 2937.2 Mixer te c h n iq u e s ...............................................................................................2957.3 Series resistance in m ix ers .................................................................................. 3027.4 Practical mixer circuits........................................................................................ 3057.5 Semiconductor converter c ircu its ...................................................................... 3117.6 Tuned narrowband small-signal a m p lif ie rs .................................................... 3147.7 Stages with double-tuned circuits ...............................................3287.8 Gain control c i r c u i t s ........................................................................................ 3317.9 Noise, distortion, and cross modulation .................................................... 336

Appendix to Chapter 7: Comparisons of -Parameters for Bipolar Transistors: Single-Ended, Differential-Pair, and Cascode ...................................345

Chapter 8 Amplitude Modulation

8.1 Amplitude modulation s i g n a l s ...................................................................... 3478.2 Amplitude modulation t e c h n iq u e s ................................... 3538.3 Practical analog modulators and m u l t i p l i e r s ...............................................3628.4 Practical chopper m odulators............................................................................ 3768.5 Square-law m o d u la to r................................... 3848.6 Tuned-circuit m o d u la to rs ............................................................................. 387

CONTENTS XI

C hapter 9 Pow er Amplifiers

9.1 “ Ideal” pow er am plifiers— class A , s i n g l e - e n d e d ...............................................4019.2 C lass B linear R F a m p l i f i e r s ......................................... 4059.3 C lass C “ linear” a m p l i f i e r s .......................................................................................4089.4 R F class C a m p l i f i e r s .................................................................................................... 4109.5 N arrow band class D pow er a m p l i f i e r s ...................................................................4159.6 B roadband class B a m p l if ie r s .......................................................................................4219.7 B roadband class D pow er a m p lif ie rs ................................................................... 4269.8 P ractical pow er a m p lif ie rs ............................................................................................. 4329.9 High-level am plitude m o d u la t io n ................................................................................ 447

A ppendix to .C h ap te r 9: Pulse T ra in E x p a n s i o n s ..............................................454

Chapter 10 Amplitude Modulators

10.1 A m plitude dem odulation te c h n iq u e s ......................................................................... 45710.2 P ractical average envelope d e te c to r s ......................................................................... 46810.3 N arrow band peak envelope d e te c to r ......................................................................... 47810.4 P ractical narrow band peak envelope d e t e c t o r ..................................................... 49610.5 B roadband peak envelope detec tor . 498

Chapter 11 Generation o f FM Signals

11.1 Frequency-m odulated s i g n a l s .......................................................................................50911.2 T ransm ission o f FM signals th rough nonlinear n e tw o r k s ........................ 51511.3 T ransm ission o f F M signals th rough linear f i l t e r s ......................................52111.4 Frequency m odulation techniques— the F M differential equation . . 52611.5 Q uasi-static frequency m o d u l a t i o n ......................................................................... 53211.6 Triangular-w ave frequency m o d u l a t i o n ...................................................................54211.7 P ractical square-wave frequency m o d u l a t i o n ..................................................... 55311.8 M iscellaneous frequency m odula to rs— the A rm strong m ethod . . . 55911.9 F requency stabilization o f frequency m o d u l a t o r s ...............................................562

Chapter 12 FM Modulators

12.1 L im ite rs .................................................................................................................57112.2 F requency-dem odulation te c h n iq u e s ......................................................................... 57712.3 D irect differentiation— the C larke-H ess frequency dem odulator . . . 58612.4 F requency-dom ain differentiation— the slope dem odulator . . . . 59312.5 T im e-delay differentiator, tim e-delay dem odulator, Foster-Seeley de

m odu la to r, and ra tio d e te c to r ...................................................................................... 60212.6 Pulse-count frequency d e m o d u l a to r ......................................................................... 61812.7 M ore exotic F M detectors— the phase-locked loop, the frequency-locked

loop, and the frequency dem odu la to r w ith f e e d b a c k ....................................... 623

Appendix: Modified Bessel Functions................................................................ 636

Answers to Selected P ro b lem s......................................................................................645

Index ................................................................................................................ 651

C H A P T E R 1

PREVIEW

The purpose of this chapter is no t to reduce the excitement o f the book by “ revealing the p lo t.” W e wish rather, by using one particular circuit as a vehicle, both to indicate a num ber of the techniques th a t we will explore later in detail and to dem onstrate to the reader the pow er o f these m ethods. W e will show how we m ay ra ther easily get a clear insight in to the design o f such apparently diverse circuits as w ideband small- signal amplifiers, large-signal narrow band amplifiers, frequency m ultipliers, active limiters, active mixers, and tuned-circuit sine-wave oscillators. By doing so, we hope to provide a fram ework for the general developm ents that follow and to share with others the enthusiasm tha t comes from being able to solve m any heretofore difficult design and analysis problems.

In this chapter, because of its nature, we cannot develop all results o r answer all questions. W e tru st tha t the unanswered questions will receive adequate treatm ent at a later point.

1.1 BASIC CIRCUIT BIASING

The circuit tha t we shall use as a skeleton upon which to construct ou r various examples is shown in F ig 1.1-1.

This circuit is shown in the m anner in which it m ight be constructed in integrated form. The sole purpose of the lower two transistors is to provide a constant current

Fig. 1.1-1 Basic junction transistor amplifier.

2 PREVIEW 1.1

bias source for transistor 1. (T ransistor 3 m ight be viewed as a d iode; however, in integrated circuits diodes are norm ally constructed as transistors.)

O ur key assum ption is that the em itter current and the base-em itter voltage of the transistors are related by Eq. ( l . l - l ) t :

iE = l ESe ^ lkT, (1.1- la )

k T ivBE = — (1.1- lb )

11 h s

where k = 1.38 x 10~23.//oK is B oltzm ann’s constant, q = 1.6 x 10~19C is the electronic charge, and ¡ES is the em itter saturation current.

Let us m ake a further set o f assum ptions: that ic = <xiE and iB = (1 — a )iE, and tha t a is both close to unity and independent of iE. (The assum ption of a constant alpha is rarely true if iE varies over a wide range; however, if alpha approaches unity, then this variation is norm ally a second-order effect.)

Since I ES is of the order of 2 x 10“ 16 A for small silicon integrated circuit transistors and since kT /q % 26 mV at norm al room tem peratures (T = 300°K), Eq. (1 .1-lb) m ay be em ployed to determ ine the required values of vBE (or for the case of a bias voltage) to produce various values of iE (or I E). Several values of Vbe vs. I E are presented in Table 1.1-1. It is apparen t that varies only slightly for large variations in 1E ; hence in many applications VBE may be approxim ated by a constant of approxim ately f V.

Table 1.1-1 Value o f VBE required forvarious values of I E

VB E,mV / E,m A

700 0.1760 1820 10880 100

As connected in Fig. 1.1-1, transistors 2 and 3 m ust have the same value forvBE (or VBE). If they occupy the same area and are on the same chip, they will havealm ost identical values f o r /ES. Therefore, iE2 = iE3 or, for biasing purposes, 1F2 = I E3. Now I = (VEE — Vbe) /R b . If VBE is approxim ated by 3/4 V (so long as Vee » Vbe this is reasonable), then IRb is known. However, l Rg = 1E3 + (1 — a)IE2 or

, _ Vee ~ 0-75 _ VEE - 0.75“ (2 - , ] R . --------- R , ’ < U - 2)

t A somewhat m ore accurate representation would be

<E = ¡es e9yVBBlkT,where j < y < 1 depending on the transistor material, i.e., germ anium o r silicon. In any situation which w arrants it y may be included w ithout affecting any of the derived results.

1.1 BASIC CIRCUIT BIASING 3

and thus

So long as Z E contains a series capacitor (no dc path), then I E1 = I C2 and the upper transistor is biased at a constant current level.

1.2 WIDEBAND AMPLIFIER LIMITS ON “ SMALL-SIGNAL” OPERATION

Let us first consider the case where Z L is a resistor R L, Z E is a capacitor CE, vt = Vl cos cut, l/a>CE approaches an ac short circuit, and o is low enough so tha t transistor reactances may be ignored. We assume that Vcc and are large enough so that the collector-base junctions o f both transistors 1 and 2 always stay reverse biased.

Since CE is an ac short circuit, appears directly across the em itter-base junction o f transistor 1. In addition , any dc voltage K.C which is developed across C E appears across the jun c tio n ; hence vBEi = v, 4- Vdi;. W hen vt is zero, iE is forced to be equal to Ic 2 ; hence

(The subscript Q denotes the quiescent value of a param eter.)For the case where u, is no t equal to zero, Eq. (1 .1 -la) may be employed to obtain

where x = V, q/k T to norm alize the drive voltage. Now from a know n Fourier series expansion,

where I„(x) is a modified Bessel function o f the first kind, of order n and argum ent x. (Properties o f these tabulated functions as well as further references concerning them will be found in the Appendix a t the back of the book.) The modified Bessel functions are all m onotonic and positive for x > 0 and n > 0 ; / 0(0) is unity, whereas all higher - order functions start a t zero. As x -> 0,

_ [Es[ev<'^¡kr]e{v ,qlkl) cos

= I ES[eVa‘:q|kT]excoso,,, ( 1.2- 1)

00

excosm‘ = I 0(X) + 2 £ In{x) cos mot, (1-2- 2)

when n is a positive integer.C om bining Eqs. (1.2-1) and (1.2-2), we obtain

4 PREVIEW 1.2

It is apparen t from Eq. (i.2 -3) th a t the average (or dc) valoe of iE is given by

iE = I ESey<icq,kT 10(x) (1-2-4)

However, the biasing circuitry dem ands th a t iE — I c2 ; hence iE m ay be w ritten in the simplified form

(1.2-5)

( 1.2-6)

Table 1.2-1 presents several sets o f da ta concerning the modified Bessel functions that will be of interest to us. F rom the first colum n of this table we see th a t if -- 260 mV, so that x = 10, then the dc voltage shifts by 206 mV from its g -p o in t value. We can also see from the o ther colum ns that the peak value of the fundam ental com ponent of the collector current o f transistor 1 is 1.9/c2, while the percentage second-harm onic distortion in this current is 85%.

Table 1.2-1

X l n / 0(x)2 / 1(x)

/„(*)

¡i(x)/,(x )

0 0.000 0.000 0.0000.5 0.062 0.485 0.1241 0.236 0.893 0.2402 0.823 1.396 0.4335 3.30 1.787 0.719

10 7.93 1.897 0.85420 17.6 1.949 0.926

A pparently a 260 mV peak sinusoidal signal is not a small signal at all from the viewpoint o f this amplifier. The limits of small-signal operation are m ade clearer by a study of Figs. 1.2-1 and 1.2-2. Figure 1.2-1 shows that the ou tpu t fundam ental is only roughly linearly p roportional to the input voltage, or equivalently x, for x < 1. However, to keep / 2(x)//i(x), which is the percent second-harm onic distortion, below .025 (2j % distortion), it is necessary to keep x below 0.1.f Consequently, for small-signal operation Vl < 2.6 mV or equivalently It^l ^ 2.6 mV.

It is apparent from Eq. (1.2-1) that the em itter current and, in turn, the collector current of transistor 1 are proportional to excm‘°,/ex for any fixed value of x. (We

In addition, Kdc may be obtained from Eq. (1.2-4) to be of the form

k T , I C2 k T I C2 k TV<* = — ln T 7 n = — ln 7^ ln Ioix)

<7 I e s I o M <1 I e s Q

k T ,— KicQ In I oM-

t F or small values of x, l 2(x)/Ii(*) * x/4 [cf. Eq. (A-2) in the Appendix a t the back of the book].

1.2 WIDEBAND AMPLIFIER LIMITS ON "SMALL-SIGNAL” OPERATION 5

x=Vxq!kTFig. 1. 2-1 Functions of modified Bessel functions vs. the normalized param eter x.

mt -Fig. 1.2 2 Normalized collector currents vs. angle for exponential junction driven by sine wave.

incorporate the ex term in the denom inator for norm alization purposes only.) Consequently, the plot of excmm‘/ex shown in Fig. 1.2-2 yields a norm alized picture of the collector current as a function of time over one cycle of the input voltage i;, = Vl cos u>t. C learly by the tim e x = 10, the collector current is flowing in narrow pulses approxim ately { cycle w ide ; hence the operation of the amplifier is certainly not linear. In fact, as x increases above one, the overall current waveshape rapidly ceases to be cosinusoidal. F o r larger values of x the dc bias shift effectively aids the signal in holding the base-em itter junction off for a good portion of the cycle.

W ith |u,| < 2.6 mV the ou tpu t voltage of the amplifier takes the form

vo(0 — Vcc ~ ‘c^-l ~ Vcc ~ c iR-lr „ 2 / i ( x )

<*ic2R L - r r v cos 0)1 ■ /o M(1.2-7)

6 PREVIEW 1.3

F or small values of x, however, 2 / 1(x )//0(x) » x = V^q/kT ; consequently,

V o ( t ) = V CC - * l C l R L - g m Q R L V l COS COt, (1 .2 -8 )

where gmQ = <xIC2q /k T is defined as the transconductance of the transistor. N ote tha t the value of gm is exactly th a t value which would be obtained as the increm ental ra tio o f collector curren t to base-em itter voltage evaluated abou t the Q-point ; that is.

d i r

dvnd i F

= ct-ÌC-0-ÌC2 dvB E ÌE — IC2 k T êmQ> (1.2-9)

where iE = IESeVBEqlkT. Thus for |t>;| < 2.6 mV, classical small-signal analyses may be employed.

As we shall see in a later chapter, one way to extend the broadband linear signal handling capacity of a transistor amplifier is to include an unbypassed em itter resistor. In the circuit under discussion this resistor R E would be placed in series with C E. Such a resistor reduces the fundam ental gain of the stage by a factor of

1 11 + R ^ J C2qlkT) 1 + g mQR E

where gmQ is the small-signal (7-point transconductance with R E shorted.The effect of this series resistance is to linearize the characteristic so that, while

it does reduce the fundam ental gain, it reduces the harm onic distortion even more rapidly.

As a practical m atter one should note tha t all of the foregoing discussion would be unchanged if the v( generator were included in series with C E while the base of transistor 1 was grounded.

1.3 NARROW BAND A M PL IFIE R S AND L IM ITER S

A different approach to utilizing the circuit o f Fig. 1.1-1 would be to again let Z E be a single capacitor C E and let Z L be a parallel R L C circuit tuned to the frequency co

Fig. 1.3-1 Circuit of Fig. 1.1-1 with Zi c ircu itwith Zl replaced by a tuned

lit/

o +v„(i) Qt=(o0RlC

9 - <D0= y / \ j L C = CO

- ± r u)0=resonant frequency•>, COS '•>>

h 2\ i ) - T 'Q (short circuitV W at «do)

1.3 NARROWBAND AMPLIFIERS AND LIMITERS 7

of the input signal as shown in Fig. 1.3-1. For a parallel R L C circuit, the m agnitude of the im pedance a t the fundam ental frequency, Z L(jcà), is, in general, greater than the m agnitude of the im pedance a t the nth harm onic, \ZL(jna>)\ ; in particular,

I Z L(jnoj)\ _ n\Z L(joS)\ ~ (n2 - 1 )Qt ' U '

where Q T is the Q of the resonant circuit. Therefore, if QT is sufficiently high, we can obtain alm ost a pure sine-wave ou tpu t voltage v0(t) in spite of large harm onic com ponents in the collector current of transistor 1.

For example, if x = 5, then Qr = 48 reduces the second-harm onic output voltage com ponent to 1 % of the fundam ental and the third-harm onic voltage com ponent to 0.31 % (the collector current has d istortion com ponents of 72 % and 40% respectively). Therefore, as a good approxim ation, the ou tpu t voltage v0(t) may be written as

!«(0 = Krr — °^C2^L“; ~r \ cosa)^ (1-3-2)I 0(x >

where R L = Z L(jw ) is the im pedance of the parallel R L C circuit a t resonance. Since the im pedance of the resonant circuit to dc is zero, no dc voltage is built up across Z L.

In this case, instead of the small-signal transconductance gmQ, it is convenient to define a large-signal average transconductance Gm which is equal to the ratio of the fundam ental collector current I c l l to the fundam ental driving voltage V1 :

r _ r — ^c i l — a^c2 2î(x) _ 2/i(x)m ^ m ( ^ ) w, I i \ ë m Q r i ( 1*3 3 )Fi Kj / 0(x) vx /0(x)

With this definition for Gm(x), t>0(i) may be written in the equivalent form

v„(t) = Krc - G J x ) R LVi cos wt, (1.3-4)

which is similar in form to the ou tpu t o f the small-signal amplifier. The basic difference is that Gm(x) is a function of V1 (or x) and no longer a constant.

Figure 4.5-6 presents values for Gm(x)/gmQ for various values of x. F rom Fig.4.5-6 we see that Gm is dow n 1 dB from its x = 0 value when x = 1 ; hence, though the harm on 'c distortion has been removed, the amplifier can operate only in an approxim ately “ linear” fashion w ith input am plitudes below 26 mV peak. By linear, in this case, we m ean that there is a constant ratio between input and output signal levels, and tha t this ratio is independent of signal level ; this is necessary if an AM wave is to be amplified. If we wish to handle larger input signals in a linear m anner, then the unbypassed em itter resistor again provides the means.

If, on the o ther hand, we want to remove am plitude variations in V1 from the output, i.e., if we wish to produce a “ lim iter,” then we need only increase x. F rom Tablé 1.2-1 or from Fig. 1.2-1 we note tha t as x increases, 2 /1(x )//0(x) approaches a saturation value of 2; hence va(t) given by Eq. (1.3-2) reduces to

vo(0 - vcc ~ <x2IC2R l cos o>t, (1.3-5)

which is clearly independent of variations in V1.

8 PREVIEW 1.4

As an example, we consider th e 'case where V'j varies between 130 mV and 520 mV (x varies between 5 and 20) because of a spurious am plitude m odulation. If we define the m odulation index as

Vi ~ Vi ,j jj 1 m a x Am i n

K + v i ,A m a x 1 m i n

then the input m odulation index is m = 0.6 (or 60%). Since for x = 5,21 ,(x)/70(x) = 1.787, and for x = 20, 2 / 1(x )//0(x) = 1.949, and since the am plitude of the ac com ponent of v0(t) is proportional to 2 /1(x )//0(x) [cf. Eq. (1.3-2)], the ou tpu t m odulation index is

m° = i! 9 ~ = 00435 (° r 435 %)•0 1.949 + 1.787 /0

A further stage driven with this signal at a norm alized level such that x > 10 could reduce the ou tpu t m odulation below 0.05 %.

1.4 FREQUENCY MULTIPLIERS

As we saw in Figs. 1.2-1 and 1.2-2, as x increases, the harm onic com ponent of the collector current increases. F o r / x = 1 0 , J2(x ) / / l (x) = 0.85, I 3(x)/I t (x) = 0.66, U x V M x ) = 0.46, and I ^ x j / I ^ x ) = 0.29. Therefore if we tune the output-tuned circuit to a harm onic o f the input, we can obtain an appreciable voltage at least up to the fifth harm onic for an input drive of 260 mV (x = 10). (For x = 2 0 ,15{x)/1{(x) has increased to 0.54.) Such circuits are know n as frequency multipliers. They are widely used to ob tain a higher frequency from a stable crystal oscillator or, in FM systems, to increase the ou tpu t F M deviation. Specifically, if the parallel R L C circuit is tuned to the nth harm onic of the input, v j t ) is given by

vo(t) = vcc - cos nu>t. (1.4-1)Io(x)

1.5 MIXERS

So far we have driven the junction of transistor 1 w ith a single-frequency cosinusoid. Let us now consider the case where i\{t) = Vt cos a ^ t + g(t) cos a>2t. The signal at frequency co1 m ay be thought o f as a local oscillator signal in a superheterodyne receiver; g(t) cos co2t may be thought of as a low-level received am plitude-m odulated (AM) signal which we wish to translate to the interm ediate frequency (IF) of the receiver. If we again note tha t for transistor 1 in Fig. 1.1-1 (with Z E = C E) the base- em itter voltage is given by vBE = v{ + Vdc, then we may write the em itter current in the form

iE = /^ ^« /k T ^x c o sm iig fo g lO /fcT lc o sw jr ( J 5_ | j

If we assum e |g(f)| < 2 .6 m V , then gi«*(‘)/*r]cos<»2r m ay jje approxim ated by 1 + [qg(t)/kT] cos co2t. In addition, if we replace excosa>'' by its Fourier series, Eq.

1.5 MIXERS 9

(1 .5 1) s im p lif ie s to

cos a jji +

cos a>2f (1 .5 -2 )

Finally, by noting th a t cos A cos B = ¿{cos (A - B) 4- cos (A + B)], we may rewrite iE in the form

Hence we have generated AM waves w ith envelopes p roportional to the input envelope g(r) at frequencies co1 — co2, + (o2, 2a>! — <y2, 2col + a)2, etc.

If we now choose Z L as a parallel R L C circuit tuned a t — a >2 with a value of Qt sufficient to remove o ther frequency com ponents from the ou tpu t [but not so large th a t the envelope inform ation of g(t) is filtered], then the ou tpu t takes the form

Clearly the input AM wave has been translated in frequency from a>2 to uj1 - to2. By choosing the oscillator frequency a jt correctly we can shift (or mix) the input AM wave to any desired interm ediate frequency.

The quantity gc, which may be interpreted as the ratio of the envelope of the collector current a t the frequency oj1 — a>2 to the envelope of the input voltage at frequency ca2, is called the conversion transconductance. Since / i(x )//0(x) increases m onotonically tow ard an asym ptote o f unity for large values of x (or equivalently Fi), it is apparent tha t gc is optim ized by choosing Vy greater than 260 mV (x > 10). For this case gc % gmQ and the m ixer not only translates in frequency but also amplifies.

As an exam ple of this fact we consider the case in which g(i) = (1 mV) (1 + cos o)mt) cos (o2t (where com « a>2), R L = 10 kQ, / C2 = 2.6 mA, Vl = 260 mV, a w 1, and Z L is a parallel R L C circuit tuned at cot - co2. Clearly then for this circuit

(1.5-3)

= *C C - g c ^ L g ( 0 CO S (C «! - (0 2 ) t ,

where

k T /„(*) 6mC/ 0(x)-

10 PREVIEW 1.6

gc = (0.1)(0.948) m ho = 0.0948 mho. Consequently, the ou tpu t voltage is

v0(t) = Vcc — (0.948 V)(l + cos w mt) cos (coj — aj2)t.

The ou tpu t signal is shifted in frequency and amplified by a factor of almost 1000.It is interesting that mixers of this form are employed in all superheterodyne

receivers, that is, in m ore than 99% of the w orld’s receivers of any kind.

1.6 SINE-WAVE OSCILLATORS

To operate the mixer we required a local oscillator; hence every superheterodyne receiver requires an oscillator. At the same time every transm itter also requires an oscillator. The first-order characteristics of an oscillator are its waveshape, its frequency, and its am plitude. Second-order characteristics are the frequency and am plitude stability with changes in time, tem perature, voltage, and physical movement.

To set the frequency of a sine-wave oscillator we connect it in to a feedback loop so that positive feedback of exactly 360° is possible only at the desired frequency. To build frequency stability into it we concentrate m ost of the phase shift vs. frequency dependence into one portion of the circuit (often a quartz crystal or a high-Q tuned circuit). The oscillator often attains its desired am plitude by reaching a balance between the feedback allowed by the passive portions of the circuit and the nonlinear gain oifered by the active portion of the circuit (the transistor in the case we are about to consider).

Figure 1.6-1 shows a sine-wave oscillator circuit constructed from the basic circuit o f Fig. 1.1-1. F o r this circuit a 360° phase shift around the loop is possible only in the vicinity of the tuned circuit resonant frequency co0 ; hence, if an oscillation occurs, it has a frequency of approxim ately ut0. Let us now assume th a t the resonant circuit has a high QT ; then, if the circuit oscillates, the voltage across it is alm ost

Fig. 1.6-1 Sine-wave oscillator.

1.6 SINE-WAVE OSCILLATORS 11

sinusoidal even if the collector current flows in narrow pulses. If we assume also that this voltage, v, = V, cos tu0f, is stepped dow n by the capacitance ratio n = C\/(Ci + C2) (cf. C hapter 2), then a sinusoidal drive voltage of the form Vl cos a>0t appears at the em itter o f transistor 1, where = nVt. This em itter voltage, in general, causes a nonlinear, pulselike collector current.

We dem onstrate in Section 5.5 tha t the loading o f the transistor em itter junction upon C 2 is equivalent to a resistance of a/Gm{x). In addition, we dem onstrate that this loading m ay be reflected across the inductor L a s a conductance of n2[Gm(x)/a], where n is again C J (C i + C 2). Consequently, the total effective conductance appearing across the inductor is

GT = G, +n2Gm(x)

According to the B arkhausen criterion, for a sustained sinusoidal oscillation at u>0, A L(jo)0) = 1, where A L(j<o) is the loop gain. To evaluate the loop gain we break the loop a t the em itter, apply a signal o f the form Vl cos co0t to the em itter, term inate the broken loop in a resistance of a/G m(x), and determ ine the signal across the term ination of the loop. The broken loop is shown in Fig. 1.6-2. The capacitor C£, which has no effect on the calculation of A t ( jw 0) since it is an ac short circuit, is incorporated to preserve the dc bias conditions.

Now, with the loop broken, the oscillator reduces to a narrow band amplifier for which we may write

v, =friGw(x) cos (Opt

G l +n G J x )

( 1.6- 1)

12 PREVIEW 1.6

Since the two capacitors act as a step-down transform er with ratio n,

n V ,G J x ) cos a>0t(1 .6 -2 )

(1 .6 -3 )

Stable sinusoidal oscillations occur at co0 for

or equivalently

Equation (1.6-4) specifies the value of Gm{x) required by the passive portion of the circuit. The am plitude m ust now adjust itself so that the transistor supplies this Gm(x). If I C2 is known, then gmQ = IC2q /k T follows, and from Fig. 4.5-6 we may determ ine the x tha t corresponds to the required Gm.

For example, if C y = 100 pF, C 2 = 11,200 pF, and R L = 13.7 kQ, then

It is quite obvious that Vcc m ust exceed 12 V if collector-base saturation is not to occur in transistor 1. So long as Vcc > 12 V, the previous am plitude is the am plitude a t which the circuit stabilizes. In addition, in this case,

n = 0.00885 and G J x ) « — = 8300 ¿¿mho.n

If, in addition, I C2 = 0 .5 mA, then

gmQ = 19,200 /¿mho and Gm(x)/gmQ = 0.432;

hence from Fig. 4.5-6,

13.7,

which is not nearly as high as we would norm ally want the Q of an oscillator to be. However, even in this relatively low -g case we have only 5% second-harm onic voltage across the tuned circuit; hence our assum ption of a pure sine-wave drive

1.7 PROBLEMS 13

was no t bad. We shall show later th a t such a circuit oscillates w ithin tw o parts in a thousand of the nom inal center frequency of the tuned circuit alone.

1.7 C O N C LU SIO N S

Now that we have seen some of the possibilities of this simple circuit, we shall go back and examine both a num ber of passive circuits and a num ber of o ther nonlinearities in detail. Then we shall return to explore in more depth each of the circuits discussed here, and also to discover m any other circuits of im m ediate interest to the com m unication or control system designer. Before we plunge into nonlinear controlled sources and then into circuits, we devote a chapter to passive transform erlike networks and a chapter to the response of narrow band filters to m odulated signals. There are no review chapters on basic electronics, for instance, on biasing small- signal amplifiers. For readers who feel deficient in such areas, some suggested background reading is listed below.

SU GGESTED BACKGROUND READING IN ELEC TRO N ICS

Angelo, E. J.„ Jr., Electronics : FET ’s, BJT's and Microcircuits, M cGraw-Hill, New York (1969).

G ray, P. E., Introduction to Electronics, John Wiley, New York (1967). A 325-page paperback developed for an introductory course in electronics. A bout half of the book is devoted to physics, diodes, and diode circuits, one-third to junction transistors, and one-sixth to field effect tran sistors and vacuum tubes.

PR O B LEM S

Problems 1.1 through 1.5 are all based on the circuit of Fig. 1.1-1.

1.1 Supposing tha t | VEE\ = 3 V, RB = 3 k fi, all alphas = 0.98 and all transistors are identical, silicon, and have / ES = 2 x 10 16 A, find / C1. If Vcc = 4- 10 V, then determ ine the value of Rl th a t can be used to replace Z L so tha t the ou tpu t dc voltage level will be + 5 V. W hat is the approxim ate power dissipation in each transistor for this case?

1.2 Suppose Z E is replaced by an ac short circuit in the circuit of Problem 1.1. Sketch v0 for the cases where vl is a pure sine wave having peak amplitudes of 1 mV, 2.6 mV, 26 mV, and 260 mV. (In Section 5.3 this case is considered in d e ta il; only a reasonable estim ate of the output is required at this point.)

1.3 Suppose Z E is replaced by a 100 i i resistor in series with an ac short circuit. Repeat Problem1.2 for the cases in which Vi has a peak am plitude of 1 mV and o f 260 mV. C om pare the

14 PREVIEW

v(/)= (260 mV) cos 107/ T = 2 T C , P=98

Y„e = 0.7 V

Figure l.P-1

results with the previous problem. (The second case is not triv ial; the solution is covered in detail in C hapter 5.)

1.4 Repeat Problem 1.2 for the case where R, is shunted by a parallel LC com bination tuned to the resonant frequency of the input sinusoidal signal. Does saturation occur?

1.5 Repeat Problem 1.2 for the case where R L is shunted by a parallel LC com bination tuned to the second harm onic o f the input sinusoidal signal and com pare the results with those of Problem 1.4.

1.6 For the circuit shown in Fig. l.P -1 , determine an expression for vj t ) (Q2 and Q3 are identical).

1.7 F or the circuit shown in Fig. l.P -2 , determ ine the quiescent values of iEl , vEBl, and v0 when IESl = 10~ 13 A, IES2 = 2 x 10~13A, and IES3 = 1.5 x 10~13A.

1.8 F or the circuit of Fig. l.P -2 , determine v0(t) where = (1 m V )cos 106f and IESl = l ES2 = IES3 = 10 13 A.

PROBLEMS 15

Figure 1 . P 2

C H A P T E R 2

BROADBAND AND NARROWBAND TRANSFORMERLIKE COUPLING NETWORKS

In this chapter we explore the similarities am ong a num ber of passive netw orks all of which have widespread practical application. All of these netw orks have the property o f being able to transform im pedance levels and hence voltage and current levels. Initially we consider a b roadband transform er, and in later sections we show how a num ber of practical circuits may be reduced to the com bination of a parallel R L C circuit and an ideal transform er. T hroughout the chapter, em phasis is placed on plausible approxim ations, usually based on a consideration of the pole-zero diagram for the circuit in question.

The reader may question the necessity of such a chapter, since he has undoubtedly already had one o r m ore courses in netw ork theory o r linear circuits. W e have included the chapter because it has been our experience tha t such courses—or the textbooks used in them — rarely bring ou t the similarities in the circuits discussed here or m ake evident the approxim ations tha t suffice to simplify them. It is ou r aim in later chapters to com bine these circuits with various nonlinear elements to m ake useful circuits. Before undertaking this com bination it seems wise to have a thorough familiarity with the individual pieces.

The reader eager to get on to com plete circuits might exam ine the equivalences shown in Table 2.5-1 and the illustrative examples at the end of Section 2.4. If these are all “old h a t,” then we urge him to push o n ; if not, we recom m end this chapter as a foundation for later work.

2.1 BROADBAND TRANSFORMER COUPLING

In this section we study the frequency and tim e-dom ain properties of a linear network consisting of a resistive load coupled to a driving voltage source by m eans of a b ro ad band transform er as shown in Fig. 2.1-1. Such networks are useful for providing dc isolation and the possibility of phase inversion between the input and the o u tp u t; they are also employed when the load resistor m ust be scaled in value to “m atch” the driver over a b road band o f frequencies. F or example, a transistor power amplifier m ight require a 200 i2 resistive load over the frequency range of 20-20,000 H z in order to deliver a required am ount of pow er w ithout exceeding its m axim um voltage, current, and pow er ratings, whereas the speaker it has to drive m ight have an im pedance of 8 Q ; hence transform er coupling is required. T ransform er coupling is also employed where the load resistor m ust “ float” referenced to the input voltage source both for dc and for ac signals.

16

2.1 BROADBAND TRANSFORMER COUPLING 17

Fig. 2.1 -1 Transform er coupling network.

Since our objective in this section is to gain familiarity with the basic operation of the transform er as a coupling element, we neglect second-order effects such as winding capacity and core nonlinearities in our analysis. In addition, we model the resistive losses in the transform er as small resistors in series with the input and output terminals. This m odel is quite reasonable where core loss is not excessive in com parison with winding loss, as is the case in m ost commercial broadband transformers.

The transform er model m ost useful for analyzing broadband coupling networks is shown in Fig. 2.1-2, where r l and r2 represent the transform er loss. The equivalence

Fig. 2.1 -2 Transform er model replacing transformer of Fig. 2.1 -1.

of this model, as well as o ther possible models, and the original transform er is explored in the appendix and in the problem s at the end of the chapter. The model explicitly indicates the cause of the loss of high- and low-frequency transm ission. In particular, at low frequencies the im pedance of L b - k 2L { approaches zero and shunts to ground the signal path to R L; and at high frequencies the impedance of L a = (1 — k1)L l approaches infinity and thus opens the signal path to R L. However, if L h » La (or equivalently k x 1), a frequency range exists where coLh is large in com parison with the im pedance it shunts, while coLa is small in com parison with the impedance in series with it. Over this range, which we call the m idband (cf. Fig. 2.1-7), the inductances La and L h may be approxim ated by short and open circuits; this yields the simplified model shown in Fig. 2.1-3 for the netw ork of Fig. 2.1-1.

18 BROADBAND AND NARROWBAND COUPLING NETWORKS 2.1

ky/Lt

Fig. 2.1-3 Midband model for transformer coupling network.

W ith the aid of the m idband model, we first observe tha t for the usual case where r2 « R l and r l « R s, an im pedance of n2R L is presented to the driving source. Therefore, by choosing n2. = k 2L 1/ L 2 appropriately, we may obtain any load resistance required by the driving source. F o r k x 1 and L l and L 2 wound on the same core,

•2 L 1 N"

where and N 2 are the num bers of turns in the windings of and L 2 respectively; hence in this case n may be related to the physical turns ra tio of the transform er.

Second, we observe th a t in the m idband n may be chosen to maximize the voltage across R L for the case where v,{t), R s, and R L are fixed. S ituations of this type arise when a transducer, such as a phonograph pickup, with a high source im pedance (R s) and a fixed developed signal (vt) m ust be coupled in the m idband to an amplifier with low input resistance (RL). W riting the m idband transfer function in the form

v M . nRL717vt(t) r l + R s + n (r2 + R L)

and equating d H J d n with zero, we obtain the value of n which maximizes H m :

(2.1-1)

nm = J{R's + r M R i . +

W ith this value of n = nm, the m idband transfer function is given by

R l

2 s / ( R s + + r2)

(2.1-2)

(2.1-3)

The value of n = nm given in Eq. (2.1-2) is intuitively reasonable as the value which produce« m axim um signal to RL since it yields a f t the tranpfom ier inputterm inals which is equal to the source resistance R s + r t . Such a m atch ensures m axim um power into the transform er and thus into R L.

It is apparent from Eq. (2.1-3) tha t the existence of transform er loss reduces the signal available to R L. To ob tain a better m easure o f this signal attenuation , we assum e th a t v,{t) is of the form Vi cos cot, where co is som e m idband radian frequency,

2.1

<20--------1 I-------- O b

BROADBAND TRANSFORMER COUPLING 19

a ' O

500 nFig. 2.1 A Typical transform er specification. / , - 2 0 Hz, f 2 15,000 Hz

and that n = n„, and then com pute the ratio of the average pow er delivered to R , to the average pow er delivered to R L with = r2 = 0. This ratio, which is a m easure o f efficiency r\, is g iv en by

Clearly then, unless r x « R s and r2 « R L, m uch of the available signal pow er is not supplied to R l .

The transform er m anufacturer usually indicates what m inim um values of R s and R l ensure R L » r2 and R s » r 1 in his specification of the turns ratio. Typically, the specification appears in the form shown in Fig. 2.1-4, which is interpreted to imply that if a 5 Q resistor is connected across term inals b and b', 500 Q is “ seen” at term inals a and a' in the m idband range extending from 20 H z to 15,000 H z; hence n = ^ 5 0 0 /5 = 10. In addition, the fact that R L = 5 Q and R s = 500 Q ensures R, » r2 and R s » r , , usually to the extent that rj > 0.8. Using smaller values of R L does not alter the turns ratio n but does decrease the efficiency rj and alter the m idband frequency range.

To extend our analysis beyond the range of the m idband, we obtain the transfer function H(p) = V0(p)IVx(p) for the circuit of Fig. 2.1-2 in the form

V \ R l2R l 4(/?s + r i }(Rl + r2)

V \ R l2R L 4R SR L

nR L

H(p) = (2.1-5)'p 2 + p

where R a = R s + R b = n2(RL + r 2), L a = (1 — k 2) ! . ^ and L b = /c2L , . Since all R L (or RC) netw orks have their poles on the negative real axis, the pole-zero


Fig. 2.1-5 Pole-zero diagram of H(p).

diagram for H(p) takes the form shown in Fig. 2.1-5, where and p 2 are the roots of the denom inator of H(p). In general, the expression for p t and p 2 is quite com plicated ; however, for the broadband transform er where L b » La, and in turn p 1 » p 2, simplified approxim ate expressions for p , and p2 may be obtained.

We note from Eq. (2.1-5) that the sum and product o f the roo ts of the denom inator of H{p) are given by

Pi + Pi = ~ and PiPiR aRbL T h

As L b increases relative to L a, p , + p2 approaches a constan t value while p tp2 approaches zero. But p ip 2 can approach zero only if one of the poles, p2 in this case, approaches the origin. Thus as L b increases relative to La the larger pole is approxim ated by the sum of the po les; that is,

Pi * Pv + Pz

and

R„ + Rh

R„Rh

+R, R a + R b

= P10

P2PlPl P10

L aL b

+ RhR„Rb 1

R„ •+ Rh L hP 20-

(2.1-6)

(2.1-7)

As the reader can readily dem onstrate numerically, if R a % R b and L b > 10L a, the approxim ations o f Eqs. (2.1-6) a n a (2.1-7) are accurate w ithin 5% . In addition , if L b > 100L„ the approxim ations are valid within 1 % for any ra tio of R a to R b. It should be noted th a t p 10 is the netw ork pole obtained with L b open-circuited and tha t p20 is the netw ork pole obtained with L a replaced by a short circuit. Figure 2,1-6 shows the two simplified single-pole circuits from which p l0 and p 20 may be obtained by inspection.

W ith the poles widely separated, H(p) is given by


and P20 •

in addition,20 log \H(ja))\

(a) (b)

20 log H m + 20 log |co/p20|

- 20 log y / l + (w /p10)2 - 20 log y / l + (m /p 20)2, (2-1-9)

where H m is the m idband transfer function. The m agnitude 20 log \H(ja>)\ and a sketch of arg H(juj) vs. a> are given in Fig. 2.1-7.

Since |p 10| » |p20|, the corrections a t each corner to the asym ptotes do not in te rac t; hence the range of frequencies over which \H(ja>)\ has not decreased by m ore than 3 d B from its m idband value is just the range between |p 10| and |p2ol- This range is conventionally defined as the — 3 dB bandw idth of the transform er coupling network. O ne should note also tha t as the poles become widely separated, arg H(jw) approaches + n/4 a t a> = |p2ol and —n/4 a t oj = |p 10|.

If the transform er coupling netw ork is now excited by a step of voltage of the form Vj(t) = K,u(t), then I«P) = V\ /P, K(p ) = F,//(p)/p, and

, _ l Vl H ( p ) _ VtnR L

K \ P i - P 2I(e - | P2|< _ e - l P'l')u(i), (2.1- 10)

Fig. 2.1-7 Magnitude and phase vs. to for transformer coupling network.


Fig 2.1-8 Step response of transform er coupling network.

where i f " 1 is the inverse Laplace transform operator. A p lo t of v0{t) vs. t is shown in Fig. 2.1-8. N ote tha t the high-frequency pole at contributes to the deterioration of the leading edge of the ou tpu t step, while the low-frequency pole at p2 contributes to the decay of the ou tpu t step to zero. This, of course, is an expected result, since a transform er does not transm it either the high-frequency com ponents of the step which contribute to its leading edge or the low-frequency (dc) com ponents which are required for a nonzero steady-state value.

If one m akes the transform er broadband by designing k » 1 (for which case Pi x Pto and P2 ~ P2o)> ^ e step response takes the form shown in Fig. 2.1-9. Since IPil >:> l?2l> the tim e duration of e ~ |pi1' becomes negligible when com pared with e ' l p2l'; thus the step response takes the approxim ate form

v M =VinRL _ | P 2 0 MO = VihR l ' \ P20 \ t u{t). (2.1- 11)^ 0P 10 R a + Rb

N ote tha t this is the response obtainable from the simplified circuit of Fig. 2.1-6b. If the fine structure o f the leading edge of the step response is required, an expanded time scale abou t the origin m ust be employed. O n such a scale e -|p21' remains essentially constan t a t u n ity ; thus the leading edge of the step response takes the approxim ate form

v„U):VinRi

R a + Rb<1 H t) , (2.1- 12)

Fig. 2.1-9 Step response of broadband transform er network.


Fig. 2.1-10 Expansion of leading edge c f Fig. 2.1- 9.

which is illustrated in Fig. 2.1-10 and which is exactly the step response obtained from the simplified circuit of Fig. 2. l - 6(a). Consequently, for the case of widely separated poles, both the step response and the frequency response may be obtained from simplified single-pole circuits.

As an application of the above analysis, let us specify the param eters of a transformer which m atches 5 fi (RL) to 500 Q (R s) in the m idband extending from 50 Hz to 5000 Hz (|pi| = 3.14 x 104 rad/sec, \p2\ = 3.14 x 102 rad/sec). W ith these values the poles are widely sep ara ted ; hence

L , - (1 - k ‘ )L, = R ’ + ' ' + " H R l + ' 2)IPlI

J i 2 r _ ( R S + r!) (RL + r2)n2L b — K L i —

[.R s + rj + n2(RL + r2)]\p2\

If the transform er is to have reasonable efficiency, we m ust have R s » r l and R L » r2. In addition, for a m atch, we need n2R L = R s ‘, therefore,

1000 250(1 - k 2)L y ^ ^ x 4 = 31.8 m H and /c2L t % — = 800 mH,

fro m w h ich w e o b ta in k = 0 .962 , L j = 832 m H , a n d L 2 = k2L i/n 2 — 8 0 0 m H /1 0 0 = 8 mH. All of these values are readily obtained in practice.

Before term inating ou r discussion of broadband transform er networks, let us consider the effect of loading the transform er in a way different from that specified by the m anufacturer. Clearly if both R s and R L are increased while their ra tio remains constant, the efficiency increases and |p 10| and |p 2ol increase in direct proportion to Rs or R L (if we assume rj « R s and r2 « R L) \ hence the transform er passband is shifted up in frequency. If R L and R s are decreased, the opposite effect results. To determ ine the effect on |p 10| and |p2ol ° f varying R L relative to R s, we plot

R J . R.20 log |p 10| = 20 log ^ 1 + - ^

andR b201o g W = 2 0 lo g j


Ra_

Fig. 2.1-11 Plot of 20 log |p10| and 20 log |p20| vs. RJR b.

vs. R J R b as shown in Fig. 2.1-11, where R /L is expressed as a dim ensionless quantity in some convenient system of units. As R a = R s + r l is increased relative to R b = n2(RL + r2), we observe tha t the high-frequency pole increases w ithout bound while the low-frequency pole approaches a constant value. Consequently, we should expect a current source drive (Rs = oo) to produce no high-frequency break point regardless of the value of k. This effect is not observed in practice, however, since an additional finite break po in t is produced by the physical winding capacity, which we have neglected.

As R a is decreased relative to R b, we observe that the high-frequency pole approaches a constant value while the low-frequency pole approaches zero. In the lim it as R a -*■ 0 we should expect the transform er to pass dc; however, this effect is also no t observed in practice because of the nonzero resistance of the driving source and the existence o f r x. A lthough the above analysis is perform ed for a transform er coupling netw ork, the same results are obtained for any broadband coupling network having a zero at the origin and two poles on the negative real axis. In particular, a netw ork of the form shown in Fig. 2.1-12, consisting of a coupling capacitor between the driver and the load with some stray capacity to ground, has a similar step and frequency response.

R, C

Fig. 2.1-12 Capacitive coupling network.

2 .2 PARALLEL RLC CIRCUIT 25

Fig. 2.2-1 Parallel RLC circuit.

L iv 'd )

In the sections which follow we shall shift our emphasis from the broadband application of the transform er to its application in narrow band circuits. We shall begin by developing the properties of simple narrow band circuits; we shall then extend these concepts to narrow band transform er-coupled networks.

2.2 PA RA LLEL R L C C IR CU IT

Before beginning a discussion of narrow band transform er networks, we shall review some of the properties o f simple narrow band resonant circuits, which are essential building blocks of the m ore com plicated transform er networks. We shall begin by considering the simple parallel R L C circuit shown in Fig. 2.2-1. If we drive the circuit with the current i,(f) and define v„(t) as the ou tpu t voltage, the transfer function (in this case, input impedance) takes the form

1 1

K(P)Z l l (p) =

* c

m 2 P 1 v R C LC

iP ~ Pi)(P - P i)’(2.2- 1)

where

PU2 2 R C+

2 R C1

L C

The poles, p 1 and p2 , m ay be real o r a com plex conjugate pair. W e consider these two cases separately.

For the case where (1/2RC)2 > 1 ¡LC (or equivalently R < co0L/2 = l/2co0C, where a>0 = 1/v/ l C ) , p x and p 2 lie on the negative real axis as shown in Fig. 2.2-2. This pole-zero diagram , which is analogous to the one for the b roadband transform er network of Section 2.1, indicates that for values of R which are small com pared with

7(0

SF 1I

Pi P2-O -

Fig. 2. 2 -2 Pole-zero diagram of Z ,,(p ) with R < w 0L/2.

2 6 BROADBAND AND NARROWBAND COUPLING NETWORKS 2 .2

3 v„(/)

Fig. 2.2-3 Simplified circuits for finding p 10 and p20.

(o0L/2, the parallel R L C netw ork functions as a broadband netw ork whose m idband frequency range extends from \p2\ to |p,|.. Physically the m idband may be interpreted as that range of frequencies over which the im pedance of bo th the inductance L and capacitance C is large, and therefore negligible, com pared with R; hence Z llm = R. At low and high frequencies the im pedances of L and C respectively approach zero, thus shunting i,(f) to ground.

F or wide pole separation the approxim ate values of and p 2 may be determ ined by inspection. In particular, for p, » p 2, P\ is approxim ated by the sum of the roots of the denom inator of Z t i (p), and p2 is approxim ated by the product of the roots divided by the sum of the roots, tha t is,

1Pi ~ — — P 10 i P 2

l /L C Rl /R C ~ L = P20'

(2.2-2)

Physically |p20| is the radian frequency a t which the im pedance of L is equal to R, and |p 10| is the radian frequency a t which the im pedance o f C is equal to R. A lternatively p 10 and p20 are the poles o f the simplified circuits shown in Fig. 2.2-3. Since the step and frequency responses of the broadband parallel R L C netw ork in terms of and p2 are identical in form with the corresponding responses of the b ro ad band transform er, we do no t present them here.

F o r the case where (1/2R Q 2 < l /L C or R > co0L/2, the expression for the polestakes the form

P 1,2 = - a ± j - J Wq - a 2 = - a ± j ß , (2.2-3)

where co0 — 1/N/ Z c , a = 1/ 2J?C, and /? = yfcol — a2. The pole-zero d iagram for this case is shown in Fig. 2.2-4. It is apparen t tha t the distance of the poles from the origin is given by a2 + ft2 = col; hence as a is increased by increasing the loadingon the circuit (decreasing R), the poles m ove in to the left half-plane along the semicircular trajectory of radius oj0 until they m eet on the real axis for a = w 0.

Physically a>0 is that rad ian frequency a t which the im pedances of the inductor and capacitor are equal in m agnitude and opposite in phase, and thus produce an open circuit when com bined in parallel. This frequency, a t which the parallel R L C circuit appears as a pure resistor R, is called the resonant frequency of the circuit.

2 .2 PARALLEL RLC CIRCUIT 27

Fig. 2.2^4 Pole-zero diagram of Z , t(p) withR > w0LI2.

SF C

■ \ r c

w0=Jl c

To obtain the sinusoidal steady-state frequency response when R > to0L 2, we express Z , {{jo) as

/(u/C KZ u ( j o j ) = — r - - - .......= ---------------------- 5-, (2.2-4)

to + 2jtoa+ jQi

to I 'ow>()

Equation (2.2-4) provides the means of obtaining an exact plot of |Z , [(yco)! and arg Z u (ja>) vs. o). O n the o ther hand, a quick sketch of \ Z l l (jto)\ and arg Z 11(jto) may be obtained from the value of Z l ,(yco0) and the asym ptotic values of Z n (ja>) as co approaches zero and infinity. The asym ptotic values vs. u> may be obtained by plotting the resistance R. the m agnitude of the capacitive reactance IA^m)! = 1/tuC, and the m agnitude of the inductive reactance \X L(o))\ = w L on the same set of coordinates vs. w as shown in Fig. 2.2-5. Clearly for to = to0 the reactances of the inductor and capacitor cancel each o ther and yield 17-11( jo))\ = R and, of course, arg Z i tO’w) = 0. As co is decreased below a>0 , the small reactance of the inductor rapidly dom inates the parallel R L C netw ork, causing \Z X i(joj)\ to approach tnL and arg Z 11(7« ) to approach n/2. Similarly, as to is increased above <o0, the small reactance of the capacitor rapidly dom inates the parallel R LC netw ork, causing \Z { ¡(jto)\ to approach 1 ¡taC and arg Z x ,(/o>) to approach — n/2, as shown in Fig. 2.2-5.

In addition, the two “ half-pow er” frequencies tol and co2 , at which |Z u (y'oj)| = R js /2 and a r g Z u (jw) = -+-ti/4 arid — 7t/4 respectively, may be found by equating the im aginary term in the denom inator of Eq. (2.2-4) to + 1 and solving for to :

1, 0) = U)] ,Qt

to20COOJn + 1, CO CO,

(2.2-5)


Direct solution of Eq. (2.2-5) yields the following relationships between at, and a >2 :

a>2 — ffli = 2a and = co0.

However, since the difference between co2 and (ot is the — 3dB bandw idth (BW) of the parallel R L C circuit, we have BW = 2a, o r equivalently

BW _ 2a _ 1

®0 «0 Qt(2.2-6)

It is interesting at this po in t to note the significance of the param eter Q T, which is referred to as the netw ork “que.” F irst of all, as QT increases, the — 3 dB bandw idth of the netw ork decreases relative to its resonant frequency. Clearly, then, if the parallel R L C circuit is used to transm it a m odulated carrier whose spectrum occupies a fixed band o f frequencies abou t co0 , Q T m ust no t be increased to a po in t where the m odulation is distorted. Second, increasing QT increases the ratio of R to co0L =1 /w 0C (note that QT = R/(o0L = Rco0C). Hence the parallel R L C netw ork greatly attenuates frequencies in the vicinity of 2<w0 (where \Z 1 ¡(Joj)! -» 1/tuC) relative to frequencies in the vicinity o f oj0 (where Z i l (jeo) x R). Consequently, if the purpose of the parallel R L C circuit is to extract the fundam ental com ponent of a periodic waveform, a QT as high as possible is desired. Third, as QT increases, the poles o f Z x i(p) approach the im aginary axis in the com plex p-plane and cause any transients induced in the netw ork to become m ore and m ore oscillatory.

If we wish, we may in terpret QT in still another way. F o r the case where ¿¡(t) = I cos (o0t, which results in v0(t) = I R cos <x>0t = Vl cos co0t, we have

27i (peak energy stored) 2n(iC V f)energy dissipated per cycle (2n/a)0)(V j/2R )

— o)qCR — Qt - (2.2-7)

2.2 PARALLEL RLC CIRCUIT 29

This in terpretation of QT in term s of stored and dissipated energy is useful in determining the Q of circuits with m ore than one source of dissipation.

In cases where QT is high, or equivalently where the poles of Z u (p) are close to the im aginary axis, a graphical approxim ation perm its us to obtain a greatly simplified expression for Z u (j(o). If we consider the phasor diagram shown in Fig. 2.2-6, we observe th a t for a> > 0 the range o f frequencies over which |Z j i(j(o)\ = (S F )p Jp PipP2 is significantly different from zero is that range in the vicinity of p, where p Pt becomes small. As a is decreased relative to co0, this frequency range decreases to the point where the phasors draw n from z and p2 = p* rem ain essentially constan t over the entire range, with the m agnitudes and angles given by

pz « to0, 6Z = t t / 2 , ppi % 2a>0, dP2 * n/2.In addition, for a « cj0,

P = J w l - a 1 % OJ0 .

Consequently, for high values of QT, Z(jto) may be clocely approxim ated hy

a '2~8 ’= exp (- jf lp .)

2CPr>

3 0 BROADBAND AND NARROWBAND COUPLING NETWORKS 2.2

Recognizing that pPl exp j0 Pi = a + j(o> — ft) % a + j(a> - co0), we may rewrite Eq. (2.2-8) in the form

1 RZ ,A jo j ) = ------- p = -------------------- id > 0. (2.2-9)

1 + / ^ *

A sim ilar form may be obtained for w < 0.It is apparen t th a t |Z n (y'co)| given by Eq. (2.2-9) is sym m etric abou t o j 0 , with

“half-power” frequencies tot and co2 given by

co 1,2 = co0 ± oc.

It is interesting that for high values of QT, u>0 approaches the arithm etic ra ther than the geometric m ean of (Oj and a>2, while the — 3dB bandw idth rem ains unchanged at (o2 — a»! = 2a.F o r (a - u)1 and for a> = u>2, pZl has increased from its m inim um value o f a to a^ /2 , thus causing \Z xl{jo))\ to decrease from its m axim um value by a factor of 1/^/2-

A lthough we have indicated tha t the simplified form o f Z u (j'a>) given by Eq. (2.2-9) is a good approxim ation for Z j ¡(jw) when’Q r is high, we have no t yet determined w hat range of Q T may be considered high. T o do this we m anipulate the exact expression for Z 11(ja)) given by Eq. (2.2-4) into the form

Z . . W - - l + 4 Q / (2.2- 10)

+ 1 2C iT4Q ~T

where i2 = (to — oj0)/a. H ere we see explicitly th a t as QT —>■ oo Eq. (2.2-10) reduces to Eq. (2.2-9). In addition , if we plo t |Z , ^ jo j^ /R and arg Z n (7'co) vs. Q as shown in Figs. 2.2-7 and 2.2-8, we observe exceptionally close agreem ent between the curves obtained for QT = 10 and Q T = oo, particularly in the vicinity of f t = 0, o r equivalently to = co0. Since the curves for QT = oo correspond to the simplified expression for Z i l (jw) given by Eq. (2.2-9), we m ay clearly use this approxim ation for Z ^ i jo j ) with confidence for QT > 10 and as a “ ballpark” approxim ation for QT as low as 5.

To obtain the response of the parallel resonant circuit driven by an impulse of current, we obtain the inverse Laplace transform of Z j ,(p) o r directly evaluate the impulse response by o ther methods. In either case, we obtain

z u (i) = ‘[Z u (p)] = ^ “ " c o s j fit + ta n " 1 u(t). (2.2-11)

For the case where QT > 10, Eq. (2.2-11) reduces to the simplified form

i(0 ~ ^ e M (cos (o0t)u(t), (2.2- 12)

since p = a>0%/ T ^ l y 4 Q f a w 0 and a//? = — 1 » 0.

2.2 PARALLEL RLC CIRCUIT 31

Q——Fig. 2.2-7 Plot of \Z{ ,{ja))\/R vs. f i = (co - co0)/a.

2.3 PARALLEL L C C IR C U IT W ITH SER IES L O SS

In this section we shall consider the parallel resonant circuit with loss in series with the inductor or capacitor. We shall show that when the series loss is small, as is most often the case in com m unication systems, the circuit behaves in the same fashion as the parallel R L C netw ork, and thus the loss may be modeled by an equivalent parallel resistor. Such a model perm its the com bination of circuit loss appearing at several points in the parallel LC circuit into a single parallel resistor for which the expressions derived in Section 2.2 may be applied directly.

We begin our analysis with the circuit of Fig. 2.3-1, in which the loss in the form of r appears in series with the inductor. We shall then generalize to the case where loss appears in series with both the inductor and capacitor and also in parallel with the com bination. For the circuit of Fig. 2.3-1, the transfer function (or input im pedance) relating r0(t) and i¡(t) is given by

Z u iP) = W l = {1/C)ip + r/L) (2.3-1)hip) (P - Pi)(P ~ Pi)

where p 1 2 = ( —r/2L) + v /(7/2L)2 — 1 /LC. We see that the poles of Z lx(p) may be either real or complex, depending on the relationship of the param eters. Since the analysis of the circuit with real poles is quite sim ilar to the real-pole analysis performed in Section 2.2, we restrict our atten tion to the case where 1 /L C > (r/2L)2 (or equivalently r < 2ca0L = 2/co0C , where to0 = 1 / ^ J l C). For this case the poles may be written in the form

P i ,2 *= ± jyjwl ~ a2 = ±jP, (2.3-2)


2 .3 PARALLEL LC CIRCUIT WITH SERIES LOSS 33

Fig. 2.3-2 Pole-zero diagram of Z n (p) for r < 2w 0L.

wherer 1

0)0 =

S F = -

2 L ' y lc’ and /? * coq — a 2.

The corresponding pole-zero diagram for Z u (p) is shown in Fig. 2.3-2. H ere again we notice thdt as a is increased by increasing r, the poles move into the left halfplane along a sem icircular trajectory of radius co0, where co0 is the frequency at which the m agnitudes of the im pedances of the inductor and capacitor are equal.

F igure 2.3-3 shows a sketch of |Zn(jco)| and arg Z n (jw) obtained graphically from the pole-zero plot of Fig. 2.3-2. Even this rough sketch indicates that, unlike the parallel R L C circuit, the circuit with loss in series with the inductor does not experience either zero phase shift o r m axim um am plitude of \ Z X 1(_/cu)| a t the frequency (o0- F o r this reason tw o distinct resonant frequencies are specified for this circuit: (1) the am plitude resonant frequency a t which the m axim um value o f |Z u (y<y)| occurs and (2) the phase resonant frequency a t which Z , ,(7(0) appears purely resistive. Fortunately, as the poles and p2 approach the im aginary axis, both resonant frequencies converge to co0 , which is sometimes defined as still a third resonant frequency.

In addition, no convenient expression exists for either the m axim um value of \Zi\(j°A\ o r the — 3 dB bandw idth of the circuit in the general case. However, for the case where a « a >0 o r QLf is high, where QL is defined as

1a)0rC'

(2 .3 -3 )2a r

certain simplifying approxim ations in the expressions for Z i l (ja)) m ay be made.

t W e shall reserve QT for “que” in parallel RLC circuits.


Fig. 2.3-3 Sketch of magnitude and phase of Z , t(joi) vs. ai.

Specifically, with the aid of Fig. 2.3-4, we write Z x in the form

ZuO'co) = exp j(02 - 6 - 0 ) (2.3-4)Pp,Ppz

and observe that, for frequencies in the vicinity of p x (the only range for a> > 0 where Z y, (jo>) is significantly different from zero), the following approxim ations are v a lid :

p z x co0 , 9Z x 7r/2, pPl % 2a>0, ep2 x n/2,

P = J a i l - a 2 « co0.

Thus Eq. (2.3-4) may be written in the form

ZnO 'to)2 CPpt

exp - j d Pl =1

2Cal 1 + /a> — to.

co > 0,

which is identical in form to Eq. (2.2-9). If, in addition, we observe that

d c ' § ¿ c = = Qlr ¥ R--then Eq. (2.3-5) may be rew ritten in the form

Z u (j(o )

1 + j.0) — O)0

oj > 0,

(2.3-5)

(2.3-6)

(2.3-7)

which is exactly the expression for the impedance of a high-Qr parallel RLC circuit


lz, lUbi

arg Z i1i7a i = e , - 0, 1 - 0 , 2

e r <*Jq

Fig. 2.3-4 Phasor diagram for evaluating Z , , (jw).

with R = /?eq = g£r. Hence, for high values of QL [specifically, QL > 10 ensures the accuracy of Eq. (2.3-7) within a few percent], the circuit of Fig. 2.3 1 may be modeled by the circuit shown in Fig. 2.3-5. W ith this equivalence in circuits, we note fromthe results of Section 2.2 that \ Z X i(./«)|max = R BW = 2a = r/L, BW /w0 !Qi ,

In terms of the pole-zero diagram of Z n {p) shown in Fig. 2.3-2, the effect of replacing the series loss by a parallel loss is the m ovem ent of the zero at —2a to the origin. This m ovem ent has little effect on Z u (p) for small values of a, which exist when Ql is high.

Fig. 2 .3-5 Equivalence o f two parallel resonant circuits.


Figure 2.3-6

In practice, the quickest way to convert series loss into parallel loss is to evaluate Ql = a>0L/r , determ ine whether QL > 10, and then form R eq — Q2Lr. N ote that if Ql > 10 and if there are no o ther losses in the network, then QL = QT. Clearly Qt — Req/ôL = Ql - If o ther losses are present in the circuit, /?eq m ust be com bined with them to determ ine QT ; thus QL ¥ QT-

Example 2.3-1 D eterm ine values for r, L, and C for the circuit shown in Fig. 2.3-6such that Z u ijco ) peaks to a value of 1000 i i at a>0 = 107 rad/sec ( / = 1.6 M Hz)with a bandw idth of 5 x 105 rad/sec.

Solution. Since the bandw idth is narrow com pared with co0 , we have a high-Q circuit for which

= ©o = ^ = 20Ql 2a BW

In addition,

1000 Q = |Z O o ) | * Q2Lr ;

hence r = 2.5 Q. F rom the relationship

— = 2a ss BW

we obtain

and finally

L =2.5 i l

5 x 10 rad/sec= 5//H ,

C = l/(o20L = 2000 pF.

If the loss in a parallel resonant circuit appears in series with the capacitor, as shown in Fig. 2.3-7, the input im pedance is given .by


o-

L

Fig. 2.3-7 Parallel resonant circuit with loss in series with the capacitor.

o

Zn(P)

Figure 2.3-8 shows a pole-zero d iagram of Z x ¡(p) for the case where the poles form a complex conjugate pair. Again, and by reasoning sim ilar to that employed when the loss is in a series with the inductor, it is readily shown tha t for Qc > 10, where Q c = cj0/2a = co0L/r = l/a>0rC, the series loss r may be replaced by an equivalent parallel loss R cq o f value g £ r with negligible effect on Z x j(p).

In addition, it may be shown that if a loss appears in series with the inductor and a loss r2 appears in series with the capacitor of a parallel resonant circuit, and if

where o 0 = 1 /y fL c , then the to tal loss may be represented as the parallel com bination of Q j / i , Qcr2, and any existing shunt resistance. We shall explore this idea further in the following example.

Example 2 .3-2 F o r the circuit shown in Fig. 2.3-9, determ ine the resonant radian frequency oo0 , the bandw idth, Q T, and Z n ( j ( o 0).

Solution. We observe first that oj0 = 1 /y/~LC = 107 rad/sec; hence

p i

// P

SF = r

P 2 = - " ± /P

-e SV orC \

\ U>n - y/EC

\ (3 = v ^ ’o2 - “ 2

Fig. 2.3-8 Pole-zero diagram of Z t l (p)." x _

P2 PÎ

38 BROADBAND AND NARROWBAND COUPLING NETWORKS 2 .4

Consequently r 1 and r2 may be replaced by equivalent parallel resistances each equal to Qcr i = 4kQ . The to tal shunt resistance is therefore the parallel com bination of 2 kQ, 4 kQ, and 4 kQ, which of course is 1 kQ = R T. Since for the equivalent parallel resonant circuit, L and C com bine to produce an infinite im pedance a t a>0, Z u ( j o )0) = Rt = 1 kQ. Also Qt = R t/ u>0L = 10 and BW = co0/Q T — 106 rad/sec.

2.4 PARALLEL RESONANT “TRANSFORMERLIKE” NETWORKS

In this section we shall consider the parallel resonant circuit in which the loss appears across only a portion of the inductor or capacitor. Again we shall show tha t when Q is high (or, equivalently, the complex poles of the circuit are close to the im aginary axis), the circuit input im pedance has essentially the same form as that o f the parallel R L C circuit. The loss may therefore be modeled as an equivalent parallel resistor for which the results of Section 2.2 are directly applicable. In addition, we shall show that the tapped and loaded inductor o r capacitor possesses m any of the properties of an ideal transform er. Specifically, if the loading is light, the load across the tapped energy-storage element may be modeled by the identical load placed across the secondary of an ideal transform er whose prim ary is placed in parallel with the to tal energy-storage element. This representation is valid no t only for the evaluation of the input im pedance but also for the evaluation of the transfer impedance.

We begin our analysis with the circuit of Fig. 2 .4 -1 , in which the loss in the form of G appears across a portion of the capacitor. W e shall then generalize to the case

2 .4 PARALLEL RESONANT “ TRANSFORMERLIKE” NETWORKS 39

where the loss appears across a portion of the inductor, and also (in Section 2.5) to the case where the loss appears across the secondary of a physical transform er with cap ac itiv e ly tu n e d p rim a ry .

For the circuit of Fig. 2.4-1 the input im pedance is given by

Z „ (p ) =VqAp )IÂP)

4 \ P + C l + C 2

2 G 1 + P — + P

(2.4-1)

L C l c , c 2

where C = C 1C 2/(C 1 -I- C 2) is the series com bination of C x and C 2. If the loading on C 2 is light, Z n (p) m ust have a pair of complex poles in addition to a real-axis pole, and thus may be w ritten in the form

Z , ,(/>) =p c \ p + c , + c 2

(2.4-2)(p + y)(p + a - jP ) { p + a +j(1)'

where p¡ = —a + jfi and p2 = —a — jfi are the complex conjugate poles and p3 = —y is the real-axis pole. Figure 2.4-2 is a typical pole-zero diagram of Z , j(p).

F rom Eq. (2.4-2) or from Fig. 2.4-2 it is apparent that the expression for Z , ,(/>) reduces to the form of the input im pedance of an equivalent parallel R L C circuit if p 3 = z2 or, equivalently, if y = G jC x + C 2. To determ ine under what circumstances

P\ X '

c, t Ci

Fig. 2.4-2 Typical pole-zero diagram of Z , ,(p).

ja

SFC

Xp i P t *


this pole-zero cancellation is achieved, and in addition the resultant values of a and fi in term s of the circuit param eters o f Fig. 2.4-1, we first write the denom inator of Eq. (2,4-2) in the form

(p + y)(p + c c - jp)(p + a + jP)(2.4-3)

= P3 + P2(y + 2a) + p(a2 + [I2 + 2ay) + y(a2 + (I2).

W e then equate the corresponding powers of p in Eq. (2.4-3) and in the denom inator of Eq. (2.4-1) to ob tain the set of equations

y + 2a = G/C2, (2.4-4a)

a 2 + p 2 + 2ay = l /L C = (2.4-4b)

y(a2 + p 2) = G /L C lC 2. (2.4-4c)

If, in addition, we define Q = coo/2ay, then Eqs. (2.4-4a, b, and c) may be com binedand rearranged in the form

n2Gl 1 - 1 /nQ l2“- - r ( - n n /n j ’ (24- 5a|

a 2 + p 2 = o)20{\ - I/O), (2.4-5b)

G 1 1 ' (2.4 -5c)' C l + C 2\ l - l / Q f ’

where n = C l/{Cl + C 2).It now becomes clear from Eq. (2.4-5c) tha t if ÍÍ > 100 then the real-axis pole

and zero of Z j ¡(p) lie w ithin 1 % of each other and, for all practical purposes, cancel to yield Z n (p) in the form of a parallel R L C circuit with a2 + ft2 x tup and 2a % (n2G /C )(l - 1/nQ).

W ith the aid of Eqs. (2.4-5a and c) we may also write

a* = Qt U \ ~ W (2.4- 6)2ay 1 - 1/n Q

where QT = co0C/n2G and QE = a»0(C! -I- C 2)/G. For large values of Í2 Eq. (2.4-6) reduces to

Q % Ot .Qe + l/n. (2.4-7)

Hence if QT QE (which may be directl) determ ined in term s of the circuit param eters) is greater than 100, we can be sure t h a t Q is sufficiently large to effect the desired pole-zero cancellation. In addition, Q , QE > 1 0 0 ensures the accuracy (within 1%) of the approxim ation of Eq. (2.4—7), from which we obtain

2.4 PARALLEL RESONANT “ TRANSFORMERLIKE” NETWORKS 41

e r .e £>.oo,

7-u (P)

Fig. 2.4-3 Parallel circuits with equivalent input impedances.

a n d in tu rn

)--------1>

- -

- Li c \

)--------1i—

= c 2 |

o - Z n (P)

Ci + Cp

Z n (P)PC

p 2 + la p + a2 + p 2

where

G „ = n2G\ 11

(2.4-9)

hQt Qe + 11

E quation (2.4-9) represents the input im pedance of the equivalent R L C circuit shown in Fig. 2.4-3.

If Qt Qe > 100 and, in addition, nQT QE is large, then Geq reduces to the simplified form

Geq = n2G, nQT QE » 1. (2.4-10)

In particular, if nQT.QE > 20, then Geq = n2G w ithin 5%, whereas if nQT QE > 100, then Geq = n2G within 1 %. (N ote that since n < 1, nQT,QE > 100 ensures Qt Qe > 100.) W ith this additional condition satisfied, we may transform the equivalent circuit show n in Fig. 2.4-3 in to an alternative and m ore useful form, shown in Fig. 2.4—4. Clearly, if G is reflected through the ideal transform to obtain

Fig. 2 .4-4 Transform er model for resonant circuit with tapped and loaded capacitor.

<?+


Geq = n2G, then the model of Fig. 2.4-3 results. However, the model of Fig. 2.4-4 is m ore versatile than the m odel o f Fig. 2.4-3, since it provides a valid approxim ation for no t only the input im pedance Z t l (p) but also the transfer im pedance Z l2(p) = Kiip)/hip) f°r the case where QE > 10 and also nQT QE > 100.

To dem onstrate this property, we note that, in general,

7 KliP) K i 2 iP) ^0liP) r y / * | _ j / \ / A J j . tZ i2ip) = -T7-T = 77- 7-7 -JT-T = Z n ( p ) H v(p), (2.4-11)

hiP) Ktip) I.iP)

Ki iP ) c ,where

H ip ) =KiiP) C l + C 2 p + G/(C, + C 2)

for the circuit of Fig'. 2.4-1. A pole-zero diagram for Z l2(p) is shown in Fig. 2.4-5. F rom this diagram and from argum ents sim ilar to those given in Section 2.3, it is clear that if œ 0/y » 1 and, in addition, o)0/a. » 1, then the zero at the origin effectively cancels the pole a t - y when one evaluates the frequency or time response of Z 12{p)

X

J W

ß

-<§>

Fig. 2 .4 -5 Pole-zero pattern o f Z 12(p). X


graphically ; hence

Z i2(P) ~ - y1— Z „ (p ) = nZu(p), C 1 + L 2

which is exactly the transfer im pedance of the model of Fig. 2.4-4. Clearly the condition nQT QE > 100 (which is required for the input impedance of the model of Fig. 2.4-4) results in co0/a % QT and cj0/y QE \ hence the com bined conditions hQt Qe > 100 and QE > 10 perm it the use of the model of Fig. 2.4-4 for obtaining both Z n (/>) and Z l2(p) for the circuit of Fig. 2.4-1. Specifically, if QE > 10 (and ,1Qt Qe > 100)’ then |Z 12(ja;)| obtained from the model is accurate within 1 % of its actual peak value and the phase of Z l2{ja>) is accurate within 6° over the passband.

If the phase angle of Z l2{j(o) is critical or if QE < 10 while Qt Qe > 100, then Z n (jw) should be obtained from the model of Fig. 2.4-3 and Z l2(ja>) should be obtained by m ultiplying Z n (jto) by H v{jco), which is determ ined exactly from F.q. (2.4-11).

At this point it is worthwhile to in terpret physically the param eters n , C, QT•, and Qe employed in the models of Figs. 2.4-3 and 2.4-4. Clearly n is the voltage division ratio of the two series capacitors with the load across C 2 removed (G = 0); that is,

v„2(t)n = -------

I ' o l U ) G = 0

The capacitor C is just the total capacitance shunting L in the circuit of Fig. 2.4-1 obtained with G = 0. In addition, Qr . = (o0C/n2G is the Q of the model of Fig. 2.4—4, whereas QE = «„(C j + C 2)/G is the ratio of the ou tput resistance (1/G) to the reactance at co0 which shunts G, evaluated with the input voltage t>0(f) of the circuit of Fig. 4.2-1 reduced to zero (i.e., with the input shorted). We shall now use the same physical interpretation to obtain the corresponding param eters for a parallel R L C circuit with a tapped and loaded inductor.

C onsider the parallel R L C circuit of Fig. 2.4—6. By a procedure sim ilar to thatemployed with the circuit of Fig. 2.4-1, we can show that the transform er model, also shown in Fig. 2.4-6, m&y be employed to obtain expressions for Z , ¡(p) and Z 12(P) provided that

h Q t Q e > 100 and QE > 10,

where in this case QT = co0C/n2G, QE = co0L lL 2/G (Li + L 2), u)0 = 1/^/LC,L = L , + L 2, and n = L 2/{L l + L 2). Here again we observe that

t’ol(f) G = 0

that L is the inductance shunting C with G = 0, that QT is the Q of the model of Fig. 2.4-6, and that QE is the ratio of 1 /G to the reactance at a>0 shunting G, evaluated with rol(f) = 0. This set of conditions is valid for any of the transform erlike networks shown in Table 2.5-1 and thus provides a handy mnem onic rule. If for the circuit of Fig. 2.4-6 Qt Qe > 100 but QE < 10, Z u (p) may be evaluated from an equivalent


Fig. 2.4-6 Parallel RLC circuit with tapped and loaded inductor and its transformer model.

parallel R L C circuit with a shunt resistance R cq = 1 /Geq, where again

G = n2c | 1 - j -\ nQT.QE + 1

In addition, Z 12(p) may be evaluated as Z l2(p) = Z n ( p ) H v(p), where

Hvip) = L tG p + (L , + L 2) /L lL 2G

(2.4-12)

(2.4-13)

Shunt Input Resistance

If an additional shunt input resistance R L is placed across the parallel R L C circuit with a tapped an d loaded capacitor, as shown in Fig. 2.4-7, it is apparen t tha t the input im pedance

z ; ,(p) = = R ^ z i a n d z '»2(p) = = z 'i i ( p ) " M*i(P) *ol(P)

where Z n (p) and //„(p) are the input im pedance and transfer voltage function of thecircuit o f Fig. 2.4-1. Consequently, if nQT QE > 100, Z i t (p) may be obtained from

2 .4

Vol<f)

PARALLEL RESONANT “ TRANSFORMERLIKE” NETWORKS 45

V„2( f )

GL+ n 2G ' " ' ’ "*-T' 'c t ' - C, + C2

Fig. 2.4-8 M odel for circuit of Fig. 2.4—7 used to obtain Z , ¡(p) and Z '12(p).

an equivalent parallel R L C circuit with a shunt resistance of 1 /n2G, and in turn Z \ l (p) may be obtained from the equivalent parallel R L C circuit w ith sh u n t resistance 1 /(n2G + Gl ), shunt capacitance C, and shunt inductance L. Clearly, for this resultant parallel R L C circuit

_ m0C co0' ¿ T — n2G + G, 2 a '’

(2.4-14)

where —a' is the real part of the poles of Z \ !(p).If both w 0/2a! = Qr > 10 and coo/y > 10, where y is the pole of H v(p), then

again Z ’l2(p) ~ Z 'u(p)n- If » S t'Q e > 100, then w 0/y « QE\ thus if nQT /QE > 100, Qe > 10, and QT > 10, then the m odel shown in Fig. 2.4-8 may be used to obtain both Z'n (p) and Z \ 2{p) for the circuit of Fig. 2.4-7. The same argum ents apply to the parallel R L C circuit with a tapped and loaded inductor.

E xam ple 2.4—1 F o r the circuit shown in Fig. 2.4—9, determ ine an expression for t)0l(t) and vo2(t) as well as the circuit bandwidth.

Solution. If we assum e nQT QE > 100 and QE > 10, we may replace the original circuit by the m odel shown in Fig. 2.4-10, for which

Ci _ 1 C 1C 22 ’ L

n =C ! + C 2

1

C ! + C 2= 1000 pF,

s/ l c= 107 rad/sec, Z n (j(y0) = R = = 2 k fi,

v„ (0

Figure 2.4-9


Figure 2.4-10

i,U) pp; looo pF

500 n v„,(/>

-o+

-o

Qt = ^ t = 20, Qe = (S O O O K tC , + C 2) = 20, and nQT QE = 200;

these values justify the use of the model.Since the current drive is a t the resonant frequency a>0, where the input

im pedance appears purely resistive [Z n (ja)0) = /?eq], voi(0 m a y he w ritten directly in the form

= nvol(t) = 1 V cos 107f.

Finally, since the only loading in the circuit is across C 2, QT = Q t and BW = w0/C t = 5 x 10s rad/sec.

E xam ple 2 .4 -2 F o r the circuit shown in Fig. 2.4—11 determ ine values for C l5 C 2, and L such that the circuit resonates at / 0 = 16M H z(w 0 = 108 rad/sec) with a bandw idth of 1.06 M H z [BW = (107/1 .5) rad/sec] and achieves m axim um signal transm ission to R L a t resonance.

Solution. If we again assum e nQT.QE > 100, QE > 10, and QT > 10, we may replace the original circuit by the m odel shown in Fig. 2.4-12. Since a t resonance the inductor and capacitor com bine to produce an open circuit, m axim um signal (or

uol(i) = I R eq cos 107i = 2 V cos 107f,and hence

Figure 2.4-11


power) is transm itted to R L in the purely resistive netw ork when n is chosen to “m atch” R l to R s, i.e., when

n = J r J r s = yi k il/9 kQ = 1/3 = C J { C , + C 2).

W ith this value of n, R L is reflected through the ideal transform er as a 9 kQ resistor; thus the total input resistance a t resonance is 4.5 kQ. Recalling that BW = 1 / R C for a parallel R L C circuit, we obtain

c , c 2 = _____ 1_ ___ = 100

Cj + C 2 (4.5 kQ)BW 3 P

Replacing C ,/(C , + C 2) by n = 5 , we obtain

C 1 1 \C 2 = — = 100 pF, c , = -----1 C 2 = 50 pF,

n \n

and finally

L = = 30 juH.0J0C

At this point the original assum ption m ust be checked. It is apparent that

and Qe = (1 kil)oj0(C 1 + C 2) = 15;

hence nQT QE = 150. In addition, we note that for the circuit in this example q t = ®0/BW = 1 5 ; thus the use of the model is justified.

E xam ple 2 .4 -3 F or the circuit shown in Fig. 2.4-13, determ ine Z n (j(u0) and

Z 12(./«o).

Solution. For the circuit shown, w 0 = I / T I C = 107 rad/sec and n = L 2/ ( L i + L 2) = 1/100. Hence

QQt - — n co0L

= 100, Q ei n

L i L 2

I Q

oj0L 2 = 1, and Qt Q e = 100.

+ L 2


v»i(0

Figure 2.4-13

C 1000 pF Z., 9.9 nHL-> 0.1 nH

S ince Qt Qe = 100 b u t nQT QE = 1,

Z u ( M ) = R eq = --------1

n GÌ 1 -1

= I B = 20 k n

nQT QE + 1 / 2

(cf. Eq. 2.4-12). O n the o ther hand, since QE < 10, the transform er model shown in Fig. 2.4-6 may no t be used to determ ine Z i2{p)- However, the exact expression for Z l2(p) is given by

1

Z 12(p) = Z l i (p)Htj(p) = Z t i (p)-L t G

P +Li + L 2l , l 2g

(2.4-15)

If we substitute ja>0 = y'107 rad/sec for p in Eq. (2.4-16), we obtain Z i2(ja>0) rh the form

Z 120 w o) = (20 kO )-100(1 + j ) '

from which we obtain200

\ Z i 2(ja)0)\ = —¡= i l and arg Z i2(jco0) = -4 5 ° .

2.5 PARALLEL RESONANT TRANSFORMERS

In applications where a narrow band circuit is needed to isolate the load from the driving source or to provide a 180° phase inversion between the input signal and the load at resonance, a physical transform er m ust be employed in the parallel resonant circuit as shown in Fig. 2.5-1. In addition , the physical transform er m ust be employed if a “ step-up” turns ratio is required between the driving source and the load at resonance for the purpose of m atching a high load im pedance to a small source impedance. The “ transform erlike” netw orks discussed in the previous section are capable of providing only a “ step-dow n” tu rn s ratio.

The physical transform er may also be em ployed in a parallel resonant circuit as an au to transform er, shown in Fig. 2.5-2. In this configuration the transform er does not provide isolation, 180° phase inversion, or a “ step-up” turns ra tio a t resonance. Its usefulness stems from the fact that it is a m ore realistic model for the inductive

2 .5 PARALLEL RESONANT TRANSFORMERS 49

M = k y /L XL2

+

V „ 2< 0

Fig. 2.5-1 TransNJcmer with tuned prim ary and loaded secondary.

“ transform erlike” parallel resonant circuit, since it includes the effect of m agnetic coupling between the tw o inductors in the form of m utual inductance. The auto- transform er representation is especially useful when the two inductors are w ound on the same core or obtained by providing a tap on a single inductor. As is shown in Ejg. 2.5-2, L i and L 2 are defined differently in this case than when the two inductors are independent. This is done so that the netw orks of Figs. 2.5-1 and 2.5-2 can be analyzed by means of the same model.

To obtain expressions for the input and transfer impedances of the two parallel resonant transform er networks, the results of Section 2.4 may be applied directly. This becomes apparen t if we replace the physical transform ers of Figs. 2.5— 1 and2.5-2 by their com m on “ tw o-inductor” term inal equivalent model (which is derived in the appendix to this chapter) as shown in Fig. 2.5-3. Clearly the ideal transform er reflects a net conductance of G /a2 .= G L 2/k 2L l across the inductor k 2L x and thus the

Fig. 2.5- -3 Circuits of Figs. 2.5-1 and 2.5-2 with transformers replaced by their “ tw o-inductor” model.


circuit becomes identical to the circuit of Fig. 2.4—6. Consequently, if

n'QT QE > lOOf and Qe > 10,where

ri = k 2, Q t 'co0CcoqC _________

k 2G/a2 ~ (M j L y f G ’and Qe —

1

co0L 2( 1 - k 2)G’

then we may replace the circuit to the left o f the ideal transform er by its equivalent transform er model as shown in Fig. 2.5-4 (cf. Fig. 2.4-6).

If we now com bine the two ideal transform ers in cascade to obtain a single ideal transform er with a transform ation ratio n = n'/a = k ^ / L 2/ L l = M / L x, the model of Fig. 2.5-4 reduces to the desired model for determ ining ,(p) and Z 12(p), shown in Fig. 2.5-5. Here again we observe for the circuits of Figs. 2.5-1 and 2.5-2 that

UO2(0

M O

that L , is the to ta l inductance shunting C with G = 0, that QE is the ratio of the resistance (1 /G) to the to tal reactance at cu0 shunting G with uol(i) = 0, and that QT is the Q o f the model of Fig. 2.5-5.

r n

resonant circuits including transformers.

Connection exists for auto transformer

t We use ri in lieu of n because n is being reserved for the overall transformer ratio.


Table 2.5— 1 Equivalent circuits and pertinent relationships for parallel resonant “ transform erlike” networks

CircuitModel for

determining Z n (p) and Z t^ p )

IC,

-pCjC,

c ,+ c 21

JL C<**qCn2G

« 0<Cl +

C i+C 2

oA

c?-o§

8 |A A

ujo* ot

c>c5cf s:

N~ N~«2 «2 C/2 V)c c.2 .2 eS coE I□ J

l 4Li+L2

n'=n

M n2G+ ^2 L2G

u •vl:n

ilc r

i-0

M¿i

«'=fc3

1JL^C n2 G

------

c ~

’ K

-¿i :- ■l A g

c i L1 :n J —

ML\

n'=k

1M n*G

Table 2.5-1 summarizes these properties that we have discussed.Since the turns ratio n o f an au to transform er in a high-Q parallel resonant

circuit may be expressed in term s of an unloaded voltage ratio, the physical construction of the au to transform er is quite straightforw ard. O ne selects (or winds) an inductor whose inductance is the desired value of L , , places a voltage source V\ cos.a>0t across the entire inductor, and moves a probe along the length of the inductor. At the point where the voltage V2 cos oj0t measured between the probe and the bottom of the inductor is equal to n V i , a lead is soldered and the construction


M k\/Z^L2

Fig. 2.5-6 Parallel resonant transform er with tuned secondary.

of the auto transform er is complete. Clearly this transform er resonates at ft>0 = 1 / y j L t C an d possesses a turns ra tio of n.

Before ending our discussion of parallel resonant transform er circuits, we should note tha t no restrictions on the coefficient o f coupling of the physical transform er were required to derive the high-Q m odel of Fig. 2.5-5. However, if k approaches unity, Qe approaches infinity and the m odel of Fig. 2.5-5 becomes an exact model for the parallel resonant transform er, regardless o f the value of G. IT we had decided to restrict ou r a tten tion to closely coupled transform ers, we could have obtained the m odel o f Fig. 2.5-5 directly by letting k = 1 in the m odel of Fig. 2.5-3 and noting that n = l /a = ± n/ L 2/L 1 = ± M / L X.

Transformers with Tuned SecondariesIn m any applications the secondary, ra ther than the prim ary, of a transform e^ is tuned, as shown in Fig. 2.5-6. Such coupling is usually em ployed to minimize the effect of any capacitance shunting i,(t) (for instance, collector capacitance) on the tuning of the resonant circuit. Specifically, if the transform er steps up im pedance by a factor of a2, the effective capacitance shunting C is 1/a2 times the capacitance shunting i¡(t). If a = 10 and the curren t source capacity is 2 pF , only 0.02 p F reflects through to load C.

The tuned secondary is also em ployed with transistor IF amplifiers to keep the collector im pedance and, in tu rn , the collector voltage small. This low im pedance level no t only reduces the “ M iller” capacitance seen at the transisto r input, bu t also reduces the effect of ou tpu t tuning on the input impedance. Such isolation is essential if stages are to be tuned independently to obtain an overall IF transfer function.

Fig. 2.5-7 Hybrid controlled-source model replacing transformer of Fig. 2.5-7.


To obtain expressions for the input and transfer impedance of the circuit shown in F ig . 2 .5 -6 , w e firs t re p la c e th e tr a n s fo rm e r b y its h y b rid c o n tro lle d -s o u rc e m o d e l (d e r iv ed in th e a p p e n d ix to th is c h a p te r) , a s sh o w n in F ig . 2.5 7. W ith th e m o d e l(which is not based on a high-Q approximation) in place, it is immediately apparentth a t Z l2(p) = Vo2(p)/Ii(p) is th e in p u t im p e d a n c e o f a p a ra lle l R L C c irc u it m u ltip lie d by M /L 2 ; specifica lly ,

M 1L PC

Z l2ip) = --------------------- 7 - . (2.5-1)1 P 1

P + R^C + T j C

In addition,

Mz n(p) = PU - k 2)Li + — Z 12(p)

l 2(2.5-2)

1 'L 2C

We therefore observe that the effective transform ation ratio is M / L 2, the factor by which i,(f) is reduced when reflected into the secondary and the factor squared by which the parallel R L C circuit is reflected into the prim ary. We also observe that if ¿¡(f) is a periodic input current o f period T = 2n/a>0, then vo2(t) will-be a sinusoid with a frequency a>0 if QT — ® o s i n c e the tuned circuit will extract only the fundam ental com ponent of ¡,(i). O n the o ther hand, because of the series inductance (1 — k 2)L , , yol(i) will be rich in harm onic content. * •

'1 -Example 2.5-1 F o r the circuit shown in Fig. 2.5-8 derive an expression for Z 12(p) and Z 13(p) with the assum ption that the circuit is a h igh-g one.

Solution. Using the hybrid controlled-source model, we may first reflect ¡,(t) through the transform er to obtain a source of (M /L 2)i,(r) shunting R L. If we now assume

hQ t Qe > 100, Qe > 10, and Qr > 10,

M= p ( l - k 2)L l + —

C

v.l(0 MÇ

v„2(0

O


where

C i oj0C co0Cn = 7T—r^Fr> Qt = Qt =C i + C 2 n2G n2G +

q e = and c = C iC 2C l + C 2’

then we may replace the tapped capacitor by an ideal transform er to obtain the model shown in Fig. 2.5-9. F rom this circuit it is im m ediately apparent that

M 1

p c l 2c

andM 1

nL ? CZ U (P) = -_2 , (nG + GL)p , 1

P I * Tl 2c

N etw orks of the form shown in Fig. 2.5-9 are usually em ployed when a single tuned response is required, as well as a low im pedance across i,(i) and an overall voltage step-down ratio. In addition, such netw orks are employed to keep stray capacitance across both and G from affecting the tuning of the parallel R L C circuit.

2.6 THREE-WINDING PARALLEL RESONANT TRANSFORMER

In this section we shall develop a m odel for the three-winding parallel resonant transform er shown in Fig. 2.6-1. Since this circuit is a fundam ental com ponent not only in the an tenna stage, bu t also in the oscillator and IF amplifier stages of superheterodyne receivers, a simplified m odel is required to facilitate its analysis and

2.6 THREE-WINDING PARALLEL RESONANT TRANSFORMER 55

Fig. 2.6-1 Three- winding parallel resonant transformer.

design. In addition, the model aids in the understanding of the physical operation of the transform er network.

We begin by presenting an equivalent circuit for the three-winding transformer. W e then incorporate this model into the circuit of Fig. 2.6-1 and, with the results of the previous section and the assum ption of high-Q, reduce the circuit to its final simplified model.

The m ost general model for the three-winding transform er is quite complicated. However, if we place the restriction on the transform er that

M 12 M u (2.6- 1)

where M u = M n is the m utual inductance between the ith and the j th windings, th e n a s tra ig h tfo rw a rd a n d u sefu l m o d e l m a y b e o b ta in e d . E q u a tio n (2 .6 -1 ) c a n be cast in the equivalent form

k i2k 23 = k l3 (2.6- 2)

by noting that k(j = M ijl%f L iL j, where k{j is the coefficient of coupling between the /th and j th windings. Although, in general, Eq. (2.6-1) is not always exactly satisfied, it is approxim ately satisfied in a larpe num ber of practical cases. In particular, if L 2 and L 3 are closely coupled (they are usually wound on the same core), then k 23 x 1 and, of course, k l2 % /c13; hence Eq. (2.6-1) is valid. In addition, if the transform er is placed in a high-Q circuit, it can be shown, by the techniques developed in Section 2.4, that even if Eq. (2.6-1) is not satisfied, the impedances obtained on the assum ption that it is are accurate within a few percent. The proof of this statem ent is left to the interested reader. With the restriction of Eq. (2.6-1), the three-winding transform er has the term inal equivalent model shown in Fig. 2.6-2. This equivalence is readily dem onstrated by showing that both the transform er and its model have the identical set of defining equations :

Vi = p L , / , + p M l2l 2 + p M i3I 3,

V2 = p M l2I x + p L 2l 2 + p M 23I 3, (2.6-3)

y 3 = p w , 3/ i + p m 23i 2 + p l 3 i 3 .

If the transform er is now replaced by its model in the circuit of Fig. 2.6-1, theequivalent circuit shown in Fig. 2.6-3 results. It is apparent that the right-hand portion of the circuit in Fig. 2.6-3 is simply the parallel resonant au to transform er

hiP)


Fig .2 .6 -2 Term inal equivalent circuit for three-winding transformers with = M 13/M 23.

considered in Section 2.5; hence we may apply the results of that section directly. Specifically, if

h ' Q t Q e > 100 («' = k 2 3 ), Q e > 10, and Q T > 10,where

_ ® o C ( M 2 1 t^ T <o0L 3(1 - k223)G 3'J

M- JL

, 0 V , Q ) i ^ v»2<,) © £ Li<

\M2i

< G 3

Fig. 2.6-3 Equivalent circuit for parallel resonant three-winding transformer.

1t If Eq. (2.6-1) is not satisfied, QE is given by: QE =

— ^23 13A l3)®3

+6

2 .6 THREE-WINDING PARALLEL RESONANT TRANSFORMER 57

andQ t

Wo CG2 + G3(M 23/ L 2)2’

then the circuit o f Fig. 2.6-3 reduces to the final h igh-g simplified form shown in Fig. 2.6-4. With the aid of this model we may immediately write

1

KiiP) _ M izU

Z u ( p ) —* c

I,ip) p. + 1 ’ L jC

where Geq = G2 + G3(M 73/ L 2)2. In addition, we obtain

(2.6-4)

Zoü-i - ~ = "ZnW - M" " "' c

I HP)

and2 p 2 + p4? + L-.C

(2.6-5)

Z l l (p) = K i (P) r ( ,1 _ t ü 7

+-2 1 2 + + J _

V C L 2C

(2.6-6)

It is interesting to note that Z i2(p) and Z 13(p) are in the form of the im pedance of a parallel R L C circuit m ultiplied by a scale factor, whereas Z ,,(p ) contains an additional term which may be m odeled as a series inductor. Consequently, if »¡(f) were a periodic waveform (of period 2n/<o0) containing several harm onics plus the fundam ental, and Qr were 10 or greater, vo2(t) and ro3(f) would be sinusoidal in form, whereas t;ol(f) would be a periodic function with considerable harm onic content. This effect is due to the series inductor o f Z , 1(p), which, unlike the parallel R L C circuit, does no t appear as a small im pedance at the harm onics of w 0.

Example 2.6-1 F o r the circuit shown in Fig. 2.6-5 determ ine the value o f M 23 w hich maximizes uo3(t). Also determ ine (with the value of M 23 found) an expression for vo2(t) and t>o3(f) and values for Qr and BW.

Fig. 2.6-4 High-Q three-winding tuned transformer model.

c m « » ==

l2

V„(/)

+ 6


Figure 2.6-5

*3 io n = 2 nH

AT,2= 10 nH ¿23 * 0 .5

Solution. If we assum e n'QT QE > 100, QE > 10, and QT > 10, we mpty em ploy the model of Fig. 2.6-4 om itting L 2 and C (which resonate in this case apco0 = 107 rad/sec) as show n in Fig. 2.6-6. N ow vo3(t) is m axim ized if n is chos6n to m atch the 10 Q resistor to the 400 kQ resistor, i.e., if

M ,n = 1 23 / i o n

400 kQ0.005.

Therefore,A f23^ 100pH x 0.005 = 3 pH , and,since fe23 = 0.5, L 3 = M \ 3f k \ 3L 2 = 0.01 pH . Consequently,

Qb = 1

and~ ^23)^3

Qt = ^ Ì ( I O Q ) = 400,

and thus n'QT QE = k 23QT QE = 80,000/3.To obtain an expression for t’o3 we reflect the input curren t source and the

400-kQ resistor across the 10 Q ou tpu t resistor. The reflected resistance is (400 kQ)n2 = 10 Q and the reflected current source is

M 12 ^2 L , M

(sin 107f)m A = 20 mA sin 107f;23

V „ 2 < 0

Figure 2.6-6

h ence

v<,3(t) = (20 mA)(5 fi) sin 107r = 0.1 V sin 107f

and

vo2(t) = = 2 0 V sin 1 0 7t ,n

Finally, by observing that R 3 reflects into the secondary as 400 kfi, we obtain

2 0 0 k f iQr = — =--------------------------= 200

(107 rad/sec)(100/iH)

(a sligh tly h igh er v alu e th an o n e w ould n o rm a lly exp ect from a n in d u cto r) and

BW = ! ° ^ - C = 5 x 104 ™ d /* c 200

S in ce n Q t Q e » 100, Q e > 10, an d Q T > 10, use o f the m o d el is ju stified .

2 .6 PROBLEMS 59

PR O B LEM S

2.1 For the transformer with the loss terms shown in Figure 2 .P-1, determine the value of L 2 required to match an 8 ft load to a 3200 Q source in the midband. W hat are the upper and lower — 3 d B frequencies for this case? W hat is the open-circuit voltage transformation ratio of this transformer? W hat is the efficiency of the unit when working under the originally specified conditions?

2.2 Assume that the transformer (with the same values of k, L , and L 2) of Problem 2.1 is used between a 3 f l load and a 1200 Q source. Find the 3 dB frequencies and the efficiency.

2.3 Repeat Problem 2.2 for the case where the load is 20 Q and the source is 8000 i l

2.4 Suppose the source output in Problem 2.1 is such that 10 mW o f'ac power reaches the 8 Q load. Plot load power vs. load impedance for the case where the load varies between 2 ft and 32 ft while the other parameters of the system are maintained constant.

100 ft k 0.50 ft

Figure 2.P-1


100 pF

2'.$' Consider the “matching network” of Figure 2 .P -2 , in which is tuned to make Z x ¡{jo>) res'onate at w0 = 2 x 106 <Ll has a Q o f 100 at oi = 2 x 106 rps). Find Z j ,(jco0) and the input Qt . W hat is the approximate phase shift between vo2(t) and voi(t) at co = 2 x 106? I f the peak sinusoid current flowing into Z u 0'co) at resonance is 1 mA, what is the peak value of i>o2(f)?

2.6 Repeat Problem 2.5 for the case where the 3 0 Q load is increased to 1000 i 2 .

2.71 A capacitor with a Q o f 200 is combined with a coil with a Q o f 80 and a 20 k ii resistor to "produce a parallel tuned circuit resonant at 60 x 106 rps. Letting C = 25 pF, find the

. bandwidth o f the resultant circuit.

2-8 Repeat Problem 2.7 for the same coil and capacitor connected in series with a 10 i2 resistor to form a series resonant circuit at the same frequency. I f the power input to this circuit at resonance is 10 W, how much power is dissipated in each series element?

2.9 For the circuit of Figure 2 .P -3 . determine v„(t) at resonance. I f the capacitors are lossless and the coil loss is included in the 10 k fi resistor, what is the circuit’s QT1 Assuming that the input current generator also produces a 0.5 mA peak sinusoidal component at both 2u>0 and 3a>0 , estimate the relative distortion (each harmonic separately) at t;„(i). (The pole-zero diagram of Fig. 2 .4 -2 provides an easy way to estimate distortion at harmonic frequencies.)

2.10 Assume that i t(t) in the circuit o f Fig. 2.P -4 is a sinusoidal current with a peak value of 15 mA and a radian frequency of co0 = 4 x 106 rps and C 2 is chosen to resonate at co0.

1013 pF

¡¡(t) = (I mA) sin w0t Figure 2 .P -3

PROBLEMS 61

M R , = 250 0 i l

Find the output Voltage across Rb. M ake reasonable assumptions. State these assumptions clearly. W hat is the Q of the circuit?

2,11 For each of the netw orks shown in Fig. 2.P-5, determ ine o j 0 , Qr , BW, Z, , { j o j 0 ), and ZnUô)-

I Ki' Ki

Ki

M

/-! = 10 jiH M = 5(xH L2= 10 jiH

lkfl

+

—o

Figure 2.P 5

APPENDIX TO CHAPTER 2

TRANSFO RM ER EQUIVALENT CIRCUITS

The two-winding transform er shown in Fig. 2 .A 1 may be described completely in term s of its term inal eq u a tio n s:

VJp) = p L J M + p M I 2(p),(2.A-1)

V2(p) = p M I i ( p ) + p L 2I 2(p).

li(p) hiP)

Fig. 2. A -l Two-winding transformer.

Consequently, any netw ork which has the same defining equations as the transform er may be substituted for the transform er in any circuit in which the transform er is placed, and this substitu tion will no t affect the voltages or currents in the overall circuit. Several netw orks having the same defining equations as the transform er (and therefore referred to as term inal equivalents of the transform er) are shown in Fig. 2.A-2. By reflecting one or m ore o f the various inductors through the ideal transform ers in the circuits of Fig. 2.A-2, one can obtain a num ber of o ther term inal equivalent netw orks for the transform er.

There are several procedures for verifying that the netw orks of Fig. 2.A-2 are indeed term inal equivalents for the transform er o r for evaluating the equivalent inductances of the new networks. O ne approach is to observe tha t Eq. (2.A-1) is equivalent to a statem ent th a t the im pedance at a-a' with b-b' open is p L t , tha t the im pedance a t b-b ’ with a-a' open is p L 2, and tha t the ratio of the voltage at b-b to a current applied a t a-a' w ith b-b' open is pM. We then choose the values of the proposed equivalent circuit so tha t these three conditions are met. The values shown in Fig. 2.A-2 do m eet these conditions.

62

APPENDIX 63

aO-

L, M 2 I------------- 1 b-TVYYV>------1-------- 1-------0i • W • i

)M

aO-

ao -

(a)¿,¿2 M 2

M

b '~o

Ideal

i------------ , hr i °

¿,¿2 a /23U M

aO-

L xL2 A/2, L , A/

ao-

oO -

(b)

(1 *2)L,

k2L,

-trz!__!rzt-Ideal

b ’-o

b-o

a - k / A V L,

b1M

vZ.iL?-bd_Jrd °

Ideal

Fig. 2.A -2 Terminal equivalent networks for transformers.

MZ-l s

6'-o

(a) (b) (cl

Fig. 2.A 3 Terminal equivalent transformers.

A transform er may have a term inal equivalent network which is also a transformer. To illustrate this, Fig. 2.A-3 shows two of the many term inal equivalent networks for an au to transform er (M is the m utual inductance between L , and L2).

Term inal equivalent netw orks for the transform er of Fig. 2.A-1 may also take the form of netw orks containing controlled sources. Three such networks are shown in Fig. 2.A-4.

6 4 TRANSFORMER EQUIVALENT CIRCUITS

The dots a t the ends of the transform er windings in Figs. 2.A -1, 2.A-2, and 2.A-3 show the relative directions of induced voltages. A current flowing into a dotted winding causes a plus-to-m inus d ro p across this winding and induces a voltage in all o ther windings such that the dotted end is positive. M oving the do t to the opposite end of the transform er winding on the secondary of Fig. 2.A-1 would reverse the signs of both of the pM -term s in Eq. (2.A-1). It would also lead to a reversal o f the secondary do t on the transform ers of Fig. 2.A-2 and to the reversal of the direction of both of the generators in Figs. 2.A-4(a), (b), and (c). For the au to transform er, placing the do t a t the bo ttom of L 2 would reverse the sign of the m utual term in Fig. 2.A-3(b) and reverse the do t on the secondary of the transform er of Fig. 2.A-3(c).

h <J> ) £>-rYW'\_

(a)

iiiP)

+ (1 ~ k 2) L t

Vi(p)

h ip )

Vl(P)

h ip )

ViiP)

Fig. 2.A -4 Term inal equivalent networks containing controlled sources.

C H A P T E R 3

TRANSMISSION OF SIGNALS THROUGH NARROW BAND FILTERS

The purpose of this chapter is to examine a simplified m ethod of determ ining the ou tpu t response of a class of “ narrow band” high-frequency netw orks when these netw orks are driven by any one of a variety of useful signals. Among these signals we include steps, impulses, and various am plitude-m odulated signals centered within the passband of the filter.

F irst we define the class of netw orks to be considered, then we define the low-pass equivalent circuit for such netw orks, and finally we show tha t the response of the original netw ork to a given signal is a function of the response of the low-pass equivalent circuit either to the signal or, in the case of am plitude-m odulated signals, to the envelope of the m odulated signal. Thus, where the m ethod is applicable, the original com plicated netw ork response problem is replaced by a simplified and reasonably accurate approxim ate solution.

It has been suggested to us that such m aterial is covered elsewhere and neied not be repeated here. O ur experience has been th a t a t least a brief review o f these co.nqepts, perhaps from a viewpoint that the reader has not encountered before, is useful in understanding their use in later chapters.

3.1 LOW-PASS EQUIVALENT NETWORKS FOR SYMMETRICAL BANDPASS NETWORKS

For a general narrow band network whose transfer function is H(p) and whose passband is centered abou t to0 , the m agnitude and phase of H(ja>) vs. oj take tfee form shown in Fig. 3.1-1. If H(joj) is indeed narrow band, then the am plitude response, \H(joj)\, falls essentially to zero a short distance on either side of co0 , as shown. In

Fig. 3.1-1 Magnitude and phase plot for a narrowband network.

65

6 6 TRANSMISSION OF SIGNALS THROUGH NARROWBAND FILTERS 3.1

addition, if H(jaj) represents the transfer function of a physical netw ork with a real (not im aginary or complex) impulse response h(t), then H (-jco ) = f which isequivalent to

m - M = \H(jw)\ (3.1-1)and

6( — co) = —6(a>),

where 6(u>) = arg Equation (3.1-1) is, of course, simply the statem ent that them agnitude of a physical netw ork m ust be an even function of co and the phase must be an odd function of co.

We define the low-pass equivalent transfer function H L(jto) for H(jw) as that function whose m agnitude has the same dependence on co in the vicinity of co = 0 as \H(ja>)\ has in the vicinity of co = co0, and whose phase angle has the same dependence on co in the vicinity of co = 0 as 8(co) — O(a>0) has in the vicinity of co0 . This definition may also be expressed in the form

H L(jco) = H (jw + ja>0)e~mo,o)u(a) + co0), (3.1-2)

where u(co + co0) is the unit step function which, in Eq. (3.1-2), has the effect of rem oving the lower portion (co < 0) of H(jw). The e~Mc>o) term removes a constant phase angle 0(coo) from the phase of H L{jw). A plot of the m agnitude and phase of H L(ja>), which correspond to the m agnitude and phase of H(jw) shown in Fig. 3.1-1, appears in Fig. 3.1-2. N ote that the phase of H L{jco) is zero for co = 0.

| / / t (0)| = |//(yw o)|= H0

(J) —

arg //¿(7w) = 0,(cj) = 0(ta»+a>o)-0(wo)

Fig. 3.1-2 Magnitude and phase plot for HL(jw).

The transfer function H(Jco) is defined as being sym m etric abou t co0 if H l ( —; co) = Hf(ju>), or equivalently

\HL(--j(a)\ = \HL(ja>)\ (3.1-3)and

0L( a») = - 0 L(-(o) .

In this chapter we shall restrict our atten tion to symmetric narrow band netw orks for two reasons. First and m ost im portant, m ost physical narrow band netw orks may

t By definition, H(jw) = J’” oo/i(f )e - -'“ ' dt. If the impulse response hit) is real, then

H(-jw) =

where H*(j(o) is the complex conjugate of H(jaj).1

3.1 LOW-PASS EQUIVALENT NETWORKS 67

be closely approxim ated by symmetric transfer functions. Second, the symmetry condition, expressed by Eq. (3.1-3), ensures that the impulse response hL(t) of the low-pass equivalent netw ork is real, which is essential if H L(jio) is to be physically realizable. If H(jco) is not symmetric about co0, the low-pass equivalent transfer function is still defined by Eq. (3.1-2); however, it may not be associated with a physical netw ork and, in addition, calculations em ploying it may become somewhat m ore involved.

In addition to relating H L(jaj) to H(jco), for the symmetric narrow band filter we may obtain the inverse relationship. Specifically, with the aid of Eq. (3.1-1) we note that

H(j(o) = H,( j(o - i(o0)eje{olo) -I- H L{ja>0 + j(o)e (3.1-4)

this equation, of course, represents H L(jai), with the appropriate phase angle added, shifted up and dow n in frequency by an am ount co0-

To solidify our ideas on the relationship between H ,(jw ) and H(jio) we shall develop low-pass equivalent netw orks for several physical narrow band networks. We consider first the parallel R L C circuit shown in Fig. 3.1-3. From Section 2.1 we recall that at resonance Z t l (./a)0) is purely resistive; hence 6((o0) = 0. In addition, we recall that w ith Q T = oj0R C > 10, which is a necessary condition for a narrow bandw idth, Z n {jco) is closely approxim ated by (cf. Eq. 2.2-9)

Z nO 'co) =R

1 + j~(Ü — 0)n

OJ > 0,

w here (o0 = l /v /L C a n d a = 1/2RC. Consequently, with 0(wo) = 0 ,E q .(3.1-2)yields

, R. . ,1 +J CJ / I X

. (3.1-5)

Kig. 3.1 3 Parallel RLC circuit and its low-pass equivalent.

u>o =v 'Tc

Q j~ u)qR C > 10

o

6 8 TRANSMISSION OF SIGNALS THROUGH NARROWBAND FILTERS 3.1

Fig. 3.1-4 N arrow band circuit and its low-pass equivalent.

which we recognize as the input im pedance of the low-pass equivalent netw ork also shown in Fig. 3.1-3.

As a second exam ple we consider the circuit shown in Fig. 3.1 4. For this circuit the voltage transfer function H(p) is given by

Hip)V2ip) w l

(3.1-6)Vi(p) p2 + lap + o>l '

where co0 = 1 /s/ l C and a = 1/2RC. A pole-zero diagram of H(p) and a sketch of

Fig. 3.1-5 Pole zero diagram o f H(p) and plot o f \H(ja>)\ and arg (Hjoj) vs. w.

3.1 LOW-PASS EQUIVALENT NETWORKS 6 9

\H(jco)\ and arg H(jco), for the case where w 0/2ct = co0R C = QT > 10, are shown in Fig . 3 .1 -5 . W h e n th e c o m p le x p o le s a re c lo se to th e im a g in a ry ax is (a n ecessa rycondition for the netw ork to have a narrow bandwidth), then H(jco) is significantlydifferent from zero only in the vicinity of ± oj0. Therefore, when graphicallyevaluating H( jco) for to > 0 from the pole-zero diagram , we can closely approxim ate the phasor pP2 draw n from p2 to the im aginary axis by

pP2 * (3.1-7)

over the significant frequency range in the vicinity of co0- 1° addition, the phasor from p l to a point co on the im aginary axis may be written as

Pp. = a + j(°> ~ P) ~ a + 7(a> - w0). (3.1-8)

Consequently, for co > 0 and QT > 10, H(ju>) may be closely approxim ated by

S F a ) fe - jinl2) QTe~ MKl2)

PpiPp 2oj0at \ 1 + j -(O co,

1 + j -C0 COn

(3.1-9)

If we now note that 0(coo) = —n/2 and employ Eq. (3.1-2), we obtain

Qt (3.1-10)

A non-unique low-pass equivalent netw ork having this transfer function is shown also in Fig. 3.1-4. The ideal transform er is required to provide the voltage amplification of Qt a t co = 0, which is the am plification of the bandpass netw ork at co = a»0. The bandw idth BW of H(jo}) is 2a, which is exactly twice the bandw idth of H L{jca).

Fig. 3 .1-6 Model for two-stage IF strip and its low-pass equivalent.

70 TRANSMISSION OF SIGNALS THROUGH NARROWBAND FILTERS 3.2

If two or m ore symmetric noninteracting narrow band filters, each of which has a center frequency of co0, are connected in cascade, then the low-pass equivalent of the com posite filter is obtained by cascading the low-pass equivalents of the individual filters. Figure 3.1-6 illustrates the low-pass equivalent netw ork com prising two cascaded, noninteracting parallel R L C circuits.

3.2 IM PU L SE AND ST E P R ESPO N SE

W ith the aid of the results of Section 3.1 we can represent the impulse and step response for a sym m etric narrow band filter as a function of the impulse response hL{t) of the equivalent low-pass filter. Such a representation perm its one to analyze the m uch simpler low-pass equivalent filter to obtain corresponding results for the bandpass filter.

If we designate h(t) as the impulse response of the narrow band filter whose transfer function is given by H(ja>), then h(t) is the inverse Fourier transform of H(ja>), that is,

h(t) = - - f H(jm)ei<ot dw. (3.2-1)2 TtJ-oo

W ith the aid of Eq. (3.1-4) we can express H(ja>) in terms of its low-pass equivalent H L(j(o) to ob tain

1h(t) = — H L(ja> - joj0)ej9(mo)ej°1' da>

2n J - x

+ J~ f H L(jw o + jco)e-j9(,ao)eJo>‘ da). (3.2-2)2 n J . . a0

If, in addition, we substitute co' — co — co0 in the first integral and co' = u>0 + a> in the second integral of Eq. (3.2-2), the expression for h(t) simplifies to the desired form

y H L( ja > y m" dc,y

= 2hL(t) cos [a>0t + 0(coo)], (3.2-3)

which directly relates h(t) to hL(t), the impulse^response of the low-pass equivalent circuit.

As an application of this result, let us evaluate the impulse response z x ,(r) of the high-Q parallel R L C circuit shown in Fig. 3.1-3. For this circuit 6(a>0) = 0. The impulse response of the low-pass equivalent circuit z llL(t) (also shown in Fig. 3.1-3) is known to be

(3.2-4)

3.2 IMPULSE AND STEP RESPONSE 71

Zll(0

Fig. 3.2 1 Sketch of z, ,(f) for a parallel RLC circuit.

C

where a. = 1/2RC. Consequently, by employing Eq. (3.2-3) we obtain

2i i (0 = ^ e " ( c o s «J0f)n(r), (3.2-5)

which is exactly the expression obtained for z n (f) in Eq. (2.2-12) with m ore conventional techniques. A sketch of z, ,(f) is shown in Fig. 3.2-1.

To obtain the step response of a(t) for a narrow band filter in terms of its low-pass equivalent circuit, we first evaluate the step response in terms of H(jw), then express H(jio) in terms of H L(ja)\ and finally find a(t) in terms of hL(t).

The step response a(t) for the narrow band filter can be written in the form

where (1 / j o ) + kS(m ) is the Fourier transform of the unit step. Since for the networks under consideration the response at dc is always assumed to be zero, it follows that H(0) = 0 and therefore the ¿(w)-term may be om itted from Eq. (3.2-6).

We now express H(jw) in term s of H L(jto) and restrict our attention to regions close enoi gh to the complex poles so that near the upper complex pole j o may be replaced by ;co0 and near the lower complex pole j o may be replaced by —ju)0 . After some rearrangem ent, one obtains the step response in terms of the product of the impulse response of the low-pass equivalent circuit and a sine wave at the center frequency of the narrow band c ircu it; that is,

(3.2-6)

(3.2-7)

Evaluating Eq. (3.2-7) for a high-Q parallel R L C circuit, we obtain

a(t) = e "(sin ai0r)w(i),oj0C

where <o0 = 1 /^ /L C and a = 1/2RC.


Physically what this means is that driving the tuned circuit with a step function causes-it to “ ring” at its resonant frequency. Since there is no continuing supply of energy at this resonant frequency, the “ ringing” decays with time. The higher the Q of the circuit, the m ore cycles it takes for the decay to fall to any given percentage of the original level.

3.3 NARROW BAND NETW O RK S W ITH M O DU LATED IN PU TS

In this section we shall apply an am plitude-m odulated signal of the form s,(f) = g{t) cos oj0t, shown in Fig. 3.3-1, to the symmetric narrow band filter and dem onstrate that the filter ou tput is of the form

s0(0 = [g(0 * hL(t)] cos [aj0f + 0(cjo)].

where * denotes convolution. This result is quite significant, since it simplifies the problem of calculating the response of a bandpass filter excited by an AM wave to the problem of calculating the response of the equivalent low-pass filter excited by the envelope waveform g(f). The resultant expression provides the m odulation for the ou tput carrier cos [co0t + 0(coo)]-

To begin our developm ent we define G(a>) as the Fourier transform of g(?) and assume that |G(co)| has some form sim ilar to that shown in Fig. 3.3-2. W ith this definition of G(w) and with the aid of the “ shifting theorem ,” the Fourier transform of

s,(f) = ^ + e - * « )

takes the formG(w + to0) G(w - c d 0 )

S ,M = -------2------- + ------- 2------- ’

which is also shown in Fig. 3.3-2. We assume throughout this analysis that the two terms in the expression for Sj(«) do not overlap. This assum ption requires that the highest-frequency com ponent com of g(f) be lower than the carrier frequency oj0.

Fig. 3.3-1 Typical plot o f am plitude-m odulated waveform.

3.3 NARROWBAND NETWORKS WITH MODULATED INPUTS 73

Fig. 3.3-2 Plot o f |G M | and |S,((u)| vs. w.

1

|-OJ0 | " U)--->

This condition is alm ost always satisfied in practice, since for m ost AM systems a>m is required to be several orders of m agnitude smaller than oj0 because of the physical lim itations of the AM m odulator and dem odulator.

For the condition a>0 » a>m, we see that the spectrum of s,(f) occupies a narrow band of frequencies centered on the carrier frequency to0. This property permits frequency division m ultiplexing (FDM ), i.e., the independent com bination of many AM signals within a single channel. This is accomplished by choosing a distinct carrier frequency for each signal in such a fashion that none of the signal spectra overlap in the frequency dom ain. To extract a desired signal at the receiving end of the channel, one need only pass the com posite signal through a narrow band filte r; the filter passes the signal of interest and attenuates all others.

The filter capable of passing the desired AM signal centered at co0 would, of course, have the form shown in Fig. 3.1-1. With s,(f) applied to this filter, whose transfer function is H(jcj), the output signal s0(t) may be written in the form

s0(r) = (3.3-1)

where 1 indicates the inverse Fourier transform operation. W ith the aid of Eq. (3.1-4), which relates H(jaj) to its low-pass equivalent, the expression for s„(f) may be rewritten as

G(w + co0) G(w — oj0)s„(0 = & - ' { [H L( j w - j c o 0)eje^ + H L(ja>0 +juj)e~m<a°>]

e j9(u>0 ) e ~ j 9 ( a > o)

= ^ ~ l [HL(ja) - j i o 0)G(co - C00)] + — 2— * + /w0)G(w + w0)].

(3.3-2)

Equation (3.3 2) makes use of the fact that the upper portion of the spectrum of S,(a>) does not overlap the lower portion of the spectrum of H{ju>) and vice versa.

74 TRANSMISSION OF SIGNALS THROUGH NARROWBAND FILTERS 3 .3

lu(i)

Mt){

+R C V»S')

Fig. 3.3-3 Parallel RLC circuit with AM input.

If the inverse shifting theorem is now employed, Eq. (3.3-2) takes the desired formg j t a o t p f f l v j o ) _ |_ g - j w o t g - j $ { i o o )

s0{t) = ----------------- ' F ~ 1[HL(ja))G(w)]

= [g(0 * hL(f)] cos [w0f + 0(wo)]. (3.3-3)

N ote carefully that l [HL(ja>)G{oL>)] = g{t)* hL{t) is simply a shorthand m athem atical expression, which we refer to as the “convolution of hL(t) with g(r).” Although this convolution can be determ ined with the aid of the convolution integral, it is in general obtained in a m ore straightforw ard fashion as the ou tpu t of the network whose impulse response is hL(t) and whose input signal is g(t), that is, as the output of the low-pass equivalent netw ork excited by g(r). A few examples should clarify this point.

As a first example, let us evaluate the voltage across a high-Qr parallel R L C circuit in which a current of the form / cos a>0t is applied at t = 0. The narrow band network shown in Fig. 3.3-3 is being excited by an AM wave with an envelope g(t) = Iu(t); therefore, Eq. (3.3-3) may be employed to determ ine v„(t). As was pointed out above, g(t) * hL(t) = v0L(t) is the output of the low-pass equivalent of the parallel R L C circuit with a current Iu(t) applied at the input. The step response of the low-pass equivalent circuit is readily found to be

iV (f) = IR( 1 - e~*)u(t);

and since d(oj0) = 0, Eq. (3.3-3) (where a = 1/2RC and oj0 = \ / s/ l C) yields

L'„(f) = 7i?(l — e “ " )c o s u>0tu(t). (3.3^1)

A sketch of v0L(t) and r0(r) is given in F ig 3.3-4. We observe that the steady-state value of v0(t) is the product of the input current / cos a>0t and the resistance R of the parallel R L C circuit. This is an expected result because at the resonant frequency Z n ( j o j 0) = R. In addition, we observe that v0(t) rises tow ard its steady-state value with an envelope governed by a single time constant r = 1/a ; thus for f > 4 t, v0(t) has attained an am plitude within 2 % of its steady-state value.

3.3 N A R R O W B A N D N E T W O R K S W IT H M O D U L A T E D IN P U T S 75

v0t (l )

/r ------------------------I R

1 ta

Fig. 3.3-4 O utput waveforms for the circuits of Fig. 3.3-3.

It is apparent that the closer the complex poles of the parallel R L C circuit lie to the imaginary axis in the complex p-plane, the longer it takes the output waveform to reach steady state. Consequently, netw orks with very narrow bandw idths (a « w 0) are capable of transm itting w ithout distortion only input AM waves with slowly varying envelopes, or equivalently, AM waves whose spectra are contained within the passband of the narrow band filter. An alternative in terpretation is that an AM wave is transm itted w ithout d istortion if the spectrum of g(f) lies within the passband of H L(jio).

As a second example, let us apply a periodically gated carrier of frequency o j 0

to a high-Q parallel R L C circuit as shown in Fig. 3.3 -5. The input current may be

1

i( r )= /S (r )

/

I-37,

II I I4

I I I/- T\ 37~| 5Tt 4 4 4

Fig. 3.3-5 Parallel R L C circuit driven by gated carrier.


represented as= IS(t) cos oj0t,

where 5(f) is a periodic switching function of period 7i that has a value of either 1 or 0. With this representation it is apparent that i',(r) is an AM wave and therefore that

|!o(f) = iv (f) cos K f + fl(co0)],

where v0l(t) = IS ( t ) * h L(t) is the ou tpu t of the low-pass equivalent of the parallel R L C circuit driven by iiL(t) = IS(t). Figure 3.3-5 also illustrates the low-pass equivalent circuit for determ ining v0L(t).

If we assume that T j/2 > 4 t = 4/a (i.e., that the circuit reaches steady state in each interval of duration Ti/2), then we can write vOL(t) in the form

vOL(t) = IR[ 1 - + _ r j A < t < T,/4,

= I R e ~x(t- Tl/4), TJA < t < 3T,/4,

= IR[ 1 - 3 ^ 4 < f < 57^ ( 3 . 3 - 5 )

and in turn v0(t) = v0L(t) cos w 0t. A sketch of v0L(t) and v0{t) appears in Fig. 3.3-5 to the right o f the corresponding circuits. N ote th a t an ou tpu t exists for the parallel R L C circuit during the intervals o f tim e during which there is no input. This phenom enon is clearly the result of the oscillatory decay of a h igh-g circuit which has acquired energy from an input signal during som e previous interval of time.

As a third example, let us apply a sinusoidally m odulated AM signal of the form s,(i) = A[l + m cos a>mf] cos u>01

to a general symmetric narrow band netw ork whose passband is centered about a>0. Figure 3.3-6 illustrates the waveform of s,(f). The param eter m for this form of AM

Fig. 3.3 6 Plot of sinusoidally modulated AM wave.

3.3 NARROWBAND NETWORKS WITH MODULATED INPUTS 77

wave is defined as the m odulation index, and A is defined as the unm odulated carrier amplitude. The m odulation index m may be related to the waveform s,(r) by the relationship

m = (13-6)

where C is the m axim um peak-to-peak value of s,(i) and B is the m inim um peak-to- peak value of s,(f). If B — 0, then m = 1 and s,(f) is said to be 100% m odulated. In general the percent m odulation is given by

% m odulation = m x 100%. (3.3-7)

W ith s,(f) applied to the symmetric narrow band network with transfer function H(jw), the ou tpu t of the netw ork s j t ) is given by

s0(f) = [,4(1 + m cos comt) * hL(t)] cos [co0t + 0(coo)]- (3.3-8)

However,

A * hL(t) = A H l(0)

and

Am cos a)mt * hL(t) = Am\HL(jmm)\ cos [ajmt + 0L(coJ],

where H L{ja)) is the low-pass equivalent transfer function of H(jtu) and dL(a>) = arg H ,( jo>). Therefore, s„(l) takes the form of a sinusoidally m odulated AM signal given by

s0(t) = / / t ( 0 ) |l + cos ["mi + ^L(wm) ] | cos [co0t + d(w0)]. (3.3-9)

We note that the m odulation index m'(a>m) of s„(t) [obtained with the aid of Eq.(3.3-6)] is given by

\ H l ( M

Ht(0)

and that the ou tpu t m odulation is shifted in phase by 9L(a)m).

Example 3.3-1 F o r the circuit shown in Fig. 3.3-7, determ ine an expression for v„(t).

Solution. For the parallel R L C circuit,

1(On =

y i c= 107 rad/sec and QT = co0R C = 10.

Since the circuit is symmetric about u>0 we can obtain the envelope v„Jt) of v0(t) by passing (5m A )(l + cos 5 x 105t) through the low-pass equivalent network, as shown in Fig. 3.3-8. The com ponent of v0l due to the 5 mA constant input is simply

78 TRANSMISSION OF SIGNALS THROUGH NARROWBAND FILTERS

Figure 3.3-7

Figure 3.3-8

5 raA x 1 k£i = 5 V, whereas the com ponent of v0L due to the (5 m A )cos 5 x 105i of input is [(5 V)/v/2] cos (5 x 105r — n/4). The attenuation of \ / s/ l and the phase shift of — n/4 are both due to the fact that the input cosinusoid lies exactly at the — 3dB point of the low-pass filter. Com bining the two com ponents of v0L(t) and multiplying by cos a>0t , we finally obtain

v0(t) = (5 V) 1 H— — cos ¡5 x 10sr - - \ 4

cos 107f.

This expression for v0(t) could also have been obtained by direct substitution into Eq. (3.3-9) of /I = 5 mA, H L(0) = 1 kQ, H L(joim) = (1 kfi)/(l + j) = (1 and 9(a>0) — 0.

A nother example exploring the m easurem ent of h igh-g filters appears in the appendix at the end of the chapter.

3.4 NARROW BAND N ETW O RK S W ITH PE R IO D IC IN PU TS

( )ne of the most efficient m ethods of amplifying a high-level sinusoidal signal is to convert the signal into a periodic train of narrow pulses in a nonlinear amplifier and then to pass these pulses through a narrow band filter to reconstruct the original sinusoid. In addition, one of the m ost fundam ental m ethods of generating an am plitude-m odulated wave is to control the am plitude of a periodic waveform (usually a square wave or a train of narrow pulses) and then to pass the wave through a narrow band filter to obtain a sinusoidal carrier. Since both of the above techniques

3 .4 NARROWBAND NETWORKS WITH PERIODIC INPUTS 79

employ a narrow band filter with a periodic input, we shall determ ine the ou tput s„(t) of such a filter centered at a>0 when the input signal is of the form

s,(f) = g(t)sp(f),

where sp([) = sp(l + T) is periodic with period T = 2ti/o)0 and g(t) is the low- frequency signal which controls the envelope of sp(t). For high-level amplifier, g(i) reduces to a constant.

Since sp(i) is periodic, it may be expanded in a Fourier series of the form•Xi

sp(t) = a0 + Ys an cos lw oi + bn sin nw0tn = 1

00= a0 + Z C n cos (nu)0t + 0„), (3.4-1)

n= 1

where C„ = a\ + bj and 0n = — tan~ l(bja„). The input signal to the filter therefore takes the form

GCS ,-(0 = a 0 g ( t ) + £ c n g ( t ) c o s ( n a >0{ + ¿ U (3.4-2)

n = 1

which is an infinite superposition of AM waves, each centered at a harm onic to0 . If the m axim um frequency com ponent a>m of g(f) is much less than co0 (which is alm ost true in practice), then the spectrum of each AM wave occupies a narrow band of frequencies of 2wm about its center frequency as shown in Fig. 3.4-1. If s,(r) is now passed through a narrow band filter for which \H(jna>0)\ » 0 for rt = 0, 2, 3 , 4 , . . . , then the ou tput s0(t) of the filter can be closely approxim ated by the response of the filter to only the fundam ental (n = 1) com ponent of sf(f); that is,

s0(t) x [C,g(f) cos (co0t + 0 ^ ] */i(r), (3.4—3)

where h(t) is the impulse response of the narrow band filter. If the filter is symmetric about o j 0 , then s 0( t ) , with the aid of Eq. (3.3-3), can be w ritten in the equivalent form

s0(t) = [C lg(f) * M O ] cos [a>0t + 6 , + 0 (« o)], (3.4-4)

where hL(t) is the impulse response of the low-pass equivalent filter and 0(a>) is the phase angle of the narrow band filter. For the case where g(i) is a constant or where

|S,(cj)|

|m

| r " - l — \ r '-U 'k

»_2 c Om

-m—

oj0

2 cl»„,

2oj0

2 cj„

3cu0

2 o j,„

(Jj

Fig. 3 .4 -1 Typical spectrum of s,(/).

8 0 TRANSMISSION OF SIGNALS THROUGH NARROWBAND FILTERS 3 .4

H L(jw) x H l (0) over the band of frequencies occupied by g(f), [i.e., where the low- pass equivalent filter passes g(t) undistorted], the filter ou tput simplifies to

s0(f) = C , WL(0 )g(f)cos [oj0c + + 0(ojo)]

= Ci\H(ja)0)\g(t)cos[oj0t + 0 i + 0(a»o)], (3.4-5)

which is in the expected form of an AM signal with envelope g(f). The constant C , , is, of course, the fundam ental com ponent of the original periodic waveform and \H(ja)0)\ = H l ( 0) is the transfer function of the filter in the vicinity of the fundam ental frequency.

As a specific exam ple of the above procedure, let us evaluate the ou tpu t o f the high-Q parallel R L C circuit shown in Fig. 3.4-2, which is driven by a periodic train of current impulses applied at t = 0. The input current /¡(i) has the form

i,(i) = q ^ S ( t - k T ),fc = 0

(3.4-6)

where q is the impulse strength (in coulom bs) and T = 2n/a>0 is the spacing between impulses. If we rewrite i',(i) in the equivalent form

/ ,(t) = qu(t)sp(t), (3.4-7)

where sp(f) = Y ”= _ ^ S(t — kT), it becomes apparent that the input current is an envelope-m odulated periodic waveform.

To begin our analysis, we expand Jhe periodic train of impulses sp(t) in a Fourier series to obtain

and in turn

1 2 xsp(t) = — + — £ cos nw0t

1 * n= 1

X

I¡(f) = I 0u(t) + 2 I0u(t) £ cos nco0t,

(3.4-8)

(3.4-9)

where / 0 = q /T is the average value of ¡¡(t) for f > 0. If

l ^ i % 0 for n = 0, 2,3 ,4 , .

/.(/) y A re a = q

I ft - ~

T

i,(t)=qu(l) I 8 ( r - kT)k = ~ao

—v— ^(0

‘R Vo (t)

Fig. 3.4- 2 Parallel RLC circuit driven by impulse tw in.

3 .4 NARROWBAND NETWORKS WITH PERIODIC INPUTS 81

or equivalentlyl Z i iO 'w a> 0 )| _ I Z u Q w t O o ) ! ^

for n 1,

th e n we n eed o n ly re ta in th e fu n d a m e n ta l c o m p o n e n t o f /¡(f) to o b ta in a n ex p re ss io n

for Physically, the com ponents of /,(/) in the vicinity of na)0 (n > 2) would be effectively sh o r te d to g ro u n d th ro u g h th e c a p a c i to r C a n d th u s w o u ld n o t c o n tr ib u te to r„(f), w hile th e c o m p o n e n t o f /¡(f) in th e v ic in ity o f co = 0 w o u ld be effectively shorted to ground through the inductor L and thus would not contribute to v0(t). W ith th e a s s u m p tio n th a t o n ly th e fu n d a m e n ta l c o m p o n e n t o f /¡(f) c o n tr ib u te s to i „(f) w e m ay w rite

In practice, if one has a very narrow pulse in the train (so that all the harm onic terms have nearly equal am plitudes) and a broad filter (so that the response is not down too much at the harmonics), then the previous approach will be som ewhat in erro r.f This error can be estim ated from the size of the network transfer function at the various harm onics and the size of the input signal com ponent at each harmonic.

+ From the time viewpoint, the addition of an impulse to a network is bound to cause a.stepwise, rather than a sm ooth, increment in the stored energy and hence in the waveshape appearing across the network.

v„(t) ^ [2I0u(t] Cos a)0t] * z x i(f)

= [2 / 0M(i)*2llL(r)]co sa j0f

= 2 /0R(l — e ~ a,)u(t) cos vj0t [cf. Eq. (3 .3-4)]. (3 .3 -10)

Fig. 3.4 3 Graphical evaluation of |Z, ,(7(0)1.

a


As an example, it is relatively simple to evaluate |Z , ,(jnoj)\ for any narrow band network from a sketch of its pole-zero diagram. Figure 3.4-3 shows the case for a parallel R L C c ircuit; \ Z n (jnio)\ is found as a scale factor (1 C in this case) times the m agnitude of the phasor from the zero to mo on the imaginary axis divided by the product of the m agnitudes of the phasors from the two complex poles to the same point. If QT > 10, or equivalently a « w0, we observe that

Pp, * (n - l)co0, p P2 ^ (/i + l)co0 , and p 2i = nio0 :

hence

|Z , x(jnuj0)\ (1C>P:‘ = — — ^ ----~7T- (3.4-11)Pp,Pp2 (Wo H >r - 1)

It follows that

¡ZÛncoo)! _ n _ n\Z u ( jo )0)\ uj0RC(n2 - 1) Q T{n2 - 1|

(3.4-12)

As we m ight expect, as Q T increases, the ra tio of \ Z X , ( y > i c o 0 ) | to \Z X 1 ( 7' o j 0 ) | decreases. Specifically for QT > 10,

Z ^ i jn o jo )< 0.067 for all n / 1.

Using these results one can estim ate the m inim um Q necessary to keep the rms contribution of the second- and third-harm onic terms of an ideal impulse train applied to a parallel tuned circuit below ! °0 of the fundam ental contribution to the output. Since all com ponents of an impulse pulse train are equal in magnitude, the desired value of Q is a solution of the equation

T p p V l + <8 x l ) ^ TT50-

or

Qt -

Since all o ther waveshapes have harm onics that fall off in m agnitude as the . frequency increases, it follows that any waveshape other than an impulse in the train will not require as high a value of Q to keep the distortion from the second and third harm onics dow n to 1 %. The distortion problem is considered again in a m ore formal and detailed m anner in Section 3.5.

3.5 TOTAL HARMONIC DISTORTION

If a periodic signal sp(t) o fperiod T drives a narrow band filter centered at co0 = 2n T. the filter essentially extracts the fundam ental com ponent of sp(r). However, if the filter transfer function is not identically equal to zero at the haj monies of w0. some

3.5 TOTAL HARMONIC DISTORTION 83

harm onic com ponents appear at the filter output. Specifically, if

sp(t) = C o + X C„ cos (na>0r + 6„)n = 1

(3.5-1)

and if H(jco) is the transfer function of the narrow band filter, then the filter output s0(t) has the form

GO

s0{t) = C0H(0) + £ C n\H(jno)0)\ cos [nw0t + Gn + 0(nwo)], (3.5-2)n= 1

where 6{na>0) = arg H {jnw0). In m any applications, in particular that of obtaining a pure sinusoid at the ou tpu t of an oscillator, the higher harm onics of s0(t) represent an unwanted distortion signal s^f)* where sî) may be written in the form

s / t ) = £ CJH(jn(o0)l cos [nco0t + 0„ + 6(nco0)].n= 2

( 3 . 5 - 3 )

[The C0H(0) term has been deliberately om itted from sjit), since it can always be removed by an RF choke or a blocking capacitor.]

To obtain a quantitative m easure of the effectiveness of a narrow band filter in reducing s / t ) relative to the fundam ental com ponent of s0(t), we define total harmonic distortion (THD) as the ra tio of the rms value of s / t ) to the fundam ental com ponent of s„(i). Thus

T H D = ( d)rmsS „)r,

t y [C „ \2 2Ln = 2 IcJ L \

( 3 . 5 - 4 )

where ¿(Cl/2)\H(jnw0)\2 is the rms value of sd(t). The quantity T H D given bythe above equation is exactly the m eter reading that would appear on a d istortion analyzer driven by s„(t). It is apparent tha t for a given input signal sp(t), the smaller the value of T H D the better the filter perform s in producing a pure sinusoidal output.

s / L C, Qt=<jjqRC> 10

Fig. 3 .5 -1 High-Q parallel RLC circuit driven by a periodic train o f impulses.

8 4 TRANSMISSION OF SIGNALS THROUGH NARROWBAND FILTERS 3 .5

T o illustrate the m ethod of obtain ing an expression for T H D , let us examine the ou tput v0(t) of the high-(?T parallel R L C circuit shown in Fig. 3.5-1. F o r this circuit the periodic input current may be expanded in a Fourier series of the form

00 00

‘¡(0 = q Z W - k T ) = C 0 + £ C„ cos na}0t,k,= — oo n = 1

where C„ = 2C 0 = 2q / T ; hence C J C X = 1 for n = 2, 3 , . . . . In addition, for a high-Qr parallel R L C circuit,

\Zu(jno}0)\ n\Z u (jco0)\ QT(n2 - 1)

(cf. Eq. 3.4-12); thus

T H D = — / z 1^5 r l - (3.5-5)1

6 rV n=~2 \w2 — 1

The series Y^,= 2 (n/ nl ~ I )2 is know n to converge to (1/16 + 7t2/ 12),f which when substituted into Eq. (3.5-5), yields

T i m 1 1 k 0.94T H D = —- /— + — * — —. (3.5-6)

Q j \ 16 12 Qj-

We now observe th a t for T H D less than 1 %, QT m ust be greater than 94. (A value of 76 was obtained in the previous section by considering only the second- and third- harm onic d istortion terms.)

If in lieu of a periodic train of impulses, ¿¡(i) in the circuit o f Fig. 3.5-1 is a squarewave with a peak-to-peak am plitude I, a period T = 2n/u>0 , and zero average value,then

00

ii(t) = z C 2n_ 1cos(2n - l)w 0i, (3.5-7)n = 1

where C 2n_! = 2 / ( - l ) " _ 1/ 7t(2n - 1). F o r this case (C2n_ i /C ,)2 = (l/2n - l)2 ; thus

« .5 -8 ,

The series

¿ [ s r r W r J

t A lthough the series in Eq. (3.5-5) has been obtained in closed form for this case, such a representation is in general no t possible. Fortunately, in most practical cases, the series o f Eq. (3.5-4) converges sufficiently rapidly so that the first two or three terms provide a reasonably good approxim ation to the entire series.

3.5 PROBLEMS 85

is known to converge to (tc2/48 - 3/16); hence Eq. (3.5-8) simplifies to

1 /* 3 0.135T H D = — / -------------% --------- (3 5 -9 )

Qt V 48 16 Qt ' 1

To yield less than 1 % T H D with a square-wave drive to a parallel R L C c ircu it jQt must only exceed 13.5, which is a factor of 7 lower than the value of QT required.to yield less than 1 % T H D with an impulse train drive. This reduction in the required value of QT results both from the fact that the even harm onic com ponents of a square wave are zero and from the fact that the higher harm onics of a square wave have am plitudes that are small com pared with the fundam ental am plitude. (The am plitudes of the harm onics vary inversely with frequency.) Since a periodic train of impulses is the only waveform whose harm onic am plitudes do not decrease with increasing harm onic num ber, we expect the T H D obtained with an impulse train drive to a particular narrow band filter to provide an upper bound on the T H D obtained with any other periodic input drive; hence QT = 94 ensures less than 1 % T H D for a parallel R L C circuit regardless of the form of the periodic input current drive.

PR O B LEM S

3.1 Find the low-pass equivalent circuit, from the y n (p) viewpoint, for a h igh-^ (Qt ^ 10) series RLC circuit.

i2(0

3.2 D eterm ine i2(t) in Fig. 3.P 1, assuming that w 0 = 107 rad/sec, r = 50 £1, QT = 20, and r,(t) is a 5 V peak cosine signal at w 0 = 107 rad/sec that is 100% amplitude m odulated by a radian frequency of 2.5 x 105 rad/sec.

3.3 D eterm ine the output voltages v0(t) when a unit impulse, a unit step, and a unit step m odulated sine wave off „ = 1.6 M H z are applied successively as i(r) in the circuit of Fig. 3.P-2.

3.4 Find the low-pass equivalent circuit of the high-Q circuit shown in Fig. 3.P-3.

TRANSMISSION OF SIGNALS THROUGH NARROWBAND FILTERS

500 pF

C

uj° = '7 lU~ q t = u « r c > \ o Figure 3.P-4

9.9 nH

9+

PROBLEMS 87

3.5 Find the response of the two-stage IF strip shown in Fig. 3.P--4 to an am plitude-m odulated sine wave at the resonant frequency of the tuned circuits. Assume an input m odulation of 100 % and that the carrier is centered in the passband of the filter. Find the output percentage m odulation for radian m odulation frequencies of 1/4RC, 1/2RC, and 1/RC. (Use the low- pass equivalent approach.)

3.6 A train of rectangular narrow pulses 0.1 /isec wide, 10 mA in amplitude, and spaced 628 nsec apart is applied as i,(i) to the circuit of Fig. 3.P 5. Estimate accurately the com ponent of yo2(f)a t2 x 107 rad/sec. (Hint: The curve for / 2/ / m»x for a rectangular pulse in the Appendix to C hapter 9 m ay prove helpful if the pulse train problem is unfamiliar.)

3.7 Determ ine v j l ) for the circuit of Fig. 3 .P-6 when i,(f) = (1 mA)(cos 2.5 x 105t) cos 107t.

Figure 3.P-6


H IG H -0 FILTER M EASUREM ENTS

The results of Section 3.3 find practical application in the m easurem ent of the m agnitude and phase angle, as a function of tom, of the transfer function of extremely narrow band filters, i.e„ filters having Q as high as 104. The frequency of a norm al oscillator often cannot be adjusted with the required precision to make such m easurem ents ; and even if it could be so adjusted, it would be unlikely to possess the required frequency stability to allow the m easurem ent to be made conveniently.

O n the o ther hand, a crystal oscillator with the same center frequency as the filter of interest can possess the required frequency stability. If this oscillator is am plitude-m odulated with the ou tpu t of a low-frequency constant-am plitude variable-frequency oscillator, the resultant signal of the form

s,-(f) = j4(1 + cos a}mt) cos a)0t

does provide the test signal with which to evaluate the high-Q filter. W ith s,(f) applied to the filter, the ou tput signal takes the form (Eq. 3.3-9)

sa(t) = /IH JO ) j l + cos [wmr + 0L(co J]j cos [co0f + 0(ojo)]. (3-A-l)

If = \HL(jv)m)\/HL{0) and 0L(u)m) are m easured as a function of cum and if H L(0)and 0{coo) are determ ined, then sufficient da ta exist to plot the m agnitude and phase of H(jco). Specifically, for w > 0,

\H(jw)\ = \HL(jw - w0)|,

where \HL(j<x>m)\ = m'(com)HL(0) and 0(oj) = 9 l ( o j — w 0) + O(a>0).Experimentally H L(0) and 0(coo) can be readily determ ined by any num ber of

standard m ethods with the m odulation on the crystal oscillator absent. In addition, and 0L(ajm) can be determ ined by forming a Lissajous pattern on an oscilloscope

face with s„(r) applied to the vertical channel and cos a>mt applied to the horizontal channel. A typical Lissajous pattern is shown in Fig. 3.A-1. Since the upper and lower envelopes of s0(t) form standard elliptical Lissajous patterns, 0L(tom) may be found from

= s in - 1 ^ , (3.A-2)E

88

APPENDIX 89

0i (<u„)=sin

Fig. 3.A -1 Lissajous pattern o f s0(t) vs. cos comt.

where E and F are indicated in Fig. 3.A-1. In addition, nt'(com) can be obtained as

m'(coJ = ^ + g (3 .A -3)

by em ploying Eq. (3.3-6). [The reader should convince him self of the validity of Eq. (3.A-2).] Consequently, by selecting a sufficient num ber of different values for (om, one can plot the m agnitude and phase angle of H( joj) to any degree of accuracy desired.

C H A P T E R 4

NONLINEAR CONTROLLED SOURCES

The norm al m ode of operation of the active devices discussed in this book is nonlinear. Thus the usual small-signal increm ental model for a device is not likely to be of much use either in designing o r in analyzing a circuit. The purpose of this chapter is to remedy this situation by providing adequate large-signal m odels for a num ber of useful devices. We shall accom plish this aim by exploring general m ethods of dealing with the nonlinearities tha t occur in a nu:r.Lci oi real-life” active devices o r circuits. In order to concentrate on basic concepts, we assum e the absence of device reactance; e.g., no charge storage is allowed in transistors. We also assume tha t only a single type of nonlinearity occurs in any given case. In tl c early sections, all sources are assum ed to be ideal in tha t series or parallel losse : are excluded. In later sections we exam ine the m odifications tha t occur through the inherent o r deliberate addition of resistive terms.

C hapter 5 will consider the com bination of nonlinear and reactive effects, while all the following chapters will be concerned with com plete circuits or systems.

4.1 GENERAL COMMENTS

O ne can break dow n the types of nonlinearities in to two broad classes. In one class, the input-output relationship is of the piecewise-linear or switched-linear segment fo rm ; in the other, the relationship varies gradually and lacks ab rup t changes of slope. Piecewise-linear types are always easily expressible in analytic form, as are certain of the gradual relationships. In some cases one can view a particu lar piecewise configuration as an asym ptotic limit for a gradual expression.

F o r som e device characteristics, simple analytic m eans of characterizing their relationships do no t seem to exist. However, in these cases one can always, a t least in theory, fit a polynom ial to any desired degree over any desired range. Although com puter program s to aid in such curve fitting are widely available, they should have to be resorted to only on rare occasions.

In general, one cannot resort to superposition in dealing with nonlinear circuits. This m eans tha t the dc (average-value) and ac (time-variable) com ponents of the ou tpu t signal are interrelated. Initially we deal with this problem by assum ing ideal voltage or current sources as bias supplies, so that average values m ay be fixed independently o f the tim e-variable terms. The circuit of Fig. 1.1-1 provides an example of such a bias circuit.

90

4 .2 PIECEWISE-LINEAR SOURCE, SINGLE DISCONTINUITY 91

In addition to the ac-dc interrelationships of nonlinear sources, there are input drive w aveshape-output waveshape relationships tha t norm ally defy generalization. This problem is circum vented in practice by assum ing one of several common driving waveshapes and determ ining the ou tpu ts for them. Fortunately , it tu rns out that dc plus a sine-wave drive or dc plus a square-wave drive suffices to provide a first-order approxim ation for the operation of alm ost all real circuits.

4.2 PIECEWISE-LINEAR SOURCE, SINGLE DISCONTINUITY

Consider the circuit o f Fig. 4.2-1. W hen v t > V0, i2 = G(vx — F0); when u, < V0, i'2 = 0 ; hence, if

Vi = Vb + v(t) and V„ - MOLax > K ,

then the operation occurs com pletely along the sloping portion of the characteristic and is increm entally linear in the conventional sense. In this case, superposition applies to the ac signal com ponents; tha t is, if i2 = Ib + i(t), then i(t) = Gv(t).

Fig. 4. 2-1 Piecewise-linear voltage-controlled current source.

As another special case, consider the previous circuit biased exactly a t the break point, so that Vb = K0. U nder this circum stance, the circuit perform s as an ideal half-wave rectifier. In this case, if v(t) = Vt cos tot, then the ou tpu t current consists of half-cycle sine-wave pulses of peak am plitude I p = GVX, while a square-wave input for t;(t) o f peak-to-peak am plitude 2Vl aqd period T = 2n/co yields a square- wave output also of peak value l p = GV1. N ote th a t in the sine-wave case the ou tpu t w aveshape is drastically different from the inpu t waveshape over one half-cycle, whereas it has the identical shape over the o ther half-cycle. O n the o ther hand, a square-wave inpu t yields a square-wave o u tp u t

In m any practical circuits the drive is periodic and the nonlinear device is followed by a narrow band filter o f the type discussed in C hapters 2 and 3. In these cases it is convenient to express the device ou tp u t in a Fourier-series form that clearly displays the various frequency com ponents and their phase relationships.

92 NONLINEAR CONTROLLED SOURCES 4.2

This is very easy to do when Vb = V0 and r(r) = Vx cos cot, since the Fourier- series expansion of a series of half-sine-wave pulses is given by

I I 21 21' 2(0 =■— + -£ cos cos -r~ cos 4cot + ■ ■ ■, (4.2-1)

n 2 371 1571

where l p = GVX is the peak value of i2. W ith v(t) in the form of a square wave,

1 21 21 21i2(t) = H cos cot — —— cos 3a>f 4- cos 5cot — ■ ■ ■, (4 .2 -2 )

2 7t 371 571

where again / p = GVX and the time origin is taken at the middle of the positive pulse.

It is worth noting at this point that if the current generator of Fig. 4 .2 -1 drives a parallel resonant circuit that is tuned to co and that has a parallel resistance R T and a high value of QT, then only the fundam ental term produces any significant voltage v2 across the tuned circuit. Consequently, with Vb = V0, v2(t) is given by

, . / v , g r t \sine-wave d riv e : v2 » I— ~ Ll coscui,

12 v xGRT\square-wave d riv e : v2 ~ 1------------ 1 coseot

\ 71 I

W ith both the square-wave and sine-wave drive we have obtained a sinusoidal ou tput voltage tha t is linearly related in am plitude to the input driving voltage am plitude VX. Thus a highly nonlinear device operation has been com bined with a narrow band filter to produce an overall linear amplifier. Such devices find wide application in the efficient am plification of AM waves. The “gain” of this amplifier is a function of the driving waveshape.

As we shall see in C hapter 9, amplifiers in which the ou tpu t current flows for exactly half of each input cycle are know n as Class B amplifiers. F rom the equations above, we see tha t a Class B R F am plifier tha t has a linear characteristic over its conducting half-cycle has an overall linear o u tp u t-in p u t characteristic in spite of its highly nonlinear in ternal behavior. This is but the first of m any examples in which we discover overall “ linearity” in spite of internal nonlinearities.

We now consider the general case of sine- or square-wave drives added to an arb itrary dc bias Vb. The square-wave case is trivial, since it always results in a square-wave ou tp u t and hence Eq. (4 .2 -2 ) is always valid. All tha t changes from case to case is the peak am plitude of the ou tpu t square wave. F o r example, if Vb ^ V0, then /„ = G(Vb + V X - V0).

In the sine-wave case the waveshape is a function of the bias Vb and the am plitude V ,; hence Eq. (4.2-1) is no t valid for any case except the half-wave rectifier. The interrelationship o f i2, Vb, V0, and VX is shown in Fig. 4 .2 -2 . H ere we see tha t i2 is in the form of a periodic train o f sine-wave tips of peak am plitude I p = (K, — VX)G, where Vx = V0 — Vb. In addition, if i2 is plotted vs. cot ra ther than t, then an entire

(4.2-3)

(4.2-4)

4.2 PIECEWISE-LINEAR SOURCE, SINGLE DISCONTINUITY 93

Fig. 4.2-2 Output waveshape for the circuit of Fig. 4.2-1.

cycle occupies a dura tion o f 2n and the sine-wave tip occupies a dura tion of 2<j>, which we define as the conduction angle, where

V<t> = cos

y i

We observe tha t for Vb > V0 + Vr, o r equivalently Vx < — Vt , the conduction angle occupies an entire cycle and increm entally linear operation occurs.

If we w rite a general expression for the Fourier series expansion of a tra in of sine-wave tips in the form f

oo

i2 = £ /„ cos ntot,n — 0

then this expansion may be used to determ ine the dc, fundam ental, o r harm onic conten t in the ou tp u t o f a piecewise-linear circuit driven by dc plus a sine wave.

T he algebraic expression for the coefficients /„ o f such an expansion and their asym ptotic values are presented in the appendix to this chapter. Figure 4.2-3 presents norm alized values for these coefficients in term s of the conduction angle 2<p and the peak pulse am plitude I p. The plus or m inus sign after the coefficient num ber indicates whether the coefficient is positive or negative.

As a practical m atter, it is often m ore convenient to have the Fourier series coefficients expressed in term s of V JV X than in term s of 4>, since in m ost p ro b lems </> is no t know n explicitly. This transposition of coordinate axes is readily

t We should recall tha t a cosinusoid applied to a nonlinear nonm em ory device produces a periodic ou tpu t which may be expanded in a Fourier cosine series with no sine terms.


1.000.800.60

0.40 0.30

■><-cj 0.20c

'5 SEg 0.10 “ 0.08 I 0.06 £-g 0.04

| 0.03

o 0.02

0.0100.0080.006 0.005

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360C onduction angle 2<p, degrees

Fig. 4. 2-3 Norm alized Fourier coefficients of a linear sine-wave tip pulse train vs. the conduction angle.

accom plished by em ploying the relationship cos 4> — VJV\ . In particular, l j l p for n = 0 ,1 ,2 is plo tted vs. VJV i in Fig. 4.2-4. N egative values of VJV X in this figure correspond to the case where Vb > V0 (note that Vx = V0 — Vb) and i2 has a conduction angle greater than 180°.

As an exam ple of the usefulness of these curves, consider a case in which Vx = 1.5 V, G = 1000 /¿mho, and Vx = 4.5 V. F rom the da ta of Fig. 4.2-4 and by m ultiplying by I p — - VJ = 3 mA, we obtain / 0 = 0.78 mA, = 1.32 mA,and 12 = 0.81 mA. If this current were passed through a tuned circuit o r its equivalent, then the various ou tpu t voltages could be found by perform ing one m ore simple m ultiplication for each current com ponent.

4.3 M U L TIPLE -SE G M E N T PIEC EW ISE-LIN EA R SO U R C ES

W ith large enough input drives, all practical devices eventually “saturate .” In many cases the saturation is sufficiently ab rup t tha t the input-output transfer characteristic,

v s .i i , can be m odeled by a m ultiple-segm ent piece wise-linear characteristic o f the type shown in Fig. 4.3-1.

Linear sine-wave tips

4 .3 MULTIPLE-SEGMENT PIECEWISE-LINEAR SOURCES 95

Linear sine-wave tips

N orm alized voltage offset - ÿ —

Fig. 4.2-4 Plot of l j l p vs. VJVX for n = 0,1, and 2.

Results for the case shown in Fig. 4 .3 -l(a) can be found as the superposition of the F ourier series for the positive and the negative sine-wave tip pulse trains. F o r the positive pulse train i2P o f peak am plitude Ipl = G,(V1 — Vx), we ob tain the Fourier coefficients from Fig. 4.2-4 by em ploying I Pi, and V JV i . The Fourier coefficients for the negative pulse tra in i2N, o f peak am plitude I Pl = G2(V{ + Vy), are obtained by em ploying I P2 and — VJVX (in lieu of VJVi) in the curves of Fig. 4.2-4. The negative pulse tra in is shifted 180° in phase; therefore, its Fourier series has the form

*2n = —Uo + I i cos + n) + 12 c o s (2(ot + 2n) + ■ ■ •

= - I 0 -(- cosftjf - I 2 c o s 2cot + I 3 cos 3tur — • • •. (4.3-1)

Consequently, when i2JV is com bined with i2P, the Fourier coefficients o f the odd harm onics of b o th pulse trains add algebraically, whereas the Fourier coefficients o f the even harm onics and the average value of the negative pulse tra in m ust be subtracted from the corresponding coefficients o f the positive pulse tra in to obtain the com posite Fourier coefficients.

9 6 NONLINEAR CONTROLLED SOURCES 4 .3

(a)

0

(b)

Fig. 4.3-1 Various multiple-segment piecewise-linear nonlinearities, dc plus sine-wave drive.

4 .3 MULTIPLE-SEGMENT PIECEWISE-LINEAR SOURCES 97

We should note that if G, = G2 and Vx = - Vy, no average value or even harmonics exist in the waveform of i2, since the m agnitude of the corresponding Fourier coefficients of both the positive and negative pulse trains are equal.

Results for the saturation model shown in Fig. 4.3—1(b) can be found by direct subtraction of the F ourier series for a sine-wave tip pulse tra in of peak am plitude IP2 = G3(K1 — Vy) and conduction angle 2<fr2[4>2 = co s- l(Vy/V l )] from the F ourier series for a sine-wave tip pulse tra in of peak am plitude Ipt = G3(Vt — Vx) and the conduction angle 2 0 1[ 0 1 = c o s '^ ^ /K ,) ] . Subtraction of the two Fourier series results in a Fourier series whose coefficients are found from the subtraction of the corresponding coefficients of the original two series. The coefficients o f each series are, of course, found directly from Fig. 4.2-3 or Fig. 4.2-4. In the limit as Vx becomes very large the curren t waveshape approaches a square wave; hence Eq. (4.2-2) again yields lim iting values.

Suppose, for example, tha t Vy = — Vx = 1.5 V, tha t the sine wave is varying around zero w ith an am plitude F, = 3 V, and that G3 = 500 /imho. Then

I pt = [500 x 10“ 6 x 4.5] mA = 2.25 mA,

and

/ p2 = [500 x 10“ 6 x 1.5] mA = 0.75 mA.

W ith the aid of Fig. 4.2-4 we obtain

I 0 = (2.25 mA) x 0.41 - (0.75 mA) x 0.226 = 0.75 mA

[which, of course, could be found by inspection from Fig. 4.3-l(b)],

/ t = (2.25 mA) x 0.535 - (0.75 mA) x 0.40 = 0.91 mA,

and

12 = (2.25 mA) x 0.093 - (0.75 mA) x 0.28 = 0.00 mA

(again as expected because of the symmetry).In general, the ou tpu t of any m ultiple-segm ent piecewise-linear characteristic

driven by dc plus a sine wave m ay be expressed as the superposition of several sine- wave tip pulse trains. A little ingenuity is sometimes required to accom plish this. O nce this representation is achieved, however, the coefficients o f the ou tpu t Fourier series may be obtained with the aid of the curves of Fig. 4.2-3 or Fig. 4.2-4.

Characteristics sim ilar to Fig. 4 .3 -l(a) occur in Class B and Class C amplifiers, whereas center-biased characteristics sim ilar to Fig. 4 .3-l(b) are used as limiters. If a characteristic like tha t o f Fig. 4 .3 -l(b ) is really biased in the middle, then there are no even harm onics a t the output. In addition, the norm alized fundam ental current ou tpu t I x/G$Vy varies only from 1.0 to 4¡n = 1.273 (the fundam ental o f a

98 NONLINEAR CONTROLLED SOURCES 4 .4

square wave of peak am plitude = 1) as the input am plitude varies from the breakpoint value, Vy = — Vx , to infinity. F or Vx = 2Vy,

/„, = 3G3Vy, 1P1 = G3Vy, ^ = 0 . 5 , -0 .5 ,

and thus

G s ty / j = 1.218.

Similarly, if Vx = 3.861^, then the norm alized current is 1.258. A plot of / ¡ /G 3Vy vs. VJVy is shown in Fig. 4.3-2. It is apparen t that if an am plitude-m odulated signal

is applied to such a characteristic so tha t the carrier am plitude is 10 times the breakpoint while the percentage am plitude m odulation is 80 % o r less, then the fundam ental voltage across a tuned circuit in the ou tpu t has less than a 4.5% peak-to-peak v aria tio n :

1.273 - 1.218 _ , --------- x 100 % 4.4.

1.24

Hence on the norm al peak basis the ou tp u t m odulation is less than 2.5 % o r equivalently, we have reduced the AM by m ore than 30 dB. Such circuitry will be discussed in m ore detail in C hapter 12.

4.4 SQUARE-LAW CHARACTERISTICS

A square-law voltage in-voltage out characteristic can be approxim ated by a network of diodes, resistors, and batteries. A square-law voltage in -cu rren t out characteristic is approxim ated quite closely by m any field effect transistors (FET) of both the

4.4 SQUARE-LAW CHARACTERISTICS 99

+

(t>¡2 V2

Fig. 4.4-1 Square-law source.

junction and the M O S types operating in their “constant cu rren t” region. Figure 4.4-1 shows a typical square-law netw ork for which

i2 = ’ ~ v )(4.4-1)

U vt < Vp,

where I DSS and Vp are constants sim ilar in no tation to those employed with junction-type field effect transistors. Clearly, as v x approaches Vp (which is sometimes referred to as the pinch-off voltage), i2 is reduced to zero. F o r = 0 , i2 = loss- 1° this section Vp is assum ed to be negative. Positive values of Vp merely shift the i2-t>i characteristic to the right.

Before proceeding further, we should note that when we refer to a square-law characteristic it is no t a “ tru e” square-law characteristic, but ra ther a “ h a l f ’ square- law characteristic. A true square law would be a parabola and therefore a doublevalued function of ¿2. Since such characteristics do not often exist in nature, we restrict our attention to those formed by one branch of the parabola, as given by Eq. 4.4-1. However, one m ust be sure that operation is entirely in the square-law region before substituting into the expression

■ - 1 h M 2l 2 — 1 D S S | 1 y J ■

Again, with the square-law characteristic, we find that a square-wave input leads to a square-wave o u tp u t; hence Eq. (4.2-2) is directly applicable to obtaining the Fourier series of i2 once the peak value of i2 is determined.

If, on the o ther hand, = Vh + V, cos cot and operation is within the square- law region, then

¿2 = — 2VXV, coscot -I- V \ cos2 <ot),' p

(4 .4 -2 )


term s; that is,

where

a Fourier series expansion for i 2 has only three

+ / , CO S (Ot + 1 2 cos 2cot,

= ^ b +y)- (4.4-3)

SI« 6.

CN1II (4.4-4)

^ D S S

~ V p 2 '(4.4-5)

Thus we find that if Vb is supplied by an ideal voltage source so tha t Vb is no t a function of / 0, then / j is a linear function of Vx and we can define a large-signal average transductance Gm,

hYi

IG “ i ^ * D S S t /m — TT ~~ — ~yT X9 (4.4-6)

that is independent of the drive voltage. We can define such a large-signal transconductance for any type of characteristic ; however, in general, it is not a constant but a function of Vt . W hen Gm is independent of Vt and i2 drives a high -QT parallel resonant circuit tuned to the fundam ental frequency, then variations in the am plitude of the ou tpu t voltage are linearly related to variations in Vi ; hence, here again, the overall transfer characteristic is linear and the device perform s as a linear amplifier for AM signals.

The square-law characteristic has the interesting property that the small-signal transconductance gm at any particular Q-point is equal to the large-signal average transconductance Gm a t th a t sam e Q-point. T o dem onstrate this property, we evaluate gm in the form

= ^ gm dv1

- 2 1 D S S {V, - v j-21D S S i

V 2v , = V b y p

(4.4-7)

which is identical to Gm given by Eq. (4.4-6). A plot of gm = Gm vs. bias voltage Vb is shown in Fig. 4.4-2. It should be m ade clear that gm is an increm ental slope at a point on the current-vs.-voltage curve, while Gm = IJ V ^ is a ra tio of fundam ental ou tpu t current to fundam ental input voltage at a particular operating point.

As an example, suppose that Vb = Vp/2, or equivalently Vx = Vp/2\ then

I ,gm

v. v p

and = I D S S Y j _

2 V.

W hen Vi is very small with respect to \VP\, then I : w ideband am plification occurs. As Vt increases, 7-

vanishes and low -distortion increases and distortionless

4 .4 SQUARE-LAW CHARACTERISTICS 101

Fig. 4.4-2 Linear transconductances for square-law current characteristic (FET).

wideband am plification is impossible ; however, if a narrow band filter is employed, then linear am plification is possible from Vx = 0 to V{ = \Vp\/2, a t which point operation exceeds the square-law region of the characteristic and Gm becomes a nonlinear function of Vt .

In the general case, the second-harm onic d istortion in the ou tput current as a function of drive voltage is given by

A4 K

K SmO 4 K gm ’

(4.4-8)

where gm0 is the value o fg m obtained for Vb = 0. Thus, if Vb = VpJ2, then 1 % distortion arises when V1 = |Kp|/50 = IKJ/25. This m eans that linear small-signal w ideband amplification is restricted to small drive voltages. For example, \f'Vp = — 4 V and Vb = - 2 V, then to keep the d istortion below 1 % K, m ust be less than 80 mV. As Eq. (4.4-8) indicates, increasing the gain by changing Vb so as to increase gm


does decrease the d istortion, bu t no t enough to m ake this type of characteristic well suited to producing wideband amplifiers with large-signal handling capabilities.

In practical devices, the characteristic is likely to depart som ewhat from square law at its extreme values; hence, if the above results are to apply, drive voltages m ust be small enough to stay out of these regions. Even if Eq. (4.4-1) does accurately describe the device, if v x = Vb + Vt cos cot exceeds Vp for some portion of a cycle the above results are not valid. In particular, the device is off for part of the cycle and square law for the rem ainder of the cycle, as illustrated in Fig. 4.4—3. Thus we see that the current is a periodic train of square-law sine-wave tips of peak am plitude

b = - K )2-V P

Fig.4.4-4 Normalized Fourier coefficients of a square-law sine-wave tip pulse train vs. the conduction angle.

4 .4 SQUARE-LAW CHARACTERISTICS 103

In addition, the conduction angle 2(f) (i.e., the angular portion of the cycle during which 12 is different from zero) is given by

2(j> — 2 co s ' 1' i

U nder this m ode of operation the Fourier series for i2 no longer consists of three term s but ra ther an infinite num ber o f terms. The coefficients of these term s of the Fourier series as functions of I p and <p are presented in the appendix to this chapter. Figure 4.4—4 presents the norm alized values for the first three coefficients in term s of conduction angle 2<p.

In addition. Fig. 4.4-5 presents the norm alized Fourier coefficients l j l p plotted vs. V JVX — cos (j> for n = 0 ,1 , and 2. Negative values of VX)VX are obtained when

Square-law sine-wave tips

¡E8

ou.

oZ

0.0100.0090.0080.0070.0060 .0 0 5

1 .0 - 0.8 0.6 0.4 0.2

Normalized voltage offset

Fig. 4.4-5 Plot /„//_ vs. VJV\ for n — 0, 1, 2.


the bias voltage Vb lies within the square-law region, or equivalently, when the conduction angle is greater than 180°. The point VJV^ = — 1 corresponds to com plete operation w ithin the square-law region and a conduction angle of 360°; consequently, for values of V JV X < — 1, the operation is well within the square-law region and the Fourier coefficients (only three) m ust be found from Eqs. (4.4-3), (4.4—4), and (4.4-5), which may be m anipulated into the equivalent forms

, = . 1 + ( v j v , ) 20 p(i - y j v . r

, = , - i j y j v i )1 pd - v j v , r

i

h = ip (i - v j v t f '

which are valid for V JVX < — 1 and for which, of course,

L = ^ ( V , - VX

It is of interest to note that for V JV t = — 1 substituted into Eqs. (4.4-9), (4.4-10), and (4.4-11), I 0/ I p = §, l j l p = i , and I 2/ I p = s- These are exactly the values shown in Fig. 4.4-5 for VJV^ = — 1; hence the Fourier coefficients /„ vary continuously as VJVX decreases beyond — 1.

These square-law sine-wave tip characteristics will be quite useful in designing self-limiting field effect transistor oscillators, as well as in explaining the action in large-signal F E T R F amplifiers.

4.5 T H E EX PO N EN TIA L C H A RA CTERISTIC

A very good approxim ation to the current em itted across a forward-biased P-N junction in adjunction diode or transistor is

i2 = I sev'qikT, (4.5-1)

where kT/q is approxim ately 26 mV for T = 300°K. If we assume tha t the transistor alpha is independent of current and ignore for the present all internal resistive voltage drops, then by choosing I s appropriately we can use Eq. (4.5-1) to represent em itter, collector, or base currents. (The equation is not “exact” in the reverse ' d irection ; however, for our purposes this “nonexactness” is a th ird-order effect.)

A typical voltage in -curren t out exponential characteristic is obtained from the circuit of Fig. 4.5-1. If i?! = Vb + v(r) is applied to the device, i2(t) has the form / dc + *(0. where / dc and i(t) are functions of both Vb and v(t). In our previous examples we assum ed tha t Vb and v(t) were supplied from independent sources and derived our results in term s of these quantities. In this case, however, we assum e tha t / dc and v{t) are supplied from independent sources and present the results in term s of these

(4.4-9)

(4.4-10)

(4.4-11)

4 .5 THE EXPONENTIAL CHARACTERISTIC 105

o-

o ■o

■O

0 >

o-f-

-o

Fig. 4.5-1 Exponentially controlled current source.

quantities. The reason for this choice is apparen t when we realize that in a well- designed transistor circuit we control the bias (or average) current and the ac junction vo ltage; hence these are the quantities we know.

W ith t’,(i) = Vh + iit) applied to the circuit of Fig. 4.5-1, i2(t) is given by

For small-signal operation, i.e., values of v(t) for which qv(t)/kT « 1, ¡2(f) reduces to the form

w h e re / dc = Is evVblkT, a n d i(t) = q Idcv(t)/kT. C le a rly i'(f) a n d v(t) a re lin ea rly re la te d by a small-signal transconductance

which is a linear function of bias current. We note that small-signal linear “gain” control can be achieved by controlling the bias current I dc. This linear gain control will be seen to have many useful applications.

Square-W ave InputIf t>(i) is a symmetrical square wave of peak am plitude K,, it is apparen t that i2 is also a square wave with an upper level

i 2 = l s g (4.5-2)

= t>(0 = qi dc gm i(t) k T

I m = I se*v»lkTe‘>v 'lkT

and a lower level


Thus the average value of i2 is given by

Jdc = / vf Im = I se“yi’lkT cosh x (4.5-3)

and the peak-to-peak value of i2 is given by

IPP = = 2IseqVb/kT sinh x, .• (4 .5 ^ )

where x = qVxj k T norm alizes the input square-wave am plitude to 26 mV (at T = 300° K). By com bining Eqs. (4.5-3) and (4.5-4), we obtain

! Pp = 2 / d c ta n h *• (4 .5 -5 )

The quantity I pp/2 Idc is plotted in Fig. 4.5-2 vs. x.

For x « 1, tanh x * x and Eq. (4.5-5) reduces to

If = I A<:x = ^ r Vl = g mVl , (4.5-6)

which, of course, is just the small-signal relationship between the peak am plitudeof the ou tput and input square waves. For large values of x (x > 2.5), tanh x is within 1 % of unity and

*~f = be- (4.5-7)

This relationship makes sense. For large values of Vt the negative portion of theinput square wave effectively cuts off i2 ; hence i2 varies from zero to I pp and thus has an average value of ¡pp/2.

It is apparent that, for large values of x, the ou tput square-wave am plitude is independent of the input square-wave am plitude and therefore the circuit functions as a limiter. Conversely, for x « 1 the circuit performs as a linear amplifier.

Regardless of the value of x, however, the peak am plitude of the ou tput square wave is linearly related to / dc; consequently, by controlling / dc we can linearly am plitude-m odulate the ou tpu t square wave. If, in addition, we pass i2(t) through a h igh-g tuned circuit, we obtain an ou tpu t voltage which is an am plitude-m odulated


sine wave. In particular, if i2(t) drives a h igh-g parallel R L C circuit which has resistance R r and is tuned to the fundam ental of i2, then the ou tput voltage v2(t) is given by (cf. Eq. 4.2-2)

v2(t) = — tanh x cos cot. (4.5-8)n

It is assumed here that if / dc is a function of time, the bandw idth of the tuned circuit is sufficient to pass the AM sideband inform ation.

S in e -W av e In p u t

If i’,(f) = Vb + Vt cos coi, then

¡2(t) = J^qVblkTpqVôsiot/kT

_ j ^ q V b l k T p X c o s tot

where again x = qVi/kT. F o r cot = 0, 2n, 4 n , . . . , i2(t) a ttains its peak value of Ip = ¡seqVh'kTex. T o observe the form of i2(t) for other values of t we normalize i2(t) to I p and plot

vs. cot with x as a param eter, as shown in Fig. 4.5-3. Note that for small values of x the ou tput current is alm ost cosinusoidal, as would be expected; however, as x increases, the ou tpu t current becomes pulselike in form.

Because of the exponential nature of i2, it is no t possible to define a conduction angle for these pulses in a conventional sense: however, we may define a fictitious conduction angle as the angular portion of the cycle for which i2(t)/lp > 0.05. This

U J t——

Fig. 4.5-3 Plot of ¡2(r)//p vs. (of with x = qVJkT = K1/(26 mV) as a parameter.


conduction angle m ay be defined as 2<j>, where 4> is the solution o f the equation_JCCOS<j>

= 0.05, (4.5-9)

</> = cos 1 1 +In 0.05

or equivalently,

x I \ x

A plo t o f this conduction angle vs. x is given in Fig. 4.5^4.

cos 1 1 (4.5-10)

Fig. 4.5-4 Plot of conduction angle 2tj> vs. x.

The pulse shape is a function only of the norm alized ac input voltage (x), and n o t o f the bias level. This, o f course, is a basic property o f an exponential characteristic.

The current i2(t) = I pexcosco'/ex m ay be expanded in a F ourier series of the formOO

i2(t) = £ C„ cos not,n = 0

where

- = rex \2 n } _ „

~ x co s 6 d 0 | = ÿ l 0(x),

21.excos9 cos nO, dfl) = —^/„(x),

e

(4.5-11)

(4.5-12)

and I n{x) is a m odified Bessel function of order n and argum ent x. The modified Bessel functions should no t frighten the reader, since they represent merely the tabulated results o f the num erical in tegrations o f the integrals of Eqs. (4.5-11) and (4.5-12) with x as a param eter. V arious useful properties o f the modified Bessel functions, including their tabulation , are presented in the Appendix a t the back of the book.

4.5 THE EXPONENTIAL CHARACTERISTIC 109

By noting that

Co — oM — ¡ic< e

we can write i2(t) as a function of / dc in the form

l„(xoW

r °° 2 1 ix)hU) = /d c 1 + Z 7 - 7 - r C O S n a j t

L n = 1 * O'

Table 4.5-1 Tabulation of modified Bessel function ratios vs. x

X 21 Ax) /„(*)

212{x) 1 o(x)

2 h(x) ¡o(x)

/ jW/l(x)

0.0 0.0 0.0 0.0 0.00.1 0.0999 0.0024 - 0.0240.5 0.4850 0.0600 0.0050 0.1241.0 0.8928 0.2144 0.0350 0.2402.0 1.3955 0.6045 0.1866 0.4333.0 1.6200 0.9200 0.3933 0.5685.0 1.7868 1.2853 0.7585 0.7197.0 1.8511 1.4711 1.0104 0.795

10.0 1.8972 1.6206 1.2490 0.85414.0 1.9272 1.7247 1.4344 0.89520.0 1.9493 1.8051 1.5883 0.926

(4.5-1 la)

(4.5-13)

Table 4.5-1 presents da ta for 21 i(x)/10{x), 2/ 2(x )//0(x), 2 I3(x)/I0(x), and I 2(x)/Ii(x) all vs. x. Figure 4.5-5 presents a plot of these same data. [These da ta have been calculated from the values of J„(x) given in the Appendix at the back of the book.] As shown in the Appendix, for small values of x, J0(x) -*• 1, /i(x ) x/2, and I„(x) -» 0

Fig. 4.5-5 Modified Bessel function ratios vs. x = qVJkT.


(actually some pow er of x greater than unity) ; hence, when x approaches zero,

<2(0 « J d c U + * cos Mt) = /d c + imY1 cos (4.5-14)

which is the expected small-signal response.For large values of x, the pulse w idth of i2 becomes narrow and the fundam ental

com ponent is seen to approach twice the dc value, as expected; thus again the exponential characteristic is useful as a lim iter so long as the m inim um am plitude does no t cause x to d rop below say 5. F o r example, w ith a norm alized carrier am plitude of x = 10, 50% am plitude m odulation in the input (m — 0.5) causes less than 4 % ou tpu t am plitude m odulation (m„ = 0.04).

W hen x is large, I 0(x) approaches ex/ s/ l n x (this equality is w ithin 6 % for x > 3 and w ithin 3 % for x > 5). H ence from Eq. (4.5-1 la) the relation between the dc com ponent and the peak curren t for x > 3 is given by

l P « I i Cy /2nx .

Thus in a circuit in which / dc is held constan t a t 1 mA (and in which resistive drops are negligible), a peak sinusoidal base-em itter voltage drive o f 260 mV will cause a current peak of 7.95 mA, while the fundam ental com ponent will have a peak value o f only 1.90 mA.

Figure 4.5-5 conveys all sorts of additional interesting inform ation abou t the exponential characteristic. If / dc is presum ed constant, §0 th a t it is no t a function o f Vlt then the 2 [ /,(x ) //0(x)] curve indicates th a t a narrow band “ linear” amplifier for the I ! com ponent is feasible only for values o f x less than unity. F rom the

curve one sees that, for w ideband linear amplifiers, x m ust be less than 0.1 ju st to keep the second-harm onic distortion below 2.5%. [N ote as x -» 0 , / 2(x)//!(x) -> x /4 ; hence x = 0.1 implies / 2(x)/7i(x) = 0.025.] Therefore, such a characteristic is useful only in an absolutely linear fashion when input voltages are below 2.5 mV. (The schemes that one employs to design amplifiers capable ofhandling large-signal inputs w ithout d istortion will be discussed in subsequentsections.)

W hen the exponential characteristic is followed by a tuned circuit which extracts the fundam ental com ponent of i2(t), it is again convenient to define a large-signal average fundam ental transconductance Gm(x) as

Gm(x) = = lf ^ (4.5-15)Vi V1 I 0(x) k T x / 0(x)

W ith this definition the ou tpu t voltage v2(t) can be w ritten as

v2(t) = ~ G m{x)ViRT cos cot, (4.5-16)

where R T is the resistance o f the tuned circuit a t resonance. If / dc is independent of V i , then

^ t i cj <?Aic

Ki — 0 and -j^Y — Sn

4.5 THE EXPONENTIAL CHARACTERISTIC 111

Thus Gm(x) as given by Eq. (4.5-15) reduces to

Table 4.5-2 presents values for Gm(x)/gm = 2 /,(x ) /x /0(x) vs. x. In addition, these data are plotted in Fig. 4.5-6.

Table 4.5-2 Tabulation of 2 /,(x ) /x /0(x) vs. x

2 / l(x) = Gj x )X

Xl oM gm

0.0 1.00.2 0.9950.5 0.9701.0 0.8932.0 0.6983.0 0.5404.0 0.4325.0 0.3576.0 0.3047.0 0.2648.0 0.2349.0 0.210

10.0 0.19015.0 0.12920.0 0.0975

XFig. 4.5-6 Plot of 21 l{x) /xl0{x) vs. x.


By the tim e x = 1, Gm(x)/gm is already dow n by 1 dB ; hence we again observe that “ linear” narrow band AM amplifiers require It lmax < 26 mV. However, the d rop in Gm(x) with increasing drive is useful in som e circuits. F o r example, it allows the ou tpu t am plitude of a sine-wave oscillator to stabilize. As the oscillations begin to grow, the loop gain decreases to the point where a further increase in the oscillation am plitude is not possible.

W e should note finally that i2(i), given by Eq. (4.5-13), is directly proportional to / dc ; hence, here again, by controlling / dc we can achieve m ultiplication or am plitude m odulation independent of the value of Vt (or x). In C hapter 8 we shall consider in detail how this control is accomplished.

-1 0 V

Figure 4.5-7

Example 4.5-1 F o r the narrow band am plifier shown in Fig. 4.5-7, determ ine an expression for v0(t). Also determ ine the peak em itter current and its conduction angle.

Solution. The base-em itter voltage consists of the input cosine plus whatever dc voltage VAc builds up across the capacitor C x; hence

V E B = Vic + ï cos 107t

and

i£ = J^qVtJkT^.ecoslOi,

0 9.6 cos 10 7r= ------


re /o (9 .6 )2/,(9.6)

+ T o fW ° s +e

T h e 2 .6 -m A cu rren t so u rce m ust eq u al th e av erag e valu e o f iE ; therefo re ,

l„lo(9-6)l Ae = 2 .6 m Ae9-6 "dc

an d the fu n d am en ta l co m p o n e n t /£1 is eq u a l to

. n . A >2/,(9.6)/F1 = (2 .6 m A )------------

£1 /0(9.6)

= (2 .6 )( 1.88) m A =, 4 .8 8 mA.

S in ce the o u tp u t-tu n ed c ircu it re so n a te s at w 0 = l , y L C = 10 7 rad/sec and h as Q r = a>0RC = 20 , o n ly th e fu n d am en ta l co m p o n e n t o f iE co n tr ib u te s to the o u tp u t; th ere fo re , if a % 1,

r 0 = (- 10 V) + (2 ki))(a)(4.88 mA) cos 107r = ( - 10 V) + (9.8 V) cos 10Y

In addition, since Ip = / dce9 6/ / 0(9.6), then by interpolating from Table A-2, o r using the relationship /„ ^ ^ / 2 n x l dc, we obtain

2.6 m A

lp = ~TnT~ =

A lso , from E q . (4 .5 -1 0 ) and by n o tin g th a t x = 9 .6 , we o b ta in a c o n d u ctio n an g le o f 96°.

A s a n a ltern a tiv e a p p ro a ch to o b ta in in g r„ , we o b serv e th a t the sm all-s ig n a l tra n s co n d u cta n ce gm is

gm = «gin = - J j T 0.1U.

C o n seq u en tly , th e larg e-s ig n al av erag e tra n sc o n d u cta n ce (cf. F ig . 4 .5 6) is given by

Gm = ( 0 . l 9 ? ) ( 0 . l ) U = 0 .0 1 9 7 15.

Therefore, the ou tput voltage has the value

<'„(0 = K c + G m(2 k Q )[ ( j V) c o s 10 7i] == ( — 10 V ) + (9 .8 V ) co s 107f,

which, of course, agrees with the previous evaluation.

1 1 4 NONLINEAR CONTROLLED SOURCES 4 .6

4.6 THE DIFFERENTIAL CHARACTERISTIC

If two transistors with characteristics of the type outlined by Eq. (4.5-1) are connected in a differential configuration so that their total em itter current, I k, is supplied by a constant current source (norm ally another transistor), then a very useful and easily integrable circuit results. The differentially connected circuit is shown in Fig. 4.6-1. This circuit has a distinct nonlinear characteristic th a t should be added to ou r collection of useful nonlinearities.

Fig. 4.6-1 Differential configuration.

To obtain an expression for the nonlinear characteristic of the differentially connected circuit, we note that

and

¿i = I SieVBEiqlkT, i2 = IS2eVBFiqlkT,

V B E 1 ~ V B E 2 — V 1 ~ V 2 -

(4.6-1)

Therefore, if / S1 = I S2 (which should be true if b o th transistors are integrated on the same “chip” and are identical in size and construction), then

i l _ g(vi~V2HlkTh

If we now set i t + i2 = I k, we obtain

f - Ik2 1 + e z

and ¿! = h1 +

(4.6-2)

(4.6-3)

where z = (vt — v2)q/kT. The norm alized nonlinear characteristics i j l k and i2/ I k vs. z are tabulated in Table 4.6-1 and plo tted inV ig. 4.6-2.

4.6 THE DIFFERENTIAL CHARACTERISTIC 115

Table 4.6-1

z h i Ik *2/Jfc !>, - r 2

+ 2.94 0.95 0.05 + 77 mV+ 2.20 0.90 0.10 + 57 mV+ 1.10 0.75 0.25 + 28.6mV+ 0.405 0.60 0.40 + 10.4 mV

0.000 0.50 0.50 0.0-0 .4 0 5 0.40 0.60 — 10.4 mV-1 .1 0 0.25 0.75 — 28.6 mV-2 .2 0 0.10 0.90 — 57 mV-2 .9 4 0.05 0.95 — 77 mV

I t is o f in te re st to n o te th a t b o th i t an d i2 p ossess odd sym m etry a b o u t th e ir av erag e valu e I J 2 . S p ecifica lly ,

h h i 1 - h , *i2 - — = — i(z) = — -------- = - ^ r t a n h “ , (4 .6 -4 )2 2 ' ' 2\ 1 + e zj 2 2

from w hich we o b serv e th a t i(z) = — i( —z). W ith the a id o f E q . (4 .6 -4 ) we can w rite i2 in th e a lte rn a tiv e form

(4.6-51

and , in a s im ila r fa sh io n , we w rite

j = 1 + tan h | j . (4 .6 -6 )

I f th is c ircu it is o p era ted a b o u t its p o in t o f sym m etry (z = 0), as it n o rm a lly is, then the d c co m p o n e n t in e ith e r cu rren t rem ain s a t I J 2 for a ll sy m m etrica l d riv in g signals (u j — v2). If, in a d d itio n , th e d riv in g sig n als a re p erio d ic , n o even h a rm o n ics are g en erated . In p a rticu la r , if t?x — v2 is a sq u a re w ave o f zero av erag e v alu e and p eak am p litu d e F , , then is a sq u a re w ave in p h ase w ith the in p u t and i2 is a sq u are

Fig. 4 .6 -2 Differential characteristics plotted vs. z = (u, - v2)q/kT.


wave 180° ou t of phase with the input. Both and i2 are sym m etric abou t I J 2 and have peak values which can be determ ined directly from Table 4.6-1 or Fig. 4.6-2. For example, if Vx = 57 mV (T = 300° K), then the peak-to-peak value of both 11 and i2 is equal to 0.8/*.

F o r small values of z (z « 1 o r equivalently |i>! — t>2l <<: 26 mV), we may approxim ate tanh (z/2) by z/2 to obtain

/fc/« A Sin/ -vh = t 1 ~ ~ = - r - - v2)

and (4.6-7)

/fc I * . Z \ I k . S in / \

2/ = ^ ^ — vwhere gin = qIJ2kT. Since g in is ju st the small-signal conductance “ seen” looking into the em itter of transistor 1 or 2 with the base grounded and with an em itter curren t of I J 2, it is clear th a t Eq. (4.6-7) represents simply the expected small-signal transfer for a differential pair of transistors. In addition, the small-signal transconductance

a i 2ac _ - « ' l a cOKI

V1 ~~ V2 V1 ~ v2

is given by

gm = (4.6-8)

where a is the ratio o f collector to em itter current.If now t), — v2 = Vt cos cot and we again define x = qVJkT, then

ij = y + i and i2 = y — i, (4.6-9)

where

i = y tanh | - cos cot | .

A sketch of i vs. cot, shown in Fig. 4.6-3, indicates how the waveform varies with x.It is quite apparen t th a t the F ourier series of i contains no even harm onics;

hence i may be expanded in the form

i = / j cos a>t + I 3 cos 3cot + / 5 cos S o t + • • •

— /k Z «2. - iW c o s ( 2 n - l)cot, (4.6-10)"M

4 .6 THE DIFFERENTIAL CHARACTERISTIC 117

Fig. 4.6-3 Sketch of i vs. cot for several values of x.

where

j tanh |— cos I

The coefficients a„(x) do not appear to be expandable in term s of tabulated functions ; however, it is a straightforw ard num erical m atter to determ ine the Fourier coefficients a„ for the fundam ental and the first several harmonics. Table 4.6-2 presents a„(x) = I J I k for n = 1, 3, 5.

Table 4.6-2 Tabulation of l j l k vs. x for n — 1, 3, 5

X a,(x) = l , / I k a 3(-x ) — ^ 3/^k a 5(x) = I J I k

0.0 0.0000 0.0000 0.00000.5 0.1231 — —

1.0 0.2356 -0 .0046 —

1.5 0.3305 -0 .01362.0 0.4058 -0.02712.5 0.4631 -0 .0435 0.002263.0 0.5054 -0.0611 0.00974.0 0.5586 - -

5.0 0.5877 -0 .1214 0.03557.0 0.6112 -0.1571 0.0575

10.0 0.6257 -0 .1827 0.0831

oo 0.6366 -0 .2122 + 0.1273

Figure 4.6-4 plots a 1(x), a 3(x), and a 5(x)vs. x and com pares the fundam ental com ponents from the differential characteristic . with those of a truncated linear characteristic (Fig. 4 .3 -lb ) tha t has the same slope at the point of symmetry and the same asym ptotic values. The values for x = c o are obtained from Eq. (4.2-2) by noting that, for very large drive voltages, the currents and i2 m ust have the


Fig. 4.6-4 Harmonic content from sinusoidally excited differential characteristic.

form of square waves. In addition, for small values of x we may expand \ tanh [(x/2) cos tot] in a M acLauren expansion of the form

k tanh I ^ cos cor ) = d x cos cot — cos3 2 2 2 2 24

cot

X X 3 \ x 3- — — cos cot — — cos 3cOf 2 32/ 96

Thus, for small x, a ,(x) % |(x — ~>c3/ l 6) and a3(x) x x 3/192. Clearly, then, a^x ) is a linear function of x, provided that x 2/16 « 1. Specifically, if x < 0.63, the nonlinear term in a ^ x ) is less than 2.5% of the linear term ; that is, x 2/16 < 0.025. It is also apparen t that for x < 1, a 3(x)/a,(x) < 0.0 2 ; consequently, the limits on linearity are determ ined not by the presence of the th ird-harm onic current com ponent but ra ther by the presence of the nonlinear term in a,(x). Hence broadband as well as narrow band linear am plification is possible for values of 11', - r 2| < 16 mV. This increase in input drive by a factor of 6.3 over the single-transistor b roadband

4 .6 THE DIFFERENTIAL CHARACTERISTIC 119

amplifier is due prim arily to the symmetry of the differential characteristic and the resultant absence of even (principally second) harm onics from the output. Intuitively what we are doing is com pensating for the nonlinear input characteristic of a single transistor by using a second transistor as a nonlinear em itter impedance.

W hen a tuned circuit which extracts the fundam ental com ponent of current is placed in the collector of either transistor, it is convenient to again define a large- signal average fundam ental transconductance as

a l i h , „ fliM „ 4a,(x)G- w = v ; = T t J ï , = 2“8‘- ~ = 8- -7 - ■

where gm is the small-signal transconductance defined by Eq. (4.6 8). With this definition of Gm(x ). if a high Q r tuned circuit with a resistance R L at resonance is placed in the collector of transistor 2, the tuned-circuit voltage vT is given by

vr (t) = G^(x)V, R l cos cut. (4.6-11)

A negative sign would be required if the tuned circuit were placed in the collector of transisto r 1. A plot o f 4a ,(x)/x = Gm{x)jgm is given in Fig. 4.6-5.

Finally, it should be noted that the differential characteristic has many of the same properties as the exponential characteristic. For small input signals the differentially connected circuit operates as a linear amplifier, for large input signals.

. Fig. 4 .6 -5 P lo t o f G m(x)/gm = 4a ,(x )/x vs. x.

¡ i / I k approaches a constant and the circuit functions as a limiter. In addition, i, and i2 are directly p roportional to therefore, linear gain contro l or am plitude m odulation can be achieved by controlling I k. In practice, as we shall see in C hapter 5, it turns out to be m ore effective to control I k for the differential pair than / dc for the single transistor.


4.7 O T H E R GRADUAL N ON LIN EA RITIES— PEN TO D ES

M any practical devices have current-source-like outputs in which the input voltage- ou tp u t curren t relationship is neither exponential nor square law. In m any cases, this relationship takes the form

' s s 1 - v, > K.„

(4.7-1)0, < Vco.

In this equation I ss is the current flowing when = 0, while Vco is a cutoff voltage (that is, i2 = 0 when = Fco).

In a pentode, for example, for which the plate-to-cathode voltage vB is kept above the knee of the vB-iB characteristic, and for which the screen-to-cathode voltage vs and supressor-to-cathode voltage vSP are held constant, the plate current iB may be related to the control grid-to-cathode voltage va by the relationship

| 3/2iH = l - r ^ l . (4.7-2)

Figure 4.7-1 shows a typical set of vG-iB characteristics for a pentode with the screen voltage as a param eter, along with the corresponding vB-iB characteristic. It should

t .V

Fig. 4.7-1 P late current characteristics for a pentode.

be noted that both I ss and Vc0 vary with the screen grid voltage. In particular, for the pentode shown, w ith vs — 150 V, / ss = 16 mA and Vco = —4.6 V; with vs = 100 V, I ss = 10 mA and Vco = — 3.2 V ; and with vs = 50 V, I ss = 4 mA and Fco = -2 .1 V.

An equation such as Eq. (4.7-1) may be derived from theoretical considerations about a device, or it m ay be m atched with the appropriate choice of n, I s s , and Vco, to a set of experim entally m easured characteristics. It turns out in practice tha t many field effect transistors have experim entally evaluated characteristics that are

4 .7 OTHER GRADUAL NONLINEARITIES— PENTODES 121

best m atched by Eq. (4.7-1) with values of n ranging from 1.5 to 2.5. The largem ajority of them, however, require n « 2, as we noted previously.

If we now apply a voltage of the form v x(t) = Vb + v(t), where v^ t) > Vco, to a device described by Eq. (4.7-1) or, as a m atter of fact, to any device whose output current is a sm oothly varying function of its input voltage, then we gain a great deal of insight into the limits of linearity, as well as the nature of the nonlinearity, by expanding i2 in a Taylor series abou t the Q-point. Specifically.

To describe i2 accurately we m ust keep all the terms, in the Taylor series; however, in practice, if the v t -i2 characteristic is sm oothly “varying, we may require only the first three o r four term s to describe the characteristic within a few percent accuracy. Any attem pt a t further resolution is usually unw arranted because of the lack of precision of the characteristic itself. (As an alternative to evaluating derivatives at the Q-point we could evaluate a2 and a3 so that the series gave exact results at two other points spaced abou t the Q-point. W ith slowly varying curves the differences between the two m ethods is small. Since this second approach involves taking the difference between tw o alm ost equal num bers, it m ust be carried ou t very carefully o r the results may be misleading.)

• t>3 + • • •

= a0 + a t v + a2v2 + a 3t>3 + • • •, (4.7-3)

where

F or the device described by Eq. (4.7-1), if we define Vb = NVco, where 1 > N > 0,then

ay = / ss( 1 - N ) \

« . = 1 - A 0 " _ l ,coCO

(4.7—4)

Clearly, for small ac input voltages v, i2 is given by

i2 = / ss(l - N )n - - N T ~ 1 v.'cnCO

(4.7-5)

Hence we observe that the small-signal transconductance gm is given by

(4 .7 -6 )

Since Vco is a negative quantity, we note that gm is actually positive.


If u(i) = Vj cos cot such tha t t>, = Vb + Vt coscot, then i2 (from Eq. 4.7-3) is given by

1*2 = a0 + a i cos cot + a2V j cos2 cot -I- a 3F 3 cos3 cot (4.7-7)

if we assume the higher-order term s to be negligible. By using the trigonom etric identities

1 cos2fl , , . 3 . 1cos 6 = - H — and cosJ 0 = - cos 6 + - cos 36,

2 . 2 4 4

we can simplify the expression for i2 into the Fourier series form

Jo h / 2 f 3

a2VÏ „ / 3a3K i\ a2V i „ a 3K?;2 = a0 -\----- I- FWa! H — I cos cot H---- — cos 2cot H — cos 3a»i.

(4.7 8)

From Eq. (4.7-8) it is now apparen t that the relative second-harm onic distortioi is simply

which for small values of Vt reduces to I/ ia2/2 a 1. W ith the aid of E q. (4.7—4) this can be expanded to

Vt a 2 Vi I — n

2^ " = W m

If n = | and N = j , then for less than 1 % second-harm onic distortion | Vi/Vco\ < 0.04. W ith VQO = - 4 V, V, < 160 mV.

The large-signal average fundam ental transconductance Gm for such a device is given by

- 7 i h ^ 3 V ' a= Oi 1 +Vi 4a,

V l I n - IU m - 2 f l(4.7-9)v ' e i / 2 t \ r _ 1Ko L 8 I/ C20 (N - l )2

The a 3-term has m ade Gm a function of a signal level. This leads to d istortion in narrow band amplifiers. To keep Gm constant within 1%, Vx m ust be restricted so that

4 .8 EFFECT OF SERIES RESISTANCE 123

U sin g th e sam e v a lu e s specified ab o v e , w e see th a t \ Vx/Vco\ m u s t be less th a n 0.283and K, m ust be less than 1.13 V. H ere again a m uch larger signal may be applied for linear narrow band operation than for linear broadband operation.

If n o w v(t) = Vx co s cot + V2 co s co2t, th e n w ith th e a id o f E q. (4.7 3) a n d severa l trigonom etric identities we can express Gm in the form

Clearly, if the V2-term were m odulated, then a distorted version of this m odulation would appear on the signal ou tpu t of our narrow band filter centered at u>. This effect of the transference o f m odulation from one carrier to another is known as cross m odulation. O ne of the advantages of a square-law (or linear) device as a narrow band amplifier is the absence of the cross-m odulation effect. (In a true square- law device, a3 = 0.) The first amplifier stage of any receiver should always have a low susceptibility to cross m odulation, since it is virtually impossible to have adequate selectivity in front of it to prevent the appearance o f some unwanted signals at its input terminals.

4.8 EFFECT O F SE R IES RESISTA N CE ON THF. EXPO N EN TIA L CHA RACTERISTIC

In many practical cases we find that the nonlinearities discussed previously are m odified by the inclusion of a resistor in series with the nonlinear element. The general effect of such a resistor is to decrease the transconductance of the com posite device (a larger input voltage is required to produce the same level of current that would exist w ithout the resistor) and to “ piecewise-linearize” the com posite nonlinear characteristic. This linearization provides the m eans o f extending the operating range of a linear amplifier at the expense of its am plification factor. O n the other hand, when the com posite device is driven with large signals, the principal effect of the series resistor is the increased drive voltage required to produce the required value of input current.

(4.7 10)

c

I

-o

o ■o« w ,

Fig. 4.8-1 A PN P transistor with resistance in series with the emitter.


Figure 4.8-1 shows a transistor w ith a resistor R in series with the emitter. A lthough such a resistor is som etim es added deliberately, in o ther instances it is inherent in the tra n s is to r; for example, it may be the bulk em itter resistance o r the intrinsic base resistance, which becomes a significant factor when the transistor carries a high current. Thus in m ost pow er amplifiers the transistor m ust be modeled as in Fig. 4.8-1.

If we assum e th a t the em itter-base junction of the transistor is described by the exponential characteristic

F o r small-signal operation the increm ental input resistance for the circuit takes the form

where r in = l /g in = kT /q lAc and I dc is the dc value of i , . C onsequently, on a small- signal basis we see that the transconductance g'm given by

decreases by a factor of 1 + ginR with increasing R. Specifically, if agin w 0.04 U, as it would for / dc = 1 mA, then a series resistor o f only 10 Q decreases g^ to 72% of its R = 0 value, while a series resistor of 100 £2 decreases g'm to 20% of its R = 0 value. In the limit as ginR becomes large com pared with unity, gm becomes equal to a/R and is independent of the change in bias current of the transistor.

To observe the “piecewise-linearization” effect of the series resistor on the v x - characteristic, we first rearrange Eq. (4.8-2) in the form

h = I Eseq,,EBlkT,

then we may relate v { to ¡\ for the circuit of Fig. 4.8-1 by the expression

• „ W , h t?i := i t R -I I n — .Q ‘ ES

(4.8-1)

(4.8-2)

(4.8-3)

q dc

where / dc is the quiescent value of ¡i and

(4.8-5)

is the quiescent value of when ij = / dc. We then norm alize vt — V0 to

K o — ^ d c ( r in + R) = ~ ( 1 + £ i n ^ )

4.8 EFFEC T O F SERIES R ESISTA N C E 125

and i , to / dc to obtain

"i ~ V° = I I 1'» + ,aKo \ l + g inR j l ic 1 + g inR ' V ' ’

The voltage VQO may be interpreted as the decrease in vv from V0 which would be required to reduce i, to zero (i.e., to cut off the transistor) if the slope of the v 1 — i, characteristic rem ained constant at its small-signal value of 1 /(rin + R).

Fig. 4.8-2 Plot of i j l dc vs. v, — Ko/Vco with g inR as a param eter.

Finally we plot i j l dc vs. (t;, - K0)/Fco with ginR as a param eter, as shown in Fig. 4.8-2. W e note from the figure th a t as ginR is increased from zero to infinity, the characteristic varies uniformly from the exponential (ginR = 0) characteristic to the two-segment piecewise-linear characteristic (ginR = oo). In addition, the ginR = 1 characteristic corresponds essentially to the half-way poin t between the tw o extremes. Thus for g lnR « 1 the circuit of Fig. 4.8-1 can be m odeled by the transistor


characteristic alone, while for g inR » 1 the circuit can be m odeled by the g inR = oc characteristic, or equivalently,

' vt ~ Vo ^ n— ^— , »i > 0,K • (4 .8 -7 )

0, t>i < 0.

It is apparent that Eq. (4.8-7) describes the circuit of Fig. 4.8-3. This circuit thus provides the model for the circuit of Fig. 4.8-1 when ginR » 1.

R-*AAA-

$ Ideal

Fig. 4.8-3 M odel for the circuit of Fig. 4.8-1 with g,„R -* oo.

We also observe from Fig. 4.8-2 that, as g¡„R is increased from zero, larger linear excursions in /, about Idc are possible w ithout gross distortion. In particular, for g,„R -* oo, the linear range extends all the way to cutoff ( i i / /dc = 0).

If the circuit of Fig. 4.8-1 is to be used as a large-signal narrow band amplifier, the analysis becomes quite difficult for an arbitrary value of giaR. Fortunately, an analysis is possible for ginR = 0 (which was accom plished in Section 4.5) and for ginR = oo. In addition, the results of the tw o analyses are sufficiently close to each other that solutions for any value of ginR may be closely approxim ated.

For the case where ginR = oo, we model the transistor-series resistor com bination as shown in Fig. 4.8-4. The current source / dc fixes the average value of the

C — oo R

Fig. 4.8-4 Model for large-signal narrowband transistor amplifier with g,„R -> oo.

4 .8 EFFECT OF SERIES RESISTANCE 1 2 7

em itter current, while the infinite coupling capacitor ensures that the entire ac voltage reaches the input terminals. (The current source could be placed on the other side of the resistor w ithout affecting the results.) Consequently, for this circuit we observe that, for Vi < h c R — VQO,

V,h = ' t = / dc + — cosw f = / dc + / , cosw r

= ^dcjl + -^coscorj, (4.8 8)

where / , = R, o r equivalently, I t/ / dc = P’or (•[ > Kco the em itter currenttakes the form of a periodic train of sine-wave tip pulses, and thus may be written as

h. = h + h cos i0t + 12 cos 2tot + ■■■

/dd 1 + v* cos a)t + ~r cos 2cot + • ■ • | , (4.8 9)¡ 0 ¡ 0

where (from the appendix to this chapter)

K, — Vx sin (f> — <() cos (j> nR 1 — cos $

V\ — Vx (¡> - cos 0 sin (ph = nR 1 — cos 4>

HVi — Vx) cos </> sin n ^nR n(n2 - I )( 1 - cos 0)

</> = c o s~ l -£ , Vx = - V c - V 0,

and Vc is the developed dc capacitor voltage. By noting that

V, — Vx = Vt ( l — co s </>) a n d / 0 = / d

we can relate VXIVC0 directly to - <j) cos (p '

In addition, i j / I 0 can be related to cos </>/ 0 sin cos 4> '

and in a similar fashion

l„ 2(cos <j) sin n<f> — n sin <j> cos n</>)

(4.8-10)

(4.8-11)

I 0 n(n2 — l ) ( s in $ — <pco$4>)n > 2. (4.8-12)


Consequently, for any value of <p, a value of V J V C0 and I J l ic may be determ ined from which a unique relationship between V J V Q0 and /„ //dc can be established. Table 4.8-1 contains values of ¡ i / I 0 = / i / / dc vs- VJVm which are plotted in Fig.4.8-5 with the designation ginR = oo. O n the same set of coordinates is plotted the ginR = 0 curve, which, of course, is just the top curve of F ig 4.5-5, since with R = 0.

Fig.4.8-5 Plot o f / , /7 dc vs. VJVa .

I i / I o = 2 /1(x)//0(x) and VJVQO = x. We observe that the two limiting curves are alm ost iden tical; hence the ratio o f / , / / 0 for any value of ginR can be closely estim ated from F ig 4 .8-5 .f F o r example, if ginR = 1 and V J V ^ = 10, it is reasonable to approxim ate I J I 0 as 1.95, which is half-way between the ginR = 0 and the ginR = oc curves. If, in addition', I ic = 2 mA, then / , = 3.7 mA, from which the ou tpu t voltage may be written as — Vcc + I tR L cos cot for the case where Z L is a high-Q tuned circuit which is resonant a t a> w ith a shunt resistance of R L.

F or the case where the circuit o f F ig 4.8-1 is driven by a sinusoidal voltage of am plitude Vt and followed by a tuned circuit resonant a t the frequency of the input

t It is no t unreasonable for the I t/Idc vs. VJVco curves to form a tight set with g¡„R as a param eter, since all the curves have the same asym ptotes for both small and large values o f Clearly,for VJVco « 1, /j/Zjc = Vi/Vce- Therefore, regardless o f the value of g inR, the curves approach the origin in an identical fashion. In addition, for large values of VJVco, if the average current is to remain constant a t / dc, the transistor m ust rem ain cut off for m ost o f the cycle. Consequently, the current flows in narrow pulses and / i / / dc -> 2. Again, all the curves approach infinity in the same fashion.

4 .8 EFFECT OF SERIES RESISTANCE 129

Table 4.8-1

<t> 1J I0 = V 'd c y j K o

1

o o 1.88 28.860° 1.80 9.1780° 1.65 4.23

100° 1.49 2.44120° 1.32 1.64140° 1.17 1.25160° 1.05 1.06

00 o 0 1 1

sinusoid, it is again convenient to define a large-signal average fundam ental trans-conductance as

n ^ l/^ d c ^ d c ^ l/ ^ d c f i A o i - y \

where g'„ is the small-signal transconductance evaluated at the Q-point. Clearly, values for

^ (4-8-14)g m V JV m

vs. Vi/VK0 may be determ ined directly from Fig. 4.8-5.

Fig. 4.8-6 Plot of G J g m VS. VJV„.


A plot o f G J g ’m vs. VJVQa for g inR = 0 and g inR = go is shown in Fig. 4.8-6. Here again we see tha t the curves are sufficiently close to perm it an accurate estimate of GJg'm for any value of ginR. It is now quite apparent that the ratio of I , / I 0 is determ ined prim arily by the ratio of VJVm and is alm ost independent of g,nR. However, the value of

k TK :o = ^dc(r in + R ) ~ -------(1 + g i n R )

increases with ginR. Consequently, as ginR is increased, the drive voltage must be increased by a factor of 1 -I- gjnR to m aintain the same fundam ental ou tpu t current.

Example 4.8-1 F o r the circuit shown in Fig. 4.8-7 determ ine an expression for

Solution. Since the tuned circuit in the collector has QT = 20 and is tuned to co = 107 rad/sec, we note that

t>0(f) = —10 V + a(2 k ii) / , cos 107f,

where is the fundam ental com ponent of the em itter current. We obtain / , by first noting that, with an average em itter current o f 2.6 mA, gin = 1/(10 £i) and therefore ginR = 1. In addition, Kco = [kT/q)(2) = 52 mV and thus Vy/V^ — 2.5. Consequently, from Fig. 4.8-5 we estim ate / i / / dc % 1.5; thus / , = 3.9 mA and

■MO = ( - 1 0 V ) + (7 .8 V ) cos 10 7r.

4.9 CLAMP-BIASED SQUARE-LAW DEVICE 131

In m any large-signal junction FET circuits the bias is established by clam ping the positive peaks of the gate to source voltage to zero (or the turn-on voltage of the gate to source diode). In particular, m ost FETs em ployed in self-limiting oscillators are biased in this fash ion ; hence we shall analyze the large-signal characteristics for this type of circuit in this section.

We begin our analysis with the clam p-biased square-law amplifier model shown in Fig. 4.9-1, which is a close first-order approxim ation to the actual junction FET amplifiers shown in Fig. 4.9-2. Clearly, in this model, if the R aC G time constant is

4.9 CLAMP-BIASED SQUARE-LAW DEVICE

Co

H f- K

v,(r) K, cos woi0 Q>

(Ideal) . 2io= /ns»0 jS^) ’ Ifcs> V„

o. %»<H

Fig. 4.9-1 Large-signal clamp-biased square-law tuned amplifier.

V„(f)

Fig. 4.9-2 Clamp-biased junction FET circuits.


m uch greater than T = 2n/co, the capacitor voltage charges to the peak value of v t(t) through the diode and rem ains constan t at th a t v a lue; hence

Vdc = K (4.9-1)

and

vGs = ^ (c o s aj0t - 1). (4.9-2)

T hus we observe th a t the inpu t voltage vGS has its peaks clam ped to zero. Consequently iD(f) has the form shown in F ig 4.4-3, with I p = / DSS, Vx = Vt + Vp, and V J V X = 1 + V J V X. Therefore, w ith the aid of Eq. (4.4-10) (for Vx/V l < - 1 or V\ < Vp/2) and Fig. 4.4-5 (for —1 < V J V i < 1 o r F, > V ^ 2) we can plot / 0//p, l j l p, and I 2/ I p vs. — V J V p as shown in Fig. 4.9-3, where I 0, h , and I 2 are the average, the fundam ental, and the second-harm onic com ponents of in respectively.

HFig. 4.9-3 Plot of I J I p vs. VJVf for a peak-clamped sine-wave drive to a half-square-law amplifier (n = 0, 1, 2).

If Z L is a high-Q r parallel R L C circuit tuned to the fundam ental o f iD(t) w ith a resistance R L, the ou tpu t voltage v j t ) is given by

v 0( t ) — V p o — R l I d s s j - c o s « V ,* p

( 4 .9 -3 )

4 .9 CLAMP-BIASED SQUARE-LAW DEVICE 133

where I J 1 P can be determ ined from Fig. 4.9-3 once - VJVP is specified. F o r example. i f / DSS = 4m A , Vp = — 4 V, Vt = 2 V, VDD = 1 0 V ,a n d /? L = 2 kfi, then - VJVp = \ and in turn / j / / p = / i / / DSS = 0.5; hence

vo(0 = (10 V) — (4 V) cos cot.

In this example the m inim um value of vB = vDS is 6 V ; since this is greater than — Vp, we can be sure tha t the F E T rem ains within its saturation region, for which the square-law m odel of Fig. 4.9-1 is valid. (In general, an FE T rem ains within its saturation region if vDG > — Vp, or equivalently, vDS — vGS > — Vp for all time.)

z hVp

G I I IFig. 4 .9-4 P lot of —— = —— for a peak-clamped sine-wave drive to a half-squkre-law

SmO \ i p)amplifier.


F o r the clam p-biased model of Fig. 4.9-1 we may also define a large-signal average fundam ental transconductance

G = — = a g 0 ^ (4 9_4)

where gm0 = 2IDSS/ - Vp is the small-signal transconductance evaluated at vGS = 0 (cf. Eq. 4.4—8). A plot of

I J 1 Pg m O 2 ( - Kj/Kp)

vs. VJVP obtained directly from Fig. 4.9-3 is shown in Fig. 4.9-4. W ith Gm obtained from Fig. 4.9-4, v0(t) may be w ritten in the equivalent form

v„ = Vdd ~ GmR LVy cos (o0t. (4.9-5)

The effect of a 0.7 V diode turn-on voltage can be included in the previous results by defining the drain current at this turn-on voltage as /£ ss and substituting ap p ro priately.

Effect of Drain ResistanceIn the analysis to this point, we have assum ed that iD is a half-square-law function of vGS and is independent of v0 = vDS. This assum ption is justified only if R L is small com pared with the effective ou tput resistance of the square-law device. If R l is not small, a m ore exact model for iD m ust be employed.

Fig. 4.9-5 Typical set of FET drain characteristics.

Figure 4.9-5 illustrates a typical set of drain characteristics for an Af-channel junction FE T which clearly indicates that no characteristic for constant vGS is - horizontal (a condition which would have to be satisfied for iD to be independent of vDS). O n the contrary, each constant-i;Cs characteristic has a slope which, when extrapolated, causes all o f the constant-yGS characteristics to meet a t a com m on

4 . 9 C L A M P-B IA SE D S Q U A R E -L A W D E V IC E 135

point Va on the negative i /)s avis I hus. i! lor i ,;s -- Vp,

<D = / DSS| l V o s > V p, (4.9-6)

then for any other value of vDS we obtain from the geometry of Fig. 4.9- 5

, 4 9 - 7 )

where 1 /r0 is the slope of the t;GS = 0 characteristic. Equation (4.9-7) includes the effect of v DS on iD.

If we now employ the expression for iD given by Eq. (4.9 7) in the model of Fig.4.9-1, in which Z L is a high-Qr parallel R L C circuit tuned to the fundam ental frequency of iD and having a resistance R L, we obtain

Vd d + V p I DlR L cosa>0t¡D — ^DSS I +

r O I D S S r 0 > D S S

x I —— -|- —— cos u>0t + y cos 2w 0t + ■ • • I , (4.9-8)\ ‘ p p p I

where l 0/ I p, I J I p, and / 2/ / p are the coefficients presented in Fig. 4.9-3, and Im isthe am plitude of the fundam ental com ponent of iD. The fundam ental com ponentof iD can also be obtained from Eq. (4.9-8) in the form

/D. = 'dss 1 + — ,----- 71 - \-r + • <49 9>Vpp + I'pj 1 1 _ I D I ^ l U o /_2_

r o I DSS I I p r o \ I p 2 / p

Hence Eq. (4.9 9) may be solved for I Di to obtain

/ 11 + Vdd +. Vp\ ! i* D S S I 1 +

/ D1 = -------' R l [ r ° lDSS. ! /p (4 .9 -10)

1 + * * i ° + h

r0 \ l P 2/„

h

— I _________Le.______“ R j l o , h

1 + — 1^ +

where

I D S S = I D SS I 1 +

ro\Ip Up]

VDD + Vpr o l D S S

i s t h e v a l u e o f iD o b t a i n e d w i t h rGS = 0 a n d rDS = VDD


Vx cos utt

Fig. 4.9-6 Plot of lDi/IDSs vs- — K/VP with as a parameter.

Figure 4.9-6 includes a plot ofhIn

r0 \ I P 2 l pfor values of R J r 0 of 1, j , and 0. The curve for R J r 0 = 0 corresponds to the / 1(/ / p curve of Fig. 4.9-3.

As an example of using the curves of Fig. 4.9-6, we consider the circuit shown in Fig. 4.9-7. We observe for this circuit that

rB!a = (2 m A )| I + i ^ J « 2.8 mA, ^ - I and - i ;

hence from Fig. 4.9-6 we obtain

I D1 = (2.8)(0.41)mA = 1.15 mA.

In addition, since the resonant frequency o f the drain circuit is 107 rad/sec and the circuit Qt = 10,

v0 = (20 V) - (11.5 V) cos 107f.

Since v„(t) > — Vp, the F E T rem ains w ithin its saturation region and the assum ed FET model is valid.

PROBLEMS 13720 V

PROBLEMS

4.1 Assume that v,{t) = 3 V sin a>0t in Fig. 4 .P-1. Determ ine the dc, fundam ental, and second- harm onic com ponents o f i2(l) for values of Vic o f 0, ± 2 V, and ± 4 V. Sketch the output waveshape for each case.

Ideal diode

4.2 Assume that the diodes in Fig. 4 .P -2 are ideal. F ind the dc, fundamental, second-harmonic, and third-harm onic com ponents of i2 k(1(r) in this circuit.

4.3 How would the results o f Problem 4.2 be changed if the magnitude of the battery in the lower branch were increased to 3 V?

4.4 Assuming that the diodes in Fig. 4 .P-3 are ideal, sketch the waveshape of v„(t) and calculate its third-harm onic component.

138 NONLINEAR CONTROLLED SOURCES

1 V

Figure 4.P-2

1 k ii

4.5 Assuming ideal diodes in Fig. 4.P 4 , sketch va(t) and determine the dc, fundamental, second- harmonic, and third-harm onic components.

(l^)cos o v

Figure 4 .P -4

PROBLEMS 139

4.6 Assume a device with the characteristic shown in Fig. 4.P 5. F o r this device,

f ("i - vn ) 2 - i’i > ^-h.

1 0, v , < KTh,

where Vlh = + 2 V and [i = 40 0 /xA/V2. Evaluate the large-signal average transconductance for this device for the case where

»i = Vic + Vy cos tot and K.c > 2 V

while

y, < IK.« - VJ-

4.7 Assuming that Vic = 0 and V, = 3 cos cot, find the dc, fundamental, second-harmonic, and third-harm onic com ponents of i2 in the device o f Problem 4.6.

4.8 Sketch the ou tpu t current waveshape for a device with an exponential current-voltage relationship

i2 = 1 x i o - ‘V l/0 026A when = 0.520 V cos cot.

Specify the “conduction angle.” Find the dc, fundamental, second-harmonic, and third- harm onic com ponents of this current. W hat is the large-signal average transconductance for this device with this driving waveshape?

4.9 In the circuit of Fig. 4 .P-6, the diode conducts the same current as the em itter of the tran sistor when vBE = t>diod(.. If / dc = 2 mA, Vcc = + 6 V, and Z L is a 2.7 k ii resistor, how large can Vx be before there is 2 % distortion in the output voltage v„(t)? If Z L is replaced by a parallel tuned circuit resonant at a>0 and with a resistive term o f 2.7 k ii, how large can Vt be before there is a 2 % departure in linearity between the peak amplitudes of input and ou tput ac voltages? (Hint: N ote the similarity o f this circuit to the differential pair of tran sistors.)

4.10 In the circuit o f Fig. 4 .P-6, assume that a 5 0 f i resistor is placed both in series with the diode and directly in series with the emitter. P lo t a curve of collector current vs. base- ground voltage over the range 1.95 > ic > 0.05 mA. Com pare this curve with the appro priate da ta from Fig. 4.6-2 and explain the significance of the resistors in extending the signal-handling capacity of the amplifier. W hat are the small-signal voltage gains (R, =2.7 kii) for the two different cases (R E = 50 fl and RE = 0)? State and explain assumptions.

4.11 Determ ine the output-tuned circuit voltage in Fig. 4.P-7. The tuned circuit is resonant at oj0 and has Qr — 20.


Z L

+y I COSOJ0 l

son

Figure 4 .P -6

. ac short circuit

Figure 4 .P -7

4.12 The FE T in Fig. 4 .P-8 is silicon and has an 1DSS m agnitude of 6 mA and a Vp m agnitude of 4 V. W hat value of K, will make the ou tpu t dc current equal to 1.8 mA? W hat value of ac voltage will result in this case? W hat is the circuit’s voltage “gain” ? How much power is being dissipated in the FE T during this operatiori? The tuned circuit is resonant at 107 rps.

4.13 a) F or the circuit o f Fig. 4 .P-9, determ ine an expression for v„(t) assuming a high value ofQt■

b) Determ ine the minim um value of Qr which keeps the total harm onic distortion below 1 %.

PROBLEMS 141

+(lOV)cos 10

" 0

f ........................ 1 < — (

^/(Vl) «

> 1

1 5kn - =;c v°

+

"Va»o=107 rad/sec

4.14 For each of the circuits in Fig. 4.P-10, determ ine and sketch v„(t) and i^t). In each case v,{t) = (260 mV) cos 107i.

4.15 F or the circuit shown in Fig. 4.P-11, determ ine v„(t) where L = 5 /xH, C = 2000 pF, andv,(t) = (5 V) cos 107f.

4.16 Repeat Problem 4.15 for L = 3.3 /jH and C = 330 pF.

4.17 Repeat Problem 4.15 with D, removed and with L = 10 and C = 1000 pF.

4.18 a) For the circuit of Fig. 4.P-12, determine the small-signal voltage amplificationA v = v j v t.

b) M ake a plot o f v„ vs. vt as i>, varies from — 8 V to + 8 V. W hat is the effect o f the emitter resistors?


PROBLEMS 143

Ideal diodes

Figure 4.P-11

Figure 4 .P-12

9 +


FOURIER EXPANSIONS

The general approach to finding the average value or any harm onic com ponent of a repetitive pulse train of period T = I n /w is to m ake a Fourier series expansion of it. In m aking such expansions, we choose the time origin so as to minimize term s and make use of various trigonom etric identities to consolidate terms. The final form of the series is quite dependent on the variables chosen to express the waveshape and on the particu lar identities used to simplify the expression.

M ost of the periodic waveforms f ( t ) that we are interested in result from passing a dc-plus-cosine wave through a nonlinear nonm em ory (no energy storage) device ; hence the ou tp u t waveforms possess sym m etry abou t the t = 0 axis and thus m ay be expanded in a F ourier cosine series of the form

It is usually convenient to change the independent variable from time to an angle in radians by defining 6 = cot and then, since the waveshapes are symmetric about 6 = 0, to m ultiply by two and integrate from 0 to n. As functions of 6, a0 and oB take the forms

00

f ( t ) = a0 + a , cos cot + a2 cos 2cot + ■ ■ ■ = a0 + Y, a«cos na)t’

where

(4. A 1 )

and

(4.A-2)

Slope G

FOURIER EXPANSIONS 145

Fig. 4. A - l T rain of sine-wave tip current pulses.

Linear Sine-Wave Tips

For the train of sine-wave tip current pulses shown in Fig. 4.A-1 we observe that i2 = 0 for |0| >\<j)\ (in the interval n < 6 < n), where , i2(t) has the value

i2(t) = G(Vl coscot — Fx);

therefore,

/ , = - ( * G(Ft cos 0 - F J cos 0 d6 n J o

2Gl V1(f> F js in 2<t>+ — Vx sin 4> (4.A-6)

By noting that G = I p/( 1 — cos <f>) and G Vx = I p cos </>/( 1 — cos <p) and employing the identity sin 2</> = 2 sin </>cos 0 , we can further simplify the expression for to obtain

I p sin (f> 1 n 1 — cos (/>

(4.A-7)

146 FOURIER EXPANSIONS

In a similar fashion, we obtain

l p sin (p — 

and

2Ip cos <f) sin n<t> — n sin 0 cos n<pn > 2. (4.A-9)

7t n(n2 — 1 )( 1 — cos (f))

Norm alized plots of I J l p vs. conduction angle (2<f>) are given in F ig 4.2-3 for values of n from zero to five. Negative values of the coefficients in the plot are indicated by a minus sign after the designation of the particular curve; that is n = 4 implies that the fourth harm onic is negative.

For small values of <f> we can approxim ate

From Eq. (4.A-10) it is apparen t th a t the ratio of /„ to / 0 approaches 2 as <j> approaches zero. This, o f course, is a property possessed by any narrow pulse train.

For a pulse as wide as 90°, however, /[//<, = 1.88 and l 2/ I 0 — 1.56; even for a 120° pulse, / j/Zq is 1.80, which is within 10% of the very narrow pulse ratio value of 2.

Square-Law Sine-Wave Tips

F or the train of square-law sine-wave tip current pulses shown in Fig. 4.A-2, we observe that i2 = 0 for |0|>|</>| (in the interval —n < 9 < n ) , where, as in the previous case, </> = co s- 1 (VJVi). In addition, for |0| < |</>|, i'2(f) is given by

to obtain

and

(4.A-10)

h(t) = coscof - Vx)2,* nP

(4.A-11)

or equivalently,

(4. A -12)

FOURIER EXPANSIONS 147

where

l — I DSSP ~ \ v 2P

(V, - K,)2

is the peak value of i2 and cos (j) = . Consequently, the fundam ental com ponent/ , of i2(f) is given by

21 C* (cos 0 — cos 4>)2 cos 0 dd2/ , rn J o (1 — cos (f>)2

2/ p I sin (f> + Y2 sin 3</>-</> cos (j>(1 — cos (j))2

(4. A 13)

In a similar fashion, we obtain

<j) — ^ sin 2</> + y cos 20

° (1 - CO S ( j ) ) 2

- — i sin 20 + sin 40 21 „ 4 6 48 r71 (1 — COS 0)

(4. A 14)

and

2 /p (4 - n 2) sin n<f> + (n — 1 )(n — 2) sin n<j) cos 20 + 3n sin {n - 2)<j> ^n (n2 — l)(n2 — 4)«(1 — cos 0 )2

148 FO U R IE R EX PA N SIO N S

N orm alized plots of I J l p vs. conduction angle (20) are given in Fig. 4.4-4 for n = 0 , 1, and 2.

F o r small values of <f>, we obtain the lim iting forms of the Fourier coefficients given by

/„ 8<b 2In 8éand (4.A-15)

n 15 7i 15

Here, as in the case of the linear sine-wave tip, the sine and cosine term s are approxim ated by the first several term s in their M acLauren expansions. Again we observe that, for small values of <j> (i.e., a small conduction angle), the ra tio of /„ to I 0 approaches 2. In addition, we observe that, when the conduction angle (2<j>) is as wide as 150°, l x/ l 0 = 1.83 ; hence for all narrow er pulses l i / l 0 ~ 2 is a good approxim ation.

C H A P T E R 5

REACTIVE ELEMENT AND NONLINEAR ELEMENT COMBINATIONS

In this chapter we study the effect of com bining nonlinear com ponents with single reactances and with tuned circuits. We begin with the problem of capacitively coupling a periodic voltage to a general nonlinear load, and show that such coupling normally results in a dc bias shift which is a function of the ac input signal am plitude. Such an interrelationship rules out the use of superposition in such circuits. We consider both steady-state and transient operations of such circuits.

We then apply the solution of this general problem to the capacitively coupled transistor amplifier, where again we investigate both steady-state operation and transient operations. In addition, we obtain results not only for the transistor amplifier with a constant current bias but also for the resistively biased transistor amplifier. In both cases we obtain universal curves which may be applied to any transistor.

We then study the effect of nonlinear loading on a tuned circuit. We show that, if the circuit Q is sufficiently high, then the nonlinear load functions as an equivalent linear load which can be readily determ ined graphically.

•Finally we consider the case where low-index AM signals are applied to a “clam ping”-type circuit, and write a transfer function which is useful in later chapters.

For the reader who has not previously been exposed to this or similar m aterial we recom m end tha t he not try to absorb this chapter completely before proceeding to .the subsequent chapters. Specifically, if he restricts himself to the steady-state phenom ena of this chapter as well as the m aterial in Section 5.5, he will be prepared to understand m ost of the m aterial of the rem aining chapters. He should then return to the transient portion of this chapter when the need arises.

5.1 CAPACITIVE COUPLING TO NONLINEAR LOAD

In m any applications we m ust capacitively couple an ac driving source to a nonlinear load as shown in Fig. 5.1-1. Such coupling is usually r'equired to block any dc com ponent of the driving source from the load. This situation arises if v^t) represents the o\itput of one stage of a large-signal amplifier while the nonlinear load represents the input to the subsequent stage.

The basic difficulty in analyzing the apparently simple circuit of Fig. 5.1-1 is that, because of the nonlinear load, superposition is no longer valid. Thus the time- varying and dc com ponents of the currents and voltages within the circuit are in general related. This interrelationship makes a general closed-form solution for

149

1 5 0 R EA C TIV E A N D N O N L IN E A R ELEM EN T C O M B IN A T IO N S 5 .1

vo(t\ MO, and ¡„(f) in term s of vt(t) practically impossible. However, if we assume that v¡(f) is periodic and that the series coupling capacitor C is an effective short circuit com pared with the nonlinear load over the frequency range occupied by t,(i) [that is, that vc(t) contains only a dc com ponent Vc , or at worst a slowly varying com ponent com pared with f,(t)], then in principle we may determ ine the steady-state expressions for u0.(i), vc(t), and ;0(r). In addition, in m ost cases, we are able to calculate the transient build-up to steady state if v^t) is applied at t = 0. Fortunately, in most cases of practical interest, the above assum ptions are valid.

In the time dom ain the assum ption tha t the capacitor is an effective short circuit corresponds to the assum ption that the circuit time constants are such that the capacitor voltage rem ains essentially constant over a cycle of v^t). Consequently, if Vj(t) is applied at f = 0, the capacitor voltage slowly adjusts itself to its steady-state value vc = Vc = - Vdc, and in turn v0(t) approaches vt{t) + Vdc. Since in the steady state the average value of the capacitor current m ust equal zero, and since the capacitor current and nonlinear load current are equal, vc = - Kc may be found as that value which causes the average value of the load current to equal zero.

Piecewise-Linear v„-i„ CharacteristicIf the vB-i„ characteristic is piecewise-linear in form, we can determ ine Vc = — Vdc by the following m ethod. We assume a value of Vc , apply t)0(t) = u;(t) — Vc to the nonlinear characteristic, and with the techniques of C hapter 4 determ ine i0(t) and the average value of i0(t), that is, i0 = l c . We then repeat this procedure until we obtain enough values to plot a sm ooth curve of Vc vs. I c . The point at which the curve intersects the I c = 0 axis determ ines the steady-state value of Vc = — Vic, from which the steady-state values of v„(t) and i0(t) follow directly.

As an exam ple of this m ethod, consider the nonlinear characteristic to be described by the piecewise-linear curve of Fig. 5.1-2 and t ,(i) = (3 V) cos wt. Figure5.1 2 also indicates two additional characteristics which com bine in parallel to yield the original characteristic. [This superposition of characteristics is perform ed only to focus attention on the structure of ¡0(r).]

If we now assum e that the capacitor C is an ac short at a> and that its voltage is given by Ff , then

v„ (t) = 3 V c o s co t — Vc

5.1 CAPACITIVE COUPLING TO NONLINEAR LOAD 1 5 1

and

where

'„(f) = 'ol(f) + 'o2(0 ,

i„(t) ~ (2 V)«01 (0 = 2 k il

and i„2(f) is a periodic train of sine-wave tip pulses of peak am plitude

(3 V) - (2 V) - Vc

and conduction angle

I p =

2(f> = 2 cos

2 k n

, vc + (2 V) 3 V

The average capacitor current l c is given by

Ic o 0 1 o2 ’\\ here .

■ Vc - 2 V 2 k f i

and io2 is the average value of the train of sine-wave tips, which can be determ ined directly from Fig. 4.2^1 once a value of Vc is assum ed (Vx = Vc + 2 V). Table 5.1-1 tabulates values for io1, io2, and I c for several values of Vc . These data are also plotted in Fig. 5.1 3, from which it can be observed that Ic = 0 for Vc = - 1.33 V. C onsequently, in the steady state

Vr = -1 .3 3 V and vQ(t) = - ( 3 V) cos coi + (1.33 V).

NL M I 1"' H

», - (3V) cos oil

Fig. 5.1 -2 Nonlinear characteristic for the circuit,of Fig. 5.1-1.

152 R EA C TIV E A N D N O N L IN E A R ELEM EN T C O M B IN A TIO N S 5.1

Table 5.1-1 Tabulation o f , io2, and Ic vs. Vc

K>V i»i,m A }o2, mA l C = '„1 + í . 2 - m A

- 5 1.5 1.5 3- 4 1 1.09 2.09- 3 0.5 0.76 1.26- 2 0 0.48 0.48- 1 -0 .5 0.26 -0.24

0 -1 0.09 -0.911 -1 .5 0 -1 .52 - 2 0 - 2

Actually, one should see by inspection that only two points need to be com puted to solve such a problem. Since the right-hand branch in Fig. 5.1-2 contains an ideal diode, i02 m ust be positive; hence for iol (in a branch with only a resistance and a battery) to ju st balance it, the current in the left-hand branch m ust be negative. Therefore, Vc m ust be greater than —2 V to m ake i0l negative. O n the o ther hand, if Vc exceeds zero, then io2 will be zero, so tha t two reasonable initial choices for Vc are — 2 V and — 1 V. Since I 0 is a relatively slowly varying function of voltage in the region of interest, straight-line approxim ations for both currents will be reasonab le ; hence, a simple interpolation is possible to obtain Vc « — 1.33 V.

This problem focuses atten tion on the clam ping phenom enon which occurs in capacitively coupled nonlinear circuits. The term clam ping refers to the shift in the average value of the load voltage v0 with drive level. W ith v,{t) = 0, in the steady state v0 = 2 V ; hence, as the am plitude of vt increases from 0 to 3 V, the average value

5.1 C A P A C IT IV E C O U P L IN G TO N O N L IN E A R LO A D 153

of v„(t) shifts from 2 V to 1.33 V. A further increase in the level of r,(f) causes the average value of v0(t) to decrease still further. We thus see dram atically tha t in such a nonlinear circuit the to tal ou tpu t voltage cannot be obtained by the superposition of independent ac and dc com ponents, as it can be in a linear circuit.

Analytic v0-i0 CharacteristicIf the v0-i0 characteristic is described by an analytic relationship, it is possible that Fc , va{t), and i„(f) may be obtained w ithout resorting to a graphical analysis. Forexample, consider the circuit shown in Fig. 5.1-4, for which

i0(t) = Ise qv°lkT - I dc. (5.1-1)

For this case, if we assume that u,(f) = F, cos a>t and that C is a short circuit at the frequency co such tha t vc(t) = VC = - V dQ, then

i„(t) = I seqVdkTe3ccos“ ' - / dc, (5.1-2)

where, as usual, x = qV^/kT. In addition,

I = Ic = I seqVclkTIo(x) - / dc, (5.1-3)

where I 0(x) is the modified Bessel function of order zero. Hence, in steady state, where Ic = 0, we have

k T , / dc k T , , ,VAc= - V c = — \ n - p -------- l n / 0(x)

q h q

k T= KicQ - — In I 0(x) = VdcQ - AV, (5.1-4)

where is the value of with Vx (or x) equal to zero and A Fis the bias depression (or the variation in clam ping voltage) induced by the driving voltage. Thus, for the given non-linearity and drive function, Vdc is a simple analytic function of / dc and x (or Fj).

Fig. 5.1-4 . N onlinear block consisting o f a current generator and a diode.

154 R EA C TIV E A N D N O N L IN E A R ELEM EN T CO M B IN A TIO N S 5.1

F o r values of x larger than 4(V1 > 104 mV), one m ay approxim ate the / 0(x) Bessel function term to w ithin better than 4 % (within 6 % for the x > 3) by

from which it follows that

k T k T— In J„(x) = F, — —- In 2nx = V, — (23.8 mV) — (13 mV) In x. q 2 q

Therefore, for x > 4,

Fdc = FdcQ - F, + (23.8 mV) + (13 mV) In x. (5.1-6)

Since the (13 mV) In x term varies by less than 32 mV as x varies from 5 to 50 (130 to 1300 mV), one sees that Vdc will vary alm ost in a straight-line fashion with F, over this region. The value of Vdc 0 will be a function of both I s (a junction param eter) and /dc (a circuit param eter). Since the natu ral logarithm of 10 is 2.3, a 10 to 1 change in one of these quantities will result in only a 60 mV change in Fdc0. Figure 5.1-5 plc*‘

vs. x and F, for the case where I s = e 30 mA = 1 0 16 A (a typical value for an integrated-circuit silicon diode) and for / dc values of 0 .1, 1.0 , and 10.0 mA.

F or / dc = 1.0 mA,

FdcQ = 30 x 26 = 780 mV.

Hence for x > 4, one can approxim ate K.c quite closely as

(780 + 24 + 26) mV - Vx o r (830 mV) - F , .

W hat this implies is that as far as calculating voltage levels is concerned, the constant current source and actual exponential diode com bination may be replaced by an ideal diode in series with a battery as shown in Fig. 5.1-6.

This ideal clam ping circuit clamps the top of F, to the positive battery vo ltage; hence Vdc = ~ V C is always just the battery voltage minus F , . Therefore, to within the precision allowed by the approxim ation, peak values of driving voltage greater than 830 mV will result in negative values of Fdc, while small values will result in positive values o f vdc. O nce Vdc = Vc is know n, it may be inserted into Eq. (5.1-2) to determ ine i0(t).

As an example of using the m odel of Fig. 5.1-6, consider the circuit shown in Fig. 5.1-7. For this circuit, we first replace the L R com bination by a current-source approxim ation and then use the model of Fig. 5.1-6 to arrive, in a simple fashion, at the ou tpu t ac signal voltages and dc ou tpu t level. O nce the ou tpu t voltages are known, one can revert to the basic diode equation or to the curves of C hapter 4 to determ ine the diode or source currents.

Consider the practical case where the reactance of L is large with respect both to the loss term and to the equivalent diode impedance, so that the ac current through the inductive branch may be assum ed to be negligible.

C A P A C IT IV E C O U P L IN G TO N O N L IN E A R L O A D 155

Then from the viewpoint of dc current, the inductor-resistor com bination may be replaced by a constant-current generator. Then the circuit of Fig. 5.1-7 may be replaced by the circuit of Fig. 5.1-4 and finally by the model of Fig. 5.1-6. In this circuit, the final value of / dc is also constrained by the additional dc circuit relationship

= vc .

If we assume that I s = e " 30mA (as in Fig. 5.1-5), then with a drive voltage Vt = 425 mV an assum ption o f / dc = 1 mA would lead to a battery voltage in the

156 REACTIVE AND NONLINEAR ELEMENT COMBINATIONS 5.1

Fig. 5.1-6 Voltage-clamping model for exponential diode and current source combination.

model of Fig. 5.1-7 of 840 mV. (This is 10 mV m ore than the value used previously, because weXised the actual value of x instead of an estim ated value.) Since Vc = 840 — 425 = 415 mV, then if r = 500 Q (a possible value for a 1 H torroidal coil) the circuit relationship would lead to / dc = 415/500 = 0.83 mA.

Now the m odel could be recalculated for the / dc = 0.83 mA case instead of for the 1 mA assumed initially. If this is done, the change in the battery voltage will be less than 5 mV out of 840 mV or much less than 1 %; hence such an additional cycle of calculation is probably not w arranted.

If r = 50 Q, then our initial calculation would lead to / dc = 8.3 mA. A recalculation would lead to an increase in the battery voltage of 26 In 8.3 = 55 mV and hence in / dc of 1.1 mA = (55/50) mA. A further recalculation would only increase the battery voltage by 26 In (9.4/8.3) = 3.2 mV or less than 0.4% and hence would rarely seem necessary.

In the real world one seldom has the simple case shown in Fig. 5.1-4. In practical circuits one often has a driving voltage source w ith a nonnegligible source impedance while a t the same tim e the diode (or transistor junction) may have either a deliberately added or an inherent series resistive com ponent.

The circuit tha t we shall consider is shown in Fig. 5.1-8. (The case of a finite source im pedance and a resistanceless diode m ay be transform ed in to the form ol Fig. 5.1-8 by adding equivalent curren t sources to “ m ove” the /dc generator to the o ther end of the source resistance. The am ount of the variation in the dc voltage across the capacitor is identical in the two cases.)

5.1 CAPACITIVE COUPLING TO NONLINEAR LOAD 157

+ R

Vx costo/ v

Figure 5.1-8

In this case

(5.1-7)

= V1 cos cot — Vc .

N ow the capacitor will no t pass dc, hence /„(i) = 0 or equivalently i^t) = Idc. The circuit will adjust itself to the value o f dc capacitor voltage tha t causes i'„(t) = /dc.

A general relationship between Vc and i0(t) does not seem possib le; however, solutions are possible for the tw o lim iting cases where IdcR « k T /q and where I dcR » kT lq . (The second case is equivalent to g in R » 1.)

O n the basis of our experience w ith the curves of Section 4.8, we plo t the bias depression A F fo r these tw o extreme cases, norm alized as in Section 4.8, and find tha t they form a tight set. We then conjecture tha t the I dcR = kT /q case falls halfway between the o ther tw o; thus we have the wherewithal to estim ate Vc = VCQ + A F for all practical cases.

W hen R is zero we revert to Eq. (5.1—4) to form A F = Vc - VCQ = (kT/q) In I 0(x) and norm alize it to Vco, which is kT /q in the R = 0 case.

W hen IdcR » kT/q , the physical diode resistor com bination may be replaced by the model shown in Fig. 4.8-3 to obtain the circuit shown in Fig. 5.1-9. F o r thiscircuit the quiescent value of — VCQ = VdcQ is I icR + K0 , and the norm alizing voltage

WV^R

5 Z (Ideal)

Figure 5 .1 -9

I 58 REACTIVE AND NONLINEAR ELEMENT COMBINATIONS 5.1

Ko = /dcK. In addition the ideal diode rem ains forward biased for VlIVco < 1, the circuit operates linearly, and no bias depression A F = VdcQ - Fdc exists.

F o r VJVC0 > 1, i0 becomes a periodic tra in of sine-wave tip pulses and Fdc adjusts itself to cause i'0 = / dc. This is precisely the problem tha t was solved in Section 4.8, from which we obtain that

Fc = FlC o s 0 - F0 = - K dc (5.1-9)

and

V*cQ ~ Vds A V V1 n ^ ^ ------- = — = 1 + TT cos <t>. (5.1-10)

CO CO 'CO

Thus Vc and in tu rn AV/Vc0 can be evaluated in term s of and Vco from the values of <j) given in Table 4.8-1.

N ow both these curves, norm alized with respect to

k TKo — ------ idc^ — idc(r in + K)>

4

are shown in Fig. 5.1-10. The do tted curve halfway between the extrem e cases is the conjectured curve for IdcR = k T /q or R = r in.

As an exam ple of the above analysis, let = 2.6 mA, R = 10 ohms,F, = 520 mV, and V0 = k T /q In (/dc/ / ES) = 0.65 V in the circuit o f Fig. 5.1-8. W ith these values we observe tha t VCQ = -0 .6 7 6 V, ginR = 1 , Vco = ldc(R + r in) = 52 mV and Fj/Fc0 = 10; hence from Fig. 5.1-10 we obtain A F/FC0 a 7, from which

K„

Figure 5.1 10

5.2 T R A N SIE N T B U IL D -U P TO STEADY STATE 159

followsvc = (-0 .6 7 6 V) + (7)(52mV) = -0 .3 1 2 V

and

v„(t) = (312 mV) + (520 mV) cos cut.

W ith this value of v0(t) one can return to the characteristics of Fig. 4.8 2 and obtain a reasonably accurate point-by-point p lo t of the diode-resistor current. For example r 0(f) - V0 has a peak value of 282 m V ; thus (vB(t) - V0)maJ V co % 5.4 in this example. This value determ ines a norm alized peak current from Fig. 4.8-2 of lop/1dc ~ 8.6 or Iop % 22.4 mA.

W ith R = 0 but with the same drive voltage, and the same /dc and IES, the capacitor dc voltage would be +193 mV, while the peak current would be approxim ately 29 mA or 30 % larger. Hence an apparently small source resistance or series resistor can have a reasonably large effect on both dc, ac, and peak values of the circuit’s voltages and currents. The key to the effect tha t the resistor will have is the size of /dcR with respect to (k T /q ) = 26 m V or equivalently the size o fg {nR with respect to 1.

5.2 TRA N SIENT BUILD -U P TO STEADY STATE

If v;(t) in Fig. 5.1 1 is periodic and applied to the circuit at t = 0, we can calculate the transient build-up of Vc and r„(f) to their steady-state values com puted above. Clearly in this case VL = Kc-(r) is a function of tim e; however, in order to obtain a reasonable solution, we continue to assume that this function of time is varying sufficiently slowly so that Vc(t) is essentially constant over any complete cycle of ¡,(f). W ith this assum ption, vjt), ijt) , and Ic = Ic(t) can be evaluated directly in term s of Kc(f) by the same m ethod employed in obtaining their steady-state values. In particular, the Vc- I c curve obtained to determ ine the steady-state value of Vc = Fcss (cf. Fig. 5.1-3) now relates the slowly varying capacitor current Ic(t) to the capacitor voltage Kc(r) by an expression of the form

M 0 = f ( V d t ) l

The capacitor itself relates I c(t) to Vc(t) by the expression

(5.2-1)

(5.2-2)

hence, by elim inating I c(t) from Eqs. (5.2-1) and (5.2-2), we obtain the first-order nonlinear differential equation for Fc(i) in the form

(5.2-3)

If./ ( Vc) is known in analytic form, we can solve Eq. (5.2-3) directly, as we illustrate in Section 5.3. O n the o ther hand, we can expand f ( V c) in a Taylor series about the

160 REA C TIV E A N D N O N L IN E A R ELEM EN T C O M B IN A TIO N S 5.2

steady-state value of Fc = Fcss to obtain

r A V c _ f i t ; \ , ^ / ( I ' c s s ) , , / T/ -, , d 2f ( V Cs s ) (V c — P c s s ) 2 i

L ~dt - j{Vcss) + ~ d V ^ Vc ~ Kcss) + ~ J v l 2 ! + " '

= —G(VC — FCSS) + b2(Vc — Fcss)2 + • • -, (5.2—4)

where I css = / ( F css) = 0, G = -d f( ,V css)/ôVc , and

, 1 S"f(Vcss)b- - J \ ^ v T f o r " a 2 '

The advantage of the Taylor series expansion is that, in general, only a few term s m ust be kept to closely approxim ate f ( V c) over the range of interest of Vc \ in addition, iff ( V c) is available only graphically, as in Fig. 5.1-3, the coefficients of the first fewterm s of the Taylor series can be determ ined by standard curve-fitting techniques to closely approxim ate the Vc- I c curve.

As a first approxim ation to the transient build-up, we keep only the first nonzero term in the Taylor expansion to obtain

C ^ = - G ( F C - Fcss), (5.2-5)

where — G is the slope of the Ic Vc curve evaluated a t Vc = Fcss . E quation (5.1 5) is linear, with the solution

VcU) = i^css + [Vc (0 + ) - Vcss] e - ‘i'}u(t), (5.2-6)

where x = C/G.F or the nonlinearity o f Fig. 5.1-2, which results in the Vc- I c curve shown in

Fig. 5.1-3, if !;,-(() = (3 V) cos wt is applied a t t = 0, then

j/c(0 + ) = - 2 V , Vcss = -1 .3 3 V, and G = 0 .72m Q ;

hence Fc(f) = [-(1 .3 3 V) - (0.67 \ > - ,/r]u(f) and v0(t) = [ - V^t) + (3 V) cos cuf]w(f), where t = (1.39 kQ)C. A sketch of v0(t) is shown in Fig. 5.2-1. Since Fc(t) varies only between — 2 V and —1.33 V, the linear approxim ation for the Fc- / c characteristic is quite valid.

W hen term s beyond the first term of the Taylor series expansion for f ( V c) are kept, then the solution to Eq. (5.2^4) is obtained in the following manner. We rearrange the equation in the form

~ Gdt _ dVc“ (Vc ~ W - (b2/G)(Vc - Vcss)2 ■ ■ •’ (5-2~?)

we expand the denom inator o f the right-hand side in partial fractions, and we integrate term by term. We illustrate this procedure for the case w here f ( V c) may be approximated by the first two nonzero term s in its Taylor expansion, i.e., where

C f = - G ( V C - Vcss) + b2(Vc - Fcss)2. (5.2-8)

5 .2 T R A N S IE N T B U IL D -U P T O STEA D Y STATE 161

. ]

I -4.33 V

t= C /G

G = 1/1.39 k il

1.33 V

'A

1.67V

Fig. 5.2-1 O utpu t transient when a 3 V peak sine wave is suddenly applied to the circuit of Fig. 5.1-2.

For this case,Gdt~C~

dVr(Ve l 'essa i (^2/G){VC ^css)]

dVc | (bj/G)dVc

K: ~ Krss css i

-t/i’

(5.2-9)

(5.2-10)

1 ~ ( b 2/G)(Vc

U pon integrating and rearranging terms, we obtain

K e ~ 11'Vr(t) = + ---------------------

fU css 1 + (b2/G)Ke

where K is a constant of integration and t = C/G. W hen b2 is zero, this equation of course reduces to the linear case. Unless K b 2/G is appreciable with respect to unity, the influence of the square term is always small. In any case, b2 does no t influence the length of the transient. In m ost cases b2 affects merely the initial slope and the values of V^t) for t < x.

A similar result can be obtained if f ( V c) consists of a linear plus a cubic term. Specifically, if Eq. (5.2-4) has the form

d t

then the solution for V^t) takes the form

V ^ t ) = V c s s +

d Vc - £ = — G {vc - Vcss) + b 3(v c - vcssy \

K e - ,/t

V'1 + ( b i /G )K e : t l x

(5.2-11)

(5.2-12)

where K is a constant of integration and t = C/G. Again, unless b3K /G is appreciable com pared to unity, the fc3-term has no influence. And again the time of the transient

162 R EA C TIV E A N D N O N L IN E A R ELEM EN T C O M B IN A TIO N S 5 .3

is set entirely by t = C/G, as it is for any polynom ial approxim ation, of whatever order, of f ( V c) o r for an exact analytic expression for f ( V c).

For the diode nonlinearity of Fig. 5.5-1, l c = f ( V c) is given by Eq. (5.1-3) and

G = d W s s ) = < lh qVcssl>T[o(x) = (5 2_ 13)Vv£ k T k T

As we shall see in the next section, the transient is governed by the time constant (cf. Eq. 5.3-11)

r , r kT C x = C/G = —— .i 'd c

W ith a knowledge of the rate a t which Vc varies, we are now in a position to determ ine the value of C which justifies our assum ption that Vc(t) is constant over a cycle o f v,{t). If the period of v^t) is T, and if t > 20T, then V(t) varies less than 5% in the time interval T. Consequently, if

C > 20G T = 4071(7co

or

coC— - > 40tt, (5.2-14)G

where to = 2n/T, then our assum ptions are justified. In many practical situations the inequality of Eq. (5.2-14) may be relaxed by a factor of 2 or 3 w ithout any noticeable effect on the expressions derived.

5.3 CAPACITIVELY C O U PLEDTRA N SISTO R A M PLIFIER —C O N STA N T C URREN T BIAS

Since a great many large-signal ac transistor amplifiers are capacitively coupled, we shall study this coupling problem in detail in this section. We begin by analyzing the capacitively coupled amplifier shown in Fig. 5.3-1, which is biased from a constant current source / dc and driven at its em itter term inal (common base) by the input voltage y,(i). We then show, in Section 5.4, that the analysis is essentially unchanged if the drive is applied to the base with the em itter capacitively bypassed (common emitter) or if the bias is developed by the m ore conventional scheme of resistively dividing - V cc and applying it to the base w ith a resistor R E placed in the em itter circuit. In all cases, we assume, however, that the input volt-am pere characteristic of the transistor may be modeled by an exponential relationship of the form

iE = l ESe*r*BikT = (5 J 1}

tha t the collector-to-em itter voltage never reaches zero and saturates the transistor, that the collector current may be modeled by a current source ai£ , where a i l . over the entire range of em itter currents, and that the internal reactances of the

5.3 TR A N SIS T O R A M P L IF IE R — C O N S T A N T C U R R E N T BIAS 163

Fig. 5.3-1 Capacitively coupled transistor amplifier.

transistor, such as charge storage, may be neglected. These assum ptions provide an . excellent first-order approxim ation to the perform ance of many transrstor amplifiers for values of \ZL\ less than 100 kQ and frequencies up to the tens of megahertz.

F o r the circuit of Fig. 5.3-1, we first determ ine its steady-state behavior with V; ( t ) a periodic drive and then determ ine how the circuit builds up to steady state. In addition, in Section 5.6 we develop the transfer function of the netw ork for the case where v£t) is an AM wave of low m odulation index. This transfer function proves to be useful for evaluating the steady-state stability of transistor oscillators and the ou tput signals of transistor limiters.

If we assume for the circuit of Fig. 5.3-1 that the coupling capacitor C is sufficiently large tha t its voltage vc rem ains constant at Vc = — Fdc over a cycle of then the model of Fig. 5.1-6 is im m ediately applicable.

F or a given input voltage, we determ ine the value of the battery and hence of the clam ping level and the capacitor voltage. We then determ ine the em itter current (the diode current in the model) from the know n dc value of current, Vc , and the driving voltage waveshape.

For example, assume that the waveshape of Fig. 5.3-2 is applied as vt{t) in the circuit of Fig. 5.3-1.

As a reasonable approxim ation to aid in fixing the battery voltage in the model, we m ight m odel the input as a 200 mV peak sine w ave; hence x % 7.7. F o r such an x, the (k T /2 q ) \n 2 n x term becomes approxim ately 50 mV. If we assume that / dc is 1 mA and th a t I s is 10“ 16 A, then

k T I— In KdcQ = 760mV.<1 I es


Fig. 5.3-2 Plot of i¿,(í)and i¿(t)/IP = w{t) vs. t.

Hence, to a very good approxim ation, the em itter voltage will be clam ped to a peak voltage of + 8 1 0 m V ; thus the capacitor voltage will be -6 1 0 m V . (If the 200 mV peak signal is to reach 810 mV, the capacitor voltage must add 610 mV to it.)

The em itter current can be expected to flow prim arily during the m ost positive interval of the driving signal.f Since this interval consumes only one-eighth of the total period, the current during this interval m ust be eight times the average value, or 8 mA.

Once the current is know n to flow prim arily in narrow pulses, it follows from C hapter 4 that the fundam ental com ponent is approxim ately twice the dc value; tha t is, IEl x 2 /dc. Therefore, if Z L is a h igh -g parallel R L C circuit tuned at (j l > = 2 n /T with a parallel resistor R L, then the output voltage v j t ) may be closely approxim ated by

v j t ) = — Vcc + 2aIdcR L cos cut. (5.3-2)If the stairstep waveshape were replaced by a 200 mV peak sine wave, then the

same clam ping level and capacitor voltages would exist. F o r this case, the ratio of the peak current to the average current is

tT he currents that flow when the input to the emitter is 810 mV and when it is 710 mV are proportional to e810/26/e710/26 = e3 8 - 45. Hence the current during the maximum voltage period is 45 times the current flow during the adjacent segments. Thus assuming that all the current flows in one flat-topped pulse will lead at most to about 3 2 % error.

5.3 TR A N SIST O R A M P L IF IE R C O N ST A N T C U R R E N T BIAS 165

for x > 4. Thus for x = 7.7 and / dc = 1 mA, IP x 1 mA. Figure 4.5-3 indicates the general shape of the current pulse, while Fig. 4.5-4 indicates a conduction angle of about 104°. Figure 4.5-5 indicates that for this value of x, / , = 1.85/dc; hence again, if the load is a tuned circuit, the ou tput voltage can be found as

v0(t) = - Vcc + 1.85a/dc/?Lcosa>£. (5.3-4)

If the load is tuned to some harm onic of the input driving waveshape, then either Fig. 4. 5-5 or the tables of 2/„(x)//0(x) that appear in the Appendix at the back of the book may be used to determ ine vjt).

A m ore general approach to the circuit of Fig. 5.3-1 is to define a waveform function W(t) as the ratio of the actual em itter current, i j t ) , to the peak em itter current I P. If the capacitor voltage has reached a steady-state value Vc = — VAc, then

¡ p = ¡ E se1v <iclliTe qvim„ lkT ^ (5.3-5)

¿¿t) = i ESe^v^ kTe9vM)lkT, (5.3-6)

and

W(t) = eqVi lkT/eqVi"'*’‘liT

= m / l r . (5.3-7)

The waveform function contains the tim e-variable portion of i^ f); hence if I P and W(t) are known, then i^ t ) is known. The average or dc value of i^ t) will be I P times the average value of W (t):

/dc = W ) = b m . (5-3-8)

Thus if / dc and v,{t) are known, one first evaluates or plots W(t), which is only a function of t?,(i), and finds its average value W(t). If H^r) and / dc are known, then IP is fixed. However, fixing I P fixes K.C and com pletes the solution.

W hen an unbypassed series resistance exists in either the base or em itter circuits, the results of Sections 4.8 and 5.1 for the added resistance case are again directly applicable.

Transient Buildup

If the periodic input r,(i) is applied at t = 0 to the circuit of Fig. 5.3-1, during the buildup to steady state the slowly varying com ponent of capacitor current is given by

/, = /,.(() = I ESe~ qVc(,)lkT eqVi(,)/kT — Jdc = C ^ . (5.3-9)at

Hence the differential equation for Fdc(t) = - Vc(t) takes the form

C— }— + W (t)= I icdt

(5 .3 -1 0 )


We may integrate Eq. (5.3-10) in a straightforw ard fashion to obtain

caV*At)lkT _ VdJlEs)e “ ' m'”‘/ (5 3-11)W(t) + K e~ ,n '

where t = C k T /q Idc = C /gin, K is an arb itrary constant of integration, and £in = i W ^ ' s the small-signal input conductance of the transistor in Fig. 5.3-1. If we assume that for t less than zero t;, = 0, then

e ~qvim,JkT _ /dcI es

and

K = e - ‘>r'™ '!kT - W(t). (5.3-12)

W ith this valué of K , Eq. (5.3-11) reduces (for r > 0) to

„qVaMIkT [láJ l ES)e qVim‘JkTW ( t )+ [e - i v' ^ kT - W{t)]e~,lx'

or equivalently,

(5 .3-13a)

I Ac

k T IVdA t ) = - v t + — l n = ---------------------------------------------(5.3— 13b)

q W + (e-*Vl™*lkT - W )e~th

Now, in the steady state, W = Iic/ I P and e~'h = 0 ; thus Eq. (5.3-13) reduces to

k T I k T ____V■ = + — \n - ^ In W. (5 .3-13c)

.9 !es <7

K i c Q

This exact form for VdQ also results if the waveform function is em ployed to evaluate the steady-state transistor response in a fashion discussed previously in this section.

By noting thatl£(f) = / £SeiFdc(,)/* V '" — "t r H/(i)

and employing Eq. (5.3-13a), we may also obtain (for t > 0)

m - U JL W )]W + (e~qVim“*lkT- W )e~,/z'

where, of course, W(t) = eqVii,)lkT leqVim‘' lkT. As t -* oo, Eq. (5.3-14) reduces to the expected form iE = ¡PW(t).

For v,{t) given by_the staircase waveform of Fig. 5.3-2, W(t) has the form shown in the same figure, W = g, and r imaj = 200 m V ; hence, for this case, with k T /q = 26 mV and I J I ES = 5 x 1012,

MdctHW]e( ) 1 + (3.7 x lO“ 3 - l ) e - " ' (5.3 15a)

5.3 TR A N SISTO R A M P L IF IE R — C O N S T A N T C U R R E N T BIAS 167

and

Fdc(r) = (614 mV) - (26 mV) In [1 + (3.7 x 10“ 3 - l)i-“,/t]. (5.3 15b)

Table 5.3-1 presents values for Fdc(f) and the peak em itter current IP(t) vs. r /t for the expressions given in Eq. (5.3-15). [IP(t) is the coefficient of W(t) in Eq. (5.3-14) or Eq. (5.3-15a).] In addition, a sketch of i^t), given by Eq. (5.3—15a), is shown in Fig. 5.3-3.

It is apparent from Fig. 5.3-3 that at t = 0, when r,(r) adds directly to ^,.(0) = — 1 (0) across the input exponential characteristic of the transistor, an exceptionally large emitter current flows with waveform W{t). The average value of this large current, flowing prim arily through the coupling capacitor, very rapidly decreases Vdc and in turn reduces the peak value of the em itter current. In particular for tjx = 1

Table 5.3-1 T abulation of values for Vdc(t) andM l ) v s . t / T

t/x h i' ) VdcUl mV

0 2200/ dc 7601 12.6/dc 6262 9.25j ic 6183 8.437dc 6144 8.15/ dc 6145 8'dc 614


the peak value of the em itter current has fallen from 2200/dc, (f/r = 0) to 12.7/dc. By the time f/r = 4, steady state has been reached for all practical purposes.

The practical difficulties with this solution are that for any appreciable v,{t) the turn-on current is so enorm ous that the bulk of the em itter current transient is over by the time t = O.IOt or less. The practical consequences of the dem and for this large current are several. F irst, C m ust be very large o r the assum ption of a small change of capacitor voltage per cycle is violated. Second, the supposedly very large instantaneous currents lead to appreciable voltage drops across internal transistor (or diode) resistances, so that we cease to have the true exponential voltage-current relationship that was assumed. We have instead a resistor-transistor com bination of the type assum ed in Section 4.8. The actual transient in a real circuit is somewhere between the extreme value suggested by Fig. 5.3-3 and the value obtained from a resistor battery model formed by the techniques of Sections 5.1 and 4.8. The reason is primarily that the presence of the resistance limits the initial em itter current am plitude for 0 < t < O .It w ithout appreciably affecting the rem ainder of the transient.

For the case where vt(t) = Vt cos cor, Wx(t) has the form shown in Fig. 4.5-3,

except that the pulse shape is given by WJ,t). Consequently, here again the em itter current starts with a very large peak value and rapidly decreases tow ard its steady- state level, which it essentially a tta ins for t > 4r. Again the internal transistor resistance or the generator impedancte limits the initial value of the em itter current.

At this point, it is apparen t that the response time of the circuit of Fig. 5.3-1 is governed by the time constant t = C /gin. Therefore, if u,(i) is an AM wave and the envelope of this AM wave varies slowly com pared with t , the circuit essentially reaches steady state for each envelope value. Consequently, any steady-state transfer function relating the peak value of v,{t) to a current or voltage in the circuit of Fig.5.3-1 can also be used to relate the time-varying envelope of r,(i) to the corresponding current or voltage.

For example, with v t(t) = F jcoscoi, the fundam ental steady-state collector current is given by

hence from Eqs. (5.3—13b) and (5.3-14) we obtain

(5.3-16)

and

(5.3-17)

A sketchS)f iE(t) in this case has essentially the same form as the sketch of Fig. 5.3 3,

'c i = a /d c -T T T c o s w f ,2 h ( x )

(5 .3 -1 8 )o'

5 .4 TR A N SIST O R A M P L IF IE R — RESISTOR BIAS 169

where x = qV JkT . However, if vt{t) = g(t) cos a>t and g(f) varies slowly in com parison with t, then

21 ¿qg(t)/kT]¡ci = « i d c , r , „ /t.x T COSOJf- (5.3-19)I 0[qg(t)/kT]

Again we note from Fig. 4.5-5 that if g(f) is large in com parison with kT /q for all values of f, Eq. (5.3-19) reduces to

icl « 2a /dc cos tot

and the circuit strips the AM inform ation from ic l , thus functioning as a limiter.N ote carefully that if the circuit of Fig. 5.3-1 is used to process AM signals, the

value of the coupling capacitor C m ust be arrived at by a compromise. The value m ust be large enough to appear as a short circuit at the fundam ental frequency of v,(t), and yet it m ust be small enough so th a t r = C /gin is small in com parison with the time over which the envelope of the AM wave varies. Such a com prom ise is possible only if there is a wide separation (of a factor of a t least 100) between the maxim um m odulation frequency and the carrier frequency.

In Section 5.2 we saw that the coupling capacitor appears as a short circuit at the frequency to if

T = — > 207,Sin

or equivalently,

— > 407t. (5 .3 -20)&in

Thus Eq. (5.3-20) puts a limit on the m inim um value of C.To obtain an inequality for the m axim um coupling capacitor value when using

the circuit of Fig. 5.3-1 to process input AM signals, it would be desirable to obtain a general expression for the capacitor voltage and the em itter current as a function of the envelope of vt(t) and the capacitor value. W ith such a general expression, the m axim um capacitor value which produces the desired em itter current could readily be determined. U nfortunately, a general relationship is m athem atically in trac tab le ; however, a small-signal transfer function relating low-index envelope variations to small variations in capacitor voltage and em itter current is possible. In addition, the m axim um coupling capacitor value obtained from the small-signal analysis again provides an excellent estim ate of the value required for large-signal operation. This small-signal transfer function will be derived in Section 5.6.

5.4 CAPACITIVELY C O U PL ED TRA N SISTO R A M PLIFIER — R ESISTO R BIAS

In this section we continue our analysis of the large-signal capacitively coupled transistor amplifier by considering the circuit shown in Fig. 5.4-1, which is biased in a conventional fashion. Here, in addition to a coupling capacitor C E, a base bypass

1 70 R EA C TIV E A N D N O N L IN E A R EL EM EN T C O M B IN A TIO N S 5 .4

y — ^2" R t+ R 2

D _ ^1 R 2- TÇ+RT

Fig. 5.4-1 Capacitively coupled transistor amplifier with conventional bias.

capacitor is included to ensure that the entire periodic ac signal v,{t) appears across the em itter-base junction. We again begin by determ ining the steady-state behavior of the circuit. This analysis is som ewhat m ore involved than the corresponding analysis of the circuit o f Fig. 5.3-1, because the depression in the dc em itter-base voltage caused by the application of results in a change in the average em itter current from its quiescent (vt = 0) value. Hence, in addition to the average em itter- base voltage, the average em itter current m ust be determ ined as a function of v/[t) before the total em itter current and in turn the ou tput voltage can be evaluated.

Fortunately , as we shall dem onstrate, the change in the average em itter current with input drive level is usually a second-order effect in m ost well-designed transistor am plifiers; hence IEQ merely replaces /dc in all previously derived results. F o r the few circuits which are exceptions to the rule, we develop the set o f universal curves for Gm(x)!gmQ shown in Fig. 4.5-5 which directly replaces the curve o f Gm(x)/gm of Fig. 4.5-6 in the analysis of narrow band circuits with sinusoidal drives.

We determ ine the steady-state average em itter current I E0 as a function of i by first com puting the depression AK of the em itter-base voltage from its quiescent value when vt is applied, and then we employ A F to com pute the shift in I E0 from its quiescent value I EQ.

W hen x > 4 for sinusoidal drives, the clam ping model of Fig. 5.1-6 may be used in two successive approxim ations to provide a very rapid answer for this circuit. (See Problem 5.5 a t the end of this chapter for an example.)

In the analysis of the circuit o f Fig. 5.4-1 we again assume tha t the transistor does not saturate, that

ic = a.i\E or iB = (1 - a)iE,that internal transistor reactances are negligible, and that

iE = I ESe ^ BlkT

5 .4 T R A N SIS T O R A M PL IFIE R — RESISTO R BIAS 171

Fig. 5.4-2 N arrow band transistor amplifier.

W ith these assum ptions and the Thevenin equivalent of the base circuit shown in Fig. 5.4-1, the quiescent em itter-base voltage Kicfl and the quiescent em itter current I eq [which exist p rio r to applying v,itj] are related by the equations

feTi J*e Vic Q — I n -r—Q I e s

(5.4-1)

and

. _ r BB r à c Q

EQ ~ R e + (1 - a )Rb(5.4-2)

In general, Eqs. (5.4-1) and (5.4-2) are difficult to solve sim ultaneously; however, if we assume

I ES = 2 x 10“ 16 A and kT /q = 26 mV

and tabulate VAcQ vs. I EQ as shown in Table 5.4-1, we find that VdcQ experiences only small variations abou t 760 mV for values of I EQ between 0.1 mA and 10 mA. Therefore, for m any practical circuits, VdcQ may be closely approxim ated by f V, and with this value of K,cQ, Eq. (5.4-2) simplifies to

Vrr - ( | V)I _ ' B B _______EQ~ R e + ( 1 - x)Rb

(5.4 3)

The value of I ES = 2 x 10“ 16 A corresponds to a silicon transistor integrated on a “chip.” F o r a nonintegrated silicon transistor, I ES x 2 x 10“ 14 A and VAqQ may be approxim ated by 650 mV. F or a germ anium transistor, / £s % 2 x 10“ 7 A and 220 mV provides a reasonable approxim ation for VdcQ.

172 R EA C TIV E A N D N O N L IN E A R ELEM EN T C O M B IN A TIO N S 5 .4

Table 5.4-1 Tabulation of VdcQ vs. l EQ with kT/q = 26 mV and l ES = 2 x 10“ 16 A

Fdce,m V IEQ, mA

700 0.1760 1820 10880 100

If we wish the quiescent point to rem ain fixed, independent of tem perature or production variations in the transistor param eters, I EQ m ust be insensitive to these variations. If I EQ does not vary, I CQ = ol1 e q rem ains fixed, since a x 1 and in turn VCEQ, which is a function of 1EQ, ICQ, and the bias circuitry, is stable. We observe from Eq. (5.4-2) that, to keep I EQ independent of transistor param eters, the following conditions m ust be satisfied:

R e » (1 - x)Rb (5 .4-te)and

VBB » VAcQ. (5.4—4b)

The greater the strength of the above inequalities, the greater the bias stability with respect to transisto r param eter variations. If R E x R B and VBB x 10 VdcQ, excellent bias stability is achieved in m ost applications.

If now the periodic input v^t) is applied and both C B and C E are assum ed to be ac short circuits over the frequency range occupied by t>,(r), the em itter current for the circuit of Fig. 5.4-1 is given by

iE = i Ese« v ^ r e M T

= I ESe‘>VdJkTe‘lv‘™JkTW(t), (5.4-5)where Vdc is the dc voltage which develops across the com bination of C B and C E. W e observe here th a t the em itter current waveform is still determ ined by eqVi(t)lkT and is therefore invariant with changes in bias configuration. In addition, we observe that, if —Vi(t) were inserted in series with C B and C E were returned to ground, Eq. (5.4-5) would rem ain unchanged; hence, if r,(f) = Vl cos cot, the only difference between driving the em itter and driving the base would be the shift in the em itter current by half of a carrier cycle.

If we again define the steady-state em itter-base voltage bias depression A F as

A V = V dcQ- V dc, (5.4-6)

then the expression for the em itter current may be rewritten as

,E(f) = I EQe- ‘‘*yikTeqVi™JkTW(t), (5.4-7)

where I EQ = l ESeqVa',QlkT In addition, the average em itter current IE0 is given by

h o = I EQe~q*vlkTeqVi™JkTV m - (5.4-8)

5.4 T R A N SIS T O R A M P L IF IE R — RESISTOR BIAS 173

In the steady state, I E0 is also constrained by the bias circuitry to have the form

j _ Vss - Fdc A F _ 7 I A V \E0 R e + ( 1 - oc)Rb eq R e + ( 1 - a)RB £Q( V j ’ '

where Vx = l EQ[RE + (1 — a)KB] = VBB — VdcQ is the sum of the quiescent voltage which appears across R B and the quiescent voltage which appears across R E. C om bining Eqs. (5.4-8) and (5.4-9), we obtain an expression for the steady-state bias depression A V :

A V1 + — e,AK/*r = eqvi™JkTW{t). (5.4-10)

B ecause o f th e tr a n s c e n d e n ta l n a tu re o f th e e x p re ss io n fo r A V g iv en in Eq.(5.4-10), it is impossible to obtain an explicit solution for A F in closed form. F o rtu nately, however, for Vx > 520 mV [which is always true in a well-designed circuit— cf. Eq. (5.4—4b)], the value of A V satisfying the equation

gtW kT = (5.4- 11)

differs by less than 5 % from the value of A Fsatisfying Eq. (5.4-10), and the agreementis closer with higher values of F^.t Consequently, with the approxim ation for A Vgiven in Eq. (5.4—11) we obtain the closed-form expression

l t ___A K = ^ m„ + — In W(t). (5.4-12)

QW ith the aid of Eq. (5.4-9) we also obtain

(qvimJ k T ) + In WI E O — I E Q \ 1 + q V J k T

(5.4-13)

By noting [from Eqs. (5.4-7) and (5.4-8)] that i ^ t ) / lE0 = W(t)/W{t), we may finally

f Assume that AF, satisfies Eq. (5.4-11). Therefore, equating Eqs. (5.4-10) and (5.4-11), we obtain

or equivalently,

q \V / k T \ ¿v/kT _ + qVJkT 6

AF, - AV _ k T I qAV/kT\ AV ~ <jAF " ( + qVJkT I

However, in (1 + x) < x ; therefore,

AV, - AV 11 <AV ~ qVJkT

AK - AF,

and, for q V J k T > 20,

AF< 0.05.


write the expression for the to tal em itter current in the form

I e M = 1 EQ(_ 4)

where I P = I E0/W . Consequently, by em ploying Eq. (5.4-14), we can determ ine the steady-state expression for ¡¿it) for any bias arrangem ent sim ilar to the one shown in Fig. 5.4-1, once the form of v,{t) is known.

As an example of determ ining the steady-state value of iE(t) as well as the steady- state ou tput voltage, we consider the circuit shown in Fig. 5.4^2 with v,{t) of the form shown in Fig. 5.3-2 applied. We assume that the transistor is silicon and not part of an integrated circuit, so tha t VdcQ st 0.65 V, and that a = 0.99. F rom the bias circuitry we observe that

VBB = (5V)(1.2kfl)/5.1 kQ = 1.18 V and R B = 1 .2 k i) ||3 .9 k fl = 916 0 ;

hence1.18 V - 0 . 6 5 V t

EQ ~ 510Q + 9 .16Q “ ' m

and Vk = 1.18 V — 0.65 V = 0.53 V. In addition, since vimitx = 200 mV, k T /q = 26 mV, and W = g, we ob tain from Eq. (5.4-13) I E0 = 1.02 mA( 1.276) = 1.3 mA, which leads directly to

¡¿it) = 8(1.3 m A)W (t) = (10.4 mA)W(i),

where W(t) is the train of narrow pulses shown in Fig. 5.3-2. Since the tuned collector circuit has Q T — 16 and is resonant at w = 107 rad/sec, the ou tpu t voltage is given by

v0(t) x (a)(2)(1.3 mA)(1.6 ki2)cos 107f - (5 V)

= (4.16 V) cos 107f - (5 V).H ere again, since the em itter current pulses are narrow , we have approxim ated the am plitude of the fundam ental com ponent as twice the average value.

In solving this problem we have assum ed that the transistor does not saturate or, equivalently, that the collector voltage rem ains m ore negative than the em itter voltage at all times. To check this assum ption we require, in addition to ¡>0, an expression for the em itter voltage vE. Clearly, vE contains the dc com ponent and the ac com ponent u,(i); therefore,

VE(t) = - (0 .6 6 2 V) + v,{t).

Figure 5.4-3 shows a sketch of u£(t) and d0 on the same set of coordinates. F rom this sketch, which also includes i^t), it is apparen t that d,, < vE a t all times and saturation does not occur.

As a second example, we consider f,(i) = V1 cos cot applied to the circuit of Fig.5.4-1. F o r this drive the bias depression A V obtained from Eq. (5.4-12) is

5 .4 TRANSISTOR AMPLIFIER— RESISTOR BIAS 175

Fig. 5 .4-3 W aveforms for the circuit of Fig. 5.4-2.

Table 5.4-2

X In / 0(x) X In l 0(x

0 0 8 6.0581 0.2359 9 6.9982 0.8242 10 7.9433 1.585 12 9.8504 2.425 14 11.775 3.305 16 13.706 4.208 18 15.647 5.128 20 17.59

I 76 REACTIVE AND NONLINEAR ELEMENT COMBINATIONS 5.4

Table 5.4—2 includes values of q A V /k T — In I 0(x) vs. x for values of x between 0 and 20. In addition, with p,(i) = VI cos cot, the average em itter current reduces to

/ - / [~i + b l M \ . I e o ~ 1eq1 1 + qVxik T y

as a result, the to tal em itter current is given by

- a - - m i " ■.?. r

(5.4-15)

/o Wcos mot

]■(5.4-16)

A plot of I e o / I e q vs x obtained from Eq. (5.4-15) with q V J k T as a param eter is given in Fig. 5.4-4. W ith the aid of Fig. 5.4-4, once the quiescent values of I EQ and Vx and the input level Vv (or x) are known, the total em itter current or any of its harm onics can be determined.

Fig. 5 .4-4 Plo t o f I E o / I e q vs- * with qVJkT as a param eter. Values of Vk shown in brackets are obtained with kTjq = 26 mV.

5.4 TRANSISTOR AMPLIFIER— RESISTOR BIAS 177

For example, consider v\ = (208 mV) cos 107f applied to the circuit of Fig. 5.4—2. Since for this circuit I EQ = 1.02 mA, Vx = 0.53 V, and K, = 208 mV, if kT/q — 26 mV, then x = 8 and qVJkT % 20. From the curves of Fig. 5.4-4 we obtain

1E 0 = 1.3■EQ

and in turn

*e ( 0 —(1.33 mA)e8

/0(8)

= (9.24 mA)l%(i).

In addition, the fundam ental com ponent of ijfr) has the value

i£1(t) = (1.33 m A ) ^ i ^ cos cot•0\v)

1 . 8 7 1

and thus vQ(t) = (4.0 V) cos cot — (5.0 V).N ote carefully that Fig. 5.4-4 provides a set of universal curves that can be used

when tj = V, cos cot for any bias circuit values and for either germ anium or silicon transistors. However, when dealing with narrow band amplifiers and oscillators, it is m ore convenient to develop a similar set of universal curves for the large-signal average fundam ental transconductance Gm(x), where

G J x ) =oclE l I

ViClVi

and / £1 and Ic , are the coefficients of the fundam ental com ponents of the em itter and collector currents respectively. W ith the aid of Eq. (5.4—16) we can express Gm(x) in the form

/ \ ^ EQnM =

SmQ

1 +

1 +

In J0(x) q V J k T

In / 0(x)

27, (x)

lo M

21 Ax)x / 0(x)’

(5.4-17)q V J k T _

where gmQ = a IEq /k T is the small-signal transconductance evaluated at the quiescent point. The norm alized curves of Gm(x)/gmQ vs. x with q V J k T as a param eter are given in Fig. 5.4—5.

Once Gm(x) is determ ined for any particular set of bias values and input voltage levels from the universal curves o f Fig. 5.4-5, the ou tpu t voltage v0 of a narrow band amplifier may be expressed directly as

v0(t) = ( V1 c o s (o t)G Jx)RL - Vcc, (5.4-18)

where RL is the resistance of the collector-tuned circuit at the resonant frequency co.

1 78 REACTIVE AND NONLINEAR ELEMENT COMBINATIONS 5.4

Fig. 5.4-5 Plot of Gm(x)/gmQ vs. x with qV JkT as a parameter. Values of Vx in brackets are obtained with kT/q = 26 mV.

For example, if we again consider i>,(i) = (208 mV) cos 107f applied to the circuit of Fig. 5.4-2, then Vx = 530 mV and I EQ = 1.02 mA. If we let kT/q = 26 mV, then x = 8, q V J k T x 20, and gmQ = q IEQ/ k T = 39.2 mmho. F rom Fig. 5.4-5 we obtain Gm(x)/g„Q = 0.307; hence Gm(x) = 12 mmho. Finally, since R L — 1.6 kQ, v0(t) is given by

v0(t) = —(5 V) + (208 mV)(cos <yf)(12 mmho)(1.6 kf2) = —(5 V) + (4 V )cos cot,

which is in exact agreem ent with the expression derived by the previous method.In most well-designed circuits of the form shown in Fig. 5.4-1, vima% is much less

than Vx or, equivalently, (cf. Eq. 5.4-4b). F o r this type of circuit,

( q v ^ / k T ) + ln W q V J k T

« 1 (5.4-19)

(since In W^must be negative and smaller in m agnitude than qvimtJ k T , the num erator of Eq. (5.4-19) is always less than qvimaJ k T ) and Eq. (5.4-13) for the average steady- state em itter current simplifies to

= I E Q - (5.4 20)

Equation (5.4-20) is intuitively satisfying, since a small ac input voltage can at most effect a small dc bias depression, this, when com pared with a large VB, has little effect

5.4 TRANSISTOR AMPLIFIER— RESISTOR BIAS 179

on the average em itter current. In such a circuit the em itter resistor R E can be replaced by a current source I EQ with no effect on any of the current or voltage waveshapes within the entire c ircu it; however, with such a current source bias, all the results of the previous section, as well as all the results of Section 4.5, are directly applicable.

For the case where t>,(r) = V, cos cot, the inequality of Eq. (5.4—19) is equivalent to (cf. Eq. 5.4-16)

In I 0(x) q V J k T

« 1.

Clearly, then, if

I T^ > 20— l n / 0(x)

q

(5-4—21)

(5.4-22)

for a particular drive level and bias configuration, I E0 may be approxim ated by I EQ within .5%. Figure 5.4-6 plots Vk, given by an equality in Eq. (5.4-22), vs. x. Thus for any circuit biased as shown in Fig. 5.4—1 having values of Vk and x which determ ine

Fig. 5.4-6 Plot o f I'- vs. a for the condition where 1E0 I EQ = 1.05 or 1 \ = 2Q( k T q ) In / 0(.v).


a p o in t a b o v e the cu rv e o f F ig . 5 .4 -6 , I E0 m ay be a p p ro x im a te d by I EQ w ith less th an 5°/0 erro r. In a d d itio n , Gm(x)/gmQ m ay be a p p ro x im a te d by H ^ / x I 0(.x) w ith less th an 5 % e rro r. T h is a p p ro x im a tio n co rre sp o n d s to em p lo y in g the q V j k T = x cu rv e o f F ig . 5 .4 -5 or, eq u iv a len tly , the cu rv e o f F ig . 4 .5 - 6 to d eterm in e Gm(x)/gmQ for an y given valu e o f x. It tu rn s o u t in p ra ctice th at th e o p e ra tin g p o in ts o f m ost c ircu its w ith s ta b le b ia s co n fig u ra tio n s and re a so n a b le d rive levels are ab o v e the cu rv e o f F ig . 5 .4 -6 .

Example 5.4-1 F o r the c ircu it show n in F ig . 5 .4 -7 , d eterm in e an exp ressio n for r 0(f).

Solution. T h e first th in g to n o te is th a t we have an N P N tra n s isto r ra th e r th an a P N P tra n s isto r an d th a t we are d riv in g the b ase and not the em itter. H ow ever, since the vBE-iE c h a ra c te r is tic fo r an N P N tra n s is to r is id en tica l w ith the vEB-iE c h a r a c te r is tic for a P N P tra n s is to r , d riv in g th e b ase o f an N P N tra n s isto r has the sam e effect on the e m itte r cu rren t as d riv in g th e em itte r o f a P N P tra n sisto r. T h e m ain d ifference b etw een the tw o cases is th a t the q u iescen t valu e o f VEB exp erien ces a ch a n g e in sign.

At q u iescen ce (v\ = 0), if we assu m e th at the tra n s isto r is s ilico n {VdcQ = 0 .6 5 V).

9 .3 5 V

I eq = 5 l k n = 183 and ^ = 9-35V -

5.5 NONLINEAR LOADING OF TUNED CIRCUITS 181

Since for this example x = 5, the operating point (x, VJ falls well above the curve of Fig. 5.4-6, and thus I E0 % I EQ.

Since the tuned circuit in the collector is resonant at w = 108 rad/sec, which is the fundam ental frequency com ponent of iE(t), and since for the tuned circuit QT = 20, only the fundam ental com ponent of iE(t) contributes appreciably to Consequently. r„ may be expressed as (cf. Eq. 5.3-12)

= (10 V) - 1.787(1.83 m A)(2kQ) cos 108f

= (10 V) - (6.5 V) cos 108f.

Alternatively we can determ ine g mQ = 70.4 m m ho, and from the V- = x curve of Fig. 5.4-5 obtain GJ5)/gmQ = 0.359, from which it follows that Gm(5) = 25.2 mmho. Therefore,

I'oit) = + Vcc - Gm(x)RLVi cos 0Jt

= (10 V) - (25.2 m mho)(2 kQ)(130 mV) cos 108r

= (10 V) - (6.5 V) cos 108t

There is a minus sign in the expression for va(t) because the collector current flows downward through the tuned circuit.

5.5 N ON LINEAR LOADING O F TUN ED C IR CU ITS

In this section we shift our attention to the properties of a parallel R L C circuitwhich is driven by a periodic current source i¡(t) and which is shunted by a nonlinearnonreactive load as shown in Fig. 5.5-1. The current source ¡¡(f) is assumed to have a period T = 2n/oj0 , where a>0 is the resonant frequency of the tuned circuit; and the loading, although nonlinear, is assumed to m aintain the Q of the circuit reasonably high. With these assum ptions, we first show that in the steady state the output voltage va(t) rem ains approxim ately sinusoidal despite the presence of the nonlinear load and that the nonlinear load may be modeled by an equivalent linear resistance.

Fig. 5.5-1 Nonlinear loading of parallel RL C circuit.


We then develop a graphical technique for determ ining the steady-state am plitude of y„(t) as a direct function of the fundam ental com ponent of i',(r).

Circuits o f the type shown in Fig. 5.5-1 arise when the ou tput of one large- signal narrow band amplifier (or limiter) drives the nonlinear input characteristic of a subsequent large-signal amplifier, o r in the case of an oscillator, when the output of a narrow band amplifier drives its own nonlinear input characteristic. Clearly the determ ination of the relationship between v0(t) and i,{t) is a necessary step in the calculation of the overall am plification of a chain of narrow band amplifiers or of the loop gain of a sine-wave oscillator.

T o begin our analyses, we observe tha t since ¡¡(i) is periodic it may be expanded in a Fourier series of the form

00

h(t) = ho + I it cos co0t + £ I in cos (na)Qt + 0in), (5.5-1)n = 2

where the time axis has been chosen in such a fashion that the phase angle of the fundam ental com ponent of i„(f) is equal to zero. In addition, because of the periodicity of i,(t), in the steady state all other currents and voltages in the circuit m ust be periodic with period T ; in particular, i„(t) m ust be periodic and therefore expressible in the form

oo

*.(0 = Z I on cos {no)0t -I- 0 J . (5.5-2)n = 0W ithout affecting the ou tput voltage vB(t), the nonlinear load may be replaced

by the infinite array of harm onically related current sources contained in i0(t), as shown in Fig. 5.5-2.

N ow the circuit is assum ed to have a high value of Q T, where for the nonlinear circuit Q t is defined as the ra tio of the to tal resistance R T seen looking into term inals a-a' (cf. Fig. 5.5-1) a t the resonant frequency <o0 to the reactance of L or C evaluated at co0, tha t is ,t

R r ' R t

s i v i t i Tv j — «' (5 .5 -3 )IX l(g>0)| | X c(co0)|

Therefore, the resonant circuit presents a low im pedance [less than | A'c(ct»0)|] a t dc and all harm onics o f co0. If, in addition, |i in — ion|max < I n for all n # 1 (a condition which is satisfied by alm ost all practical periodic waveforms), then the dc and h a rm onic com ponents of it{t) and i0{t) are essentially shunted to ground and contribute very little to the waveform of Thus v„(t) may be closely approxim ated by a sinusoid of the form

v0U) = K cos (a>0t + 0ol). (5.5-4)

The phase angle of v0(t) m ust of course be identical with the phase angle 0ol of the fundam ental com ponent of since the nonlinear device possesses no energy storage. (The reader should convince himself o f this fact.)

t This definition is clearly consistent with the definition given in Section 2.2.


( ,! = / , ! COS W oti'i, = /* cos (n<iJof + 0»,)1«, = /«. cos (ncuo/f 0 „ )

Fig. 5.5-2 Current-source representation of non-linear load.

A simplified circuit for determ ining v0(t), from which the dc and harm onic com ponents of i',(f) and i„(t) have been om itted, is shown in Fig. 5.5-3.

The problem in determ ining v0(t) in term s of i,it) is now reduced to evaluating Vj and 60l. At resonance, however, it is apparent from Fig. 5.5-3 that

/¡, cos aj0r = (K ,G + /„ ,) cos (to0f + 0O|); (5.5-5)hence 60l = 0 and

I ox = In - VXG. (5.5-6)

To evaluate Vl we employ a m ethod similar to the one used in Section 5.1 to determ ine Vc . We assume a value for which, of course, specifies v jt) = Vx co sw 0f. W ith this value of v0{t) applied across the nonlinear load, we determ ine i0(t) and, either analytically or graphically, determ ine the am plitude of its fundam ental coefficient I0 l . This procedure is repeated for a sufficient num ber of different values of V{ until a sm ooth curve of Ioi vs. Vx is obtained. This curve represents the relationship between Vx and Iol required by the nonlinear load. The linear portion of the circuit, however, requires that Vx and I ol be related by Eq. (5.5-6). Therefore, if the “ load line” of Eq. (5.5-6) is plotted on the same set of coordinates as the Vl-IBl curve, as shown in Fig. 5.5-4, the intersection of the “ load line” and the curve determ ines the steady-state values of V, and I o l . W ith Vt known, /„(t) can readily be com puted.

v J l ) = V, cos (oj0/ + 0„i)


F ig . 5 . 5 - 4 G r a p h i c a l m e th o d fo r d e t e r m in in g th e s te a d y - s ta te v a lu e s o f Vj a n d l o i .

It is desirable to write an expression for QT in term s of the steady-state values of V{ (F1SS) and I0l (I o1Ss) such that, once K1SS is found, the assum ption of high Q T can be checked. To accom plish this goal, the model of Fig. 5.5-3 is further simplified by replacing the ou tpu t current source by an equivalent linear conductance,

GN, =v , '

which m aintains the same currents and voltages throughout the network. This simplified model is shown in Fig. 5.5-5, from which it is immediately apparent that

Q t —w 0C 1

G + Gnlss u>0L(G + Gnlss)(5.5-7)

where

' iss

M ore im portant, however, is the fact that nonlinear loading across a high-Q tuned circuit has the same effect as linear resistive loading. We now dem onstrate the above m ethod with several examples.

Fig. 5 .5 -5 Simplified m odel with nonlinear load replaced by equivalent linear conductance.


Example 5.5-1 For the circuit shown in Fig. 5.5-6, find r0(t).

Solution. We assume Q T > 10 and write ^„(f) = V, cos 107f. Applying this voltage to the nonlinear load, we observe that i0(t) is a square wave, with a peak-to-peak am plitude of n, which may be expanded in a Fourier series of the form

i0(f) = (2 mA)(cos 107r — j cos 3 x 10 7r + 5 cos 5 x 107r — • • •)•

The resonant circuit, with a high value of Q T, appears as an effective short circuit to the com ponents of /„(f) at oj = 3 x 107 rad/sec, 5 x 107 rad/sec, etc; hence. r„(f) is obtained by m ultiplying the input current, less the fundam ental com ponent of /„(f), by the tuned-circuit im pedance evaluated at to = 107 rad/sec, that is,

vB(t) = (4-2 mA)(2 kQ) cos 107f = (4 V )cos 107f.

It is apparent that the same value of v0(t) would be obtained by replacing the nonlinear load by a resistor of value

1 V1SS 4 V ____1SS = 2 k i l

Gnlss K iss 2 mA

Consequently,

2 k f i ||2 kQ 7 ( 107 rad/'sec)(5/iH)

and our assum ption is justified.

Example 5.5-2 For the circuit shown in Fig. 5.5-7, determ ine an expression for r0(f).

Solution. We again assume Q r > 10 and write v j t ) = V{ cos 107f. Applying this voltageto theseriesd iode-battery-resistorcom bination ,w eobservethat,for Vx < 2.5 V, i j t ) = 0 ; thus /„, = 0. For V, > 2.5 V, the waveform of i j t ) is a periodic train of sine-wave tip pulses of peak am plitude Ip = (V, — Vx)/R and conduction angle 2<j> = 2 c o s ' X(V JV X), where Vx = 2.5 V and R = 10kQ. Table 5.5-1 includes values of 1P and l 0l for values of between 2.5 V and 5 V (cf. Fig. 4.2^4), and Fig. 5.5-8 includes a plot of ¡ol vs. F , .


Table 5.5-1

y „ v IP,HA h i

2.5 0 03.0 50 123.5 100 304.0 150 524.5 200 755.0 250 98

Figure 5.5-7

In addition, since K irchhoff’s current law m ust hold for currents at the resonant frequency o>0 flowing into the tuned circuit of Fig. 5.5-7, we obtain the load line

which is also plotted in Fig. 5.5-8. The two curves intersect at Iolss = 67 ¿¿A and îss = 4.3 V. Consequently,

_ L _ _ « i r O n 65 kQ|| 10 kQ Gnlss 1 Qt (107 rad/sec)(5^H )

(which justifies the h igh-g assum ption), and finally,

v„(t) = (4.3 V) cos 107i.

Example 5.5-3 F o r the circuit show n in Fig. 5.5-9, calculate u0(f)-

Solution. We note that this example is identical to the previous example except that the resistance in series with the diode is equal to zero. Clearly, then, when m aking a table for Iol sim ilar to Table 5.5-1, we find that Jo1 = oo for all values of Vt > 2.5 V. Consequently the curve for Iol vs. Vt takes the form shown in Fig. 5.5-10.

5.5 NONLINEAR LOADING OF TUNED CIRCUITS

Again plotting

on the same set of coordinates, we obtain

Vlss = 2.5 V, I giSS = 250 nA, - i — = 10 kQ,V N L S S


500 n /

250

2.5 V 5 Vs V.

Figure 5.5-10

and

H ence the m ethod of solution is justified.N ote tha t in this problem , although a battery and diode are placed directly across

the tuned circuit, the voltage rem ains sinusoidal of the form v„ = (2.5 V )cos 107f ; hence the m ajor effect o f the diode is to clam p the peak-to-peak am plitude of the resonant circuit voltage to twice the battery value. The diode does not, as is sometimes erroneously believed, clip the top o f v0(t) and leave the bottom unchanged. Such a waveform obviously cannot exist, since its average value would be different from zero and thus could not be sustained across an inductor.

The circuit of this example finds application as an FM limiter. Clearly, variations of the input current abou t the j mA point result in the same peak output voltage of F, = 2.5 V. Only when the input current drops below | mA does Vx d rop below its2.5 V level.

Example 5.5-4 F or the circuit shown in Fig. 5.5-11, assum ing that no collector saturation occurs and tha t both input and outpu t tuned circuits have high values of Q t and both resonate a t co0 , determ ine an expression for u,(i) and t>„(l). Also determ ine the effective linear loading the transistor applies to the input parallel resonant circuit.

Solution. Since QT for the input tuned circuit is assum ed to be high,Vi(0 = K cos o>0t and vBE = F, cos co0t + Fdc.

Thus if an exponential relationship is assum ed to exist between iE and vBE, i^ t ) is given by

and in the steady state the am plitude of the fundam ental com ponent of em itter current I El is given by

Í ¿ t ) = l ^ V ö J k T ^ cosoot

j 2 / j ( x )

5.5 N O N L IN E A R L O A D IN G OF T U N E D C IR C U IT S 189

9+ K,

/c o s a >0tQ-*~oo

Figure 5.5-11

Since ic = aiE and iB = (1 — a)/£, the am plitudes of the fundam ental com ponents of the collector and base currents are given by

Ic 1 = <*Ic2 /,(* )

and I bi — (1 a)Idc2/,(x )

/«(*) 01 ac I„(x)respectively.

N ow to obtain an expression for Vt (or x ) we equate the fundam ental com ponents of the currents entering the input-tuned circuit to obtain

, „ , 2 / >WR , /dc(1 a) 70(x) ’

or equivalently,

I Vl I x 21 ,(x)

(i - a)/dc “ a - a)/dcR, “ (\ ~x)!dc ~ 0 - ijginK, ~ l o M 'where gin = qIAJkT. By plotting the “ load line”

I x(1 - a )/dc (1 - a)ginR ,

on the same set of coordinates as 21 ¡(x)/10(x) vs. x, as shown in Fig. 5.5-12, we obtain x ss and 21 ¡(xss)/I0(xss) as the coordinates of the intersection of the two curves; thus


kTxv. = ss cos a)0t.

Specifically, if 1/(1 — a)/dc = 2 and (1 — a)gin/?i = 6, by plotting 2 — x/6 on the coordinates of Fig. 4.5-5 we obtain

xss — 2,21,(2)/o(2)

= 1.35, and v^t) = (52 mV) cos <y0f.

With 2 1 i(x ss)/10(xss) known and the output-tuned circuit Q assumed to be high, v0(t) can immediately be written as

v0(t) = Vcc ~ Ici 1 2 c o s co0t

= Vcc - a /dc ir r ^ T ^ 2 cos g)0t.i dcIo(xss)

The equivalent linear loading on the input-tuned circuit is given by

r — — I Cl — « o\NL~ J ï ~ W l ~ ~ T ~ ' ( }

where p = a/(l — a) and Gm is the large-signal average fundamental transconductance of the transistor, which is plotted as a function of x and normalized to gm in Fig. 4.5-6. For this bias configuration, gm = aIdcq/kT ; GNL may also be expressed in the form

G dmN L ---

Smß gm'

(5.5-9)

where g J f i is the small-signal input conductance looking into the base with the emitter grounded.

If the emitter, rather than the base, is driven as shown in Fig. 5.5-13, the equivalent linear loading on the input-tuned circuit is

G j SL = £ i n ~ - ( 5 . 5 - 1 0 )gm

If either the base or emitter is driven and the bias is developed as shown in Fig. 5.4-1, the algebraic expressions for GNL remain unchanged; however, gm = a.lEQq/kT and G J g m must be obtained from Fig. 5.4-5 rather than Fig. 4.5-6.


Ct —— oo

s

Figure 5.5-13

Example 5.5-5 For the circuit shown in Fig. 5.5-14, determ ine the ou tput voltage v0(t) and the resistance seen by the input current in the steady state. Assume that Qt > 10.

Solution. For the circuit of Fig. 5.5-14 we first note that, since Q T is high and the circuit is driven by a current source a t its resonant frequency, then v0(t) m ust have the form u,(r) = Vt cos a>0f. We then observe that, since C is large, it appears as a battery Vdc in the steady state. Consequently the ideal diode-battery com bination limits Vt to Kdc (cf. Example 5.5-3) and in turn

Now, to evaluate V, as a function of I, we note that with v0(t) clamped to zero the diode conducts only at the peak of v0(t) and thus i'D(f) flows in extremely narrow pulses; hence iD(t) may be represented as

(For a periodic train of very narrow pulses, the am plitude of the harm onics is twice the average value.) However, since the average value of the capacitor current i„(f)

v„(t) = V\ ( c o s o v - 1).

QOiD(0 ~ JDdc 1 + 2 X cos na)0t\.

Vi(t)= V\ COS Cll0i oo

I-o+

Figure 5.5-14


must be zero in the steady state, then

, _ Î d*' - - « * - - J - z -

Thus the fundam ental com ponent of iD(t) has a m agnitude o i2 V x/R. W hen 2 V J R is added to the fundam ental com ponent of iR, Vl/R , the sum is equal to the input driving current, / = 3 VJR. Consequently we finally obtain

I R ,v0 = — (cos 0)0t - 1).

In addition, the equivalent resistance seen by ¡'¡(f) is given by

R - V l - R K nl ~ i ~ y

U p to this point we have assum ed that a high value of the loaded QT in the circuit of Fig. 5.5-1 keeps va(t) sinusoidal of the form v0(t) = K, cos u)0t. We shall now treat the problem more quantitatively by obtaining an expression for the total harm onic d istortion (THD) of vB(t) as a function of the loaded QT and the nonlinear load. Thus, once a specific nonlinear load is specified, we may determ ine QT to keep the T H D below a desired level.

At the outset o f the analysis we assum e tha t the T H D is sufficiently small (say below 5 %) so that the current waveforms obtained assuming v j t ) = Vx cos oj0t in the circuit of Fig. 5.5-1 very closely approxim ate the actual current waveform. With this assum ption the expression for i0(t), and in turn the expressions for the ion{t) sources of Fig. 5.5-2, can be obtained using the steady-state value of K,. Consequently, the harm onic d istortion com ponent of v j t ) can be obtained by determ ining the contributions of the various harm onic current sources of Fig. 5.5-2 flowing through the parallel R L C circuit.

Specifically, if we define

I n — i i in hinl max

(/„ is the am plitude of the total m h-harm onic current com ponent flowing into the parallel R L C circuit) and

/ . = | / , 1 - / o i l

then from Eq. (3.5-4)

T H D = Yn = 2

Z u ( jn u ) q)

Z 11 (ju>o)(5.5 11)

where Z , ,(./to) is the input im pedance of the parallel R LC circuit. Since the Q of the parallel R L C circuit is assum ed to be high, then from Eq. (3.4-12)

Z u U n w o ) ^ C, n ZnOcoo) ~ co0C n2 - T


Equation (5.5-11) can therefore be rew ritten in the form

2

(5.5-12)

It is apparent, however, that

hence

h i = G + Gnl 11 G

and the T H D reduces to the desired form

(5.5-13)

where QT = co0C/{G + GNL). For any nonlinear load, the ratio I J I n may be determined in the steady state. If the maxim um tolerable T H D is specified, then Eq. (5.5-13) may be solved (employing the first few term s in the series) to yield the minimum permissible value of QT.

As an example of this technique, let us determ ine the m inimum value of QT which keeps the T H D less than 0.01 (1 %) for the circuit in Fig. 5.5-9. In this circuit the steady-state value of v„(t) is given by v„(t) = (2.5 V )cos 107t; therefore, the ideal diode conducts only on the positive peak of v„(t), and i0(t) is constrained to flow in extremely narrow pulses. However, for narrow pulses, /„„ = /„ = /„ ,. In addition, since

then I0J I n = j . Substitution of this ratio into Eq. (5.5-13) yields (cf. Eq. 3.5-5)

It is now apparent that, to keep the T H D below 0.01, Qr m ust be greater than 47. A value of QT = 10 yields a T H D = 0.047 (* 5 % ).

Smoothly Varying Nonlinearities

If the nonlinear load shown in Fig. 5.5-1 has no abrupt changes in slope, we can often approxim ate its v0-i0 characteristic by a polynom ial com prising the first few

Iol = In - V & = (\ mA) - | mA,


term s of its M acLauren expansion, tha t is,

L(t) = a0 + a ^ t ) + a2[v0(t)]2 + a3[u0(t)]3 + • • •. (5.5-14)

If the v0-i0 characteristic is known analytically, then the coefficients a„ may be evaluated in the form

a" n ! dv,(5.5-15)

I\> = 0

However, if the v0-i0 characteristic is available only graphically, then a polynom ial of the desired order can be fitted to the characteristic by standard curve-fitting techniques.

Once a polynom ial form for the v„-i0 characteristic is obtained, it is a simple m atter to determ ine the equivalent linear loading on a high-Q tuned circuit and the resultant level of v„(t). Specifically, if u0(t) = Vl cos oj0t and we approxim ate i„(t) by a polynom ial of order 3, then

i0(t) = a0 + ajK , c o s o v + a2V \ cos2 cu0t + cos3 co0t. (5.5-16)

E quation (5.5-16) may be arranged in Fourier-series form by use of the trigonom etric identities

cos3 co0t = | cos a)0t + 5 cos 3<y0i and cos2 to0t = j + j cos 2u>0t.

Thus

a2Vi , „ L , 3fl3 I/2|i0(t) = a0 + — + — - V j j c o s co0t

+ ~ ^ V \ cos 2co0i + ~ V i cos 3<w0t, (5.5-17)

from which we observe that

/„i = VM x + & 3V \) (5.5-18)and

Gnl = ^ = a , + & 3V l (5.5-19)

Hence, if I n and G are known, then

/ . , = GVl + Vl(al + | a3.V2) (5.5-20)

and a cubic equation for obtaining V, is obtained which may be solved graphically or analytically. F o r example, if

^ 4 m m hoa t — G = 1 m m ho, a3 = — — , and I n = 5 mA,

then the steady-state value of p„(i) is (1 V) cos co0i and the equivalent linear loading o f the nonlinear load is GNL = 4 mmho.

5 .6 TRANSFER FUNCTION FOR LOW-INDEX AM INPUT 195

5.6 TRANSFER FUNCTION FOR LOW-INDEX AM INPUT

In this analysis we apply a low-index AM signal of the form

v,{t) = [V, + Ui(f)] cos 0)0t,

where ^ ( t) to the generalized capacitively couplea input circuit shown inFig. 5.6-1 and determ ine the resultant AM variations on the envelope of the fundam ental com ponent of i',(t). We then adapt the analysis to the specific transistor amplifier shown in Fig. 5.3-1.

v , ( f ) = V\(t) cos u ) / = [ Vx + V |(/> ] cos uM

i.(0 = /o (0+ A (0 cos u*+ • ■A»i0 =l4c+kAt)/j(0 =/i + ii(i)

K (0 = K + n>(0

Fig. 5.6-1 G eneralized capacitively coupled nonlinear input circuit.

Because of the nonlinear load, variations in the envelope of u,-(t) produce variations not only in the envelope of i,(f) but also in Vc and the average value of i',(r). The form of these variations, which we assume to be small in com parison with the quiescent values of the respective param eters [i.e., the values obtained with v t(t) — 0], is shown in Fig. 5.6-1. If small variations are assum ed, it is apparent that

M O = G n tM 0 + G i0v0(t),

i0(t) = G0lv l(t) + G00v0(t),

and, in addition,

where

*o(0Jvo(t)

dt

G „

G 0, — dio8V,

d± LSVX

Q-pt

Q-pt

and

_ d i ,dVc »

Q-pt

d l0IIOoo

W c 0-pi

(5.6-1)

(5.6-2)


[Q-pt denotes Vc = K.C and V^t) = KJ. Thus, by expressing Eqs. (5.6-1) and (5.6-2) in Laplace transform no tation and elim inating /0(i) and v0(t), we obtain

Yl(p) = Ji (P)K (p )

G „l / r GioGo,

P + C c "» - ~ G î(5.6-3)

P +

where / t(p) = and Fi(/>) = i?[>j(r)]. The transfer function Yj(p) perm its thedeterm ination of ^(r) once the form of v t(t) is known.For the capacity coupled transistor amplifier of Fig. 5.3-1,

where

and

hence

and

/.(t) = iE(t) — [ESeqVci,)lkTeqV'(1),kT cos cat

= W 1 + 2w i COS(Ut +

x(i) = qVAt) k T '

I Eo(t) = i ESi 0[ m e qVc(mT,

S I EO q l d e£ in^ 0 0 svc Q -pt = kT

r ' S I EO

O1 dx^ 0 1 SV> Q-pt dx 'SVx

= g

G io —

i iW

’ fo to

S I E l

Q - pt

[ n o k : ^ = /,< *)],

2/i(x )

e-p, g in / 0 t o ’

G „ =a / £1 d/ F1 dx

s Vi le-pt Sx dV¡ Q - pt

(cf. Eq. A -l 1).

(5.6—4)

(5.6-5a)

(5.6—5b)

(5.6-5c)

(5.6-5d)

5.6 TRANSFER FUNCTION FOR LOW-INDEX AM INPUT 197

Consequently, Y,(p) is given by

*i (P) = 2gin

w h ere r = C /g in a n d

1 -W

x l 0(x) P +A(x)

(5.6-6)

A(x) = 1 -

P +

/ i(x)Jo(x )

1 /.(* )x l 0(x)

(5.6-7)

A plo t of A(x) vs. x is given in Fig. 5.6-2. It can be shown (by differentiating Eq. (5.6-8)) that

¿(x) =

ddx

h ( x )Iq(x )

1 I l ( x ) ’

x / 0(x)

thus, for small values of p (or u>), Y¡(p) is given by

(5.6-8)

Y M * ^ ( 0 ) = g ind 2/,(x )

(5.6-9)’d x I 0(x) ’

which is, of course, the admittance which relates small static variations in K, to / £1.From the pole-zero diagram for Yt(p) it is apparen t that the m axim um frequency

com ponent ajm of ^ ( i ) or V,(t) m ust be a t least an octave below A (x)/t, for any given value of x, to perm it the use of the static transfer function relating I E1 and Vx for the case of dynam ic envelope variations. This restriction may be expressed in term s of the inequality

co_t to ~C- j f \ = 2- (5.6-10)/(x) ginA(x)

Since A(.x) decreases with increasing x, the maximum permissible value of the coupling capacitor which satisfies Eq. (5.6-10) also decreases. For example, if x = 10 ,

A(x) = 0.0059 andSi.

< 0.0118.

H o w e v e r, fro m E q. (5.3 20), u>C/gin > 4 0 7t ; th e re fo re , fo r c o rre c t o p e ra t io n o f th e circuit of Fig. 5.3-1, we require

u>> 1.06 x 104.

198 REACTIVE AND NONLINEAR ELEMENT COMBINATIONS 5 .6

Fig. 5.6-2 Plot of A(x) vs. x.

Even though this restriction may be relaxed somewhat, we see that if the circuit is to function as a narrow band lim iter, which necessitates large values o f x for its correct operation, there m ust exist a large separation between the m axim um m odulation frequency and the carrier frequency. If this separation is not achieved, the circuit fails to limit properly, even though its static transfer characteristic indicates tha t it should. Consequently, in lim iter applications, the single-ended transistor is not particularly satisfactory and the differential pair, which may be operated w ithout a coupling capacitor, finds m ore widespread use.

PROBLEMS 199

5.1 Show that a reasonable approximation for the “steady-state” values of the source current, /s(i), and the ou tpu t voltage, vjt) , for the circuit o f Fig. 5.P 1(a) is given by the results shown in Fig. 5. P -1(b). Assume that C is an ac short circuit.

5.2 Find the steady-state value of v„(t) in Fig. 5.P-2. How much average power is consumed in the 1 k ft resistor after the capacitor voltage reaches equilibrium ?

5.3 The repetitive waveshape shown is applied to the circuit in Fig. 5.P-3. Find the dc capacitor voltage and the shape and am plitude of the silicon diode current.

5.4 Sketch the collector current waveshape and find the collector voltage and the dc capacitor voltage in Fig. 4.P-4.

PROBLEMS

30 usee +

0 V

260 mVO J L T 0H

3 usee

iD = Ise" ‘m,kT /s = 2 x è- » mA

(a)

2 mA

18 mA

30 usee

I T ¥3 usee is(t)

30 usee

- 580 mV

840 mV H J — L TI— I

3 usee ?„(/)

(b)

Figure 5.P-1

200 REACTIVE AND NONLINEAR ELEMENT COMBINATIONS

1000 IiF

Figure 5.P-4

5.5 Use the model of Fig. 5.P-5(b) with the assum ption that all of the 5 mA flows through the diode to determ ine an initial value for in the circuit of Fig. 5.P—5(a). Then approxim ate Ir — Vi J R and refine /„ = (5 mA) — /„ to find a second (and final) value for Kdc. Find the output collector voltage for this final case. C om pare the results with those obtained employing Fig. 5.4-5.

5.6 W hat is the em itter-current conduction angle in the circuit of Fig. 5.P-5(a)? W hat is the approxim ate value of the peak em itter current in this circuit? How large m ust the capacitor be if co0 = 1 x 10® and the capacitor voltage should vary less than 2 % over the period of a sine wave after equilibrium is reached?

5.7 F or the circuit of Fig. 5.P-6, find the dc voltage across the 10,000 pF capacitor, the fundam ental com ponent of the diode current, the equivalent linear loss resistance of the diode, and the ac power loss in the 1 k il resistor. Explain and justify your assumptions.

5.8 Assume that the voltage source of Fig. 5 .P-6 is replaced by a coil of QL = 150 at 108 rad/sec an d :a sinusoidal current source of radian frequency 108 rad/sec and peak am plitude of f mA: The coil is tuned to resonance with the effective input capacitance of the two capacitors. W hat is the fundam ental diode current? W hat is the loaded Q of the tuned circuit?

PROBLEMS 201

ac short circuit

(a)

C

(b)

Figure 5.P 5

100 pF

Figure 5.P-6

2 0 2 REACTIVE AND NONLINEAR ELEMENT COMBINATIONS

5.9 Explain how the results o f Problem s 5.7 and 5.8 would be modified if the radian frequency were reduced to 107 rad/sec.

5.10 Find the output voltage, v0{t), and the fundam ental and third-harm onic diode currents for the circuit of Fig. 5.P-7.

5.11 Find the to tal harm onic distortion in v„(t) in the circuit of Fig. 5.4-7.5.12 F or the circuit shown in Fig. 5.P-8, determ ine an expression for v(t) and vjt).

ac short circuit

(1 m A) cos Wo I ^ ^ : C

=2000pF

:2 k ii 45 pH =200 CD 4

v„(0

W o-l/VZCFigure 5.P-7

L = 2 x e -30 mA

PROBLEMS 203

5.135.14

5.15

F or the circuit shown in Fig. 5.P- 9, find u,(f) and v0(t). Also sketch iE2{t).F or the circuit shown in Fig. 5.P-10, determ ine an expression for v„(t) and the equivalentlinear loading across the tuned circuit for each of the following nonlinear loads and input current levels / , . In each case find the T H D present in i'Jt):

a) 11 = 4 mA, iD = l se‘,,’“,kT (see Fig. 5 .P -lla ) ,

b) / 1 = 2 mA (see Fig 5.P-1 lb),

c) / , = 2 mA, i„ = (20/jV/V3)^ + (IO/iV/V5)^5,d) / j = 2 mA (see Fig. 5.P—1 lc),

e) / , = 1.5 mA (see Fig. 5 .P -lld ).F ind the dc voltage across the input capacitor in the circuit of Fig. 4.P-11. Estimate the shape (conduction angle and peak value) o f the collector current pulse for this case. Repeat for the case in which the source im pedance is reduced to 5 Q. W hat is the form and m agnitude of v j t ) for this case? Does the transistor saturate?

204 REACTIVE AND NONLINEAR ELEMENT COMBINATIONS

' 1 m A

(b)

Ideal Ideal

20 k n

Figure 5.P-11

C H A P T E R 6

SINUSO IDAL OSCILLATORS

In this chapter we com bine the results of all the previous chapters to explore the design and analysis of a num ber of sinusoidal or alm ost sinusoidal oscillator circuits.

Once the waveshape of the output of such a circuit is specified as sinusoidal, then the two properties that remain to be defined are the operating frequency and the ou tput am plitude. A good portion of this chapter will be devoted to exam ining the frequency- and am plitude-determ ining m echanisms of various circuits, the stabilities of these mechanisms with variables such as time, tem perature, or supply voltage, and the interrelationships between am plitude and frequency.

There are m ethods of producing sinusoidal signals by filtering separately p ro duced square waves or impulse chains or by the nonm em ory shaping of triangular w aves; however, the discussion of such circuits will be reserved for C hapter 11.

Both from the viewpoint of the techniques employed and from that of the practical division of circuits, it is convenient to consider m ost of our circuits as belonging to one of two classes. In the first class we com bine a tw o-port active device with a two- port passive netw ork in a feedback configuration. In the second type we com bine a one-port active device in parallel with a one-port passive network. Obviously some circuits may be considered from either viewpoint. In spite of this, the division is still useful. We shall first consider the feedback type of circuitry. In subsequent sections we shall consider the one-port or “ negative-resistance” circuits.

All sine-wave oscillators must, at the m inim um , contain

a) an active device with power gain at the operating frequency (the circuit must supply not only an ou tpu t signal but also its own input driving signal),

b) a frequency-determ ining elem ent or network, andc) an am plitude-lim iting and stabilization mechanism.

In practical circuits these functions may become quite intermingled. However, initially we shall consider these operations separately so that the necessary or desirable properties of each one may become evident.

6.1 O PER A TIN G FREQ U EN CY AND M IN IM U M GAIN C O N D ITIO N S FOR LINEAR-FEEDBACK O SC ILLA TO R S

In order to sustain sinusoidal oscillations, a netw ork must have a pair of complex conjugate poles in the right-half complex plane when power is applied a t t = 0 . These right-half plane poles, when excited by therm al noise or the step generated by closing the pow er switch, give rise to a sinusoidal ou tput voltage with an expon

205

2 0 6 SINUSOIDAL OSCILLATORS 6.1

entially growing envelope. If such a netw ork has been designed to sustain a constant- am plitude sinusoidal ou tp u t voltage, then, as the envelope o f the noise-induced sine wave increases, it causes a change in the value of one or m ore of the network param eters (usually the amplification) in such a direction tha t the complex conjugate poles move tow ard the im aginary axis. Finally, a t some predeterm ined am plitude of the growing sinusoid, the poles reach the im aginary axis and a constant-am plitude sinusoidal ou tpu t is realized. If for some reason the ou tpu t increases further, the poles move into the left-half plane and the ou tpu t decreases tow ard the desired level, where again the poles lie on the im aginary axis.

W e now see th a t the basic requirem ent for a sinusoidal oscillator consists of a pair of small-signal complex conjugate right-half plane poles, which determ ine the frequency of oscillation, plus a m echanism for moving these poles tow ard the

im aginary axis as the envelope of the ou tpu t waveform increases. This m echanism determ ines the steady-state oscillation am plitude. In order to obtain right-half plane poles, we require some form of feedback. Figure 6.1-1 illustrates a generalized feedback amplifier whose transfer function is given by

V0{p) AH^jp) AH,(p)

V&p) 1 - A H M H M i -where A L{p) = A H i(p)H2(p) is defined as the “ loop gain” of the amplifier. It is apparent that the poles of the feedback amplifier are the zeros or roots of

l - A L{p) = 0. (6.1- 2)

Thus if Eq. (6.1-2) has a pair o f com plex conjugate roots in the right-half plane, v0(t) will be a growing sinusoid even with ^(i) = 0.

The process of designing a sinusoidal oscillator now becomes clear. We select a suitable pole-zero pattern for

A L(p) = A H 1{p)H2(p)

which causes one pair o f complex conjugate roots of Eq. (6.1-2) (and no other roots— otherwise unw anted oscillations might occur) to cross the im aginary axis at a predeterm ined frequency w0 as A (or — A) is increased. We also determ ine the minimum value / lmin of A which places the roots on the imaginary axis, and choose A somewhat larger than this value to ensure self-starting. We then incorporate some nonlinear

6.1 FREQUENCY AND GAIN CONDITIONS FOR LINEAR-FEEDBACK OSCILLATORS 207

mechanism which reduces A to A min as the ou tput oscillation grows tow ard a desired am plitude. Finally, we choose a netw ork having the desired A L{p) and com bine with it the desired nonlinear am plitude limiting.

To arrive at pole-zero patterns for

A ¿P) = A H t(p )H 2(p)

which are capable of producing oscillations, we start with the simplest pole-zero configurations and plot the loci of the roots of Eq. (6.1-2) as A is increased from zero to infinity and is decreased from zero to minus infinity. We then increase the com plexity of the pole-zero diagram and repeat the roo t locus plot until a num ber of pole-zero diagram s capable of producing complex conjugate right-half plane roots are obtained. This approach im m ediately indicates that A L{p) m ust have two or m ore poles to cause Eq. (6.1-2) to have right-half plane complex conjugate roots. For example, with

. , , Acüiip + w 2)A l(P) — : ;------r

ío2(p + a>!)

the roo t locus is shown in Fig. 6.1-2 as A is increased (a) and then decreased (b) from zero. We see from the figure that a pole does indeed enter the right-half plane as A is increased. However, since it is a real-axis pole, the result is a growing exponential waveform and no t a growing sinusoidal oscillation.

, Au>i(p+ui) l (p) u> ¿ p + u, ¡ )

J CÜ

¿ > 0

— W 2 —tL»| <T

Locus withincreasing A

j i i i

A< 0

— Ul2 — U| CT

Locus withdecreasing A

(a) (b)

Fig. 6.1-2 Root locus of AL(p) with a single pole.

We also find tha t if A L{p) has two poles and no zeros,“oscillations are impossible. The simplest pole-zero com bination for A L(p) which is capable of producing right-half plane roots in Eq. (6.1-2) is the com bination of two poles and one zero. The root locus for this case with

A { ) = A(° lP L (p + W,)(p + 0)2)

is shown in Fig. 6.1-3, from which we observe that positive feedback, A > 0, is required for oscillations. It is apparent that for A > 0 the zero may move somewhat into the left-half plane w ithout destroying the possibility of oscillation.


AÁP)=Au)\p

(P + cdiKP+cdî)

-to, u 2

JW

A< 0

Fig. 6.1 -3 Root locus of AL(p) with two poles and a zero.

To determ ine the frequency of oscillation a>0 (i.e., the point at which the roots cross the im aginary axis) and the m inim um gain A for oscillations (i.e., the value of A which just places the roots on the im aginary axis), we assume A = A min. With -4 = A min, p = jco0 m ust be a solution of Eq. (6.1-2). Hence

or equivalently,

A L(j(O0) = 1,

A L(ja)0) = 1.

J m A L{jCÛ0) = 0.

(6.1-3)

(6.1 4a)

(6.1 4b)

Equations (6. l-4 a and b) provide a necessary condition for oscillation (this condition is known as the Barkhausen criterion) and thus a set of two equations for solving for A min and io0. The B arkhausen criterion makes sense intuitively, since it states that at the frequency of oscillation co0 the signal m ust traverse the loop with no a ttenuation and with no phase shift.

W ith A L{p) = A w xpj{p + (ü,)(p + co2).

„ . , ■ V '4mina ,la,o(°-, la ,2 Wo)A l(jojo) = 2-

cooicji + a>2) + ((0¡cl>2 — w0)= 0 .

from which it is apparent that co0 = y/co]u>2. In addition, Eq. (6. l-4a). with o j 0 =

s / c l> 1 u>2 , yields

J)t A L(jco0) =„ W t

CÜ, + co21,

from w h ic h w e o b ta in

^min (O,


F ig u re 6 . 1 4 il lu s tra te s a sm a ll-s ig n a l o r lin e a r m o d e l fo r a c ircu it w ith th e p o le - zero d iag ram show n in F ig . 6 .1 -3 . F o r th is c ircu it A L(p) is given by

th u s A = A '2. o ;, = R L. a n d oj2 =__\;RC. If 1/ R C = R L, th e n we h av e th e specia l case w h ere w , = w 2 = w 0 = l / v L C a n d A min = 2 o r 4 min = N 2.

A n o th e r w ay to c o n s id e r th e c irc u it o f Fig. 6. 1 3 is to a sk a t w h a t s in u so id a l freq u en cy th e p h a se shift a ro u n d th e lo o p is ze ro . T h e first s tag e h as a s ing le p o le a t - R /L : hence its p h a se g o es fro m z e ro to —90° as oj g o es fro m ze ro to in fin ity . T h e seco n d s tage h a s a z e ro a t th e o rig in a n d a p o le a t - 1 R C ; h en ce its p h ase goes from -(-90° to z e ro a s io goes fro m ze ro to in fin ity . If 1 /K C = R L. th e n th e o v e ra ll z e ro p h ase -sh if t p o in t is a t « = 1 RC. (A t th is p o in t th e m a g n itu d e o f th e p h a se shift from each c irc u it is 45°.) A t th is freq u en cy each n e tw o rk h as an a t te n u a t io n o f 3 dB : hence a g a in p e r s tag e o f v 2 is re q u ire d to ra ise th e lo o p g a in b ack to un ity . T h is g a in d o e s n o t h av e to be sp re a d th ro u g h th e c ircu it, b u t can be c o n c e n tra te d in o n e p o in t.

In s te a d o f tw o real-a.xis p o les a n d a z e ro a t th e o rig in , A L(p) c an be c h o se n to h av e a p a ir o f co m p lex c o n ju g a te le f t-h a lf p la n e p o les a n d a ze ro a t o r n e a r th e o rig in . T h e ro o t lo cu s fo r th is case w ith

A L(p) =A p w 0

p + l a p + Mq(6.1 5)

is sh o w n in Fig. 6 .1 -5 . H e re a g a in w e o b se rv e th a t o sc illa tio n s a re p o ss ib le o n ly for 4 > 0. H o w ev er, th is p o le -z e ro c o n f ig u ra t io n h a s tw o a d v a n ta g e s o v e r th e c o n f ig u ra tio n w ith re a l-a x is poles. F ir s t, a sm a lle r / l min is re q u ire d to p ro d u c e o sc illa tio n s. Specifically, for this pole-zero pattern we find , employing the Barkhausenc r ite r io n , th a t

2 a 10)0 = w 0 a n d A min = —r = — ,

io0 Qt

210 SINUSOIDAL OSCILLATORS 6.1

jw

AlÍp) p2 f 2a/H A< 0 A> 0

av Wo w0V y

Fig. 6.1-5 Root locus for AL(p) with two complex conjugate poles and a zero at the origin.

where Q T is the Q of the passive circuit w ithin the feedback loop. Clearly, then, as Qt increases, the required am plification o f the active element within the feedback loop decreases.

A second advantage o f the complex conjugate poles of A L(p) is increased frequency stability. If spurious poles and zeros should appear in the pole-zero pattern o f A L(p) because of parasitic capacitance and inductance, the roo t locus is modified somewhat. However, if the poles of A L(p) are sufficiently close to the im aginary axis to start with, then the modified loci are still constrained to cross the im aginary axis relatively close to co’0 .

This frequency stability m ay be observed from a different point o f view. If the passive netw ork has a high Q r , its phase shift varies rapidly with frequency in the vicinity of co'0 (cf. Fig. 2.2-7). Therefore, spurious phase shifts introduced by parasitic elements w ithin the feedback loop require only a small change in frequency from oS0 to produce a com pensating phase shift from the high-Q r netw ork and a net zero phase shift around the loop. Thus the higher the value of QT, the smaller the deviation from oi\o to the frequency co0 a t which zero loop phase shift exists and oscillations are possible.

F o r example, with A L(p) given by Eq. (6.1-5), the phase (f> of A L(joj) in the vicinity o f oj'0 is given by

(6.1-6)w'0

and

d ÿ 2Qt

d a > tOb (O'o '

Thus, with Q = 104 (typical of a quartz crystal), a 1° (0.0176-rad) shift in the overall loop phase shift leads to only an 87 rad/sec or a 13.8 Hz shift in frequency in an <±t0 = 108 rad/sec oscillator.


Fig. 6.1-6 Oscillator circuit with A,_(p) having a pair of complex conjugate poles and a zero at the origin.

F ig u re 6.1 6 illu stra tes o n e o f m an y sm all-s ig n a l o sc illa to r c ircu its w hose lo o p tran sfer A ,(p ) c o n ta in s a pair o f co m p lex co n ju g a te poles and a zero at the origin . F o r th is c ircu it. A L(p) is given by

, , , (gmn /G T)(2ap)•1/.'/>> = ----- — -,2 i ' (' l I

p + 2ocp + oj0

w here co0 = 1 N a = G r / C , and G T = G L + n 2G in. T h is c ircu it cou ld be realized by any o f the tra n sfo rm erlik e n e tw o rk s show n in T a b le 2.5 1 ca scad ed w ith a co m m o n b ase tra n s is to r , a co m m o n grid trio d e , o r a c o m m o n g ate F E T . R eg a rd less o f how the c ircu it o f F ig . 6 .1 -6 is im p lem en ted , th e m in im u m gm req u ired for o sc illa tio n at(Uq == is

= 7 - (6 -1 -8 )

w hich d ecreases w ith d ecrea sin g G T o r, eq u iv a len tly , w ith in crea sin g Q r = co0C /G T .F ig u re 6 .1 - 7 illu s tra te s th e ro o t lo c i fo r sev eral o th e r p o le -z ero p a tte rn s for

A L(p) w hich a re c a p a b le o f o sc illa tio n s. In clu d ed in the figure a re th e co rre sp o n d in g values o f w 0 an d A min. T h e n e tw o rk s w hich yield lo o p g a in s w ith such p o le -z ero d iag ra m s are exp lo red in su b seq u en t sec tio n s and in th e p ro b lem s a t the end o f the ch ap ter. It sh o u ld be n o ted in passing , h ow ev er, th a t b eca u se o f th e c lo sen ess o f th e p o le and zero in F ig . 6 .1—7(b ) (th e M e a ch a m brid ge), sp u rio u s p o les an d zero s have very little effect o n the y-ax is c ro ss in g o f th e r o o ts ; thu s th is co n fig u ra tio n should su rp ass the o th e rs in its a b ility to m a in ta in a s ta b le frequ ency .

In n o ca se is th e ro o t lo cu s p lo t n ecessa ry to so lv e an o sc illa to r p ro b lem . I t d o es h elp in u n d erstan d in g the a c tio n o f th e c ircu it and a lso in d eterm in in g w h eth er it is p o ssib le for a c irc u it to o sc illa te a t a seco n d un d esired freq u en cy . An a ltern a tiv e m eth od o f e xp lo rin g the o sc illa tin g p o te n tia l o f a c ircu it is to p lo t the p h ase an g le o f A L(ja j) vs. o) and to d eterm in e w h eth er 0 o r ± 2 n o f p h ase ex ists a t a freq u en cy w here \A,(j(i>)\ exceed s u n ity (cf. E q . 6 .1 -3 ) .


A (n)= ¿(P2 ~ 2aP+<*>lU2)

Â Cüi -f*ü>2Ya

tUO=tUÓa vûiüb

Wien bridge

A > 0(a)

Al(pY- A ( p 2 - 2 a \ p f c » ó 2)

(p2 + 2a2p + ó2)

4 = —' “mm n .02«i

ü>0 =

AL(pY

A m,n =

(1»0

Meacham bridge

J0 - VCJ, ÜJ2 -CÜ2CÜ3 Í 01,0),

AL(p) = j £ Í(p+ u iX p+ti)2)(P+W3)

X - - 0

f A>0/u'o

a2

j 10

a i

(^+<Ul)(p+W2) (P + U>3)

_ (2 + j £ + £ + £ + £ - # -

Cü] >

Phase shift 1

X/

CÜ3 CÜ3 CÜ2 Cl>2 Cüj — x v x --CÜ2\ ÜJ0Ji

V

JU1

A< 0

Phase shift 2

/

<3 < 3 < 2 ^ 1 - UJ¡Í w2 ^

JtU

+y i )

ÜJo =

Fig. 6.1

oil u>2 a>3

^ ^ < 0

(b)

(c)

(d )

W, +W2 + Cl>3

•7 Root loci for several A,(p) capable of producing oscillations.

6.2 A M PLITU D E-LIM ITIN G M EC H A N ISM S

So far we have ignored the problem of the determ ination of the am plitude at which the oscillations stabilize. At least four different m ethods or com binations of m ethods of am plitude stabilization are employed in practical circuits.

a) O ne or more tem perature-sensitive resistors (a therm istor, a low-voltage light bulb, or a sem iconductor resistor) are used to build an attenuator. This power- sensitive a ttenuato r is constructed so tha t its a tten tuation X increases with increasing drive level. It is inserted in the feedback loop in which the small-signal amplification

6 .2 AMPLITUDE-LIMITING MECHANISMS 213

_______A~UVl)Ao________I- Attenuator Amplifier "]

V\ COS U V

Total loop amplifier

Input operating amplitude

Fig. 6. 2-1 “ N onlinear” attenuator as loop amplitude stabilizer.

exceeds the m inim um required for oscillation. The result is a curve of loop gain scale factor A vs. a ttenuato r input drive such as the one shown in Fig. 6.2-1.

The thermal time constant of the a ttenuato r is made long in com parison with the period of the lowest frequency expected from the oscillator; hence in the steady state the a ttenuato r is “ linear.” If, in addition, the signal levels in the active device are kept within the lineat range of the device, low-distortion sinusoidal signals exist throughout the loop.

The long therm al time constant imposes a lim itation both on the lowest frequency of operation and on the rate at which such oscillators can be swept or changed in frequency. This approach is widely used in audio- and video-frequency laboratory signal generators, particularly those of the RC network types. It is particularly applicable for adjusting R , an d /o r R 3 in the Wien bridge and M eacham bridge oscillators shown in later sections and in the problem s at the end of the chapter.tb) A second technique, which also produces low-distortion sinusoidal oscillations at all points th roughout the feedback loop is the use of an am plitude detector (any of the types discussed in Section 10.2 or Section 10.4) to provide a control signal, p roportional to the am plitude of the sinusoidal output, to inversely control the gain of some amplifier within the oscillator loop. Again the gain should decrease with increasing am plitude, so that a characteristic similar to Fig. 6.2-1 results. There is

+ Hundreds of commercial therm istors (negative-temperature-coefficient resistors) exist. Resistance values for 25°C are available from I i i or less up to 1 M ii, with thermal time constants from 2 sec up to 200 sec or more. N ot all com binations are available, nor are all units satisfactory from the reactive viewpoint for blind insertion into any network. M any units have only 20% tolerance and all are subject to variations with ambient temperature.

Any tungsten filament lamp with a rating of 6 W or less may be used as a positive-tem perature- coefficient resistor. H ot resistances may be ten times as great as cold resistances. If the operatingtem perature is considerably above the am bient (but below incandescence), then the ambient variations will be much less significant than for a thermistor. Cold resistance (25°C) values range from below 1 O for 1 V flashlight bulbs to perhaps 400 i i for a 3 W, 120 V light bulb.

Several of the references in Section 6.7 contain extensive data on such units. Later examples will contain illustrative data on some units.


no necessity for either the am plitude detector or the gain-controlled amplifier to have a linear characteristic with respect to the envelope of the detector drive signal. The gain-controlled amplifier, however, must have a linear transfer characteristic if the oscillations th roughout the loop are to have low distortion. As in case (a), the m ore steeply the variable-gain curve cuts through the required gain line, the better the am plitude stability will be.

c) A third and very com m on form of am plitude stabilization uses the nonlinearity of the active device to provide the am plitude-lim iting feature for the circuit. In such cases the active device m ust be followed by a narrow band filter, which passes only the oscillation frequency, in order to remove the harm onics generated by the non linear active element and thus m aintain a sinusoidal ou tpu t signal. M ost tuned oscillators sim ilar to the one shown in Fig. 6.1-6 do provide the necessary filtering a t the ou tput of the active device.

If the nonlinear device is a com m on base transistor, then its effective transconductance at the oscillation frequency is Gm(x), which is plotted vs. x in Fig. 4.5 6. Clearly, if for x = 0 (Kt = 0) and Gm(x) = gm0 the small-signal value of A L {jcL>0 ) is some num ber M which is greater than unity, then when pow er is applied to the oscilla tor at t = 0 the am plitude of the sinusoidal input to the transistor grows until it reaches a value Ki0 (or x 0) for which A L ( j a ) 0 ) [or equivalently, Gm(x)] has decreased

Fig. 6.2-2 Characteristic for self amplitude stabilization.

by a factor of M from its small-signal value. Therefore, x 0 and in turn Vl0 — k T x J q may be obtained by determ ining a t w hat po in t Gm(x)/gmQ = 1/M as illustrated in Fig. 6.2-2. F o r example, if M — 2, then from Fig. 4.5-6 x 0 = 3.3.

As we shall see, any of the nonlinear characteristics of fundam ental current vs. sinusoidal drive voltage of C hapter 4 may be used as the basis for an am plitude stabilization scheme.

d) A fourth m eans of am plitude stabilization is presented by the problem of Example5.5-3, where a diode-battery com bination is used to limit the ou tpu t sine wave from a tuned circuit.

While it is always possible to add an external diode and power supply, such additional circuitry may become available naturally in transistor or junction FET

6.2 AMPLITUDE-LIMITING MECHANISMS 215

9 Vrc= 12 V

% ^ ; v

Fig. 6.2 3 C ollector-base limiting.

circuits when ¡.he collector-base or drain-gate diode is driven to the edge of conduction once per cycle. Figure 6.2 3 illustrates the situation.

From Eq. (4.5-13) and Table 4.5 1 we can evaluate the peak amplitude of the fundamental component of the collector current, / , , in terms of / dc and the peak amplitude of r,ii). In this case,

i , 0.98 x 1.79 = 1.75 mA.

•iovv if i-jr were large enough we would have a peak tank voituge of R Tl t — 17.5 V'. Since I ( ( - 17.5 V. the collector-base diode of the upper transistor} conducts when ,(0 exceeds 12 7 V ; this limits the output tank voltage to a sinusoidal peak amplitude

; f 12 7 V provided that Q T > 10. Note that for the oscillator is derived from . ,it): hence onot' limiting from collector saturation occurs. wc l ;.ow the peak tuned- •. irc’.’i voltage i.nd m turn rhe feedback network yields iiv va'ii-.- of rM} With -ti> ’•; !i; .* -.A tan then determine /¡ . The ratio t J; I , _ ;dd the equ,valent

: near loading of the tuned-collector circuit, from which we can determine Q f and, in turn. ihe iiis;orlk r? of the output sinusoid.

2 1 6 SIN U S O ID A L O SC IL LA TO R S 6 .3

6.3 FREQ U EN CY STABILITY

O ne of the m ore im portan t attributes of an oscillator is its ability to m aintain an ou tpu t frequency which is independent of changes in tem perature, supply voltage, circuit loading, hum idity, etc. U ndesired frequency variations can, in general, be broken into two broad categories: “d irect” frequency variations and “ indirect” frequency variations. D irect frequency variations result from changes in the param eters which directly control co0. F o r example, for the oscillator of Fig. 6.1-6 co0 is given by co0 = 1 /^ /L C ; hence, if L or C varies w ith tem perature, a>0 also varies with tem perature. Q uantitatively this variation in a>0 may be expressed in the form

^ - - L f e 4 L + ^ ? A c ) (6.3-1,to0 co0 \ o L oC I

_ AC'2 \ L + ~C

where Aco is the variation of the oscillation frequency resulting from variations of AL and AC in the inductance and capacitance respectively. C learly a 1 % increase in L or C results in a j % decrease in co0 . Therefore, for the case of an L C oscillator of the form of Fig. 6.1-6, the m ain frequency-determ ining elements, the inductance and the capacitance, m ust be stabilized against tem perature variations, stray loading, etc.

Tem perature stability is usually accom plished by trying to choose an inductor and capacitor (or resistors and capacitors in an R C oscillator) with equal and opposite tem perature coefficients, thereby causing the variations to cancel in Eq. (6.3—l).f Stray capacitive o r inductive loading is usually minimized by coupling the L C tuned circuit to the active elem ent by m eans of a step-down transform er, as shown in Fig. 6.4—9. Still further frequency stability against tem perature may be accom plished by placing the m ain frequency-determ ining elements in an oven. This is often done when a crystal is em ployed in lieu of the L C tuned circuit in the oscillator.

In general, direct frequency variation of co0 may be expressed as

A<a 1 " dw0— = — 2. Aai> (6.3-2)ft)0 a>0 .= i

where the a /s are the various circuit param eters on which o;0 depends. Clearly the highest direct frequency stability is achieved by choosing an oscillator configuration for which a)0 depends on the fewest num ber of param eters (a;) and then stabilizing these param eters against environm ental changes. Alternatively the param eters may be chosen in such a way that their net variation with am bient changes results in a small value Aco.

t Typical L and C tem perature coefficients are tabulated by Eisenberg: B. R. Eisenberg, “ Frequency Stability of a Clapp V C O ,” IEEE Transactions on Instrumentation and Measurement, IM -18, No. 3, pp. 221 ff. (Sept. 1969).

6 .3 FR EQ U E N C Y STA B ILITY 2 1 7

The oscillator whose pole-zero d iagram is shown in Fig. 6.1-5 and the M eacham bridge oscillator (Fig. 6.1-7b) are two examples of oscillators with values of co0 which depend on a small num ber of param eters. Consequently these pole-zero diagram s represent optim um configurations for achieving a high direct frequency stability.

Indirect frequency variations are a result of parasitic reactances which introduce additional poles and zeros, with unknow n locations, into the pole-zero diagram of A L(p), thereby changing the frequency of oscillation. U nfortunately m any of these reactances are functions of tem perature and supply voltage (e.g., ou tput capacitance of a transistor is a function of the collector-to-em itter voltage and therefore the supply voltage).

To see how these parasitic poles and zeros affect oj0 , we recall that a>0 is the radian frequency at which the net phase shift of A L(p) is zero. Since the parasitic poles and zeros introduce additional phase shift (let us say A<j>), the frequency of oscillation m ust shift by an am ount Aco which causes the m ain frequency-determ ining elements of A L(p) (the know n poles and zeros) to supply a phase shift o f -A(f> and thus m aintain zero phase shift around the feedback loop.

F o r example, if we have a crystal oscillator that operates near the nom inal series resonant frequency, co0, of the crystal, then if the rest of the circuit suddenly in tro duces an additional phase shift of + 1 deg, the operating frequency shifts until the transm ission through the crystal produces an additional com pensating — 1 deg of phase shift. It is apparen t that if the crystal has a variation of phase angle with frequency of 1 deg/H z, then only a 1 Hz shift in frequency results.

Thus the greater the phase variation A(f> of A L{j(o) for a change in frequency A co from <y0 , the greater the indirect frequency stability. Consequently we define the indirect frequency stability S F as

A<f> d<!>SF — — - « co0—

L A O j j COq (ICO(6.3-3)

where A0 is the phase variation of A L(ja>) due to a change in frequency Aw from co0. The larger the value of SF, the better the circuit functions in com bating parasitic changes in frequency. Specifically, if parasitic elements in the feedback loop change the phase by A<f>, the resultant change in frequency from to0 is given by

Aco = -------— . (6.3-4)

N ote, however, th a t SF predicts variations in a>0 only for those cases where the main frequency-determ ining poles and zeros of A L(p) are stable; i.e., direct' frequency stability has been accomplished.

In order to com pare circuits, it is instructive to evaluate SF for the several oscillators for which the pole-zero diagram s of A J jj) were presented in Section 6.1. In each case we m ust evaluate


however, since

<t> = I * . .i — 1

where </>,-’s are the angles contributed by the individual poles and zeros of A L(p), then

S F — w o ¿¡=i dm

(6.3-5)

Consequently, by evaluating

S F¡ — w 0dw

for a single real-axis pole and zero, a pair of complex conjugate poles, and a pair of complex conjugate zeros, we may apply superposition to obtain SF for any of the pole-zero diagram s for A L(p).

Fig. 6.3-1 Pole-zero diagrams of a single real-axis zero and a pair of complex conjugate zeros.

The phase angle for the real-axis zero at — cu, shown in Fig. 6.3- l(a) is given by

o í(6.3-6)

thus

Sp i — (o, d<t> 11 ' dw !,

«O/Wi+ («o/'w i )2

(6.3 7)

A real-axis pole - w , would have the same m agnitude of SF¡ but a negative sign. Is is apparent that, if the real-axis pole or zero lies at the origin (w, = 0), its net contribution to the indirect frequency stability is zero. This is so because the phase

6.3 FREQUENCY STABILITY 219

o f th e p o le o r z e ro re m a in s c o n s ta n t a t + j i / 2 o r — n/2 respec tive ly , in d e p e n d e n t o f th e v a r ia tio n in <o. In a d d it io n , it is a p p a re n t th a t if w , is c lose to in fin ity its c o n tr ib u tio n to S F is sm all.

The phase angle 4>3 of the pair of complex conjugate zeros shown in Fig. 6.3 1 (b) is given by

, j i <*> + li$ 3 = + $■> ~ ,an lan — - • (6.3 -8)

a x

t hus

d<j>3 | 2<X(o0{(o'o + <4)(y0 7 ! / "<2 2\2 , 1 2 ( )

« « U K - Mo) + 4 a z('J5

w h ere ui'q ~ or + [i2. F o r th e u su a l case w h e re oj'0 = o>Q (the W ien b rid g e a n d th e M e a c h a m b ridge), S F3 sim plifies to

S Fi = — = 2Qt ,. (6.3 lU)a

where 0 !: -■ (o0/2a is the effective Q of the zeros. Again, for a pair of complex conjugate poles, the magnitude of SF3 is unchanged but the sign is reversed. Similarly the sign would be reversed if the zeros were in the right-half complex plane with the same Q i z .

If we apply the above results to the oscillator for which A L(p) consists of two negative real-axis poles at — a>, and —<o2 and a zero at the origin as shown in Fig.6.1 3, we obtain

('JoS r = - - . - — ; "Tir — r ," 7 .’ ¡f + 0. (6. .M l )

! + (cij0 'f.'j,) 1 +

if we introduce the fact that = <0 ii0 2, then S F simplifies to

S t — ,6 , , , ,(I) I + W2 1 + lOfiOJ 2

■ he ro.iiiu- .-.ign merely implies that <j> is decreasing with increasing o>; hence a ¡os!;: vc ' \ 6 from the rest of the circuit requires a negative A<p from the frequency- leic' miiiing element-, and a positive shift in frequency A p h i of - S F v* 'Vo, ¡s given hi Fiii 6.3 2; from this we observe that S, reaches its maxim a :n value 1 ' lie; the tw<> poles of A r(p) lie at the same point, li; adt'j'o>.<ii. the maximum is ■ n!>] hr-. sd so that even if the pole p o r t ions dit’-r b • i f i d o r of !4. ha*-iecrcase-.i or:!', 'r>y a factor of 2 from its maximum vaiuc

t or the oseihatoi lor which 4,(p) has a pair «<> • ompie'- conjugate poles at - x ■ ;/■ (where or f{2 - u)t: <,>2) and a zero at th: or s r r.iay he written

•'iirctfiy ¡¡on) Lq. (6.3-10,1 as

S F = - 2 Q r , (6.3 13)

2 2 0 SIN U S O ÏD A L O SC IL LA TO R S 6 .3

U>2Fig. 6.3-2 Plot of - SF vs. tûl/œ2 for A L(p) = A(oj)/{p + w¡)(p + w2).

where Q T is the Q of the passive netw ork yielding A L{p). Clearly, as anticipated in the previous section, the higher the value of QT, the greater the im m unity to changes in frequency due to parasitic elements.

F or the Wien bridge oscillator for which A L(p) is shown in Fig. 6.1-7(a) and for which col = (o'o = (0 , 0)2 , SF may be written directly as the superposition of Eqs. (6.3-10) (with a negative sign) and (6.3-12); thus

SF = - 2y / ^ 2 - 2QTz. (6.3-14)+ CO 2

It is quite apparen t that the indirect frequency stability of the Wien bridge oscillator exceeds that of the oscillator with tw o negative real-axis poles by 2QTz, while its frequency stability is essentially the same as that of the parallel R L C oscillator with com parable Q T.

F or the M eacham bridge oscillator for which A L(p) is shown in Fig. 6.1-7(b) and for which w 0 = co'0, S F m ay be im m ediately written as

SF = - 2 ( g r + QTz), (6.3-15)

where QT and QTz are the Q ’s of the poles and zeros respectively. As we shall see, inm any cases

QtA

where A is the voltage am plification of the active device within the oscillator loop. Hence with A = 100 we can obtain an SF 26 times that of the pole pair alone. Thus,

6 .3 FR EQ U E N C Y STA BILITY 221

if the poles of A L(p) are the result of an R L C netw ork with Qr = 100, then SF % - 5200. If the poles are due to a crystal with QT — 5000, SF can be increased to SF x 260,000.

F or either of the phase shift oscillators for which A L(p) is given in Fig. 6.1-7 (c and d), SF may be immediately written as

3 O)0M= - 1 , - r ? 1 ; \2 ’ (6.3-16)

1 = 1 1 + ( a > o M )

which, for the special casef wherewo —a»i = ft>2 = ia3 = —= or a)l = co2 = a>3 = y/3o) 0v 3

[Figs. 6.1 7(c) and (d) respectively], reduces to

y l3This value is significantly less than that obtainable with the Wien bridge or M eacham bridge oscillator o r with the oscillator shown in Fig. 6.1-6 with a high value of g r in addition, for the phase shift oscillators a>0 depends on a>1,(o2, and a>3 ; hence these oscillators have poor direct frequency stability.

Internal phase shifts are caused not only by reactances within active devices o r amplifiers but by dc blocking capacitors, by transform er couplings, and by the presence of harm onic term s in the active device voltage or current ouput. In all cases, the higher the S F, the less troublesom e these effects will be.

As we proceed through this chapter it will be useful to bear in m ind that it is generally desirable in any oscillator circuit to provide the m axim um possible isolation between the device and the network. F o r one thing, this isolation will increase the circuit’s frequency stability, since the frequency-affecting device reactances are norm ally functions bo th of the Q-point and of tem perature. M ost useful oscillator circuits achieve this isolation by w hat am ounts to a gross im pedance m ism atching between the device input and the netw ork ou tpu t im pedance and between the network input im pedance and the device ou tpu t impedance.

F rom this po in t of view oscillator circuits can be classified into the four forms shown in Fig. 6.3-3.. In this figure the device is shown as driving the network, which in tu rn works into the input im pedance of the device. The term “current-driven” implies that the netw ork input im pedance is small in com parison to the ou tput im pedance of the device; the term “voltage-driven” implies that the netw ork input impedance is large in com parison to the ou tpu t im pedance of the device.

All such classifications are relative. Thus a 50 f l device input im pedance may be considered as an open circuit if the netw ork has an ou tput im pedance of 5 Q.

M ost of the circuits of the next three sections will be of the type shown in Fig.6.3-3(a). The bridge circuits o f Section 6.8 are norm ally of the form shown in Fig.6.3-3(c), while certain of the crystal oscillators, as well as some of the circuits of Section 6.10, illustrate the o ther two farms.

t This case results in the maximum value of SF.

222 S IN U S O ID A L O SC IL LA TO R S 6.4

Gv¿t) v„(t) (a)

(b)

(c)

Fig. 6 .3-3 (a) C urrent-driven network w orking into an “ open” circuit, (b) Current-driven network working into a “sh o rt” circuit, (c)Voltage-driven network working into an “open” circuit, (d) Voltage-driven network working into a “ short” circuit.

T h is g en era l c la ss ifica tio n is o ften useful w hen one is co n sid er in g th e use o f a n ew d ev ice in an o sc illa to r c irc u it o r w hen o n e is co n sid erin g a p o ssib le new c ircu it co n fig u ra tio n . In such ca se s it is usually d esira b le to go b a ck to the b a s ic p rin cip les ra th e r th a n m erely to try to “ c o n v e r t” som e ex istin g circu it.

6.4 SELF-LIM ITIN G SIN G LE-TR A N SISTO R O SCILLA TO R

T h e m a jo r ity o f a c tu a l o sc illa to rs in co m m o n a p p lica tio n are self-lim itin g s in g le tra n s is to r o sc illa to rs o f the form show n in F ig . 6 .4 -1 . In th is sec tio n we shall ex am in e the p ro p erties o f th is type o f o sc illa to r , and in su b seq u en t sectio n s we shall extend ou r an a ly sis to th e cases w here the a ctiv e e lem en t is rep laced by a d ifferen tia l pair o r a ju n c t io n field effect tra n sisto r.

6.4 SELF-LIMITING SINGLE-TRANSISTOR OSCILLATOR 223

Fig. 6.4-1 Prototype single-transistor Colpitts oscillator.

A lthough the frequency-determ ining network in this (Colpitts) oscillator is a tapped capacitive transform er, any of the transform erlike networks shown in Table2.5-1 can be employed w ithout altering the analysis. Some of the other configurations are shown in Fig. 6.4—2.

For the circuit of Fig. 6.4—1 the quiescent em itter current is given by

, _ VEE — VBEq _^ ' r e - ~ R e

0.65 V for a silicon transistor and 0.22 V for a germ anium transistor.where V,B E Q

Thus the small-signal input conductance at the em itter is given by

g i n Q k T ’

(a) (b)

Fig. 6.4-2 Other configurations for single-transistor oscillators, (a) Hartley oscillator, (b) Tuned- collector oscillator.

2 2 4 SIN U SO ID A L O SC IL L A T O R S 6 .4

nQr'Qt >100 C = C ,C 2/(C ,+ C 2) n = C,/ (C, + C2)g£>io gm= g ja iy /n r

Fig. 6.4-3 Small-signal model for single-transistor oscillator.

and the small-signal transconductance has the form

SmQ ^SinQ‘

If we assume QT > 10, QE > 10, and nQT QE > 100, then the capacitive transform er m ay be replaced by the transform er model of Table 2.5-1 to yield the small-signal oscillator circuit shown in Fig. 6.4—3. Clearly the small-signal model is identical to the model shown in Fig. 6.1-6, from which we observe tha t growing oscillations occur with a frequency of a>0 = 1/^ /L C provided that

= Gl + + grnQm, » ;

or equivalently,

SmQ > SmQm¡n

Gl + n2GESmQ > gmQmin = n(1 _ n/a) • (6.4-2)

If the inequality of Eq. (6.4-2) is satisfied, the oscillation grows until the transistornonlinearities reduce A L(ja)0) to unity, at which level the oscillation stabilizes.

To determ ine at w hat am plitude (V,) the ac collector voltage v,(t) stabilizes, weassume1) that collector saturation does no t occur and2) that the passive circuit QT is sufficiently high so that for large-signal operation

the second and higher harm onics of r,(r) and v,(t) are negligible, and consequently the capacitive transform er can be replaced by its ideal transform er m odel given in Table 2.5-1.

W ith these assum ptions, on a large-signal basis,

v,(t) = V, cos ft)0î

a n d

Vi(t) = Vi c o sc ü q t + Vdc, (6 .4 -3 )

6 .4 SE L F -L IM IT IN G S IN G L E -T R A N S IST O R O SC IL L A T O R 225

| Ideal J

C — C 1C 2A C 1 + C 2) , n = C i / ( C i +■ C 2) , ciô 1 / V L C

Fig. 6.4-4 Large-signal model for the circuit of Fig. 6.4-1.

where V, = nV, and Vic is the average value of the em itter-to-base voltage. In addition, the circuit for obtaining v,(t) may be modeled as shown in Fig. 6.4-4. In this model the nonlinear transistor input characteristic has been replaced by its equivalent linear conductance Gm(x)/a(cf. Exam ple 5.5-4) and only the fundam ental com ponent of collector current G ^x )!7! cos co0r has been retained, since all o ther harm onic com ponents are effectively shunted to ground by the h ig h -g r tuned circuit. The transconductance Gm(x) is given by Eq. (5.4-17).

It is apparent that the large-signal model of Fig. 6.4-4 is identical to the small- signal model of Fig. 6.4—3 except that Gm(x) replaces gm(2; therefore, A L(jco0) = 1 for that value of x = qV1/ k T for which

Gm(x) = gmQrGl + n2GE n(l — n/a) '

o r eq u iv a len tly ,

G J x ) 21 ¿x)gmQ X10(X)

By p lo t tin g th e c o n s ta n t

1 +In Jq (x )

qVJkT_Gl + n2GE

gmQn( 1 - n/a)

(6.4-4)

(6.4-5)

Gl + n2GE

gmQ«(l - »/«)

on the characteristic o f Fig. 5.4-5, we obtain (with the appropriate value of Vx) the value of x, and in tu rn Vt , for which the oscillation stabilizes. W ith Vx and n known, V, = VJn is readily determ ined and the problem is solved.

In alm ost all self-limiting oscillators, QT is high (to keep the ou tpu t voltage sinusoidal) and n « 1 (since the ou tpu t voltage am plitude is usually required to be large in com parison with V,). W ith these two conditions (QT » 1 and n « 1), certain simplifications result. The first is that Eq. (6.4—5) may be closely approxim ated as


The second is that the total conductance GT shunting the tuned circuit of Fig. 6.4-4 may be approxim ated by GL. The to tal conductance Gr is given by

Gt = Gl + n2GE + n2Gm(x)/cc. (6.4—7)

W ith the substitution of Gm(x) given by Eq. (6.4-4) into Eq. (6.4- 7), Gr simplifies to

Gl + n2GEGt = (6.4-8)1 — n/a

which is the desired result. The im plication here is that if n « 1 the total loss reflected through the transform er is negligible in com parison with GL. Similarly,

(o0C oj0C* G , 'Qt — (6.4 9)

In addition, for « « 1 it is readily dem onstrated that

Qe ~ Qt ~ nQr ,where

+ C 2)

(6.4 10)

Qe — and nQr ■ =a) 0C

GE + Gm(x)/oc " "*■' n[GE + G J x ) /a ] ’

hence QT » 1 and n « 1 ensures not only that v,(t) and vt(t) are sinusoidal but also that the capacitive transform er may be replaced by its ideal transform er model. [The reader should convince himself of the validity of Eq. (6.4-10).]

Example 6.4-1 F o r the Colpitts oscillator shown in Fig. 6 .4-5, determ ine an expression for v0(t).

6 .4 SELF-LIMITING SINGLE-TRANSISTOR OSCILLATOR 227

Solution. For the oscillator shown in Fig. 6.4-5 we note that

C = r = 1000 PF<1 + 2

u)0 = \__= 107 rad/sec,J L C

C , 1” C 1 + C 2 8 0 ’

VA = 9.3 V,

9 3V‘ 2 0 k n ‘ a 4 ( '5 m A -

_ _ _L ~g i n Q ~ k J - 56 Q ~ & " C ’

and

QT * co0C R l = 100.

Since Qt » 1 and n «: 1, i>,(f) and are sinusoidal and the model of Fig. 6.4 4 is va lid ; thus the oscillation grows at a>0 = 107 rad/sec and stabilizes at that value of x for which Gm(x)/gmQ = 0.448 [cf. Eq. (6.4-4) or Eq. (6.4-6)]. Therefore, from Fig.5.4 5 (Vx = 9.3 V sr o>), we obtain x % 3.5 and in turn V, = 99 mV, V, = VJn = 7.9 V, and finally

Vu = (10 V) + (7.9 V) cos 107t,

which does not saturate the transistor.

Total Harmonic O utput Distortion

To obtain an expression for the total harm onic distortion (THD) of v,(t), we assume that the d istortion of v,(t) and i\(t) is sufficiently small so that in the circuit of Fig. 6.4—1 has the form

+ ) cos « V. f . I oW

We then replace the transistor input by a current source iE(t) and reflect iE(t) through the capacitive transform er [by taking a N orton equivalent of the circuit to the right of Gl and assum ing that GE is negligible in com parison with <0(^1 4 C 2) over the frequency range, oj > a>0 , occupied by iE(t) - I E0] to obtain the circuit modelshown in Fig. 6.4-6. For this circuit the am plitude of the fundam ental com ponent of v,(t) is given by

21,(x)

2 2 8 S IN U S O ID A L O SC IL LA TO R S 6.4

Fig. 6.4-6 Circuit for obtaining THD of v,(t).

whereas the am plitude of the fcth harm onic com ponent of v,(t) is given by

(6.4-12)

If the QT of the circuit of Fig. 6.4-7 is high, then

iZuUkcoo)

a>0C(k - 1)

(cf. Eq. 3.4-11). In addition, for n « 1,

therefore (cf. Section 3.5).

Gl + n2GE ~ GL\

thdn JJ¥ a)0Ci k y r /» M i 2

¿ 2 I *2 * 1/ l h ( x ) ] f • < M -» >

where

U w l 2

I U i W j

A plot of D(x) vs. x obtained by num erical evaluation of the significant term s of D(x) is presented in Fig. 6.4-7. F o r large values of x,

h ( x )1

and thus D(x) approaches the upper asym ptote

6 .4 SE L F -L IM IT IN G SIN G L E -T R A N S IST O R O SC IL LA TO R 229

For small values of x,

/ 2(x) x I k(x)j - — % - » —— for k = 3 ,4 , . . ./ i(x) 4 /,(x )

(cf. Appendix at back of book); therefore, D(x) approaches the lower asym ptote

2 / 2(x) x3 /[(x) 6

These asym ptotes are also illustrated in Fig. 6.4-7.From Fig. 6.4-7 it is apparent that, if the oscillator stabilizes with x = 10

(V, = 260 mV), then T H D = 0.642/Qr . Consequently Q T m ust exceed 64.2 to keep the harmonic distortion below 1 %. Clearly the smaller the value of x at which the oscillator stabilizes, the smaller the distortion. U nfortunately, a small steady-state value of x often results in poor am plitude stability, as we shall see in th‘e following paragraphs.


Amplitude StabilityThe am plitude stability of an oscillator is the sensitivity of the ou tput oscillation am plitude V, to variations in supply voltage, tem perature, etc. In particular, the am plitude sensitivity factor is defined quantitatively as

. S , - ^ . (6.4-14,AH/H

where /j is the param eter within the oscillator which is subject to variation, Afi isthe am ount of the param eter variation, and AV, is the corresponding variation in V,.Small values of are desirab le ; the sm aller the value of , the smaller the change in V, w ith changes in fi.

W ith this definition we may evaluate the various am plitude stability factors for the oscillator of Fig. 6.4-1 by first writing the ou tp u t voltage in the form

v„(t) = - Vcc + V, cos co0t, (6.4-15)

where V, = V Jn = k T x /q n and x is the solution of the equation

GJjx) = 2I J x ) I" l n / 0(x)~l = Gl + n2GE

gmQ */oWL <IVx/kTJ "(1 - n/<x)gmQ

We then vary the param eter of interest (p.) the desired am ount (A/i) and determ ine the resultant value x from Eq. (6.4-5) or Fig. 5.4-5. This value of x immediately yields the new value of V, = k T x /q n from which AV, may be com puted and in tu rn Su evaluated.

Consider, for example, the evaluation o f S„ for the oscillator of Exam ple 6.4-1 as replacem ent of C 2 causes the transform er ratio n to increase from 1/80 to 3/160 (an increase of 50% — that is, An/n = j). W ith the new value of n = 3/160,

Gm(x) Gi* — - = 0.298;

EmQ ft&mQ

therefore, from Fig. 5.4-5 (Vx x oo) we obtain x ~ 6.1, from which it follows that V, = 8.45 V, AV, = 0.55 V, and Sn = 0.13.

O n the o ther hand, if GL is increased from 100 fil3 to 150 /¿O in Example 6.4-1 (again a change of 50%), then V, = 4.7 V, AV, = 3.2 V, and SGl = —0.81.

Several facts becom e apparen t a t this point. If we wish to minimize SGl or S ,EQ = SgmQ, we m ust minimize the variations of x with GL or I EQ. W e accomplish this by choosing the param eters of the oscillator so tha t the oscillation am plitude stabilizes in the vicinity of x * 2, a t which value the slope of the Gm(x)/gmQ curve is steepest.

O n the o ther hand, if we wish to minimize Sn for n « 1 and Vx x oo, we choose the param eters o f the oscillator so tha t the oscillation am plitude stabilizes for the largest possible value of x such tha t 2 / 1(x)//0(x) « 2. F o r this condition,


and

v ^ = 2/eqQ Gl Gl '

which is independent of n ; thus Sn —► 0.Similarly, if we assume that gmQ = q IEQ/ k T is the prim ary source of param eter

variation with tem perature, we obtain S T —► 0 if the param eters of the oscillator are chosen so that the oscillation am plitude stabilizes for x as large as possible.

Let us now try to consolidate the above inform ation in the form of an example dealing with the design of an oscillator.

Example 6 .4-2 Design a Colpitts oscillator which provides an 18-V peak-to-peak, 107 rad/sec sine wave, having a total harm onic distortion of 1 %, to a load resistor of 3 kQ. The oscillator should m ake use of available +10-V power supplies and provide reasonable first-order am plitude stability with tem perature variations.

Solution. We first draw the schematic diagram of the Colpitts oscillator we intend to employ, as shown in Fig. 6.4—8. The design entails the choice of C , , C 2, L , R E, and the appropriate transistor to meet the required specifications.

T o achieve good am plitude stability with tem perature variations, we require the oscillator to stabilize with a large value of x or (Kt ). O n the o ther hand, a very large value of x requires a very large value of QT to achieve the desired TFID ; thus we

2 3 2 S IN U S O ID A L O SC IL LA TO R S 6 .4

com prom ise by designing the oscillator to stabilize for x = 10, which is achieved if (cf. Fig. 5.4-5 with Vk = 9.3 V)

Gm(10) Gl + n2GEgmQ ngmQ(l - n/x)

= 0.19. (6.4-16)

W ith x = 10, from Fig. 6.4-8 we obtain Z)(x) = 0.642; thus to obtain T H D = 0.01, we m ust have

Qt ~ ^ C R l = 64.2,

or equivalently

C ,C 2 64.2 l 2 7140 n FCi + C 2 (3 k i2)(107 rad/sec)

Therefore, L = 1 ¡<»\C = 4.67 /iH. (Slug tuning of L is usually necessary to obtain the exact required value of co0.) In addition,

n = „ = - g j - = = 0. 029 V, C , + C 2 9 V

(which ensures V, = 9 V with x = 10). F rom the above values of C and n we obtain

C 2 = - = 0.074 n F and C, = = 2200 pF.n 1 — n

W ith a value for n, with GL specified, and with the knowledge that a « 1 and n2GE « Gl , we finally observe from Eq. (6.4-16) that

EQ Q(VJRe) « n o t , ,= 1 ( 7 ^ = k T = ’

or equivalently,R e = 5.75 k i l

At first glance the choice of the dc bias resistor R E might seem u n im p o rtan t; however, this is not the case, since it directly determ ines I EQ, which in tu rn sets the value of gmQ and thus the steady-state operating point of the oscillator.

In choosing a transistor, two key factors m ust be considered: (1) the collector-to-base breakdow n voltage and (2) the inherent frequency-limiting param eters of the transistor. In this example the m axim um value of vCB is 19 V ; thus a transistorwith a breakdow n voltage in excess of 19 V m ust be chosen.

In general, if the transistor is chosen with an ou tput capacitance small in com parison with Ci and an input capacitance small in com parison with C 2, the transistor has little effect on the oscillator design. In general, low-power transistors can be found with sufficiently small values of inpu t and ou tpu t capacity so th a t designs sim ilar to the one in this exam ple are valid for oscillation frequencies in excess of 100 MHz.

6 .4 SE L F -L IM IT IN G S IN G L E -T R A N S IST O R O SC IL L A T O R 233

Variable-Frequency OscillatorIf a single-transistor oscillator is to be used as a local oscillator or a local oscillator- m ix e r c o m b in a tio n in a su p e rh e te ro d y n e (A M , F M , o r T V ) rece iv e r, th e o sc illa to ris usually constructed in the form shown in Fig. 6.4-9. The basic reason for this three-winding transform er configuration is to isolate the input and outpu t capacity of the transistor from the tuning capacitor C by providing a step-down transform er turns ratio both to the collector and to the em itter from the L 2-C tuned circuit. Such an arrangem ent prevents changes in the transistor capacitance, which is quite sensitive to small variations in supply voltage, from affecting the frequency of oscillation, particularly at the high-frequency end of the tuning range where C is usually quite small.

Variable

v(/) V¡ cos u>0t

Fig. 6 .4-9 Schematic diagram of local oscillator.

If the Q t of the tuned circuit is assum ed to be high, the oscillator of Fig. 6.4-9 has the large-signal model for determ ining v,(t) as show n in Fig. 6.4-10; therefore, the oscillation has a radian frequency of a>0 = l / ^ f L 2C and stabilizes at that value

of x = q V J k T for which(M 12/ L 2)nGm(x)

Gl + n2[Gm(x)/oc + Ge]= 1,

or e q u iv a le n t ly .

Gl + n2GEGm(x) _ ___________________gmQ ngmQ( M l2/ L 2 - n/x)'

(6.4-17)

(6 .4 -1 8 )

2 3 4 S IN U S O ID A L O SC IL LA TO R S 6 .4

v,(t)=Ifcos (U p/

V\ cos u>0tG M

a

w o - l / v ^ , n—Mn/Li

Fig. 6.4-10 Large-signal local oscillator model.

Again x is determ ined from Fig. 5.4-5, and in tu rn V, is obtained as Vt = kTx/qn. A lthough v,(t) and vt(t) are reasonably sinusoidal, we should recall that the collector voltage is not sinusoidal because of the presence of the series inductor (1 — k \ 2)L i (cf. Section 2.5).

Collector Saturation

If the value calculated for the am plitude of the fundam ental com ponent of collector current I cl on the assum ption of no collector saturation exceeds VtS/ R T = (Vcc + 0.7)/R t (in Fig. 6.4-1), then collector saturation does occur and the tuned- circuit voltage is lim ited to Vcc + 0.7. In this case the em itter-base driving voltage is n times the know n tuned-circuit voltage and the am plitude of the fundam ental collector curren t is

where x = qn(Vcc + 0.7)//cT in the saturation case. The difference between this curren t and the allowable net fundam ental curren t into the tuned circuit is the com ponent rem oved by the collector-base satu ration pulse train.

It is apparen t that, if the loaded QT of the tuned circuit is high, the effect of collector saturation may be represented as an equivalent linear conductance shunting the tuned circuit to yield a to tal shunt conductance GTS given by GTS = VlS/ I c l , which for the circuit of Fig. 6.4-1 reduces to

N ote that this equation is valid when Gts > GT, tha t is, only so long as sa tu ra tion does occur on the peak of each cycle.

Thus, if collector saturation occurs, the ou tpu t am plitude is directly p roportional to Vc c , while the effective loaded Q of the circuit depends on Vc c , l EQ, and n. Specifically, Q T = o^C !G TS.


E xam ple 6 .4 -3 F o r th e C o lp i t ts o sc i lla to r sh o w n in F ig . 6 .4 -1 1 , d e te rm in e a n e x p re s sion for v0(t).

Figure 6.4-11

Solution. F o r the o sc i l la to r sho w n in Fig. 6 .4 -1 1 , we n ote that

c = - C lC l = 1000 p F ,C ] + C 2

1CO n 107 rad/sec,

» — = 1 , C , + C 2 80

Vx = 9.3 V,

I eq = 0 .9 3 m A ,

an d

f!m0 = 1/28 n .

2 3 6 SINUSOIDAL OSCILLATORS 6 .5

Consequently,

SmQ ^SmQ

and, from Fig. 5 . 4 - 5 , x » 8 . 5 ; thus Vt = 221 mV and V, = 17.7 V.Since Vt > Vcc + 0.7, it is apparen t tha t collector saturation does occur on each

negative peak of v,(t); hence Vt is constrained to a peak value of K,s = Vcc + 0.7, or 10.7 V in this case.

N ow V! = 1 0 . 7 m = 1 3 4 mV and x = 5 .1 4 . F rom Eq. ( 6 . 4 - 1 9 ) and Fig. 4 . 5 - 5 we note tha t the collector saturation has increased Gr from 1 0 0 /im ho to 1 5 6 /im ho and has reduced the loaded QT from 1 0 0 to 6 4 . A dditional calculations of the same kind lead to the values shown below for the elfects o f varying n in this circuit.

n X Q t

1/40 10.3 611/80 5.1 641/160 2.6 761/240 1.7 90

6.5 SELF-LIMITING DIFFERENTIAL-PAIR OSCILLATOR

In any of the oscillator circuits discussed in Section 6.4 the differential pair may be employed in lieu of the transistor as the active element. The result is a self-limiting differential-pair oscillator sim ilar to the one shown in Fig. 6.5-1.

Two distinct advantages result from em ploying the differential pair as the active element. The first is the fact tha t the ou tpu t sinusoid v0(t) may be obtained at the collector o f transistor 1, which is external to the oscillator feedback loop, by choosing Z L as a parallel R L C circuit. Consequently, variations in the load have little effect on the frequency or am plitude of oscillations provided tha t transistor 1 does not saturate.

The second advantage is that, for a given value of Q T for Z L, the to tal harm onic d istortion o f the ou tp u t sinusoid is m uch less than it would be for a single-transistor oscillator. This lower T H D is a direct consequence of two facts: (1) tha t no even harm onic com ponents exist in the collector currents of either transistor in the differential p a ir; and (2) that, for a given input sinusoidal drive level, the harm onic am plitudes o f the collector cu rren t are significantly sm aller than they are for a single transistor.

T o determ ine an expression for v,(t), we require a large-signal m odel for the differential-pair oscillator. By proceeding in a fashion identical to tha t of the previous section, we arrive a t the large-signal m odel shown in Fig. 6.5-2, which is a valid representation o f the oscillator o f Fig. 6.5-1 provided th a t Qr « qj0C/Gl > 10, tha t n = M /L , « 1, and that neither transistor saturates. W ith this m odel, which is

6.5 SELF-LIMITING DIFFERENTIAL-PAIR OSCILLATOR 237

Fig. 6.5-1 Transform er-coupled differential-pair oscillator.

Fig. 6.5-2 Large-signal model for self-limiting differential-pair oscillator.


identical to the large-signal model of Exam ple 6.4- 3 (except for two com pensating phase inversions), we conclude that oscillations occur a t a radian frequency of co0 = l / y / Z ^ C and stabilize a t tha t value of x for w hicht

G J x ) 4 aj(x) G, G,

gmn( 1 - n/0) ng„(6.5-1)

In this case, x is found by plotting G J g mn{ 1 — n/fi) on the curve of Fig. 4.6-5. W ith the resultant value of x, we obtain = k T x / q and Vt = VJn.

For the self-limiting differential-pair oscillator, the to tal harm onic d istortion (THD) is readily found, by m ethods sim ilar to those used for the single-transistor oscillator, to be of the form

T H D =D(x) ^ D(x)

Q tÍ T V

where QTU = oj0C/Gl and D(x) is given by

*>(*> = J I2n — 1 2

’«2. - lW "(2n - l )2 - 1 ai(x)

(6.5-2)

(6.5-3)

The Fourier coefficients a„(x) are defined by Eq. (4.6-10). A plo t of D(x) vs. x obtained by num erical evaluation of the significant term s of D(x) is presented in Fig. 6.5-3. N ote that, for a given value of x, D(x) for the differential-pair oscillator is lower, by m ore than a factor of 7, than the corresponding D(x) for the single-transistor oscillator. This, o f course, is an anticipated result.

The upper asym ptote Dv(x) shown in Fig. 6.5-3 is obtained by noting that as x -> oo, iFA and iE2 approach square waves and thus

2*2 n lM 1

(2n - l )2Therefore,

Dv{ x ) = J ZI

(2n - l )2 - 1= l ^ - V6 = 0.135. (6.5-4)

For small values of x,

a,(x)x2 „ <*2 n-l(x) 48 fli(x)

for n = 3,4, 5,.

t The approximation of GJgmn{\ — n/P) by GJgmn is valid even if n is not much less than 1, provided that fi is large. This is a direct result of the fact that the base, rather than the emitter, is being driven and thus the reflected loading across GL is reduced by a factor of fi from that obtained with a corresponding emitter drive. Therefore, in the differential pair, QT k u>0C/Gl » 1 and fi » 1 replace the conditions of QT » 1 and n « 1 required of a single transistor as sufficient conditions to ensure the validity of the ideal transformer model, to ensure the validity of assuming i’,(r) to be sinusoidal, and to ensure Gr w GL.

6.5 SELF-LIMITING DIFFERENTIAl.-PAIR OSCILLATOR 239

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

D(x)D t {x ) j i 0.135 (U pper asym ptote)

x2 /- D , ( x ) (Low er asym ptote) /

2 n - \2

« 2 ,- lW(2n - 1)2 1 a ,(x )

Total harm onic distortion

X D(x)1 0.00732 0.002505 0.007857 0.0978

10 0.1130

4 6 8 10 12 14 16 18 20 22 24 26 x

Fig. 6.5-3 Plot of D{x) vs. x for the differential pair.

(cf. Section 4.6); therefore, D(x) approaches the lower asymptote

3 x 2 8 48

,x

128’(6.5-5)

which is also plotted in Fig. 6.5-3.

Example 6.5-1 For the oscillator shown in Fig. 6.5-4, determine an expression for („(f). Also determine the T H D of vjr).

Solution. For the circuit of Fig. 6.5 4 ,

Mn = - = 0.02 « 1, o j 0 = 10 rad/sec,

L i

and

Q t ~ Q r v — a)o — 50 :

therefore, if we assume that neither transistor saturates, then the large-signal model of Fig. 6.5 2 may be employed. With the use of the model we conclude that the oscillation frequency is 107 rad/sec and that the oscillation am plitude stabilizes for that value of x = q l \ / k T for which

Gm(x) _ 4fl1(x)

gm X n( 1 - n/ß)g,r ngm


Figure 6.5-4

For the circuit of Fig. 6.5—4,

gh= av

where

_ qI eq _ ^ 1 .gin “ k T ~ 2 k T ~ 2 6 i2 ’

hence Gm(x)/gm = 0.52, which, with the aid of Fig. 4.6-5, yields x = 4.2 and in turn

^ c ™ , = 545V

Consequently,

»„(t) = (10 V) + (5.45 V) cos 107f ;

this value of v„ does no t saturate the collector of transistor 2.Since x = 4.2, from Fig. 6.5-3 we obtain D(x) = 0.065 and in turn

this value is quite small, as expected.

TH D = ^ = 0.0013 (0.13%);

6 .6 SELF-LIMITING JUNCTION FIELD EFFECT TRANSISTOR OSCILLATORS 241

6.6 SELF-LIM ITIN G JU N C T IO N FIELD EFFEC T TRA N SISTOR O SCILLA TO R S

Like the differential pair, the junction field effect transistor may be used as the active element in any of the self-limiting oscillators discussed in Section 6.4. In addition, the FET may be employed as the active element of the tuned-gate oscillator shown

Fig. 6 .6 -1 Tuned-gate junction FET oscillator.

T h e tu n e d -g a te o sc i l la to r is p ro b a b ly th e m o s t log ica l c o n f ig u ra tio n fo r a self-limiting FET oscillator, since in this configuration the tuned circuit is virtually not loaded at all by the FET and therefore it is possible to m aintain a high value of Q T with its accom panying good frequency stability. Obviously Rc , which is usually chosen anywhere between 1 M il and 10 MQ, does not appreciably load the tuned circuit. In addition, with M / L 2 « 1, only an exceptionally small am ount of loading due to the ou tpu t im pedance (both capacitive and resistive) of the FET is reflected through the transform er. The ac com ponent of vDS is also kept small with M / L 2 « 1, the result being a negligibly small “ M iller effect” ; therefore, the capacitive loading of the FET input directly across the tuned circuit is minimized. The Colpitts, Hartley, and tuned-drain configurations possess none of these advantages and therefore do not find as wide application as does the oscillator shown in Fig. 6.6-1.

N ote that the FET oscillator of Fig. 6.6 -1 is clam p biased ; i.e., the com bination of the capacitor C G (which is an ac short circuit at the oscillation frequency) and the gate-to-source junction diode clam ps the positive peak of the gate-to-source voltage essentially to z e r o j Thus if c, = V, cos w0i, then vGS = ^ (c o s w0f - 1). This form

t Actually vos is clamped to K0, the turn-on bias o f the diode. Since V0 is a function of the average diode current (cf. Fig. 5.1-6) and since an R(i of several megohms requires the average diode current to be quite small ( ^ ¿(A or less), V0 is usually required to be less than j V, even for silicon; thus as a good first approxim ation we may usually assume V0 % 0.


of bias, in contrast with tha t obtained by inserting a negative voltage source in series with R g , thus applying a fixed gate-to-source bias, has the advantage that the oscillation am plitude may be stabilized with the FE T operating completely within its square-law region. This property is quite desirable when the circuit of Fig. 6.6-1 is employed as an “ oscillating m ixer” (C hapter 7), in which case square-law operation minimizes undesired interm odulation products. W ith a clam ped bias, the FE T transconductance decreases with increasing sinusoidal input voltage am plitude within the square-law region (a necessary condition for am plitude stabilization—cf. Fig. 4 .9-4); with a fixed bias, on the o ther hand, the FE T transconductance rem ains independent of input voltage am plitude for operation com pletely within the square- law region (cf. Eq. 4.4—6).

N ote also that the transform er in Fig. 6.6-1 with M / L 2 « 1 keeps the reflected im pedance in the drain cricuit quite small ; hence the ou tput resistance r0 of the FET m ay be neglected in obtaining the FE T model, or equivalently the drain current iD may be expressed in the form

'd = ^Dssjl jrrj > (6.6-1)

where I DSS is the value of iD when vGS = 0 and vDS = VDD, and Vp is the pinch-off voltage of the FET. In addition, M / L 2 « 1 ensures that the ac drain voltage is quite small, causing, in general, operation com pletely within the saturation region of the FET.

From the above considerations and the assum ption that Q T of the tuned circuit is sufficiently high to keep vt{t) sinusoidal (that is, v£t) = V, cos oj0t), we may obtain the large-signal m odel shown in Fig. 6.6-2 for the oscillator of Fig. 6.6-1. In this model, because of the assum ed high Q T, only the fundam ental com ponent o f drain current is reflected through the transform er, in the fashion of Fig. 2.5-7, to obtain the driving current source

M r v- r G mV\ cos (O0t,L 2

where Gm is the large-signal fundam ental transconductance of the clam p-biased FET. (A plot of Gm/gm0 vs. — VJVp is shown in Fig. 4.9-4.) In addition, because of

Fig. 6 .6-2 Large-signal model for the tuned-gate FET oscillator.

6.7 CRYSTAL OSCILLATORS 243

the assum ption of high Q T, the clam p-biasing circuit is replaced by its equivalent conductance G NL = 3/ R G, which is developed in Example 5.5-5.

With the model of Fig. 6.6-2, we observe that

and that

-1,1 /(■)(,) = 0 forsJ l 2c

^ l ( M ) = J - r Gfn, = 1 (6.6- 2)L-2 U, + ÎKJr,

for that value of V{ for which

G m = ¿2 Gl + 3Gc 8m0 M gm0

(6.6-3)

where gm0 = 2 I DSS/ — Vp is the small-signal transconductance of the FET with v g s = 0- Clearly, once Vp, / DSS, GL, and Gg are specified, — Vi:/Vp, and in turn K ,, may be determ ined from Fig. 4.9^4.

As an example, consider

L 2 = IOjiH, M = l / i H , C = lO pF = CG,

R g = 3 MQ, R l = 50 kQ, Vp = - 4 V,

and

Therefore.

I — 4 mA.

co0 = 10 rad/sec, QT = 47.5 > 10,

gm0 = 2000 /imho, and = 0.105.^mO

From Fig. 4.9-4 we obtain — VJVp ~ 1.67; hence Vx = 6.67 V and

v,(t) = (6.67 V) cos 108r.

6.7 CRYSTAL O SCILLA TO R S

Since 1921 there have been hundreds of articles, tens of chapters in textbooks, and several books abou t crystal oscillators and piezoelectric resonators. G erber and Sykesf have provided a recent sum m ary of the state of the art with respect to crystals, while H afnerJ has written extensively on crystal unit models and the m easurem ent

t E. A. G erber and R. A. Sykes, “State of the Art—Q uartz Crystal U nits and O scillators,” Proc. IEEE, pp. 103-116 (Feb. 1966). Summary of data and characteristics; 63-item bibliography.J E. Hafner, “The Piezoelectric Crystal U nit— Definitions and M ethods of M easurem ent," Proc. IEEE, pp. 179-201 (Feb. 1969). Includes a 46-item bibliography.


of model param eters. Sections 6.12 and 6.13 and C hapter 9 of E dsont and Sections 8.1,8.2, and 8.3 of G roszkow skiJ provide textbook sum m aries of m any q uartz crystal properties and also discuss vacuum -tube crystal oscillator circuits in detail.

It is not our purpose here either to provide design inform ation abou t piezoelectric resonators o r to catalog all possible crystal oscillator cricuits. We wish simply to point ou t the basic properties of these resonators and to explain how they may be exploited to produce satisfactory circuits.

The reason why one employs a piezoelectric resonator in place of a conventional L -C com bination is tha t the available Q in these mechanical vibration devices may be up to 1000 times greater than that available with conventional elements. As we have seen, indirect frequency stability is directly proportional to Q ; hence crystal oscillators are used as fixed-frequency, highly stable frequency or tim ing sources.

The principal m aterial used as a m echanical resonator for oscillators is crystalline quartz. The properties of this m aterial differ in different directions through the crystal; in addition , each different “cu t” may be m ounted and vibrated in several different ways. These different “cu ts” have different tem perature properties, different ranges of possible operation, and different relations between their dim ensions and their m echanical and electrical properties.'

Table 6.5 in Edson sum m arizes the properties of twelve of the m ore useful “cuts.” Some “cuts,” for example the G T cut, have frequency shifts of considerably less than one part per million per degree centigrade over a 100°C range. O ther “cu ts”

have less desirable tem perature properties bu t may have higher Q’s, lower series resistances, o r m ore desirable properties a t harmonics. The basic electrical model for a properly m ounted quartz crystal and its holder is shown in Fig. 6.7-1.

As we shall see, the crystal is norm ally operated within 1 % of the series resonant frequency of one o f the shunt b ranches; hence the circuit norm ally is reduced to C 0 in shunt with a single series resonant circuit. The m ultiple branches result from

+W. A. Edson, Vacuum Tube Oscillators. John Wiley, New York (1953). î J. Groszkowski, Frequency of Self Oscillations. Macmillan, New York (1964).

6 .7 CRYSTAL OSCILLATORS 245

m echanical vibrations a t approxim ately the odd harm onics (usually called overtones) of the fundam ental frequency. Since the mechanical frequency of the fundam ental vibration is proportional to the crystal dim ensions, practical considerations usually limit fundam ental operation to the neighborhood of 20 M Hz or less. By operating on an “overtone” frequency, crystal oscillators up to the neighborhood of 200 M H z are possible. For higher frequencies we m ust em ploy harm onic m ultiplication from a lower-frequency circuit.

If operation at a particular overtone is desired, then this should be specified when ordering the crystal. If a “cu t” and m ounting are used that minimize overtone effects, then sinusoidal operation may be possible in circuits with no filtering elements except the crystal itself.

Co

Case electrode capacitance

2 Ql

Fig. 6.7 2 M odel for a crystal, its adm ittance equation, and its pole-zero diagram.

Figure 6.7-2 repeats the simplified version of the model together with its adm ittance relationship and pole-zero pattern.

In order to understand the particu lar properties of crystals it is useful to consider typical numerical values for the crystal param eters, for example,

(M0 = 107 rad/sec,

C C 0 = 1/100,

C0 = 4 p F ,

a n d

Ql = 20,000,

which lead to C = 0.04 pF, L = 250 m H , and r = 125 Cl For this typical case, the vertical spacing between the pole and the zero in the vicinity of a >0 is approxim ately


co0/200, while the horizontal distance from the j-axis to a complex pole or to a complex zero is co0/40,000, o r 1/200 times the vertical spacing between the complex pole and zero locations. Thus near a complex pole one has in effect an isolated very high-Q pole, while near a complex zero one has in effect an isolated very high-Q zero.

Two choices rem ain before we can select a circuit configuration. We must decide whether we w ant to use the crystal as a low im pedance [near a zero in Zip) o r a pole in 7(p)] or as a high im pedance [near a pole in Zip) or a zero in Y(p)], and we have to choose a m ethod of am plitude limiting for the circuit.

Series resonant m odes of operation usually place the crystal directly in the feedback path or use it to shunt a bias resistor. In both cases, the object is to keep the loop gain below the m inim um required for oscillation except at the series resonant frequency of the crystal. Since the pole o f Yip) [zero o f Z ip )] is independent o f C 0 such circuits should not be affected bv circuit-caused variations in the crystal's shunt capacitance.

A capacitance Cx in series with the crystal modifies the scale factor of Y (p) from C 0 to

C 0 + Cx

and shifts the poles of Y(p) from a>0 to

jc + C q + Cx V c0 + cx aj°-

W hen C 0 » C, this capacitance changes the approxim ate vertical pole-zero spacing from Ccjq/2C0 to

C j i j0 C x

2 C0 C0 + Cx

T hus the possibility of a slight adjustm ent o f the series resonant frequency is purchased at the expense of a narrow ing of the pole-zero spacing.

If C q = 100C and Cx varies from C0 to C 0/5, it is possible to have a frequency tuning of 0.166% of co0 while reducing the m inim um pole-zero spacing by only j .

A great m any vacuum -tube oscillators employ the crystal operating in a high- im pedance mode. Field effect transistor oscillators tend to follow this same pattern. In all these circuits the operating frequency tends to be below the zero of Yip) [pole of Z{p)] a t a distance co0/QL o r less from it. At these frequencies the crystal looks like a parallel inductance-resistance com bination and is used to replace an inductance in a Colpitts-like o r Hartley-like configuration. Such arrangem ents can be “ tuned” by varying the shunt capacitance, C 0.

F o r operation at higher frequencies and higher overtones, the pole-zero spacing may be increased by adding a shunt inductance L s that resonates with C0 at the operating frequency. W ith such an arrangem ent a new pair of complex poles in Zip) is fo rm ed; these poles lie between the complex zeros of Zip) and the origin. For


C0L S = L C and C 0 » C, the a p p r o x im a te vertical spacings o f the two c o m p le x poles a b o u t the u n ch anged zero p o s it io n is

which for C0 = 100C leads to +(21/400)<uo and — (19/400)a>o . We should note that (hese spacings are ten times the spacing for the unshunted case. Since C / C 0 tends to decrease as the square of the overtone multiple, the shunt inductance allows us to keep a nearly constant percentage pole-zero spacing at the higher frequencies.

Varying C 0 (or the shunt inductance) provides a means of varying both of the complex poles. Operation of the crystal as an “ inductance” is possible just below either pole, but is not in general desirable, since the pole positions depend strongly on the external shunt inductance.

A well-designed crystal oscillator circuit should stop oscillating if the crystal is removed and should minimize the crystal driving voltage and current. The last requirement is desirable not only to avoid physical damage through overdriving, but also to minimize temperature shifts due to self-heating For example, if the thermal resistance o f !h.:: mounted unit is 5!)0°C W. then an internal dissipation o ! 2 mVV •:j(k to iK/" te ;T pera t in c rise; this temperature use may lead, in some ciystai

, ,i :!i:it'iem temperatures, to frequency shifts o! several par.s per .niiiion.

i;i.ji or- iilator circuits separtne I e hequencv ¿ '¡ r r ; :

ii :i - bv employ ing -.ep,.:' i iii'.c. ¡ai- i


r = series-resonant resistance of crystal, oj° — scries resonance of crystal,

n= C ,/(C , + C2), C= CtC2 /{Ci + C2)

Fig. 6.7-3 Series mode crystal oscillator.

where Vebi is the am plitude of the fundam ental em itter-base voltage. N ow ÊBl is a function of I El and I EQ which can be found by assum ing that an ac current source and a dc current source drive the em itter-base junction and calculating the fundam ental com ponent of the em itter voltage. [N ote tha t the em itter-base voltage is no longer sinusoidal, since the crystal separates the junction from the assum ed sinusoidal voltage Vi (r).]

I f

then

*£ ~ I E S e V E B q l k T ~~ I E Q + I E l COS cot,

k T I EQ k T I I E1 \= — In —-- H------- In 11 + -— cos cot

Q * E S <1 \ I E Q I

(6.7-2)

(6.7-3)

Hence, if the last term in Eq. (6.7-3) is expanded in a Fourier series, we may determ ine VEB1 to be of the form


where y = 1Ei/ I Eq \ in addition,

v b » = — \ W - 2 +q \_y

(6.7-5)

F o r this case we define a large-signal average fundam ental transistor input conductance as

q . a _ I e i _ I e i I eq _ 4 I e q _______ >,x_________

¡EQ ^ E B l k T 2 (1 — ^ / l — y 2)

y 2 I y 2

= ginQ2(i - * M 1 “ t

for y < 0.6. A p lo t o f Gin(y)/ginQ is shown in Fig. 6.7-4. If we now assume tha t thereflected loading o f the crystal resistance r in series with the transistor input resistance

y G,nj&in Q0.0 1.0000.2 0.9900.3 0.9770.4 0.9610.5 0.9370.6 0.9000.8 0.8000.9 0.7251.0 0.500

y = h \ l h QFig. 6.7-4 Normalized input conductance for a current-driven junction.

1/Gin(_y) across R L is negligible, then the loop gain for the circuit of Fig. 6.4-3 may be written as

> M M > ) = * n „ L( v ? ( 6 . 7 - 6 )r + R J y )

hence the oscillation stabilizes for that value of y = I e1/ I eq for which A L(jco0) = 1, or equivalently,

= ------------- , ( 6 . 7 - 7 )ginQ (m R L - r)gmQ

where r is the series resistance of the crystal, R L is the load resistance, and n = C ,/(C , + C 2).


I t is a p p a ren t fro m Fig . 6.4—4 th a t for re a so n a b le am p li tu d e stabi l ity I El should be betw een 0 .4 a n d 1 t im es 1EQ o r that

0 .5 0 < -------- 4 - --------- < 0.95. (6 .7 -8 )g>nQ( x n R L - r)

In o rd er to c a lc u la te the o p e ra t in g a m p litud e , o n e ca lc u la te s Gm(y)/ginQ from Eq. ( 6 .7 -7 ) an d uses Fig. 6 . 7 - 4 to find y. T h e n , s ince V, = a I E i R L , o n e finds that

V, = x y h a R f

As a n exa m p le , we m igh t m odify the c ircu it o f Fig. 6 . 4 - 5 to include the crystal, as in F ig . 6 .7 -3 . I f we increase n fro m 1/80 to 1/50 while m a in ta in in g all the o th e r q u a n tit ies at their prev ious values, then for a = 0 .98 and r = 125 £i,

g in C 71also , from Fig. 6 .7 - 4 , y = 0 .82 , so that IEl = 0 .8 2 1EQ and Vt = 3.7 V or

vB(t) = (10 V) + 3.7 V cos car.

The effective loaded Q of the crystal is a)0L divided by r + R in = 125 + 71 ^ 196i2; hence the crystal Q has been reduced from 20,000 to 12,800.

The crystal dissipation for this case i s / | , r / 2 = 8.5 / i W -this am ount of dissipation will certainly not cause excessive frequency shifts.

In making the previous calculations wc assumed that the tuned circuit consisting of L, R, , and C ,C 2/(C , + CY) was tuned to the series resonant frequency of the crystal [the pole of T(/j)] in Fig. 6.7 7. We might ask what happens to the amplitude and frequency of the oscillations as the tuned circuit is detuned or mistimed.

On a small-signal basis, in the vicinity of the pole in Yip) of the crystal, Y{ jot) is closely approximated by

Y (je>) - - W()C ; w . -2(jd<o -r <v0/2Qi )

»»here

s , -I n ,da) = u) - oj0, <')0 f <0 2i'jn , anu =nQ

! tie n o rm alized form o f the v a r ia t io n s in the m a g n itu d e and phase o f this function>.>iven by the Q = x_> curves o f Figs. 2 . 2 - 7 an d 2.2 8. The same n o rm alized curves

j i also valid for ¡he impedance of the timed circuit n e a r ,

A h ’A = y - ~on y * ,0oR i.c <' (6 J 9)2C,(jdo) + (oa/ 2 Q r )

where the subscript t is used to differentiate the parameters of the tuned circuit fromthose of the crystal.

6 .7 CRYSTAL OSCILLATORS 251

If we detune the tuned circuit by co0/8Q T (2 kHz in the example where Q r = 100 and m0 = 107). we reduce the magnitude of Z{joj) from R L to 0.970R L and introduce a phase shift of 14°. To cause a return to zero net phase shift, the operating frequency must shift until the crystal phase shift supplies the compensating 14°. Since the effective crystal Q is much higher than the tuned-circuit Q, the frequency need only shift by 6m == <o0/8Q'L (16 Hz when co0 = 107 and Q'L = 12, 800)t; however, this frequency shift causes an equivalent normalized decrease in the magnitude of Y(jio). Thus, if we let

■which is the normalized decrease in the magnitude of the tuned-circuit impedance ;,s well as the decrease in the magnitude of the crystal admittance, then Eq. (6.7 7) becomes

i f (V-.. i' . •.oc.-sive. then (r ;n!.v),. q exceeds unity and o -lolkti ir-;v; ct\:se ('„ f * it5.. • > 7- ---, I'.'•1 s iletuning the tuned circuit causes o n h h s’.v-i- ,h:n in !'u: jiscnc>

u! ■ :i,:.r- in ¡’ppreeiabk’ change in amplitude. With ,V

:c : . j ’ “ V Th. i- ;■■■ 'si :in =;■:1,¿¡I :<■;c;c.} .it.ib.Miy ir. il.ev, -I ; i ■ :hc-; nir- ,1

i ■ ■ ; ■: " t ' r • ; ■: i; v,.‘I, ' ' h.C înpii tudc■a; ■. of ibis basic c m „ o c

- V ■ r • n-” ö n ü h ' i b y -.in e r r c .. - ^

Nv 1 + 4 Q t ( S o j J ( o 0 )2

(6.7-10)

0.58. and V - y

^ O-K ■ - !->i i

; : V,f(5 - '

•;n -r

' ' - ; I <;• .1 n.- n . ill,1V" : i !: i c n : ! : f i- K - in g le e ; . I


Fig. 6. 7-5 Differential-pair crystal oscillator.

L —10 nH A/ = 0.156 nH L2= 1 |iH

If the crystal overtone frequencies are properly suppressed, then operation of this circuit is possible with com m ercial integrated-circuit differential pairs in which a load resistor replaces the tuned circuit and the com bination of the crystal series resistor and the base input resistor replaces the capacitive attenuator.


(a)

C Inductive

Fig. 6.7-7 (a) Pierce circuit (Colpitts-like). (b) Miller circuit (Hartley-like).

It is apparent that, for satisfactory operation in such a circuit, we require a much higher crystal series resistance than in the previous cases. This is true because the effective Q'L is reduced to r/{RL + R B + r) o f its initial value, where r is the crystal series resistance and R L and R B are the circuit ou tpu t and input resistances respec


tively. If R l = 4kQ , R B = 500 Q, and r = 10 kQ, then only a § degradation of Ql results.

A nother variation of the single-ended circuit is shown in Fig. 6.7-6. If n = M / L = 1/5Q and r = 125 Q, then this circuit has the same output am plitude as the circuit of Fig. 6.7-3. The calculation of this am plitude is performed in exactly the fashion outlined previously. Intuitively we see that the circuit can have adequate loop gain to cause oscillations only when a> is near the series resonant frequency of the crystal (i.e., when the low series crystal resistance bypasses the large em itter impedance).

Figure 6.7-7 illustrates two possible high-im pedance m ode FE T (or vacuum -tube) crystal oscillator circuits. In both circuits the crystal behaves like an inductor and the operating frequency is near, but slightly below, the pole in Z(p). For the Pierce circuit,

Gm « ^ G g + C/ G d, (6.7-12)' - ' D

which for CD % C a and R Cl » R D leads to Gm as 1 / R D, where R D is the load resistor. The required Gm is much smaller than for the series mode transistor; however, since the available Gm from an FET or vacuum tube is also m uch smaller, the com bination is reasonable. T hat the operating frequency must lie slightly below the pole frequency is seen by cons ider ing the crystal as an equivalent inductance which resonates with the series com bination of CD and C c . Only within several multiples of <ovoh/Q, below « k. will the crystal have a sufficient induct ive react:?nee.

6.8 SQUEGGING 255

Similar argum ents hold for the M iller circuit, where C resonates with an equivalent inductance from the crystal in series with an equivalent inductance from the drain-tuned circuit. The algebra is m ore com plicated in this case and lends little insight into the circuit o p e ra tio n ; hence it is not presented here.

Figure 6.7-8 shows a series m ode crystal oscillator circuit which is a variation of the FET circuit of Fig. 6.6-1. In this case the small-signal gm, with the crystal removed, is reduced to gm/(l + gmR s)- If R s is sufficiently large so that Eq. (6.6-3) cannot be satisfied even for zero gate bias, then, with the crystal replaced, oscillations can occur only near the crystal’s series resonant frequency.

As in the b ipolar case, the source current is forced to be sinuso idal; hence again a complete calculation necessitates finding the fundam ental source voltage as a function of the ac and dc currents.

6.8 SQ UEGG IN G

In self-limiting sine-wave oscillators where the feedback loop is closed with a coupling capacitor, it is possible to have an interaction between the time constants of the bias and coupling circuits of the loop and the time constants of the high-frequency tuned circuits of the loop such that a self-produced am plitude m odulation of the high- frequency oscillation occurs. Such self-modulated behavior is known as squegging. The low-frequency variations of the envelope of the high frequency may be sinusoidal or exponential.

In this section we indicate an approach that either allows us to design for deliberate squegging or, as is m ore often the case, guarantees that it will not occur. First we shall outline a general m ethod, and then we shall apply it to a specific transistor circuit to obtain definite num erical results.

In our analysis we assume that conditions exist for norm al high-frequency oscillations to occur and that these oscillations have been set up. We then examine the small-signal low-pass transfer junction tha t governs the am plitude of the envelope of these oscillations. If the low-pass transfer function has only left-half plane poles, then after an initial transient at t = 0, the am plitude rem ains constant a t its desired value. However, if this low-pass transfer function has right-half plane complex conjugate poles, then sinusoidal oscillations of the am plitude occur; and if it has a right-half plane real-axis pole, then exponential buildups and decays of the am plitude occur.

Since we want some inform ation about the onset of squegging, we are interested in the perturbation of the param eters relating to the envelope around their norm al steady-state values. Consider first the generalized self-limiting capacitively coupled oscillator shown in Fig. 6.8-1, for which the self-limiting transistor oscillator shown in Fig. 6.8-2 is a special case. If for the circuit of Fig. 6.8-1 we assume a small perturbation r,(r) on the envelope of v¡(t), that is,

= [^1 + t’i(f)]C O Sft)0f,

then a small perturbation j j(i) results in the envelope of the fundam ental com ponent of (,(f). We note, however, that the right-half portion of the circuit of Fig. 6.8-1 is

256 SINUSOIDAL OSCILLATORS 6 .8

identical to the circuit of Fig. 5.6-1; thus ^(f) may be related to t>,(i) by the adm ittance Yi(p) given by Eq. (5.6-3). In addition, if Z i2(p) is a sym m etric narrow band filter centered about co0 (which it alm ost always is), then the results of Section 3.3 tell us that u \ ( t ) is related to the envelope variation of the fundam ental com ponent of the

\Zi2( j u ) x .

tu0

/ (/) - /o t /] c o s <±>Qt + I 2 C O S 2u>0/+ • • ■A , ( j u i n) 1

Fig. 6.8-1 G eneralized self-limiting capacitively coupled oscillator (nonlinear load).

Fig. 6 .8-2 T ransistor oscillator. (It is assumed that collector saturation does no t occur and that norm al high-frequency oscillations are possible.)

6 .8 SQUEGGING 257

a i'i(r) CD Z n L(P)'i U)

Vl(0 YAp)

Z\2l (p) low-pass equivalent of Z l2(p)

YAP); " [p + c ( (roo

GiqGqiGu )]

p+%-0

A , cm(p) = aYAp)Zn / {p)

Fig. 6.8-3 Envelope feedback circuit.

current driving the filter by the low-pass equivalent transfer function Z l2l (p) of the filter. The com plete low-frequency feedback circuit relating i,(f) and u1(i) is shown in Fig. 6.8-3, from which it is apparent that

¿ L e M = <x Yi (p )z 12M (6 .8- 1)

If 1 — A Lcnv(p) = 0 contains roots in the right-half complex plane, then squegging results. Even if no right-half plane roots exist, the positions of the left-half plane roots give us a good idea of the envelope transients when power is applied to the oscillator at f = 0.

To carry the analysis further, we require expressions for y^p) and Z l2 l (p). For the transistor oscillator of Fig. 6.8-2 (or any of the self-limiting transistor oscillators discussed in Section 6.4), y,(p) is given by (cf. Eq. 5.6-6)

2gi,

(P) =

1 M *)x / 0 ( x )

/(x)I

p +1

where t = C FJ gin, gin = q l dc/kT\ and A(x) is given by Eq. (5.6 -7) and plotted vs. x in Fig. 5.6-2. In addition, if Q T = to0C R T > 10, then

yiR y

Zl2l(P) =P +

1(6 .8-2 )

where xL = 2R TC, R T is the total loading across the tuned-collector circuit, and n = M / L x. As is required of a low-pass equivalent, Z 12,(0) = Z i2(jo)0) = n R T.

Now with the specific values for Z l2lip) and ^ (p ), ^Lenv(p) reduces to


However, since the circuit is assumed to be oscillating at a>0, A L(ja>0) = 1, or equivalently,

A pole-zero diagram of A Lenv(p) is presented in Fig. 6.8-4, in which the root locus of 1 — ^Lenv(p) = 0 with increasing A(x) is indicated. Clearly the possibility of squegging exists.

The roots of 1 — A Lenv(p) = 0 reach the im aginary axis when the coefficient of the p-term is zero. Therefore, to keep the roots in the left-half plane and thus to prevent squegging, we require

Figure 6.8-5 shows a plot of A{x) — 1 = x /0(x)//,(x) — 2.For example, if an oscillator is designed to operate at a given x, with a given

tuned circuit capacitance C and load R T, and a t a given dc bias which fixes g in, then there will be a m axim um value of CE a t which squegging may be avoided.

From Fig. 6.8-5 we see that for small x, say less than 2, then A(x ) — 1 is small and CE may be relatively large w ithout any danger of squegging. As x is increased, however, we may have difficulty in sim ultaneously satisfying Eq. (6.8-5) and the condition a»0C£/g in > 10. (The latter condition is necessary if C E is to be considered as an ac short circuit.)

r t \ » 2 I^ d 1Gm(x)nRT = gm n R T = 1x / 0 x

W ith this additional restriction, ALenv(p) may be expressed as

/l(x)| A(x)P +

a i \ L t^Lenv(P) = 1------ l \ j f (6.8-4)

where

1 1 A(x)(6.8-5a)

or

T C £/g in 0}0C E/gin(6.8—5b)

or

(6.8-5c)

6.8 SQUEGGING 259

Fig. 6 .8-4 Pole-zero diagram of AL(p) and root locus of 1 - Atenv(p) = 0.

Fig. 6.8-5 Plot of A(x) — 1 vs. x.

For example, if co0C £/g in = 20 and Q r = 50, then from Eq. (6.8-5b) and Fig. 6.8-5 we see that a design value of x = 6.4 will cause squegging. For proper non- squegging operation of such a circuit, either x m ust be kept small (below 6.4). or Q T must be increased.

If squegging is allowed to occur, then its initial frequency of sinusoidal variation will be the frequency at which the roots of 1 — A,(p) = 0 cross the 7-axis or

11 — A(x)X(x) (o0sf \ — A{x)X{x)

x‘-t v /2 2 r ft)0C £/g in(6.8-6)


By com bining term s we can write

1 — A{x)X(x) = 2 +x l i(x) x /0(x)I 0(x) /^ x )

Table 6.8-1 lists several values of , / l — A(x)X(x).

Table 6.8-1

X J 1 - A(x)X(x)0.0 0.001.0 0.462.0 0.733.0 0.866.0 0.95

10.0 0.9700 1.00

By using the large-x expansions for I ^ x ) and / 0(x), one can show that, to with.:’ 5 % for x > 3,

J 1 - A(x)X(x) « 1 -

Therefore, since the exact frequency of squegging is seldom of great im portance and since squegging is m ost likely for x > 3, it seems reasonable to rewrite Eq. (6.8 -6) as

o)0(l — 5/16x)

v / 2Qr (a)0C £/gïIJ

For the previous example, where QT = 50, co0C£/g in

(6.8-7)

20. and x = 6.4. we findthat

to,,“'sq - <u0/47.5.In actual circuits the values to be used for C and for CE are usually obvious from

inspection. For example, in the Colpitts-type circuit C = C l C 2/ (C1 + C 2) and C £ = C 1 + C 2. I f a bypass capacitor C B is placed from the base to ground (as shown in Fig. 5.4-1), then CB/( 1 — a) m ust be added in series with CE to determ ine the total coupling, capacitor.

A similar technique, starting with the results of Fig. 6.8-3 and evaluating the four partial derivatives, either analytically or experimentally, will lead to a sim ilar set of results for vacuum -tubef or FET oscillators. In all cases squegging can be stopped,

fT w o sources that consider squegging in the vacuum tube case in some detail are: (1) W. R. M acLean, “C riteria for the Amplitude Stability of a Power O scillator,” Proc. IRE , 42, pp. 1784- 1791 (Dec. 1954); and (2) B. G. Dammers, J. H aantjej, J. O tte, and H. Van Suchtelen, “ Squegging Oscillators," in Application of the Electronic Valve in Radio Receivers and Amplifiers, Book IV. N. U. Philip’s G loeilampenfabrieken, Eindhoven, N etherlands (1950), C hapter II E, pp. 227-250. In the vacuum tube case, it is not possible to arrive at nearly such a com pact nor universal result as we have obtained here for the transistor.

6 .9 BRIDGE OSCILLATORS 261

if it occurs, by some com bination ofa) re d u c in g th e size o f th e c o u p lin g c a p a c i to r (if o th e r re s tr ic tio n s w ill a llo w

th is re d u c tio n ) ,b) ra is in g th e tu n e d -c irc u it Q, a n dc) reducing the active-element driving voltage.

6.9 BRIDGE OSCILLATORS

Bridge oscillators utilize the transfer function of a bridge network to produce a pair of high-Q right-half plane complex zeros. As was pointed out in Section 6.3, such a set of zeros contributes directly to the indirect frequency stability of the circuit.

In this section two basic bridge circuits are discussed. The first is the fixed- frequency M eacham bridge, in which the already high Q of a crystal resonator is multiplied by several additional orders of m agnitude to yield a very stable circuit. The second is the wide-range variable-frequency circuit known as the Wien bridge oscillator, f In the Wien bridge oscillator the aim is not to achieve extreme frequency stability, but rather to achieve the stability and waveform purity of an L C circuit of reasonable Q while retaining the wideband tuning capability of RC networks.]:

A b rid g e o sc i lla to r co n s is ts o f th re e m a in e le m e n ts :a) the bridge circuit,b) a lin e a r a m p lif ie r , a n dc) a b r id g e -b a la n c in g c ircu it o r e lem en t.

As w e shall see. th e d e s ira b le p ro p e r tie s o f th e b rid g e o sc illa to rs a re o b ta in e d o n ly if th e b rid g e is o p e ra te d a s c lose to b a la n c e as is possib le . S ince th e am p lif ie r ga in tim es th e b rid g e e r ro r sig n a l m u s t be a c o n s ta n t if s tab le o p e ra t io n is to be ach iev ed , a n d s ince th e am p lif ie r g a in A c a n n o t be a ssu m ed to be a c o n s ta n t, we ach iev e a m p litu d e c o n tro l by s a m p lin g th e o u tp u t a m p litu d e a n d u sin g th is sig n a l to p ro v id e ex ac tly th e n ecessa ry b rid g e u n b a la n c e .

F igure 6.9 1 illustrates both bridge circuits under discussion. With vacuum -tube circuitry the self-balancing effect is often achieved with a positive-tem perature- coefficient resistor (a light bulb) in the upper half of the negative feedback arm (element R }). In addition to being som ewhat sensitive to am bient tem perature, such elements often require m ore power for their operation than the total power available from a small sem iconductor operational am plifier; hence in sem iconductor circuitry it is usually more reasonable to obtain the self-balancing effect with an FET, a light- sensitive resistor, or perhaps a diode acting as a controlled resistor.

While we could operate a bridge oscillator with a simple differential-pair circuit as an amplifier, the desirable properties of the circuit require high voltage gains;

+ The literature is full of RC feedback oscillators that are misnamed Wien bridge circu its; these circuits do not contain any bridge structure and hence do not have the properties described here. I Wien bridge oscillators are comm on with 10/1 variations in frequency per dial turn and overall coverage of 10b/ l . say from 2 Hz to 2 MHz. W ith an RC network a 10/1 variation of R or C gives a 10/1 variation of frequency, while in the LC case it yields only a 3.16/1 variation. In addition, for frequencies below several kilohertz, linear inductors become both unwieldy and expensive.


Model for

(a) (b)

Fig. 6.9-1 (a) Meacham bridge, (b) Wien bridge.

hence we assume an ideal operational amplifier with a sufficiently high-im pedance differential input that bridge loading may be neglected and with a sufficiently low ou tpu t im pedance tha t the bridge does not load the amplifier. So long as we restrict the ou tput signal level to several volts and the frequency range to below, say, 10 M Hz, then such amplifiers are widely available at relatively low prices. In practical circuits, particularly at low frequencies where im pedance levels are high, one often uses an FET pair a t the input to the differential amplifier to further reduce the effect of loading.

M eacham Bridge

If there is zero phase shift through the amplifier, then the M eacham bridge oscillator of Fig. 6 .9 -la operates extremely close to the series resonant frequency of the crystal, and thus A L(jca0) is given by

A L(jco0) = + A \ —--------------—— — I. (6.9-1)l \ j o ; ^ + R 3 R 2 + rJ

W e define N = R 2/r, M = R 3/r, and 5 = N M — R J r . If (N + 1 )M » \d\, then stable oscillations occur [AL( jw0) = 1] when

* = (6.9-2)A

[N ote: (5 = 0 corresponds to perfect bridge balance, while a slightly negative value of 5 is obtained when u^f) is a small negative voltage relative to its defined polarity.]

For example, if the bridge arm s are nearly equal, so tha t M = N = 1, then d = —A/A\ hence if A > 100, certainly |<5| « (N + 1 )M and ou r assum ption is justified. It will be the jo b of ou r bridge-balancing circuitry to adjust 5 so that Eq. (6.9-2) is always satisfied regardless of variations in A.

6.9 BRIDGE OSCILLATORS 263

T h e c o m p le te exp ress io n for the lo o p gain o f the M e a c h a m bridge is given by

- w = + „ R,A "I ~ y + i . « . « IR i + R 3 p + 2 a 2p + m0

where

1 ¡R 2R 3 1 , 1i = r ’ ar,d a 2 = , 7 (r + i?2 ).

J ° - ~ r l c ~ " ° ’ = 2 l | ~ r T " ’ ) ' a ' l u " 2 “ 2 V " 1 X 2 1V

T h e p ole-zero d ia g ra m o f A L(p) is given in Fig. 6 .1 -7 (b ) . [T h e reader should c on v in ce h im se lf tha t the e q u a t io n for A min in the figure is identical to Eq . (6.9 2).] It is ap p aren t that the effective Q o f the zeros o f A L(p) has the form

Q--= I t2a i

while the Q o f the poles is given by

Q = —2 a 2

I f we define the Q o f the crysta ls as Q L = to'0L / r , then we o b ta in the fo llow ing in te rest ing r a t i o s :

& _ a 2 _ R i ( R 2 + r) _ ( N M - S ) ( N + 1) ^ M N ( N + 1)

Q P R 2R i ~ r R x <5 ~ S

N Afor A > 100,

N + 1

w hich is c o m p le te ly ’independent o f the purely resistive side o f the bridge, and

Q, N M - ô N M N Ab) ■— = --------- = ---------------- % ---------- = — -T . (6 .9 -4 )

Q, R 2R 3 - r R j <5 ô ( N 1 1'2

C learly , for a given crystal, the greater the ra t io Q J Q i , the greater the indirect frequency stabil ity (see Eq . 6 .3 -1 5 ) .

T h e ra t io QJQL is a function o f N which is g eo m etr ica lly sym m etr ica l a b o u t N = 1, w here it o b ta in s its m a x im u m value o f ,4/4. By m a k in g N = 4 or J , we reduce Q:/Q, to 6 4 of its m a x im u m v a lu e ; by m a k in g N = 10 or ,‘0 , we reduce the ra t io to 33 o f its m a x im u m value.

I f we wish to vary N from unity , we suspect th a t it would be desirab le to increase it, s ince this w ould raise the bridge im p ed an ce level ; a higher im p ed ance level would in turn m a k e the bridge easier to drive, reduce the crystal cu rren t , and reduce the crysta l dissipation.

F r o m Eq . ( 6 .9 - 4 ) we see th a t for N = 1 a 6 0 d B am plifier gain can lead to a frequency stabil ity e n h a n ce m en t 2 5 0 times that o b ta in a b le with the crystal alone. O f course , to benefit from this increased indirect frequency stabil ity we m ust ensure


a high degree of direct frequency stability; i.e., we must m ake sure that a>0 of the crystal rem ains stable. This is usually accomplished by placing the crystal in an oven which m aintains its tem perature constant within a few degrees Celsius.

A beautiful feature of the M eacham and Wien bridge oscillators is that the same factors which lead to a large indirect frequency stability factor also cause the output oscillation am plitude V, to be insensitive to variations in the amplifier gain A ; that is, they cause the am plitude sensitivity function SA to be small (cf. Section 6.4). Specifically,

(N + l )2S A =

N A(6.9-5)

hence is m inimized with N = 1 and A as large as possible.To dem onstrate the validity of Eq. (6.9-5), we assume that R , is the bridge-

balancing element which decreases with increasing output voltage am plitude V,, as shown in Fig. 6.9-2. A lthough a hyperbolic relationship is shown between Vt and R !, the same results may be obtained for any sim ilar relationship between V1 and R t . However, it is apparen t tha t such a voltage resistance relationship does lead to am plitude stability. If Vt should decrease for some reason, R ! increases and in turn causes A L(ja>0) [given by Eq. 6.9-1)] to increase, thus restoring the ou tput oscillation amplitude.

Fig. 6.9-2 Relationship between the bridge balancing resistor R , and the output voltage amplitude V,.

The am plitude Flss a t which K, stabilizes may be Found directly from the curve of Fig. 6.9-2. If we assume tha t A is large (which it m ust be for a well-designed circuit), then S is small, since Eq. (6.9-2) m ust be satisfied in the steady state. A small value of 6 implies bridge balance or, equivalently, R lss « R 2R 3/r; hence the value of Vt which results in R t m R 2R 3/r on the curve of Fig. 6.9-2 is îss- Clearly, if we desire a specific ou tput am plitude, we m ust choose our voltage-controlled resistance to be equal to R 2R 3/r for the desired o iltput am plitude. Or, equivalently, for a given

i - Kj characteristic, R 3 m ust be chosen such th a t R 3 = R J N for the desired ou tpu t

6 .9 BRIDGE OSCILLATORS 265

am plitude. For the optim um case of N = 1, R 3 ~ Riss- From the above reasoning it is apparen t that, if A is sufficiently large to keep the bridge balanced and R 2, R 3,r,and the characteristic are stable, then the ou tpu t am plitude is quite stable.

We now evaluate S,, quantitatively by noting that

dVl A _ A 8Vi d R x 8SA ~ dA Vx V, dRt dS d A '

Since V, = 1 / ( R , ,

d V l __ 1

d R [ ~

In addition since R , = r ( N M - ¿),

SRt

and, from Eq. (6.9-2),

85 = “ r ;

86 _ (N + 1 )2M dA A 2

Consequently,

„ (N + 1 )2M r (N + l )2S , = ----------------- w -------------- (for large A).

A A R i A N B

Since ( drops out of the expression for Sx , we see that the exact form of the K,-/?! characteristic is not critical so long as R lss produces the desired îss-

Two practical M eacham bridge oscillator circuits in which Ri is controlled inversely with the ou tpu t am plitude are shown in Figs. 6.9-3 and 6.9-4. In Fig. 6.9 2 the output voltage drives a small light source which illuminates a light-sensitive

Fig. 6 .9 -3 M eacham bridge controlled by light-sensitive resistor.


Fig. 6 .9-4 M eacham bridge controlled by enhancem ent mode insulated-gate FET.

sem iconductor resistor. As the ou tput voltage and, in turn, the power to the light source increase, the sem iconductor resistance decreases. M any sealed units containing both the light source and the sem iconductor resistor are available com mercially with nom inal resistance values (with no illum ination) from several hundreds of ohm s to several hundreds of kilohms. The Raytheon C orporation produces a line of such elements under the trade nam e of Raysistors.t

In Fig. 6 .9-4 the ou tput voltage drives a peak envelope detector of the form discussed in Section 10.5 to develop a dc voltage proportional to . This dc voltage in turn inversely controls the drain-source resistance of the enhancem ent m ode insulated-gate FET. Since the drain-source resistance in this circuit rem ains linear only for drain-source voltages under 200 mV, there will be some distortion in the voltage across the FE T if Vx is designed to stabilize at too high a level. Fortunately, however, the M eacham and Wien bridge circuits possess a large am ount of negative feedback (as the interested reader may calculate) at the harm onics of oj0 ; therefore, very little of the harm onic com ponents appearing across the FET appear at the output, especially if A is large. Thus, for large A, nonlinearities introduced by R x do not appreciably destroy the sinusoidal purity of the output.

A lthough m axim um indirect frequency stability and m inimum am plitude sensitivity both dem and that N = R 2/r = 1, the permissible crystal dissipation sometimes requires us to select an N different from 1 and m ake up the deficiency in SF and SA with increased amplifier gain.

Excessive crystal current may cause the crystal to fracture. F lat statem ents about the size of a destructive current are difficult to make, since it may exceed 25 mA for a physically robust low-frequency unit and it may be less than 1 mA for a fragile

t A possible disadvantage of the Raysistor as a control element is that it requires a relatively large am ount of input current and also tends to be som ewhat expensive.

6.9 BRIDGK OSCILLATORS 267

high-frequency unit. E ven w ithout a p p r o a c h in g the level o f d estru ct io n , o ne norm ally

wishes to keep the crystal d issip ation below one m illiw att so as to m inim ize internal heating. S u ch heat ing usually causes shifts in frequency o f the crystal resonances. T h e a m o u n t and d irec t io n s o f such shifts will be a function o f the " c u t " o f the crystal.

F o r o p e ra t io n at series reso nance , the crystal pow er is 1't 2 r( l + .V)2 ; hence, for r = 100 Q and a m a x im u m a l lo w a b le crysta l pow er o f 1 m W . we have

K1 + N

< 0.445 (6.9 7)

or

V\ , V

1 1.252 3.54 8.0

Sm a lle r values o f r will p lace eve’n m o re severe l im ita t io ns on N.

Wien Bridge

F o r the W ien bridge circuit o f F ig. 6.9 1(b) the lo o p gain is given by

2 I 1 1 \ 1n + p\ ---- + ---------------------- ;—A/ [p) R , A ‘ \R4C 4 R 2C 2 R . R . C , r 4r 2c 4c 2

1 1 1n I ----------------- I _j_

‘ \r 4c 4 r 2 c 2 r 4c 2 ) r 4r 2c 4 c 2

(6.9 8)

T h e p ole-zero d ia g ra m o f A, ( p ) is show n in Fig. 6.1 —7(a).T h o u g h there m ay be cases where the R C a rm s should not be identical, it is

usually d esirable to im p o se this restrict ion . W ith the special case where R 2C 2 = R 4 C 4 = R 4C 2 = I/cOq and the d efinition o f R 3 as R , = (2 4 - S ) R t . Eq. (6.9 8) reduces to

A p 2 — pd(o’0 + wq¿ l(p ) = 3 + Ô p2 + 3pcj'0 + uï0

T h is o sc i l la to r has j - a x i s poles at u>0 = a>'0 when

A — — + 3 d

o r Ô =A - 3

T h is value for 6 leads to a “ q u e ” for the c o m p le x zeros of

Q z = 5 ~ 9 * 9

(6.9 9)

(6.9 10)

(6.9 Hi

when A » 3 : hence, to ach iev e S F % 2 Q . = 100, we m u st have A = 450.


In this case the limits on the allowable im pedance levels for the bridge are set by the m axim um desirable capacitance and the problem of the amplifier input im pedance loading the bridge. These problem s become extreme at low frequencies; at high frequencies, amplifier ou tpu t im pedance and input shunt capacitance may be limiting factors.

In general, the bridge input im pedance can be kept to the order of several hundreds of o h m s; hence the ordinary operational amplifier should drive it w ithout difficulty. Any of the control circuits discussed for the M eacham bridge case is an appropriate replacem ent for R i in the W ien bridge.

6.10 T H E O N E -PO R T A PPR O A C H T O O SCILLA TO R S

In this section we examine another approach to the analysis and design of sine-wave oscillators. C onsider the four circuits in Fig. 6.10-1. As we shall see, all of them have the potential to produce nearly sine-wave oscillations. They are representative of a class of circuits— or of a way of looking at circuits— in which the amplifying device and its nonlinearition are lum ped into a single, reactance-free, controlled source.

C2- ) h

---------------1> i >--------------- 1

> * 4

> r 2 ;^ c

— 1 m— i >----- H ► ------(b)

<c) (d)

Fig. 6.10-1 Four “one-port” oscillators.

) f ( 0

All of these circuits contain a filter, an ac energy source, and some form of positive feedback; hence, if the loop gain is high enough, all o f them are potential oscillators.

In this section we are interested no t in proving that oscillation is possible, o r in determ ining the exact frequency or waveshape of oscillation, but in seeing how possible nonlinearities in the controlled source may limit the am plitude of the oscillations.

F o r the purposes o f our basic explanation it is im m aterial whether the nonlinear device is constructed of separate elements such as F E T ’s or transistors or w hether it is a single one-port device such as a tunnel diode. F o r example, the voltage- controlled current source in Fig. 6.10-1 (d) m ight be a single bipolar or F E T transistor,

6 . 1 0 THE ONE-PORT APPROACH TO OSCILLATORS 269

Fig. 6.10-2 Possible im plem entation of Fig. 6.10-1 (a).

while the current-controlled voltage source in Fig. 6.10- 1(a) might be constructed of a current-sensing resistor and an operational amplifier as illustrated in Fig. 6.10-2.

Figure 6.10-3 shows one of many ways to construct a circuit of the type shown in Fig. 6.10-1 (b).

It is not our purpose at this point to develop the nonlinear characteristics of the various active “devices,” but rather to calculate the oscillator perform ance given a calculated or m easured “device” characteristic. ’

As a first example consider the circuit of Fig. 6.10-2, where the com bination of operational amplifier and r2 is assum ed to have a characteristic of the form shown in Fig. 6.10-4.

2 7 0 SINUSOIDAL OSCILLATORS 6 .1 0

Fig. 6.10 4 Assumed characteristic for device in Fig. 6.10-1 (a) or Fig. 6.10- 2.

First consider the small-signal situation, where the controlled source always operates in its linear region, |i| < 1 mA. In this region / ( / ) may be replaced by a linear voltage source, R Di. t W ith this source the loop gain A L(p) becomes

R dPC= A L(p),

p L C + prC + 1

where r = r, + r2.W hen R d = r, the circuit has j-axis poles a t oj0 = +1 L C ; hence sinusoidal

oscillations build up at this frequency. W hen R D < r, the poles are in the left-half plane and, while transients may cause ringing, free oscillations do not occur. For R D > r the poles are in the right-half plane and the oscillations grow. In fact, in Fig. 6.10- 1(a), if we merely replace the g en e ra to r/( i) by the negative resistor — R D (which is permissible, since the voltage source is linearly related to the current through it), we can perceive the previous results in an even m ore straightforw ard manner. W hen the net circuit series resistance is positive, the oscillations decay; when this resistance is negative, they grow. (Since for |/| < 1 mA any increase in / causes an increase in /( i) , the equivalent resistance - R D must indeed be negative.)

If initially R D = Ar2 > + r 2, then certainly oscillations build up until eventually at least the peaks of / exceed the linear region in Fig. 6.10-4. For the given characteristic, when |/| > 1 mA the / ( / ) generator ceases to be a negative resistance and becomes a 6 V battery whose polarity depends on i. This action causes limiting in the buildup of i, and eventually an equilibrium condition is reached. It is this equilibrium state that we shall now find.

In general, when operation of the circuit shown in Fig. 6.10-2 extends beyond the linear region,/ ( / ) becomes a distorted sinusoidal voltage which may be written in the form

/ ( ') = F0 + F, cos u)0t + V2 cos w 0t + • • • + .

However, if the Q T of the tuned circuit is high enough that even the distorted voltage /( / ) still produces an alm ost sinusoidal current through r, then we may assume sinusoidal input currents of the form /*, cos co0f for several values for I , and utilize

t In the circuit o f Fig. 6 .1 0 2. RD = Ar2.

6.10 THE ONE-PORT APPROACH TO OSCILLATORS 271

Pulses to be subtracted

6yv[i2oj~—| 120° f - \ /

- 1 2 V

Fig. 6.10-5 Sinusoidal current drive.

Fig. 4.2 -3 to calculate the harm onics in the subtracted sine-wave tips in order to calculate the ou tpu t fundam ental voltage Vt . Figure 6.10-5 illustrates the technique.

The fundam ental ou tpu t voltage equals the extrapolated linear transfer value (neglecting the break points) minus the fundam ental com ponents in the sine-wave tip pulses that m ust be removed from the extrapolated value to obtain the actual characteristic. For the case shown.

By applying this technique several times we obtain a plot of F, vs. / , for the device. Now since the passive circuit also imposes a relationship between V, and / , of the form / j = VL/r, we see that, as in previous sections, the operating am plitude is fixed by the sim ultaneous satisfaction of the active device and of the passive circuit requirements. Figure 6.10-6 illustrates several ways to present the results o f this calculation.

The (4/ 7t) x 6 limit of Vl in Fig. 6.10-6(a) is the limit of the fundam ental value of a square wave (see Eq. 4.2-2). W ith r = 2 kQ, this circuit will stabilize at about3.8 mA = 11 , for which Vt = I¡r = 7.6 V. Now for this am plitude we can return to Fig. 4.2 -3 and calculate the am plitudes of the first several harm onics in order to see how high Q m ust be for the assum ption of a sinusoidal current to be valid in the first place. In this sym m etric case, the second-harm onic voltage is zero, while the th ird-harm onic voltage is 2.4 V ; hence, to reduce the third-harm onic current to 1 % of the fundam ental current, we m ust have

K, = [12 - 6 x 0.39 - 6 x 0.39] V = 7.32 V,

where 0.39 is the function of 2 6 kQ, then no intersection occurs in Fig. 6.10-6 and operation ceases. In this case the net circuit resistance is always positive.

2 7 2 SINUSOIDAL OSCILLATORS 6 .1 0

/i, mA /,, mA(a) (b)

Fig. 6.10-6 (a) V, vs. 1, (from Fig. 6.10-5). (b) V,/I, (from Fig. 6.10-5).

As we pointed out previously, Fig. 6.10-6(b) allows us to visualize the nonlinear controlled source as a negative resistor {remember that i flows up th ro u g h /(i) hence the generator is supplying rather than consum ing ac power]. F rom this viewpoint the circuit will stabilize when the passive circuit resistance just balances the "average” negative resistance — R D = V J I , from the active circuit. N ote that since all “negative-resistance” oscillators may be approached from the controlled-source viewpoint, there is no need to isolate them as a special case.

In this approach the feedback loop employed previously does not always appear explicitly; hence the previously employed criteria for oscillation (the loop gain being one) have been replaced by the criterion of zero net resistance [or j = axis poles— which is, of course, exactly what was achieved when A L(j(o0) reached unity].

A num ber of questions may arise at this point. W hat does one do when Q is not high? W hat sort of transients occur during the buildup of oscillations? W hat is the actual frequency of oscillation? (It will not be 1 /^ /Z c in Fig. 6.10-2, for example.) How does one deal with the case of sm oothly varying device characteristics?

W hen the Q is not high enough to remove essentially all the harm onics, two approaches are possible. O ne is a perturbation technique whereby we initially assume a sine-wave drive and then find an approxim ate level of operation. Then, near that level, we determ ine the first several harm onics assum ing a sine-wave drive, and after filtering these harm onics through the circuit we use the approxim ation to the actual ou tpu t waveshape as a new driving function. W ith piecewise-linear characteristics this approach is not appealing.

F o r the sm oothly Varying case, we may approxim ate the function over what is estim ated to be the useful region by as low an order of polynom ial as appears plausible, and proceed. The iterative technique is m ore reasonable in this case.

A nother approach which attacks the low-Q problem directly and which also yields results for transient buildups is the phase plane m ethod described in Section 6.11. Section 6.12 then considers the effect o f harm onic com ponents on the operating frequency and thus relates waveshape distortion to variations in operating frequency.

6.11 THE PHASE PLANE APPROACH 273

Though the literature is full of presentations of the “ phase p lane” m ethod, m ost of these general approaches tu rn out not to be easily applicable to practical oscillator problem s of the type presented in Section 6.10.

This section presents a modified version of the Lienard m ethod that leads to rapid solution of this type of problem. The specific m ethod presented here is applicable to all second-order systems with nonlinear damping, that is, to systems in which the small-signal transfer function from the controlled source to the controlling variable has a single zero at the origin and only two poles (real or complex). Though the m ethod as presented will not work directly for such circuits as the M eacham bridge, it is directly applicable to all the circuits of the previous section.

C hapter 2 of H ayash if outlines m ore sophisticated graphical m ethods that are applicable to m ore com plicated circuits.

The m ethod presented here provides a graphical solution for the initial transient buildup and for the fihal operating waveshape of the oscillator. Its use implies that the reactances of the controlled-source (if any) have been separated out and combined with the circuit reactances. It also assumes that the circuit's dc (?-point is known and that it does not shift appreciably between the no-oscillation and full-oscillation cases. Once the ou tput waveshape has been determ ined, it will be clear w hether an average value of current or voltage does exist. If so, and if such an average value will have caused a £>-point shift (this will be a function of the bias circuitry), then the result is only an approxim ation. In such cases it is possible to pursue a perturbation approach through several steps tow ard a better estim ate of the result. This is seldom undertaken, since the main purpose of the phase plane m ethod is to provide rapid insight into the type of d istortion likely to occur in oscillator circuits with “ low-(>" filters.

The initial estim ate, even if not exact, is usually sufficient for design purposes. The m ethod handles both piecewise-linear and gradual nonlinearities with equal ease. In circuits with highly selective filters, however, one of the analytic m ethods described previously will give both quicker and m ore accurate results.

The result to be achieved is a graphical construction of a contour in a “phase p lane” that will indicate w hether oscillations are possible (a small-signal calculation will, of course, indicate this fact also) and, if so, how they build up from any particular set of initial conditions. A further construction derived from the first one gives a reasonable estim ate of the circuit’s waveshape and fundam ental frequency.

In order to achieve these aims, a certain set of algebraic and trigonom etric m anipulations m ust be presented. The new reader would probably do well to skim through these once to see the conclusion and then return to unravel the details.

Basically w hat we are trying to do is obtain a solution of a nonlinear differential equation of the form

d2x(t) dx(t) d~ w + “- ¡ r + w , ) = 161

t Chihiro Hayashi, Nonlinear Oscillations in Physical Systems. McGraw-Hill, New York (1964).

6.11 THE PHASE PLANE APPROACH


where x may be a voltage or a current a n d /(x ) is some known single-valued function of x. The m ethod involves the m anipulation of the equation so as to remove time as an explicit variable and so as to obtain a relationship for dy/dx or dy/drj, where t] is some linear m ultiple of x and y = (b/c)J'x(t) dr or equivalently x = (c/b)(dy/dt).

In obtaining Eq. (6.11-1) from a specific circuit, it is often convenient from the viewpoint of m anipulation to write the small-signal transfer function from the controlled-source output, say X 2(p), to the controlling variable, say Xx(p), and then, after m ultiplying out, to convert this to its equivalent large-signal nonlinear differential equation. A subsequent illustrative example will m ake this point clear. This transfer function should have the form

where X ,(p) = if[x(f)] and X 2(p) = i f [ / ( x ) ] . Taking the inverse Laplace transform we immediately ob tain Eq. (6.11-1).

Once Eq. (6.11-1) is obtained, we integrate once and divide through by c, the multiplier o f/(x ) , to obtain

term has a unity m ultiplier and so that t] and y have the same dimensions. We now write the modified version of Eq. (6.11-6) as a solution for dy/drj,

* i (p ) = x 2(p )pc

(6.11- 2)p2 + ap + b

or

p2X l(p) + ap X ¡(p) + b X x(p) = c p X 2(p),

(6.11-3)

We now remove the integral term by defining a new variable,

(6.11-4)

The dx/dt term is now removed by noting that

dx dx dy dx bxdt dy dt dy c

(6.11-5)

therefore.

(6.11- 6)

Now unless b/c2 = 1, we define a new variable r\2 = (bjc2)x2 so that the slope

dydr¡ y - [ / (* ) - {a/c)x}'

(6.11-7)

W ith this m ethod it is convenient always to factor out a minus sign and to place it as shown in Eq. (6.11-7).


Now Eq. (6.11-7) presents the increm ental change in y for an increm ental change in t] in terms of y, x, and f (x) . Since rj is a linear function of x and y is an integral of x, we have as variables the controlling variable (or a linear m ultiple of it) and the integral of this variable (or a m ultiple of it). The two final variables are always connected via an integration or a differentiation and always have identical units.

At this point we lay out a set of axes with y as the ordinate and tj as the abscissa and with equal increm ents in each direction. The coordinate system is known as the “ phase p lane.” Now t] and y in Eq. (6 .1 1 - 7 ) are merely distances in this plane. If we assume values for x (and hence t]), we may find a corresponding value for j \ x ) - {a/c)x treated as a function of r]. This function may be plotted in the y vs. tj

plane. Figure 6 . 1 1 - 1 illustrates Eq. (6 .1 1 -7 ) .

From any arb itrary point (»71 y i ) in the phase plane, say in Fig. 6.11-1, we d rop a perpendicular until it crosses the / (x) — (a/c)x curve. At this point we travel horizontally until we reach the _y-axis. From Fig. 6.11-1 it is clear that

tan _ *1 iy i - LA*i) - (a/c)x,]

= dy i d*i i '

However, tan (180° - a) = - t a n a ; hence

^ - = tan (180° - a). dtjl

This means that the gradient of y with respect to t) at any point in the pha*’ plane is given by the perpendicular to the radius from the equivalent point Q x to the point P, in question. Thus from any arb itrary starting point we may trace out a path by connecting successive arcs of circles swung about successive points on the y-axis.

Several points are w orth noting before an illustrative example is undertaken. Since we are w orking around an assumed g-po in t, the / (x) — (a/c)x curve can

always be norm alized so that it passes through the origin. Furtherm ore, unless it


passes through the origin with an upw ard slope to the right, tha t is, u n le ss /(x ) > (a/c)x for small positive values of x, there is no sense in proceeding, because oscillations are no t possible [ / (x) > (a/c)x for small positive values o f x implies th a t the loop gain exceeds unity and tha t the right-half p lane poles exist in the system].

Because the construction is bo th Q -point and circuit sensitive, any change in the circuit o r the Q-point requires a new construction.

As an illustrative example consider the circuit of Fig. 6 .10-l(b) with R 2 = R* and C 2 = C 4 and th e /( i) vs. i characteristic as shown in Fig. 6.11-2.

-0 .2 5 mA 0 +0.25 mA

Fig. 6.11-2 Controlled-current-source characteristic.

From the circuit,

* i(p ) h ip) Y,{p)* 2 ip) hip) y 2 ( p ) + y4(p)

(6.11- 8)

F or the specific case where R 2 = R 4, C 2 = C 4, and co0 = l / R 2C 2, the transfer function reduces to

(61I_9)h ip) P + 3 co0p + col

hence a = 3a>0, b = col, and c = to0.Since Eq. (6.11-9) is of the form of Eq. (6.11-2) and since b/c2 = ojqjcog = 1, we

may write dy/di directly in the form of Eq. (6.11-7):

T ------------ r7T)— v rdi y - [/(O - 3i]

We now assume variations of i about the Q-point and calculate [ /( i) — 3i] vs. i as plotted in Fig. 6.11-3.

It is intuitively obvious tha t this circuit should osc illa te : from the adm ittance transfer function a t p — jco0 it follows tha t the netw ork loss is 5 , while the small- signal current gain around the Q-point is 4 ; hence the loop gain exceeds unity.

Figure 6.11-3 also indicates the closed trajectory of the steady-state operating path. This pa th was sketched ou t with a com pass in abou t three m inutes from the arb itrary starting point of y = 0,1 = 0.50 mA.


<E

Arbitrarystartingpoint

I, mA-0 .5 0

Fig. 6 .1 1 3 Phase plane trajectory for circuit of Fig. 6.10—1(b) with /(¿) of Fig. 6.11-2.

Once we have this phase plot, the question arises as to what we can say about the waveshape of / o r y vs. t. Since y and i are connected via an integration or differentiation, if one of them is sinusoidal then the o ther m ust be cosinusoidal. Therefore, since the scales are identical in bo th directions, it follows from what we know of Lissajous figures, tha t if i were sinusoidal then the path would be circular. Therefore, the extent to which the actual trajectory departs from a circle gives us an indication of the nonsinusoidalness of the waveshape.

Before we look for further waveshape inform ation we should consider the possible differences am ong the various available signals. Since y = oj0J idx, then y is proportional to the voltage across C 4. Certainly the effect of the integration (a low-pass operation) should reduce the harm onic content in y with respect to that in Therefore, grounding the capacitor C 4 and taking the ou tpu t across it should lead to a “b etter” waveshape than grounding R 4 and taking the ou tput across it.

W hatever waveshape we desire, if we had the slope with respect to time for every point (as well as the corresponding value at that point), then we would be able to m ake a reasonable sketch of the waveshape with respect to time.

From our original equations we may write or derive both dy/dt and di /dt :

These equations m ean that the slope of y with respect to time is always p ro p o rtional to the corresponding value of i or to the horizontal distance from the vertical

j t = - u 0{y ~ [ / ( ') - 3/]}.


axis to the point on the trajectory. At the same time, the slope of i with respect to time is p roportional to the negative of the vertical distance between the trajectory and the / ( / ) — 3/ characteristic. W hen the trajectory crosses th e / ( i ) — 3i characteristic, di/dt goes through zero; when the trajectory crosses the >’-axis, dy/dt goes to zero.

If one is attem pting to plot y(t), then the easiest approach would seem to be to draw a circle with center at the origin and radius equal to ymax. Then sketch out carefully on a separate sheet o f graph paper about l j cycles of a cosine wave of arb itrary period. So long as the trajectory lies on the circle, then y(t) is following the sine wave. As the trajectory falls away from the circle, the slope of y(t) decreases from the slope of the cosine wave. If the slope of the cosine as it passes through zero is taken as the standard , then the slope of y(r) at any point is merely ipomi/ymax times the standard slope.

The slope-tracking approach will break dow n near the negative peak in y. However, even if the negative m axim um is different in am plitude from the positive maximum, the trajectory in the vicinity of the peak is still nearly an arc of a c irc le ; hence the negative tip of y(t) will be sinusoidal in shape. W ith this inform ation it is a straightforw ard task to sketch ou t y(t). Since vC4(t) = R 4y(t), this sketch provides the

outpu t voltage w ithin a constant. F igure 6.11-4 com pares y(t) with a standard sine wave of the same peak am plitude and of period a>0. N ote that, in spite of the reasonably strong overdrive, the waveshape across C4 is a good approxim ation of a sine wave, but a sine wave of a frequency at approxim ately 80 % of co0.

As the overdrive is reduced, the slope o f/ ( / ) — 3i through the origin is reduced until in the limit it becomes horizontal. At this point i would be sinusoidal with a

6.12 THE DISTORTION-OPERATING FREQUENCY RELATIONSHIP 279

peak am plitude equal to the breakpoint value of / (assuming a symmetrical characteristic is m aintained as the slope is reduced). Any further reduction in the slope will cause the oscillations to cease.

As the overdrive is reduced, not only do the waveshapes become m ore sinusoidal but their frequencies increase tow ard w 0. In the next section we shall see how the frequency of an oscillator is affected by d istortion in its waveshape and hence why the overdrive (with its large distortion terms) should cause a frequency depression.

6.12 T H E D IST O R T IO N -O PE R A T IN G FREQ U EN CY R E L A T IO N SH IP!

Almost any practical sinusoidal o r nearly sinusoidal oscillator may be considered in a form similar to one of the circuits shown in Fig. 6.10-1. T hat is, it may be considered as a com bination of a netw ork and a controlled source. In general, we have been neglecting active circuit reactances; however, even if they occur it is usually possible to segregate them and then to treat them as part of the network. Thus if we wish we can consider all of our oscillations as a network plus a nonreactive controlled source.

W hatever the com bination of variables,J a nonreactive active element driven by a periodic waveshape (we assume steady-state operation, but m ake no initial assum ptions about the waveshape other than that it is periodic) must have zero net area when integrated around a com plete cycle. Figure 6.12-1 illustrates this property for the general controlling variable q and controlled v a r ia b le /^ ) .

If the active device were reactive, then for any given am plitude swing for q the result of the hysteresis in the characteristic would be a constant area and our approach would still be valid although slightly m ore complicated. Therefore, as a simplification we separate the device reactances initially and assume no device hysteresis.

+ The relationship between distortion and operating frequency of an oscillator was first presented by Groszkowski in the 1920's. His more recent book. Frequency o f Self Oscillations (Macmillan, New York. 1964), summarizes his earlier work and presents results for more complicated cases than are presented here.J Controlled-source possibilities include current-controlled voltage or current sources and voltage-controlled voltage or current sources; network functions include impedances, admittances, and voltage and current ratios.

/(<?)

<j> q d l f i q )] 0

Fig. 6.12-1 General nonreactive controlled-source characteristic.

280 SINUSOIDAL OSCILLATORS 6 .1 2

Now instead of taking the integration from qmin to qmax and back, we could take it over a com plete period of the input cycle, in which case we would find

Furtherm ore, since q and hence f ( q ) are periodic variations about a Q-point, each of them can be expanded in a Fourier series starting from a fundam ental com ponent and going as high as is necessary:

If Eqs. (6.12-3) and (6.12-4) are substituted in Eq. (6.12-1) and the integration is performed, then all term s except those containing sin (ij/n — </>m), where m = n, vanish and we obtain

E quation (6.12-5) indicates an active-device-imposed relationship between the

The passive circuit also imposes a different relationship between these quantities. These two relationships m ust coexist. In general, this m eans that the operating frequency of the oscillator m ust shift until the two relationships can be m utually satisfied.

If the transfer function between f ( q ) and q through the passive netw ork of the oscillator is H(p), then it is readily shown that

Fn S m H( jna>) = Qn sin (ip„ - </>„),

where J m H(jnaj) is the im aginary part of H(jnoj). W ith this result, Eq. (6.12-5) simplifies to

(6.12 1)

oo<? = £ Qn sin (nwt + I/o, (6 .12-2)

(6.12-3)

00

f(.q) = Z Fm sin (mwt + <f>J. (6.12-4)m = 1

(6.12-5)

harm onics of the controlling and controlled quantities and the phase angle between them.

¿ n f I J m H(jna>) = 0 vo. 1 2 -6 )

6 .1 2 THE DISTORTION-OPERATING FREQUENCY RELATIONSHIP 281

Here we see that the operating frequency m ust shift from a >0 [the point at which H(jco) is real] to a point where H(joj) is sufficiently reactive to satisfy Eq. (6.12-7). It is also apparen t tha t this shift in frequency is minimized if the harm onic com ponents in / (q) are small.

Up to this point we have not made any approximations whatsoever. In m any cases, it is convenient to m ake two types of simplifying approxim ations. In one, we assume a simple driving function for q(t) in order to ease the calculation of the harm onics o f f(q). In the other, we m ake some simplifying assum ptions about the netw ork pole-zero patterns in order to facilitate the calculation of the J m H(jnw).

As a num erical example, consider the circuit of Fig. 6.1 0 -1(a) as expanded in conjunction with Figs. 6.10-4, 6.10-5, and 6.10-6. F o r the given drive, let us find the first-order effect of the circuit Q T on the operating frequency. To accomplish this we assume tha t Q T is sufficiently high so that i rem ains essentially sinusoidal as Qt varies.

F o r this case the variable Q„ becomes the variable F„ becomes V„, and thevariable J m H( jmo) becomes J*n Y(jnco), where

Y{p) = ---------- P / L —_ r = r , + r 2,„2 ^

and

thus

Y(jmo) as Y(jmo0) & — — — for n > 2 and high QT \co0Mn - 1)

y(jn(a) ^ — it.co0L(n2 - 1)

In addition,

(a)/L)(a> o — ct>2) ^ 2AcoQ]( « o - o)2)2 4 - (cor/L)2 W o Ln j o > ) = , , , 2 . * — I T 1 for hi8 h Q t .

where Aco = co - co0 is the departure in operating frequency from to0 . Thus for this example we obtain, from Eq. (6.12-7),

— * - A I {— ) 2 (6-12“ 8)0>0 2Ql-n=2" ~ l \ V l l

Now (V„/K j)2 may be evaluated for any particular steady-state operating point by using Fig. 4.2-3. Table 6.12-1 lists the approxim ate harm onic ratios for the first five harm onics on the assum ption of operation around the operating point found in Section 6.10.


Table 6.12-1

Vi/Vt * 0.0F3/F , % 0.32 VJVX * 0.0 f 5/ f , « a i s

/ , « 3.8 mA, V, 7.6 V

W ith these values of VJVX and with QT = 12, we find that

— ~ JW8 ~ (I x 0.322 + x 0.152)a )0

% 1.5 x 10“ 4,or that the effect o f the harm onics has been to decrease the operating frequency by 0.015%. W ith a smaller Q T there would be m ore harm onics in i and thus m ore uncertainty in estim ating V„. A sm aller QT would certainly also lead to a greatly increased Aco.

E quations (6.12-7) and (6.12-8) indicate that the operating frequency of a ■ oscillator may be shifted by injecting harm onics of the oscillator frequency or by varying the phase of these injected harm onics. This leads directly to the possibility of synchronization with an external signal either a t its fundam ental or any of its harm onics.

F rom this po in t of view we can see why the oscillator explored in Section 6.11 operated so far below its nom inal frequency of operation. To begin with, the oscillator was badly overdriven^ so tha t the harm onic content was h ig h ; but to m ake m atters worse, the passive circuit transfer function

rr/_\ _ ______ PWO_____•*\P) 2 i -j i 2p + 3 pa>0 + a>o

neither attenuated the harm onics rapidly with respect to the fundam ental nor provided a rapid phase shift with frequency in the neighborhood of co0. Thus a large decrease in frequency was necessary to bring Eq. (6.12-7) back into balance.

PROBLEMS

6.1 a) In the circuit of Fig. 6 .P-1, A x = A 2 = / I3 and all sources are linear. Find the values of C and Amin required if this circuit is to be on the verge of sinusoidal oscillation at 20 kHz. How would the circuit be affected if the signs of one, of two, o r of three of the generators were reversed?

b) W ith the sam e value for C, assume that A 2 = A 3 = 4 while amplifier 1 has the following nonlinear relationship between its input rm s sinusoidal voltage and its output sinusoidal voltage:

Find the sinusoidal ou tpu t voltage v3{t). (The nonlinear relationship is assumed to have a time constant that is long in com parison to a cycle o f the sine wave.)

PROBLEMS 283

Figure 6.P-1

6.2 If all of the lO kfl resistors in Problem 6.1(b) increase 10°/„ with age, what would be the new values of frequency and ou tpu t am plitude? If R did not vary, bu t A 2 decreased from 4 to 3, w hat would be the effect on frequency and amplitude?

6.3 A possible sine-wave oscillator circuit is shown in Fig. 6.P 2. Will it work? If so, find the frequency and the am plitude of vx. If not, explain why not. Assume that the A VC block does not load Qi and that a * 1 for the transistors.

+ 0

284 SINUSOIDAL OSCILLATORS

6.4 F or the circuit o f Fig. 6.P-3, find for oscillations. F ind L for 100 kH z oscillations.

6.5 F or the circuit o f Fig. 6 .P-4, indicate the loci o f the poles of K(P)/Is(P) as t* varies, assuming identical n 's and that a current generator, is , is applied at the term inals at v„. W hat is the minimum value of n for sinusoidal oscillations? W hat value of C will cause sinusoidal oscillations at 10 kHz?

/> = 20 k f i

Figure 6.P 4

PROBLEMS 285

>------o+

S.*’. Q ) :* 10ki2 <>R vb

20 kO-VvV

: 1000 pF 5 k ü -

g„ 1000|iö with vh — Vb sin lüt

Figure 6 .P 5

6.6 In the circuit of Fig. 6 .P-5, R is a tem perature-sensitive resistor with a long time constant. Find vb. W hat is the Q of the tuned circuit? W hat are the form and m agnitude of va>

6.7 In the circuit of Fig. 6.P-6(a), assume that the box B has Rin = 25 k£i, no phase shift, and the am plitude characteristic shown in Fig. 6.P—6(b). Letting N = 5, find the frequency and am plitude of v v (Assume identical tubes and the model shown in Fig. 6.P-4.)

6.8 W ith a pole-zero pattern o f the type shown in Fig. 6.1—7(c), produced by cascading three identical isolated RC circuits, show that co0 = ^ / l / R C and Ami„ = — 8. Indicate a possible circuit configuration for this case.

6.9 Assume that three identical, isolated RC circuits are used to produce a network for a feedback oscillator with a root locus pattern as shown in Fig. 6.1 7(d). Prove that for sinusoidal oscillations co0 = 1/^f lRC and that Amin = — 8. Indicate a possible transistor circuit configuration for this case. W hat values would a>0 and A have if the isolated center section were readjusted to have twice the time constant of the two outside sections?

6.10 Derive Eq. (6.9-8). Assuming tha t Ri = R 2 = R*. = R, C t = C 2 = C, and R } = 2.010i?, relate the transfer function of the Wien bridge to th e pole-zero pattern of Fig. 6.1 7(a) and determine the sign of A and the m agnitude of / tmin necessary if the circuit is to oscillate. Repeat, letting R} = 2.0010R. W hat is the SF for each case?

6.11 Derive Eq. (6.9-3) for the M eacham bridge oscillator shown in Fig. 6.9- 1(a). Let w 0 = 106 rad/sec, R 3 = 100fl, R 2 — 5 0 Q, R x = 200£2, and Q = 104 (a quartz crystal is used as a series tuned circuit). Find r for oscillation if the available A is 200. W hat is the sign of A"! W ith these values, assume that the phase shift in the amplifier shifts by 0.10 rad. Find the shift in frequency from w 0.

6.12 For the circuit shown in Fig. 6.P-7, determ ine the output collector current com ponents at 1050 kH z and at 2100 kHz. Find the voltage across L2 at 1050 kHz. The transistor is silicon and a = 0.99; t 2/C j = 40, L2 = 150/iH, QL2 = 50, and M l2 = 4/^H. The tuned circuit is tuned to 1050 kHz.

6.13 Find the frequency and am plitude for the voltage v0{t) for the circuit shown in Fig. 6.P-8. Find the total harm onic distortion in the output. The transistor is silicon with a = 0.98 and 1ES = 3 x e “ 30mA; L , = 10/iH, L2 = 250nH, M i2 = 2 5 /jH, C , = 200pF, C 2 = 0.04 fiF, and R 2 = 100 kQ.

2 8 6 SINUSOIDAL OSCILLATORS

(b)

Figure 6.P-6

Figure 6.P -7

PROBLEMS 287

Figure 6.P 8

14 Is squegging possible in the circuit o f Fig. 6 .P-7? If so, at what sinusoidal frequency will it occur? If not, then would it be possible if the tuned circuit's Q were reduced to 20?

.15 A simplified version of a commercial RC oscillator is shown in Fig. 6.P 9. The network transfer function is most easily found by superposition, that is, by (1) calculating the voltage at the input to amplifier 1 as a function of r„(r) assuming that amplifier 2 has zero output impedance and zero output voltage, and then (2) calculating the voltage at the input to amplifier I assuming that amplifier I has zero output impedance and voltage while the voltage output of amplifier 2 is N i „(t). These two results are then combined to produce the input to amplifier 1 (assumed as having an infinite input impedance) as a function of [„(f). If the two operational amplifiers have equal voltage gains of A, then show that stable sinusoidal oscillations occur with w 0 — 1 /RC and N — 2 + (hiA).

If A = 100, r = 300 £}, and r„ = 450 exp [ —65P], where P is the ac power dissipated in r„, what is the value of v0(t) a t equilibrium ? W hat is the value of rv for this case? If A were increased to 400 for each amplifier, how much would v j t ) increase? How does the indirect frequency stability SF of this circuit com pare with that of a Wien bridge oscillator'’

.16 All transistors in the circuit of Fig. 6 .P -1 0 are silicon with a > 0.99, and Q s are identical, Ql i = Q a — 100, w0L , = 1000Q, M / L 1 — 1/20, C ,/[C , + C 2] = 1/40, and L }C i is tuned to the same frequency as L , , ,an d C 2. The total load on the L}C} tuned circuit including its own losses is 20 kfi. Plot Gm(x) vs. x for this circuit, assuming an ideal envelope detector. Find the stable oscillation am plitude existing across ¿ 1. Find Ik for this amplitude. Find the changes in this am plitude caused by a + 10% change in the + 6 V supply. Find SF for this oscillator.

2 8 8 SINUSOIDAL OSCILLATORS

PROBLEMS 289

Figure 6.P-11

6.17 For the circuit of Fig. 6.P-11, find frequency and amplitude of va assuming that the loss in the 1 Mi2 resistor is negligible, that Cx is an RF bypass, and that the grid diode is ideal as shown.

6.18 For the circuit of Fig. 6. P -12, find the voltage magnitude and frequency across RL. (Neglect losses in coil.) Repeat, reducing Vcc to 12 V. Is squegging possible in the circuit shown in Fig. 6.P -12? At what frequency does it occur? The transistor is silicon.

100 pF

Figure 6.P-12


6.19 For the circuit shown in Fig. 6.P-13, determ ine an expression for v0(t).

6.20 A sine-wave oscillator constructed from a commercial differential pair-curren t source com bination is shown in Fig. 6.P-14.

a) Find Ik.b) Choose R 2 so as to make vB2(t) = 0.104 cos w 0t + K c-c) Find the voltage across each tuned circuit and i„(t).d) Find the T H D in r0(t).

6.21 For the circuit of Fig. 6.P-15, R = 10ft, C = 2 0 0 pF, L = 100/xH, and vD = (11 — iD + ~‘d + 3i’d) V, where vD is m easured in volts and iD is m easured in mA. Find the frequency and the am plitude of the fundam ental com ponent of vcit). W hat is the total harm onic distortion in r c(f)? How much frequency depression is caused by this distortion?

6.22 Assume vD = V, cos w 0t for the circuit shown in Fig. 6.P-16.a) Plot l Dl vs. Vy for the device shown.b) Determine a numerical expression for the steady-state value o f vD(t).

6.23 Find the approxim ate am plitude of vD(t) in the circuit shown in Fig. 6.P 17.

6.24 Plot the phase plane characteristic for the device characteristic and circuit shown in Fig. 6 .P -18. Start the plot at = 0.0 V, iT = 0.50 mA, and let r = 16 kft, w 0L = 32 kQ, and 2 = 2.

6.25 Plot the output waveshape for v^t) in Fig. 6.P-18. Estim ate the frequency depression for the actual output in com parison to a>0 = I /^/l C.

acshort

PROBLEMS 291

6 V0

6 Vo

M 10 nH

14 HH j - - ■4 k n

2000 pF

o--)

v„(i)

X Qi

RFC

* 9 k O < 3 k £ ì

6 V

5 k i ì -

10 nH,.120 nH

c ,

C2 = 59C,ui0 = 6 x 10 rad/sec

a »0 .99

ac_short 5 k i l

1.5 kO

V

6 V

6 V

Figure 6.P-14


10co0= 6 x 106 rad/sec

«0 = (-0 .2 x 10_3vd+ 0.75 x I0-*vo3)î

C H A P T E R 7

MIXERS; RF AND IF AMPLIFIERS

7.1 T H E SU PER H ETERO D Y N E C O N C EPT

In designing a receiver, one norm ally starts from the detector or dem odulation circuitry and works in both directions. In this chapter we are interested in the circuits that lie in front of the dem odulators.

As we shall see in subsequent chapters, m ost detector circuits do not work well in the presence of noise or interfering signals, and many of them do not work well, if at all, with input signal levels below several volts in am plitude. Since the desired signal may have a receiver input field strength in the m icrovolt/m eter range while the total rms noise and interfering signal strengths available to the antenna may m easure in the volt/m eter range, it is apparent that one needs both gain and selectivity in front of the dem odulator.

The real problem s in designing a carrier frequency or RF amplifier for a fixed- carrier-frequency receiver are the following :

a) to control its front-end noise so as to keep it an adequate distance below the incoming signal level;

b) to control the active device nonlinearities so as to prevent signal distortion or unwanted signal interactions [cross m odulation, for exam ple]f;

c) to keep the resultant high-gain narrow band amplifier from becom ing an oscillator. [If the amplifier gain is 120 dB (106 in voltage), then a feedback of 1/106 of the output, with the proper phase relationship, to the input will cause a loop gain of unity and hence will produce an oscillator.]

U nfortunately for receiver designers, m ost receivers are not fixed-frequency units; hence one is forced to cope with the previously m entioned problem s while sim ultaneously tuning this high-gain m onster over some wide frequency range. In addition, one m ust be able to solve the difficult problem of designing the dem odulator to have adequate and reasonably constant perform ance over the frequency band(s) in question. (N orm al AM broadcasting has a 3/1 band, norm al V H F television in the U nited States has a 4/1 band, and a “go o d ” com m unications receiver may be expected to cover a range of m ore than 10Q/1 in frequency.)

t As we sh#U see shortly, cross m odulation is the transfer o f the m odulating signal, o r a distorted version of this signal, from one carrier to a neighboring carrier. In order for such a phenomena to be possible, certain types of device nonlinearities must be present.

293

294 m ix e r s ; r f a n d i f a m p l i f i e r s 7.1

Very early in the developm ent of the radio com m unications business some people decided that this situation was ridiculous and tha t the way to simplify things was to continue to design both the detector and the bulk of the gain and selectivity on a fixed-frequency basis and to shift or translate the m odulation from all desired incoming signals to this new, fixed, “ interm ediate frequency” or IF. As with m ost new ideas, it took some time for this innovation to be accepted ; however, it has since so completely dom inated the field tha t “ stra igh t-through” receivers are rarely seen except in museum s and an occasional very special situation.

Figure 7.1-1 illustrates the block diagram of a “ superheterodyne” receiver. This is the com m on nam e for the system containing an IF amplifier and a fixed-frequency detector.

Fig. 7.1-1 Superheterodyne receiver.

The “m ixer” utilizes the trigonom etric identity that expands the product of two cosine term s into sum and difference frequencies :

[a(i) cos At][b{t) cos Bt] = [cos (A — B)t + cos (A + B)t]. (7.1-1)

Thus if a(i) cos At is the desired incom ing signal while b(t) cos Bt is a constant- am plitude cosine signal supplied by the local oscillator, and if the selective IF amplifier is tuned to a radian frequency of A — B, then the IF ou tpu t signal will be a frequency-translated version of the incom ing signal. So long as everything is done properly, such a frequency translation is m odulation insensitive. T hat is, it is just as effective for AM , FM , SSB, or any other type of m odulation.

Like m ost innovations the superheterodyne concept does add some additional problem s to those already listed for the case of the fixed-frequency receiver. Some of these additional problem s are as follows :

d) the mixer and local oscillator m ust be designed and the local oscillator must be m ade to track all tuned circuits in front o f the mixer ;

7.2 MIXER TECHNIQUES 2 9 5

e) since m ix e rs a lm o s t a lw ay s g e n e ra te m o re n o ise th a n am p lif ie rs a n d since by th e ir very n a tu re th ey m u s t c o n ta in n o n lin e a ritie s , o n e m a y find th a t c r i te r ia (a) a n d (b) still re q u ire a s tag e (o r s tag es) o f R F a m p lif ic a tio n in fro n t o f th e m ixer.

f) Some new types of interference are generated by the local oscillator-m ixer c o m b in a tio n th a t a re n o t p re s e n t in a s tra ig h t- th ro u g h o p e ra tio n .

T h e nex t se c tio n w ill d e a l w ith sev era l g e n e ra l te c h n iq u e s fo r a n a ly z in g n o n lin e a r c ircu its as m ixers. S u b se q u e n t sec tio n s w ill d ea l w ith specific m ix in g a n d “ c o n v e r t in g " c ircu its (a se lf-o sc illa tin g m ix e r is ca lled a c o n v e rte r) a n d w ith th e d esig n a n d an a ly s is o f R F a n d I F am p lif ie r stages.

7.2 M IXER T E C H N IQ U ESAs w as p o in te d o u t in c o n n e c tio n w ith Eq. (7 .1-1), a n y m u ltip lie r fo llow ed by a p ro p e r b a n d p a s s filte r w ill fu n c tio n as a m ixer. O n the o th e r h a n d , since th e loca l o sc illa to r in p u t h a s a c o n s ta n t a m p litu d e , it is n o t n ecessa ry to h av e a n id ea l m u ltip lie r in o rd e r to m a k e a useful m ixer.

C hapter 8 will consider several general m ultiplier circuits. The emphasis in this section will be on techniques applicable directly to specific mixer circuits.

T h e tw o m o s t c o m m o n m ix ers in use to d a y a re th e sim p le field effect tr a n s is to r a n d th e b ip o la r tr a n s is to r . In b o th cases o n e a p p lie s th e in c o m in g sig n a l a n d th e lo ca l o sc illa to r v o lta g e s so th a t th ey effectively a d d to th e d c b ia s v o lta g e to p ro d u c e th e to ta l g a te -so u rc e o r b a se -e m itte r v o ltag e . T h is signal is th e n p assed th ro u g h th e dev ice n o n lin e a r i ty to c re a te th e d e s ire d sum a n d d ifference frequenc ies.

F E T M ix e rs

Figure 7.2 1 illustrates several possible FET mixer circuits.If a ju n c t io n o r M O S F E T is b ia sed so th a t th e to ta l e x c u rs io n s a ro u n d th e

(? -p o in t n ev e r c a u se it to leave th e " c o n s ta n t c u r r e n t” ( s a tu ra t io n reg io n ) o r to tu rn on the gate-to-source junction, then the drain current is always approxim ately re la te d to th e g a te -so u rc e v o lta g e by a “ sq u a re - la w ” c h a ra c te r is tic , a n d th e d ra in c u rre n ts a re g iven by

l'i (0 + r 2(0 + Kis[D — I D S S 1 - (7.2-1)

and

!D - - [ l ’ l(0 + l '2 ( 0 + K j.S “ ^ThY- (7.2-

resp ec tiv e ly . W h ere vus = r , + r 2 + Fc s , a n d v2 a re th e loca l o sc illa to r a n d R F in p u t s igna ls re sp ec tiv e ly , a n d K , s is th e d c g a te - to -so u rc e b ia s v o ltag e . T h ese e q u a tio n s a re i l lu s tra te d in F ig . 7 .2 -2 . If E q . (7 .2 -2 ) fo r th e M O S u n it is ch o sen as an ex am p le a n d th e v o lta g e te rm is e x p a n d e d , th e n

[f’liO]2 , ta W ]2 , (Kjs ~ VTh)i d — ß Y ' l l O l ^ U ) + (^ G S ~ ^T(i) [ l ' l ( f ) + ^------ h ------- ---------- 1-

(7 .2 -3 )

2 9 6 m ix e r s ; r f a n d if a m p l if ie r s 7 .2

Tuned to o>0 —ws

Fig. 7 .2 -1 (a) F E T mixer.

Fig. 7 .2 -1 (b) FET mixers (both signals to gate circuit).

7 .2 MIXER TECHNIQUES 297

Fig. 7.2-2 FET characteristics.

I f v2(t) = vs(t) co s ftjsi w hile ^ ( t ) = Vt c o s c o 0 f, an d if we assu m e I F filterin g a t the ou tp u t an d co n sid er on ly th e freq u en cy co m p o n e n t o f the o u tp u t cu rren t in th e v icin ity o f a>0 — u>s , we m a y d efine a la rg e -s ig n a l c o n v ersio n tra n sc o n d u cta n ce , Gc, a s the en v elop e o f the o u tp u t cu rren t a t the d esired d ifference freq u en cy d ivided by the

en v elo p e o f the inp u t s ig n a l voltage.E q u a tio n (7 .2 -4 ) p re s e n ts th e c o n v e rs io n tr a n s c o n d u c ta n c e fro m tos to co0 — tos

as w ell a s th e o rd in a ry la rg e -s ig n a l a m p lif ie r tr a n s c o n d u c ta n c e :

Gc = Gm = P(V0S - VTh). (7.2-4)

H en ce G J G m = V J 2 (V GS - VTh). N ow sin ce vs(t) + Vt m ust be less th an Vas ~ Vtii if o n e is n o t to exceed th e “ sq u a re law ” reg io n , th e co n v ersio n tra n sc o n d u cta n ce can n o t exceed o n e -h a lf th e tra n s c o n d u cta n ce o f th e d evice w hen it is em p lo y ed as a n ord in a ry am p lifier. N o te th a t if Vt is tru ly c o n sta n t, th en as lo n g as the p rev iou s a ssu m p tio n s are m a in ta in e d , G c is in d ep en d en t o f vs(t) and d is to rtio n le ss co n v ersio n results.

A sim ila r e x p a n sio n fo r th e ju n c t io n F E T yields

Gc = n , Gm = 2I §HVp - VGS). (7.2-5)V P V P

I f th is d ev ice is b iased m id w ay b etw een Vp and zero an d if Vl » |t>s(i)|, then the m ax im u m o b ta in a b le Gc is o n e -q u a r te r o f the sm all-s ig n a l gm ev alu ated a t VGS = 0 o r o n e -h a lf o f the G m — g m for th e l^ s = V J 2 b ia s po in t.

Bipolar Transistor MixersF ig u re 7.2-3 illu s tra te s a p o ssib le b ip o la r tra n s is to r m ixer.

W hen tw o tim e -v a ry in g v o ltag es, (t) an d v2(t), a re ap p lied a cro ss an ideal b ip o la r tra n s isto r ju n c t io n , then

i E ( t ) = [ E s e 9 y d o / k T e v u t ) « / k T e v 2(,)q/ k T ( 7 . 2 - 6 )

If again v2(t) = vs(t) cos o)st and 11,(1) = Vx cos u>0t while y(t) = vs(t )q /k T and x = V, q/"k T. then

iE(t) = [!ESe“v^ kT] I 0(x ) + 2 £ /„(*) co s na}0t 1

¡o(y) + 2 'Lhiy) cos na>st 1

(7 .2 -7 )


Fig. 7 .2-3 Bipolar transistor mixer.

I f we c ro ss -m u ltip ly term s, we o b ta in

'e (î) = UEseqVaJk1"iUo(y)I o(x) + 2 / 0(.x )/,(> ')cos cost

+ 2/0 (y )/ j(x ) c o s co0t + 4 / 1 (x)/,(>»)[cos cost c o s a>0t

from w hich

/d c = l Ege qV*Jk T I 0{y)Io{x\

h (y) / ¡(x )I — I f 7 1 — 11 ____d c / o ( y ) ’ r o o ~ d c / „ ( * ) ’

, ? / t W h(y),/ 0(x) / 0(j>) dc'

(7 .2 -8 )

(7 .2 -9 )

(7 .2 -1 0 )

(7 .2 -1 1 )

In o rd e r to hav e a “ lin e a r” m ixer, o n e m u st hav e /O0_ lin early p ro p o r tio n a l to y a n d in tu rn vs(t). I f F , a n d h e n ce x a re assu m ed to b e c o n s ta n t a n d i f Vt » |us| so th a t v a ria tio n s in vs d o n o t effect /dc, th en w ill v ary w ith vs a s I i (y ) /1 0(y)varies w ith vs.

F r o m th e e x p a n sio n s fo r th e m o d ified B essel fu n ctio n s w ith sm all a rg u m en ts, we m ay w rite

ii(y)_i 0 00

l z ! + 4 8 16

(7 .2 -1 2 )

w hich is w ith in 2 % fo r y < 1 .F r o m E q . (7 .2 -1 2 ) w e see th a t fo r h ( y ) / I 0(y\ an d h en ce I mo- as, to b e lin ea r w ith

7.2 MIXER TECHNIQUES 299

resp ect to y to w ith in 2 % th e c o n d itio n y 2 /8 < 0 .0 2 o r y < 0 .4 an d h e n ce |t>J < 1 0 .4 m V a p p lie s ; fo r lin ea r ity to w ith in 0 .5 % , th e a p p ro p ria te c o n d itio n is y < 0 .2 o r |rs| < 5.2 m V . T h e re fo re , w hen |t)J < 10 .4 m V w e m ay re p la ce Ii(y)/I0(y) b y y/2 and from E q . (7 .2 -1 1 ) o b ta in

w h ere g in = q I ic/ k T a n d vs(t) = Fs .fT h u s in th is ca se G c m ay never exceed Gs but m ay exceed G m if x > 2. S in ce

/,(x)/70 (x ) is w ith in 7 0 % o f its a sy m p to tic v alu e o f unity w hen x = 2 an d w ithin 8 6 % o f this v alu e w hen x = 4 , n o t m u ch is to be gained fro m using valu es o f x exceed in g 6.%

T h e re a re tw o useful w ays to c o n n e c t th e sig n als to a d ifferen tia l-p a ir c irc u it to m ak e a m ixer. In o n e ca se th e re la tiv e ly large o sc illa to r v o ltag e is fed in to o n e (or a cro ss b o th ) o f the d iffe re n tia l-p a ir b a ses , w hile the re lativ ely sm all sig nal v o lta g e is fed a cro ss the e m itte r-b a se ju n c t io n o f the “ c o n sta n t c u rre n t” tra n s is to r . F ig u re 7 .P - 3 illu stra tes a sim p lified v ersio n o f su ch a c ircu it.

In the seco n d ca se th e o sc illa to r v o lta g e is used to c o n tro l th e “ c o n sta n t c u rre n t” tra n s isto r , w hile the signal is fed in to (o r acro ss) the d ifferen tia l-p a ir b ase c ircu its . F ig u re 7 .4 -4 p ro v id es an exam p le o f a b a la n ced v ersio n o f such a c ircu it.

A s we sh a ll see in S e c tio n 7 .4 , th e m o st d esira b le way (fro m the d is to rtio n view p o in t) to o p e ra te a d ifferen tia l-p a ir m ixer is so th a t b o th th e o sc illa to r an d the “ s ig n a l” o p e ra te th e m ixer lin early w hen they a re co n sid ered as sep a ra te inputs. U n d er th ese c ircu m sta n ce s , the m ix in g a c tio n ta k e s p lace b e ca u se the “ g a in ” from the d ifferen tia l-p a ir b a se -b a se v o lta g e to the o u tp u t is d irectly p ro p o rtio n a l to the cu rren t from the “ c o n sta n t c u rre n t” tra n s is to r ’s co lle c to r .

O v e ra ll system c o n s id e ra tio n s w ill d ecid e w hich typ e o f d rive is m o st su ita b le for a p a rticu la r ca se . W h e n th e m ixer g en era tes its ow n o sc illa tio n (serves as a co n v erter), th en o n e w ould tend to c o n fin e the o sc illa to r to the d ifferen tia l p a ir an d feed the s ig n a l in to th e “ c o n s ta n t c u rre n t” tra n s isto r .

T h e resu lts o f S e c tio n s 4 6 an d 4 .8 a re d irectly a p p lica b le to m ixer ca lcu la tio n s . F o r exam p le, T a b le 4 .6 - 2 and F ig . 4 .6 - 4 in d ica te the lim its on th e drive i f lin ea rity is to be m a in ta in ed , an d p ro v id e th e valu e o f the fu n d am en ta l te rm for th e ca se in w hich the d ifferen tia l p a ir is d riven d irectly by a n o sc illa to r signal th a t is large en o u g h to

t Since the resultant conductances are independent of they may be calculated with a constant for tis(i) rather than the m ore complicated function of time,

t It is true that g,„ may increase slowly with x since

(see Fig. 5.4-4, for example). For x = 6 and a 1 V or more dc drop across an em itter resistor, this increase will be less than 10%.

I. (7 .2 -1 3 )

G J G m = x/2, G J G S = /i(x)//0 (x), (7 .2 -1 4 )

300 m ix e r s ; r f a n d if a m p l if ie r s 7.2

ca u se th e re la tio n sh ip to b e co m e n o n lin e a r . F r o m T a b le 4 .6 -2 , w e see th a t an o sc illa to r d rive o f 78 m V b etw een th e b a ses lead s to a fu n d a m en ta l te rm very c lo se to j th e valu e o f Ik. H en ce the c o n v ersio n tra n sc o n d u cta n ce fro m a sig nal th a t in flu en ces Ik to th e o u tp u t cu rre n t o f o n e sid e o f the d ifferen tia l p a ir w ill b e | (a n o th e r \ co m e s fro m the sep a ra tio n o f the o sc illa to r-s ig n a l p ro d u ct in to sum an d d ifference term s) o f the sm all-s ig n a l tra n sc o n d u cta n ce (fro m b a se -e m itte r sig nal v o ltag e to c o lle c to r cu rren ts) o f the “ c o n s ta n t c u rr e n t” tra n s is to r .

T h e te ch n iq u e s o f S e c tio n 4 .8 m ay b e used to lin earize e ith er the d ifferen tia l-p a ir c h a ra c te r is t ic o r the c h a ra c te r is t ic o f th e “ c o n sta n t c u rre n t” tra n s isto r .

Generalized MixersW h ile th e F E T , b ip o la r tra n s isto rs , an d d ifferen tia l p a ir a c c o u n t fo r m o st p ra c tica l m ixers, it is useful to g en era lize the a p p ro a c h , b o th as a m ean s o f d ea lin g w ith o th er d ev ices— v acu u m tu b es, for exa m p le— an d as a m ean s o f e x a m in in g th e p rev io u s d ev ices w hen th e ir c h a ra c te r is tic s d ep a rt fro m the usual shap es.

O n e gen era l a p p ro a c h is to co n sid er th e m ix e r as a g a in -c o n tro lle d am p lifie r in w h ich th e lo c a l o sc illa to r v o lta g e c o n tro ls th e g ain w hile th e s ig n a l v o lta g e is am p lified . F ig u re 7 .2 -4 illu s tra te s o n e ex tre m e ca se , w here th e g a in h a s o n ly tw o valu es an d w here the lo ca l o sc illa to r is b iased so th a t th e d ev ice sp end s h a lf the tim e in e a ch reg ion .

K C O S l lV

©

-of

d >■S[v,(/)]— 1. v ,(/)> 0

S[v,(i)]=0. v ,(r)<0

v ,(0 = K COSWo< /„= vs (iK?5[v1(0]

Fig. 7 .2-4 O n-o ff amplifier or switched amplifier as a mixer.

S in ce th e o u tp u t cu rre n t m u ltip lier is c o n tro lle d by u t(r), it is p e rio d ic an d can be exp an d ed in a F o u r ie r series w ith fu n d a m en ta l tu0 :

G 2 G 2 GG S f t ^ f ) } = — H c o s (o0t — — c o s 3 co0t +

2 n 3n+ .

S in ce the o u tp u t c u rren t is the p ro d u ct o f v2(t) an d GS[i>i(f)], it is a p p a ren t th a t

71

T h is te ch n iq u e o f sp ecify ing th e a m p lifie r “ g a in ” o r tra n sfe r te rm w ith resp ect to the in p u t s ig n a l and th en exp a n d in g the c o n tro lle d “ g a in ” in a F o u r ie r series is a co m m o n one. U su a lly th e gain o r tra n sfe r fu n ctio n is a c o n tin u o u s fu n ctio n o f th e

7.2 MIXER TECHNIQUES 301

o sc illa to r v o ltag e an d it is ev a lu a ted by ta k in g th e p a rtia l d eriv ativ e o f i w ith resp ect to the sig nal v o ltag e. (T h is p ro ce d u re is based o n the a ssu m p tio n th a t th e signal term is sm all en o u g h th a t it d o es n o t in flu en ce the gain . In an y p a rticu la r ca se the lim its on vs b e fo re in to le ra b le d is to rtio n sets in m u st be d eterm in ed .)

F o r exam p le, if we d ifferen tia te the b a s ic tra n s isto r ju n c t io n e q u a tio n , E q . (7 .2 -6 ) , w ith resp ect to v2 an d assu m e th a t v2 a p p ro a ch e s zero ,

diEdv2

wherev z ~ * 0

J L j ESe ‘>y^ kTe v'(,)qlkT, (7 .2 -1 5 )

diEdv2 V2~*0

is the tra n sfer co n d u cta n ce fro m v2(t) to the o u tp u t cu rren t. T h e righ t side o f E q .(7 .2 -1 5 ) m ay be exp an d ed as th e first tw o b ra c k e ts on the rig h t side o f E q . (7 .2 -7 ).A gain we find

= /i(x ) q/dc = /i(x )

c /„(x) k T /0 (x )gin'

In th is exa m p le the in itia l a p p ro a c h is p re ferab le , sin ce an e x a c t exp ressio n is p o ssib le for I ao- a, and h e n ce the a p p ro x im a tio n s a re m ad e a t the last step ra th e r th a n a t the first s te p .f

A s a seco n d ex a m p le c o n sid er the field -effect tra n s is to r for w hich th e “ g a in ” term is sh o w n in F ig . 4 .4 -2 . S in ce th is te rm v aries lin early as lo n g as the o sc illa to r v o ltag e stays a b o v e Vp an d b elo w the ju n c t io n tu rn -o n p o in t, the co n v ersio n tra n s co n d u cta n ce is d irectly p ro p o r tio n a l to th e o sc illa to r v o ltag e w ith in th is reg ion . W h en th e s ig n a l ex ten d s o u ts id e th e lin e a r reg io n , then th e s in e-w av e tip fu n ctio n s o f F ig . 4 .2 -3 o r F ig . 4 .2 - 4 m ay b e used to c o m p u te th e fu n d a m en ta l te rm o f th e “ g a in .” T h is te rm is d irec tly p ro p o r tio n a l to th e sm a ll-s ig n a l co n v ers io n tra n sco n d u cta n ce .

F ig u re 7 .2 -5 illu s tra te s th e resu lts fo r a p e a k c la m p in g n -ch a n n el F E T c ircu it

s im ila r to th a t used in P ro b le m 7.5 (w ith th e 100—Q so u rce -to -g ro u n d re s is ta n ce set to zero).

T o a c c o u n t fo r th e d rive up to + 0 .7 V, I DSS is rep laced by I%ss = y i - o.7/kp) 2

an d Vp is rep la ced b y V * — \ Vp\ + 0 .7 , an d c la m p in g is n o w p resu m ed to o c c u r a t zero v o lts o n th e sh ifted cu rv e. (A s d iscu ssed in C h a p te r 5, th e a c tu a l tu rn -o n v o ltag e is a fu n ctio n o f the g a te -g ro u n d re s is to r . W h e n th is re s is to r is sev era l m eg o h m s, th e tu rn -o n v o lta g e m ay b e o n ly a b o u t 0 .3 V .)

A s we sh a ll see w hen {V l / V * ) exceed s 0 .5 , m an y o f th e d esira b le lo w -d isto rtio n p ro p e rtie s o f th e F E T m ixer a re lo st. O n e p o ssib le a rg u m en t for o p e ra tio n a b o v e th is p o in t is th a t th e d c c u rren t is red u ced , w hich lead s to less d issip a tio n an d usually less d ev ice n o ise .

t Yet another approach to this problem would be simply to expand eqvA,yiT and retain only the first several terms, and then to expand the remaining expression as in the first two terms of Eq. (7.2-7) and take the resultant product.

302 MIXERS ; RF AND IF AMPLIFIERS 7.3

k =\ k \ + m

0.50

0.25

0 0.50 1.0

Figure 7.2-5

1.5

W h ile we hav e d o n e th e tw o exam p les, th e usual d ifficu lty w ith th is a p p ro a ch is in e v a lu a tin g th e F o u r ie r co e ffic ien ts o f th e “ g a in ” fu n ctio n . In the g en era l c a se this c a n n o t be d o n e in a sim p le a n a ly tic m a n n e r ; it m u st be d o n e g ra p h ica lly , by a c o m p u ter a n a ly sis , o r ex p erim en ta lly . O f co u rse , if th e p ro b le m is a tta ck e d exp erim en ta lly o n e will lo g ica lly m easu re the co n v ersio n tra n sc o n d u cta n ce d irectly in stead o f fiddling a ro u n d w ith in te rm e d ia te steps.

7.3 SERIES RESISTA N CE IN M IX ER S

In o rd er to o p e ra te , a m ixer m u st hav e a n o n lin ea rity . As we m igh t exp ect (see S e c tio n 4 .8 , fo r exam p le), ad d in g a re s is to r in series w ith th e em itter o f a b ip o la r tra n s isto r , w ith the so u rce o f an F E T , o r w ith the em itte rs o f a d ifferentia l p a ir a lw ays tends to lin earize th e ch a ra c te r is tic . T h e re fo re , su ch a re s is to r w ill a lw ays red uce th e c o n v ersio n tra n sc o n d u cta n ce (if a c o n s ta n t lo ca l o sc illa to r v o ltag e is assu m ed ). S in ce all rea l d ev ices h av e a t least a m in im u m a m o u n t o f in h eren t se r ie s res is tan ce , to say n o th in g o f g e n e ra to r o u tp u t im p ed an ce, o n e w ill find th a t th e c a lc u la tio n s o f the p rev io u s sec tio n p ro v id e an up per lim it on the exp ected p e rfo rm a n ce o f real m ixers.

T h e effect o f a series re s is ta n ce in m o d ify in g th e in cre m e n ta l s lo p e o f th e i , vs. v 1 c h a ra c te r is tic is c le a rly in d ica ted by F ig . 4 .8 -2 . I t is a s tra ig h tfo rw ard m a tte r to solve for 8i\/8vi from th e e q u a tio n s o f S e c tio n 4 .8 :

w here g in = q I AJ k T an d i , is the cu rren t w hich flow s w ith = Vt co s co0t + K c- W h en ginR » 1, E q . (7 .3 -1 ) red u ces to 1/R for u, — V0 > 0 an d to zero for

i>i — V0 < 0. F o r g inR -> 0 , E q . (7 .3 -1 ) red u ces to

k T /q + i iR I dc + g inR i i '1 (7 .3 -1 )

i 1! _ / „<lVö J kT „ q V , cos m ot/* /77F “ i.t Es ek T k T

7 .3 SERIES RESISTANCE IN MIXERS 303

w hich is o f co u rse id en tica l w ith E q . (7 .2 -1 5 ) . F o r in te rm e d ia te valu es o f g inR , n o sim p lified form fo r the gain fu n ctio n is p o ssib le .

A p o ssib le , and re la tiv e ly sim p le, a p p ro a ch is first to find a re a so n a b le re la tio n

sh ip betw een Gc an d Gm fo r the ex tre m e ca ses o f zero res is ta n ce and d o m in a n t re s is ta n ce , an d th en , u sin g F ig . 4 .8 - 6 , to d eterm in e G m vs. F , and e x tra p o la te to a p p ro x im ate Gc vs. V1. T h e resu lts w ill n o t be e x a c t, how ever, the series re s is ta n ce is rarely kn ow n e x a ctly e ith er. T h e best th a t o n e ca n d o h ere is to get a feel fo r w hat is go in g

on. I f “ e x a c t” resu lts a re d esired , th en m easu rem en ts are so s tra ig h tfo rw ard th a t pages o f a n a ly tic m a n ip u la tio n a re co m p le te ly un justified .

W h en th e ju n c t io n is d o m in a ted by R, the slo p e o f the o v era ll c h a ra c te r is tic is

l /R w hen r,. > V0 = (k T /q )\ n (I dc/ I ES) (see F ig . 4 .8 -3 ) and is zero otherw ise. T h is m ean s th a t the “ g a in ” fu n ctio n is a re c ta n g u la r w ave o f p eriod T = 2 k /oj0, as show n in F ig . 7 .3— 1. I f Idc is assu m ed to be c o n sta n t, then the c o n d u ctio n an g le 0 C = 2<j) is re la ted to F,//dcR by T a b le 4 .8 -1 . T h e c o n d u ctio n an g le th en tells us the rela tiv e n o n z ero w idth o f the “ g a in ” fu n ctio n fro m w hich the fu n d am en tal co m p o n e n t o f the F o u rie r series m ay be co m p u ted . T h e c o n v ers io n tra n sco n d u cta n ce is then o n e -h a lf o f th is fu n d a m en ta l co e ffic ien t. T a b le 7 .3 -1 in d ica tes rep resen ta tiv e values.

Table 7 .3-1

ec = 2 <t> V j l dcR GCR GJGm

360° 1 .0 0 0 . 0 0 0 . 0 0

300° 1.063 0.167 0.17270° 1.333 0.217 0.242 1 0 ° 1.667 0.29 0.36fe00 3.1416 0.317 0.63

00

1 0 .0 0.305 1.790° 20.5 0.23 2.4

F ig u re 7 .3 -2 p lo ts G J G m vs. V1 / I ic( r in + R ) fo r the tw o cases R = 0 and R » l/gin [the R = 0 exp ress io n co m es fro m E q . (7 .2 -1 4 )] .

F r o m th is figure it is h y p o th esized th a t fo r valu es o f g inR n e a r u n ity it w ill be p la u sib le to re la te G J G m to th e n o rm a liz e d v alu e o f V J l Ac(r in + R ) by a fa c to r o f 3 . F o r valu es o f ginR betw een 0 and 1 th is fa c to r should lie betw een \ and 3 ; fo r valu es

jG a in function

------------------------ .-I/ / ? .....

-<J) 0 <t> n 27t — <t> 2 jt

Fig. 7 .3 -1 GJGm for a bipolar transistor mixer with added series resistance.


Fig. 7.3-2 D ata for Eq. (7.3-2).

betw een 1 and in fin ity it sh o u ld lie b etw een ^ an d N o rm a lly o n e w ould e x p e c t to m in im ize th e res is ta n ce , an d h e n ce th e g inR < 1 reg io n w ould b e exp ected to b e the in terestin g one.

F o r exam p le, i f /dc = 1 m A , R = 2 6 0 , and V1 = 6 x 52 m V = 3 1 2 m V , then from F ig . 7 .3 - 2 G c % 2 G m, w hile fro m F ig . 4 .8 - 6 G m = 0 .2 9 5 (5 2 Q ) = 5 7 0 0 /imho. W ith R = 0, /dc = 1 m A , and h a lf the o sc illa to r drive (th a t is, V, = 156 m V ), then x = 6 and Gc = 3 4 ,5 0 0 /imho, o r a p p ro x im a te ly th ree tim es as m u ch as in the p rev ious ca se (a g la n ce a t F ig . 4 .8 - 5 w ill in d ica te th a t m a in ta in in g th e 3 1 2 m V d riv e w ould n o t in crea se th e G c a p p rec ia b ly fo r th is seco n d case).

A s im ilar a p p ro a ch is p o ssib le for F E T ’s and d ifferen tia l p airs. In e ith er case , w hen R tim es the q u iescen t sm a ll-s ig n a l tra n sc o n d u cta n ce b e co m e s a p p rec ia b le , th e c o n v ersio n tra n sc o n d u cta n ce w ill fall sharp ly .

V o g e lf h a s co n sid ered th e effect o f serio u s so u rce re s is ta n ce o n th e p ea k -c la m p in g F E T o f p ag es 301 a n d 3 0 2 (F ig . 7 .2 -5 ) . W h e n c e r ta in o f V o g e l’s resu lts a re sim p lified an d tra n s la ted in to th e p ea k c la m p in g term s V * an d I%ss o f F ig . 7 .2 -5 , th en th e (G CF *)/ / JSS vs V J V * cu rv e o f F ig . 7 .2 -5 is fo u n d to b e d ep ressed by th e a d d itio n o f a series so u rce re s is ta n ce , R s.

A s a re a so n a b le firs t a p p ro x im a tio n to th e e ffect o f th e series re s is ta n ce , o n e ta k e s th e v alu e o f (G CV */I% SS) fro m F ig . 7 .2 -5 , fo r th e p a rtic u la r v alu e o f V J V * o f in terest, an d d iv ides it b y (1 + q )3, w here q = (2l%ssR sz ) /V * an d z is a te rm th a t a c co u n ts fo r th e sh ift in th e d c b ia s cu rre n t w ith in cre a sin g d rive ; z is p lo tted vs. V J V * in F ig .7 .3 -3 .

S u p p o se th a t ( ¡ dss/ V * ) = 2 0 0 0 /im ho, th a t ( V J V * ) = 0 .5 , an d th a t R s is 2 0 oh m s. In th is c a se z w ill b e 0 .5 , q = 0 .0 4 , an d (1 4- q ) 3 w ill b e 1 .125 . T h e re fo re G c w ill be red u ced fro m 1 0 0 0 /imho fo r R s = 0 to 8 8 9 /imho fo r R s = 2 0 o h m s. [F r o m F ig .7 .2 -5 , (G cV*p/I*DSS) = 0 .5 w hen ( V J V * ) = 0 .5 .]

f J. S. Vogel, “N onlinear D istortion and M ixing Process in Field Effect Transistors.” Proc. IEEE, 55, No. 12, pp. 2109-2116 (Dec. 1967).

7 .4 PRACTICAL MIXER CIRCUITS 305

y,/K

Figure 7.3-3

D o u b lin g th e d rive v o lta g e so th a t (F j/ K * ) = 1 w ill red uce z to a p p ro x im a te ly0 .3 2 , q to 0 .0 2 5 6 , an d (1 + q )3 to 1 .079 . In th is c a se G CV */I% SS is a g a in 0 .5 ; h en ce now Gc is 9 2 7 /imho in stea d o f 1 0 0 0 /¿mho.

I f R s is m ad e m u ch la rg er, say 2 0 0 o h m s, th en fo r th e ( V J V * ) = 0 .5 ca se , o n e w ill find th a t z = 0 .5 , q = 0 .4 , a n d (1 + q )3 = 2 .7 4 , so th a t now Gc is red u ced fro m 1000 /imho fo r R s = 0 d o w n to o n ly 3 6 5 /¿mho.

7.4 PRACTICAL M IX ER C IR CU ITS

In th is sec tio n we sh a ll d iscu ss the in terferin g -s ig n a l p ro b lem e n co u n tered in b ip o la r tra n s isto r , d ilfe re n tia l-p a ir , and F E T m ixers. In the n ext sec tio n we w ill in d ica te how to c o m b in e th e m ix in g and o sc illa tin g fu n ctio n s to p ro d u ce “ co n v e r te rs ” from e a ch o f these d evices.

B efo re we p ro ce e d w ith p a rtic u la r c ircu its it is d esira b le to reco n sid er the p ro b lem s th a t a re lik e ly to b e en co u n te re d , b o th so th a t we ca n av o id th em o r c o m b a tth em an d so th a t we c a n c o m p a re d ifferen t c ircu its w ith resp ect to them .

Mixer Problems1) O u tp u t s ig n a ls a t th e I F freq u en cy a ris in g from o th e r th a n th e desired inp u t

signal.2) D is to r t io n o f the m o d u la tio n o f th e d esired in p u t signal.3) T ra n sm iss io n o f th e lo c a l o sc illa to r freq u en cy to th e input c ircu it. ( I f the lo ca l

o sc illa to r s ig n a l re a ch e s th e a n te n n a , it m ay be rad ia ted an d serve as an in te rferin g sig n a l to o th e r receiv ers .)

4 ) N o ise g en era ted in th e m ix e r stage.5) In a d e q u a te g a in in th e m ix e r stage.

F ig u re 7 .4 -1 illu stra te s a freq u en cy sp ectru m show ing a n u m b er o f p o ssib le sig n als th a t m ay ca u se un w an ted co m p o n e n ts a t th e m ixer o u tp u t’s I F frequ en cy ,a) I f a sig nal a t ct>image = o j0 + colF re a ch e s the m ixer, th e d ifference frequ en cy w ill be a»,F. T h e o n ly rem ed y is a d eq u a te filte rin g in fro n t o f th e m ixer. In a new

3 0 6 m ix e r s ; r f a n d if a m p l if ie r s 7.4

Fig. 7.4-1 Desired and possible interference signals in a superheterodyne receiver.

system the c h o ic e o f the h igh est p o ssib le I F freq u en cy w ill ease the im a g e -re je c tio n p ro b lem .

b) A signal a t a>J2 w ill cau se tro u b le if the R F stage h as en o u g h d is to rtio n to p ro d uce a seco n d -h a rm o n ic te rm fro m th e sig nal o r if th e m ixer p ro v id es a b e a t freq u en cy betw een th e co0-te rm and th e seco n d h a rm o n ic o f the in co m in g signal. F ro m E q . (7 .2 -8 ) fo r th e -b ip o la r tra n s is to r ,

, , / i M /„(*) .

ra° - - /n / 0(x) / 0(z) dc’

w here z is the n o rm a liz ed en v elo p e o f th e (ojs/«)-term . W h en z is sm all, /„ (z ) / /0(z) m ay b e a p p ro x im a te d as zn/2 nn !, so th a t

G c K ) _ 2 " - lnl

Gc( w j n ) ~ z " - 1 '

A d eq u a te filte rin g to rem ov e the h a lf-freq u en cy term s m u st be p rov id ed in fro n t o f F E T o r sq u a re -la w R F a m p lifie rs ; o th erw ise , th ese d ev ices w ill d o u b le th e inp u t freq u en cy an d n o la te r s tage w ill rem o v e it. W ith “ l in e a r” R F am p lifiers a ll p rem ixer filte rin g is effective in red u cin g su b h a rm o n ic term s.

c) T h e sig nal a>x is a t a >0 + cos. I t w ill cau se tro u b le in an y system w here a b e a t freq u en cy w ith 2co0 is p o ssib le . H en ce it w ill ca u se tro u b le in tra n s isto r m ixers b u t n o t in tru e sq u a re -la w F E T c irc u its o r in d ifferen tia l-p a ir c ircu its , p rov id ed th a t th e o sc illa to r d riv in g v o ltag e is free o f seco n d h arm o n ics .

d) T h e sig n a l coy falls a t coimage/2; h en ce an F E T R F am p lifie r w ill sh ift it to a>image an d , unless the filte rin g betw een th is R F am p lifie r an d th e m ixer rem ov es th is d is to rtio n term , th ere w ill b e a n un w anted o u tp u t term . A sin gle-en d ed b ip o la r tra n s isto r m ixer w ill p ro d u ce a n o u tp u t s im ila r to th e c o j2 case .

O b v io u sly th e re are m an y o th e r p o te n tia l so u rces o f in te r fe re n c e ; how ever, the a b o v e term s suffice to illu s tra te the p ro b lem . A s a n u m erica l exam p le, co n sid er a ca se w here

u)lF = 4 5 5 k H z, cos = 6 0 0 k H z, <x>0 = 1055 k H z

and h en ce

œ s/2 = 3 0 0 k H z , coy = 7 5 5 k H z , (omage = 1 5 1 0 k H z,

an d ojx = 1655 kH z.


A sin g le-tu n ed c irc u it in fro n t o f a sim p le m ixer th a t is sup p osed to h an d le 5 k H z m o d u la tio n sh ou ld n o t h av e a Q m u ch a b o v e 8 a t 6 0 0 k H z ; if it does, tra ck in g p ro b

lem s w ill b e co m e d ifficu lt an d o v era ll system sid eb an d cu ttin g w ill b e co m e excessive. S u ch a tuned c irc u it w ill red u ce coy by a fa c to r o f o n ly 4 w ith resp ect to the tra n sm is sio n o f cos. S in ce th e s ig n al a t coy m ay b e fo u r o r m o re tim es as b ig as ws to s ta rt w ith

the R F tu n e d -c ircu it o u tp u ts m ay b e eq u al. I f so , an d i f b o th sig n als hav e am p litu d es o f 10 m V , th en in a b ip o la r tra n s is to r m ix e r ’s o u tp u t the u n w anted term w ill b e 25 % o f the d esired sig nal term . In a n F E T m ixer o r a d ifferen tia l p air, th e m ain d is to rtio n term from a 7 5 5 k H z inp u t w ould b e a t 3 0 0 k H z an d w ould b e re jec ted by th e I F filter.

T o s u m m a riz e : A n F E T o r d iffe re n tia l-p a ir m ix e r should be su p erio r to a b ip o la r m ixer from th e v iew p oin t o f un w anted signals. A t least tw o stages o f filterin g a re d esira b le in fro n t o f th e m ixer. A lin ea r , a u to m a tic a lly g a in -co n tro lle d R F am p lifie r is d esira b le in fro n t o f the m ixer. T h e gain red u ces the im p o rta n ce o f the m ixer no ise ,

w hile the gain c o n tro l a llo w s the c ircu it to k eep excessiv e signals from rea ch in g th e m ixer. E xcessiv e s ig n a ls in ev ita b ly lead to in creased d isto rtio n . In sp ite o f o u r o u tline o f th e ideal s itu a tio n , m an y sim p le rece iv ers hav e the m ixer as the first stage and em p lo y a b ip o la r tra n s is to r as the a c tiv e d ev ice in th e c ircu it. F ig u re 7 .4 -2 illu stra tes

such a c ircu it.It is p resum ed th a t the im p ed a n ce fro m th e b ase to grou n d is sm all a t b o th co0

and co1F and th a t the im p ed a n ce fro m e m itte r to g ro u n d is sm all a t a>s and co,F . F a ilu re to satisfy these a ssu m p tio n s lead s to red u ced co n v ers io n tra n sc o n d u cta n ce as w ell as the p o ssib ility o f o sc illa tio n a t the I F freq u en cy . I f o sc illa tio n o ccu rs , it sh ou ld be

Fig. 7.4-2 Bipolar transistor mixer.

308 m ix e r s; r f a n d if am plifiers 7.4

c u ra b le by a re d u ctio n o f th e b a se im p ed a n ce a t a>,F . I f n ecessa ry , a series tra p tuned to co,F co u ld b e c o n n e cte d fro m th e b a se to ground .

B o th in p u t an d o u tp u t tra n sfo rm ers a re tap p ed dow n n o t o n ly fo r th e p rev io u s rea so n s, b u t a lso so th a t th e tra n s is to r im p ed a n ces w ill n o t ca u se excessiv e d etu n in g o f the c ircu its .

A s h as b een p o in ted o u t, it is h igh ly d es ira b le to p rev ent larg e s ig n a ls o f any k ind (excep t th o se fro m th e lo ca l o sc illa to r) fro m re a ch in g th e m ix e r inpu t. H en ce it is n o t su fficien t to c o n tro l th e m ix e r-s ta g e gain o r co n v ers io n tra n sc o n d u cta n ce , say, by c o n tro llin g I EQ. O n e sh o u ld c o n tro l a n a tte n u a to r in the in p u t c ircu it a lso . F ig u re 7 .4 -3 illu s tra te th e key p o rtio n s o f th e a d d itio n o f such a c irc u it to F ig . 7 .4 -2 .

F r o m th e resu lts o f C h a p te r 4 an d C h a p te r 5 w e m ay w rite th e im p ed a n ce o f the d io d e to the s ig n a l freq u en cy as

G A =I kq 2 /t (x)

( 7 - 4 - 1 )k T x l 0(x )

I f we k eep th e in p u t s ig n a ls a t o r b e lo w 2 6 m V (as fro m S e c tio n 7 .2 we k n o w we should), then 2 I t( x ) /x I 0(x ) is n early un ity and Gd is d irectly p ro p o rtio n a l to I k. H en ce in cre a sin g I k w ill sh u n t th e in p u t an d red u ce th e m ix e r d riv in g sig n a l. In cre a s in g I k w ill a lso red u ce I EQ o f th e m ix e r an d h e n ce w ill red u ce Gc , y ie ld in g fu rth er A V C a c tio n . T h e d riv in g so u rce fo r I k w ill b e d eriv ed fro m o n e o f th e a m p litu d e d e te c to r c irc u its o f C h a p te r 10. T h is e ffectiv e d io d e sh u n tin g c o n d u cta n ce red u ces th e Q o f


th e in p u t-tu n ed c ir c u its ; ho w ev er, it o n ly d o es so w hen th e I F o u tp u t is la rg e and h en ce w hen the d esired s ig n al is p resen t, o n e h o p es , a t a re a so n a b le strength .

Differential-Pair MixersF ig u re 7.4—4 sh o w s a p o ss ib le c ircu it.

I f Q 2 and Q 3 a re tru ly id en tica l an d if the inp u t and o u tp u t tra n sfo rm ers are tru ly b a la n ced , th en n o o sc illa to r v o lta g e w ill a p p e a r a cro ss e ith er th e inp u t o r the o u tp u t tra n sfo rm er. R em o v a l o f th e o sc illa to r v o ltag e from th e inp u t rem ov es the


rera d ia tio n p r o b le m ; in a d d itio n , rem o v in g the large o sc illa to r cu rren t co m p o n e n t from the o u tp u t red u ces the s tra in on th e I F tran sfo rm er. (In the s in gle-en d ed stage, the first I F tra n sfo rm er m u st red u ce th e large o sc illa to r co m p o n e n t sufficien tly th at it d o es n o t ca u se n o n lin e a r o p e ra tio n o f th e first I F am p lifier .)

In a d d itio n , b e ca u se o f th e sy m m e trica l c h a ra c te r is tic s o f the d ifferen tia l p air, seco n d -h a rm o n ic c ro ss p ro d u cts shou ld n o t b e gen erated by th is c ircu it. T o av oid even h a rm o n ic term s co m p le te ly , the o sc illa to r v o ltag e m ust be a true sine w ave and the v a ria tio n s in th e c o lle c to r cu rren t o f Q x m ust be kept sm all en o u g h th a t n o o s c illa to r h a rm o n ics are g en erated . M ea su red d a ta a re q u o ted by R C A fo r such a d ifferen tia l-p a ir c ircu it in w hich , for V, = 141 m V and an untuned input (so th at n o in terferin g sig nal re d u ctio n from inp u t filte rin g is p resent), on ly th ree in terferen ce term s (except the im age freq u en cy and a co m p o n e n t at the I F itself) a re w ith in 70 d B o f the desired signal. T h e first o f these in terferen ce term s co rre sp o n d s to ajy in F ig .7 .4 -1 ( 2 o)y - o j0 = a > I F ), th e secon d tw o to a»x and its tw in ( 2 a » 0 - w x = c o I F ,

o)'x — 2 co0 = a>lF). F o r I k0 = 2 . 5 m A all o f these th ree term s w ere red uced by a fa c to r o f 2 5 0 o r m o re b elo w the level o f the desired o u tp u t term . T h e rem ov al o f the w y in terferen ce term req u ires ex a ct sym m etry in the d’lfferen tia l-p a ir ch a ra c te r is tic , w hereas the rem ov al o f th e cox-term s req u ires an a b so lu te ly pure o sc illa to r v o ltage. N e ith e r o f these id eals w ill ev er be ach iev ed in p ractice . T h e a y t e r m is the m o st tro u b le so m e , sin ce it is the c lo sest o n e to the desired sig nal and h en ce the m o st d ifficu lt to filter o u t in itia lly . T h e 60/1 red u ctio n in the a y t e r m th a t is o b ta in e d by using a p ra c tica l d ifferen tia l p a ir ra th e r th an an ideal sin g le-en d ed b ip o la r stage is im pressive.

F E T M ix e rs

F ig u re 7 .4 -5 show s a n o th e r F E T m ixer c ircu it. In th is ca se th e h ig h -im p ed an ce g ate ca n n o rm a lly b e co n n e cte d d irec tly a cro ss the in p u t-tu n ed c ircu it. T h e o sc illa to r v o ltag e feed b ack w ill o cc u r via the g a te -so u rce c a p a c ita n c e , and h en ce w ill be m in im ized if the m in im u m v alu e o f C , is m u ch g re a te r th an C GS. I f th e d ev ice is b iased so th a t th e p eak s o f the o sc illa to r plus sig nal v o ltag es never sw ing it o u t o f the sq u are - law reg ion an d if the o sc illa to r v o ltag e is a p u re sine w ave, then th e o n ly th e o re tica l in terferen ce term (b esid es the im age freq u en cy an d the I F freq u en cy ) will b e a t h a lf th e I F frequ en cy . A gain , if th e seco n d h a rm o n ic is p resent in the o sc illa to r v o ltag e, then th e a>x- and o ^ -te rm s w ill co m e th ro u g h w ith re lativ e valu es p ro p o rtio n a l to the a m o u n t o f th is seco n d h a rm o n ic . S o lo n g as the o sc illa to r v o lta g e is restric ted from sw inging to Vp/2 (th is sa crifices som e co n v ersio n tra n sco n d u cta n ce ), an in creased sig nal level d o es n o t lead to d is to rtio n in the F E T c a s e ; h en ce in p u t A G C (o r A V C ) is n o t necessary . V a r ia tio n o f the co n v ersio n tra n sco n d u cta n ce is ach iev ed by varying the o sc illa to r v o lta g e (see E q . 1 .2 -4 o r E q . 7 .2 -5 ) .

M a n y o th e r m ixer p o ssib ilitie s ex ist, but they w ill n o t be exam in ed in d etail. M o st o f them , b e they m u ltig rid v acu u m tu b es, b ea m -sw itch in g vacu u m tu b es, o r d u a l-g a te M O S tra n s isto rs , ca n be an a lyzed in a stra ig h tfo rw a rd fash io n , using the g a in -co n tro lle d am p lifie r a p p ro a ch o f S e c tio n 7.2.

7 .5 SEMICONDUCTOR CONVERTER CIRCUITS 311

7.5 SEM IC O N D U C TO R CO N V ERTER C IR CU ITS

T h o u g h a sep a ra te o sc illa to r and a sep arate m ixer c ircu it can n o rm a lly be d esigned so th a t each d o es its ow n jo b b est, it is p o ssib le to c o m b in e th e tw o fu n ctio n s in a single a c tiv e d evice. T h is c o m b in a tio n is kn o w n as a co n v erter. H ig h -q u a lity re ceiv ers usually k e e p these tw o fu n ctio n s sep a ra te , w hereas m o st m ass-p ro d u ced

receiv ers c o m b in e them .F ig u re 7 .5 1 show s a ty p ica l b ip o la r tra n s isto r co n v erter stage. (A tw o-pow er^

supply v ersio n is show n to red u ce the co m p le x ity o f th e d raw ing slightly . T h e read er

shou ld have no d ifficu lty in v isu aliz in g th is c ircu it in a sin g le-p ow er-su p p ly form .)In such a c irc u it we d esign the o sc illa to r c ircu it to give the d riv in g level the valu e

o f x = q V J k T d esired for the m ixer operation. In d esign ing the 9Scillator we w ould like to b e a b le to n eg lect th e inp u t s ig n a l c ircu it an d th e o u tp u t I F c ircu it. N o rm a lly o n e d o es n eg lect th em , and th en ch e ck s up by sh ow in g th a t the v o ltag e d ro p s a cro ss these im p ed an ces cau sed by the ca lcu la te d cu rren ts a re tru ly neg lig ib le.

A s in all m ix in g s itu a tio n s , o n e m u st w orry a b o u t o sc illa to r a m p litu d e v a ria tio n a c ro ss the b a n d , s ig n al an d o sc illa to r c irc u it tra ck in g , and m ixer in terferen ce and d is to rtio n p ro b lem s. As u su al, co m p ro m ise s w ill b e n ecessary . F o r exam p le, the

a m p litu d e s ta b ility o f th e c ircu it m ay be im p rov ed by in crea sin g the am p litu d e o f


R F signal input

th e o sc illa tio n s , w hile th e in terferin g o u tp u t term s from o sc illa to r h a rm o n ics o r the d ifficu lties w ith excessiv e o sc illa to r v o lta g e in e ith e r th e I F o r th e in p u t c ircu it a re a ll m in im ized b y d ecre a sin g th e a m p litu d es o f th e o sc illa tio n s to th e ir sm allest p o ssib le values.

A ny o f th e p rev io u s m ixer c ircu its m ay b e tu rn ed in to a co n v erter by co m b in in g it w ith the a p p ro p ria te o sc illa to r c ircu it fro m C h a p te r 6 .

E xam p le 7 .5 -1 F o r th e c o n v e r te r sh o w n in F ig . 7 .5 - 2 , find an exp ressio n fo r v0(t).

Solution. I f a t th e o s c illa to r freq u en cy co0 = l / sj L 3C 3 = 1.5 x 1 0 7 rad/sec th e im p ed a n ce o f th e in p u t-tu n ed c irc u it is a n e ffectiv e sh o rt c irc u it co m p a re d w ith the b a se -e m itte r im p ed a n ce o f th e tra n s is to r , th en th e o sc illa tio n a m p litu d e an d freq u en cy fo r th e co n v e r te r m ay b e foun d b y g ro u n d in g th e b a se o f th e tra n s is to r an d em p lo y in g the resu lts o f S e c tio n 6 .4 . S p ecifica lly ,

a >0 = 1.5 x 1 0 7 rad/sec,

G m(x) ^ G l _ G l L 3 _ Q J Ç

S m Q ^ S m Q 34

7.5 SEMICONDUCTOR CONVERTER CIRCUITS 313

Figure 7 .5 -2

an d fro m F ig . 4 .5 - 6 x = 10( = 2 6 0 m V ). In a d d itio n , sin ce th e lo o p g a in is m u chless th a n un ity a t a n y o th e r freq u en cy , sp u rio u s o sc illa tio n s a t th e R F o r I F freq u en cies a re n o t p o ssib le .

I f w e a lso a ssu m e th a t th e in p u t R F - tu n e d c irc u it is n o t lo ad ed by th e tra n s is to r , then th e tra n s is to r b a se v o lta g e is given by

= 1 r f ^ - 1 ^ 1 2 = 1 .57 m V (l + co s 10 3 t ) c o s 1 0 7 r,

s in ce th e b an d w id th o f th e in p u t-tu n ed c irc u it (BW RF = 1 / R l C l = 5 x 1 0 5 rad/sec) is su fficien t to pass iRF u n d istorted .

N ow w ith the a id o f E q . (7 .2 -1 3 ) , we o b ta in th e I F co m p o n e n t o f c o lle c to r

3 1 4 MIXERS; RF AND IF AMPLIFIERS 7 .6

cu rren t in th e form

'/fW = - ctVg{t)Gc = - a t ’B(f)£in/i(*)//oM -

T h is I F c o m p o n e n t o f c o lle c to r c u rre n t is e x tra c te d by th e o u tp u t-tu n e d c irc u it (a>IF = \ /y j L sC 5 = 5 x 106 rad/sec, B W ,F = 2.5 x 106 rad/sec) to yield

va(t) = ( —iO ,V ) - R 5 /1F(t),

w hich w ith x = 1 0 red u ces to

, 0(r) = ( - 1 0 V ) - ( 2 8 6 m V ) ( l + co s 103 f) co s 5 x 106 t.

T o c h e ck th e a ssu m p tio n s m ad e in o b ta in in g v0(t), we first o b ta in |ZB(7'c<;0)|, w here Z B(p) is th e im p ed a n ce o f th e in p u t c irc u it “ seen ” a t th e b a se o f th e tra n s isto r . A t co0 , R , m ay b e n eg lected c o m p a re d w ith L j a n d to yield

^ 1 2 )(Wq/îDrf) —1 12,

* 48 Q,

w hich is ind eed n eg lig ib le co m p a re d w ith ( 1 + P )/g in = 5 .2 k i i , w hich is the tra n s isto i im p ed an ce a t its b ase te rm in a l. H en ce o u r c a lc u la tio n s o f th e o sc illa tio n freq u en cy an d am p litu d e n eg lec tin g th e in p u t c irc u it is ju stified .

W ith th e te ch n iq u e s o f S e c tio n 5 .5 , th e eq u iv a len t “ lin ea r lo a d in g ” o f th e tr a n s is to r on th e in p u t-tu n ed c irc u it is read ily sh o w n to be

G nl = ~ = g JO + P):kr f

G nl m ay b e re flected to th e in p u t o f th e R F -tu n e d c irc u it as G s d M ^ / L , ) 2 ^ 1/208 k Q ; h en ce as a first a p p ro x im a tio n th e tra n s is to r lo a d in g m ay b e neg lected . A ctu a lly th e tra n s is to r in th is e x a m p le d o es d ecrease the im p ed a n ce o f th e in p u t- tun ed c ircu it by 5 % , a n d th u s th e I F co m p o n e n t o f v0(t) w ill b e 5 % b e lo w th e valu e p rev iou sly ca lcu la ted .

7.6 TUN ED NARROW BAND SM A LL-SIGN AL A M PLIFIER S

T h e usual rece iv er req u ires selectiv ity and gain b o th b e fo re an d a fte r th e m ixer o r co n v e r tçr c ircu it.

T h e sec tio n s in fro n t o f th e m ixer a re kn o w n as R F stages. T h e se stag es should be lin ear, to p rev ent the g e n era tio n o f c ro ss m o d u la tio n o r the g e n era tio n o f d is to rtio n p ro d u cts th at w ill in te ra c t in the m ixer to ca u se s ig n als in the I F band . In the fre q u en cy b an d s w here rece iv er n o ise is a lim itin g system p a ra m e ter , these stag es should have a d eq u a te gain to p rev en t th e h ig h er n o ise o f th e m ix e r s ta g e fro m lim itin g the re ce iv e r’s p erfo rm a n ce . In a d d itio n , th ese R F stages should p ro v id e the desired selectiv ity to p rev en t un d esired s ig n a ls from re a ch in g the m ixer.

T h e fixed tuned stages b etw een the m ixer an d th e d em o d u la to r c irc u it a re kn o w n a s th e I F stages. In A M system s th ese I F stages m u st be lin ea r , w hile in F M system s

7 .6 TUNED NARROWBAND SMALL-SIGNAL AMPLIFIERS 315

their am p litu d e c h a ra c te r is t ic s m ay be m ark ed ly n o n lin ea r. T h e I F stages should have h igh selec tiv ity w ith o u t ca u sin g d is to rtio n in the desired p assb an d . In gen eral, these fixed tuned stages p ro v id e th e b u lk o f the gain betw een the inp u t and the d em o d u lato r .

T h u s the R F stages a re n o rm a lly lo w -gain , re la tiv ely b ro a d b a n d tu n a b le stages,

w hile the I F stages a re h ig h -g a in , re la tiv e ly n a rro w b a n d , fixed tuned stages. I f o n e keeps these d ifferen ces in m ind , the sam e gen eral tech n iq u es m ay be u tilized for d e signing b o th c ircu its .

S in ce o n e o f th e p u rp o ses o f these stages is selectiv ity , the design o f th e ir selectiv e n etw o rk s is im p o rta n t. H ow ever, such d esign is n o t the m ain focu s o f th is b o o k .

F ig u re 7 .6 1 illu stra te s the selectiv ity o b ta in a b le w ith a co m m e rc ia l sev en -p ole m ech a n ica l f i lte r ,t and F ig . 7 .6 -2 illu stra tes a p o ssib le d riv ing and iso la tin g c ircu it for a 4 5 5 k H z version o f such a filter. A g reat v ariety o f such filters is a v a ila b le , som e at relatively low prices.

T h e c o n ce n tra tio n o f se lec tiv ity in o n e “ b o x " and b ro a d b a n d gain in a d ifferent " b o x ” is one re a so n a b le w ay to d esign a receiver. T h e field ad ju stm e n ts req u ired to p lace seven sep a ra te p oles in a B u tte rw o rth o r T sch e b y sch e ff a rra n g e m en t are n o t d esira b le under an y c irc u m s ta n c e s ; h en ce , even if th e filter w ere to be bu ilt up o f lum ped e lem en ts, o n e m igh t w ell c h o o se to have the filter “ co m p u te r d esig n ed ” and p laced in the c ircu it as a unit.

W h a t we will p o in t o u t h ere is the b asic ap p ro a ch to d esign ing n a rro w b a n d sm all-s ig n a l a m p lifiers th a t w ork in to sin gle- o r d o u b le-tu n ed n etw o rk s and are driven from sin g le- o r d o u b le-tu n ed c ir c u its .!

W e shall use th e n a rro w b a n d y-m odel for the a ctiv e c ircu it to p rov id e an an alysis th at w ill be valid fo r single v acu u m tu b es, F E T ’s, o r b ip o la r tra n s isto rs , as w ell as for d ifferentia l p a irs , ca sco d es , o r o th e r d iscre te o r in tegrated c ircu it c o m b in a tio n s .

F ig u re 7 .6 -3 show s th e b a s ic a rra n g e m en t fo r a single stage. T h e y -p aram eters are defined at a sin gle 0 -p o in t by the e q u a tio n s

h ( p ) = y u (p )V t(p ) + y n (p W 2(p) (7 .6 -1 )

andh i p ) = y 2 i(p )v i(p) + y 22(P)V2(p). (7 .6 -2 )

A lth ou g h these p a ra m e ters m ay be ca lcu la te d from som e o th e r d ev ice m o d el, they

t Mechanical filters are mechanically resonant elements whose inputs are driven by, and whose outputs are picked up by, piezoelectric, magnetostrictive, or other electromechanical transducers. The quartz crystal is a mechanical filter; however, as commonly used, the term refers to m ultielement (hence multipole) ceramic elements, usually with flat passbands and sharp falloffs in response outside the passband. C hapter 5 of Solid Slate Magnetic and Dielectric Devices, edited by H. W. K atz (John Wiley, New York, 1959), discusses the physical and electrical properties of a num ber of such devices in some detail.t For those readers with no experience at all with double-tuned circuits, C hapter 18 of E. J. Angelo’s Electronic Circuits (M cGraw-Hill, New York, 1964, second edition) will provide a worthwhile background.

OdBFig. 7.6-1 The am plitude response of a seven-pole commercial mechanical filter in a circuit similar to that of Fig. 7.6-2.

Fig. 7.6-2 O verall IF stage with mechanical filter and preliminary broadband filter to remove spurious response possibilities. [Overall gain can be controlled by varying R, and by capacitively shunting all or p art o f the 200 f l in the last emitter. As shown, with R, = 30 k il, the circuit will handle 1 V peak input with an overall gain of 5.]


l l ( P ) h(P )

Active device

Fig. 7 .6-3 Basic ^-param eter stage.

a re n o rm a lly m easu red , s in ce it is d es ira b le th a t they inclu d e all p a ra s itic re a c ta n ce s as w ell as such d ifficu lt-to -m o d e l p a ra m e ters a s excess p h ase sh ift in tra n sisto rs .

T h e y -p a ra m e te rs a re u sually fu n ctio n s o f b o th Q -p o in t and frequ en cy . S in ce they a re fu n ctio n s o f freq u en cy , o n e m u st exerc ise som e ca re w hen a tte m p tin g to co m b in e , say, y 2 2 from th e d ev ice an d YB from the c ircu it.

In essence, w hat o n e m u st d o is m a k e a “ sm a ll-b a n d ” a p p ro x im a tio n in a n o rm a l m od el. (T h e “ s m a ll-s ig n a l” a p p ro x im a tio n is u sually a lread y im p licit in the y-

p a ra m e ter valu e.) F o r exam p le, a t freq u en cies n e a r w 0 o n e m ig h t a p p ro x im a te y 22 as

y 22 = g 22 + jo j qC 22 + jS(oC'2 2 , (7 .6 -3 )w here Sto = to — to0 .

N ow if Yb w ere a single p a ra lle l R L C c ircu it th a t co m b in ed w ith y 22 to tu n e to oi0 , o n e w ould hav e

Yt = YB + y 22 = (G fl + g 22) + ju ) 0( C 22 + C B)

1+ jSco(C'22 + C B). (7 .6 -4 )

H o w ever, if Sto « co0 , th en 1 /(to0 + ¿to) h o o d o f a>0 ,

(co0 + Scj)Lb

( l / t o o ) — (Sto/coo) so th a t, in th e n e ig h b o r-

Y, — (G b + g 22) + jStol C'22 + C B +1

(l)o L b(7 .6 -5 )

H ere , in stead o f ju s t ta k in g th e valu e o f th e im a g in a ry p a rt o f y 22 a t to0 a s ja>0C 2 2 , we h av e a lso ta k e n th e s lo p e term jdtoC'2 2 . In g en era l, th ere is n o re a so n w hy C 22 should eq u a l C 2 2 , ju s t as th e re is n o re a so n w hy the in cre m e n ta l slo p e o f the d iod e e q u a tio n should eq u a l the ra tio o f the d c ju n c t io n v o ltag e to the d c ju n c tio n cu rren t. W h en th e slo p e o f bxx w ith resp ect to freq u en cy is sm all, then C xx x C'xx. C ases d o arise w here d ifferen ces o f tw o o r th ree tim es o cc u r betw een the tw o term s. (O n e co u ld a lso ta k e a s lo p e term fo r the real p a rt o f yx x ; how ever, s in ce reso n a n ce d o es

n o t ca n ce l g 22 o r g , , , th is re fin em en t is se ld o m n ecessary .)F ig u re 7 .6 -4 show s typ ical 10.7 M H z and 100 M H z co m m o n e m itte r >’-p a ra -

m eters vs. c o lle c to r cu rren t for a g o o d N P N silico n b ip o la r tra n s isto r. F ig u re 7 .6 -5

ou

i11 k nui

318 m ix e r s ; r f a n d if a m p l if ie r s 7 .6

(a) yu : input adm ittance (output short circuit)

0.10

0.08

0.06

0.04

0.02

o!

/ = 10.7 mHz VCE= 10 V

0 10

l.Or-

0.5

0.2

0 . 1

0.05

0.02

0.01

V = 5V7- 'c e jN ^ = 10V

/ = 100 mHz

& - 1 0 V

“ ¿M2 Vce- 5 V

1 1 1 1 10

/c ,m A4 6Ic , raA

10

(b) y : reverse transfer admittance (input short circuit)

QU

I '

7.6 TUNED NARROWBAND SMALL-SIGNAL AMPLIFIERS 319

(c) y2] '■ forward transfer adm ittance (output short circuit)

(d) y 22 : ou tpu t adm ittance (input short circuit)

Fig. 7 .6-4 Com m on em itter y-parameters vs. collector current at two different frequencies. 2N918. Reproduced by permission of Fairchild Semiconductor.


/ , mHz

(a) y u : input adm ittance

/ , mHz

(b) y u : reverse transfer adm ittance

(c) y2l: forward transfer adm ittance (d) y n : output adm ittance

Fig. 7.6-5 Com m on em itter ^-param eters vs. frequency. 2N918. Reproduced by permission of Fairchild Semiconductor.

show s the c o m m o n e m itte r y -p a ra m e te rs vs. freq u en cy for the sam e tra n s is to r (I c = 5 m A , VCE = 10 V). I t is a p p a ren t th a t th ese p a ra m e ters d o indeed vary w ith b o th freq u en cy an d Q -p o in t, b u t th a t o n e ca n m a k e a re a so n a b le m o d el o v er a t least som e small range o f b o th quantities.

C o n sid e r th e p ro b le m o f find ing Yin = Vi(p)/I\(p) in the n e ig h b o rh o o d o f o j 0

w hen b o th th e o v era ll in p u t an d o u tp u t c irc u its a re tuned to co0 and w hen b o th Ya an d YB a re sim p le p ara lle l R L C c ircu its .


S in ce V2(p) = — >'2 i(p )l/i(p)/[>’2 2 (p) + *«(?)]> the effect o f th e fe ed b ack g e n era to r y l 2 V2 is to p la ce an im p ed a n ce in p a ra lle l w ith YA an d . T h e re fo re , we first c a lculate this “ reflected” impedance and then add the y 1 r and J^-terms:

Yu = Y a + y i l - / ' j ' 21 = Y< + y n - y t2y 21Z „ (7 .6 6)Yg + y 22

w here Z , = \/Yt.

I f th e last term o f Yin has a large en o u g h negativ e real p art, then o v er som e range o f freq u en cies, Yin m ay have a n eg ativ e real p art and the c ircu it w ill o sc illa te in th e m an n er o u tlin ed in S e c tio n 6 .10 .

W h a t we w ish to d em o n stra te is a s im p le g ra p h ica l a p p ro a ch th a t n o t only lets us e v a lu a te Yin(jo j0), b u t a lso lets us see how Yin(joj) varies n e a r <d0 an d e x a c tly w hat we sh ou ld d o to c o n tro l th is v a ria tio n .

In gen eral, the p ra c tica l p o ssib ilitie s in c o n tro llin g Yin in c lu d e :

a) Reduction of y 12- T h is is a lw ays d esirab le . W ith a single given F E T , vacuum tube, o r tra n s isto r it ca n be a cco m p lish e d by ad d in g an a d d itio n a l o u t-o f-p h a se feed b ack to ca n ce l som e o r a ll o f th e d evice feed b ack . S u ch a feed b ack c ircu it is kn o w n as a n e u tra liz in g netw ork .

An a ltern a tiv e is to c o m b in e several d ev ices in to a c ircu it such as a ca sco d e c ircu it o r a d ifferen tia l p a ir and then to co n sid er the > -p a ra m eters for the co m p o site c ircu it. As we shall see, y i2 for such a c ircu it m ay be reduced by a fa c to r o f several hu nd red o r m o re fro m y 12 for a sin gle stage.

b) Increasing G B. T h is co rre sp o n d s to lo ad in g th e o u tp u t c ircu it. It w ill red uce the v o ltag e gain o f the c ircu it. I f the lo a d in g is d o n e by the p ro p er co u p lin g o f the im p ed an ce from th e n ex t stage, then it d o es n o t n ecessarily lead to excessiv e p ow er loss.

c) Increasing G A- T h is co rre sp o n d s to lo a d in g the input c ircu it. A gain o n e is trad in g gain for c irc u it s ta b ility an d sym m etry .

F ig u re 7 .6 -6 p lo ts Y,(ja)) fro m E q . (7 .6 -5 ) vs. co in the n e ig h b o rh o o d o f a>0 . F ig u re 7 .6 -7 p lo ts Z,(ju>) = 1/Y ,(ju)).

T h e stra ig h t lin e in F ig . 7 .6 -6 is tra n sfo rm ed in to a c irc le w hen o n e ta k es its re c ip ro ca l to find Z , = 1 / Y t. It w ill be a p p a ren t th a t th e larger G B is m ad e, the sm aller the c irc le o f F ig . 7 .6 -7 w ill b eco m e. In te rm s o f a m ag n itu d e an d an ang le,

e -je(iu>)Z ,[j(o j0t ¿m )] = ,

G B + g 22

w here

6(3a>) = ta n "S oj

5w 3 d B

T h e — 3 d B p a ssb a n d o f the a m p lifie r o ccu p ie s o n e -h a lf o f th e c ir c le ; a ll o th e r freq u en cies fro m m in u s in fin ity to p lu s in fin ity o ccu p y th e o th e r half. A ctu a lly w hen


1G> + 822

Origin^

A

| Increasing w

W =UJo

j8 w (c '22 i c . i

Y,(yojo

Fig. 7 .6-6 Combined circuit and device admittance out at the output o f Fig. 7 .6 -3 . Parallel resonance at u>0 is assumed. is shown in the neighborhood o f resonance.

dm

/ \ ÜJ =Wo

X 1 / )+ £ 2 2 ^ /

Increasing ui

Fig. 7 .6 -7 Reciprocal o f the admittance shown in Fig. 7 .6 -6 .

3(0 is larger, th e a p p ro x im a tio n s b re a k d o w n and y 22 v aries w ith freq u en cy , so th a t th e lo cu s o f Z , w ill p ro b a b ly cea se to b e e x a c tly c ircu la r. S in ce th e a p p ro x im a tio n is very g o o d in th e reg io n o f in te re st an d is b o th a d eq u a te an d c o n v en ie n t e lsew here, we assu m e th a t it h o ld s fo r a ll v alu es o f a).

S in ce re so n a n c e is n o t inv olv ed w ith e ith e r y 12 o r y 2 1 , we w ill a p p ro x im a te b o th o f th em by “ c o n s ta n t” v alu es a c ro ss th e freq u en cy b a n d in q u estio n . F u r th e r m o re w e w ill co m b in e th e re a l an d im a g in a ry p a rts o f th e cu rv es o f th e typ e show n in F ig . 7 .6 - 4 an d 7 .6 -5 so as to rep resen t y 12 a n d y 2 1 a s m ag n itu d es a n d an g les. W ith th is re p re se n ta tio n th e resu lt o f th e m u ltip lica tio n sh o w n in E q . (7 .6 -6 ) w ill b e to ro ta te th e c irc le o f F ig . 7 .6 -7 by th e sum o f th e a n g les o f y , 2 an d y 2 x an d to ch a n g e th e d ia m e te r o f th e c irc le to |yi2 y 2 il/(GB + g 22).

F o r exa m p le , i f y 2l = gm an d y 1 2 = - j ( o 0C r , then

^ 1 2 ^ 2 1 _ % C r S m ^/fTt/2

Y B + y 2 1 G g + g 2 2(7 .6 -7 )


w hich is c irc le 1 in F ig . 7 .6 -8 . N o w a d d in g G A + g n to th e re flected a d m itta n ce w ill shift the c irc le to th e rig h t by th is a m o u n t, in a d d itio n , i f the im a g in a ry p a rt o f YA + y , ! is eq u al to

~ jto 0 C rg m

G b + §22

th en th e c irc le w ill be sh ifted d ow n in to p o s itio n 3 in F ig . 7 .6 - 8 .C u rv e 3 is n o lo n g e r q u ite a c irc le , s in ce its e q u a tio n is g iven by

C u rv e 3 = G A + g u + “ ’° C rKm sin Q(Sco)G B + g l l

+ J \ ^ c o s O(daj) - co0 ( C n + C A) + - - - -L G B + g 2 2 G>0 L A _

+ j d < v (c A + C ’u -i 2T ~ | - (7 .6 -8 )\ o) I L a!

Fig. 7.6-8 Input adm ittance of Fig. 7.6-3 for specific y-param eter and circuit conditions, in particular for y 2i = gm, y !2 = —7 W0 Cr , YB adjusted to resonate with y 22, and YA adjusted to resonate with y , , and the reflected impedance.

T h e m ag n itu d e an d p h ase o f the o v era ll in p u t im p ed an ce are given by the m ag n itu d e an d an g le o f th e v e cto r fro m th e o rig in to a p a rtic u la r p o in t o n cu rv e 3. (U su ally c irc le 3 is a su fficien tly g o o d a p p ro x im a tio n fo r first-o rd e r d esign p u rp oses.)

It is a p p a ren t th a t, to a v o id o sc illa tio n s , cu rv e 3 m u st lie co m p le te ly w ith in the righ t h a lf o f th e a d m itta n ce p lane. H o w ever, in o rd er to have a stage w ith a re a so n a b ly sy m m etrica l in p u t im p ed a n ce an d w ith an inp u t im p ed an ce th a t d o es n o t b eco m e b ad ly m istu ned w h en ev er a n a d ju s tm e n t is m ad e in th e o u tp u t, it is n ecessary fo r the


rad iu s o f c irc le 1 to b e sm all in co m p a ris o n to th e d is ta n ce th a t c irc le 2 is to th e rig h t o f the im ag in ary axis.T h is ra tio o f th e rad iu s o f c irc le 1 to G A + t is ca lled th e a lig n a b ility fa c to r , k :

2^211 (7 .6 -9 )2(Gb + g22)(GA + guY

I t is g en era lly a cce p te d th a t fo r w ell-d esign ed c irc u its k < 0 .20 .A ctu a lly k a lo n e is n o t rea lly a su fficien t s p e c if ic a tio n ; fo r c ircu its w ith eq u a l

valu es fo r k, th o se in w hich c irc le 1 lies a lm o st co m p le te ly w ith in the le ft-h a lf p lan e w ill b e w orse b y a fa c to r o f f th a n th o se c irc u its in w hich c irc le 1 lies a lm o st co m p le te ly in th e r ig h t-h a lf p lane.

T o rep eat o u r e a r lie r co n clu s io n s , o n e c a n red u ce k e ith e r by re d u cin g |_y12| o r by in cre a sin g G B o r G A . (R e d u c in g |_y21| is n o t re a so n a b le , s in ce it red u ces the g a in d irectly .) O n e red u ces \yi2 \ by sw itch in g d ev ices o r Q -p o in ts , o r b y a n eu tra liz in g feed b a ck , o r by c o m b in in g d ev ices in to a c o m p o s ite d evice. W h a te v e r m eth o d is used , o n e m u st c h e ck fo r th e w o rst-ca se co n d itio n s , b e ca u se n e u tra liz a tio n can n o rm a lly b e ach iev ed o n ly to w ith in a c e r ta in to le ra n ce , w hile the _y-paramete> vary w ith te m p e ra tu re and as the Q -p o in t is sh ifted by A V C (o r A G C ) c o n tro l signals.

In o rd er to n eu tra liz e a stage w ith a p u rely ca p a c itiv e in te rn a l feed b a ck , o n e req u ires a p u rely c a p a c itiv e fe e d b a ck o f o p p o site p h ase, a s illu s tra te d in F ig . 7 .6 -9 .

N o rm a lly C r is n o t k n o w n e x a c tly a n d v aries fro m u n it to u n it, w ith tem p era tu re , an d w ith Q -p o in t. In a d d itio n , o n e seld o m uses b e tte r th a n a 1 5 % to le ra n ce c a p a c ito r fo r C N. T h u s it m ay b e re a s o n a b le to red u ce \y12\ to p erh ap s | o f its o rig in a l value, b u t it is u n likely th a t in a p ro d u c tio n ru n o n e w ill a tte m p t to d o b e tte r th a n this. (In h ig h -p o w er a m p lifie rs o n e o ften d o es a tte m p t to m a k e a d ju stm e n ts o n ea ch am p lifie r so th a t |y12| is red u ced b e lo w th is value.)

F o r the p u rp o se o f n u m e rica l co m p a ris o n w e shall c o n sid er a n am p lifie r c o n sis tin g o f a sin gle b ip o la r tra n s is to r o f th e typ e illu stra ted in F ig s . 7 .6 - 4 an d 7 .6 -5 , as w ell as a c a sc o d e an d a d iffe re n tia l-p a ir c o n n e c tio n o f a co m m e rc ia l in teg ra ted tria d such a s th e R C A C A 3005 . In e a ch ca se we a rb itra r ily set G A = g n . T h e co m m o n p ro p e rtie s a re listed in T a b le 7 .6 -1 . T a b le 7 .6 -2 lists th e y -p a ra m e te rs and v a rio u s d erived q u a n titie s fo r a ll th ree cases.

Table 7.6-1 Com m on properties of three amplifiers

/ dc = 2.5m A , |Kcc| = |Ve e I = 6V

Single-tuned, directly connected ou tpu t circuit

O utput coil with La = 8 ¿iH, Q = 100

O utput circuit loaded to obtain + 4 0 0 kH z bandw idth at w 0 = 6.72 x 107 H z; that is, 6 ,oaded = 13.4

Total shunt load from coil losses and device output loss is 7.2 k il ; that is,Gb + g22 = 139 /im ho for each case

The tuning capacitor for each case is CA = (28 pF) — C 22

7.6 TUNED NARROWBAND SMALL-SIGNAL AMPLIFIERS 325

Table 7.6-2 List o f -parameters and various derived quantities for three amplifiers (10.7 M Hz values; all admittances in /imho; 2.5m A in single or cascode stages; 1.25mA in each half of differential pairs)

Single unit Cascodef Differential pa irt

>11 950 + /700 2000 + j3000 600 + /800C „ 10.4 pF 48 pF 12.8 pF

10.4 pF 45 pF 19 pF>12 1 0 1 1—

00

0 0 . 2 2 - /0.08 2 - j2

>2 1 75,000 -715 ,000 6 8 , 0 0 0 - / 1 2 ,0 0 0 -2 0 ,000 + ;500>2 2 60 + 7 170 1 0 + ./150 1 0 + ./150C'22 2.5 pF 2.4 pF 2.4 pF

t'22 2.5 pF 4.8 pF 4.8 pF>12 81 Z _ -97° 0 .2 4 ^ - 2 1 ° 2.8 /_ — 45°>2 1 7 6 ,0 0 0 ^ .-1 1 ° 69,000 — 10° 20,000 /_ + 178.5°

>' 1 2 >' 2 1 6 . 2 x 1 0 " 6 /i,ho2 0.016 x 1 0 - 6 //m ho2 0.056 x 10~ 6 nm ho21_ + 1T Z_+149° ¿1 — 46.5°

( B 29.5 pF 29.6 pF 29.6 pFi ' B 10 1 151 151~ >12>2 1/ (Gjj + 822) 38,500/_ + 72° 1 0 0 Z J4 9 0 348 /_ —46.5°

1 > 1 21 l>2 112 [GB + g j j lP g , , ]

1 0 .1 0.0125 0.146

tT h e y-parameters of the CA 3005 for the stated conditions are given in Figs. 125 and 126, pp. 176 and 177, of the booklet “RCA Linear Integrated Circuit Fundamentals,” Technical Series IC-40, RCA, Harrison, N.J. (1966).

T h e last e n try in T a b le 7 .6 -2 is the a lig n a b ility fa c to r k fo r th e ca se w here G A = g i ! . N o te th a t w hen G A — g t t the sin g le-en d ed ca se is u n stab le , w hile b o th the o th e r stages have k < 0 .1 4 6 . T h e a c tu a l d ifferen ce b etw een the c a sco d e and d ifferen tia l- p a ir c ircu its is n o t as g re a t as th e valu e o f k w ould in d ica te , s in ce c irc le 1 in the ca sco d e ca se is a lm o st co m p le te ly in the le ft-h a lf p lan e w hile in the d iffe re n tia l-p a ir case it is a lm o st en tire ly in the r ig h t-h a lf p lane. T o b e sp ecific , w hen G A = g M and b o th in p u ts a re tun ed to b e res is tiv e a t <w0 , the net in p u t c o n d u cta n ce G,„ is 38 1 4 / im h o in th e c a sco d e c a se an d 1448 /imho in th e d iffe re n tia l-p a ir case , th e d ifferen ce b e in g o n ly 2.65/1.

T o aid in see in g th e d ifferen ce b etw een th e tw o ca se s as w ell as in u n d erstan d in g the m eth o d , it is d es ira b le th a t th e read er tra ce o u t cu rv e 3 fo r b o th cases. F ro m th is cu rv e o n e c a n get an idea o f the in p u t im p ed an ce sym m etry fo r e a ch c ircu it.

I t is so m etim es useful to c o m p a re am p lify in g stages in te rm s o f th e ir pow er gain s, s in ce p o w er d o es n o t have th e in d efin iten ess that is so m etim es a sso c ia ted w ith v o ltag e o r cu rren t gains.

A stra ig h tfo rw a rd a n a ly sis o f F ig . 7 .6 -3 w ill show th a t th e re so n a n t pow er

gain d efined a s th e lo ad p o w er d ivided by th e p o w er supplied by the / ', g e n era to r is

326 m ix er s; r f a n d if am plifiers 7 .6

Fig. 7 .6 -9 Possible neutralization circuitry when y l2

g iven b y E q . (7 .6 - 1 0 ) :

-j(oCr.

Ap =\y2 t\2G B

(&A + S l l + g R) ( g 22 + G b)(7 .6 -1 0 )

T h is e q u a tio n p resu m es th a t th e in p u t an d o u tp u t a re b o th tun ed to r e s o n a n c e ; g R is the resistiv e p o rtio n o f the im p ed a n ce reflected fro m th e o u tp u t th ro u g h th e y 12 g e n era to r , an d G B is assu m ed to b e th e d esired lo ad resisto r.

F o r th e p u rp o se o f c o m p a rin g stag es, we c o n sid er a sp ecia l fo rm o f A P in w h ich g R = 0 a n d b o th th e in p u t an d o u tp u t a re “ m a tch e d ” resistiv ely . F o r th is ca se , A P b e co m e s A Pmiit an d E q . (7 .6 -1 0 ) red u ces to

a \y2\\2 ,n cA PM = — • (7 .6 -1 1 )

o£ u £22

N o w we c o m p a re A Pm^ fo r th e th re e stag es an d A P fo r th e la st tw o cases, a s s u m in g g , i = G A, b u t a l lo w in g g R, G B, a n d g 22 to have th e ir p rev io u s valu es.


(S in ce th e sin g le-en d ed tra n s is to r w ith o u t n e u tra liz a tio n w ill o sc illa te , a c a lc u la tio n for A P is m ean in g less .)

Single unit Cascode Differential pair

41.1 dB 44.5 dB 39.2 dB__ _ — 38.5 dB 35.3 dB

F ro m these resu lts we see th a t, w hile n e ith er o f these co m p o s ite stages yields its “ m a tc h e d ” p ow er gain , tw o stag es o f e ith e r c o m p o s ite type w ould yield a th e o re tica l gain m o re th a n 7 0 d B and a p ra c tica l ga in o f a t least 6 0 dB.

W e m igh t a tte m p t to sa lv age th e sin gle-en d ed stage by n eu tra liz in g it. I f we assu m e th at we c a n n eu tra lize th e rea ctiv e p o rtio n o f y 12 to w ith in 1 0 % , then

& c [ y l2 ] = — 1 0 , -j- 8 > \ i \ ^

from w hich, for the tw o ex tre m e cases, we o b ta in the fo llow ing valu es :

y l2 = - 1 0 + j8 .V1 2 = - 1 0 - 7 8

12.8 / _ + 141.5° 1 2 . 8 Z_— 141.5°0.98 x 10~ 6 /imho2 / L —49.5° 0.98 x 10~b /imho2 ¿_ + 27.5°

+ £ 2 2 ) 6070 —49.5° 6070 /_ -+ 27.5°

I f one co n stru c ts the typ e 1 c irc le s in d ica ted by the values a b o v e, o n e will see th a t the y 12 = - 10 + jS ca se is the w orst ca se b u t th a t, even here, ad d in g 110 0 //mho o f real input c o n d u cta n ce (via g n and G A) w ill yield a b so lu te s ta b il ity ; h ence, if £ 1 1 = G a = 9 5 0 /imho, then the type 3 c irc le s w ill lie co m p le te ly w ithin the righ t- h a lf p lane. T h e a lig n a b ility values will n o t be less th an 0 .20 . H ow ever, sin ce the c irc les are n o t ro ta ted so far from th e real axis to begin w ith , the v a ria tio n a cro ss the band m ay not be excessive. I f th is v a ria tio n still seem s to o m u ch , the next step w ould p ro b a b ly b t to red uce the lo ad seen a t the o u tp u t. T h is can be d on e w ithou t ch an g in g the ban d w id th by using a step -u p tra n sfo rm er betw een the tra n s is to r o u tp u t and the L C c ircu it. F o r exam p le, w ith an effective tu rns ra tio o f 1 :2 we m ay in crease G„ to 6 3 4 /¿m ho; this red u ces the d ia m e te r o f c irc le 1 to 1517 /imho and m ak es k = 0 .40 . T h e resu lt, w ith the sm all an g le o f ro ta tio n o f c irc le 1, should be a u sab le stage.

N o te th a t th is d elib e ra te m ism a tch in g a t th e o u tp u t has d ecreased the p ow er gain w hile in crea sin g the a lig n a b ility . T h is trad e is o ften a re a so n a b le o n e to m ake. F o r th is n eu tra lized , heavily load ed stage, as specified for the y ,2 = - 10 + jH / i m h o case, the a c tu a l re so n a n t p ow er gain is 3 1 0 0 or 3 4 .9 d B ; h ence the sin gle-en d ed ca se is still co m p e titiv e w ith the c a sco d e and d ifferen tia l-p a ir ca ses if ga in a lo n e is c o n sidered. T h u s by p artia l n e u tra liz a tio n and the use o f sufficien t inpu t o r o u tp u t m ism a tch in g we ca n o b ta in re a so n a b ly a lig n a b le s in g le-u n it stages w ith pow er gains o f the o rd er o f 3 0 d B per stage.


7.7 STAGES WITH DOUBLE-TUNED CIRCUITS

F ig u re 7 .7 -1 illu s tra te s th ree p o ssib le typ es o f co u p lin g betw een sm all-s ig n a l d o u b le tuned c ircu its . O b v io u sly a g re a t m an y v a ria tio n s are p o ssib le w ith su ch c ircu its . T o illu stra te th e b a s ic co n ce p ts we sh a ll re s tr ic t o u rselves to th e sp ecific cases, o u tlined b e lo w :

a) e a ch h a lf-c ircu it (u n co u p led ) id e n tica l an d tun ed to a>0 ;b) o n ly a sin gle typ e o f co u p lin g in a g iven c ir c u i t ;c) o v era ll b an d w id th s o f 5 % o r less o f co0 .

F o r all such ca ses w here Y , = Y, t ,

Vi(p) . YJp)I lip) YAp )[2YJP) + YAp)]'

Vi(P) _ Yt(p) + YJp) l i tp) Yl(p)[2YJp) + YAP)]'

T h e tran sfer im p ed a n ce fo r su ch ca ses a lw ays has tw o sets o f co m p le x poles. O n e set is a t th e z ero s o f Yt(p) an d h en ce h as a rea l p a rt o f — l / 2 R lC l and a re so n a n t freq u en cy o f l / s/ L 1C l . T h e q u a d ra tic fo r th e o th e r set o f p o les is th e o n e resu ltin g from th e [ ^ ( p ) + 2 y „ (p )]-te rm ; h e n ce it w ill v ary w ith the typ e o f co u p lin g . T a b le 7 .7 -1 lists th e resu lts fo r th e th ree ca se s u n d er co n sid era tio n .

Table 7.7-1 Comparison of three types of coupling

Inductive M utual inductive Capacitive

Q uadratic

Scale factor Zeros at origin

2 P 1 + 2 / c 2 P 1 D2 1 PP R lC i LC

L J L m ko>l!C,1

v R , C , (1 + k)LC

MIL,wJ/cL,/(l - k2)1

v + 2k)

1+ LC{ 1 + 2k)

c j c {k/( 1 + 2k)Ct 3

A ctu ally fo r m u tu a l in d u ctiv e co u p lin g th e co n v en ie n t fo rm u la tio n p u ts the d e n o m in a to r in a fo rm in w hich b o th sets o f p o les are fu n ctio n s o f k. T h e n o rm a l a p p ro x im a tio n fo r sm all v alu es o f k is to ig n o re the v a ria tio n s in Q cau sed by k ( th a t is, to n eg lect th e h o r iz o n ta l m o tio n o f th e p o le in th e ca p a c itiv e ca se ) an d to co n sid er the v ertica l m o v em en t. F o r k « 1,

1 i * 1, « 1 — - an d x 1 — k.y i + k 2 + 2k

t E. J. Angelo (Electronic Circuits, McGraw-Hill, New York, 1964, second edition) illustrates the straightforward extension to the case where Y2(p) = aYt(p).

7 .7 STAGES WITH DOUBLE-TUNED CIRCUITS 329

L m

Fig. 7.7-1 (a) Inductive coupling, (b) M utual inductive coupling, (c) Capacitive coupling.

T h e p ole m o tio n fo r th e up p er set o f co m p le x p o les for the th ree cases is show n in F ig . 7 .7 -2 . I t is a p p a ren t th a t, e x ce p t for the skew ness caused by the v a ria tio n s o f Q w ith k, a ll th ree cases m ay be used to o b ta in m a x im a lly flat o r o th e r tw o -p o le n etw o rk a d ju stm en ts . T h o u g h the in d u ctiv e ca se has the easiest a lg eb ra , the o th e r tw o cases turn o u t to be m u ch m o re p ra c tica l. B o th o f these cases are w idely used. F o r m a x im a l fla tn ess in th e ca p a c itiv e case , a>0C mR = 1 ; in the m u tu al in d u ctiv e ca se , o)0 M R = 1.

T h e q u e stio n th a t n a tu ra lly a rise s i s : H ow d o es the inp u t im p ed an ce o f a d o u b le tuned c ircu it a ffect the s ta b ility and a lig n a b ility ca lcu la tio n s o f the p rev io u s sec tio n ? F ro m F ig . 7 .6 - 6 it is c le a r th a t th e a d m itta n ce reflected in to the inpu t side o f the y -p a ra m e te r m o d el is a lw ay s - y ^ y n Z , , w here Z , is th e to ta l im p ed a n ce a cro ss the y 2 1 g en era to r. T h a t is,

Z , = ------- 1------- ..F22 + 5b

330 m ix e r s ; r f a n d i f a m p l i f i e r s 7.7

Lku> o

3

u* " L

L 4 1 1

i_2 * ,C , “ " 1

i L I==^=ta»0 *

*w 0

r(a) (b) (c)

Fig. 7 .7 -2 (a) Inductive coupling, (b) M utual inductive coupling, (c) Capacitive coupling.

I f w e a ssu m e th a t y 22 is a b s o rb e d in to th e p rim ary tu n ed c irc u it , th en Z , is th e in p u t im p e d a n ce fo r th e d o u b le -tu n e d c ircu it. E q u a tio n s (7 .7 -2 ) an d (7 .7 -3 ) list Z , fo r th e m u tu a l in d u ctiv e an d c a p a c itiv e co u p lin g c a s e s :M utual inductive coupling

P\P +

Z , =« 1 C ,

+col

c A p 2 + +Oil

P2 +w l

R i c , l

(7 .7 -2 )

C apacitive coupling

Z, =1 + k \

1 + 2k

P2 + +col

R i C t i 1 + k) 1 + k

clP2 +R i c 1

+ col p 1 +col

+ 2 k ) I + 2 k

(7 .7 -3 )

T h o u g h th ese e q u a tio n s lo o k ra th e r fo rm id a ble, b o th ca n be g reatly sim plified if we co n sid er o n ly th e reg io n n e a r co0 = 1 /y /L ^ C i an d if we m a in ta in o u r a ssu m p tio n o f k « 1. In th is ca se b o th e q u a tio n s h a v e tw o p o les th a t a re a p p ro x im a te ly eq u ally sp aced a b o u t th e zero . I f we c o n sid er th e m a x im a lly flat tra n sfe r fu n ctio n ca se , then Z , a t th e c e n te r freq u en cy [co0 in th e m u tu a l in d u ctiv e ca se an d co0( l — k/2) in th e c a p a c itiv e c o u p lin g ca se ] is R /2 , w hile th e a p p ro x im a te d n o rm a liz ed valu es o f th e im p ed a n ce in te rm s o f a = 1/ 2 / ijC j = c a J 2 Q T a re g iven b y T a b le 7 .7 -2 . T h e cen ter-freq u en cy tra n sfe r im p ed a n ce fo r the sam e case is R /2(1 — k 2), w hich is a p p ro x im a te ly R /2 and thu s a p p ro x im a te ly the sam e as Z ( .

F ig u re 7 .7 -3 p lo ts the n o rm a liz ed d a ta o f T a b le 7 .7 -2 an d c o m p a re s th em w ith a c irc le o f rad iu s 5 . T h a t is, F ig . 7 .7 -3 c o m p a re s th e d a ta o f T a b le 7 .7 -2 w ith the inp u t im p ed a n ce to b e e xp ected fro m a s in g le-tu n ed c irc u it h av in g th e sam e ce n te r freq u en cy an d a lo a d R. H o w ever, th is is ju s t th e c a se o f th e sin g le-tu n ed c ircu it.

T h e use o f th e sam e g ra p h ica l a p p ro a c h to ca lc u la te th e co m p le te reflected im p ed a n ce an d the re su lta n t a lig n a b ility is a p erfectly s tra ig h tfo rw a rd p roced u re. H o w ever, it is a p p a ren t th a t i f the s in g le-tu n ed ca se w ith lo ad R is s tab le , th en , b e ca u se the new im p ed a n ce lies w ith in th e c irc le o f the sin g le-tu n ed case , th e d o u b le tu n ed , m a x im a lly fla t ca se w ith a lo ad o f R per side w ill a lso be stable .

7 .8 GAIN CONTROL CIRCUITS 331

Table 7.7-2 Norm alized input im pedance for identically tuned coupled circuits (R , on each side)

ôw/ct 9te(ZJR) J ^ Z J R ) | ZJK Angle

0 0.50 0.00 0.50 0°+ 1 0.60 +0.20 0.635 + 18.5°+ 2 0.259 + 0.40 0.50 + 53°± 4 0.070 + 0.25 0.26 + 74°

Fig. 7.7 -3 Plot of the norm alized Z jn for a maximally flat double-tuned identical set of tuned circuits.

7.8 GAIN C O N T R O L C IR CU ITS

S in ce th e level o f th e sig n als p resen t a t the inpu t o f a receiv er ca n vary by m o re than 1 0 6 /1 w hile the “ l in e a r” ra n g e o f o p e ra tio n o f m o st am p lifiers is 1 0 3 /1 o r less and the p ra c tica lly useful ra n g e o f m an y d em o d u la tio n c ircu its is on ly a b o u t 1 0 /1 , it is im p erativ e th a t m o st rece iv ers hav e a b u ilt-in a u to m a tic gain c o n tro l c ircu it.

S u ch a c irc u it sh ou ld sense th e lo n g -te rm av erage (lo n g -te rm w ith resp ect to the low est m o d u la tio n freq u en cy ) valu e o f th e sig nal an d a d ju st th e am p lifie r gain in such a m an n er th a t a ll s tages o p e ra te w ith in th e ir o p tim u m signal level ranges. Id eally , as th e in p u t sig nal level in crea ses , the gain should rem ain a t its m ax im u m valu e un til th e o p tim u m d em o d u la to r in p u t level is reach ed , and then for fu rth er in crea ses in th e in p u t th e g a in sh o u ld d ecre a se in su ch a fash io n th a t the o u tp u t ca rrie r

level rem ain s co n sta n t.F ig u re 7 .8 -1 illu s tra te s the ideal s itu a tio n fo r the ca se w here the o p tim u m

d em o d u la to r v o lta g e in p u t is 1 V and th e m ax im u m am p lifier gain is 8 0 d B . P ra c tic a l

332 m ix e r s; r f a n d if am plifiers 7 .8

10,000

® 1000

.£3 100

10 0.00110 100 1000 10,000

Input, |iV (a)

10 100 1000 10,000 Input, |iV

(b)

Fig. 7 .8 -1 (a) Ideal gain vs. input signal for A G C amplifier, (b) Ideal v0(t) vs. vt(t) for A G C amplifier.

a m p lifiers u su ally d ep a rt fro m th e id ea l s itu a tio n b o th a t low in p u t levels, w here the g a in b eg in s to d ecrea se so o n e r th a n it sh o u ld , an d a t h igh levels, w here som e am p lifiers b e co m e serio u sly o v erlo ad ed .

A n o th e r p ra c tica l p ro b le m w ith m o st g ain c o n tro l sch em es is th a t th ey m od ify the a c tiv e d ev ice ’s in p u t an d o u tp u t im p ed a n ce an d h en ce cau se d etu n in g and/or b an d w id th ch an g es. In a d d itio n , m an y o f th e sch em es m od ify th e c ir c u it ’s n o ise and d is to rtio n p ro p e rtie s in w h at m ay tu rn o u t to b e an u n sa tis fa c to ry m an n er. T h u s th e b e st gain c o n tro l c irc u it is u sually a co m p ro m ise a m o n g a n u m b er o f c o n flic tin g req u irem en ts.

B efo re we ca n in te llig en tly d iscu ss o v era ll system s, we need to o u tlin e th e p o ssib le gain c o n tro l m ech an ism s. A s we sh a ll see, th ere m ay b e sev eral p o ssib le m ean s o f v ary in g the g a in o f a given d evice. T h e s e m ay v ary w ith th e freq u en cy o f o p e ra tio n . In so m e ca se s sev era l id e n tic a l- lo o k in g c irc u its m ay in fa c t o p e ra te in so m ew h at d ifferen t m a n n ers , s in ce they o p e ra te a t w idely d ifferent cen te r freq u en cies.

T h e m o st s tra ig h tfo rw a rd m e th o d o f gain c o n tro l is to vary g m in th e d ev ice m o d el show n in F ig . 7 .8 -2 . S u ch a sim p lified m o d el can rep resen t a v acu u m tu b e , a ju n c t io n F E T , a M O S F E T , o r a b ip o la r tra n s is to r in th e freq u en cy ra n g e w here b o th th e b a se sp rea d in g re s is to r rBBa n d th e b a se re c o m b in a tio n re s is to r r B C o r r n m ay b e n eg lected a s w e h av e b een d o in g in p rev io u s section s.

V a cu u m tu b e p en to d es e x ist in w hich gm v aries sm o o th ly (b u t n o t lin early ) fro m , say, 5 0 0 0 /imho a t z e ro g r id -to -ca th o d e v o lta g e to zero m ic ro m h o s a t a Vgk va lu e o f — 3 0 V . In o th e r tu b e s th e sam e g m ra n g e m ay b e co v ered in o n ly a 5 V sw ing o f th e b ia s v o ltag e.

T h e w id e-ran g e o r “ re m o te -c u to ff” d ev ice is m ad e by effectiv ely p a ra lle lin g (in the sam e en v elo p e) sev eral tu b e stru ctu res . O n e stru ctu re h a s a h igh gm a t zero v o ltag e b u t cu ts o ff a t a re la tiv e ly low g r id -to -ca th o d e v o lta g e , w hile a n o th e r h a s a low g m a t zero v o lta g e an d d ecreases o n ly slo w ly w ith in cre a sin g b ias. T h e a d v a n ta g e o f su ch a c o n tro l c h a ra c te r is t ic is th a t, a t large b iases a n d low gm’s an d h en ce low gain s, th e g m is n early c o n s ta n t o v er a w ide v o lta g e r a n g e ; h e n ce th e d ev ice w ill be a b le to h an d le la rg e sig n als w ith o u t d is to rtio n .

F ig u re 7 .8 -3 illu s tra te s th e g m vs. K ;k fo r a p e n to d e v a cu u m tu b e , a s w ell a s the g a in fo r a tw o -sta g e ta n d em o f such p en to d es in w hich R t = 1 0 k i2 is assu m ed fo r

7 .8 GAIN CONTROL CIRCUITS 333

'o ^ (P)

Fig. 7 .8 -2 (a) 7t-model for a device, (b) Equivalent y-model for the device of Fig. 7.8—2(a).

ea ch stage. T h e am p lifie r illu s tra te d d o es hav e a ran ge o f c o n tro lle d gain o f m o re

th an 106 /1. W h e th e r th is w h o le ra n g e is u sab le is a q u e stio n we m u st now exam in e.E ven if gm is set to zero in the m o d els o f F ig . 7 .8 -2 , the rem a in in g v o ltag e “ g a in ”

o f a single stage has a m ag n itu d e o f l/^/l + [(g0 + GL)/w C 2] 2 w here G L is the load

co n d u cta n ce . F o r large valu es o f to th is term a p p ro a ch e s unity . I f C 2 = 1 p F ,a) = 60 x 10^, an d g 0 + G L = 1 0 0 //mhos, th en th e re su lta n t “ g a in ” is 0 .51 ra th er th a n zero . I f tw o stag es a re ca sca d e d , th en the o v era ll g a in d ep en d s n o t o n ly o n how the g m o f e a ch s tage is c o n tro lle d , b u t a lso on th e co m p o s ite m od el. T h e o v era ll resu lt is th a t th e ca p a c itiv e feed th ro u g h term p rev en ts o n e fro m rea ch in g the low er p o rtio n s o f th e c h a ra c te r is t ic o f F ig . 7 .8 -3 a t h igh freq u en cies. T h is ca p a c itiv e effect is red uced in th e sam e w ay th a t it w as red u ced in S e c tio n 7 .6 , th a t is, by n eu tra liz a tio n .

In a lm o st all A G C c ircu its , p ro p er n e u tra liz a tio n w ill ex ten d th e lo w er-g a in end o f th e cu rv e by the o rd e r o f 2 0 d B . T h e lim it to su ch ex te n s io n s is th e a ccu ra cy o f the n e u tra liz a tio n o f a p a rticu la r stage. In a d d itio n to the n o rm a l to le ra n ce p ro b lem s, the A G C a c tio n is likely to vary the in tern a l feed b ack p a ra m e ters , and a t low gains th e large in p u t sig n als p resen t m ay effectively vary the “ sm a ll-s ig n a l” m od el p aram eters.

In b o th ju n c t io n and M O S F E T ’s, gm is p ro p o rtio n a l to yJTD\ h en ce o n e cano b ta in gm vs. ^ 1D (o r g„ vs. Vas - Vn cu rv es for M O S tra n s is to rs o r gm vs. V„ — Vccu rv es for ju n c t io n F E T 's ) . In e ith er ca se it is p o ssib le for the m a n u fa ctu rer to bu ild “ re m o te -c u to ff" d ev ices o r for th e user to p a ra lle l several u n its to o b ta in th is effect. In th eo ry , o n e ca n o b ta in resu lts s im ila r to th o se for vacu u m tu b es, a lth o u g h p ro b a b ly

3 3 4 m ix er s; r f a n d if am plifiers 7 .8

Grid-cathode dc voltage, V Fig. 7 .8 -3 g m and two-stage gain vs. Vg k (vacuum-tube gain-controlled amplifier).

w ith a som ew h at red u ced g a te -so u rce range. A gain n e u tra liz a tio n w ill be n ecessary to reach the low er gain levels.

F o r b ip o la r tra n s isto rs o p era ted in the freq u en cy ran ge w here F ig . 7 .8 -2 is a valid rep resen ta tio n (c o C jrBT » 1 ) and w ith a low en o u g h b ias cu rren t th a t the em itter- b ase tra n s itio n reg io n ca p a c ita n c e d o m in a te s C t (C , = C„ = C B C), g m is p ro p o rtio n a l to I E and thu s th ere is a lin ea r re la tio n sh ip betw een gain an d e m itte r cu rren t. F o r m an y h ig h -freq u en cy tra n s isto rs w ith sm all effective d iffu sion c a p a c ita n c e s , b ias cu rren ts b e lo w 1 m A lead to th e d esired o p e ra tio n , t

t For higher currents C , becomes a linearly increasing function of /£ ; hence the current gain becomes a "constant” as the decrease in the input impedance tends to compensate for the increase in gm. In addition, the variation o f C , with IE may cause serious detuning problems.

7.8 GAIN CONTROL CIRCUITS 335

A n u m b er o f o th e r typ es o f gain c o n tro l c ircu its are p o ssib le an d a re in use.A ll tra n s is to rs have a lp h a s an d b e ta s th a t a re to som e e x te n t fu n ctio n s o f th e d c

b ias cu rren t. N o rm a lly the tra n s is to r d esign er tries to m in im ize th ese c h a n g e s ; how ever, in an A G C s itu a tio n he m ay try to u tilize them .

O n e a p p ro a ch is rep resen ted by the “ te tro d e ” tra n s is to rs , in w hich a seco n d

b ase co n n e c tio n is ad ded to the d evice. T h e seco n d b ase cu rren t now c o n tro ls the a lp h a o r b eta o f th e “ n o rm a l” tra n s isto r . B e ta v a ria tio n s o f 20/1 o r m o re sh ou ld be p o ssib le for a 1 0 / 1 v a ria tio n o f c o n tro l cu rren t.

A second a p p ro a ch uses th e fact th a t th e gain o f cer ta in b ip o la r tra n s is to rs falls off, n o t o n ly as l E in crea ses , b u t a lso as Ktb d ecrea ses, p a rticu la rly w hen th e a b so lu te valu e o f K n is sm all. In such tra n s isto rs (a 2 N 2 4 1 5 a t 7 0 M H z , for exam p le), by v arying VCB from 1.2 V to 2 .5 V (th is is eq u iv a len t to a 1.3 m A ch a n g e a c ro ss a 1 k f i res is to r) o n e ca n ch a n g e th e gain by a fa c to r o f 100 (4 0 d B ). F o r larg er valu es o f VCB the gain will be re a so n a b ly in d ep en d en t o f K : b (o r VCE). A c ircu it using this p h en o m en o n for gain c o n tro l p u rp o ses is kn o w n as a “ fo rw a rd -a ctin g A G C ” c ircu it.

S in ce v a ria tio n o f VCB, esp ec ia lly for sm a ll'v a lu es o f cau ses a large v a ria tio nin C B C, it w ill b e d ifficu lt to a ch iev e n e u tra liz a tio n w ith “ fo rw a rd -a c tin g ” A G C o f the v a ria b le -v o lta g e type.

A ll b ip o la r gain c o n tro l c irc u its in w hich l E is varied a c tu a lly o p e ra te by m ean s o f a c o m b in a tio n o f the m e ch a n ism s o u tlin ed p rev io u sly plus m ism a tch in g effects at b o th the inp u t and o u tp u t o f th e d evice. F ig u re 7 .6 -4 c le a rly in d ica tes such input

and o u tp u t a d m itta n ce v a ria tio n s a t 10.7 M H z and 100 M H z. A t low er freq u en cies the v a ria tio n s in o u tp u t im p ed a n ce m ay b e co m e less im p o r ta n t ; how ever, the v a ria tio n o f inp u t im p ed a n ce w ith I E co n tin u e s d ow n to dc.

In the usual tuned a m p lifier , o n e d o es n o t w ant to d ep end on m ism a tch in g for gain c o n tro l, s in ce m ism a tch in g a lso ca u ses d etu n in g an d se lec tiv ity c h a n g e s ; h ence o n e d elib era te ly tap s the inp u t an d o u tp u t tra n sfo rm ers dow n to th e p o in t w here the c ircu it n o t o n ly is s ta b le and a lig n a b le but is a lso re lativ ely im m u n e to shifts in tu n in g caused by A G C a c tio n .

T h e g a in s o f c a sco d e c o n n e c tio n s are varied as a co m m o n em itte r stage, w hile the gm o f the d iffe re n tia l-p a ir co n fig u ra tio n is c o n tro lle d by v ary in g I k.

A n o th er a p p ro a ch to th e A G C o r A V C p ro b le m is to use a sep arate a tte n u a to r sec tio n o r to shu n t o n e o f the tuned c ircu its by a v a ria b le a tte n u a to r . P o ss ib ilitie s inclu de an F E T o r lig h t-co n tro lle d re s is to r as illu stra ted in F igs. 6 .9 -3 and 6 .9 -4 or a d iod e a tte n u a to r as illu stra ted in F igs. 5 .5 -7 , 5 .5 -9 , o r 7 .4 3.

In any ca se , o n e m ust d erive a fe ed b ack v o ltag e (o r cu rren t) to drive the A G C c ircu it from the sig nal itself. T h is is n o rm a lly d o n e by m ean s o f o n e o f the d e te c to r c ircu its o f C h a p te r 10. O n e should n o te th a t the desired re la tio n sh ip betw een the A G C v o ltag e (o r cu rren t) an d an in crea se in the s ignal am p litu d e d ep en d s on th e type o f A G C em ployed . It should a lso be a p p a ren t th a t the d esira b le fea tu re o f a “ d e la y ” in the s ta rt o f gain red u ctio n until a d esired m in im u m o u tp u t is o b ta in e d m ay be in co rp o ra ted e ith e r in th e d e te c to r c irc u it o r in th e g ain c o n tro l c ircu it. F o r exam p le, b ip o la r c ircu its th a t dep end on a re d u ctio n o f VCB d o n o t p ro d u ce large ch a n g es in gain until VCB falls below a b o u t 2.5 to 3 V. H en ce if Vc c is 9 V , if th ere is a 1.5 k i i


d ro p p in g re s is to r, if VB = 0, an d if 1EQ = 3 m A , then the first 1 m A ch a n g e in I E w ill n o t cau se m u ch ch a n g e in g ain , w hile th e seco n d 1 m A ch a n g e in I E w ill ca u se a large ch an g e in the gain.

It should a lso be a p p a ren t th a t, s in ce the A G C system is a fe ed b ack system , the lo o p filterin g m u st b e p ro p erly d esigned o r o sc illa tio n s m ig h t o ccu r. In p a rticu la r, excessiv e lo w -freq u en cy filterin g m u st n o t be added to rem ov e the m o d u la tio n sig nal from the c o n tro l sig nal path .

7.9 N O ISE , D IST O R T IO N , AND C R O SS M O D U LATIO N

N o ise and cro ss m o d u la tio n are p ro b le m s on ly in the early stages o f a re ce iv e r ; if these p ro b le m s are n o t solved there , no la ter stage ca n c o r re c t them . N o n lin e a r d is to rtio n m ay o cc u r in an y stage, b u t o n ce a d eq u a te filte rin g has rem ov ed any in terferin g ca rr ie rs , th is ty p e o f d is to rtio n is m o st lik e ly in th e la st sta g e o f the a m p lifier w here th e signal level is a m ax im u m .

T h e tech n iq u es o f C h a p te r 4 in d ica te , for v a rio u s d ev ices, w hat sig nal levels are p o ssib le w ith o u t such n o n lin e a r d is to rtio n an d w hat form it w ill ta k e w hen it o ccu rs. W e sh a ll n o t d iscu ss it fu rth er here.

N o ise has several p rim e sou rces. O n e is the a n te n n a ; n o ise arr iv in g fro m th is so u rce m ust b e rem ov ed by a d eq u a te filtering . I f a ll th e e a rlie r stages o p e ra te lin early and if n o ise term s a t a ll th e p o te n tia l in terferin g freq u en cies are ad eq u a te ly rem ov ed , then the I F selectiv ity w ill rem ov e the o u t-o f-b a n d p o rtio n s o f such no ise . T h e secon d co m p o n e n t o f n o ise is in -b a n d n o ise gen erated in th e rece iv er itself. Su ch n o ise is p ro d u ced in the F E T , th e b ip o la r tra n s isto r, o r th e v acu u m tube. W h ile every d ev ice has its ow n p ecu lia rities , the n o ise n o rm a lly has b o th an inp u t c ircu it and a n o u tp u t c ircu it co m p o n e n t and it n o rm a lly in creases w ith the b ias cu rren t. In g en era l, a n te n n a n o ise d o m in a te s b e lo w a b o u t 3 0 M H z and fro n t-en d rece iv er n o ise d o m in a te s a t h igh er freq u en cies.

O n e w ay to co m p a re am p lifiers w ith resp ect to their n o isin ess is by the n o ise figure, F :

P S ig n al/ N o ise R a t io a t In p u t

S ig n al/ N o ise R a t io a t O u tp u t

F o r an ideal n o ise less a m p lifier , F is u n ity o r zero d B , sin ce the am p lifie r ad ds no n o ise o f its ow n. P ra c tic a l am p lifiers a lw ays hav e F > 1. (O b v io u sly b o th inp u t and o u tp u t signal/noise ra tio s need to be m easu red in the sam e b a n d w id th ; o therw ise, valu es o f F a p p a ren tly less th a n o n e ca n be o b ta in e d .)

In a d d itio n to the n o ise p ro d u ced in the d esired ban d itself, b o th a n te n n a and receiv er n o ise o u ts id e th e b an d m ay get tra n sla ted in to the ban d by d irec t m o d u la tio n o f an am p lifier, by cro ss m o d u la tio n in an am p lifier , o r by b e in g a t the a p p ro p ria te m ixer in terferen ce p o in t. M o d u la tio n by lo w -freq u en cy n o ise (w hich is o ften an o rd er o f m ag n itu d e larg er th a n in -b a n d no ise) is avoid ed by p ro v id in g a d eq u a te low - freq u en cy b y p assin g and by o p e ra tin g th e in itia l a m p lifier in a lin ea r m an ner.

7 .9 NOISE, DISTORTION, AND CROSS MODULATION 337

S in ce the a m p lifie r n o ise figure is re la ted to th e a m p lifie r ’s p ow er g ain , it w ill be a fu n ctio n o f th e so u rce im p ed a n ce seen by th e d evice. N o rm a lly th ere is a level o f so u rce im p ed an ce th a t is o p tim u m fro m th e v iew p oin t o f th e n o ise figure. F o r tu n ate ly , th is m in im u m in F a s th e so u rce im p ed a n ce is varied is fa irly b r o a d ; h en ce 2/1 n o ise m ism a tch in g sh o u ld ca u se less th a n a 1 d B in crea se in F . F o r a ju n c t io n o r M O S tra n s isto r th e o p tim u m so u rce im p ed a n ce is o f the o rd er o f 2 k i i , fo r a b ip o la r tra n s is to r 2 0 0 i 2 is a p t to be m u ch m o re rea so n a b le .

S in c e th e e rra tic d iv isio n o f c u rren t b etw een th e grid s m ak es m u ltig rid tu b es m u ch n o is ie r (p erh a p s five tim es n o isier), o n e uses trio d e v a cu u m tu b es as R F am p lifie rs in freq u en cy reg io n s w here rece iv er n o ise is im p o rta n t. O f co u rse , a “ g o o d ” p en to d e m ay still b e b e tte r th a n a n o isy trio d e , so d iscr im in a tio n m u st still b e exercised here.

A g o o d R F p en to d e a t 6 0 M H z m igh t hav e an o p tim u m so u rce im p ed a n ce o f 3 k Q an d a n o ise figure o f 4 d B , w hile th e sam e tu b e co n n e cte d as a trio d e m ig h t have an o p tim u m so u rce im p ed a n ce o f 1.4 k Q and a n o ise figure o f 2 dB.

C ro ss m o d u la tio n ca n o cc u r w hen ever th e series e x p a n sio n for the o u tp u t cu rren t a s a fu n ctio n o f th e in p u t v o lta g e c o n ta in s a cu b ic term . F o r exa m p le , i f th e o u tp u t cu rren t o f a n a m p lifie r has th e form

'„W = /dc + gm!’(f) + a 2v2(t) + rt3 i-3 (f), (7 .9 -2 )

w here v(t) is assu m ed to have th e form

v(r) — Vl c o s a j 1t + V2[l + m 2f(t)'\ co s w 2t (7 .9 -3 )

[\f(t)\ < 1 an d th e h igh est ra d ia n freq u en cies in f ( t ) a re m u ch less th a n to, o r ca2], then th ere w ill be o u tp u t term s w ith ce n te r freq u en cy c«! as show n in E q . ( 7 .9 -4 ) :

L W = {S n ,K + + m 2f ( t i ] 2a 3} co s c o ^ (7 .9 -4 )

T h u s even if th e o rig in a l signal a t a ), is u n m o d u lated , an in terferin g signal a t co2 m ay tran sfer its m o d u la tio n to th e d esired signal frequ en cy .

I f f (t) = co s fi2t, then th e tran sferred m o d u la tio n w ill hav e s id eb an d c o m p o n en ts a t b o th n 2 and 2 ¡x2 . I f m 2 is sm all en o u g h , say 0 .1 0 , o n e m igh t ign o re the seco n d , h a rm o n ic term an d ask fo r th e size o f V2 th a t w ould cau se 1 % sin u so id a l m o d u la tio n on the d esired ca rrie r . F r o m E q . (7 .9 -4 ) th is v alu e w ould be

V 23m 2— = 0 .01 , (7 .9 -5 )g m

w hich is in d ep en d en t o f V , and o f co2 - T h u s, u n less a 3 can b e m ad e very sm all w ith resp ect to gm, a d e q u a te selectiv ity m u st be prov id ed to keep V2 a t th e d ev ice inp u t b e lo w th e d esired level. F o r exa m p le , if |a3/ g j = 1 a n d m 2 = 0 .1 0 , then V2 < 180 m V .

A n ideal F E T a m p lifie r , w ith n o series res is ta n ce , has a 3 = 0 and h en ce n o cro ss m o d u la tio n p ro b lem s. Id ea l d ifferen tia l p airs a lso have a 3 = 0 an d a g a in should n o t ca u se c ro ss m o d u la tio n . P ra c tic a l F E T ’s and d ifferen tia l p airs w ill hav e n o n zero b u t sm all valu es fo r a 3 ; h en ce they sh o u ld have o n ly very sm all c ro ss -m o d u la tio n effects.


T h e c ro ss -m o d u la tio n te rm fo r th e b ip o la r tra n s is to r c a n b e ca lcu la te d fro m E q . (7 .2 -7 ) i f we assig n cos to th e d esired sig n a l an d a>0 to th e in terferin g sig n a l ; then

ias(t) = 2 I ESe qVdJkTl 0(x )I ,(>’) co s wst, (7 .9 -6 )

w here y = Viq /k T is th e n o rm a liz ed v ersio n o f the d esired sig n a l an d

. q V zV + / (*)]X k T

is th e n o rm a liz ed v ersio n o f th e u n d esired sig n a l. F o r sm all v a lu es o f x an d y we can exp an d I 0(x) a s /0 (x) x 1 + (x 2 /4) an d / ,(y ) as / j(y ) » y /2 so th a t ^ ( r ) b eco m es

I f f ( t ) = m 2 c o s n 2t, th en for sm all valu es o f m 2 th e valu e o f V2 th a t w ill cau se 1 % c ro ss m o d u la tio n o f th e d esired c a rr ie r is a p p ro x im a te ly

lq V 2\2 m 2„ < 0 .0 1 . (7 .9 -8 )

k T I 2 ’

F o r m 2 = 0 .1 0 , V2 m u st be less th a n 8 . 6 m V .In a d d itio n to c ro ss m o d u la tio n , the cu b ic term a lso ca u ses d is to rtio n o f the

m o d u la tio n o rig in a lly on th e d esired ca rrie r . F o r exam p le, if

i’i(t) = + g W Jc o s a v ,

then the c u b ic term cau ses a cu rren t co m p o n e n t a t to , o f

| K ? {1 + 3 g ( i ) + 3 [g (t ) ] 2 + [g (f)]3} c o s o j , f . (7 .9 -9 )

O b v io u sly su ch a te rm w ill ca u se d is to rtio n in a single s in e-w av e m o d u la tio n andin te rm o d u la tio n in any m u ltifreq u en cy m o d u la tio n .

A s a p ra c tica l m a tte r , 1 % c ro ss m o d u la tio n is b a re ly v is ib le in a te lev isio n p ictu re an d ea sily to le ra b le in v o ice c o m m u n ica tio n . W ith 5 % c ro ss m o d u la tio n a te lev isio n p ic tu re is bad ly d is to rte d , a lth o u g h v o ice c o m m u n ica tio n is s till p o ssib le w ith o u t undue d ifficu lty .

PR O B LEM S

7.1 Assume a mixer of the type shown in Fig. 7.2—1(a) with a silicon FET having IDDS = 4 mA, Vp = — 4 V, and an ac bypassed source resistor o f 2 kii. The oscillator peak voltage is1.8 V. Evaluate the conversion and amplifier transconductance from the gate-ground terminals to the drain current. F o r an input sinusoidal signal of 1 mV peak amplitude, determine the drain current com ponents at the input signal frequency, at the oscillator frequency, and at the difference frequency.

7.2 In the mixer of Problem 7.1, w hat is the limit on the size of an interfering signal input before nonlinearities result in the transfer o f am plitude m odulation from this signal to the difference frequencies? Explain.

PROBLEMS 339

7.3 Consider a bipolar mixer of the type shown in Fig. 7.2-3. Assuming that the dc com ponent of the em itter current is 1 mA and that the oscillator drive voltage (across the em itter side of the driving transformer) is 100 mV peak, determine Gc and Gm. Assuming that the input signal has a peak am plitude of 2 mV, evaluate the collector current com ponent a t the signal, the oscillator, and the difference frequencies.

7.4 Estimate the effect on the results for the mixer of Problem 7.3 of adding an unbypassed 20 Q resistor in series with the transisto r’s em itter while leaving all other quantities unchanged.

R esonant at 3x10* rad/sec

v(/) = 4Vcos 9 x 10*/

Figure 7 .P -1

7.5 Estimate the conversion transconductance of the mixer shown in Fig. 7.P-1. Use the same transistor param eters as in Problem 7.1. The oscillator peak voltage is 4 V.

7.6 Consider the differential-pair mixer shown in Fig. 7.4-4. How large can V, be w ithout introducing m ore than 7 % second-harm onic distortion in Ik (assuming a pure sine-wave drive from the V, cos co0t generator)? W hy is it desirable to keep the second-harmonic com ponent in Ik small? W hy is it desirable tha t the voltage-current characteristic o f the differential pair be symmetrical?

7.7 For the mixer circuits shown in Fig. 7.P-2, show that the equivalent linear loading the transistor places across the input-tuned circuit is given by

In I q(x )

qVJkT '

where Vx = VEE- V0, x = qVJkT, and g in = qIEQ/kT.

7.8 For the mixer circuit shewn in Fig. 7 .P-3, determ ine an expression for t)0(i) for the case where g(r) = (1 mV)[l + mf(t)], a>s = 9 x 107 rad/sec, and the output-tuned circuit has sufficient bandw idth to pass the modulation. (Assume Q i , Q i , and Q3 to be identical and that a ^ ' l . l

Ggp —1 + 0

340 m ix e r s ; r f a n d if a m p l if ie r s

7.9 For the converter shown in Fig. 7.P-4, determ ine an expression for t;0(r). Justify all assumptions. ws = 9 x 107.

7.10 F or the “ front end” shown in Fig. 7.P-5, determine an expression for vjt). Justify all assumptions and do not neglect the loading of L 2 by Q , (cf. Problem 7.7). Assume that L6C6 passes the modulation. At the frequencies of operation, transistor reactances are negligible.

7.11 A nonlinear device has an input voltage-output current transfer characteristic given by

i'o = a0 + a t v, + a2vj + a4v -

If v t = t>s(i) cos a>st + V, cos a>0r, where |us(i)l « v „ show that the voltage-controlled transconductance of such a device for vs cos cost is given by

g JK , cos w Qt) = a , + 2a2V, cosa>0t + 4aJ Vx cos a>0f)3.

Also show that the conversion transconductance between cos and a>0 — ws is given by

G = a2Vt + 3/2 a4V],

PROBLEMS 341

Figure 7 .P -3

7.12 It is possible to choose an IF frequency at either a>0 + cos or |<w0 — oij. Explain in terms of the local oscillator tuning complexity why |co0 — a>s| is alm ost always chosen for the interm ediate frequency.

7.13 Show that for the A VC circuit of Fig. 7. P -6 the envelope detector output is given by

1 _ 0 1 vr CO

Also show that the envelope detector output approaches

Kco[l + mf(t)]

for A 0Vi » | Vm\. In this case the output level of the envelope detector is insensitive to input variations.

7.14 W ith the assum ption that the m odulation is not fed around the AVC loop o f Problem 7.13, show that the poles of the closed-loop system are given by the roots of

Ko ~ = o,

342 m ix er s; rf a n d if am plifiers

where H lFL(p) is the low-pass equivalent transfer function of the IF filter and H,(p) is the transfer function of the low-pass filter. Using a three-pole IF filter, show by means of a root locus plot how oscillations are possible as A0V\IVm is increased to provide an output level which is insensitive to input-level variations. Show also how increasing the RC time constant of H L(p) reduces the possibility of oscillations at the expense of a “ sluggish” A VC loop.

7.15 A simplified version of a 10.7 M H z IF amplifier is shown in Fig. 7.P-7. The transistor is a 2N918 biased at 2 mA with = 10 V. At the collector, the adm ittance seen by the tran sistor consists o f a parallel com bination of 8 kO, 3 /¿H, and C„, where C t is to be adjusted, in conjunction with C 22 of the transistor, so as to produce resonance at 67 x 106 rad/sec.

20 V

Figure 7 .P -4

PROBLEMS 343

Figure 7 .P -5

Choose values of input conductance and susceptance (to be presented at the base tap) so that the overall base adm ittance will be real at w = 67 x 106 rad/sec and so that the alignability factor at this point will be 0.15. Do the numerical values that result seem realistic? Explain.

7.16 Redistribute the loss between the input and ou tpu t circuits of Problem 7.15 so that the bandw idths are roughly equal and neither Q is larger than 30, while at the same time keeping k < 0.15. Sketch the shape of the bandpass (to the ± 3 d B points) that would be seen at the base circuit under this new arrangement. (Assume a current source for the signal supply at the base terminals.)

7.17 Reexamine Problem 7.16 assuming that a 0.9 pF “neutralizing” capacitor is properly connected (via a unity voltage gain arrangem ent) across the transistor.

7.18 For the circuit of Problem 7.17, find the effect on the base circuit passband of varying the dc em itter current to 1 mA and then to 3 mA, while leaving all circuit com ponents at the values found for the 2 mA case.

344 m ix e r s; rf a n d if am plifiers

Low-pass filter removes modulation

Figure 7.P -6

Figure 7 .P -7


C O M P A R I S O N S O F ^ - P A R A M E T E R S F O R B I P O L A R T R A N S IS T O R S : S IN G L E -E N D E D , D IF F E R E N T IA L -P A IR ,

A N D C A S C O D E

If o n e assu m es th a t id en tica l tra n s is to rs a re used in a ll th ree co n fig u ra tio n s , if o n e n eg lects v ario u s p a ra s itic re a c ta n ce s an d re s is ta n ce s, an d i f o n e a ssu m es sim plified m o d els fo r the tra n s isto rs , th en o n e c a n ca lc u la te th e y -p a ra m e te rs fo r e a ch ca se in a relatively straightforward fashion. The purpose of this appendix is not to do this a lg e b ra , b u t m erely to p o in t o u t th e a p p ro x im a te c o m p a ra tiv e valu es to b e exp ected from th e th ree cases . S in ce th e y -p a ra m e te rs a re a fu n ctio n o f freq u en cy an d sin ce w e a re o n ly try in g to give o rd e r-o f-m a g n itu d e resu lts, o n e m u st n o t a tte m p t to app ly these resu lts to p recise c o m p a riso n s b etw een c ircu its .

I f we assu m e eq u a l p o w er-su p p ly c u rren ts fo r a ll th ree co n fig u ra tio n s , then the d ifferen tia l-p a ir tra n s is to rs e a ch h av e o n ly h a lf th e d c cu rren t o f a ll o th e r tra n s is to rs , h en ce th e ir valu es fo r r E w ill b e d o u b led an d th e ir valu es fo r g m w ill b e halved. In a d d itio n , th o u g h th e ir valu es fo r a lp h a o r b e ta m ay b e ch an g ed , we sh a ll ig n o re such chang es.

W ith th ese a ssu m p tio n s an d th e re a liz a tio n th a t fro m th e sm all-s ig n a l v iew p oin t the d ifferen tia l p a ir c o n sis ts o f a n e m itte r fo llo w er d riv in g a co m m o n b a se stage, w hile th e c a sco d e c o n n e c tio n co n sis ts o f a c o m m o n e m itte r s tage d riv in g a co m m o n b ase stag e , we ca n m a k e th e c o m p a riso n s show n in T a b le 7 .A -1 .

Table 7.A 1

Differential pair com pared to comm on em itter stage with twice the dc bias current of each transistor of the differ

ential pair

Cascode connection compared to comm on em itter stage with

the same dc bias current

> u 1/4 1y i2 1/30 to 1/200 1/200 to 1/2000>21 1/4 1>221 1 to 1/3 1 to 1/3

t For u>CB ErBB « 1, y 21 s pCs c( 1 + g mrB B ), while for the two configurations with common base output stages it is approximately p C B c . (Ideally the cascode case should have an even smaller y 2 2 ', in practice, parasitic terms tend to keep it from being much smaller.)

In sp ec tio n o f th e resu lts in T a b le 7 .6 -2 w ill in d ica te th a t th e resu lts show n in T a b le 7 .A -1 a re g en era lly s im ila r to th o se re p o rte d earlie r . T h e y n resu lts fo r th e

345

346 m ix e r s ; r f a n d if a m p l if ie r s

d ifferen tia l p a ir an d th e c a sc o d e c o n n e c tio n show th e exp ected 4/1 d iffe re n tia l; how ev er y t , fo r th e s in g le-en d ed stag e (n o t th e sam e tra n s is to r in th is ca se ) is c o n sid erab ly sm a ller th a n th e c a sc o d e valu e. T h is im p lies th a t th e 2 N 9 1 8 used fo r the sin gle-en d ed ex a m p le is a “ b e tte r” h ig h -freq u en cy tra n s is to r th a n th o se used in the C A 3 0 0 5 in th a t it h a s a sm a ller to ta l in p u t c a p a c ita n c e (by a fa c to r o f a b o u t four) an d a h ig h er b e ta (b y a fa c to r o f a b o u t three). T h e s e c o n clu s io n s fo llo w fro m th e fact th a t, o v er th e ra n g e o f freq u en cies b e in g co n sid ered , th e c o m m o n e m itte r s tage has an in p u t a d m itta n ce

y n ~ S b 'c + P (C B'c + C b e ) o r J>1 1 « g„ + P (C B'c + C n),

d ep en d in g o n o n e ’s c h o ic e o f n o ta tio n .

C H A P T E R 8

AM PLITUDE M ODULATION

8.1 A M PLITU D E M O D U LA TIO N SIGNALS

I f th e am p litu d e , freq u en cy , o r p h a se o f a h ig h -freq u en cy sin u soid is forced to vary in p ro p o rtio n to a d esired lo w -freq u en cy s ig n al f ( t ) , a m o d u la ted sig nal is gen era ted w hose freq u en cy sp ectru m is co n ce n tra te d in th e v ic in ity o f the freq u en cy o f th e u nm o d u la ted h ig h -freq u en cy sin u soid . T h e m o d u la ted signal, u n lik e th e m o d u la tio n f ( t ) , m ay be freq u en cy -d iv is io n m u ltip lexed w ith o th e r s im ilarly m o d u la ted sig n als a t d ifferent cen te r freq u en cies (cf. C h a p te rs 3 an d 7 ) ; it c a n a lso b e tra n sm itted effic ien tly by a n te n n a s n o t n early so larg e as th o se w hich w ould be req u ired to tra n sm it / (f) d irectly . S p ecifica lly , i f f ( t ) w ere a s in g le to n e a t a freq u en cy o f 1 k H z , a 3 w avelength a n te n n a to tra n sm it f ( t ) in to the a tm o sp h e re w ould b e 75 k m lo n g , w hereas if / (f) m o d u la ted a 100 M H z sin u so id th e co rre sp o n d in g a n te n n a len g th w ould be o n ly | m.

In th is c h a p te r we co n sid er c ircu its th a t ca n a m p litu d e -m o d u la te a s in u soid al c a r r ie r ; in C h a p te r 11 w e sh a ll fo cu s o u r a tte n tio n o n c ircu its w hich m o d u la te the freq u en cy an d p h a se o f a s im ila r ca rrie r. T h e n in C h a p te rs 10 an d 12 resp ectiv ely , we e xa m in e th e te ch n iq u e s by w hich am p litu d e- an d freq u en cy -m o d u la ted ca rrie rs a re p ro cessed to re co v e r / (i) a t th e receiv er.

A ty p ica l a m p litu d e-m o d u la ted c a rr ie r h as th e form

u(r) = A[\ + m f(t )] c o s (a>0r + 0 o) = g(r) co s (a>0r + 0 o), (8 .1 —1)

w here A is th e c a rr ie r a m p litu d e , m is the m o d u la tio n ind ex, / (f) is a s ig n a l w hich is p ro p o rtio n a l to th e m o d u la tio n in fo rm a tio n an d w hich has th e p ro p erties

1 / ( 0 1 max = 1 and 7 ( 0 = o, (8 . 1 - 2 )

(o0 is th e c a rr ie r freq u en cy , 0 O is th e ca rr ie r p h ase an g le (w hich is u sually ch o sen as zero w ith o u t lo s s 'o f g en era lity ), an d g (i) = A [l + m f(t )] is th e en v elo p e fu n ctio n o f th e A M w ave. F o r th e c a se w here g(t) > 0 fo r a ll tim e , o r e q u iv a le n tly m < 1 (w hich w e refer to a s norm al A M ), g (i) fo rm s th e u p p er en v e lo p e o f v(t) w hile - g(t) form s th e lo w er en v elo p e o f u(r), as illu stra ted in F ig . 8 .1 -1 . F o r n o rm a l A M th e m o d u la tio n ind ex c a n b e o b ta in e d d irec tly fro m th e w av efo rm o f v(t) by m e a n s o f th e re la tio n sh ip

m - (8.1-3)

w here C is th e m a x im u m p e a k -to -p e a k v alu e o f v(t) an d B is th e m in im u m p e a k -to - p eak valu e o f v(t). O n th e o th e r h an d , i f g(f) d o es n o t rem a in p o sitiv e fo r a ll tim e,

347

348 AMPLITUDE MODULATION 8.1

g(t) not always greater than zero

(b )

F ig . 8 .1 - 1 T y p ic a l A M s ig n a l s f o r w h ic h (a ) g( t ) > 0 f o r a l l t im e a n d (b ) g( t ) d o e s n o t r e m a in g r e a t e r t h a n z e r o f o r a l l t im e .

|g(i)| an d - |g(r)| fo rm th e u p p er an d lo w er en v e lo p es o f v(t) resp ectiv ely , a s illu stra ted in F ig . 8 .1 -1 . F o r th is c a se m c a n n o lo n g e r b e o b ta in e d fro m E q . (8 .1 -3 ) . A s we shall see in C h a p te r 10, o n ly n o rm a l A M ca n b e d em o d u la ted by s im p le en v e lo p e d em o d u la to rs (o r d etec to rs). In a ll o th e r ca ses sy n ch ro n o u s d ete c tio n , w hich is m u ch m o re co m p lica te d to im p lem en t, m u st be em p loyed . C o n se q u e n tly a lm o st a ll c o m m e rc ia l A M s ta tio n s tra n sm it n o rm a l A M to k e e p th e rece iv er as s im p le a s p o ssib le .

A s d eriv ed in C h a p te r 3 , th e F o u r ie r tra n s fo rm K(co) o f v(t) h a s th e fo rm

I ' M = i [ G ( c t ) + a>0 ) + G (ft) - a»0 ) ] , ( 8 - 1 - 4 )

w here G(oj) = A[2nô{u>) + m F(aj)] is th e F o u r ie r tra n sfo rm o f g(f) an d F(tu) is th e F o u r ie r tra n sfo rm o f f ( t ) . A sk e tch o f |F(oj)| vs. a> is a g a in p resen ted in F ig . 8 .1 - 2 fo r th e ca se w here F (a >) an d in tu rn G(co) a re b a n d -lim ite d to com < co0. A s illu stra ted in th e figure, th e p o rtio n o f th e sp ectru m ly in g a b o v e to0 is referred to a s th e up p er s id eb an d o f w h ereas th e p o rtio n o f the sp ectru m ly in g b e lo w co0 (an d a b o v e œ = 0 ) is referred to a s th e lo w er s id eb a n d o f K(eo).

8. 1 AMPLITUDE MODULATION SIGNALS 349

|G M l 1 dc component

f " 1-Olm (*)»> ^

|G(w -a*,) | 2

I— T l

|K(W)| ^ / C a r r i e rA G(u>~ iüq) I

w 0 <M* ~<uo w0+u)„, j oj0 to« (jj0 f Upper sideband

Lower sideband

Fig. 8 .1 -2 Frequency spectrum of y(i).

I f a n o rm a l A M sig n al v(t) is p laced a c ro ss a 1 Q res is to r, th e p ow er d elivered to the re s is to r is g iven b y f

/) i ii = /l c o s 2 co0 f [ l + 2 m f(t) + m 2f 2 (f)]

= A 2 c o s 2 u>0t[ 1 + m 2f 2(t)]

A 2 A 2m 2 —

= T + — f { t )

= P C + P m, (8 .1 -5 )

w here P c = A 2¡2 is th e c a rr ie r p ow er, P m = (A 2m 2/ 2 ) f 2{t) is the m o d u la tio n pow er, and f 2 d en o tes th e tim e av erag e o f f 2 o r e q u iv a len tly th e m ean sq u a re v alu e o f f ( t ) . S in ce |/(f)| < 1, then f 2(t) < 1 ; h en ce the ca rr ie r p ow er is g re a te r th a n o r eq u a l to the m o d u la tio n p o w er (o r th e s id eb an d pow er) e ven w ith m = 1 ( 1 0 0 % m o d u latio n ). F o r exa m p le , i f / (f) = c o s a>mt and m = 1, then / 2 (f) = \ an d P c = 2 P m. I f f ( t ) = co s u>mt and m = th e m o d u la tio n p ow er h as d ecreased to o n e -e ig h th o f th e ca rr ie r pow er. F o r e ffic ien t tra n sm issio n o f n o rm a l A M , it is c le a r th a t we w ant th e m o d u la tio n ind ex to be as c lo se to u n ity as p o ssib le .

B eca u se o f th e larg e a m o u n t o f c a rr ie r pow er, w hich c a rrie s n o in fo rm a tio n , th at m u st be tra n sm itte d in a n o rm a l A M sig n al, the ca rr ie r is so m etim es rem oved (o r n o t gen era ted in the first p lace ) b e fo re tra n sm ittin g the A M signal. T h e resu lt is a suppressed c a rr ie r A M sig n al o f the form

A'

v(t) = A m / (f) c o s co0t

= g(f) co s (u0 (f), (8 . 1 —6 )

w here g(f) = A f ( t ) and g(f) = 0. In th e tim e d o m a in a sup pressed c a rrie r sig nal has the form show n in F ig . 8 .1 --1 (b), w h ereas in the freq u en cy d o m a in th e sp ectru m is

+ In evaluating P |fl we use the relationships A + B = A + B and AB = AB. The second relationship is valid only if A and B are independent as / (r) and cos a>0t are.


id e n tica l w ith th a t sh o w n in F ig . 8 .1 -2 e x ce p t th a t the im p u lses a t ± c o 0 a re ab sen t. T h e p rice th a t m u st b e p aid fo r th e red u ced p o w er a t th e tra n sm itte r is in creased rece iv er co m p lex ity .

S till a fu rth er sav in g in tra n sm itte d p o w er, a s w ell as a re d u ctio n in th e freq u en cy sp ectru m , c a n be a ch iev ed by rem o v in g th e u p p er o r low er s id eb a n d o f a supp ressed c a rr ie r A M sig n al to fo rm a single-sideband A M signal. A filte rin g te ch n iq u e , to fo rm a s in g le-s id eb an d sig n al, illu s tra te d in F ig . 8 .1 -3 , is ju stifie d by th e fa c t th a t a ll the in fo rm a tio n n e ce ssa ry to re c o n stru c t g(t) is co m p le te ly c o n ta in e d in th e p o sitiv e (o r n eg ativ e) p o rtio n o f G(a>). I f g (t) is re a l (w h ich it a lw ays is), th en G(qj) = G *( — oj).

Sidebandfilter

( 0H(jw)

SSB (0

v(0=g<f) cosou0(0

|G(w+w0)| ; 2 \ 1

1 K(tu) | |G(tü—iuo)|

2/ V V

-«Md- w», <*'0 -«‘'o+H ,1

cüq—w,„ w0 ai0+<jj,„1

Upper-sideband

filter

-Wo w„

J 1

Upper-sideband

SSBK » M |

A<*k>—<M„ 1H)

Lower-sideband

filter

Cl

|«(/w )|

'n w 0 + o j ,

- w 0.

11 Lower- 1 sideband 1 SSB

r v

u

|K s„(i»)|

n

>0

—O J q U ÎQ -OJm CÜQ

Fig. 8.1 - 3 Generation o f upper and lower single-sideband signals.

8.1 AMPLITUDE MODULATION SIGNALS 351

Therefore, the positive and negative portions of G(a>) contain identical in fo rm a tio n ;

i.e., th e n eg ativ e sp ectru m c a n be co m p le te ly co n stru c te d i f the p o sitiv e sp ectru m is know n.

T o e v a lu a te an e x p ress io n fo r th e s in g le-s id eb a n d signal vSSB in th e tim e d o m a in , we o b ta in the inverse F o u rie r tra n sfo rm o f w hich for u p p er-sid eb an d S S B is

given by

K;SB(w ) —

G(o) + a>0) G(a> — a>0)

l o ,

M >

| Cü| < w 0 .

(8 .1 -7 )

A ssu m in g th a t g (i) is b a n d -lim ite d to a>m 0 , we o b ta in

«’ssbM —- - f2« J

1 G(co + c %e1“ ' doj +

G(<o - oiQ)

COS O)0t

G(w)eian dm

2 I

sin co.

\ h f G^ ' do}

l r

]

■ e^ d o )

g(0

2

COS OJrtt

“ [ ¿ j:g(0

G(a>)( —j sgn oj)eJl"' da>

sin u>0t, (8 .1- 8 )

w here g(t) = & i [G (oj)( —j sgn oj)] is referred to a s th e H ilb e rt tra n sfo rm o f g(t) and sgn co is given by

1 . o j > 0 ,

sgn oj 0, o j = 0 , (8 .1 -9 )

- 1, o j < 0 .

In a similar fashion, lower-sideband SSB can be shown to have the form

c ii)«WO - ~ Y c o s u>0t + — sin U)0t. (8.1 -1 0 )

N o te th a t g (0 is o b ta in e d by p la c in g g(t) th ro u g h a lin ea r filter h av in g th e tra n sfer fu n ctio n H H1(ja j) = —y 'sg n c j. T h e filte r //HT(ya>), w hich is referred to a s a H ilb e rt tra n sfo rm er, has th e m ag n itu d e an d p h ase c h a ra c te r is tic show n in F ig . 8 .1 -4 . C le a rly by sh iftin g a ll th e p o sitiv e freq u en cy co m p o n e n ts o f g (f) by — 7t/ 2 we o b ta in

g(0-In a d d itio n to ch a ra c ter iz in g S S B in th e tim e d o m a in , E q s. (8 .1 -8 ) an d (8 .1 -1 0 )

suggest a n a ltern a tiv e m eth o d o f g e n era tin g S S B w hich is sh o w n in b lo ck d iag ram

352 AMPLITUDE MODULATION S.l

\HhA M

Hilbert transformer

arg Hm(jw) n2

nou—

2

Fig. 8 .1 -4 Plot of magnitude and phase of the Hilbert transformer.

fo rm in F ig . 8 .1 -5 . H e re g(t) is su p p ressed -ca rrie r-m o d u la te d (o r m u ltip lied ) by c o s ojnt w hile g(f) is su p p ressed -ca rrie r-m o d u la te d by — s in cu 0 i. T h e resu lta n t sum (o r d ifference) y ields E q . (8 .1 -8 ) o r E q . (8 .1 -1 0 ) .

A lth o u g h th e b lo c k d ia g ra m o f F ig . 8 .1 -5 a p p e a rs to be a very re a so n a b le m eth o d o f g en era tin g S S B , it e n c o u n te rs the sev ere p ra c tica l p ro b le m th a t th e H ilb e rt tra n s fo rm er is a n o n re a liz a b le filte r (its im p u lse resp o n se hHT(t) = l /n t is n o n cau sa l). C o n se q u e n tly its resp o n se m ay a t b e st b e a p p ro x im a te d o n ly o v er a re a so n a b ly n a rro w b an d o f freq u en cies. In a d d itio n , su ch a p p ro x im a tio n s u su ally p ro d u ce a p h ase o f the fo rm

■\ + 0 M ,

w h ich m u st b e c o m p e n sa te d fo r by p la c in g g(t) th ro u g h a filte r w ith c o n s ta n t m a g n itu d e an d w ith a p h ase o f 0(co) b e fo re m u ltip ly in g it by c o s a>0 t. In g en era l, n o rea lly sa tis fa c to ry S S B m o d u la to rs o f th e fo rm sh o w n in F ig . 8 . 1 - 5 h av e b een prod u ced .

Multiplier

Fig. 8.1 - 5 Alternative m ethod o f generating a single sideband.

8.2 AMPLITUDE MODULATION TECHNIQUES 353

E v e n d irec t s id eb a n d filte rin g o f su p p ressed c a rr ie r A M to p ro d u ce S S B h a s its p ra c tica l lim ita tio n s . I t is a d ifficu lt m a tte r to d esign a s id eb a n d filter w hich cu ts o ff su fficien tly rap id ly to a tte n u a te o n e sid eb an d w hile n o t d is to rtin g th e o th e r s id eb an d e ith e r in m a g n itu d e o r in phase. F o r S S B v o ice m o d u la tio n , m e ch a n ica l, c ry s ta l, o r c e ra m ic b a n d p a ss filters a re u sually em p loyed to rem ov e th e undesired s id eb an d (See F ig . 7 .6 -1 fo r a n exam p le). E v en th o u g h th ese filte rs hav e co n sid era b le rip p le in m a g n itu d e as w ell as s ig n ifica n t n o n lin e a rity in p h ase in th e p assb an d , th e ir effect on th e in te llig ib ility o f av erag e sp eech is neg lig ib le. F o r o th e r fo rm s o f m o d u la tio n th e s id eb an d filte r u sually h as to be h a n d -ta ilo re d to the m o d u la tio n to m in im ize d is to rtio n . In a ll ca ses , h ow ev er, th e sh a rp c u to ff req u ired o f th e b a n d p a ss s id eb an d filter is p o ssib le o n ly i f th e filte r ce n te r freq u en cy is n o t to o h igh (th e req u ired Q o f th e tu n ed c irc u its in th e filte r to m a in ta in a fixed B W is d irectly p ro p o rtio n a l to a)0). C o n se q u e n tly , in a lm o s t a ll S S B tra n sm itte rs , th e in fo rm a tio n is su p p ressed -carrier- m o d u la ted a t a re a so n a b ly low c a rr ie r freq u en cy (5 0 k H z to 5 0 0 k H z fo r v o ice m o d u la tio n ) w here s id eb a n d filte rin g is a cco m p lish e d , a n d th en th e resu lta n t S S B signal is h etero d y n ed to th e d esired c a rr ie r freq u en cy co0 .

In a d d itio n to th e co m p lic a tio n s th a t S S B c re a tes a t the tra n sm itte r , its d em o d u la tio n is p o ssib le o n ly by sy n ch ro n o u s d ete c tio n , w h ich req u ires a re feren ce o sc illa to r a t th e rad ian freq u en cy co0 . S in c e it is im p o ssib le to d erive th is freq u en cy fro m the

S S B signal itself, a sm all p ilo t c a rr ie r is u su ally tra n sm itte d a lo n g w ith the S S B sig nal to p ro v id e th e re feren ce a t th e receiv er. T h e d em o d u la tio n o f S S B w ill be p u rsued in m o re d eta il in C h a p te r 10.

In th e su b seq u en t se c tio n s o f th is c h a p te r we shall co n sid er th e th e o re tica l m eth o d s by w hich am p litu d e m o d u la tio n (o r m u ltip lica tio n o f tw o sig nals) m ay be a cco m p lish ed . W e sh a ll th en ex a m in e so m e p ra c tica l c ircu its w hich im p lem en t th e th e o re tica l m eth od s.

8.2 A M PLITU D E M O D U LA TIO N T EC H N IQ U ES

In th is se c tio n w e in v e s tig a te th e th e o re tic a l m e th o d s b y w h ich w e c a n m u ltip ly o r m o d u la te co s a>0t by g(t) to o b ta in th e A M s ig n a l

v{t) = g(t) co s w 0t

= A [1 + m f (t)] co s co0t. (8 .2 -1 )

In g en e ra l, th e re a re fo u r b a s ic m e th o d s by w h ic h a m p litu d e m o d u la tio n c a n be a c c o m p lish e d :

a) a n a lo g m u ltip lica tio n ,b) ch o p p er m o d u la tio n ,c) n o n lin e a r d ev ice m o d u la tio n , a n dd) d irec t tu n e d -c ircu it m o d u la tio n .

A s w e sh a ll see, a ll th ese m eth o d s c a n b e em p lo yed to g e n era te n o rm a l A M , w hereas on ly d irec t a n a lo g m u ltip lica tio n a n d ch o p p e r m o d u la tio n c a n b e em p lo yed to g e n e ra te sup pressed c a rr ie r A M . In a d d itio n , w ith th e e x ce p tio n o f th e d irec t tu n ed - c irc u it m o d u la to r , m o d u la tio n is a cco m p lish e d a t low p ow er levels an d am p lified

3 5 4 AMPLITUDE MODULATION 8.2

(c lass B — see S e c tio n 4 .2 an d 9 .2 ) to th e d esired o u tp u t level. T h e d irec t tu n ed -c ircu it m o d u la to r d irec tly m o d u la tes th e am p litu d e o f a h ig h -p o w er c a rr ie r w hich h as been am p lified by m o re e ffic ien t c la ss C am p lifie rs to th e d esired level.

Analog ModulationA n a lo g m o d u la tio n (o r m u ltip lica tio n ) is a cco m p lish ed in a n y d evice w hose o u tp u t 0 o(r)] is d irec tly p ro p o rtio n a l to tw o in p u ts [i>,(f) an d i>2 (0 ], th a t is,

v„(t) = K v i(t)v2{t). (8 .2 - 2 )

C lea rly , i f v t(t) = c o s oo0t an d v2(t) = g(f)> then v0(t) = K g (t ) c o s u>0t, w hich is the d esired A M w ave. In su ch d ev ices n o th e o re tica l l im ita tio n s ex ist ; how ever, p ra c tica l d ev ice lim ita tio n s u su ally im p o s e 'lim its on b o th th e am p litu d e and freq u en cy o f vt (t), v2(t), and v0(t) in o rd e r to m a in ta in th e valid ity o f E q . (8 .2 -2 ) .

A n a lo g m o d u la tio n c a n a lso b e a cco m p lish e d w ith tw o sq u are-law d evices as

Fig. 8.2-1 Analog m odulator constructed with two square-law devices.

show n in F ig . 8 .2 -1 . In th is sy stem th e o u tp u t o f th e to p sq u a re -la w d ev ice , v3 , is given by

v3 = K g{v 2i + 2 v iv 2 + vl)-, ( 8 .2 -3 )

th e o u tp u t o f the b o tto m sq u a re -la w d ev ice , v4 , is given by

v4 = K giv] - 2 v tv2 + vl). ( 8 .2 -4 )

T h e system o u tp u t v„ = v3 — v4 is thu s given by

v0{t) = 4 K sVl(t)v2(t), (8 .2 -5 )

w hich is the form o f th e o u tp u t o f a n a n a lo g m o d u la to r.I f th e sq u a re -la w d ev ices a re “ h a lf sq u a re la w ” ra th e r th a n fu ll sq u a re law , i.e., if

f K s t ; ? , Vi > 0 , }

1 0 , Vj < 0 ,

th en t;, + v2 an d v, — v2 m u st b e co n stra in e d to be g rea ter th a n zero to p ro d u ce the

8 .2 AMPLITUDE MODULATION TECHNIQUES 355

d esired o u tp u t. T h u s i f = A [ 1 + m f{t)] a n d v2 = Kj c o s co0 i> w here f ( t ) is the m o d u la tio n in fo rm a tio n (cf. E q . 8 .1 -2 ) , then

S in ce Vx > 0 , we o b serv e th a t, w ith h a lf-sq u a re -la w d evices in th e c irc u it o f F ig .8 .2 - 1 , th e m o d u la tio n in d ex is co n stra in e d to b e less th a n un ity an d thu s th e c ircu it c lea rly c a n n o t b e used to g e n era te su p p ressed c a rr ie r A M .

E v en th o u g h th e c irc u it d o es n o t p ro d u ce a 100 % m o d u la ted A M w ave (w hich is d esira b le fo r e ffic ien t tra n sm issio n ), the m o d u la tio n ind ex a t th e o u tp u t c a n be in crea sed by su b tra c tin g so m e o f th e excess ca rr ie r fro m the u n d erm o d u lated signal.

T h is te ch n iq u e is re ferred to as ca rr ie r cancellation. F o r exa m p le , i f we su b tra c t D c o s o>0t fro m v(t) --- A [ 1 + m f(t)] c o s co0t, we o b ta in v„(t) in th e form

from w hich it is a p p a re n t th a t we c a n in crea se th e resu lta n t m o d u la tio n ind ex m' to u n ity by c h o o s in g D su ch th a t {A — D )/A = m o r D /A = 1 — m.

Chopper ModulationC h o p p e r m o d u la tio n is a cco m p lish e d by ch o p p in g g(t) a t th e c a rr ie r freq u en cy ra te an d p la c in g th e resu lta n t s ig n al th ro u g h a b a n d p a ss filter cen tered a t th e c a rr ie r

frequ ency . T h e b a s ic sk e le to n c irc u it o f th e ch o p p e r m o d u la to r is show n in F ig .8 .2 - 2 , in w h ich th e sw itch , w hich is c o n tro lle d by cosa> 0 i, rem ain s o p en for A c o s u)0t > 0 a n d clo sed fo r A c o s a>0t < 0. T o d em o n stra te th a t th is c ircu it a cco m p lish e s a m p litu d e m o d u la tio n , we first w rite va(t) in the form

Vo(0 = 4 K s Fi/ 4[l + m f(t)] c o s coQt, (8 .2 -7 )

provid ed th a t ,4(1 — m ) — Vx > 0 o r eq u iv a len tly

« < 1 Vl m 0t > 0 ,

c o s a)0t < 0 .(8.2- 11)

C le a rly , th en , S (i) is a sq u a re w ave w ith u n ity a m p litu d e w hich m ay b e exp an d ed in a F o u r ie r series o f th e form


Fig. 8 .2 -2 Chopper modulator.

to yield

Va(t) = ^ + M } COS(O0t- . M l c o s 3 œ 0t + ■ • • • (8 .2 -1 3 )2 n 3 t i

F r o m E q . (8 .2 -1 3 ) w e n o te th a t va(t) is th e su p e rp o sitio n o f A M w aves cen tered a t co0 , 3<w0 , 5<w0 , . . . . I f th e b a n d p a ss filte r H (ja j) a tte n u a te s th e lo w -freq u en cy c o m p o n en ts o f va(t) a s w ell as th e A M co m p o n e n ts o f v jt ) in th e v ic in ity o f 3<»0 , 5co0

th en the o u tp u t v„(t) is g iven b y

* M o J c o s [to0t + 0(coo)]. (8 .2 -1 4 )

w here hL(t) is th e im p u lse resp o n se o f th e lo w -p ass eq u iv a le n t o f th e b a n d p a ss filter an d #(co0) is th e p la n e a n g le o f H (jo i) a t o> = co0 . I f th e lo w -p a ss e q u iv a le n t filte r is


flat over the b and o f frequencies occu p ied by g(t), then va(t) sim plifies to the desired form

v0(t) = # ¿ (0) cos [<u0i + 0(« o)], (8.2-15)n

where H L(jio) is the Fourier transform of hL(t).N ote that chopper m odulation is possible only if the spectrum of the desired

AM wave does no t overlap the spectra of any of the other com ponents of u0(f)- A typical sketch of \Va(w)\, where Va(oj) is the Fourier transform of va(t), is presented in

IK M l 1, , /|G (uH-cjo)| G(uj—3cmo)|

s ^ r |C M ! ! / * 3 * \ ,

0 °« |cù0 wo f tum 3wo — 3w0 cu„

Fig. 8 .2-3 Spectrum of va(t).

Fig. 8.2-3 for the case where g(t) is band-lim ited to u>m. It is apparen t from the figure that, unless

wm < 2, (8.2-16)

then the spectra overlap and chopper m odulation is impossible. In addition, the closer iom is to co0/2, the m ore complex the bandpass filter m ust be to effect the frequency separation.

N ote that no restriction has been placed on the m odulation index of g(t); therefore, norm al AM with a m odulation index of unity as well as suppressed carrier AM may be generated with the chopper m odulator.

In order to relax the inequality of Eq. (8.2-16) and in tu rn m ake the design of the filter H(jto) simpler, the single-pole voltage-controlled switch of Fig. 8.2-2 may be replaced by the double-pole vollage-controlled reversing switch shown in Fig.8.2-4. The reversing switch has the effect of m aking va(t) symmetrical abou t zero.

COS W 0 l

Fig. 8.2-4 Balanced chopper modulator.

thus elim inating the low-frequency com ponent of va(t). Consequently, with the chopper m odulator of Fig. 8.2-4, m odulation is possible if

œ m < co0. (8.2-17)

E quation (8.2-17) ensures tha t the spectra of the desired AM wave centered at a>0and the AM wave centered a t 3w0 do not overlap.

O n a rigorous basis we observe tha t for the balanced chopper m odulator of Fig. 8.2^4

®.(0 = g(i)S'(i) (8.2-18)

where

f 1, cos oi0t > 0,[ — 1, cos œ 0t < 0.

Since S'(t) is a square wave with a peak-to-peak am plitude of 2 and zero averagevalue, S'(t) may be expanded in a Fourier series of the form

4 4S'(t) = - cos ai0t — — <?os 3co0i + • • •. (8.2-20)

n in

Hence

4 g(t) 4g(i)va(t) = cos co0t ---------- cos 3w 0t + ■ • -, (8.2- 21)

n 3n

which, as expected, has no low-frequency com ponent. E quation (8.2-21) also indicates that the reversing switch perm its us to obtain twice as much output from the m odulator of Fig. 8.2-4 as from the m odulator of Fig. 8.2-2.

In m any practical cases the switch S(f) in the chopper m odulator of Fig. 8.2-2 has a resistance r in series with it which prevents com plete attenuation of g(t) whenS{t) is closed. Consequently, va{t) is switched between g(i) and rg(t)/(r + R) in lieuof g(t) and 0. This may be expressed m athem atically as

^ , S M 1 ~ S(t)]rva(t) = g(0 S(0 + -----------—5— ■ (8 .2 - 2 2 )r + R

It is apparent that the fundam ental com ponent of va(t) is given by

1 -------'— I cos w 0t = ^ cos co0t, (8.2-23)7t \ r + RJ n R + r

and in turn va(t) is given by

[2g(t)


^ cos [co0t + 0 K ) ] . (8.2-24)R + r

Thus we see tha t the only effect of the series resistance r is to a ttenuate the ou tpu t by the factor R/(R + r). If r = R, the ou tpu t level is reduced by a factor of 2.

8.2 AMPLITUDE MODULATION TECHNIQUES 359

Nonlinear Device ModulationN onlinear device m odulation is accom plished by sum m ing the m odulation and the carrier, applying them to a nonlinear device, and then passing the device output through a bandpass filter centered at tu0 to extract the desired AM signal. A block d iagram of a nonlinear device m odulator is shown in Fig. 8.2-5. As we shall see, the nonlinear device m odulator has m ore restrictions for its p roper operation than any

v$t)=g(t)=A[\ + m /(i) 1o

I « 0 )1

»2(0= V\ cost«*/

Fig. 8 .2-5 Block diagram of nonlinear modulator.

of the previously considered m odulators. F irst o f all the nonlinear device m ust have no greater than a second-order (square-law) nonlinearity. Second, the m axim um m odulation frequency u>m m ust be less than a>0/3 ; and third, if the nonlinear device contains a “ half-square-law ” term , 100% m odulation or suppressed carrier m odulation is not possible.

To determ ine the reason for these restrictions as well as an expression for the ou tpu t signal v0(t), let us first express the ou tpu t of the nonlinear device v j t ) in the form of a M acLauren series,

va = a0 + a tVi + a2vj + a3vf + • • -,

where

_ 1 dva n \ dv;

(8.2-25)v, = o •

W ith Vi = v x + v 2 , v a reduces to

Va = ao + Oi(vi + V 2 ) + a2(vl + 2 v xv 2 + v$

+ a 3(t;f + + 3vtvl + r^) + • • •• (8.2-26)

A little thought indicates that, if u2(t) = Vx cos co0t and t;,(f) = g(t), the following com ponents of va (as well as a num ber of o ther com ponents) have frequency spectra

3 6 0 AMPLITUDE MODULATION 8 .2

in the vicinity of w 0 :

a ,v 2 = c o s co0t,

l a 2v l v2 = 2a2Vlg(t)cos w0t,

3a3v2lv2 = 3a 3K!g2(i)cos ai0t,

na„v"~1v 2 = '(f) cos a>0i- (8.2-27)

It is apparent tha t any filter which extracts the desired term 2a2K1g(f)cos <o0t from va(t) also extracts the rem aining term s of Eq. (8.2 27) ; however, the term s with the coefficient a„ for n = 3,4, 5 , . . . possess envelopes which are nonlinear functions of g(i). C onsequently for p roper operation the M acLauren expansion of the nonlinearity m ust have a„ = 0 for n > 3.

W ith this restriction, va(t) simplifies to

Since g2(i) has a spectrum which is band-lim ited to 2com if g(f) is band-lim ited to a>m, it becomes apparen t th a t the spectra of the desired com ponent of va(t) and the low-frequency com ponent of va(t) overlap unless

However, if Eq. (8.2-29) is satisfied and the nonlinearity is “ square law ” (a„ = 0 for n / 2), then v j t ) reduces to

or, for the case where H L(jaj) is flat over the band of frequencies occupied by g(f) =

which is the desired AM wave.If the nonlinear device is “ half square law ” rather than full square law, which is

alm ost always the case in practice, then the argum ents leading to Eq. (8.2-8) apply in this case and the m odulation index is lim ited to m < 1 — VJA. H ence again neither 100% AM m odulation no r suppressed carrier m odulation is possible.

low-frequency lerm desired term

second-harmonic term+ y V \ CO S 2 (O0t.

(8.2-28)

(8.2-29)

v„(t) = 2a1Vl [g(t) * hL(t)] cos [co0t + 0(coo)], (8.2-30)

A '

v0(t) = 2VIH L(0)a2A [l + mf(t)\ cos [co0t + 6(oj0)], (8.2-31)


D irect T uned-C ircuit M odulation

Direct tuned-circuit m odulation is effected by em ploying g(t) = A[\ + m/(f)] to directly control the voltage across a parallel resonant circuit tuned to the carrier frequency and driven by a periodic current source. Figure 8.2-6 illustrates such a m odulator. If the loaded Q T of the tuned circuit is sufficiently high, it is apparent that v0(t) contains only a fundam ental term. In addition, from Example 5.5-3 we recallthat the peak value or envelope of v0(t) m ust be g(t); consequently the high-level m odulator produces the desired AM signal.

%(/)

Fig. 8.2-6 Direct tuned-circuit modulator.

Like all the o ther m odulators previously considered, the high-level m odulator has a limit on the m axim um m odulation rate com. If g(t) increases too rapidly, the norm al increase in the envelope o f v0(t) cannot keep p?ce, the diode rem ains open, and the envelope of v„(t) is independent of g(t). The resultant distortion is called failure-to-follow distortion. An exact relationship specifying the maxim um perm issible w m to prevent failure-to-follow distortion is obtained in Section 8.6.

In the direct circuit m odulator, g(t) is constrained to be greater than or equal to zero [a negative value of g(t) would result in an infinite dc current through the d io d e]; therefore, suppressed carrier m odulation is not possible even though norm al AM with a m odulation index of unity may be obtained. Since an exact expression for v0(t) as well as all the properties of the circuit of Fig. 8.2-6 are obtained in Section 8.6 and in C hapter 9, no further general analysis is attem pted at this point.

M easurem ent o f E nvelope D istortion

W hen any of the m odulators discussed in this section are im plem ented by practical circuits, the possibility exists that the circuits will introduce envelope distortion. For example, if f ( t ) is the m odulation inform ation, then the m odulator output might have the d istorted form

v(t) = A[l + mfd(t)] cos co0t, (8.2-32)

362 AMPLITUDE MODULATION 8 .3

where f d(t) is a nonlinear function o f /(f). In general, from direct observation of v(t) the distortion is not a p p a re n t; however, if a Lissajous pattern o f v(t) vs. Af( i) is formed on the face of an oscilloscope, a trapezoidal pattern results which m akes any envelope nonlinearity clearly evident. A typical trapezoidal pattern is show n in Fig. 8.2-7.

If no envelope distortion exists, then f d(t) = f ( t ) and the upper and lower edges of the trapezoid are straight lines. Any departure of these edges from straight lines quickly makes evident the nature and m agnitude of the nonlinear envelope distortion.

8.3 PRACTICAL ANALOG M ODULATORS AND MULTIPLIERS

A lthough the need for analog m ultipliers seems to be universal, the existing circuits tha t are capable o f accom plishing the task are few and far between. In addition, m ost practical m ultipliers suffer from either significant d istortion in one or both input channels, which limits the dynam ic range, o r a poor frequency response in one or both channels. F o r example, a H all m ultiplier, which is shown in Fig. 8.3-1, p ro duces a transverse voltage Vt which is directly proportional to the longitudinal current

Fig. 8 3 -1 Hall multiplier.

8.3 PRACTICAL ANALOG MODULATORS AND MULTIPLIERS 363

iL and the m agnitude of the norm al m agnetic induction field Bn, that is, V, = kiLBn. A lthough perfect m ultiplication seems possible, the norm al B field is severely limited in its m axim um frequency. This lim itation is due to the fact that the voltage am plitude required to produce the necessary B field increases linearly with frequency; thus unless special high-voltage circuitry is employed, B„ m ust be limited in frequency to under 10 kH z for norm al Hall multipliers.

As a second example, an N -channel junction FE T with a small drain-to-source voltage may be closely modeled as a voltage-controlled conductance of the form

gDS = ” ( l - ^ r ) , \vDS\ < 100 mV, t;GS > VP, (8.3-1)

where gDS is the drain-to-source conductance, I DSS is the drain current with vGS = 0 and vDS = — VP, and VP is the pinch-off voltage. If this FET is placed in the input of

Rr

Fig. 8 3 - 2 F E T multiplier.

a high-gain operational amplifier as shown in Fig. 8.3-2, then the output voltage va(t) may be w ritten as


which is p roportional to the product of v ^ t ) and v2(t). However, this FE T m ultiplier is restricted in dynam ic range. The voltage |t>,(i)| = \vDS\ m ust be kept less than 100 mV to limit d istortion, and v2(t) is restricted to positive values less than | VP\ + 0.7 V. N egative values of v2 would cut off the FET, while positive values in excess of | VP\ + 0.7 V would tu rn on the gate-to-source diode.

An even sim pler version of the m ultiplier of Fig. 8.3-2 may be obtained by replacing the operational amplifier by a single transistor, as shown in Fig. 8.3-3. In this circuit C E is chosen sufficiently large to be a short circuit for the frequency com ponents o fu^t), and / dc is chosen sufficiently large so tha t gin = qIdJ k T » gDSmax; hence ^ ( f ) appears directly across the FET. In addition, no dc voltage appears across the F E T because of C E; thus

vDs = vi(0 ,

‘d ( 0 — 8 d s v i ( 0

' c i ( f ) = a [ / dc + *d ( 0 ] ,

8 .3 PRACTICAL ANALOG MODULATORS AND MULTIPLIERS 365

and finally,

vo(t) = Vcc ~ a ^dc^L ~ a gDS^Lv i(f )

T/ I D 2(xR lI dSS~ K :C — — y V l ( 0 y ■ ^ )

It is apparen t tha t v j t ) contains a term which is directly proportional to the product o f i>i(i) and v2(t) ; however, as in the case of the operational am plifier-FET multiplier, |u,(t)| = |dos| m ust be kept less than 100 mV while v2(t) is restricted to positive values less than | Fp| + 0.7 V.

M ultipliers which employ m echanical shaft ro ta tion of a linear potentiom eter to produce m ultiplication, as shown in Fig. 8.3-4, are also limited by the maxim um frequency at which the shaft can ro tate and by the fact that x is constrained to be positive.

Fig. 8 .3-4 Potentiometer multiplier.

The advent of integrated circuits, however, has now m ade possible an analog m ultiplier in which both inputs have a linear range which is a good fraction of the supply voltage and also have a frequency characteristic which extends into the gigahertz region. A typical integrated m ultiplier, introduced by G ilb ertf and m aking use of the differential pair as its basic building block, is shown in skeleton form inFig. 8.3-5. F o r this circuit, if I ES1 = IES and I ES3 = IES4 (a condition which requiresthe transistors to be integrated on the same chip and to have the same geometries), then from Eq. (4.6 3)

± L = iA = _ _ L _ a n d h . = !± = _ ! _ ( 8 . 3 _ 4 )

H h 1 + ez h h 1 + e

where z = <2(1 — v2)kT and — i t + i2 . In addition , since i\ = ij + (1 — oc)j3,

., _ Ik + (1 _ a) h _ h /0 , ^

i + e - " I T ? ’ (° " 5)

t B. Gilbert, “A Precise Four-Quadrant Multiplier with Subnanosecond Response.” IEEE Journal o f Solid State Circuits, SC-3, No. 4, pp. 365-373 (Dec. 1968).


JT

Q >Q}

(1 - a ) i j

-J , ( t )

a

04

(1 a) i4

d >

Qi

Q i and Q j are active, since vc£ s ; ^ > 0

/, + i 2 = /t"

i ' x + i i = / * '

/;= /."+( i - o ) / t

---------<►'1 ‘2

Fig. 8.3-5 DifTerential-pair multiplier.

where l k = I'k + (1 — a.)Ik = i\ + i'2 . Therefore, com bining Eqs. (8.3 4 ) and (8.3-5), we observe that

and in a similar fashion

¡1 - ' 4 1 (8.3-6)

Ù - (8.3-7)* It

I t is apparen t that, when the tw o differential pairs are com bined as shown in Fig. 8.3-5, 13 varies in p roportion to while i4 varies in proportion to i'2 . In addition, both i3 and iA vary directly w ith Ik; thus if we constrain I'k to be a constant (we can accomplish this by developing i\ and i'2 from the collectors of another differential pair biased with a curren t source ¡¡Jo), then i3 is p roportional to the product of i\ and l k while i4 is the proportional to the product o f i'2 and I k.

An alternative, but som ew hat m ore practical, form of the differential-pair m ultiplier is shown in skeleton form in Fig. 8.3-6. As can readily be seen from the figure, i'f and i'2 are outward-flowing current sources which, unlike the corresponding sources of Fig. 8.3-4, may be developed from an N P N differential pair. The ability

8.3 PRACTICAL ANALOG MODULATORS AND MULTIPLIERS 367

Fig. 8 .3-6 Alternative form of differential-pair multiplier.

to employ all N P N transistors in a circuit greatly simplifies the m ass integration process. T he p en a lty w h ich m ust b e paid for th is ad van tage is that the m ultip lier requires large values of fi = a /(l — a), usually in excess of 150, for its p roper operation.

For the differential-pair m ultiplier o f Fig. 8.3-6 it is again readily dem onstrated that if I ESl = I Es2 and 1ES3 = I Es4 i then

« l = h = 1 h _ U _ 1rk i k l + ^ rk i k l +

where z = q(vt - v2)/kT. In this configuration it is the currents in the diagonally opposite transistors which are proportional. Now if (1 — a)i4 « i , and ( 1 — a)i3 « i2 (this is the high-j9 requirem ent), then

<1 = '"l- <2 = »2. <3 = ~jr> and I4 = (8.3-8)Ik

These equations are identical to Eqs. (8.3-6) and (8.3-7). Here again, if / k = /, -t- i2 = i\ + i'2 is constrained to be a constant, i3 is p roportional to the product of I k and .

Since in both differential-pair m ultipliers i\ , ï2 , and I k m ust be positive (to prevent any transistor from being cut offt), the m ultipliers in their present form produce outputs lying w ithin only one quad ran t of a cartesian coordinate plane and are therefore referred to as a “ one-quadrant m ultipliers.” However either multi-

t In addition, both and i'2 must be less than I'k to prevent either Q, or Q2 from being cut off.


plier can be extended to a “ tw o-quadrant m ultip lier” by taking advantage of the balanced nature of the differential pair.

If we define

Ifl(r) = 2i; ir2, (8.3-9)

or equivalently,

•/ Ik hi j Ik hih = — 2---- and '2 = — 2— ’

then ¡a (i) has a dynam ic range between ± I 'k and therefore occupies two quadrants. (This definition merely implies a dc shift in i[, which occurs naturally if a differential pair develops i\ and i’2.) In term s of /n (i), Eqs. (8.3-6) and (8.3-7) (or Eq. 8.3-8) reduce to

= and u = £ ( 5 f + l ) . (8.3-10)

By subtracting i4 from ¡3, which is readily accom plished a t the collector o f the differential pair, we obtain

ia = «(*3 - U) = «77'n, (8.3-11)Ik

w hich is a tw o-quadrant m ultiplication of I k(t) and in (f). A voltage va = i„R can be developed differentially by placing resistors of value R in the collectors of Q3 and Q4 in a fashion similar to that shown in Fig. 8.3-8 or using the scheme illustrated in Problem 8.13.

In general, tw o-quadrant m ultiplication is sufficient for generating norm al AM, since g(t) is always greater than zero and can be applied to the I k input. However, for suppressed carrier AM, a “ four-quadrant m ultiplier” is required.

A four-quadrant m ultiplier can be constructed from two tw o-quadrant multipliers by connecting them as shown in Fig. 8.3-7. A corresponding four-quadrant differential-pair m ultiplier is shown in Fig. 8.3-8. F o r the m ultiplier of Fig. 8.3-8, both the 6 3 -6 4 pair and the Q 5~Q6 pair share com m on base voltages with the Q x Q2 p a ir ; hence if ¡3 is high both i3 and i5 are p roportional to i\ as well as their respective em itter bias suppliers, and i4 and i6 are p roportional to i'2 as well as their respective bias supplies. Therefore, from Eq. (8.3-8) we obtain

PRACTICAL ANALOG MODULATORS AND MULTIPLIERS

h 0<‘,2 </«.

Fig. 8 .3 -7 Synthesis of four-quadrant multiplier.

Fig. 8 .3 -8 Four-quadrant differential-pair multiplier.

3 7 0 AMPLITUDE MODULATION 8.3

and in turn

r , • \ “M , , 'il'«*'oi — a(>3 + 'e) - ^ 11 + 777Z \ l v l .jkJkOj

¿.2 = «(¿4 + ¿5) = ^ ( l - (8 .3 -13 )\ 1 k1kO/

Finally, provided that i0lR < Vcc — VBB + 2V0 and io2R < Vcc - VBB + 2V0 to keep the differential pairs from saturating (V0 x | V for integrated silicon),

Vo = W . 1 - i . 2 ) = ^ i n / , 2 , (8 .3 -14 )*k

where — l'k < in < /¡c and — / k0 < ii2 < i k0; thus we see th a t the circuit of Fig.8.3-8 functions as a four-quadrant m ultiplier w ith a scale factor K = xR/I'k.

Since (, ! may be as large as I'k and ii2 may be as large as I k0, we observe fromEq. (8.3-13) that iol has a m axim um value of oclko. Similarly io2 has a maxim umvalue of a/jto- Therefore, a sufficient condition to ensure that the differential pairs do not saturate is that

a I k0R < Vcc - V bb + 2V0. (8.3-15)

Figure 8.3-9 illustrates a typical circuit for developing the required differential currents to drive the m ultiplier o f Fig. 8.3-8. f In this circuit, if R E » rin = 2 kT /q Ik, which is usually the case, then

»ci = “ '£1 = <*

for

Ate , ^<01 , f AieT + — j and y - ^ J (8.3-16)

In addition, if Q 3, Q4, and Q s are identical in geometry, then I E3 = l E4 = 1ES = I E and hence a net current of (3 — 2a)IE flows through R B. Thus

= ai E = a(VEE ~ Vo) « Vee ~ V° . (g.3-17)2 E R ^ 3 - 2a) R B K ’

Figure 8.3-10 illustrates a com plete integrated m ultiplier which includes the differential current source drivers. In this circuit (if R E » rin) we can identify

2<xvn 2ai)i2*¡1 = ~ 5 > l i2 = . *kO ~ a ^ d c 2 > a n c * I'k = a ^ d c l J

1 K E 2

t An alternative circuit form would have two series emitter resistors of value RFj2 with a single current source driving their common node. This approach suffers from two disadvantages in comparison with the circuit of Fig. 8.3-8. First, an additional resistor is needed which requires more area on an integrated chip than the transistor it replaces; second, the bias current IdJ2 flows through the resistor, making the Q-point a function of RE.


R'f -I- 2rm

2kT o—+

MO

o—(3 2a)/£

r ^ n03Ir

I “

hi- _L 2 R'c

¥ \

Re-AAA-

02*£"2

04/f

A:. I2 I

05

- o v.

1 f S .3 I e s 4 ~ I e s S ‘ ¡E 4 I e 5 I e

Xo_ . B > IQ( EE V0) VçE (1 2 (3 -2 a)Ä s ~

Fig. 8.3-9 C urrent drive for differential-pair multiplier.

hence, provided tha t all transistors rem ain in their active region, from Eq. (8.3-14) the four-quadrant m ultiplier ou tpu t becomes

I'oiO =4 a 2R

ÊlÊ2^d(8.3-18)

which is the desired product of the two input voltages i’n (i) and vi2(t).To ensure the validity of Eq. (8.3-18), the following restrictions on the m agnitudes

of Vn(t), vi2(t), and I dc2 m ust be observed :

I) ~ ^££ + < ui 1 < ^BB

^ E E + ^ 0 < V i2 < V B B ~ V o

- I .d e l ^ ^¿1 ^ I del

to keep Q-i active,

to keep Qg active,

to keep Q 7 and Q8 from being cut off,

to keep Q9 and Q i0 from being cut off,

to keep Q3, Q4 , Q 5, o r Q6 from saturating (cf. Eq. 8.3-15).

^ In m ost integrated analog m ultipliers of the form shown in Fig. 8.3-10, the transistors are integrated on one chip ; however, the resistors R, R B1, R B2, R El > Ê 2>and R bb m ust be supplied externally by the engineer m aking use of the multiplier.

2 )

3)

4)

^£1

~ I d c 2 < V i2 < I d c 2

R<

5) (x2IAc2R < Vcc - VBB + 2V0


Fig. 83-10 Complete integrated multiplier.

In general, R is chosen sufficiently small so tha t it is no t appreciably loaded by either the input im pedance of the next differential pair or stray ou tpu t capacitance. For example, if 20 p F of ou tpu t capacitance exists and the m ultiplier is to operate up to frequencies of 20 M Hz, then

1R <

2 n ( 2 0 x 1 0 6 ) ( 2 0 x 1 0 ~ 12)= 400 a

O n the o ther hand, R is not chosen any smaller than necessary; otherwise, enorm ous currents would be required to develop a reasonable ou tpu t voltage.

8 .3 PRACTICAL ANALOG MODULATORS AND MULTIPLIERS 3 7 3

O n the assum ption that Vcc is specified a t 10 V, — VEE is specified at — 5 V, and

l ^ i l l m a x I ^ ¿ 2 1 ma x 4 V ,

is chosen to be greater than 4 V + 0.75 V = 4.75 V to avoid saturating Q9. A choice of 5 V in this case would be reasonable. W ith R = 400 Q and VBB = 5 V. restriction (5) limits / dc2 to

1 6.5 Va 2 4 0 0 Q

16.3 mA.

Probably a value of 16 mA would be selected, since a smaller value would require a larger value of R E2 [restriction (4)] and thus a smaller output voltage.

W ith / dc2 = 16 mA the m inim um value of R E2 is

Re2 = 2 !^L ax = 500 Q‘ d c 2

(a standard value of 510 i i would be chosen). Now for maximum output voltage we also desire IdciR Ei to be as small as possible; however, IdclR Ei is constrained to be greater than 2|i>n |max x 8 V. A choice of / dcl = 9 mA and R El = 1 kQ is reasonable. W ith these values we can be sure that all transistors operate in their active region and that

v o U ) = “ y i ’. i W M f ) - (8.3-19)

Finally, we choose

5 V~ 9m A ~ 5-6

to yield the desired value for BB’

4.25 V ~ ~ 470 Î2

9 mA

to yield the desired value of / dcl, and

4.25 V

to yield the desired value o f / dc2.Now, for this multiplier, if

»nit) = (4 V )/(f) and vi2{t) = (4 V) cos 108f,

then the output r„(f) takes the form

u0(t) = (5.57 V )/(i) cos 108t,

which is a suppressed carrier AM signal. Clearly the scale factor could be decreased to any value less than 5.57 V by increasing either R Ei or R E2 o r by decreasing R.


Analog Multipliers with One Nonlinear Input ChannelM any m ultipliers exist in which one input channel is linear while the o ther input channel is highly nonlinear. An exam ple of this type of m ultiplier is a single differential pair for which the collector curren t is directly proportional to I k and a highly nonlinear function of u i — v2 (Eq. 4.6-3). A lthough m ultiplication in such a device is not ideal, am plitude m odulation can be accom plished by applying the carrier to the nonlinear input channel and the m odulation to the linear input channel while placing the ou tpu t through a bandpass filter centered abou t the carrier. The bandpass filter removes the harm onics of the carrier generated by the nonlinear channel.

Such a m odulator em ploying an integrated differential pair is shown in Fig.8.3-11. For this m odulator, Q3 and QA act as a current source to develop a current drive I k(t) to the differential pair of the form

hU) = /«>[ 1 + (8.3-20)where I k0 = <x(VEE - V0)/(2 — a.)RB and m = V2/{ VEE - V0). This assumes, of course, tha t Q3 and Q4 are integrated on the same chip and with identical geometries, so that /£53 = I ES4.) In addition, from Eqs. (4.6-9) and (4.6-10),

>E2 = AkWDz — a i ( x ) c o s a>0t — a 3(x) c o s co0t — ■■■], (8 .3 -2 1 )


where x = q V J k T and a ^ x ) is the Fourier coefficient plotted in Fig. 4.6-4. Therefore, the fundam ental com ponent of the collector current of Q2 is given by

'c 21 = - / M« iW [l + mf(t)] cos co0t, (8.3-22)

which results in

v0(t) = vcc + W *i(* ){[l + " i / ( i ) ] c o s a v } *Z n(f)

= Vcc + /k o a iM U l + mf(t)] * z, 1L(f)} cos w0f, (8.3-23)

where z ^ i t ) is the impulse response of the ou tpu t tuned circuit and Zj lL(t) is the low- pass equivalent of z , , (r) (cf. Fig. 3.1-3). Im plicit in the expression for vB(t) given by Eq. (8.3-22) is the fact that the tuned circuit has a sufficiently high Q to remove the low-frequency and higher-harm onic com ponents of ic2(t). If, in addition, the output filter is flat over the band of frequencies occupied by iC2 i(r), then v0(t) simplifies to

vo(t) = Vcc + hoRL<ii(x)[l + cos a>0t, (8.3-24)

which is clearly the form of the desired AM signal.As a num erical example let

Vcc = 12 V, VEE = - 12 V, Vi = 1.56 mV, V2 = 5.63 V,

R b = 6kQ , Rt = 5 k il , C = 1000pF, L = 10#iH,

io0 = 107 rad/sec, and /( f ) = cos 105t,


11 25 Vm = h / t0 = — —— = 1.875 mA, x = 6,

6 ki2

and from Fig. 4.6-4,

a t(6) = 0 .6 .

Therefore

‘ci'Jt) = (1.25m A )(l + \ cos 105f)cos 107i

and

v jt) = (12 V) + (1.125 mA)[(l + ¿co s 105r) z t ¡ it) cos 107r]

= (12 V) + (5.63 V) 1 -i t= cos ( 105r — —2 ^ 2 \ 4

cos 107i.

The attenuation of the cos 105f term by l / y /2 and its phase shift by —n/4 results directly from the fact that the low-pass equivalent of the output-tuned circuit has a — 3dB bandw idth of 105 rad/sec.

N ote that the m inim um value of v j t ) is 4.4 V ; hence Q2 does not saturate.


8.4 PRACTICAL CHOPPER M ODULATORS

The key com ponent in a chopper m odulator is the voltage-controlled switch which opens and closes at the carrier rate. Therefore, in this section we shall look at two voltage-controlled single-pole single-throw (SPST) switches—one em ploying a diode bridge and the o ther em ploying a single FET—and then we shall consider the problem of em ploying these single-pole switches to construct a voltage-controlled reversing switch. The SPST switch is used in the single-ended chopper m odulator shown in Fig. 8.2-2, whereas the reversing switch is employed in the balanced chopper m odulator shown in Fig. 8.2—4.

N ote that, although the voltage-controlled switch is being discussed in conjunction with chopper m odulators, it functions equally well as a synchronous dem odulator or a mixer in the same configuration as Fig. 8.2-2. For a synchronous detector, however, the ou tput filter m ust be replaced by a low-pass filter, whereas for a mixer the ou tput bandpass filter m ust be tuned to the interm ediate frequency.

Diode-Bridge M odulator

Almost all single-ended chopper m odulators in use today employ the diode bridge as the voltage-controlled switch. Figure 8.4-1 illustrates a typical diode-bridge m odulator in which a positive value of v^t) causes all the bridge diodes to conduct thereby bringing va(t) close to ground potential, and in which a negative value of reverse-biases all the bridge diodes, thereby perm itting va(t) to follow g(r).

It is apparent that the am plitude F, of r , (i) must be sufficiently large when u, is negative to keep all the diodes reverse-biased. A little thought indicates that for g(t) > 0, and with i',(f) = — V,, D2 and D3 are on the verge of conduction when g(t) = V, + 2V0 ; and that for g(t) < 0, D, and Z)4 are on the verge of conduction for |g(i)l = K + 2V0 . Hence to ensure tha t all the bridge diodes rem ain reverse-biased for r , = — K,, we require that

It is also apparent that Vv must be sufficiently large when = + Vy to keep all the diodes forward-biased so that the bridge presents a low impedance to g ro u n d ; that is, va{t) should be a small voltage with v, = V To determ ine the required m agnitude of K, in this case, we define the current leaving the r , source as I 2 and the current leaving the g(f) source as i^t). Since, in general, the four bridge diodes are integrated on a single chip with identical geometries, the bridge is balanced and ^(r) and l 2 split equally between the two bridge arm s; thus

V, > g(t) - 2V0 (8.4-1)

for all f.

(8 .4 -2 )

8.4 PRACTICAL CHOPPER MODULATORS 377

V |( 0

Fig. 8.4-1 Chopper modulator employing diode bridge.

If we assume that each diode is characterized by the volt-am pere relationship

then

iD = l s( e ^ lkT - 1 ) % l seqr” /kT,

k T l I 2 + ii(t) i 2 - /,(i)\l D2 Vn i = ----1 In

<z 2/c' - I n

2IS

(8.4-3)

(8.4—4)

k TIn

Q \ l - i iU V h l

By expanding va(t) in a M acLauren series in ¡ i / / 2, we obtain

‘Ì312, ' 5I t

+ (8 .4 -5 )

where rd = {kT/q)(2/12) is the small-signal diode resistance with / 2 2 as a bias current. It is apparent that, if we wish to keep nonlinear com ponents of /, [which is p ro p o rtional to the m odulation g(f)] out of the output to avoid envelope distortion in va(t), then ¡ i /3 /2 « 1. W ith this restriction t>a(f) = rdi,(t) and the forward-biased diode


bridge may be modeled as a single resistor of value rd shunting va ; hence the diode bridge takes the form of an ideal voltage-controlled switch in series with a resistancerd-

W ith this modelg(f)

ii(0 =R i + ra

while

¡2 =Vi - 2V0 and

2 RirA =

q(V, - 2V0)/kT

thus to keep if /3 /f < 0.01,

5.8g(f)R 2

f * > r 7 T T 7 + 2Ko(8 .4 -6 )

for all (. The inequalities of Eqs. (8.4-1) and (8.4-6) may be satisfied sim ultaneously by choosing Vt > g(t) — 2V0 and choosing R 2 o f the o rder of r o R j . F o r exam ple, if |g(t)lmax = 10 V, then the bridge rem ains open with u, = — Vx for > 8.5 V (Vo = ?V). ^ we select Vx = 9 V, then if

+ r d ^

7.75 ~ 7 .7 5 ’

the bridge appears as a resistor rd with v t = Vt .A com plete diode-bridge m odulator which incorporates the floating source

v t(t) as well as the ou tput filter is shown in Fig. 8.4-2. In this circuit the transform er

Fig. 8.4-2 Practical balanced modulator.

8 . 4 PRACTICAL CHOPPER MODULATORS 3 79

is a closely coupled transform er operating in its m idband range, and therefore functions as an ideal transform er. In addition, (1 + fi)RE is large in com parison with R, so that the transistor does not load the bridge. Consequently, if the output-tuned circuit is broad enough to pass the m odulation and yet narrow enough to remove the low-frequency and higher-harm onic com ponents of va(t \ then from Eq. (8.2-19)

v o U ) = — c o s o V + vcc • (8.4-7?n R E R , + rd

Equation (8.4-7) assumes, of course, that Eqs. (8.4-1) and (8.4-6) have been satisfied and that the transisto r rem ains in its active region.

The control voltage ^ ( r ) may be supplied by a sufficiently large sine wave of radian frequency o»0 instead of a square wave. If, as shown in Fig. 8.4-3, K, is large in com parison with Va and Vf (Eqs. 8.4—1 and 8.4—6), then the sine wave functions in essentially the same fashion as the square wave in controlling the states of the bridge.

One main advantage of a sine-wave drive is that the transform er coupling v^t) to the diode bridge need not be nearly as broadband. On the other hand, the larger value of V, with a sine-wave drive requires a much higher breakdow n voltage for the bridge diodes.

W hether r,(f) is a sine wave or a square wave, in practical diode bridges short- duration transient “ spikes” appear on v j t ) in the vicinity of the bridge transitions from open to closed because of parasitic capacitance and diode charge storage. These spikes are, in general, of little consequence, since they contain sufficiently high- frequency com ponents so that they are not transm itted to the output through the bandpass filter H{ jw).

FET M odulator

A junction or an insulated-gate FET may be employed instead of the diode bridge as the voltage-controlled switch in a chopper m odulator. Figure 8.4-4 illustrates a


Fig. 8.4-4 N-channel junction FET chopper modulator.

typical N-channel junction F E T chopper m odulator. In this circuit, with v^ t) = — Vx < Vp(Vp is the pinch-off voltage of the FET), the FET opens, perm itting va(t) to follow g(r). O n the other hand, with t>,(i) = 0, then the FE T functions as an ohmic conductance gDSS of value

S o » = ^ . (8-4-8)Vp \ P I v c s = 0 V P

provided that I t^ l = \va\ < 100 mV. Consequently the FET may be modeled as an ideal voltage-controlled switch in series with a resistance rDSS = l /g DSS. For typical junction and insulated-gate F E T ’s, rDSS varies from several ohm s to several thousand ohms.

W ith y,(f) = 0,

" r D SS

thus to ensure tha t |i’DS(f)i rem ains less than 100 mV for all t we require to be sufficiently large so that

I IgWInR l > rDSSl i o ô ^ v

(8.4-9)

For example, if |g(t)Lax = 5V and rDSS = 500Q, then R, > 24 .5kQ. To avoid loading by the ou tpu t transistor when the FE T is reverse biased, the resistor R ,

8.4 PR A C T IC A L C H O P P E R M O D U L A TO R S 3 8 1

should not be chosen too much greater than this value. If, on the o ther hand, R, is chosen to be less than 24.5 kQ, then va(t) exceeds 100 mV, rDSS becomes nonlinear, and v j t ) is no longer a linear function of g(t); consequently nonlinear envelope distortion begins to appear on the ou tpu t AM wave.

If R e is sufficiently large so that transistor loading can be neglected, then v j t ) is given by (cf. Eq. 8.4—7)

v„(t) = 2i§ - ) cos co0(t) + Vcc- (8.4-10)n R E R , + r DSS

However, if R h is not sufficiently large, then the loading must be incorporated with the ^(i)-K i network as shown in Fig. 8.4-5 by forming a Thevenin equivalent ne twork. Clearly g(r) is decreased by a factor of r\ because of the loading; however, in addition, a dc bias V is added in series with g(r). If g(i) = 0, as it is for suppressed carrier m odulation, then the presence of V produces an average com ponent in the m odulation voltage being chopped and thus a nonzero carrier at the output. To eliminate this undesired carrier com ponent, either R E must be increased relative to R { or an isolation stage such as a source follower must be inserted between the chopper stage and the ou tpu t transistor.

Fig. 8.4-5 Effect of transistor loading.

In addition to the diode bridge or the FET chopper, a bipolar transistor being switched between saturation and cutoff may be employed as the voltage-controlled switch. However, when saturated, the transistor may be modeled as a resistor in series with a dc voltage source of approxim ately 100 mV (for silicon). This voltage source has the effect of introducing a carrier com ponent at the m odulator output, which is quite undesirable if suppressed carrier A M is being generated. This saturation voltage may be largely balanced out by placing two transistors in series (emitter to em itter) and placing the switching voltage between their bases.


Balanced Chopper Modulator

Figure 8.4-6 indicates how two diode bridges can be employed to alternately apply + g(i) and — g(t) across and thus produce the effect of a reversing switch. N ote that the bridges are arranged so tha t one bridge is open when the o ther is closed. It is apparent tha t the closed bridge is unaffected by the open bridge and thus Eq. (8.4-6) still determ ines the value of Vx required to ensure th a t the bridge rem ains

Fig. 8.4-6 Reversing switch for balanced chopper modulator.

closed for all t. O n the o ther hand, the closed bridge does affect the open bridge in that it increases the voltage across the open bridge to

hence with the argum ents em ployed to obtain Eq. (8.4-1) we require

V7, > g(t)\ 1 +R 1 + rc

2V0 (8.4-11)

for all t to ensure that all the diodes in the open bridge rem ain reverse biased.Figure 8.4-7 illustrates a practical chopper m odulator in which both g(t) and

i>i(r) are supplied from grounded sources. The transform er Tt is a closely coupled center-tapped audio transform er with a m idband frequency range sufficient to pass the frequency com ponents of g(f) (see Section 2.2), while the transform er T2 uses a

8.4 PRACTICAL CHOPPER MODULATORS 383

closely coupled transform er with a m idband capable of passing the main frequency com ponents of i;,(f). If v^ t) is a large-am plitude sine wave, then the restrictions on T2 are nominal. [A lthough unity turns ratios are indicated for the two transform ers, other turns ratios merely introduce a scale-factor change in g(f) and i’i(f).] If we assume that is not loaded by the transistor and that Eqs. (8.4-6) and (8.4-11) are satisfied, then va(t) may be expressed as

I ' J i t ) = (8.4-12)K i + r d

where S'(i) is given by Eq. (8.2-19). And if we assume that the transistor does not saturate, we may write

ic = ccva(t)/RE


and in turn

4 a g ( f ) R i

'«(0 = K rn R E K , + r d _

z U L ( t ) c o s w 0 t (8 .4 13 )

where z, , L(i) is the low-pass equivalent impulse response of the ou tpu t parallel R L C circuit. If the ou tpu t filter passes g(/) undistorted while removing the 3w0 com ponent of u„(r), then v0(t) reduces to the desired form

4 ocR Rvo(t) = J 'c c „ L „ ‘ KU) CO S (Jj0t. (8.4-14)n R h: + rd

8.5 SQUARE-LAW MODULATOR

The square-law device, although quite attractive as a mixer, finds very little application as an am plitude m odulator. The basic reason for this is that most physical devices have half-square-law characteristics rather than full-square-law characteristics. As we saw in Section 8.2, unless a full-square-law characteristic exists, not only is suppressed carrier m odulation impossible but also norm al AM with 100% m odulation is impossible. Consequently, unless a “quick and d irty” low-index m odulator will satisfy the requirem ents of the situation, the other m odulators discussed in this chapter are usually employed. Therefore, we shall look only briefly at one square- law m odulator constructed with a junction FET operating within its saturation region.

A typical square-law FET m odulator is shown in Fig. 8.5-1. If for this circuit we assume that the FET operates within its saturation (square-law) region and that R L is much less than the ou tput im pedance of the FET, then we may approxim ate the drain current as

1d = I d s s 11 j ^ | ’ ( 8 . 5 - 1 )

where VP is the pinch-off voltage and I DSS is the drain current with rGS = 0 andvDS = — VP. For the bias arrangem ent shown in Fig. 8.5-1, iD reduces to

l v ‘ + v A 2 , I K cos co0t + A[i + m f( t)] \2l m ‘ ) — ' D S S I y I — ' D S S I y I - ( O . J - Z )

Since the com ponent of iD(t) centered about a>0 is

2!dS' V2 ' A{ \ + c o s a)0t ,VP

then with the assum ption tha t the ou tpu t filter removes the low-frequency and second- harm onic com ponents of iD, v„(t) is given by

2 /n s s Mv„(t) = VDd -------- 772— [1 + * z n J i) c o s co0t. (8.5-3)

8 .5 SQUARE-LAW MODULATOR 385

If, in addition, the ou tput filter is flat over the band of frequencies occupied by the AM signal, va(t) simplifies to the desired form

21 ossV\ARLr

~V\l ' o U ) = V D D ------------ Ï72--------[! + w / ( f ) ] cos w 0f. (8.5-4)

To realize v0(t) in the form of Eq. (8.5-4) we must restrict va(t) > — | VP\ for all time to ensure that operation remains within the saturation region. This may be accom plished for any FE T param eters by choosing a sufficiently small value of R , .

In ad d itio n ,

t’i(i) + v2(t) > 0

for all time to keep the FET from being cut off and

ih(t) + 1.-2« <\Vp\ + V0= VP for all tim e to k eep the g a te -to -so u rc e d iod e from tu rn in g on. T h e se tw o re s tr ic tio n s


imply that A, Vx, and m m ust satisfy the following inequalities:

A( 1 — m) - Vt > 0,

A(\ + m) + Vi < V'P. (8.5-5)

If we desire the largest possible ou tpu t with the smallest value of R L (to ensure R l « /•„), then we satisfy Eq. (8.5-5) with an equality to obtain

y V'A = - y a n d v i = - y ( l - m).

The output voltage given by Eq. (8 .5 ^ ) then reduces to

vo(0 = (1 - [1 + mf(t)} cos a)0t. (8.5-6)

Here again we see explicitly that we can achieve a high m odulation index only at the expense of reducing va(t) in am plitude or increasing R L to the point where it is anappreciable fraction of the FE T ou tpu t resistance r„. If R L ^> rB, then square-lawoperation no longer exists.

As a num erical example let

VDD = 12 V, VP = - 4 V, V0 = 0.7 V, I DSS = - 4 mA,

co0 = 107 rad/sec, and a>m = 2.5 x 105 rad/sec.

[<Bm is the m axim um frequency com ponent of /(f)]- Let us now choose param eters for the circuit of Fig. 8.5-1 to achieve a m axim um output voltage with m =

Since K (t) |max < VDD + VP to ensure saturation-region operation, then fromEq. (8.5-6) we obtain R L < 3.87 Q ; thus a reasonable choice would be R L = 3.6 k ii,which is much less than the typical values of r0, say 100 k£X W ith this value R L and a choice for the bandw idth of the ou tput-tuned circuit of 5 x 105 rad/sec (the — 3dB point corresponds to tom),

^ (3.6 kft)(5 x 105 rad/sec) ^ ^

and in tu rn L = 1 ¡(o\G = 18 //H. Finally, A — 2.35 V, Vt = 1.175 V, and

v0 = (4.96 V)[l + m/(t)] cos co0f. (8.5-7)

The square-law m odulator with the above param eter values is shown in Fig.8.5-2. The gate bias circuitry is obtained by first noting from Fig. 8.5-1 that

% s = VP + A[ 1 + ¿ /(f)] + Vi cos 107f

= ( - 1.65 V) + (1.175 V )/(f) + (1.175 V) cos 107f (8.5-8)

and then applying the — 1.65 F to the gate by bleeding down the — 12 V supply and coupling the ac sources to the gate through a capacitor.

8 .6 TUNED-CIRCUIT MODULATORS 387

8.6 TUNED-CIRCUIT MODULATORS

As we shall see in C hapter 9, it is generally difficult to provide linear, high-power ou tpu t amplifiers for AM signals; therefore, one usually tries to accomplish this m odulation at as high a power level as is possible. In vacuum tube transm itters such m odulation is alm ost inevitably performed in the last stage. To the extent that the transistor behaves as a current source, the triode vacuum -tube-type m odulator cannot simply be “ transistorized.” As we shall see in Section 9.9, it is possible to construct efficient transistor power amplifiers if one both drives them with a m odulated signal and sim ultaneously m odulates the collector supply. It is the purpose of this section to present an idealized version of a circuit that can be used to perform the initial m odulation of the carrier signal so th a t this prem odulated signal may then be used to drive a power-amplifying stage. As we shall see, a t first glance both the m odulating and the power-amplifying stages look identical, but they do indeed operate in somew hat different fashions.

The basic m odulating circuit being considered is shown in Fig. 8.6 1. The operation of this circuit requires the collector-base junction of the transistor to become saturated (or to tu rn on) a t the peak of every driving carrier cycle. The current pulse that flows as a result of this saturation has two effects. The direct effect in the output circuit is to cause the ou tput tuned-circuit voltage am plitude to follow the variations


L

Fig. 8.6-1 Saturating collector AM modulator.

in f( t) . The indirect or reflected effect is to increase the loading in the base circuit, thus effectively causing the input driving voltage also to follow the variations in f( i) .

In order to concentrate our a tten tion on one problem at a time, we assume for the present that the input voltage is supplied by a voltage source. That is, we neglect any possible variation of the input driving signal. Also, in order to be able to m ake very simple calculations, we m ake the further assum ption that the voltage drive is

-sufficiently large (x » 10) so tha t the transistor collector current may be assumed to flow as a train of impulses of strength q.

If the transistor is m odeled as a current source in parallel with an ideal diode and a battery ^CE.at’ then by splitting the current source into two parts [and then removing the one in parallel with the + Vm f (t) voltage generator] and redrawing

L

(a) (b)

Fig. 8.6-2 Two equivalent forms for the model o f the circuit of Fig. 8.6-1.


the circuit, one can reduce the circuit from the form of Fig. 8.6-2(a) to the form of Fig. 8.6-2(b). If m is defined as

Km - ---------------Vcc — Vce.„

and g(i) is defined as

g(0 = (Vc c - VCE. J [ 1 + mf(t)lthen this circuit reduces to the form shown previously in Fig. 8.2-6.

The initial buildup of the tuned-circuit voltage in such a circuit is shown in Fig. 8.6-3. This buildup continues until the step on the tip of one of the carrier cycles exceeds g(t) and the tuned-circuit voltage is caught at this value. Thereafter, so long as g(t) does not vary too rapidly, the diode will conduct on every peak and the circuit will “ ring” between the peaks at its natural frequency, w0.

<<('} Strengy

th of q

tT 2 T 3 T 4 T 5 T 6 T I T l— —

ic(i) q Z 8(r — nT). T=2n/ujnit = — a.

Fig. 8.6-3 Transient buildup of the tank voltage in a tuned-circuit modulator.

If the tuned-circuit QT is high enough (say m ore than 30) so that the energy decay between resettings can be considered linear, then for any constant value of g(t) the size of the step at each peak will be approxim ately g(t)n/QT. Therefore, the higher the value of Q r , the smaller the d istortion at each cycle peak. In a practical circuit we do not drive with im pulses; hence the output voltage does not have steps. For example, with current pulses that are 90° wide the voltage transient in the tuned- circuit voltage will be nearly imperceptible, even though the fundam ental current into the tuned circuit is essentially the same in the case of impulse, and in the case of wider current pulses.

F ailure-to-Follow D istortion

If g(i) increases too rapidly in a cycle, then the maximum step in the tuned-circuit voltage, A V — q/C. will be insufficient to cause the diode to conduct and the tuned- circuit voltage will not track the m odulating signal.


Jt';gan be shown for the impulse drive case that the necessary condition to prevent upward-going failure-to-follow distortion is the satisfaction of the condition

[ g ( i o ) / « ] + g ( t 0 ) ^ I ( 8 . 6 - 1 )

where a is the real part of the tuned-circuit pole, t 0 is the tim e a t any cycle peak, and / 1 = 2q /T is the fundam ental com ponent of the collector impulse train. A simple proof of Eq. (8.6-1) requires only (1) the assum ption of a high enough value for QT that the envelope decay during a cycle is linear and (2) the assum ption of a^low rate of Ghange for g(i). Further investigation indicates that the result is valid for all values of Qt and of g(i0).

. Intuitively it is obvious that the input drive m ust always be such that I {R L exceeds the peak value of g(t), otherwise, failure-to-follow distortion will certainly result. E quation (8.6-1) establishes a further relationship am ong the drive signal, the circuit, and the rate of change of the m odulation.

Phase DistortionIf g(t) decreases m ore rapidly than the natural envelope decay of the tuned-circuitvoltage between current pulses, then the period between diode conductions will fallbelow T = 2n/m0 and phase m odulation of the carrier com ponent of the tuned- circuit voltage will result. To prevent such distortion,

g(t0 + T ) > g ( t 0)e - ‘r (8.6- 2)

is a necessary and sufficient condition.W hen QT is high and the rate of change of g(i) is low, this equation may be approxi

m ated by[«(to)/«] + g U o ) > 0. (8.6-3)

As in the case of failure-to-follow distortion, a m ore involved analysis shows that Eq. (8.6-3) is in fact valid w ithout restrictions on either g(i) or QT.

To prevent either type of d istortion from occurring, the two previous restrictions may be com bined to yield

I J 2 C > g{t0) + ag(f0) > 0, (8.6-4)

where excessive positive values for g(t0) lead to the violation of the left-hand inequality and excessive negative values of g(t0) lead.to the violation of the right-hand inequality.

Sinusoidal ModulationFor the specific case of sinusoidal m odulation where

' g(i) = (Ice - J 'ce .JO + m cos wmt),

one may evaluate Eq. (8.6^4) in order to show the restrictions necessary on m to avoid either type of distortion. F or dow nw ard-going m odulation the restriction is

m < . 1 = ; (8.6-5)J 1 + (com/a)2


for upw ard-going m odulation the restriction is

m < V M V c c Z W 1 - 1 . (8.6-6)y j 1 + ( tü ja )2

Hence if ¡¡R L is greater than 2(VCC Vc£,„)> then the dow nw ard-going restriction governs in all cases. F o r 100% m odulation this restriction on the m inim um size for 11 is imposed at any rate ; hence no additional hardship is added by the ra te of variation of the m odulation.

One way to ensure that m always satisfies the necessary restrictions is to place the m odulating signal th rough a low-pass filter with a — 3 dB bandw idth lower than a and to obtain g(i) a t the filter output. Specifically, if the filter transfer function is H(jaj), then

m = (VCC- V c i j m o ) j l + COS [0)mt + arg H ( M J ] J . (8.6-7)

Since the resultant value of m from Eq. (8.6-7) is \H(ja>m)\/H(0), then, if this function is always equal to o r less th an l / ^ / l + (00J o ) 2, Eq. (8.6-5) will always be satisfied. Figure 8.6-4 illustrates this situation.

Fig. 8.6-4 Prefiltering of the modulation to ensure no distortion in the modulated output[IxRl > 2(Vc c - FC£.J ] ,

F o r a practical circuit the phase-distortion restriction should be reduced, since with a spread-out current pulse an earlier conduction time for the diode conduction should lead to a less ab rup t phase shift in the output. Certainly, if Eq. (8.6-5) is satisfied, then distortion should be avoided.

To sum m arize: We have seen that, if the collector-base diode of a transistor conducts on every cycle, then the circuit of Fig. 8.6-1 performs as an AM m odulator. As was pointed ou t in a previous section, it is possible to obtain 100% m odulation from such a circuit w ithout the strain of driving the tuned circuit between zero and 2( K:c — ^ one merely m odulates to a peak value of, say, 70 % and then subtracts a carrier com ponent sufficient to bring the valleys down to zero.


O nce again we shall point out that, though the m odulator o f this section looks like the m odulated power amplifier of Section 9.9 the operation of the two devices is not identical. The m odulated power amplifier can operate perfectly satisfactorily even if the collector-base diode never goes into conduction, whereas the whole basis of operation for the m odulator of this section is the conduction of this diode.

PROBLEMS

8.1 Find the amplitude and frequency of the spectral components of v(t) = g(i)cosoj0f for each of the following cases:a) g(r) = (5 V) cos wmt, com « co„,b) g(t) = (5 V) + (3 V) cos a>mt, wm « u>0,c) g(t) is a symmetric square wave which varies between 0 V and 5 V with a period T =

l/wm » l/(t)0.8.2 The voltage i>(f) given in Problem 8.1 appears at the input of an antenna having an input

resistance of 100 i) in the neighborhood of a>0. Determine the carrier power and the sideband power delivered to the antenna for each g(t) given in Problem 8.1.

8.3 The voltage v(t) given in Problem 8.1 is applied to a single tuned circuit having a center frequency of a>0 , a bandwidth of 2a>m, and unity transmission at o»0 ■ Find the voltage at the filter output for each g(r). Sketch the frequency spectrum in each case.

8.4 The modulation g(r) = (5V)cosco,f + (10V )cos«2i (wi < w2 « w0) is applied to theSSB modulator shown in Fig. 8.1-5. Determinea) g(t), the Hilbert transform of g(r),b) an expression for vSSB(t),c) the spectral components of i'SSB(r), andd) the power fSSB(f) delivers to a lOOil resistor when placed across it.

8.5 Show that, when g(r) is placed through two Hilbert transformers in cascade, the output ofthe second Hilbert transformer is — g(t); that is, H[Hg(t)] = — g(r).

8.6 Determine the amplitude of the carrier which must be subtracted from t>(r) = {10 V)[l +0.2/(i)] cos cu0f to yield a modulation index of 0.9.

8.7 In the circuit shown in Fig. 8.2-2, g(t) = (5 V)(l + cos 104t), oi0 = 107 rad/sec, and H(ju>) is a single tuned filter having a bandwidth of 2 x 104 rad/sec and a center frequency of 3 x 107 rad/sec. Assuming that the filter has unity transmission at resonance, find an expression for

8.8 In the circuit shown in Fig. 8.2-5, A = 10 V, F, = 5 V, and

where va and u, are in volts. Calculate the m aximum output m odulation index which can be achieved w ithout causing envelope distortion in v0(t).

8.9 F or the circuit shown in Fig. 8 .P-1, find v0(t). W hat is the m odulation index? Sketch i f / ( 0 = cos 103f.

Vi < 0, t), > 0,

lO k iiPROBLEMS 393

10 V

Figure 8 .P -2

394 AMPLITUDE MODULATION

8.10 For the circuit shown in Fig. 8.P-2, calculate i!„(?) if/(<) = cos 1.66 x 105f. Repeat, assuming that f( t) is a square wave with transitions between +1 and — 1 and with a 100 ^sec period.

8.11 a) For the circuit shown in Fig. 8.P-3, determine an expression for v0(t) assuming that none of the transistors saturate, that all the transistors are identical, and that

|t’,(f)| < lOOmV and 0 < v2(t) <\VP\ + V0.

b) Evaluate v0(t) for the case where

¡ d s s = 4 m A , K P = — 2 V , R L = 2 k O , ± I /c c = ± 1 0 V ,

RB = 3.9kfi, t’, = lOOmV cos 107f, and D2(f) = (1 V)( 1 + 0.6cos 103f).

Does any transistor saturate?

Figure 8 . P3

PROBLEMS 395

? Vcc ? V c c

II o Vo(0

Ro V v V+■

V|(f)

Qi

rll /?

Figure 8 .P -4

.12 For the two-quadrant multiplier shown in Fig. 8.P-4, show thata qRLr

vJt) = 2kT(R + r) provided thata) Qi and Q2 are identical with sufficiently high values of fl so as not to load r,b) neither Q , nor Q2 saturates,c) Ik(t) > 0, andd) |t’ir/(R + r)| < 16 mV.

GivenR = 1 kfi, r = io n , R,

D,(t) = 1 V cos 107t, and ¡k(t) = (2.5 mA)[l

find an expression for t;0(i). Are the assumptions justified?

10 V,

mi {t)],

.13 Figure 8.P-5 illustrates a circuit for unbalancing a differential output. Show that if Q, and Q'i are identical transistors, i = 1,2, 3, with high values of fl, then

* (<2 - h)Rl-

(This is the same voltage that would be obtained differentially between the collectors of Q 5 and Q i if R , were placed in the collectors of both Q 5 and (?4.)

Show how this unbalancing network may be used in conjunction wiui me circuits of Figs. 8.3-10, 8.P-3, and 8.P 4. Write an expression for i) for each case.


8.14 For the circuit shown in Fig. 8.P-6, determine an expression for v0(t) as well as t-0l(r) andvo2(t). Assume that all transistors are identical with [i > 200 and VBE v 0.7 V.

8.15 Determine an expression for vjt) in the circuit shown in Fig. 8.P-7. Assume that ji = 99,VBE st 0.7 V, and that the tuned circuit passes the modulation.

8.16 For the circuit shown in Fig. 8.4-1,

g(i) = (10 V)cos 103f, io0 = 107 rad/sec, R x = 1 kii, RL = 1 kfi,

C = 1000 pF, L = 10 fiH, and gm = 10,000 ¡M.

Determine the minimum value of Vt and the maximum value of R 2 which permit proper circuit operation. With these values write an expression for va(t).

8.17 For the circuit shown in Fig. 8.P-8, calculate v„(t) if IDSS = 4 mA and VP = —4 V.8.18 Derive Eqs. (8.6-1) and (8.6-3).8.19 For the circuit shown in Fig. 8.P-9,

i , ( r )= X Q S it- n T ) ,n - - qo

where T = 2n x 10“ 7 sec and Q = 30rc pC. Make an accurate sketch of v0(t). Indicate regions where failure-to-follow distortion occurs.

PROBLEMS 3 9 7

8.20 For the circuit of Problem 8.19, tit) = (10 V)(l + m cosl0 ;’r). Find the maximum value of in before either phase distortion or failure-to-follow distortion occurs.

8.21 For the circuit of Problem 8.19, v(t) — [10 — 9 nil — r j] V. Assuming that r, occurs on a negative peak of r„{t), write an expression for (or plot) ¡ „(i) for two carrier cycles beyondis there any phase distortion?

Figure 8.P-6


10 Vo

v, “ (100 mV) cos 10 /(0l l

Q3

V )/(0

6 - 1 0 V - ± - 5 V

Figure 8.P-7

PROBLEMS 399

v„(/)

C H A P T E R 9

POWER AMPLIFIERS

In previous chapters we have been concerned with perform ing some operation upon a signal, rather than with the efficient conversion of supply pow er into signal power. In this chapter we shall concentrate upon this power conversion problem. Again we shall begin by considering the basic problem s in a device-free context, and then we shall modify these general results by taking into account the additional lim itations imposed by specific types of devices.

Intuitively we see tha t the efficient conversion of supply pow er into signal power requires that the losses in the conversion device be minimized. If we consider the frequency ranges and circuit adjustm ents where device voltage drops and device currents are in phase, then it follows that device power is minimized if maximum current flows during m inim um voltage drops while m axim um voltage drops are accom panied by m inim um current flows.

In addition, if the signal to be amplified is either absent or small for a substantial part of the time (speech or music, for example), then it is clear that the long-term average device dissipation will be m inimized by minimizing the “ standby” or Q-point power through the device.

Power amplifier ou tput for a given device is norm ally limited by one of two fac to rs:

1) excessive distortion or2) some device and /o r circuit lim itation (other than distortion).

Examples of the device lim itations include: (a) voltage breakdow n, (b) current handling limit, and (c) m axim um allowable device tem perature. Examples of com bined circuit and device lim itations inc lude: (a) operating path lim itations (e.g., violation of the “ safe operating region” or SOA Rf) and (b) therm al runway

t A device manufacturer may specify regions of the characteristics of a certain device in which it is “safe” to operate. These regions may be functions of the type of output circuitry and the shape and frequency of the driving waveshape. Most high-power transistors have such regions as a portion of their specifications. Part of the design procedure for any power amplifier is to ensure that the operating path (see Fig. 9.4-3, for example) lies completely within the “safe” operating region.t Thermal runaway is a phenomenon wherein device self-heating causes more dc current to flow; this leads to more device heating, and the process continues until the device is destroyed. This positive feedback problem is peculiar to transformer- (or choke-) coupled circuits, since with dc resistive loads the increased device current must decrease the device dc voltage and eventually the circuit will reach equilibrium. The problem is controlled by a combination of

400

9.1 “ IDEAL” POWER AMPLIFIERS— CLASS A, SINGLE-ENDED 401

9.1 “ IDEAL” POWER AMPLIFIERS— CLASS A, SINGLE-ENDED

Initially we shall concentrate on the lim itations imposed by distortion caused by the drive waveshape and the circuit configuration. T hat is, we neglect gradual internal device distortion's by assum ing that until the device reaches cutoff o r saturation the output param eter (current o r voltage) is a faithful copy of an input param eter (current or voltage).

We further simplify the situation by assum ing that our devices are “current- source-like” or “ voltage-source-like.” F o r example, with a “current-source-like” device the ou tput current depends only on the input param eters and is independent of the output voltage.

Figure 9.1-1 shows a b roadband version of such an idealized device together with a possible circuit for its use as a power amplifier. If such a circuit is operated in such a way that id always exceeds, zero, i.e., the device never reaches cutoff, it is called a Class A am plifier.!

(a) (b)

Fig. 9.1 -1 Device characteristic and possible Class A power amplifier circuit, (a) /, as a parameter. (b) Idealized circuit.

reduced thermal resistances, increased bias stability, and the use of external temperature compensation elements. For details, see Chapter 3 of Transistor Circuit Analysis by M. V. Joyce and K K. Clarke (Addison-Wesley, Reading, Mass., 1961).t Power amplifier classes are defined in terms of the device conduction angle and/or the type of device operation:

Class A ; "Linear" operation, 360° conduction angle Class B: “Linear” operation, 180° conduction angle Class C: Fixed drive, less than 180" conduction angleClass D : Switched operation, conduction angle may vary with time from0° to 360°

or may be fixedClass AB: "Linear" operation, conduction angle less than 360° but more than 180°

In certain cases a particular circuit may be viewed as belonging simultaneously to several different classes.

4 0 2 POWER AMPLIFIERS 9.1

It is apparen t th a t am ong the variables to be chosen in this circuit are Vcc , n , R L, the Q-point, the waveshapes of i, , and the peak value of the driving waveshape. The choice of these variables indicated in Fig. 9.1-1 is the optim um one assum ing a given Vc c , a given I inm, and the desire to operate in a Class A fashion with the m axim um possible efficiency: i.e. vd = Vmin — Ksa, when id = / max.

If it is symmetrical abou t /¡max/2 and never passes cutoff o r / imax, then the ou tpu t current will be symmetrical abou t I Q, which will be equal to / dc. At the sam e time, since we are assum ing m idband operation where the transform er reactances may be neglected and the upper and lower breakpoints are widely separated, the device voltage will be swinging symmetrically abou t Vcc ■

In some cases, certainly with very low-frequency signals o r with devices that have very short therm al time constants, one wishes to consider instantaneous ra ther than average device power. However, it norm ally suffices to average the-pow er over a cycle of the lowest-frequency signal to be amplified. To locate the circuit o f Fig. 9.1-1 in the pow er amplifier hierarchy, let us begin by calculating its pow er output, pow er input, pow er dissipation, and pow er conversion efficiency q, all as a function of input waveshape.

where P dc is the pow er supplied by the battery and P ac *s the pow er consum ed by the load. (Actually P ac is the pow er available a t the input term inals of the transform er. Some sm all-output pow er transform ers have transfer efficiencies as low as 0.60; hence all of P ac m ay not actually reach the load.)

W hen this circuit has no sinusoidal drive, all the pow er supplied by the battery o r by the pow er supply m ust be dissipated by the device. As the drive increases, so does t], and the device dissipation falls. W ith the m axim um drive allowable before cutoff or saturation occurs and gross d istortion results, one finds

where I t R L = Vcc - Fs>land l y = / dc. Suppose th a t Ksat = 0.05 Vcc ; then f/max = 0.475

'</ = I q + 11 cos cot, IQ + / , < / max,

I q Aic> R l = n2R,

V4 = Vcc + h R L cosœ t, / ,K t < Vcc - Vsal, (9.1-1)

and

device Pic - Pac = Pdcd - ri) = pj~- (9.1-2)

(9.1-3)

P1 device min

(Vcc + V sJhc 2

9.1 “ IDEAL” POWER AMPLIFIERS CLASS A, SINGLE-ENDED 403

while P device falls from VccI dQ with no sinusoidal drive to 0.525VccI dc with full sinusoidal drive and P ac climbs from zero with no drive to 0.475 VccIdQ at full drive.

Figure 9.1-2 indicates the relative values of P dc, P ac, P device, and r] as a function ofi !//dc or VJ(VCC - Fsat). In all cases the normalization is with respect to P dCmsj<.

From Fig. 9.1-2 it is obvious that, if the drive signal varies in am plitude and is occasionally zero, then the device m ust be capable of dissipating / dcFct w atts and the overall long-term efficiency may be 10 % or less.

A/4Fig. 9.1-2 Normalized powers and efficiency for a sinusoidally driven Class A amplifier (Vm = 0.05 Vcc).

Square-W ave Drive— Broadband LoadIf we assume that the device and the ou tput transform er will both handle square waves w ithout d istortion, then, if the peak-to-peak value of the square-wave current is lp, where Ip < 2/dc, it follows that

p4 ’

I 2pR l n = —£—t - 4 /dcFcc

!p \ 2 Vcc2 /h

V

Vn(9.1 —4)

In this case rjmdX = 1 - Fsat/F c c , while P dcvice decreases from Pdc with no drive to dc at m axim um drive.

Figure 9.1-3 indicates the norm alized variation as a function of / p/2 /dc or FP/(FCC — Ksat). If the drive ever drops to half its maxim um value, the efficiency will

/„/2/dcFig. 9.1 -3 Normalized powers and efficiency for a square-wave-driven Class A amplifier (Fsal = 0.05 Vcc).

404 POWER AMPLIFIERS 9.1

drop from 95% to 24% while the device dissipation will rise from 50% of the dc Q-point power to 76 % of this power. Hence, except for the special case of continuous m axim um drive, the device m ust be prepared to dissipate I iQVcc and the average overall efficiency, while higher than in the sine-wave case, may easily be 20 % or less.

Class A Narrowband OperationAs we have seen, a pow er amplifier consists of a controlling device, a load, a basic pow er source, and a driving waveform. We have also seen tha t the results obtained with power amplifiers are quite sensitive to the type of driving waveshape. The results are also a function of the load circuit. In general, there are two useful types of loads. In the b roadband case one wishes the load to look “ resistive” and “constan t” over a frequency range of a t least 10/1 and m ore norm ally 100/1 or even 1000/1 or more. In the narrow band or high-(? case one is often satisfied if the 70% variation points of the load im pedance are within several percent o f the center frequency. Again one would norm ally like the im pedance to be resistive at the center frequency.

O ne basic difference between the two types of loads is tha t in the broadband case the load cannot be counted on to remove signal distortion, while in the narrow band case the filtering properties of the load may lead to ou tput voltages or currents that are quite different from the driving waveshapes.

If we replace the broadband transform er and load com bination by a high-Q parallel R L C circuit, then the circuit of Fig. 9.1-4 will result. In order to see the different effects of driving waveshapes and to com pare the broadband and narrow band cases, let us calculate the results for this circuit with sine- and square-wave output device currents. (Again we assume that id contains a dc com ponent of half its maxim um value, / max, R slne wave = 2(VCC - Vsai)/Imix while tfsquare wavc equals ( Vcc - Ksal)/ 2 /max. We also assume that the driving waveshape has a fundam ental radian frequency of co0 .) The results are shown in Table 9.1-1.

The results for the case of sinusoidal drive are identical to those for the broadband case as sum m arized in Fig. 9.1-2. O n the o ther hand, the case of the square-wave

9 .2 CLASS B LINEAR RF AMPLIFIERS 405

Table 9 .1 -1

Sine wave

‘d = he + cos w0tPac = K c l dc

/, = / ,

I , = / , / < < Vcc - K„

l\R=

2 \id J \

i ’de.ke = Pd c(! - I )

Square wave

‘d - ^dc + j

Pdc = ^-c/dc

11 Ip' Ip maxn

V - - f - < V - Vyi — ' k(( Val71

=2 HR

n =1 i L V i V c c - K,

2 n \ l

device ^dci V)

driving waveshape now has a m axim um possible conversion efficiency of 0.635 (practical efficiency of 0.60 if Ksat = 0.05 Vcc) in com parison to a broadband practical efficiency of 0.95. F o r a case where P dc = 10 W the practical device dissipation is \ W for b roadband m axim um drive and 4 W for narrow band m axim um drive. This eightfold increase in actual dissipation with a m axim um am plitude square-wave drive is the significant difference between the broadband and the narrow band cases. The im portant difference between the sine- and square-wave current pulse drives into a tuned-circuit load is tha t for the same supply voltage and the same maximum allowable device current the square-wave drive case yields Ajn times as much ou tpu t power.

9.2 CLASS B LINEAR RF AMPLIFIERS

W hen the signal to be power-amplified is am plitude-m odulated in one of the forms discussed in C hapter 8, there m ust be a linear relationship between the envelope of the power amplifier ou tpu t and the envelope of the input driving signal.

This desired linearity of the envelope transfer function does not preclude the proper piecewise-linear operation of the device and does not require “ undistorted” current or voltage waveshapes th roughou t the system. W hat is required is that there be a linear relationship between the am plitude of the fundam ental com ponent of the device ou tpu t current pulses and the input envelope. W ith such a relationship and with a narrow band output filter, one can obtain a useful power amplifier for SSB, suppressed carrier o r norm al AM signals.

By using a current pulse tra in instead of keeping the device current always on, one expects to reduce device dissipation and increase conversion efficiency. This increase in efficiency should occur because current flow will be stopped during the


part of the period when the device voltage is high and/or the device voltage will be reduced during the part of the cycle when the current is high. Two possible current pulse shapes are half sine waves and a train of rectangular pulses of width r(t 7 where a>0 = 2n/T).

Figure 9.2-1 indicates two idealized possibilities for a com bination o.r device characteristic and driving waveshape that, when followed by an appropria te narrow band filter, will allow “ linear”- RF power amplification.

The fundam ental com ponent of the half-sine-wave tips is equal to /p/2; hence,

(b)

Fig. 9 .2 -1 Two possibilities for device characteristics and driving waveshapes that will allow “ linear” R F amplification.

if the device characteristic is straight all the way down to the origin, one may “ linearly” amplify a 100% m odulated AM wave in a single-ended Class B amplifier with a sinusoidal driving waveshape. F or any particular value of I p for which

, „ VCC - ^sa, , _ /■Ip S ^ , <Hr —R l

I = -£ 1 2 ’

and

so that

VrrIC C ‘ p I 2P ac = TT*. and = Î Æ *

Vrr 8’and since R L = 2[VCC - Kat]/^max> to achieve optim um efficiency

n =Vr , K a .U

/ I V 14 ’1 max \ V CC I H(9.2-1)

which has a m axim um value of (n/4)(VCC — VSM)/VCC when ¡p = 7max. Figure 9.2-2 indicates the operating path for this case and com pares it with the Class A case (shown dashed).

9 . 2 CLASS B LIN EA R RF A M PLIFIER S 407

It is a p p a ren t from this figure w hy the C la ss B device d issip a tio n should be reduced w ith resp ect to the C la ss A d issip a tio n . F o r the h ig h -v o lta g e h a lf o f r d the

d evice ca rr ie s n o cu rren t a t all. T h is cro w d in g o f the cu rren t in to the lo w -v o ltag e h a lf o f the cy cle is in tu itiv e ly e x a c tly w hat o n e w ants in o rd e r to red u ce the p ow er loss inthe device.

It is a lso a p p a ren t th a t m ax im u m device d issip a tio n will n o lon ger o cc u r a t the m axim u m d rive p o in t b u t a t som e in te rm e d ia te p o in t w here the p ro d u ct o f the d evice cu rren t and v o ltag e d ro p go es th ro u g h a m axim u m . T o find th is m ax im u m d issip a tio n p o in t, o n e d ifferen tia tes P device = P dc — P ac w ith resp ect to / p and sets the resu lt eq u a l to zero :

^ P r n d d - V c c p __ V c c ' l max K '( IQ 1 7 1i 1 / t / 1 d e v i c e ir> iS x 2 • i / i 7 ‘ 1 ** ma x ^ *C C ' s a t ^ * CC *sat

T h u s for eq u al valu es o f Vcc and /max the C la ss B stage h as a m ax im u m d evice d iss ip a tion o f a p p ro x im a te ly I th a t o f a C la ss A stage.

P u t a n o th e r w ay, the ra tio o f the m ax im u m load pow er. P l t . to the m axim u m d evice d issip a tio n is

r o r 0 l i t s s B .

l ; - ' , 1 , 9 . : 3 ,

. = 1 Vci Z for Class A.

Fig. 9.2 -2 Class A and Class B operating paths (sinusoidal input signals).

4 0 8 POWER AMPLIFIERS 9 .3

or, when Vsat = 0.05 Vcc , the ra tio for the sinusoidally driven Class B stage is 2.2 com pared to 0.475 for the Class A stage. Thus to supply a m axim um load pow er of 100 W via a Class A stage requires a device capable o f dissipating 210 W, while a Class B stage requires only a 45 W device.

9.3 CLASS C “LINEAR” AMPLIFIERS

W ith a sinusoidal driving signal, when one attem pts to increase the conversion efficiency beyond n/A by reducing the conduction angle of the current pulses to less than 180°, one loses the linear relationship between the fundam ental com ponent of the pulses and their peak value. The reason is tha t in this case the conduction angle, and hence the ratio of the fundam ental current to the peak current, becomes a function of the amplitude.

O n the other hand, for a rectangular pulse train of any finite w idth the fundam ental am plitude is related to the peak am plitude by

21 . mI , = —-s in —

1 n T(9.3-1)

so that a linearly m odulated pulse AM signal at the input of a zero-biased piecewise- linear stage would yield a linearly m odulated AM signal across its ou tpu t tank circuit.

-- --h

4t»- I —-----►

Fig. 93-1 RF amplifier with pulse drive.

If one considers the case of a fixed-amplitude pulse train into an output-tuned circuit as shown in Fig. 9.3-1, then, for / lmax R L = Vcc ~ âi»

R, =y CC *satK., n 1

max , 2

L

2 sin (nz/T)'

sin — ) —cc ~ V" ^ In T j n

I XP — p — I Vdc j rp max *XC ’

* max *a n d

Vcc - sat sin jnz/T)^ max Vcc n x /T

(9.3-2)

(9.3-3)

(9 .3 -4 )

9.3 CLASS C “ LINEAR” AMPLIFIERS 409

F or a given value of I p/ I ma% the available ou tpu t power increases as a sinusoidal function as x /T increases from zero to j . At the same time, the circuit conversion efficiency decreases as a s in x /x function. Table 9.3-1 indicates the relative Pac, efficiency, and in 1 — (sin x/x) the relative device dissipation for a.pulse-driven Class C amplifier with the conduction angle as a variable. N ote that increasing the conduction angle from 120° to 180° increases the output power by only 15% but increases the device dissipation 1.56 times. F rom these da ta one would norm ally expect to operate with x/T in the range from £ to y (120° > 6 > 60°).

Table 9.3-1

0 TT

. nr sin jr sin (nx/T) nx/T

1 sin xX

180° 12 1.00 0.635 0.265120° 13 0.866 0.83 0.17o©O' 14 0.707 0.90 0.1060° i6 0.500 0.955 0.045

N ote that these data are perfectly valid if the bias point is chosen anywhere to the left o f th e b re a k p o in t in F ig . 9 .2 - l ( b ) . T h e o n ly tim e th a t b re a k p o in t b ias o p e ra tio n is necessary is when “ linear” envelope am plification is required.

As in the Class B case of sine-wave drive with a constant supply voltage, when the device operates in an on-off fashion the m axim um device dissipation does not occur at the maximum drive but a t some interm ediate level. Again this m axim um level isfound by writing = P.dc Pac, differentiating with respect to / , and setting theresult equal to zero. W hen this is done for the case of rectangular pulses the point of

, is found asmaximum device dissipation, /

Vr,

L1

2 Vcc - K*

sin (n x /T )n x /T

At this drive level the actual efficiency is

1n =

sin (nx/T) n x /T

(9.3-5)

(9 .3 -6 )

which for 8 = 90° and Fsal = VCC/2Q is down from 0.855 at m axim um drive to only0.405 of the m axim um device dissipation point of / Pmdd/ / max = 0.475.

Figure 9.3-2 indicates the operating paths for no m odulation and for 100% positive peak m odulation for a 90° rectangular pulse drive Class C “ linear” amplifier.

It will be dem onstrated in Section 9.9 that the overall efficiency can be increased if the ou tput voltage is varied in step with the input. The general results of that section can also be applied to the pulse-driven linear Class C amplifiers of this section.

F or ordinary power am plification of unm odulated signals, Class C is preferred over Class B. At higher frequencies the pulse-like drive signals are difficult to produce


No modulation

Fig. 9.3-2 Operating paths for 90° “ linear” Class C amplifier.

and one tends to use sinusoidally driven Class C amplifiers. Such amplifiers are discussed in Section 9.4.

W ith a triode vacuum tube the ou tput current pulse is a function of the output voltage; hence direct ou tput m odulation of such stages is possible. Furtherm ore, they can be adjusted for reasonably linear operation up to nearly 100% m odulation. Therefore, vacuum tube amplifiers are m odulated in the final power amplifier whenever possible. N orm ally single-sideband and suppressed carrier operations are performed at low power levels. These signals do require linear power amplifiers for all subsequent stages.

T ransistors would seem to lend themselves quite readily to the m odulated Class B amplifier scheme of Section 9.9. Since the best-m odulated Class C amplifier seldom has an overall efficiency of m ore than 80%, one is really sacrificing almost nothing in this case.

O f course, FM , PM , binary AM, and the various industrial uses o f CW are com pletely indifferent to the “ linearity” of the power amplifier employed.

9.4 RF CLASS C AMPLIFIERS

The norm al RF power amplifier does not have a piecewise-linear device characteristic, nor is it driven by a “square” pulse. It is norm ally operated into a tuned-circuit load, so that no m atter what the shape of the input current pulse one can begin by assuming that the output voltage is sinusoidal. If one knows the actual input driving waveshape, if one knows that the ou tput voltage is sinusoidal, and if one has the proper device characteristic, then one can always calculate the ou tput current waveshape.

In some cases this current pulse will have a known analytic form, and the dc and fundam ental term s may be w ritten by inspection. Even if the analytic form is not known, one can norm ally approxim ate any symmetrical pulse by one of several forms such as rectangular, trapezoidal, triangular, cosine, cosine squared, or sine-wave tip. (If the output circuit is tuned to be resistive, then a symmetrical input pulse will yi^ld a symmetrical ou tput pulse.)

9.4 RF CLASS C AMPLIFIERS 411

T o aid in the a n a ly sis o f th e cu rren t pu lses, th e ap p en d ix to th is c h a p te r ta b u la te s

the e q u a tio n s fo r v ario u s pulse fo rm s and p resen ts n o rm alized cu rv es fo r /dc, l { , and I2 vs. the pulse w idth o r c o n d u ctio n angle. A s these cu rv es in d ica te , th e v a ria tio n s am o n g the d ifferen tia l w av esh ap es are n o t la rg e ; co n seq u en tly an e x a c t fit is not

n ecessary in o rd er to o b ta in a re a so n a b le e stim a te o f the fu n d am en tal and d c values.

Fig. 9 .4 1 Triode Class C amplifier

The circuit of Fig. 9.4-1 provides an example of the use of these curves as well as i f the determ ination of the current waveshape for an idealized triode vacuum tube m which the plate current is a function of both vPk and v0K. in order to simplify our I-.xatuple slightly, lei us assume that the device characteristics are as shown in Fig. l».4-2. (As shown in Fig. 9.4-2, this tube has a constant /; = 10 and a constant ; p — 1.8 kQ In a real tube the curves would be neither straigh! nor equally spaced.) The bias has been chosen so that with a sine-wave drive of 5.4 kV peak and with a Knm + 2 kV for the plate-cathode voltage, the conduction angle will be 120°.+

For com parison purposes let us assume three different driving waveshapes, in each case the bias and supply voltages are constant but R , is changed, so that we always drive up to vGk — vPK = 2 kV in the upper left-hand corner:1. Sinusoidal: 5 .4 k V ,0 p = 120°2. P u ls e : 5 .4 k V ,0 p = 120°3. C o m p o s ite : ( 5 .4 k V ) ( l . l c o s u>t — 0.1 c o s 3 a » ) , 6 p ~ 140°

T h e o p e ra tin g p a th s fo r the th ree ca ses a re show n in F ig . 9 .4 -3 . T h e se p a th s are co n stru cted by d raw in g the d riv in g v o lta g e a lo n g the v ertica l a x is cen tered at - 3 .4 k V ,

+ For idealized tubes the relation among and the conduction angle 0p may be writtenas

VccKb IVPKmin + COS (9p/2)'A M /1 - cos (0J2)

(9.4-1)

hence for VBB = 12 kV, n = 10, vPK = 2 kV, vGK = 2kV, and 9p = 120°, it follows that Vcc = - 3400 V.

4 12 POWER AMPLIFIERS 9 .4

Fig. 9.4-2 Mythical triode characteristics (¿1 = 10, rp = 1800 fi).

drawing a 10 kV sine wave along the horizontal axis centered around 12kV, and plotting the resultant Lissajous pattern. All the circled points in any vertical line represent the same angular distance m easured from the peak of the current pulse (or from the valley of the ou tpu t tank circuit voltage). Note, as in previous cases, the extreme dependence of the operating path on the driving waveshape.

Once the operating path is sketched, the current pulse is draw n alm ost by inspection, since each circle indicates the plate or grid current for its particular angular displacement. The respective current pulses are shown in Fig. 9.4—4.

In order to com pare the three cases let us approxim ate the various current pulses as follows.

1. Sinusoidal d r iv e :/'„—cosine pulse of 120° width, / , = 3.8 A, /dc = 2.1 A;iG— triangular pulse of 102° width, I G1 = 0.85 A, I G0 = 0.45 A.

9.4 RF CLASS C AMPLIFIERS 413

2. Pulse drive:iB— square pulse of 120° width m inus a triangular pulse of 120° w idth and 0.2 A height, / , = 5.5 - 0.6 = 4.9 A, /* = 3.33 - 0.33 = 2.0 A; zG—square pulse of 120° w idth and 2.2 A height plus a triangular pulse of 120° width and 1 A height, / G1 = 1.2 + 0.30 = 1.5 A, I G0 = 0.733 + 0.167 = 0.90.

3. C om posite d riv e :iB— symmetrical trapezoid of 140° width, / 1 = 5 A, / dc = 2.0 A;iG—cosine p.ulse of 120° width, I Gl = 1.14 A, l GJ = 0.42 A, I G0 = 0.63 A.

Fig. 9.4-3 Operating paths for three different driving waveshapes and load resistors (Vcc = 3400 V, V „ = 1 2 lcV).

4 1 4 P O W E R A M PL IFIE R S 9.4

12.5

60° 0 60' (a)

60° 0 60‘ (b)

60" 0 60(c)

Fig. 9.4-4 Plate and grid current pulses for the three operating paths of Fig. 9.4-3. (a) Sinusoidal drive, (b) “Pulse” drive, (c) Composite drive.

From these current waveshapes and their harm onic analyses in addition to the assum ption of a known driving waveshape and a known output waveshape (10 kV peak sinusoid), we can derive values of i„ and iG for the necessary value of R, as well as the various powers listed in Tables 9.4-1 and 9.4-2.

The grid circuit calculations are explained in Section 9.8.W hich driving waveshape is allowable will depend on the device limitations.

For example, if the peak allowable cu rren tf is 10 A, then the pulse drive would not be satisfactory although the o ther two would be. (Actually the peak cathode currents do not differ too drastically, since for the pulse case the valley in iB corresponds to the peak in iG.) If the peak plate dissipation is 7 kW or the peak grid dissipation is 900 W, then only the sinusoidal case would be allowed. The com posite drive does yield the same output power as the “ pulse” drive, but does it at a higher efficiency, with 20% less peak current, with 76% of the grid dissipation, and with a waveshape that is much easier to generate than is the pulse w'aveshape. The last column of Table 9.4-2

f The maximum cathode current should be supplied by the manufacturer of the device. If it is not, then the most reasonable procedure is to find the peak allowable cathode current in milliamperes per watt of filament power for another tube having the same type of filament material and roughly the same power rating. This "constant” can then be used to determine for the tube in question.

For pure tungsten filaments run at their rated power input, this value is usually about 5 mA/watt. As the filament power decreases, the emission per watt falls but the life increases. Thus reducing the filament power to 90% of its rated value (normally this means reducing the filament voltage to about 95 % of its rated value) will approximately double the life of the filament while reducing the maximum emission per watt to perhaps 75% of its original value.' The lives of thoriated tungsten filaments cannot be extended by low-power operation. Normally the filament voltage for these tubes should be regulated to within 5 % of the nominal value. Such filaments are much more efficient emitters than pure tungsten. However, they should not be operated within a factor of 2 or 3 of their temperature saturation currents; hence one has a nominal mA/watt figure in the region of 15 to 17 for this type of filament.

Oxide-coated cathodes are generally too delicate to withstand the ionic bombardments of high-voltage operation. With small tubes that have oxide-coated cathodes one can assume that peak cathode currents of 40 mA/watt will not cause damage.

9 .5 N A R R O W B A N D CLASS D PO W E R A M PLIFIER S 415

indicates the power gain of the stage. From this and from the driver power figure it is apparent that a pow er amplifier will be needed to drive this stage.

Table 9.4-1 Plate circuit values

Drivingwaveshape Ri, n Pdt. kW Pplat.: - kW------ ---- :----- ---------- --------- -------

Sinusoidal 2620 19 25.2 6.2Pulse 2040 24.5 36 11.5Composite 2000 25 35.5 10.5

Table 9.4-2 Grid circuit values

Drivingwaveshape P f r o n , d r i v e r - W P W‘ lo b a t t e r y o r b i a s c i r c u i t - ' T w J’.c/Pdhv.r

Sinusoidal 2300 1530 770 8.3Pulse 4850 3050 1800 5.05Composite 3400 2253 1147 7.1

It is apparent that one could generate a series of curves of Pac, rj, and Pdcvice vs. conduction angle for different assumed device current waveshapes. In general, the results will all be similar to those of Table 9.3-1 for the rectangular pulse. T hat is, the power output will rise as the conduction angle increases, but the device dissipation will rise even more rapidly; hence norm ally the conduction angle must be held below 180°.

In order to increase the efficiency while m aintaining the power output, one wants to operate the device completely as a switch so that the device voltage drop remains at or near the m inimum value over a half-cycle of current flow. The next section examines such switched-device or Class D circuits.

9.5 NARROWBAND CLASS D POWER AMPLIFIERS

The two basic forms of the R F Class D power amplifier are shown in Fig. 9.5-1. In both cases the switch is driven back and forth at the resonant frequency of the tuned circuit. Initially the time when neither contact is made is assumed to be zero.

V oltage-Sw itch ing C ase— Ideal (100% E fficiency)

Since the switch spends half its time in each position, the voltage r,-(f) is a square wave of am plitude 2VCC. This square wave can be expanded by means of Eq. (4.2-4) to obtain a dc value of Vcc and a fundam ental value of V1 = AVcc/n. Since the circuit is resonant at the fundam ental frequency of the square wave and is assumed to have Qr > 5, the current is essentially a sinusoid of peak am plitude / , = VJR. Since this current must also flow through the switches, it follows that the current through each

4 1 6 POWER AMPLIFIERS 9 .5

/Sw itch

(b)

Fig. 9.5-1 (a) Voltage-switching Class D circuit, (b) Constant-current-switching Class D circuit.

switch is a half sine wave. However, once the current through the battery is known, then P dc can be calculated and the analysis problem is solved.

V oltage-Sw itch ing C ase— P ractica l

The efficiency of this circuit departs from 100% because of losses in the switches and in the inductor an d /o r capacitor. Since the device is assumed always to be saturated when on, the closed switches may be modeled as shown in Fig. 9.5-2.

-V W -Vo

• ° jh

Fig. 9.5-2 Device model in saturated region.

N ow if rL is the com bined effective series loss element of the inductor and the capacitor and if R z = rs + rL + R, then the effective peak-to-peak input voltage square wave is reduced in m agnitude to 2( Vcc — V0), from which

4(KCc - V0)Vl = and

' - 7 -nObviously the devices m ust not break dow n at a voltage of 2VCC; on the o ther hand, if they are to be used to the limits of their current, then the peak current / , will be set at /max (certainly 1 x < / max).

W ith these assum ptions,

P d c = 2 V C C I d c , P a c ~

P„ c Vcc ~ V0

V J i 4 (VCC - K0) I iQn

(9 .5- 1)

device dc VCC

9 .5 NARROWBAND CLASS D POWER AMPLIFIERS 4 1 7

and

R Vcc - V 0Coverall “ D T/ ( *5Vcc

I f / , = /max’ then4(KCC - |/0)

R Imax

=

7t/„

_ 2(Fcc - K0) R^dc ^ •* ac load _ 'max rjKy

(9.5-3)

and

K „ / / 2 rP _ r 0* max , max* 5 ,q r a v^perdevice ' ,« * Lr)

For example, if / max = 2 A, Kcc = 20 V, F0 = 0.4 V, rs = 0.25 Q, rL = 0.125Q, and = 100, then -R£max = 12.12 i2 and the loaded Q of the series circuit is approxim ately

10. Thus

and

P dc = 25.4 W, P acload = 24 W, n = 0.95.

= 0.255 + 0.25 = 0.55 W..

Hence the ratio of ac load pow er to to ta l device dissipation (two devices) is nearly 22/1 ; this value should be com pared to the ratio o f 2.2/1 for the Class B sinusoidally driven stage.

Figure 9.5-3 illustrates two transisto r arrangem ents that would be possible as switches.

As we shall see in a subsequent section, it is a straightforw ard m atter to build a transform erless driver for the com plem entary symmetry version of the switch. The driving circuit m ust supply enough turn-on current to keep the on transistor saturated during its conducting half-cycle, and then enough turn-off current to get it out of saturation rapidly. If the base current is sinusoidal and /i is constant as the current varies, then I Bl > /[//?, so tha t if / 1 = 2 A and [3 % 10, then I Bi > 200 mA. The turn-off problem is im portan t because if one unit stays saturated while the second unit comes on, the result is alm ost a dead short from 2 l/cc to ground. Unless the pow er supply has a current-lim iting feature, the transistors may be destroyed by these repeated surges.

If a square-wave drive is employed, then a “ speed-up” capacitor m ight be added in parallel with the base resistors to increase the turn-off and turn-on times.

As the carrier frequency increases, the assum ption of negligible transition times for the switch will no longer be valid. Since a heavy current flows during this interval while the switch is going off, the rise tim e does no t have to become appreciable before the transition-region losses are equal to the saturation-region term s and the very

418 P O W E R A M PLIFIER S 9.5

O 2 Vcc

- W v -

- A W

• • -

(a) (b)

Fig. 9.5-3 Two possible voltage-switching circuits.

low ca lcu la te d v alu es fo r P device are d o u b led o r trip led . In e stim a tin g th e tra n s itio n losses, we c a n a ssu m e lin ea r rise an d fall tim es, e a ch o f d u ra tio n t,. T h e n the to ta l tra n s itio n -re g io n p ow er lo ss w ill be a p p ro x im a te ly (f,/T)Pdc. T h u s i f th e d evice p ow er fo r o u r p rev io u s exam p le is n o t to b e m o re th a n d o u b led , we need t J T < 0 .02 . T h is m ean s th a t fo r o p e ra tio n a t 5 0 0 k H z , t, < 4 0 nsec, an d fo r o p e ra tio n a t 1 M H z , t, < 2 0 nsec.

S in ce th ese n u m b ers rep resen t a p p ro x im a te ly th e lim its o f p resen t-d a y pow er tra n s is to rs , o n e c a n h o p e to a ch iev e ra tio s o f lo ad p ow er to to ta l d ev ice lo ss o f a b o u t 10/1 o n ly up to ce n te r freq u en cies o f a b o u t 1 M H z . F o r lo w er freq u en cies the gains o v er the C la ss B o r C la ss C stag e ca n be im pressive.

C u rren t-S w itch in g C a se — Id eal

F o r th e c ircu it o f F ig . 9 .5 —1 (b) the c u rren t th ro u g h ea ch o f th e tw o sw itch es has th e fo rm o f a sq u a re w ave o f am p litu d e 2 Ip . T h e se sq u a re w aves m ay b e exp an d ed to p ro v id e a d c v alu e o f Ip th ro u g h e a ch sw itch and a fu n d am en ta l co m p o n e n t 4 I p/n w hich , b e ca u se o f th e p h ase rev ersa l b etw een the tw o sw itch cu rren ts , m ay b e view ed as flow in g o u t o f th e b o tto m o f th e ta n k c irc u it an d in to th e top . T h a t is, fro m th e v iew p oin t o f cu rren ts , th e c ircu it o f F ig . 9 .5 —1(b) m ay b e rep la ced by th a t o f F ig .9 .5 -4 .

S in ce th e tu n ed c ircu it is p resu m ed to h av e a re a so n a b le Q, we n eg lect the h a r m o n ic g e n e r a to r s ; an d s in ce the sw itch is p resum ed to sp end eq u a l tim es o n ea ch

9 .5 N A R R O W B A N D CLA SS D P O W E R A M PL IFIE R S 419

- k

Fig. 9 .5 -4 Equivalent circuit (from the current viewpoint) for Fig. 9.5-1 (b).

4 LR

LYJX. Switch open

, Switch closed

21,

Fig. 9 .5-5 Device current and voltage in the current-switching circuit.

half, we c o m b in e th e fu n d a m en ta l g e n era to rs . N o w K, = 4 I pR/n and th e ta n k c ircu it v o ltag e is s in u so id a l. T h e v o lta g e a c ro ss e a ch d ev ice is n o t s in u so id a l b u t ta k e s th e

form of a train of half-sine-wave pulses as shown in Fig. 9.5-5.

C u rren t-S w itch in g C a se — P ra c t ic a l

In a rea l c ircu it th e c o n s ta n t cu rre n t supp ly m u st b e a b le to a d ju st itse lf to th e needs o f th e c ir c u it ; h e n ce it w ould b e m e ch a n iz ed by an in d u cta n ce , L c c , th a t is a t least five tim es as g reat as L in series w ith a supply v o ltag e Vc c . T h e n , s in ce P dc = 2 Ip Vcc w hile th e a c p ow er rem ov ed is 8/j;K/7i2, it fo llo w s th a t, fo r th e id ea l ca se w here th ere is n o d ev ice loss, P dc = P ac an d

2 h = <9 -5- 5)

I f w e w ish to set 21 p a t /max fo r e a ch sw itch , th en we m u st set R a t a p a rticu la r value. O n c e R is set, a ll th e p ow ers an d effic ien cies c a n b e ca lcu la te d im m ed iately .

In th is c a se th e cu rren t d u rin g c o n d u ctio n is c o n s ta n t ; h en ce fo r a n y p a rticu la r /max o n e c a n sim p lify th e m o d el o f F ig . 9 .5 - 2 b a c k to a single b a tte ry o f valu e A g ain th e L an d C lo sses co u ld b e rep resen ted as a n eq u iv a len t lo ss R t ; thu s the ta n k c irc u it e ffic ien cy is /?,/(/?, + R) an d th e p o w er in to th e ta n k is

m-


w here

Ru — R R ,/(R + R ,) = t]tankR ,

= 2 I pVcc, _ ,^p\2^nk^ (9 5_6)p v CC> ^ a c load ' tc I 2 *

f each device — I p K at» (9 .5 -7 )

and R is ch o se n to lim it the valu e o f 21 p to a valu e th a t the d ev ice ca n h a n d le :

K, > n- (V^ r V l (9 .5 -8 )max

R a is in g a b o v e th is v alu e will red u ce th e p o w er a v a ila b le to the load .In o rd er to c o m p a re the v o lta g e- an d cu rren t-sw itch in g ca ses we m igh t ca lc u la te

n u m erica l valu es fo r a set o i p a ra m e ters s im ila r to th o se used in th e v o lta g e-sw itch in g case . S u ch valu es a re show n in T a b le 9 .5 -1 .

Table 9.5 I

Voltage switch

vccVo

= 20 V, /max = 2 A 0.4 V, rs = 0.25 O

rL = 0.125 k i) , Ql * 100

Loaded Q

perdevice

C urrent switch

Vcc = 2 0 V ,/mai = 2AKsat = 0.9 V

R, = 4.9 k n , Q, s 100

10 Loaded Q » 1012.1 Q R = 48.5 i225.4 W P dc = 40 W24 W P,oad = 37.8 W0.55 W p _ o o wperdevice y ”

0.95 '/overall * 0.95

T h e cu rren t sw itch is n o t su b je c t to th e p ro b lem o f excess c u rren t w hen b o th sw itches are c lo sed s im u ltan eo u sly , but it d o es suffer from excess d ev ice v o lta g e if b o th sw itches try to o p en a t o n ce . I t a lso suffers fro m the p ra c tica l d ifficu lty th a t th e tu rn -o ff sw itch ing s ta rts from th e full on cu rren t ra th e r th a n fro m the th e o re tica l valu e o f zero d ev ice cu rren t th a t sh o u ld be p resen t in the v o ltag e sw itch a t th e in sta n t o f sw itching.

I f th e peak cu rren t o f the d ev ices is lim ited , then the sq u a re cu rren t pulse o f the cu rren t sw itch w ill p ro v id e m o re lo ad p ow er th a n th e h a lf sine p u lse o f th e v o lta g e- sw itch ing case. O n th e o th e r h an d , i f the c irc u it is dev ice d iss ip a tio n lim ited , th en in th eo ry th e c irc u its can yield e x a c tly eq u a l o u tp u ts. In p ra c tice , th e a c tu a l effic ien cy o f the cu rren t-sw itch in g c ircu it ten d s to b e low er th a n th e effic ien cy o f th e v o lta g e- sw itch in g c ir c u i t ; c o n se q u e n tly , i f th e d ev ice d iss ip a tio n is lim ited , the v o lta g e sw itch is th e b e tter ch o ice .

9.6 B R O A D B A N D CLASS B A M P L IF IE R S 421

O th e r fa c to rs to co n sid er w hen m a k in g a c h o ic e betw een th e c ircu its hav e to do w ith th e p o te n tia l d ev ice fa ilu re m e ch a n ism s an d w hether excess cu rren t o r excess

v o ltag e seem s m o re d an g ero u s. M o re d eta ils on b rea k d o w n and losses in th is typ e o f c irc u it ca n b e foun d in a p ap er by C h u d o b ia k an d P a g e .f

9.6 BROADBAND CLASS B A M PLIFIER S

T o sup p ly th e b a seb a n d p ow er n ecessa ry to d rive a m o d u la ted C la ss B o r C la ss C stage o r to d rive th e lo u d sp ea k ers o f a p u b lic ad d ress system o r to d rive th e lo u d sp ea k er o f a ra d io o r te lev isio n set, o n e w ould lik e a b ro a d b a n d , h ig h -g a in , d is to r t io n less am p lifie r b u ilt o f the least exp en siv e p o ssib le devices.

In S e ctio n 9.1 we in v estigated C la ss A am p lifiers and found th a t, th o u gh th eir d is to rtio n co u ld b e low , th e ir o v era ll av erag e e ffic ien cy w as very low and , in g en era l, th e ir requ ired d ev ice d issip a tio n w as m o re th an tw ice th e ir p eak a c load pow er.

In S e ctio n 9 .2 we co m b in e d a d ev ice h av in g half-sin e-w av e o u tp u t pulses o f cu rren t w ith a filter to o b ta in a “ l in e a r” am p lifie r fo r the “ e n v e lo p e ” o f n a rro w b a n d signals. W ith b ro a d b a n d s ig n als w e c a n n o t use a f i l t e r ; how ever, we can c o m b in e tw o d evices so th a t they a lte rn a te ly fo rce h a lf-sin e-w av e pulses th ro u g h the lo ad in o p p o site d irec tio n s. S u ch a sch em e w ill give us th e ad v a n ta g es o f C la ss B o v er w ide- freq u en cy ranges. F ig u re 9 .6 -1 illu stra te s o n e o f the b a s ic form s o f a C la ss B b ro a d b an d am p lifier.

I f th e cu rren t w av esh ap es o f g e n era to rs 1 and 2 a re as show n in F ig . 9 .6 -2 , w here

-f

‘d.

R

Fig. 9 .6-1 Class B amplifier configuration.

is the m ax im u m valu e fo r Ipl and Ip2, then fo r th is m ax im u m drive

P( Vcc K a tK m a x (9 .6 1)ac 2

t W. J. Chudobiak and D. F. Page, “Frequency and Power Limitations of Class D Transistor Amplifiers,” IEEE Journal of Solid State Circuits, SC-4, pp. 25-37 (Feb. 1969).

422 P O W E R A M PL IFIE R S 9.6

A A A.\ r A a j

Fig. 9 .6-2 C urrent waveshapes for Class B amplifier with sinusoidal drive.

andn V rr ~ V„

( 9 .6 -2 )

T h e m a x im u m d rive e ffic ien cy in th is c a se is 5 0 % h igh er th a n it w as in th e c a se o f C la ss A sin u so id a l drive. A s w as tru e fo r n a rro w b a n d o p e ra tio n , su ch a c o m p a riso n is n o t su fficien t to give a g o o d in d ic a tio n o f the re la tiv e d ev ice d iss ip a tio n p o ssib ilities in th e tw o cases. A s b e fo re , th e p e a k d ev ice d issip a tio n d o es n o t o cc u r a t p ea k d rive bu t, fo r a s in u so id a l s ig n a l, o ccu rs w hen

rcc ^CC^max

* Vcc - K .per devicemax 2 n 2

(9 .6 -3 )

A g ain w e find th a t fo r C la ss B th e d ev ice (th e re a re tw o d ev ices n o w ) m u st d issip ate o n ly a b o u t y a s m u ch p o w er in th e w o rst ca se as d o es a C la ss A stag e th a t p ro v id es th e sam e useful p e a k p o w er to th e lo ad . W ith v a ria b le s ig n a ls su ch as sp eech o r m u sic , th e ra tio o f n ecessa ry d ev ice d iss ip a tio n is m o re n early 20/1 b etw een C la ss B an d C la ss A stages.

B eca u se o f th is e n o rm o u s d ifferen tia l in d ev ice d issip a tio n , a lm o st n o C la ss A stages a re used a s p ow er am p lifiers. E v e n in sm all tra n s is to r ra d io s w here the p eak o u tp u t p ow er m a y b e o n ly o f th e o rd e r o f 5 0 m W , th e low a v era g e d iss ip a tio n an d low av erag e p o w er supp ly cu rre n t d ic ta te th e use o f C la ss B stages.

O n e b a s ic d ifficu lty w ith th e cu rre n t-so u rce C la ss B s ta g es is so u rce m atch in g . F o r exam p le, we m ay lo o k o n th e lo a d cu rre n t a s b e in g m ad e up o f tw o tra in s o f h a lf sin e w aves a s sh o w n in F ig . 9 . 3 - 2 E a c h o f th ese tra in s h a s a d c an d an in fin ity o f even h a rm o n ic v alu es, a s w ell as a fu n d a m en ta l valu e o f I J 2. I f /pi = I p2, th en th e d c an d a ll th e h a rm o n ic s e x c e p t th e fu n d a m e n ta l c a n ce l e a c h o th e r an d w e o b ta in a p u re s in e-w av e o u tp u t. I f /pl / 1p2, th e n th is c a n ce lla tio n fa ils an d d is to rtio n results.

F r o m E q . (4 .2 -1 ) th e resu lta n t seco n d - an d fo u rth -h a rm o n ic d is to rtio n s a re seen to be-

seco n d h a rm o n ic

fu n d a m en ta l4 /p. ~ K a

3 * /pi + I p i '

fo u rth h a rm o n ic

fu n d a m en ta l^ lpl (9.6-4)

15ti Ipl + Ip2

9.6 B R O A D B A N D CLASS B A M PLIFIER S 423

T h u s if the cu rren t so u rces a re tra n s is to rs w ith a 2/1 m ism a tch o f /Ts (n o t a t all

u n likely w ith u n selected un its), o n e w ill h av e 1 4 % seco n d -h a rm o n ic d is to rtio n and 2 .9 % fo u rth -h a rm o n ic d is to rtio n fro m th is cau se a lo n e . T o k eep the se co n d -h a rm o n ic d is to rtio n term fro m m ism a tch to b elow 1 % , the so u rces sh ou ld b e m a tch ed to

w ith in 2 .5 % .M a n y sm a ller (less th a n 1 W ) tra n s is to rs fro m th e sam e b a tch (certa in ly fro m the

sam e ch ip ) w ill m eet th is m a tch in g c r ite r io n w ith o u t undue selectio n .F ig u re 9 .6 - 3 show s fo u r p o ssib le b ro a d b a n d C la ss B stages. T h e tw o em itter-

fo llo w er stages a v o id the ^ -m a tch in g p ro b le m a t the exp en se o f p ro v id in g o n ly u n ity v o lta g e gain. (W e assu m e th a t the s tage d riv in g th e stages show n h as a n o u tp u t im p ed an ce th a t is sm all in c o m p a riso n to P R L. A n o th e r em itte r fo llo w er ca n usually p ro v id e such a d riven s tage.) I f re a so n a b ly sy m m e trica l P N P and N P N tra n s is to rs ca n b e fo u n d ,f th en , a t th e exp en se o f tw o sep a ra te p ow er sup p lies, the c ircu it o f F ig . 9 .6 -3 (a ) av o id s a ll tra n sfo rm ers an d co u p lin g ca p a cito rs .

Figure 9 .6 -3

t The problem here is that high-power N P N transistors tend to be silicon, while high-power P N P units tend to be germanium. Even with different turn-on biases it is normally difficult to make such units track in ou tpu t current over a wide range. The emitter-follower configuration eases these difficulties by minimizing the effect of the transistor.

424 P O W E R A M PL IFIE R S

Figure 9.6-3

9 .6 B R O A D B A N D CLASS B A M PLIFIER S 425

T----

V #

<d)

Fig. 9 .6 3 (a) C om plem entary symmetry Class B em itter follower, (b) Transform er-coupled Class B em itter follower, (c) Transform erless current-driven Class B stage, (d) “Classic” Class B current-driven stage. Except for the biasing arrangements and impedance levels, vacuum tubes could be substituted directly in this circuit.

T h e o th e r th re e c irc u its o f F ig . 9 .6 - 3 suffer from p o o r b ias s ta b ility u n less som e em itte r res is ta n ce is inclu d ed . [In F ig . 9 .6 -3 (b ) the w ind ing re s is ta n ce o f the tra n s fo rm er m ay serve th is p u rp o se .] T h e p ro b le m is th a t, unless R E is sm all w ith resp ect to th e lo ad (o r th e reflected load ), it b o th co n su m es pow er an d red u ces the a v a ila b le o u tp u t v o ltag e sw ings. W ith a 5 A peak cu rren t, a 0 .2 i ) re s is to r co n su m es 1 V o f the a v a ila b le sw ing as w ell as 1.25 W o f the o u tp u t pow er.

A ll four c irc u its in d ica te a “ tu r n -o n ” b ias fo r th e tra n sisto rs . T h is b ias is n ecessary to av oid a d ead zo n e o r c ro ss -o v e r z o n e w here w ith sm all inpu t s ig n als b o th u n its w ould be o ff o r so n early o ff th a t the gain w ould be red uced . T h e trick h ere is to ju s t m in im ize th is d is to rtio n w ith o u t ca u sin g excessiv e s tan d b y p ow er d issip atio n .

T h e c ircu it o f F ig . 9 .6 -3 (c ) a v o id s an o u tp u t tra n sfo rm er by using a large co u p lin g ca p a c ito r . (T h e lo w er 3 d B p o in t is a t l / R C ; h en ce i f R = 4 Q an d if /low — 20 H z is d esired , then C = 2 0 0 0 p F .) T h e c a p a c ito r ch a rg es up to Vc c (we assu m e m atch ed tra n sisto rs), an d th e c irc u it is s im ila r to th e b a s ic m o d el o f F ig . 9 .6 -1 . E v en th o u gh the to p tra n s is to r h as its em itte r co n n e cte d to th e lo ad , it is the c o lle c to r cu rren t th a t flow s th ro u g h th e lo a d ; h en ce /^-matching is v ita l for th is c ircu it. (T h e tra n s isto r o p e ra te s as a cu rren t so u rce , n o t as a n em itte r fo llow er.)

T h e c ircu it o f F ig . 9 .6 -3 (d ) show s a c la ssic tra n sfo rm er-co u p le d C la ss B stage. Its lo w -freq u en cy resp o n se is go v ern ed by the tra n sfo rm er-lo a d c o m b in a tio n , w hile its h ig h -freq u en cy lim its m ay be set by tra n s is to r p h ase-sh ift m ism a tch o r by tra n s fo rm er reactan ces .

426 P O W E R A M P L IF IE R S 9.7

9.7 BROADBAND CLASS D PO W ER A M PLIFIER S

T h o u g h m o st p resen t-d a y p ow er am p lifie rs a re C la ss B d o u b le-en d ed un its, it is tem p tin g to e x a m in e the p o ssib ilitie s o f ex ten d in g th e sw itch in g m o d e o f o p e ra tio n to th e b ro a d b a n d case .

In th e n a rro w b a n d C la s s D ca se , th e o u tp u t c irc u it w as tu n ed to th e fu n d am en ta l o f th e sw itch in g freq u en cy . F o r b ro a d b a n d o p e ra tio n we m u st sw itch a t a freq u en cy m u ch h igh er th a n th e to p o f o u r d esired freq u en cy band . T h e sw itch in g m u st b e d o n e in such a fash io n th a t th e lo w -freq u en cy av erag e valu e is th e d esired o u tp u t signal. A n o u tp u t w id eb an d b a n d p a ss filter w ill th en rem o v e th e u n w an ted h ig h -freq u en cy ca rr ie r term s (and th e ir s id eb an d s) and th e d c co m p o n e n t.

Fig. 9.7 -1 G eneration o f “ naturally sam pled” pulse-width m odulation (trailing edge m odulation). Time origin a t zero crossing o f the saw tooth’s downgoing ram p.

F o rtu n a te ly , a sw itch in g s ig n al o f th e p ro p er typ e d o es ex ist. It is kn o w n as a“ n a tu ra lly sa m p le d ” p u lse -w id th -m o d u la ted signal. A s d ev elo p ed in F ig . 4 - 1 2(p. 2 6 0 ) and E q . 2 2 0 (p. 2 8 4 ) o f R o w e ,f th e s ig n al illu stra ted in F ig . 9 .7 1(d) h as thefo llo w in g freq u en cy c o m p o n e n ts :

V 21 21¿Wpwm = ! p + E s in " o f ------~ sin w 0[t + t s ( f )]. (9 .7 -1 )

’ % n 71

It a lso h as h ig h er h a rm o n ics o f a>0 and s id eb an d s a ro u n d th em w here

2 k i ™ ^ V^ T(oo = y , |s(t)| < 1, and r = -y ^ -.

T h u s i f s(r) is b a n d -lim ite d an d i f u>0 is su fficien tly larg er th a n th e h igh est freq u en cy in

+ H arrison E. Rowe, Signals and Noise in Communications Systems. Van N ostrand, New York (1965).

9.7 B R O A D B A N D CLA SS D P O W E R A M P L IF IE R S 427

s(t), th en it w ill b e p o ssib le to sep a ra te o u t an am p lified signal p ro p o rtio n a l to s(t) w ith a filter th a t re je c ts b o th d c and th e “ c a r r ie r ” and its h arm o n ics .

I f s(f) = co s fit, th en the s in a>0[f + rs (i)] term w ill exp an d in to a c a rr ie r a t w 0 an d s id eb an d s w ith am p litu d es p ro p o rtio n a l to J n{a>0z) sp aced a t npi a ro u n d w 0 [ J „ (c o 0 t ) ] is an o rd in a ry B essel fu n ctio n . S in ce V m < V s h as b een im p osed as a re s tr ic tio n , it fo llow s th a t (u0i < n an d h en ce th e v alu es o f J„(n ) w ill give an in d ica tio n o f the m ax im u m am p litu d e to be exp ected fro m th e e x trem e s id e b a n d st :

J i(7 t) * 0 .2 8 5 , J 2(n) x 0 .4 8 5 , J 3(jt) * 0 .3 4 , J 4(ji) % 0 .0 1 6 ,J 5( n ) x 0 .0 5 6 , J 6( n ) x 0 .0 1 6 , J-,(n) w 0 .0038 .

T h u s if fi is a t th e to p o f th e a llo w a b le “ m o d u la tio n ” ban d and to0 is eq u al to o r g rea ter th an 8/i, th en th e sev enth s id eb an d w ill in terfere d irectly w ith the low -p ass sig nal b u t th e re la tiv e d is to rtio n w ill be on ly

2 2-J-,( o>0t) = —J 7 (71) = 0.0024, n n

o r o n e -q u a r te r o f o n e p ercent.A ctu a lly , if th e c ir c u it ’s lo w -p ass filter d o es n o t fall o ff rap id ly en o u g h , the

lo w er-o rd er h a rm o n ics m ay b e the lim itin g facto r . F o r exam p le, ~ 1 0 0 J 7(7t); h en ce w ith /0 = 8 f m th e lo w -p ass filter m u st p ro v id e an a tte n u a tio n o f 1/100 o r 4 0 d B o n ly tw o d eca d es a b o v e th e m a x im u m m o d u la tio n freq u en cy . T o d o th is and s im u ltan eo u sly av o id a tte n u a tin g th e d esired sig nal w ill req u ire a very so p h istica ted lo w -p ass filter. T h e p ro b le m is eased by ra isin g th e c a rr ie r frequ en cy . T h e lim its on su ch in crea ses are d ev ice freq u en cy lim ita tio n s th at p rev en t sq u a re -co rn e re d o p e ra tio n an d h en ce in cre a se d ev ice d iss ip a tio n an d th e d ifficu lty o f co n ta in in g the R F in terferen ce cau sed by th e h a rm o n ic -r ic h m o d u la ted ca rrie r. (F o r exa m p le , o n e sh o u ld av oid su b m u ltip les o f th e co m m o n I F freq u en cy o f 4 5 5 k H z.) A 120 k H z c a rrie r shou ld be sa tis fa c to ry fo r o rd in a ry n o n -h ig h -fid e lity a m p lifica tio n w ith s(r) n o rm a lly b elow 6 kH z.

A co m p le te C la ss D a u d io a m p lifie r is illu stra ted in F ig . 9 .7 2. A s in the n a rro w b an d ca se we m ay co n sid er c o n stru c tin g a sw itch in g c ircu it to sw itch a c o n sta n t cu rren t betw een tw o end s o f o u r filter, o r we m ay co n sid er a sw itch in g c ircu it to effectively sw itch th e in p u t o f th e filter betw een tw o d ifferent v o ltages.

F ig u re 9 .7 -3 in d ica tes the b a s ic v o lta g e-sw itch in g case. A t first g la n ce th is c ircu it ap p ears to be id en tica l to th e n a rro w b a n d c ircu it o f F ig . 9 .5 - l ( a ) . T h is is a ca se w here lo o k s a re d ece iv in g ; sin ce th is filter m u st fu n ctio n as a b ro a d b a n d b a n d p a ss filter, n o t as a n a rro w b a n d filter, th e sw itch is n o lo n g er d riven in a sq u are-w av e fash io n , and , as we shall see, the tu rn -o n req u ire m e n ts fo r the sw itch in th e b ro a d b a n d ca se a re m u ch m o re s trin g en t th a n they a re in th e n a rro w b a n d case.

W h en R /L » 1 /R C , th e p o les in th e inp u t a d m itta n ce o f th e filter a re w idely sep arated and its a sy m p to tic m a g n itu d e-v s .-freq u en cy p lo t is as show n in F ig . 9 .7 -4 .

t For smaller values of <a0i the higher-order Bessel functions decay much more rapidly than linearly ; hence the maximum percentage distortion will occur at maximum m odulation. See Chapter 11 for details.

P O W E R A M PLIFIER S

Fig. 9.7-2 Class D PW M broadband power amplifier.

Fig. 9 .7 -3 Basic voltage-switching broadband Class D power amplifier.

Fig. 9 .7-4 Magnitude of the admittance of the filter of Fig. 9.7-3 (R/L » l/RC).

9.7 B R O A D B A N D CLA SS D P O W E R A M PLIFIER S 429

I f the sw itch is d riven b a c k and fo rth betw een the tw o te rm in a ls by a s ignal o f the form show n in E q . (9 .7 -1 ) , then we m ay w rite

V 2 V 2 Vl\ (0 = J'cc + Vc c ~ s ( t ) -------— s in w 0f --- - s in a > 0 [i + x.s(f)]. (9 .7 -2 )

K 71 n

I f s(i) is restricted to lie b etw een l /R C and R /L and if (oQ » R /L , then the c a p a c ito rw ill ch a rg e to Vc c and the n etw o rk cu rren t w ill be

<»7-3,

w here Vm < Vs is a n a d d itio n a l re s tric tio n . F o r exa m p le , i f Vm = Vs and s(t) = co s pit, then

. Vcch = — cos/rt.

A s show n in F ig . 9 .7 -3 , th e sw itch h as n o d is s ip a t io n ; h en ce all the inp u t pow er is co n v erted in to a c o u tp u t pow er.

I f th e d ev ice (sw itch ) h a s a p eak cu rre n t lim ita tio n /max, then

R 5: Vc c / I max.

I f R = Vcc/1 max, then P ac = Vc c I mJ 2 an d P dc = 2Kr r /dc; h en ce

dc = /max/4-

I f th e sw itch lo sses ca n be m o d eled co m p le te ly as res is to rs R s , th en th e efficien cy b eco m es

In th is ca se the p rev io u s R is now R s + R L , w here R L is th e a c tu a l lo ad resisto r.F ig u re 9 .7 5 in d ica te s th e filte r in p u t v o ltag e an d th e cu rre n t th ro u g h Vc c fo r the

ca se w here s(t) is a sin e w ave th a t lies betw een \ /R C an d R /L . A s th e figure c learly in d ica tes, the sw itch m u st c a rry c u rren t in b o th d ire c tio n s ; h e n ce fo r e ffic ien t o p e ra tio n w ithou t excessiv ely large d riv in g sig n als, th e sw itches sh ou ld be sy m m etrica l d ev ices (n o t v acu u m tu b es) c a p a b le o f ca rry in g cu rren t in the reverse d irectio n .

I f th e signal is s in u so id a l and d riven so th a t the peak cu rren t th ro u g h th e sw itch is /max, then fo r th e ca se w here the sw itch is rep laced by a b a tte ry , Ksat. t

P — ^ c c ~ K a l K m a x p _ I/ I p _ [ / ( I Q 1 - A )* ac ^ , d c d c * p e r s w i l c h ' s a t ' d c - V ' ^ 7

S in ce P j c P ac "f" 2 i5per SWjtCh»

/dc = and n = (9.7 5)4 vcc

t A battery is not really as good an approximation to the switch as a resistor is, since the switch current is not constant but varies over the cycle.


m)

J 7 U U 'v,(0

2 Vcc

0 t

Fig. 9 .7 -5 Battery and upper switch current and filter input voltage when 5(f) is a sine wave(Vm * K i

T h is e ffic ien cy is, o f co u rse , e x a c tly th e v alu e we ach iev ed p rev io u sly fo r a m a x im u m - a m p litu d e sq u a re -w a v e drive. T h e d ifferen ce is th a t now we h av e a s in u so id a l o u tp u t. E v en th o u gh o u r b a seb a n d m ay o n ly ex ten d to 5 kH z, we sh a ll w ant a sw itch in g freq u en cy o f 100 k H z o r m o re , an d h e n ce the p ro b lem o f tra n s itio n -tim e d issip a tio n as d iscu ssed in S e c tio n 9 .5 is still p resent.

A ctu a lly th e s itu a tio n is c o n sid e ra b ly w orse in th is ca se , s in ce now th e sw itch m u st b e turned on an d o ff w hile th e c u rren t is a t n early its m a x im u m value, w hile in th e n a rro w b a n d ca se th e o re tica lly th e cu rren t w as g o in g th ro u g h zero a t every v o ltag e-sw itch in g in terval.

F ig u re 9 .7 - 6 sh o w s a p o ssib le m e ch a n iz a tio n o f F ig . 9 .7 -3 . W h e n Q t is sa tu ra ted , then Q 2 is o n an d Q 4 is off. W ith Q 2 sa tu ra te d , Q 3 is turned o n w ith (fo r th e resistiv e valu es show n)

^ 2 (VC C - V SJ ~ V BE

B ~ 66 Q

o r w ith Vc c = 10 V an d Ksa, = 0 .5 V fo r e a ch o f (? , and Q 2,

17.2

6i n =

I f ocR > 0 .8 , th en th e tra n s is to r @ 3 c a n b e k e p t o n fo r rev erse c u rren ts o f 1.3 A , s in ce l B > (1 — otR)ICR m a rk s th e edge o f rev erse sa tu ra tio n .

W h en Q i g o es off, Q 2 an d Q 3 g o o ff an d Q 4 is d riven o n w ith e ssen tia lly the sam e b a se cu rren t a s Q } h ad o n th e o th e r p o rtio n o f th e cycle . W ith th e v alu es show n in F ig . 9 .7 - 6 th e lo ad re s is to r sh o u ld b e 10/1.3 = 7 .7 £1. F o r a sm a ller lo ad th e tu rn -o n d rive o r th e sw itch in g tra n s is to r rev erse a lp h a s w ould need to b e increased .

S in c e th e c irc u it a lw ays req u ires 2 6 0 m A o f b a se c u rren t, it rea lly w ould seem fa ir to ch a rg e it w ith 2VCC1B ~ 5 W o f a d d itio n a l p o w er d ra in . It is tru e th a t th e C la ss

9.7 B R O A D B A N D CLASS D PO W E R A M PLIFIER S 431

Fig. 9.7 6 Practical Class D pulse-width-modulated power amplifier.

B stage a lso d raw s b a se c u r r e n t ; how ever, in th a t ca se the b a se cu rren t is sm aller and tra c k s the sig nal in am p litu d e in stead o f b e in g co n sta n t.

C u rren t-S w itch in g C ircu its

It w ould a p p e a r th a t o n e cou ld c o n stru c t a b ro a d b a n d cu rren t-sw itch in g c ircu it as th e d u al o f th e v o ltag e ca se ju s t d iscu ssed . T h e p a ra lle l R L C filter in th is ca se w ould hav e c j1ow = R /L a n d <ohigh = 1 /R C . F o r a s in u so id a l o u tp u t fro m a co n fig u ra tio n sim ila r to F ig . 9 .5 - 4 o r 9 .5 —1(b), th e ind iv id u al " s w itc h ” cu rren ts w ould have the sh ap e o f r , in F ig . 9 .7 -5 an d a p ea k v alu e o f I p (assu m in g a c u rren t so u rce o f 2 I p). T h e “ sw itch ” v o ltag es w ould h av e a s in u so id a l “ e n v e lo p e ,” but w ould b e z e ro w hen ever the

sw itch w as clo sed an d l p w as flow ing. T h is c irc u it w ould seem to hav e th e ad v a n ta g e o v er th e v o lta g e sw itch o f h av in g to ca rry cu rre n t in o n ly o n e d irec tio n . It h as the d isa d v a n ta g e o f h av in g to h o ld th e tra n s is to r o ff fo r a reverse c o lle c to r v o ltag e equal to th e p eak o u tp u t v o ltage.

T h e p ra c tica l d ifficu lty w ith su ch a c irc u it is th a t th e tru e d u al o f th e b a tte ry has n o t yet b een inv ented . T h u s o n e is fo rced to use a b a tte ry Vc c in series w ith a n in d u c to r L c c a s a n a p p ro x im a tio n fo r a cu rren t sou rce .

432 P O W E R Â M PLIFIERS 9.8

O n e n ow h as d ifficu lty p ro p erly re la tin g L c c , c a rr ie r freq u en cies (sw itch in g rates), an d a llo w a b le m o d u la tio n freq u en cies. N ow , in a d d itio n to h a v in g th e p ro b le m o f sep a ra tin g th e c a rr ie r freq u en cy rip p le fro m the m o d u la tio n freq u en cy in the o u tp u t p a ra lle l R L C filter, o n e m u st sep a ra te th ese freq u en cies so th a t L c c c a n be c h o se n to be an “ o p en c ir c u it” a t the c a rr ie r freq u en cy an d a “ sh o rt c ir c u it” a t th e h igh est m o d u la tio n freq u en cy . T h is a d d itio n a l re s tr ic tio n tend s to d em an d th a t th e sw itch in g freq u en cy be ra ised to b e a t least 100 tim es the h igh est m o d u la tio n freq u en cy , thus in crea sin g th e d ifficu lty o f o b ta in in g a re a so n a b le a p p ro x im a tio n to an ideal sw itch in g d evice.

E x c e p t in sp ecia l ca ses in w hich o n e w ants a co n sta n t-a m p litu d e , h igh -effic ien cy , v a ria b le -fre q u en cy p ow er a m p lifie r , the cu rren t-sw itch in g b ro a d b a n d o r C la ss D typ e o f c ircu it w ould seem to be p rim a rily o f a c a d e m ic in terest.

9.8 PRACTICAL PO W ER A M PLIFIER S

A n u m b er o f p ra c tica l p ro b le m s a rise w hen o n e a ttem p ts to co n stru c t a pow er am p lifier. T h e se p ro b le m s in clu d e the p ro v isio n o f d riv in g sig n a ls, the p ro v isio n o f p ro p e r 'b ia s , th e d esign o f c irc u its th a t p ro te c t them selves ag a in st a cc id e n ta l d estru c tio n , and th e “ m a tc h in g ” o f the a c tu a l lo a d s to the o p tim u m im p ed an ce req u ired by the d evice fo r effic ien t o p era tio n .

L et us c o n sid er first a fixed -am p litu d e C la ss C v a cu u m -tu b e p o w er a m p lifie r o f the typ e co n sid ered in S e c tio n 9 .4 an d illu stra ted g en era lly in F ig . 9 .4 -1 . In th is ca se we assu m e a b ia s an d a d rive and ca lc u la te the d esired h ig h -freq u en cy res is tiv e -lo ad im p ed an ce. In the p ro cess o f th ese c a lc u la tio n s we a lso c a lcu la te th e w av esh ap e o f the grid c ircu it cu rren t w avesh ap e. F r o m th ese w av esh ap es w e ca n c a lc u la te th e fu n d am en ta l and d c valu es o f cu rren t. N o w w ith the assu m ed valu e o f d riv in g v o ltag e we c a n c a lcu la te th e e ffectiv e inp u t im p ed a n ce o f the c irc u it as w ell as a b ias re s is to r th a t in c o n ju n c tio n w ith /G0 will supply the d esired b ias. I f the b ias is supplied co m p le te ly by a n /C0-K C c o m b in a tio n , then w hen the d rive c ircu it is tu rned o ff the b ias is lost. F o r th e case o f F ig . 9 .4 -3 th e tu b e d iss ip a tio n a t VBB = 12 k V and Vc c = 0 is 6 6 k W , o r m o re th an six tim es the “ n o rm a l” o p e ra tin g d issip atio n . Su ch a d issip a tio n w ill u n d o u b ted ly lead to d a m a g e o r d estru ctio n . O n e m igh t c h o o se to o b ta in en o u g h o f the b ias fro m a fixed supply to p rev en t d ev ice d estru ctio n and th e rest fro m a “ grid le a k ” co m b in a tio n .

T h is self-b ia sin g a rra n g e m en t h a s the ad v a n ta g e th a t the b ias a d ju sts itse lf to the d riv in g sig nal in su ch a fash io n as to tend to k eep the o u tp u t co n sta n t. T h a t is, if the d rive signal d ecreases, then so d o es th e b ia s , o r v ice v e rsa ; h en ce th e o u tp u t cu rren t p u lse sh ap e ch a n g es m u ch less th a n it w ould h av e if the b ia s w ere fixed and the sam e ch a n g e in in p u t level o ccu rred .

F ig u re 9 .8 -1 illu s tra te s tw o p o ssib le b ia s arra n g em en ts. In e ith e r ca se C G sh o u ld h av e a n im p ed a n ce a t th e o p e ra tin g freq u en cy th a t is sm all en o u g h so th a t

flow in g th ro u g h it w ill ca u se a v o ltag e d ro p o f 1 % o r less o f the d esired d riv ing v o ltag e. A lso , in e ith er ca se , Fpartia, + I(‘; 0R G sh ou ld eq u al th e d esired b ias v o ltage. N o rm a lly the c irc u it o f F ig . 9 . 8 - 1(a) is to b e p referred , s in ce th e o th e r a rra n g e m en t

9.8 P R A C T IC A L PO W E R A M P L IF IE R S 433

Fig. 9 .8 -1 Class C bias arrangements.

h as an a c p o w er lo ss in R 0 th a t is n o t n eg lig ib le and w hich m u st b e supplied by the d river stage.

A s a n e x a m p le , fo r the s in u so id a l d rive pulses illu stra ted in F ig . 9 .4 -4 (a ) , if ^partial = 120 0 V , th en R g = 2200/ 0.45 = 4 .9 kQ . T h e d c p ow er lo ss in th is re s is to r is 2 2 0 0 x 0 .4 5 = 10 0 0 W . In the c ircu it a rra n g em en t o f F ig . 9 .8 1(b) th ere w ould be an a d d itio n a l a c lo ss o f 3 0 0 0 W in R G. T h e pow e; th a t m u st be supplied by th e d river fo r F ig . 9 .8 - l ( a ) is (5 4 0 0 x 0.85)/2 = 2 3 0 0 W , o f w hich 5 4 0 W go es to “ c h a rg in g ” ^partial a n d 2 3 0 0 — 5 4 0 — 1 0 0 0 = 7 6 0 W is d issip ated in heat at the grid c ircu it. S in ce the d river m u st sup p ly 2.3 k W in to a 5400/ 0.85 = 6 3 5 0 Q lo ad , it m u st a lso be a p ow er am p lifier.

D riv e S ig n als

S in ce iG flow s in pulses, it will c o n ta in a th ird -h a rm o n ic c o m p o n e n t ; hence o n e cou ld

g en erate the req u ired n o n sin u so id a l d riv in g w av esh ap e to supply th e p ro p o sed cu rren t w avesh ap es o f F ig . 9 .4 - 4 by in c lu d in g a seco n d ta n k c irc u it in series w ith th o se show n in th e grid c ircu its o f F ig . 9 .8 -1 . I f th is seco n d ta n k c ircu it is tuned to th e th ird h a r m o n ic o f oj0 and is load ed so th a t l Gi cau ses a 5 4 0 V d ro p a cro ss it, then the desired grid w av esh ap e w ill resu lt ( /C3 flow s in the o p p o site d irec tio n to the in p u t fu n d am en tal cu rren t from the d riv in g stage). I f o n e assu m es th a t the cu rren t from the d river is purely fu n d am en ta l by th e tim e it re a ch e s the seco n d a ry , then all the p ow er supplied by th e d riv in g stag e is a t th e fu n d am en ta l frequ ency .

A “ C la ss C ” tra n s is to r stage is u sually ru n w ith zero b ias o r w ith a t m o st a sm all e m itte r r e s is to r ; h en ce the in p u t c ircu it c a lc u la tio n s a re m uch sim p ler than th o se o f the vacu um tube case . In g en era l, w ith tra n s is to r c ircu its the fu n d am en tal input im p ed a n ce o f a p ow er a m p lifier w ill b e very sm all, p erh ap s o n ly a few o h m s ; thu s an

im p ed a n ce step -d o w n c irc u it o r tra n sfo rm er w ill be n ecessary b etw een th e d river an d th e p ow er am p lifier .

C oupling N etw orks

It is re la tiv e ly sim p le to say w hat the co u p lin g n e tw o rk should d o , but som ew h at

m o re cu m b e rso m e to e x p la in h ow to m a k e it d o w hat is requ ired . Id eally , th e co u p lin g


n etw o rk sh o u ld “ tra n s fo rm ” the a c tu a l lo ad to the d esired resistiv e v alu e at the o p e ra tin g freq u en cy w hile p resen tin g zero inp u t im p ed an ce a t all h a rm o n ic freq u en cies. A t th e sam e tim e the n e tw o rk sh ou ld hav e a tra n sfe r fu n ctio n th a t p asses the d esired m o d u la tio n p a ssb a n d w ith o u t ca u sin g am p litu d e o r p h ase d is to rtio n w hile offering in fin ite a tte n u a tio n to a ll h a rm o n ic freq u en cies. T o so m e e x ten t th ese req u irem en ts m ay be co n tra d ic to ry . C e rta in ly they m u st a ll be ch eck ed in d iv id u ally , sin ce satisfy in g o n e o f th em d o es n o t g u a ra n te e th e sa tis fa ctio n o f th e o thers.

h (p ) r2 C2

M L,L2 - M2n — — , R2 - R l + r2, L} — -

L i L j

(a) (b)

Fig. 9 .8-2 O utput coupling circuit, (a) Coupling circuit, (b) Possible circuit model.

T h e c la ss ic co u p lin g n e tw o rk fo r v a cu u m -tu b e p o w er am p lifiers p ro v id es an ex a m p le o f a n a lm o s t id eal co u p lin g s itu a tio n . T h e c ircu it an d a useful m o d el fo r it a re show n in F ig . 9 .8 -2 . T h is p a rtic u la r m o d el is ch o sen from the m u ltitu d e o f p o ssib le m o d els b e ca u se it lea d s to a sim p le p re se n ta tio n o f b o th Z in an d the tra n sfer im p ed ance.

I f th e a d m itta n ce to th e left o f the tra n sfo rm er in F ig . 9 .8 -2 (b ) is ca lled ^(/ j) w hile th e a d m itta n ce to th e rig h t is ca lle d Y2(p), th en it is a p p a ren t th a t

7 I \ a Fz(p) _ nY2(p)R2m P Yi(p ) + n2 Y2{p) a n I t (p) Yt (p) + n2 Y2{p)

B o th th e in p u t an d th e tra n sfe r im p ed a n ce have id en tica l p o les. T h e tra n sfe r im p ed an ce h as tw o z e ro s a t the o rig in and a sca le fa c to r o f ojqM R 2/ L 3, w hile the inp u t im p ed an ce h a s co m p le x z e ro s a t th e ro o ts o f p 2 + p (R 2/ L 3) + 1 / L 3C 2 an d a sin gle zero a t th e o rig in . T h e sca le fa c to r fo r th is ca se is 1/C,.

T h e p o les fo r e ith e r fu n ctio n m ay b e fo u n d by th e ro o t lo cu s tech n iq u e . F r o m E q . (9 .8 -1 ) th e p o les o f e ith e r fu n ctio n a re a t th e ro o ts o f

9.8 P R A C T IC A L P O W E R A M PLIFIER S 435

In o rd e r to c o n c e n tra te o n th e im p o r ta n t a sp e c ts o f th e c irc u it, w e m a k e th e fo llo w in g very re a s o n a b le a s s u m p tio n s o r a p p ro x im a t io n s :

a) S in ce the u n lo ad ed Q o f the p rim a ry a lo n e should be eq u a l to 100 o r m o re, weassu m e, for the p u rp o se o f lo ca tin g the first p o le pair in th e d e n o m in a to r o f

E q . (9 .8 -2 ) , th a t R , a p p ro a ch e s in fin ity . (T o ca lc u la te the n etw o rk loss and the n etw o rk tra n sfe r e ffic ien cy , we m ust re in s ta te the a c tu a l R ,.)

b) W ith resp ect to the up per co m p le x p ole p a ir we neg lect the an g les from the

low er p o les and zeros. I f R 2/ 2 L 3 > co0/ 4 , th a t is, if the lo ad ed Q o f the m y th ica l seco n d a ry exceed s 2 , and if we a re in terested in the lo c i from and a fter the po lesco a le sce , then the m ax im u m a n g u la r e rro r is 14° o r 7° from e a ch up p er pole. C e rta in ly in an in itia l c o n sid e ra tio n this n eg lect is ju stified in term s o f o b ta in in g a sim p ler e x p la n a tio n .

W ith th ese a ssu m p tio n s , if the up per co m p lex p o les are in itia lly lined up h o r izo n ta lly , that' is, if

w here co = 1/L, C then as n in crea ses they w ill m o v e tow ard e a ch o th e r an d co a le sce at K 2 /4 L 3 from the j - a x i s ; they w ill then m o v e a p a rt a lo n g a line p a ra lle l to the j-a x is . F ig u re 9 .8 -3 illu stra te s the sim p lified locus.

I f we re stric t o u rselves to v alu es o f n th a t cau se c o a le sce n ce o r v ertica l sp littin g , then from the ro o t lo cu s m ag n itu d e c o n d itio n and from E q . (9 .8 -2 )

L 1 C 1 \ 2 L 31 L 3C 2(9 .8 -3 )

or

(9 .8 -4 )

*j- axis

rtR 2I2L 3

Fig. 9.8-3 Simplified locus of upper pole pair.

n(9 .8 -5 )

2 J ~ L 3C td■ V 3 1

w here d is the d is ta n ce from th e o rig in a l r ig h t-h an d p o le (actu a lly from eith er o rig in al p ole) to the d esired cou p led p ole p o sitio n s.


T w o p o s itio n s w ould a p p e a r to b e o f p rim ary in terest. In o ne, the p o les have ju s t co a lesced . In th e seco n d s ig n ifica n t p o s itio n , the p o les h av e m oved a p a rt v e rtica lly by 4 5 ° so as to yield a m a x im a lly fla t tra n sfer fu n ctio n . T h e resu lts fo r th ese tw o cases are su m m arized in T a b le 9 .8 -1 , w here R in is th e d esired input im p ed an ce at

»0 -Table 9.8-1

M /L ,

L,

^2

C l

ni

— Z 2 ¡ ( j vJo)

Coalescence Maximally flat

R2/4L3

4 L 3/C ,

R 2/ 2 ^ 2 L }

2 L J C ,

\/ Rl/Rm

1 /«¿c, l 3 + m 2/ l ,

l

(OqL 3 1 +R 2

R iR l + Rin

/R^Ry

T a b le 9 .8 - 2 in d ica te s th e re la tiv e m ag n itu d es o f Z in an d Z 2 j as a fu n ctio n o f 6u>, w here Sco is m easu red fro m a>0 = 1/X/ L 1C 1. N o te th a t b y th e seco n d h a rm o n ic we a re assu m in g th a t th e resp o n ses fo r th e tw o ca se s a re id en tica l. T h is is n o t a n ex a ct resu lt, b u t is a very g o o d in itia l a p p ro x im a tio n . I t is a p p a ren t th a t the m a x im a lly fla t c a se is to b e p referred w hen ev er it is p o ssib le , sin ce it gives a b e tte r “ in -b a n d ” ch a ra c te r is t ic w ith o u t ca u s in g any a p p re c ia b le d ifferen ce in its resp o n se to h a rm o n ics .

T o p lace th ese resu lts in p ersp ectiv e , let us co n sid er a n u m e rica l exam p le. A ssum eth a t

R m = 2 0 0 0 f i , R 2 = 5 0 Q , (o0 = 1 0 7, and Q , = Q 2 = 100.

N o w to k eep th e p rim a ry c irc u it lo sses dow n to 5 % o f the p o w er p assin g th ro u g h it,

„ RJ ~r = 0 .9 5 o r R 1 = 2 0 R in = 4 0 k i l tK l + K in

t Actually R m should include /? ,. So long as R , » Rin, the separation shown is reasonable.

9.8 PR A C T IC A L P O W E R A M PLIFIER S 437

Table 9.8-2

4SwL }/R 2Coalescence Maximally flat

\Z J /R in |Z 21|/Z 21(0) IZ J /R in |Z 21|/Z 21(0)

0.0 1.00 1.00 1.00 1.00+ 0.5 0.83 0.80 1.02 0.99± 1 .0 0.56 0.50 1.00 0.89+ 1.5 0.38 0.31 0.83 0.67+ 2.0 0.28 0.20 0.63 0.44

+ nth n 1 / * I* n 1 / « yharm onic n2 - 1 oj0C tR,„ I " 2 - l) n2 - 1 u>0C iR tn k - «)

1 2 V - i 2 )|®o C t Rin) P o C l^ jn )

I f Q , = 100 and R , > 4 0 i l ,

L , > 4 0 fiH fo r u)0 = 1 0 7.

I f we ch o o se L , = 4 0 /iH, then

C i = 1 / ojqL i = 250 p F

and

M = L XSJ R 2/ R X = y /4 0 / iH = 6.35 ¡iH.

F r o m T a b le 9 .8 -1 (th e seco n d lin e) we see th a t the d ifferen ce b etw een the tw o p ole p o s itio n s rests in L 3 (and a lso , th ere fo re , in L 2 and C 2). T h e resu lts fo r th e tw o cases are show n in T a b le 9 .8 -3 .

T h e m a x im a lly fla t ca se h a s a s lig h tly low er c ircu it effic ien cy (^circuit = thtl2), how ever, it a lso h a s a low er co e ffic ie n t o f co u p lin g , k, and a sm a ller valu e fo r C 2 A p ra c tica l p ro b le m rem a in s as to w h eth er th e d esired co il c o m b in a tio n is p hy sica lly p ossib le . In th is ca se , i f L 2 is p laced o v er the end o f L , , then th e req u ired a m o u n t o f c o u p lin g shou ld b e p ossib le .

Table 9 .8 -3

Coalescence Maximally flat

L , 6.25 /j H 12.5 /iHl 2 7.25 /¿H 13.5 /iHc 2 1380 pF 770 pFk 0.374 0.2751z 0.985 0.973'/circuit 0.935 0.925

438 PO W E R A M PLIFIER S 9.8

W e m ay now e x a m in e th e re la tiv e in p u t an d tran sfer im p ed an ce a t th e second h a rm o n ic to see how effectiv e th e n e tw o rk is in su p p ressin g h a r m o n ic s :

|Z J 2 M _ 2 1 _ 2 - \Z2 l{2jo))\ 4 4 _ 16

R m 3 5 1 5 ’ \Z2 l ( M 9 25 2 2 5 '

T h e se resu lts in d ica te th a t in a c a sé s im ila r to th e o n e sh o w n in F ig . 9 .4 -4 (a ) , w here/2//, = 0 .7 2 , the s e c o n d -h a rm o n ic v o ltag e a cro ss th e in p u t o f th e ta n k c ircu it willbe a p p ro x im a te ly 1 0 % o f th e fu n d am en ta l v o ltag e, w hile th e seco n d -h a rm o n ic o u tp u t v o ltag e w ill b e 5 % o f th e fu n d am en ta l v oltage.

S in ce the in p u t h a rm o n ic v o lta g e is sh ifted in p h ase by n early 9 0 ° w ith resp ect to th e fu n d a m en ta l, o n e ca n a p p ro x im a te th e in p u t tan k v o lta g e (th is v o lta g e adds to Vc c to m a k e th e d evice v o lta g e) by

Fi c o s co0t + 0 .1 Vi s in 2 a>0t.

T h e effect o f such a ta n k v o lta g e o n th e o p e ra tin g p ath , w ith a p u re s in u so id a l d riv ing w avesh ap e, is show n in F ig . 9 .8^1.

S e v era l th in g s a re a p p a ren t fro m th is new p ath . O n e is th a t th e o u tp u t cu rren t p u lse sh ap e is d ifferen t fro m th a t found p rev io u sly an d is n o n sy m m etrica l. B eca u se the p u lse sh ap e is d ifferent, the p rev io u s c a lc u la tio n s w ill be som ew h at in erro r . B eca u se th e c u rre n t p u lse is n o t sy m m e trica l, th e re w ill b e a p h a se d ifferen ce betw een th e assu m ed ta n k v o lta g e an d th e resu lta n t fu n d am en ta l fro m th e cu rren t pulse. N o rm a lly o n e d o e s n o t w an t to p u rsu e th is c a lc u la tio n an y fu rth er a ro u n d th e loop . W h a t o n e d o es w an t to d o is re s tr ic t th e p rin c ip a l in p u t h a rm o n ic v o lta g e to the o rd er o f 1 0 % o f th e fu n d am en ta l.


I f o n e w an ts to red u ce th e h a rm o n ics , one sh ou ld in crea se C , ; how ever, for a

fixed u)0 , in crea sin g C , w ill red u ce L x an d ca u se th e p rim ary c ircu it e ffic ien cy to suffer. F o r th e ca se show n, we h av e p resented a re a so n a b le co m p ro m ise betw een th e tw o p o sitio n s.

T h e key term in h a rm o n ic red u ctio n is w hich is the “ lo a d e d ” Q o f thep rim ary circu it. W e h av e used a valu e o f 5. A valu e o f 10 w ould red u ce th e o u tp u t h a rm o n ic v o ltag e by a fa c to r o f 4 , but w ould d o so a t th e exp en se o f n e tw o rk efficien cy .

F o r lo w -p o w er c irc u its o n e o ften d o es sa crifice the o u tp u t p ow er fo r the sak e o f im p roved filtering .

I f a p u sh -p u ll o u tp u t s tag e is em p loyed , th e sec o n d -h a rm o n ic term s should id eally can cel. U n d er these c ircu m sta n ce s o n e m igh t halve w ith o u t m a k in g th e h a rm o n ic sp rea d in g o f th e o p e ra tio n p a th m u ch w orse. T h is w ould a llo w o n e to d o u b le L ; an d , if Q x ca n be m a in ta in ed , to h alv e the p rim ary losses. (In a m eg aw att tra n sm itte r 5 % is 5 0 k W ; h en ce red u cin g th e p rim a ry losses fro m 5 % to 2 .5 % is a w orthw hile en d eav o r.)

In c lo s in g the d iscu ssio n o f th is n etw o rk , it shou ld b e p o in ted o u t th a t the values for th e m id b an d Z in and tra n sfer im p ed a n ce ca n be o b ta in ed a lm o st by in sp ectio n if b o th sides o f th e tra n sfo rm er a re assu m ed to b e tuned to reso n an ce . T h is a p p ro a ch , co m b in ed w ith th e id ea th a t th e h a rm o n ic c o n te n t o f b o th th e in p u t v o ltag e and the o u tp u t v o ltag e d ep en d s essen tia lly on the lo ad ed Q o f the p rim ary c ircu it, i.e., on cj0C i R in, h as b een in use fo r p erh ap s fifty years. W h a t has n o t b een w idely a p p re c ia ted is th a t a te ch n iq u e lik e th e ro o t lo cu s m eth o d lets o n e sh ap e the “ in -b a n d ” resp o n se in a re la tiv e ly sim p le fash ion .

P i N etw ork

A n o th e r n etw ork th a t is w idely used in the o u tp u t o f p ow er am p lifie r stages is the

pi n etw o rk o f F ig . 9 .8 -5 . In th is figure R c j , R C2, and r are a ll lo ss e lem en ts th a t m ust be co n sid ered in ca lc u la tin g n etw o rk effic ien cy b u t th a t m ay, o n e h o p es , b e n eg lected in e stim a tin g the n etw o rk freq u en cy response.

U n fo rtu n a te ly , th e pi n e tw o rk is n o t as a m e n a b le to a g en era l tre a tm e n t as is the coupled coil circuit just considered. The basic conceptual problem is that the input im p ed an ce o f th e pi n e tw o rk h as a p a ir o f co m p le x zero s and a p a ir o f c o m p le x p oles ra th e r th an th e tw o sets o f co m p le x p o les foun d in th e cou p led co il case . As we shall

o t — — AAA -o

o- o

Fig. 9.8-5 Pi coupling network.


see, the up p er p o le an d zero a re u su ally c lo se en o u g h to g e th e r a n d low en o u g h in th e ir valu es o f Q th a t th e s im p le-m in d ed a p p ro x im a tio n s n o rm a lly used fo r d ea lin g w ith such c irc u its tend to b re a k d ow n. In a d d itio n th e p resen ce o f b o th a co m p le x p ole an d a co m p le x zero n e a r th e o p e ra tin g freq u en cy m ak es th e p a ssb a n d n o n sy m m e trica l and fo rces o n e to reexa m in e such co n ce p ts as “ b a n d w id th .”

W e sh a ll see th a t the tra n sfe r im p ed a n ce h a s the sam e p o les a s th e in p u t im p ed an ce, b u t is la ck in g the co m p le x zero s. H ow ever, sin ce th e c e n te r o f th e p assb an d is rarely s itu ated d irectly “ u n d er the p o le ,” th e tra n sfer im p ed a n ce , Z 21(p), is a lso n o n sy m m e trica l a b o u t the cen te r freq u en cy .

B efo re we co n sid er th e g en era l ca se , le t us e xa m in e a n a p p ro a ch to ca lcu la tin g th e inp u t im p ed an ce o f th e c irc u it o f F ig . 9 .8 - 5 in w hich we assu m e th a t a ll th e c o m p o n en ts hav e kn o w n valu es. T h e m e th o d c o n sis ts o f c o m b in in g im p ed an ces in an a lte rn a tin g p a ra lle l and series fa sh io n as o n e s ta rts w ith the lo ad an d w o rk s b a ck to w ard the in p u t o f the n etw o rk . T h a t is, o n e c o m b in e s R C2 an d R L to fo rm R 2 and th en , a t th e o p e ra tin g freq u en cy , co n v erts the p ara lle l C 2-R 2 co m b in a tio n s in to an e q u iv a len t series R SC S fo rm :

Q i + 1 R 2C S = ~ ^ C 2 , = Q 2 = u)0 C 2R 2 . (9 .8 -6 )

F o r exam p le, if to0 C 2R 2 = 3, then

C , = (10/ 9)C 2 an d R s = R 2/ 10.

O n e th en c o m b in e s R s and r an d jcoL an d l /jw C s. I f th e in p u t im p ed a n ce is to be resistiv e , then a>L > 1/ ujC s is n ecessa ry so th a t th e resu lt o f th e c o m b in a tio n is an e q u iv a len t series L -R c o m b in a tio n . T h is c o m b in a tio n is th en co n v erted b a c k in to p a ra lle l fo rm an d the in d u ctiv e p a rt is re so n a te d o u t, w ith C t leav in g th e p ara lle l co m b in a tio n o f R CI an d R p as R in :

RP = (1 + Qs)rs, Lp = Qs = —(9 .8 -7 )

w here

r , = R s + r, L = L - 1

F ro m th is a p p ro a c h we see th a t the n e tw o rk pow er tra n sfer e ffic ien cy w ill be

= R c l + R p ^ T r R C2 + R l (9 8 “ 8)

H en ce, fo r high effic ien cy , R C2 > 1 0 0 f lL , R s > lOOr, an d R CI > 1 0 0 R p a re a ll d esira b le re la tio n sh ip s.

N ow th a t the lo ss term s have b een co n sid ered , let us n eg lect th em in o rd e r to sim plify the a lg e b ra w hile we ex a m in e th e n etw o rk freq u en cy resp onse. C o n sid e r

9.8 PR A C T IC A L P O W E R A M PL IFIE R S 441

tw o n u m e rica l e x a m p les (in b o th ca ses C , = C 2):

(OqC 2R 2 = 3, o)qC 2R 2 — 3,

22 = 3, Q2 = 3,R s = i? 2/10 R s = i ? 2/10,

co0L/R2 = 6/10 m0L/R2 = 1/3,

Qs = 3, Qs = 1/3.

Rp = R 2 > R P = R 2! 9,

(OoCji?in = 3, = 1/3.In o n e ca se we hav e n o im p ed a n ce ch a n g e, w hile in th e o th e r we hav e a 9/1 im p ed an ce step -d o w n . I f we s ta rted w ith a 5 0 Q lo a d , then th e step -d ow n ca se w ould be useful

fo r a tra n s is to r p o w er am p lifier.As we shall see sh o rtly , by c h o o sin g a la rg er valu e fo r io0L / R 2 in the first ca se , and

by ch a n g in g C , in a n a p p ro p ria te fa sh io n , we co u ld hav e o b ta in ed a large im p ed an ce step -u p ra tio ra th e r th a n the 1/1 ca se o b ta in ed .

In o rd er to see h ow su ch a w ide ra n g e o f in p u t im p ed an ces is p o ssib le , as w ell as to b eg in to u n d erstan d th e freq u en cy resp o n se o f th e n e tw o rk , let us w rite th e e x p re ssio n s fo r Z in and Z 2 1 , assu m in g lo ssless e lem en ts in b o th c a s e s :

2 P 1P + i T T r +

ZiB(p) = — ------- ;----R f 2 , ;LC----------r , <9.8-9)+ -^2^2 i-C 1 I

Z 2 ,(p ) = -------------- , 1 z 7 T v (9 .8 -1 0 )sam e p o les as Z in(p)

I f we d efine u>j = 1 /(L C 2), C 2 = N C l , an d Q 2 = a>0C 2R 2 th en Z in(p) m ay be rew ritten as

z , v = ________________ p 2 + PM p/Qi + oil______________________ (9 8 -1 1 )C i(p 3 + P2o) 0/Q 2 + p [l + N ]a )l + œ lN cü 0/Q 2)'

w here « j0 is th e d esired o p e ra tin g freq u en cy — th a t is, the freq u en cy a t w hich Z in(jœ )

is rea l an d h as th e d esired m ag n itu d e.I f o n e k n o w s th e c irc u it e lem en ts, th en th e c a lc u la tio n o f Z jn( joy) is s tra ig h t

forw ard , a lth o u g h ted io u s if d o n e by han d ra th e r th a n by a sto red p ro g ra m in a

m in ico m p u ter . The problem is that— ra th e r th a n k n o w in g the c ircu it e lem en ts on e kn o w s th e freq u en cy an d th e im p e d a n ce — tra n sfo rm a tio n ra tio , and w ishes to c h o o se th e e lem en ts so as to p erfo rm th e tra n s fo rm a tio n in som e o p tim u m m an ner.

T o get so m e fu rth er feeling fo r th e p ro b le m , we m igh t let Q 2 a p p ro a ch infin ity (th a t is, le t th e effectiv e lo ad re s is ta n ce R 2 a p p ro a ch in fin ity ) in E q . (9 .8 -1 1 ) . T h e


resu lta n t e q u a tio n w ou ld h av e co m p le x z ero s a t ±ja > 2 an d co m p le x p o les at

± /v/1 + N cj2 , a s w ell as a p o le a t th e o rig in . N ow from ro o t- lo c u s co n sid e ra tio n , it m ay b e sh o w n th a t as R 2 d ecrea ses fro m in fin ity th e a c tu a l p o les an d z ero s w ill m ove in to th e left-h an d p lan e, in itia lly a lo n g p a th s th a t are p erp en d icu la r to th e j-a x is . T h e Q o f th e co m p le x z ero s is ju s t Q 2 , w h ile— a g a in from ro o t-lo c u s c o n s id e ra tio n s— the Q o f the co m p lex p o les m ay b e show n to b e on the o rd e r o f 1.5 to 2 .5 tim es Q 2 .

I f o n e is try in g to o b ta in an im p ed a n ce step -u p , then he w ould e x p e ct to o p e ra te in th e v ic in ity o f th e up per co m p le x p o le , w hile if he w ishes a n im p ed a n ce step -d ow n , he w ould ex p e ct to o p e ra te m o re n e a rly “ u n d er the z e ro ” o f Z in.

A s we shall see, fo r the valu es o f Q 2 in th e ra n g e from 1 th ro u g h 5, n e ith e r th e step - up n o r th e step -d o w n ca ses w ill b e p a rticu la rly sy m m etrica l w ith resp ect to frequ en cy . In th e im p ed a n ce step -d o w n case , o n e h a s an a d d itio n a l c o m p lic a tio n in th a t— unless th e valu e o f N = C 2/ C l is c h o se n ca re fu lly — o n e w ill find th a t th e co m p le x p o le in th e im p ed a n ce fu n ctio n s w ill end up n e a r o n e o f th e h a rm o n ics o f th e d esired o p e ra tin g freq u en cy . S in ce , in p o w er-am p lifie r u sage, th e n e tw o rk in p u t cu rren t is n o rm a lly very r ich in h a rm o n ic co m p o n e n ts , th e resu lt w ill be a larg e v o lta g e co m p o n e n t a t the u n d esired h a rm o n ic freq u en cy . F o r e x a m p le , if N = 3 (C 2 = 3 C ,) , th en th e co m p lex p o le o ccu rs n e a r 2 a>0 an d b o th the in p u t im p ed a n ce and th e tra n sfe r im p ed a n ce m ay h av e seco n d h a rm o n ic te rm s th a t g re a tly exceed the fu n d a m en ta l m ag n itu d es. As we saw in c o n n e c tio n w ith F ig . 9 .8 -4 , su ch a s itu a tio n lead s to u n d esirab le o p e ra tin g c o n d itio n s fo r th e o u tp u t d evice. I t a lso lead s to undesired in terferen ce term s in th e n e tw o rk ’s o u tp u t v o ltage.

T h is p ro b lem o f the larg e h a rm o n ic co m p o n e n t is red u ced by g e ttin g th e co m p lex p o le aw ay frp m th e h a rm o n ic freq u en cy . T h is end is a cco m p lish e d by red u cin g the valu e o f N ( th a t is, by in cre a sin g C , w ith resp ect to C 2). F o r an y given Q 2 th ere is a lim it to th e a m o u n t th a t C , c a n be in crea sed an d still a llo w o n e to o b ta in a real in p u t im p ed a n ce a t th e d esired o p e ra tin g freq u en cy . I f Q 1 = co0 C l R 2 , th en th is lim it m ay b e exp ressed as

Q i + 1 2 0 ,Q 1 < Z ,^ 2 Qj +Q l + 1 '

W h ile fo r m o st c o m b in a tio n s o f Q 2 an d N th ere a re tw o o p e ra tin g freq u en cies a t w h ich Z in is resistiv e— th e step -u p an d th e step -d o w n ca ses— in th e lim itin g ca se th ese freq u en cies c o a le sce in to a single p o in t.

T a b le 9 .8^ 4 re la te s sev eral o f th e c irc u it p a ra m eters , a t th is lim itin g valu e o f N , to v a rio u s v alu es o f Q 2 .

T h e v alu e o f th e inp u t im p ed a n ce a t th e seco n d h a rm o n ic is fo u n d by ev a lu a tin g the in p u t im p ed a n ce w ith th e valu es o f c o m p o n e n ts an d freq u en cies specified by the e a rlie r valu es in th e tab le . W h ile th e e x a c t im p ed a n ce exp ress io n is so m ew h at c u m b e rsom e, fo r a ll valu es o f Q 2 > 1 it m ay b e c lo se ly a p p ro x im a te d by

(98~12)

9.8 P R A C T IC A L P O W E R A M PL IFIE R S 443

Tablé 9 .8 -4 Pi netw ork param eter when N — C J C 2 has its minimum allowable value

Q 2 — o j 0 C 2 R 2 o j 0 L / R 2 Q l — w 0 ^ 1 ^ 2 R J R 2 £ O ÌT

J

I Z i J 2 j ( ü 0) l / I Z J j c o

1 1.0 1.0 1/1 1.0 0.724

2 0.60 2.5 1/2.5 1.10 0.635

3 0 .4 0 5.0 1/5 1.10 0 .590

4 0.29 8.5 1/8.5 1.085 0.575

■ 5 0.23 13.0 1/13 1.075 0 .5606 0.19 18.5 1/18.5 1.062 0 .550

F o r Q 2 = 1 th e d ifference b etw een th e e x a c t and th e a p p ro x im a te exp ressio n s

is less th a n 5 d egrees in an g le and less th a n 3 .5 % in m agnitu d e.F ro m E q . (9 .8 -1 2 ) an d T a b le 9 .8 -4 , we see th a t even if Q 2 a p p ro a c h e s in fin ity ,

the m ag n itu d e o f th e seco n d h a rm o n ic te rm w ill still be o n e -h a lf o f th e m ag n itu d e o f th e term a t th e o p e ra tin g freq u en cy . F r o m o u r p rev io u s c a lc u la tio n s we are aw are th a t th is m ay w ell lead to a d ev ice o u tp u t v o ltag e c o m p o n e n t a t th e seco n d h a rm o n ic eq u a l to 2 0 - 3 0 % o f th e fu n d am en ta l c o m p o n e n t. E v en if the d evice o u tp u t v o ltag e has n o c o n tro llin g effect on th e o u tp u t cu rren t p u lse (an ideal tra n s isto r , fo r exam p le), th is large h a rm o n ic co m p o n e n t w ill sh ift the d evice o p e ra tin g p ath in to a h ig h er d issip a tio n reg io n , as in d ica te d b y F ig . 9 .8 -4 . H en ce it is n o t d esirab le .

A ra tio n a l w ay to rem ov e th e seco n d h a rm o n ic term fro m b o th th e in p u t and the tra n sfer im p ed a n ces is to p a ra lle l Q by a series-tu n ed , h igh -Q , lo w -lo ss L r C c ircu it th a t is re so n a n t a t 2 co0 . T o b e effective, th is c ircu it m u st h av e a re so n a n t im p ed a n ce a t 2 co0 th a t is sm all w ith resp ect to the desired o u tp u t im p ed an ce a t co0 . In m an y h igh -p o w ered tra n s is to r c ircu its , th is d esired im p ed a n ce is o n ly a few o h m s ; h en ce th e series tra p m u st h av e a n im p ed a n ce less th a n a few te n th s o f a n ohm .

In c lu sio n o f su ch a series c irc u it w ill m ov e th e p oles o f Z in as w ell as in tro d u cin g a new set o f p o les a b o v e 2co0 . I f th e Q o f th e tra p is h igh en ou g h (Q > 100 is d esirab le), th en th ese new p o les w ill b e c lo se en o u g h to th e 2o>0 so th a t they w ill n o t cau se p ro b le m s a t th e th ird h a rm o n ic , w hile th e ir effect a t th e o p e ra tin g freq u en cy will be o n ly seco n d ord er.

T h e p ro b le m re m a in s o f o b ta in in g en o u g h o f a han d le on the “ b a n d w id th ” o f th e c irc u it to b e a b le to sp ecify it in a n in te llig e n t m an ner.

T o see th e effect o f v a r ia tio n s in freq u en cy , we p lo tted th e m ag n itu d e an d p h ase o f Yjn(ja>) vs. cu/w0 fo r th e c a se in w hich Q 2 = 3 , N = 0 .60 , and h e n ce (u>0 L / R 2) = 0.4. T h a t is fo r th e lim itin g 5/1 step -d o w n ca se o f T a b le 9 .8 -4 . W ith th ese v alu es, a 5 0 -o h m lo a d is tra n sfo rm ed in to a 10-Q lo ad w hen w = co0 , in to 7 .45 QZ_ — 6 ° w hen oj =0.95coo , an d in to 13.8 - 2 .5° w hen to = 1.05co0 - T h u s a 5 % shift in frequ en cycau ses ro u gh ly a 3 0 % shift in th e m ag n itu d e o f th e in p u t im p ed a n ce , an d o n ly a re la tiv e ly sm all sh ift in th e p h a se ang le .

H ig h er valu es o f Q 2 w ill m o v e th e co m p le x z ero s c lo se r to th e 7 -a x is , an d h en ce w ill a llo w larg er step -d o w n ra tio s , w hile k eep in g th e im p ed an ce a t 2co0 fro m b eco m in g


excessiv e. F o r a given v alu e o f Q 2 , red u cin g a)0L /R 2 fro m the valu es in T a b le 9 .8 -4 w ill in cre a se th e s tep -d o w n ra tio a t th e exp en se o f a n in crea se in the re la tiv e seco n d - h a rm o n ic co m p o n e n t. I f a very larg e step -d o w n is req u ired , then a c o n flic t m ay arise b etw een b a n d w id th an d step -d o w n ra tio . T w o ca sca d ed pi sec tio n s m ay p ro v id e a so lu tio n to th is p ro b lem . H o w ever, th e e x tra re a c ta n ce s in th e c irc u it w ill a lso p rov id e e x tra p o les th a t m u st be steered c le a r o f the h a rm o n ics o f th e o p e ra tin g freq u en cy .

Sm ith C h a rt

A p ra c tica l d ev ice th a t y ields so m e in sig h t in to this m a tch in g p ro b lem is a m o d ified v ersio n o f th e tra n sm iss io n line c h a r t k n o w n as th e S m ith ch a rt. I f o n e n o rm a liz es all re a c ta n ce s (o r co n d u cta n ce s) w ith resp ect to th e lo ad , w hich in o u r case is R 2 , then th e ch a rt, illu stra ted in F ig . 9 .8 -6 , a llo w s o n e to see by in sp e ctio n w hat c o m p o n en t ra tio s a re a llo w a b le . It a lso a llo w s the sim p le in c lu sio n o f losses from the n e tw o rk ’s in d u cta n ce s and ca p a c ita n c e s , as w ell as a llo w in g o n e to ca lc u la te the e ffects o f v a r ia tio n s in freq u en cy o n th e im p e d a n c e ..

T h e m e ch a n ism o f the c h a r t th a t is useful is th a t a “ re fle c tio n ” o f a n o rm alized p o in t th ro u g h the c e n te r o f th e c h a r t p erfo rm s the p a ra lle l-se fie s an d series-p ara lle l co n v ersio n s o f E q s . (9 .8 -6 ) and (9 .8 -7 ) .

T h e usual V ersio n o f the S m ith c h a r t h as resistiv e (o r co n d u ctiv e ) c irc le s cen tered o n a v ertica l b ise c to r o f th e ch a rt an d re a c tiv e c irc le s cen tered o ff to the sid es o f the c h a rt. T o be useful fo r o u r p u rp o ses th e c h a r t need s several “ re fle c tio n ” c irc le s added, a s show n in F ig . 9 .8 -6 . T h e c irc le lab eled 1 is a re flectio n in to th e up p er h a lf o f th e c h a r t o f th e (R / Z 0) = 1 c irc le fro m th e lo w er h a lf o f th e ch a rt. S in ce we h av e n o rm a lized co0C 2 w ith resp ect to G 2 the first p o in t on th e c h a r t a lw ays lies in th e lo w er rig h t- h an d q u a d ra n t a t th e in te rsec tio n o f th e 1.0 “ resistiv e” c irc le an d the “ re a c tiv e ” c irc le eq u a l in n u m e rica l valu e to Q 2 = (co0C 2/G 2). T h e le ft-h a n d h a lf o f th e reflected (R / Z 0) = 1 c irc le a lw ays c o n ta in s th e seco n d p o in t as the “ re fle c tio n ” o f th e first p o in t th ro u g h th e ce n te r o f the ch a rt. N ow fo r the step -d ow n ca se we p lo t in the u p p er left q u a d ra n t th e re fle ctio n h a lf-c irc le (s) co rre sp o n d in g to the d esired step -d ow n ra tio (s). T h a t is, fo r a 5/1 im p ed a n ce s tep -d o w n the 5 c irc le ta n g en t to th e b o tto m o f the c h a rt is re flected to the to p o f th e ch a rt.

N ow u nless we ca n tra v e l a ro u n d (to the righ t) on the resistiv e c irc le fro m the first re fle ctio n p o in t in th e up per le ft-h a n d q u a d ra n t and in te rsec t the reflected step - dow n c irc le , th en th is c o m b in a tio n o f Q 2 an d step -d ow n are n o t a llo w a b le . I f the s tep -d o w n valu e c a n n o t b e red u ced , th en e ith e r Q 2 m u st be in creased o r th e step -d ow n m u st be o b ta in e d as a tw o -sta g e p ro cess.

A n u m erica l ex a m p le m ay h elp to c la rify th e use o f th e m o d ified ch a rt. F irs t we n o rm a liz e th e co m p o n e n ts o f F ig . 9 .8 - 5 to R 2 o r G 2 .

o)C'2 = a )C 2R 2 , coL ’ = a>L/R2 , coC lR 2 = a)C\,

Gin = G J G 2 , and G\ = G 2/G 2 = 1.

As a n u m e rica l ex a m p le re la ted to o u r p rev io u s c a lc u la tio n s , c o n sid er th e case in w hich Q 2 = to0R 2C 2 = a>C'2 = 3 an d fo r the ca se o f lo ssless co m p o n e n ts find the valu es (n o rm a liz ed ) o f L an d C , th a t w ill p ro v id e a 4/1 im p ed a n ce step -d ow n.


Fig. 9 .8 -6 Modified Smith chart. A dded reflection circles aid in the easy design o f pi m atching networks.

T h e n o rm a liz ed a >0C 2 — G 2 co n d u cta n ce is 1 + )3 , w hich is show n as p o in t a in F ig . 9 .8 -6 . T h e p a ra lle l-to -ser ie s c o n v ers io n is p erfo rm ed by re fle ctio n th ro u g h the c e n te r o f th e c h a r t to p o in t b, w h ich h as the n o rm a liz ed series im p ed a n ce o f 0.1 y'0.3. N ow we p ro ceed to th e righ t a lo n g th e 0.1 resistiv e c irc le until we s trik e th e 4/1 step- d ow n c irc le a t p o in t c. W e th en reflect th ro u g h th e c e n te r to p o in t d (4-J4 .9 ). N ow , by a d d in g a n o rm a liz ed v alu e o f a>C\ = 4 .9 , we a rr iv e b a c k a t a p u rely resistiv e ca se an d a 4/1 im p ed an ce step -d o w n . T h e d ifferen ce b etw een the re a c ta n ce a t p o in t b and at

point c is 0.423, which is, of course, the normalized value of a>L2AR2 •I.f in stea d o f a 4 / l im p ed a n ce s tep -d o w n w e h a d w anted a 20/1 im p ed a n ce step -u p ,

we w ould hav e co n tin u e d fro m p o in t c to th e in te rsec tio n o f th e 0.1 resistiv e line and the 0 .0 5 “ co n d u ctiv e lo ad c ir c le .” T h is in te rsec tio n o ccu rs a t 0.1 + j 1.41 (p o in t / ),


w hich reflects in to 0 .0 5 — j0 .7 0 5 (p o in t g ) ; h en ce ad d in g = 0 .7 0 5 G 2 w ill yield th e desired 20/1 im p ed a n ce step -u p (p o in t h).

S ev era l th in g s a re a p p a ren t a t th is p o in t. O n e is th a t co m p o n e n t lo sses m ay be easily han d led sin ce, o n ce n o rm a liz ed , th ey m erely add in to th e a p p ro p ria te sec tio n o f th e o p e ra tio n . A n o th e r fact is th a t o n e is a b le to w o rk in e ith e r d ire c tio n on th e c h a r t ; thu s, fo r exam p le, o n e ca n assu m e b o th C j and C 2 and read o ff the req u ired n o rm a lized valu e o f L to tie th em to g eth er.

T o re la te T a b le 9 .8 —4 to the ch a rt, it is help fu l to n o te th a t th e valu es o f the ta b le co rre sp o n d to th e ca se in w hich th e p a th fro m th e eq u iv a len t o f p o in t frjust b ise c tsa step - dow n re flectio n c irc le in th e u p p er r ig h t-h a n d q u a d ra n t. R e fle c tin g th is b ise c tio n p o in t b a c k in to th e lo w er le ft-h a n d q u a d ra n t in d ica te s th a t e ith e r a la rg er o r a sm all step - d ow n ra tio w ill lead to a sm aller valu e o f C x an d h en ce to a la rg e r valu e o f N = C 2/ C , .

T h e c h a r t a lso m ak es it c le a r th a t as Q 2 is in crea sed th e v a ria tio n in th e in p u t im p ed an ce w ith freq u en cy o r w ith te m p era tu re -in d u ced v a ria tio n s in L and/or th e C ’s will, in crease .

In A ddition, th e m od ified c h a r t m ay b e used to find the effect o f v a r ia tio n s o f freq u en cy o n Yjn. O n c e a so lu tio n is o b ta in e d , a 1 0 % in cre a se in th e th re e reactiv e term s co rre sp o n d s to a 1 0 % in crea se in fre q u e n cy ; h en ce th e n o rm alized resu lt is th e in p u t a d m itta n ce a t th e new freq u en cy . N o rm a lly sev eral p o in ts suffice to show the p a ttern o f b o th the rea l an d th e im ag in ary p a rt o f Yjn n e a r reso n an ce .

O n c e re a so n a b le valu es fo r L , C x, an d C 2 have b een lo ca te d , th en o n e can retu rn to E q . (9 .3 -9 ) o r E q . (9 .3 -1 0 ) an d an a ly z e th e b e h a v io r o f th e pi n e tw o rk in m o re d etail.

T h e m od ified S m ith c h a rt is w ell a d ap ted to d ea lin g w ith ad d ed co m p o n e n ts such as a c a p a c ito r in series w ith L. (T h is c a p a c ito r m igh t p ro v id e d c b lo c k in g and/or m igh t a llo w L to b e c o n stru c te d in a m o re co n v en ie n t size. B o th o f th ese uses are im p o rta n t a t h igh freq u en cies.) I t is a lso p erfec tly c a p a b le o f h a n d lin g a tw o -sectio n filter. S u ch a filte r w ill h av e five p o les in its tra n sfe r im p ed an ce and , in a d d itio n to th e p o les, w ill h av e fo u r co m p le x zero s in its in p u t im p ed an ce. In th is ca se it is even m o re im p o rta n t th a t o n e o p e ra te c lo se to th e h ig h est-freq u en cy p o le o r zero so as to av o id p o les a t the h a rm o n ic freq u en cies. B e ca u se th e c h a r t is a re la tiv e ly fast w ay o f o b ta in in g d a ta , o n e ca n use it to c h e ck th e v a ria tio n s in Z in vs. a> to m a k e sure th a t th e c ircu it is o p e ra tin g n e a r the u p p erm o st p o le o r z e ro .f

P o le -Z e ro P lo ts

A n o th e r w ay to g ain so m e in sig h t in to th e p assb an d p ro p erties o f the pi n e tw o rk is to p lo t th e p o le -z ero p a tte rn s o f Z in (an d thu s the po le p a tte rn s o f Z 21) fo r several d ifferent cases. I t m u st be p o in ted o u t th a t th is m eth o d is n o t a sa tis fa c to ry w ay to m ak e an in itia l design , sin ce to m a k e th e p lo ts o n e m u st first c h o o se th e c o m p o n en t valu es ra th e r th a n b e in g a b le to w ork b a ck w a rd fro m a d esira b le p a ttern to co m p o n e n t values.

t W ith the middle C equal to twice the outside ones and with equal inductances, such a two- section filter has unloaded poles in its Z in a t the origin, a t ± j u t 0 , and a t ± j ^ / 2 o j 0 , unloaded zeros in its Z in at ±0.54oj0 and at ±/1.31w 0, and u>l = l/L C outside.

9.9 H IG H -L E V E L A M P L IT U D E M O D U L A T IO N 447

As an exam p le co n sid er th e tw o ca se s in w h ich N = 1 ( C t = C 2) an d in w hich Q 2 = 3. As we saw in the p rev io u s n u m e rica l exa m p le , th ere a re tw o d ifferent valu es o f (o0 a t w hich Z in(ja>0) is pu rely resistiv e . A t o n e freq u en cy <w0 = a>2 , w hile a t the o th e r LL>0 = 1.342o»2 . T h e first ca se c o rre sp o n d s to the 9/1 im p ed a n ce step -d ow n case , w hile the seco n d co rre sp o n d s to th e 1/1 im p ed an ce tra n sfo rm a tio n . In th e first ca se the z ero s o f Z in(p) o cc u r a t ( — 0 .1 6 6 7 + 7'0.986)co0 , w hile in th e seco n d case

they o cc u r at ( - 0 . 1 6 6 7 + ;0 .7 2 6 )a > 0 . In b o th ca ses the n eg ativ e rea l p o le o ccu rs q u ite c lo se to — 0.17coo , w hile in the co0 = co2 ca se the co m p lex p o les lie n ear ( —0 .0 8 2 ± j/1.402)a)o an d in th e seco n d case they lie n ear ( — 0 .0 8 1 5 + j\.042)a>0 .

S in ce in b o th cases the cen te r freq u en cy is a>0 , a sk etch o f the p o le -z ero p a ttern w ill in d ica te th a t n e ith er Z in(p) n o r Z 2 l(p) is to o sy m m etrica l a ro u n d o j 0 fo r e ith er the 9/1 step -d o w n o r fo r the 1/1 case.

As N is red u ced (for a given v alu e o f Q 2), th e up per co m p le x p o le and zero a p p ro a ch ea ch o th e r un til a t the valu e o f T a b le 9 . 8 - 4 th ere is o n ly a single freq u en cy at w hich Z in ca n be real. F o r Q 2 = 3 th e m in im u m v alu e o f N is 0.6. F o r th is valu e the co m p le x zero s lie a t (0 .1 6 6 7 + 7'0.8975)g)o , w hile the real p o le is n ear —0 .1275coo and the co m p le x p o les are n ear ( — 0 .1 0 2 5 ± 7'1.1375)a>0 . In th is ca se the p assb an d is even m o re n o n sy m m e trica l th a n in th e N = 1 case .

9.9 H IG H -LE V E L A M PL IT U D E M O D U LA TIO N

S in g le -sid eb an d an d su p p ressed ca rr ie r o p e ra tio n s a re n o rm a lly d o n e a t low signal levels an d th en am p lified up to th e d esired fin al p ow er levels in C la ss B “ l in e a r”

R F am p lifiers.W h ile n o rm a l am p litu d e m o d u la tio n c a n be d o n e a t lo w er p o w er levels, it is

u su ally a cco m p lish e d as c lo se to th e fin a l s tage as p ossib le . W h a t we sh a ll try to d o in th is sec tio n is e x p la in th e id eal g o a l o f a ll h ig h -lev el m o d u la tin g c ircu its an d how o n e go es a b o u t a ch iev in g th is goal. W e sh a ll n o t be ab le to p ro v id e sp ecific d eta ils a b o u t p a rticu la r d esigns. I t is h op ed th a t, if th e read er kn o w s w hat he is try in g to do, it w ill be e a sie r fo r h im to d o it p ro p erly . '

B efo re o n e tr ie s to m o d u la te th e s ig n a l, it is w ise to co m p a re the o v era ll e ffic ien cies o f C la ss A an d C la s s B R F stag es w hen e a ch is am p lify in g a 100 % s in u so id a lly m o d u lated signal. T h e resu lts a re as fo llo w s:

3 1 Vc c — VSM

8 2 Vcc

3 n Vc c - Fsa,

4 4 Vc c

for C la ss A ,

fo r C la ss B.

(9.9-1)

W ith th e sam e c a rr ie r b u t n o m o d u la tio n , th e e ffic ien cies d ro p to

4 4 8 P O W E R A M PLIFIER S 9 .9

T h u s fo r n o rm a l v a ria b le am p litu d e m o d u la tio n th e effic ien cy o f th e C la ss B a m p lifier lies b etw een 5 0 % an d 7 5 % o f its p eak effic ien cy , w hile th e C la ss A stage ach iev es o n ly 25 % to 37 .5 % o f its p e a k effic ien cy . F o r e ith er no m o d u la tio n o r fu ll m o d u la tio n , the C la ss B e ffic ien cy is n tim es th e C la ss A efficiency .

A lo g ica l first step to w ard an id eal h igh -lev el m o d u la to r w ould b e to try to ra ise the av erage e ffic ien cy o f th e C la ss B am p lifie r fro m the 3 5 - 5 5 % ra n g e found here b a ck to th e 7 5 % p o ssib le fo r u n m o d u la ted signals.

F ig u re 9 .9 -1 illu s tra te s a c irc u it th a t a llo w s us to h a v e lin ea r en v e lo p e am p lific a tio n a t a c o n s ta n t th e o re tica l effic ien cy o f n e a rly tc/4. In th is c irc u it C BP is presum ed to b e a sh o rt c ircu it w ith resp ect to R F freq u en cies and an o p en c ircu it w ith resp ect to the “ m o d u la tin g ” sig nal / (f).

(z — [ 1 +/(0] - f t + cos ‘•to* + ■ • •

g(t)=m[vcc~ & , ]

Fig. 9 .9-1 O utput-m odulated Class B amplifier for a m odulated signal.

F ig u re 9 .9 - 2 sh o w s th e o p e ra tin g p a th s fo r valu es o f / (f) o f - 0 . 9 , 0 . 0 , an d + 0 .9 fo r a d ev ice w ith a lin ea r sa tu ra tio n c h a ra c te r is t ic th a t p asses th ro u g h th e origin . F o r th is d ev ice c h a ra c te r is t ic (w hich is a g o o d a p p ro x im a tio n fo r m an y tra n sisto rs), th e c irc u it is a lw ays o p e ra tin g as a fu lly d riven C la ss B a m p lif ie r ; h e n ce its effic ien cy is a lw ays

9.9 H IG H -L E V E L A M P L IT U D E M O D U L A T IO N 449

w here R sal is the slo p e o f th e d ev ice sa tu ra tio n ch a ra c te r is tic and I p is the p eak c u rren t a t 1 0 0 % m o d u la tio n . S in ce p eak a c p ow er o u tp u t fo r th is ca se is (2VCC — I pR sat)(Ip /4), the ra t io o f p eak a c p o w er o u tp u t to p eak d evice d issip a tio n b eco m es even la rg er th a n th e C la ss B resu lt o f E q . (9 .2 3) fo r u n m o d u lated carriers . W h en IpRs = Kat = (0 .0 5 )2 Vc c , then

7t V rrI 2 V rrlr, = 0 .9 5 - , P ac = 0 . 9 5 - ^ p , P dc =

4 2 n

F device = 0 .162K c c /p, an d = 2 9 5 . (9 .9 -4 )' d e v i c e

H en ce fo r a 100 W o u tp u t o n a m o d u la tio n p eak o r a 25 W u n m o d u lated o u tp u t all o n e need s is a d ev ice c a p a b le o f d issip a tin g a p eak p o w er o f 100/2.95 = 34 W . I f the p eriod o f th e lo w est-freq u en cy m o d u la tin g sig nal is su fficien tly sh o rte r th an the th e rm a l tim e c o n s ta n t o f the d ev ice so th a t o n e ca n assu m e a v erag in g o v er a m o d u la tio n cycle , th en fo r a 100 % s in u so id a lly m o d u la ted sig nal a 12.8 W d ev ice d issip atio n ca p a b ility sh o u ld b e sufficient. (T h is is in fa c t th e n o rm a l s itu a tio n .) T h u s , by “ m o d u la tin g ” the p o w er supp ly o f a n am p lifie r d riven by a m o d u la ted sig nal, we have

in crea sed th e lo n g -te rm av erag e effic ien cy b y a fa c to r o f a b o u t 2/1 an d red u ced the d ev ice ’s p o w er-h a n d lin g re q u ire m e n ts by m o re th a n 3/1.

(a) (b) (c)

Fig. 9 .9 -2 Operating paths for a modulated Class B amplifier, “variable supply.”

N o te th a t, fo r a n u n m o d u la ted in p u t to th e c h a ra c te r is tic o f F ig . 9 .9 2(b), in c rea sin g Vc c [ l + / (f)] up w ard fro m Vc c w ould n o t lead to an y v a r ia tio n in th e o u tp u t w hatsoever. R e d u c tio n o f Kc c [ l + / (f)] d ow nw ard fro m Vcc w ould lead to th e c o m b in a tio n o f e ffects d iscu ssed in S e c tio n 8.6.

A s w as exp la in ed in C h a p te r 8, w ith the d rive p ro p erly in creased , th e c o lle c to r m o d u la tio n and its a c co m p a n y in g “ re fle c t io n ” in to th e b a se c ircu it c a n lead to a lin ea r m o d u la to r.

A ll in a ll, the d es ira b le o p e ra tio n o f a tra n sisto riz ed A M tra n sm itte r w ould seem to req u ire th a t th e o u tp u t s tag e b e o f th e typ e illu stra ted in F igs. 9 .9 1 an d 9 .9 -2 . F o r h ig h -p o w er o u tp u ts o n e w ould e x ten d th is rea so n in g b a c k to in c lu d e th e d river stage. A t som e p o in t w here th e d river p o w er level is re a so n a b ly low , o n e m u st gen era te the A M e ith e r by a c o lle c to r sa tu ra tio n te ch n iq u e o r p o ssib ly b y o n e o f th e o th e r m e th o d s o u tlin ed in C h a p te r 8.

O n e fu rth er m e th o d th a t c a n b e used to im p ro v e the q u a lity o f a m o d u la tio n system is to em p lo y o n e o f th e A M d ete c to rs from C h a p te r 10 to d ete c t the final


m o d u la tio n and to ap p ly it via a n eg ativ e feed b ack c irc u it to c o r re c t th e m in o r d is to rtio n s o f th e w hole m o d u la tio n ch a in .

M od u lated V acu u m -T u b e A m p lifiers

I f a m o d u la tio n tra n s fo rm e r w ere in clu d ed in series w ith VBB in F ig . 9 .4 -1 (and if it w as a d eq u a te ly b yp assed fo r R F sig n als), th en th e effect o f a p p ly in g slow m o d u la tio n w ould b e the sam e as th e effect o f v ary in g Vb b - A ctu a lly , i f we w ished to m o d u la te th is am p lifier , we w ould first co n v ert it to th e fo rm o f F ig . 9 .8 -1 . F u rth e rm o re , we w ould c h o o se the tim e c o n sta n t o f R G an d C G so th a t the b ia s c ircu it w ould be a b le to tra c k the h ig h est m o d u la tio n frequ en cy . W ith these ch a n g es , as w ell as a red u ctio n in th e d rive sig nal and a sligh t re d u ctio n in th e load im p ed an ce, we w ould hav e a m o d u la ted C la ss C am p lifier.

I t is n o t a t a ll o b v io u s in tu itiv e ly w hy such a m o d u la to r sh o u ld b e lin ear. I t is a fact th a t re a so n a b le lin ea r ity ca n be o b ta in e d th ro u g h th e ju d ic io u s c o m b in a tio n o f th ree e f fe c ts :a) T h e fin ite rP o f the tu b e m a k es iB a fu n ctio n o f vPK.b) T h e grid cu rren t is a lso a fu n ctio n o f vPK; hence th e grid c ircu it lo ad in g , and

th ere fo re the drive v o ltag e, is a fu n ctio n o f the p la te -ca th o d e voltage.c) T h e grid c irc u it b ia s is a fu n ctio n o f I G0 an d hence o f vPK.C a lc u la tio n o f th ese e ffects is co m p lica te d by the fact th a t now th e ta n k lo ad im p ed an ce m u st b e fixed , it c a n n o lo n g e r be left free to b e d efined as V J l , . T h u s o n e is fo rced to a ssu m e a ta n k v o lta g e , a d rive v o lta g e , and a b ia s fo r a g iven c o m b in a tio n o f VBB an d m o d u la tio n an d to d erive an o p e ra tin g p ath and set o f cu rren t pulses. T h e se cu rren t p u lses m u st now yield valu es fo r l G x , / ,, and /G0 th a t a re c o n sis te n t w ith the assu m ed values. O b v io u sly w ith so m an y v a ria b les th is is n o t a s im p le p ro ced u re .

In tu itiv e ly w h at o n e w ants to h ap p en is th is : As the m o d u la tio n d ro p s in to a valley , th e b ias sh ou ld in crea se , an d th e lo a d in g should red u ce th e drive so th a t the o u tp u t sw ing falls b u t th e effic ien cy h o ld s c o n sta n t. T h e n as th e m o d u la tio n sw ings to w ard a p eak , th e drive sh o u ld in crea se , an d th e b ias sh ou ld fall so th a t a g a in the effic ien cy re m a in s re a so n a b ly c o n sta n t.

In effect, we a re try in g to re flect th e m o d u la tio n in to the grid c ircu it. T h e fact th a t th is “ re fle c t io n ” c o m b in e s w ith th e n o n lin e a rity in th e re la tio n sh ip betw een th e cu rren t p u lse size and sh ap e an d its fu n d a m en ta l valu e to yield an o v era ll lin ea r resu lt is ju s t a fo rtu ito u s c irc u m sta n ce th a t we exp lo it.

M a n y c o m m e rc ia l h ig h -p o w er m o d u la tio n s a re c la im ed to h av e p la te effic ien cies up to 8 0 % . S in ce th is is o n ly 5 % b e tte r th a n is p o ssib le w ith th e “ v a ria b le c o lle c to r v o lta g e ” C la ss B m o d u la ted a m p lifie r , th e v a cu u m tu b e h as very litt le th e o re tica l ad v a n ta g e as a h igh -lev el m o d u la to r. A ctu a l o v era ll tu b e e ffic ien cies are a lw ays low er th a n tra n s is to r e ffic ien cies, s in ce th e v acu u m tu b e req u ires a n a d d itio n a l p ow er so u rce o f 1 0 % to 2 0 % o f its a v a ila b le o u tp u t p ow er ju s t to h ea t its filam ents.

M o s t h ig h er-lev el tra n sm itte rs cu rren tly a v a ila b le em p lo y v acu u m tu b es b eca u se o f th e tu b e s ’ h igh v o lta g e ra tin g s an d h en ce th e ir a b ility to p ro d u ce h igh p ow er at re a so n a b le cu rren t levels ra th e r th a n b e ca u se o f any th e o re tica l su p erio rity in c o n v ersio n efficien cy .

PROBLEM S 451

9.1 In the circuit o f Fig. 9 .P -1 . i,(r) = (0.1 + 0.09 cos aie) A and the transistor ß is constant at 20 for 4 A > ic > 100 mA (this is unlikely in practice). Find the power delivered by the battery , the power consumed in the 3 f l resistor, and-the power dissipated in the transistor.

PROBLEMS

Figure 9 .P t

9.2 Assuming that the transistor in Problem 9.1 has a therm al resistance of 4°C /W from its junction to its case, find the tem perature rise of the junction over the case both for the signal of Problem 9.1 and for the case where the ac portion of the drive signal is removed.

9.3 Assume that the 3 Cl resistor of Problem 9.P-1 is replaced by the prim ary of a lossless transformer that reflects a 3 Si load a t the operating frequency co. Repeat Problem s 9.1 and 9.2 for this case. M ake the additional assum ption that the 12 V supply is reduced to 6 V.

9.4 Repeat Problem 9.3 assuming that the 0.09 peak am plitude sine-wave drive is replaced by a 0.09 A peak am plitude square wave.

9.5 C om pare the conversion efficiencies of the cases of Problems 9.1, 9.3, and 9.4. Explain physically the great difference in these values.

9.6 Repeat Problem 9.4 assuming that the broadband transformer-coupled load is replaced by a narrow band tuned circuit that presents the same impedance at the fundamental frequency.

9.7 In Problem 9.4 assume that the square-wave drive is not constant at 90 mA but that its am plitude varies so that the percentage of time at each amplitude is as listed in Table 9. P 1. Find the long-term average values o f collector dissipation, load power, and power supplied by the battery.

Table 9 .P-1

Percent of time Peak amplitude, mA

10 9020 8030 7020 5010 3010 10

452 POWER AMPLIFIERS

9.8 Find the “long-term average" pow er supplied to a 50 f t load by each of the following signals. (“ Long-term average” means averaged over a complete cycle o f the longest period in the modulation.)

a) A triangular m odulated 10 kH z sine wave as shown in Fig. 9.P-2(a).b) A repetitive wave as shown in Fig. 9.P-2(b).c) A 50 V rms, 5 M H z carrier tha t is frequency-modulated to an rm s deviation of 50 kH z by

G aussian noise.d) A square-wave-and-generator com bination as shown in Fig. 9.P-2(c).e) A tone burst controlled triangular signal as shown in Fig. 9.P-2(d).

1 2 0 V

50 V2 m sec

50 ft

(d)

Figure 9 .P -2

9.9 A certain device produces an ou tpu t current that flows in half-sine-wave pulses o f peak am plitude equal to 288 A and peak spacing o f 1 /¿sec. The m inim um allowable output voltage o f the device w ithout excessive distortion is 1 V. Choose values for , C2, and L (assume L is lossless) that will yield maximum power to the 72 f t load while preventing excessive distortion across C , o f Fig. 9 .P-3 w 0C 2 = 0.07 mho. Calculate the size of the second harm onic com ponent o f the voltage across both C, and across the 72-ft load. Sketch the passband for ± 1 0 % around the center frequency for transmission from the device current pulses to the 72-ft load.

9.10 Consider the circuit shown in Fig. 9.3-1 with a 120° wide pulse train wherein Ip = 2 A. If the resonant im pedance of the tuned circuit is 3 k f t and the allowable minim um device voltage is 200 V, what value m ust Vcc have? F ind the fundam ental ou tpu t power for this case. If the tuned-circuit Q is 10, calculate the m agnitude and phase o f the ou tpu t second- harm onic voltage. W hat is the conversion efficiency of this device under these conditions? W hat is the device dissipation?

PROBLEMS 453

Currentpulses

72 Q

9.11 The circuits of Fig. 9.5- 3(a) and 9.6 3(c) are both driven by a 10 kHz, 1 V peak square wave. The positive supply in each case is 20 V, while the load resistor is 10 i2 {RE = 0.25 fl and the coupling capacitor is 100 in Fig. 9.6-3c). D raw the transistor collector current waveshapes for both cases and calculate the power delivered to the lO il load. [The series L C in Fig. 9.5—3(a) is resonant at 10 kHz.] Define or specify any other param eters as necessary to complete the problem.

9.12 In the circuit of Fig. 9.6-3(a), Vc c = 20 V, p ^ 50, R L = 100 ii , R t = 25 kii, R 2 = 1 kO, and the input voltage is successively a train of 13.5 V peak sine, square, and triangular waves, each with zero average value. Calculate the load power and the transistor dissipation for each case. Make, and state, any assum ptions that are necessary in working the problem.

9.13 For the circuit of Problem 9.12, find the am plitude of each driving waveshape that results in maximum transistor dissipation. Assume a zero signal em itter current of 2 mA per transistor.


PULSE TRA IN EXPANSIONS

In gen era l it is p o ssib le to a p p ro x im a te th e p la te , grid , c o lle c to r , o r b a se c u rren t w av esh ap es th a t a re p ro d u ced in C la s s C a m p lifie rs b y o n e o f sev era l ta b u la te d w av esh ap es (o r by som e c o m b in a tio n o f th ese w aveshapes).

T o a id in such c a lc u la tio n s , th is ap p en d ix ta b u la te s th e g en era l resu lts o f I J l p vs. t/ T fo r six w avesh ap es. In a d d itio n , it p resen ts g rap h s o f I dJ I p , I J I P, an d I 2/ I p a ll versu s t/ T o r th e to ta l c o n d u ctio n an g le , 6.

F ig u re s 4 .2 -3 an d 4 .2 - 4 p resen t s im ila r d a ta fo r s in e-w av e tip curves. In p ra c tice , th e s in e-w av e tip an d th e c o s in e p u lse d a ta a re a lm o s t in te rc h a n g e a b le in th e reg io n w ftere t/ T < j .

Table 9.A -1

Pulse shape Idc h

Rectangular VT

21. . rmx — s in — - nn T

Triangular V2 T

IpT sin (nnz/lT)1 T nnx/2T

Symmetrical trapezoidal3 y4T

3¡px sin(nnx/4T) s in (innx/4T) 2 T nnx/AT 3rmx/4T

Cosine pulse 2 ‘p tn T

4 r x cos (nnx/T) n "T 1 - (2nx/T)2

Cosine-squared pulse V4 T

/ p sin (nnx/2T) nn 1 — (nx/2T)2

T h e co s in e p u lse is n o t a s in e-w av e tip (e x ce p t w hen t/T = j ) , a lth o u g h it c lo se ly a p p ro x im a te s o n e ; i = l p c o s (nt/z) i f |i| < t , i = 0 o therw ise. T h e co sin e-sq u a red p u lse is eq u a l to I p c o s 2 (nt/z) w hen |t| < t/2 an d z e ro o th erw ise. A t t = ± t/ 4 , th is pu lse h a s a v alu e o f J p/2 w hile th e c o s in e p u lse h a s a v alu e o f I J ^ J 2 . T a b le 9 .A -1 an d F ig . 9 .A -1 sh o w th e w av esh ap es an d th e exp ressio n s fo r 7dc an d /„. F ig u re s 9 .A - 2 ,9 .A - 3 , and 9 .A - 4 sh o w the n o rm a liz ed fu n d a m en ta l, d c , an d se co n d -h a rm o n ic co m p o n e n ts vs. th e co n d u ctio n angle.

454

PU L SE T R A IN EX PA N SIO N S

Square

Trapezoid

Cosine

Triangle

Cosinesquared

0 30" 60" 90° 120° 150° 180° Degrees

0.25 0.50 i /TFig. '.'.A 2 / , / / p vs. t/T or total conduction angle for repetitive pulse trains.

_ n _ Square

■Trapezoid -Cosine • T riangle

" Cosine squared

0 30° 60° 90° 120° 150° 180" Degrees0.25 0.50 x/T

Fig. 9.A 3 I j J I vs. x/ T or total conduction angle for repetitive pulse trains.

P O W E R A M PL IFIE R S

0.25 0.50 i /T

Fig. 9.A 4 / 2/ / p vs. i / T o r total conduction angle in degrees for repetitive pulse trains.

C H A P T E R 10

AM PLITUDE DEMODULATORS

In th is c h a p te r we co n sid er th e th e o re tica l lim ita tio n s on th e am p litu d e d em o d u

la tio n o f s ignals, as w ell as the d eta ils o f a n u m b er o f p ra c tica l c ircu its to a cco m p lish th is d em o d u latio n .

T h e m o st w idely used p ra c tic a l c irc u it is th e n a rro w b a n d en v elo p e d ete c to r o f S e c tio n s 10.3 and 10.4. T h o u g h sim p le to c o n stru c t, th is c ircu it h as d efin ite lim ita tio n s th a t m ay be c ircu m v en ted by th e use o f sy n ch ro n o u s o r av erage d etec to rs . In o th e r sec tio n s o f th e c h a p te r we e x a m in e th ese c irc u its in som e detail.

10.1 A M PL IT U D E D EM O D U LA TIO N T E C H N IQ U E S

In th is sec tio n we co n sid er th e b a s ic th e o re tica l p rin c ip les inv olv ed in th e d em o d u la tio n o f A M an d S S B signals. In th e su b seq u en t sec tio n s o f th is c h a p te r we shall co n sid er the p ra c tica l c irc u its by w hich th ese p rin c ip le s a re im p lem ented . In g en eral, th ere a re o n ly th re e b a s ic m eth o d s o f am p litu d e d em o d u la tio n o r d e te c tio n : (a)

sy n ch ro n o u s d ete c tio n , (b) av erage en v elo p e d e te c tio n , an d (c) p eak en v elo p e d ete c tion . T h e b a s ic id ea b eh in d a ll o f th ese m eth o d s is to re c la im th e m o d u la tio n in fo rm a tio n g(r) fro m th e m o d u la ted c a rr ie r w h ich h as th e fo rm

w here g(t) is th e H ilb e rt tra n sfo rm o f g (i) (see C h a p te r 8). A s we sh a ll see , sy n ch ro n o u s d ete c tio n m u st a lw ay s b e em p lo y ed to d em o d u la te S S B o r su p p ressed ca rr ie r A M , w hile a n y o f th e th re e d em o d u la tio n m eth o d s m ay b e em p lo yed to d em o d u la te a n o rm a l A M sig nal w hose m o d u la tio n in d ex d o es n o t exceed unity .

Synchron ous D etectio n

T h e b lo c k d ia g ra m o f th e sy n ch ro n o u s d e te c to r is show n in F ig . 1 0 .1 -1 . F o r th is

d e te c to r , if t¡¡(t) h a s th e fo rm given b y E q . (1 0 .1 -1 ) , th en w ith th e a id o f th e id en tities

U;(f) = - y - c o s œ 0t + - y sin œ 0t, (S S B ) ( 10.1- 1)

co s x c o s y = j [ c o s (x + y) + c o s (x - y)]

and

sin x c o s y = K sin ( * + y) + sin (x ~ y)] 4 5 7

4 5 8 A M P L IT U D E D E M O D U L A T O R S 10.1

Multiplier constant : K

MO o vjt)

B cos aiqf (Reference signal)

v„U)

\ « L \

| H M

Fig. 10.1-1 Block diagram of synchronous detector,

the m u ltip lier o u tp u t vm c a n b e w ritten as

g (0 , g (0v j t ) = K B

vm(t) = K B

2 + ^ c o s 2 w 0i

g (0 , g(0 » , , g (t) • » t—— I— — c o s 2 a)r,t ± — sin 2a)0t

4 4 4

(n o rm a l o r su p p ressed ca rr ie r A M )

(S S B ) (1 0 .1 -2 )

I f the lo w -p ass filte r rem ov es th e co m p o n e n ts o f vm(t) w hich are co n ce n tra te d a b o u t th e rad ian freq u en cy 2to0 , th en th e d e te c to r o u tp u t ta k e s the form

(1 0 .1 -3 )

w here I = 2 fo r n o rm a l A M , X = 4 fo r S S B , an d hL(t) is the im p u lse resp o n se o f the lo w -p ass filter. I f th e filter h as a w ide en o u g h b an d w id th to p ass g(t) u n d isto rted , th en E q . (1 0 .1 -3 ) sim p lifies to th e d esired form

v M = (1 0 .1 -4 )

w here H L(ju>) is th e F o u r ie r tra n sfo rm o f hL(t). N o te th a t i f the re feren ce sig nal has the fo rm B co s (<x>0t + 9), th en v0(t) is given by (for n o rm a l A M )

v0(t) =K B g (t )H L(0) c o s 0

w hich is a iv a tte n u a te d v ersio n o f E q . (10.1^4). H en ce fo r m a x im u m d e te c to r o u tp u t, 6 m u st b £ d esigned to eq u a l zero.

F o r sy n ch ro n o u s d ete c tio n to be a cco m p lish e d it is a p p a ren t th a t the sp ectra l co m p o n e n ts o f vm(t) in th e v ic in ity o f 2a>0 m u st n o t o v erlap the sp e ctra l co m p o n e n ts o f vm(t) in the v ic in ity o f the orig in . F o r n o rm a l o r sup pressed ca rr ie r A M th e F o u r ie r

10.1 A M P L IT U D E D E M O D U L A T IO N T E C H N IQ U E S 459

tra n sfo rm Vm(a>) o f vm(t) is g iven by

K B G ico) K BVm(a>) = ------+ ~ [ G ( a > - 2co0) + G(co + 2co0) ] , (1 0 .1 -5 )

w here G(co) is th e F o u r ie r tra n sfo rm o f g(t). A p lo t o f | Vm{w)\ vs. a> for th e ca se w here

g(t) is b a n d -lim ite d in freq u en cy to o)m is show n in F ig . 1 0 .1 -2 . In o rd er to sep arate the desired o u tp u t s ig n a l fro m the d o u b le freq u en cy term , th e in eq u ality

aim < c o 0

m u st be s a tis f ie d ; i.e., th e m ax im u m m o d u la tio n ra te m ust be less th a n th e ca rrie r frequ ency . I f a>0 is n o t g re a te r th a n com, th en sy n ch ro n o u s d ete c tio n (or, fo r that m a tte r , any o th e r fo rm o f d e te c tio n ) is im p o ssib le . In a d d itio n , th e c lo se r com is to

a»0 , the m o re co m p le x th e lo w -p ass filte r o f the sy n ch ro n o u s d ete c to r m ust be to e x tra c t the d esired o u tp u t signal.

F o r an S S B sig n a l w ith g(t) b a n d lim ited to a>m, | Vm{o>i vs. co has th e fo rm show n in F ig . 1 0 .1 -3 . H e re we see th a t fo r lo w er-sid eb an d S S B , sy n ch ro n o u s d ete c tio n is p o ssib le o n ly i f a>m < a>0 , w h ereas for u p p er-sid eb an d S S B sy n ch ro n o u s d etec tio n c a n b e a cco m p lish e d o n ly i f com < 2co0 .

T o im p lem en t th e sy n ch ro n o u s d ete c to r , a m u ltip lier an d a re feren ce-sig n al so u rce o f th e fo rm B c o s co0t a re req u ired . A ny o f the m u ltip lier c irc u its d iscu ssed in C h a p te r 7 o r 8 is su fficien t a s a m u ltip lier . A fter a ll, sy n ch ro n o u s d e te c tio n is essen tia lly m ix in g d ow n to dc. T h e re feren ce-sig n a l so u rce a t co0 m ay b e o b ta in ed b y p la c in g v,{t) th ro u g h a v ery n a rro w b a n d filter w hich e x tra c ts th e c a rr ie r c o m p o n e n t o f v,{t) (if it ex ists ) an d rem o v es th e s id eb an d in fo rm a tio n . S p ecifica lly , if

i\{t) = /4[1 + mf(t)\ c o s co0t,

w here th e F o u r ie r tra n sfo rm V^to) o f v jt) an d th e sp ectru m F(a>) o f / ( ( ) a re as show n in F ig . 1 0 .1 -4 , th e n th e o u tp u t o f th e n a rro w b a n d filter cen tered a t a>0 w ith a b a n d w idth less th a n 2 u>L is given by

A c o s m 0t,

w h ich is the d esired re feren ce signal. S u ch an ex trem ely n a rro w b a n d filter is usually im p lem en ted w ith a c ry sta l o r a m e ch a n ica l filte r o r w ith a p h a se -lo ck e d lo o p (P L L ). I f a cry sta l filte r is em p lo y ed , th e cen te r freq u en cy co0 o f t!,(i) m u st a lso b e crysta l

460 A M P L IT U D E D E M O D U L A T O R S 10.1

Upper-sideband SSB

1 1—cu„, 2 alp

(a)

CÜ —►

|K («)|Lower-sideband SSB

\ i-------------1oJm 2cl>o 2u>o <JJ —^

(b)

Fig. 10 .1-3 Plot of | Vm(w)\ vs. a) for S SB modulation with g(t) band-limited to wm.

|F(w)| |K(W)I C arrier

I— 2 c jl

1 r ~ i n r iU JL u ) „ , O f — OlQ — CÜ0 OJb + KJm UJ—» -

Fig. 10 .1-4 Plot o f |F(a>)| and | V oi)\ vs. w.

co n tro lle d . O n th e o th e r h a n d , i f a P L L is em p lo y ed , a>0 n eed n o t b e (b u t u su ally is) c ry s ta l c o n tro lle d , s in ce th e p h a se -lo ck e d lo o p is c a p a b le o f tra c k in g slow v a ria tio n s in co0 w hile s till a c tin g a s a n a rro w b a n d filte r a b o u t co0 .

In th e c a se o f S S B an d su p p ressed c a rr ie r A M , a ca rr ie r co m p o n e n t d o es n o t u su ally e x ist in u/t)- F o r S S B s ig n a ls a low -lev el o r p ilo t ca rr ie r is u su ally ad ded to th e tra n sm itte d s ig n a l an d e x tra c te d a s th e re feren ce sig n a l a t th e rece iv er w ith a n a rro w b a n d f i l te r .t H o w ev er, w hen th e S S B m o d u la tio n is a n a u d io s ig n a l, th e re feren ce s ig n al m ay b e o b ta in e d fro m a n in d ep en d en t o sc illa to r . T h is o sc illa to r g e n era tes a s ig n a l o f th e fo rm B c o s [ajnt + 0 (f)], w here 0(f) is a s lo w ly v a ry in g ra n d o m

f In many voice communication systems a large number o f voice channels (BW < 3 kHz) are SSB-modulated and frequency-division-multiplexed at 4 kHz intervals from dc to 4n kHz, where n is the total number o f channels. This composite signal plus a 4 kHz reference signal then frequency-modulates a high-frequency carrier. At the receiver, after frequency demodulation the various SSB signals must be synchronously demodulated. The approximate reference signal for each SSB channel is generated by extracting the 4 kHz reference signal and then distorting it and extracting the desired harmonic by appropriate filtering.

10.1 AMPLITUDE DEMODULATION TECHNIQUES 461

p h ase (o r freq u en cy ) e rro r b etw een th e o s c illa to r a t th e tra n sm itte r an d th e o sc illa to r a t th e receiver. W ith th e in d ep en d en t re feren ce s ig n a l, the o u tp u t o f th e sy n ch ro n o u s d e te c to r is given by

v„(t) - ~ [ g ( t ) CO S 0(f) ± g(f) sin 0(f)]. (1 0 .1 -6 )

S in ce it h as b een ' o b serv ed th a t the h u m a n e a r c a n n o t read ily d istin g u ish betw een g(f) and g(f), th e o u tp u t s ig n al o f th e d e te c to r is a p erfectly a c ce p ta b le re p ro d u ctio n o f g(f) even w hen 0(f) d rifts th ro u g h n/2 . I f 0(f) = et + 0 , th a t is, i f th e re feren ce o sc illa to r d iffers in freq u en cy fro m th e tra n sm itte r o sc illa to r , the e ffect o n th e e a r is to ra ise o r low er the p itch o f the d em o d u la ted sig n a l w hile m a in ta in in g its in te llig ib ility . T h e eye, h ow ev er, is n o t a s fo rg iv in g as the e a r ; h en ce an in d ep en d en t reference o s c illa to r c a n n o t b e used fo r d em o d u la tin g S S B v id eo in fo rm a tio n .

F o r supp ressed c a rr ie r A M th e re feren ce sig n a l ca n b e gen erated b y p assing v,{t) th ro u g h a n e tw o rk o f th e fo rm show n in F ig . 1 0 .1 -5 . T h e o u tp u t o f th e sq u are- law d evice, w hich m ay b e im p lem en ted w ith a field effect tra n s isto r , is given by

v2(t) = k g 2(t) c o s 2 (o0t

k g 2{t) , k g 2(t) ,, ,= ------------1------ — c o s 2a )0f. (1 0 .1 -7 )

2a OJ0

(Reference signal)

Fig. 10.1-5 Network for extracting reference carrier from suppressed-carrier AM.

S in ce g 2(f) is a lw ays g re a te r th a n zero , its av erag e value, u n like th e av erag e valu e o f g(f), is g re a te r th a n zero and thu s v2(t) h as a c a rr ie r freq u en cy co m p o n e n t a t 2u>0 w hich is in tu rn e x tra c te d b y the n a rro w b a n d filter cen tered a t 2 a>0 . T h is filte r m ay a lso b e im p lem en ted as a cry sta l filter o r as a p h a se -lo ck e d lo o p . T h e filte r o u tp u t is th en co u n ted d o w n in freq u en cy b y 2 a n d filtered to p ro v id e th e d esired referen ce s ig n a l B co s co0t. T h e co e ffic ie n t B is, o f co u rse , a fu n ctio n o f th e in d iv id u al dev ice

an d filte r sca le fa c to r s w ith in the n e tw o rk o f F ig . 1 0 .1 -5 .

A verage Envelope D e te ctio n

T h e b lo c k d ia g ra m o f th e av erag e en v e lo p e d ete c to r is show n in F ig . 1 0 .1 -6 . I f i ;,( t) is a normal AM input of the form

»i(t) = g (i)co sw 0i,

462 A M P L IT U D E D EM O D U LA TO R S 10.1

vAD Low-Half-wave pass

rectifier filterHL(jw)

l« ,l

A h l{o)

Fig. 10 .1 -6 Block diagram of average envelope detector.

j L

w here g(i) > 0 , an d i f th e tra n sfe r c h a ra c te r is t ic o f th e half-w av e re c tifie r is g iven by

vt > 0,Vi,

0 , Vj < 0 ,

as show n in F ig . 1 0 .1 -6 , then

v jt ) = g (f)(co s (O0t)S(t),

( 10.1- 8)

(1 0 .1 -9 )

w here S (t) is a sw itch in g fu n ctio n w ith th e p ro p erty S(t) = 1 fo r co s a>0t > 0 and S(t) = 0 fo r c o s co0t < 0. A sk e tch o f v,{t), va(t), and S (t) is show n in F ig . 10 .1 -7 .

I t is in tu itiv e ly o b v io u s fro m th e sk e tch th a t by e x tra c tin g th e av erage v alu e o f v jt ) w ith the lo w -p ass filter, a n o u tp u t s ig n a l p ro p o rtio n a l to g(t) is o b ta in ed . T o d em o n stra te rig o ro u sly th a t v j t ) is p ro p o r tio n a l to g(r), we e xp a n d S (t) in its F o u rie r series

1 2 2S(t) = — I— c o s a>0t ---------- c o s 3cot +

2 n i n( 10. 1- 10)

and su b stitu te it in to E q . (1 0 .1 -9 ) to o b ta in

M , , 2 g(r) 2 2g(i)v a ( 0 = C O S W o 1 "I C O S w 0 1 -------- ^ -----2 n i n

c o s 3a>0t c o s w 0t

— + ^ c o s (o0t + h ig h er h a rm o n ic A M signals.71 2

(10.1- 11)

I f the lo w -p ass filte r (cf. F ig . 1 0 .1 -6 ) rem ov es th e freq u en cy co m p o n e n ts o f va(t) cen tered a b o u t to0 , 2co0 , e tc ., th en v jt ) is g iven by

p i t )v„(t) = — * M O ,

n( 10.1- 12)

w here hL(t) is th e im p u lse resp o n se o f th e lo w -p ass filter. If, in a d d itio n , th e lo w -p ass filter h a s a w ide en o u g h b an d w id th to pass g(t) u n d istorted , then E q . (1 0 .1 -1 2 )


V ,(0 g ( l ) COSCJ0 t

/ 1 \

\ I \ 1 y ^ \ s• t

vAO

' Average value :g(t)/x

SU) I

"i a n n n n2nU)0

Fig. 10.1-7 Plot of iJ,(r), v j t ) and S(t) vs. t.

sim p lifies to th e d esired fo rm

g(t)vB(t) =

n(1 0 .1 -1 3 )

w here ag a in H L(j(o) is th e F o u r ie r tra n sfo rm o f h, (t).I f we p lo t th e m ag n itu d e o f the F o u r ie r tra n sfo rm Va(<x>) o f va(t) fo r the case w here

g(i) is b a n d -lim ited to a>m a s sh o w n in F ig . 1 0 .1 -8 , we o b serv e th a t th e in eq u ality

com <w n

(1 0 .1 -1 4 )

m u st b e satisfied if th e lo w -p ass filte r is to b e ca p a b le o f e x tra c tin g g(t) from v jt ) .

\VA<*>)\

I G(ut) | 11

|G(w cun) / 4

t1 \ r \ / — t _W„, Cil0-tLl„, 0% U o to i , 2(1*) w

Fig. 10.1-8 Plot of | VJo))\ vs. to.


I f th e h a lf-w av e re c tifie r o f F ig . 1 0 .1 -6 is rep la ced by a fu ll-w av e re c tifie r , n o t o n ly d o es th e o u tp u t o f th e lo w -p ass filte r d o u b le , b u t n o term cen tered a b o u t co0 ex ists in va(t). C o n se q u e n tly g (i) m a y b e e x tra c te d fro m v jt ) , p rov id ed th a t <»m < eo0 , w hich is th e sam e freq u en cy c o n s tra in t im p o sed b y th e sy n ch ro n o u s d etecto r.

I t is a p p a ren t fro m F ig . 1 0 .1 -7 th a t th e o u tp u t sig nal v j t ) m u st a lw ays b e p o s itiv e ,f i.e., th e a v era g e en v e lo p e d e te c to r e x tra c ts th e p o sitiv e env elo p e . I f g (i) c o s co0 t w ere rep laced by — g(i) c o s m 0t = g(r) c o s (co0t + n ), the o u tp u t w ould b e u n ch an ged , s in ce th e av erag e en v e lo p e d ete c to r is c le a r ly in sen sitiv e to th e p h ase o f th e c a r r ie r ; h en ce th e o u tp u t o f th e a v era g e en v e lo p e d e te c to r is w ritten m o re p recise ly as

t;0(i) = — H l(0). (1 0 .1 -1 5 )7t

T h u s i f a su p p ressed c a rr ie r w ave fo r w h ich g(f) h as b o th p o sitiv e an d n eg ativ e valu es w ere d em o d u lated b y a n en v elo p e d ete c to r , g ro ss d is to rtio n w ould a p p e a r a t th e o u tp u t. S p ecifica lly , th e o u tp u t w ould a p p e a r as i f the m o d u la tio n had b een passed th ro u g h a fu ll-w ave re c tifie r as illu s tra te d in F ig . 1 0 .1 -9 . S im ila rly , i f th e m o d u la tio n in d ex o f a .n o rm a l A M sig n al exceed s un ity , th e av erag e en v elo p e d e te c to r in tro d u ces d is to rtio n and sy n ch ro n o u s d ete c tio n m u st b e em p loyed .

B y w riting the e x p re ssio n fo r th e S S B sig nal given by E q . (1 0 .1 -1 ) in th e eq u iv a len t form

v,{t) = is/g2(t) + g2(tj co s

we o b serv e th a t the o u tp u t o f an av erag e en v elo p e d ete c to r w ith an S S B in p u t w ould ta k e th e form

v„(t) = ¿ v V W + g2(t)HL(0), (1 0 .1 -1 7 )

t If the half-wave rectifier characteristics are reversed, that is, if va = for v, < 0 f i d va = 0 for V, > 0, then v jt) must always be negative.

u)0t + ta ni| ( i )

g(f)(1 0 .1 -1 6 )

10.1 A M P L IT U D E D E M O D U LA TIO N T E C H N IQ U E S 465

w hich is a lso a h igh ly d is to rted o u tp u t signal. H ere again we see th a t sy n ch ro n o u s d ete c tio n is requ ired .

H ow ever, i f a la rg e c a rr ie r co m p o n e n t A co s co0t is added to the tra n sm itted S S B sig n al, the resu lta n t sig nal m ay b e d em o d u lated by en v elo p e d etec tio n . S p ecifica lly , if

= V/(t) + A co s co0t

A + CO S a ) n ig(i) .— sin a>0t 2 °

A -L. g ( t ) 1 -L. ^4~ cos

co0t + tan "g(t)

2 A + g(t)_

is d em o d u lated by an en v elo p e d ete c to r , the o u tp u t is p ro p o rtio n a l to

(1 0 .1 -1 8 )

A + g(t) g 2(t) ^ A + g(t)

' 4 % ' 2

provid ed th a t A » |g(f)/2| an d A » |g(t)/2|. S u ch a signal is rarely tra n sm itte d , since m o st o f th e tra n sm itte r pow er m u st be exp en d ed in p ro v id in g a large ca rrier.

T h e b asic co n ce p t, how ever, p ro v id es the b a sis fo r v estigial s id eb an d tran sm issio n .

M o st video sig n als hav e th e p ro p erty th a t th e ir freq u en cy sp ectra are c o n ce n tra ted at low freq u en cies, as sh o w n in F ig . 1 0 .1 -1 0 . C o n seq u en tly , w hen a c a rr ie r a t oj0 is a m p litu d e-m o d u la ted by a v id eo sig n al, the resu ltan t signal v ,(t) has a ban d w id th o f 2 co2 . T h is b an d w id th m ay be red u ced co n sid era b ly to W j + a>2 by p lacin g the A M signal th ro u g h a filter H (ja>) o f the fo rm show n in F ig . 1 0 .1 -1 1 , w hich rem ov es the low -level p o rtio n o f th e low er- (o r u p p er-) s id eb an d in fo rm a tio n and d o u b les the low -level p o rtio n o f the u p p er- (o r low er-) s id eb an d in fo rm a tio n to p ro d u ce a v estig ial s id eb an d sig nal v2(t). T h e b a s ic p ro p e rty o f th e resu ltan t v estig ia l s id eb an d signal is th a t it req u ires less b an d w id th fo r its tra n sm issio n and still is ca p a b le o f bein g d em o d u lated by a n en v elo p e d etec to r.

|FM|\

k.r \

— Olj ÜJ, ÜJ, U>2 (Jj—

11 M l

-w, w,IF2M I

OJ -►

f 1 \OJ2 -w, w, ÜJ2 CÜ —

F(oj) F x (ui) + F2(oj)

Fig. 10.1-10 Frequency spectrum of typical video signal f(t) .

4 6 6 A M P L IT U D E D EM O D U L A T O R S 10.1

I h M li C arrier

% - w2 / ^0 \ tU0 + w2

Fig. 10.1-11 C onstruction of a vestigial sideband signal.

T o d em o n stra te th is fa c t, we w rite th e v id eo sig n a l / (f) in th e fo rm

m=m+Mt), do.i-19)

w here / ¡(f) c o rre sp o n d s to th e h igh -lev el, lo w -freq u en cy p o rtio n o f / (f ) and f 2(t) c o rre sp o n d s to th e low -lev el, h ig h -freq u en cy p o rtio n o f / (f). T h e resu lta n t A M signal th ere fo re ta k e s th e fo rm

v t(t) = A [l -(- m fiit) + m f2{t)] co s a>0f, (1 0 .1 -2 0 )

w here A is the c a rr ie r a m p litu d e an d m is the m o d u la tio n index. T h e v estig ia l sid eb an d sig nal (w hich c o n sis ts o f n o rm a lly m o d u la ted an d f 2 s in g le-s id eb a n d - m o d u la ted an d in crea sed b y a fa c to r o f 2 re la tiv e to / ,) is g iven by

f 2(f) = A[1 + m f j t ) + m f2(t)] co s co0t — A m f2(t) sin co0t. (1 0 .1 -2 1 )

S in ce |/2(f)|2 « 1, t>2(f) is eq u iv a le n t to a n S S B sig n a l w ith a la rg e ad ded c a rrie r , and thu s h a s an en v elo p e o f th e form

+ m f j t ) + m f2(t)] = A [ 1 + w /(f)], (1 0 .1 -2 2 )

w hich is id en tica l w ith the en v elo p e o f the n o rm a l A M signal u^t).In p ra ctice , th e am p litu d e o f th e u p p er-sid eb an d in fo rm a tio n is n o t d o u b led a t

th e tra n s m itte r ; ra th e r , the h ig h -freq u en cy p o rtio n o f / ( f ) is e n h a n ced b y a fa c to r o f 2 by the rece iv er a fte r th e s ig n al h a s b een d em o d u lated . T h is n o t o n ly s im p lifies the b an d p ass filte r a t th e tra n sm itte r , w hich rem ov es th e lo w er-sid eb an d in fo rm a tio n , b u t a lso p erm its fa ith fu l en v e lo p e d em o d u la tio n o f th e v estig ia l s id eb a n d s ig n al w ith th e w eak er re q u ire m e n t th a t |/2(i)/2|2 « 1.


P e a k Envelope D etectio n

A n id eal p eak en v elo p e d ete c to r is a d ev ice w hich sam p les the p eak o f ea ch p ositiv e (o r neg ativ e) c a rr ie r cy c le and h o ld s th e p eak v alu e un til the n e x t ca rr ie r cy cle occu rs. F ig u re 1 0 .1 -1 2 i llu s tra te s a typ ical set o f in p u t and o u tp u t w av eform s fo r an ideal p eak en v elo p e d ete c to r . I t is a p p a ren t fro m F ig . 1 0 .1 -1 2 th a t a c o n sid era b le a m o u n t o f rip p le a p p e a rs o n th e o u tp u t s ig n a l v0(t) u n less th e c a rr ie r freq u en cy g rea tly exceed s th e m ax im u m freq u en cy co m p o n e n t com o f g(f). C o n seq u en tly , u n less su b seq u en t filterin g is em p lo y ed , the use o f th e en v elo p e d e te c to r is re stric te d to s itu a tio n s w here a very w ide se p a ra tio n e x ists b etw een a>m an d co0 . H ow ever, w hen a w ide sep a ra tio n betw een com an d co0 ex ists , it is a p p a ren t th a t v0(t) c lo se ly a p p ro a ch e s g(i) fo r g(t) > 0.

Ideal peak envelope detector ’

v„(0—o

Fig. 10 .1-12 Ideal peak envelope detector.

M o st p ra c tic a l p ea k en v e lo p e d e te c to rs em p lo y a d io d e to d rive the h o ld in g n etw o rk (usu ally a re s is to r in p a ra lle l w ith a c a p a c ito r ) to th e p e a k valu e o f ea ch c a rr ie r cycle , as sh o w n in F ig . 1 0 .1 -1 3 . O n c e i>0(t) h a s reach ed th e p ea k v alu e o f u,(r), th e d io d e b e co m e s rev erse b iased an d v jt ) d eca y s s low ly to w ard z e ro w ith a tim e c o n s ta n t z = R C u n til, n e a r th e p e a k o f th e cy c le , vt = v0, w hich a g a in tu rn s th e d io d e o n an d b rin g s v0(t) to th e p e a k v alu e o f u^r). T h e re s is to r R in the h o ld in g n etw o rk o b v io u sly h as th e e ffect o f in cre a sin g th e r ip p le ; h ow ev er, it is req u ired in m o st p ra c tica l d e te c to rs to en su re th a t v0{t) d ecay s m o re rap id ly d u rin g every h o ld in g

p erio d th a n th e en v e lo p e o f U;(t). I f th e d eca y in v0(t) is in su fficien t, th e d io d e d o es n o t tu rn o n a t th e p e a k o f every cy c le o f t>;(f), an d “ fa ilu re -to -fo llo w ” d is to rtio n resu lts. C le a rly th e tim e c o n s ta n t t m u st b e c h o se n to m eet a co m p ro m ise b etw een ripp le and. “ fa ilu re -to -fo llo w ” d is to rtio n .

I t is a p p a ren t th a t th e p e a k en v e lo p e d ete c to r , lik e th e av erag e en v e lo p e d ete c to r , p m d u ces a n o u tp u t p ro p o r tio n a l to |g(t)| w h ich resu lts in d is to rtio n i f g(t) is n o t a lw ays p o s itiv e ; h en ce su p p ressed c a rr ie r A M d em o d u la tio n and S S B d em o d u la tio n are

4 6 8 A M P L IT U D E D E M O D U L A T O R S 10.2

Fig. 10 .1 -13 Practical peak en velope detector.

im p o ssib le w ith th e p e a k en v e lo p e d ete c to r. O n the o th e r h an d , th e p eak en v elo p e d ete c to r n icely d em o d u la te s n o rm a l A M sig n als, as w ell as v estig ia l s id eb an d signals. S in ce e x a c t e x p ressio n s fo r v j t ) w ill b e o b ta in e d in S e c tio n s 10.3 and 10.5 fo r sp ecific p ra c tica l p eak d ete c to rs , n o fu rth er g e n era l a n a ly sis is a ttem p ted a t th is po in t.

10.2 PRACTICAL AVERAGE EN V ELO PE D ETECTO RS

In th is sec tio n we c o n sid er so m e o f th e p ro b le m s en co u n te re d in im p lem en tin g c ircu its w h ich a ch iev e a v era g e en v e lo p e d ete c tio n . U n fo rtu n a te ly , th e id eal d io d e d o es n o t ex ist in n a tu re ; th e re fo re , the g re a te s t p ro b le m in d esig n in g a n av erag e en v elop e d e te c to r is th e sy n th esis o f th e h a lf-w av e re c tifie r p o rtio n w ith p h y sica l d iod es. W e first show the e ffects o n th e d e te c to r o f em p lo y in g a p h y sica l d io d e in p la ce o f a n ideal d io d e in the h a lf-w av e re c tifie r c ir c u i t ; th en we e x p lo re a m eth o d b y w hich a p h y sica l d io d e ca n b e m ad e to fu n ctio n as a n id ea l d io d e in th e d e te c to r c ircu it. T h is m eth o d u tilizes feed b a ck to rem ov e th e n o n lin e a ritie s crea ted b y th e n o n id e a l d io d e fro m th e half-w av e re c tifie r ch a ra c te r is tic .

T o o b serv e th e b a s ic p ro b le m w h ich arises w hen p h y sica l d io d es a re em p loyed in p la ce o f id eal d io d es, we c o n sid er th e d e te c to r show n in F ig . 1 0 .2 -1 . I f th e d io d e D

is id eal, then

1D = 'r f o r Vi > °

10.2 P R A C T IC A L A V ER A G E EN V ELO PE D ET EC TO R S 4 6 9

Half-wave rectifier Low-pass filter

Fig. 10.2-1 Im plem entation o f average envelope detector.

andiD = 0 fo r v, < 0 ;

h ence iD is a n e x a c t half-w av e rectified v ersio n o f v JR . T h is c u rren t, m u ltip lied by a, is th en p laced th ro u g h a lo w -p ass filter (R 0 in p a ra lle l w ith C 0) to e x tra c t its av erage value. C o n seq u en tly , i f t>,(r) = g(f) c o s wt, then

v0(t) = zL(t), (1 0 .2 -1 )nR

w here z ,(t ) is th e resp o n se o f o0(f) w hen aiD(t) = S(t) (a unit im pulse). I f th e filter is w ide enou g h to p ass g(t), th en , o f co u rse ,

V„U) = CC~ g ( 0 - (1 0 .2 -2 )n R

E q u a tio n s (1 0 .2 -1 ) an d (1 0 .2 -2 ) a re ju s t the resu lts o f the p rev io u s sec tio n applied to the c ircu it o f F ig . 1 0 .2 -1 .

H ow ever, if th e d io d e is n o t id eal, th e exp ressio n s fo r iD and , in tu rn , v„ are m od ified . In p a rticu la r , i f th e d io d e is m od eled by the re la tio n sh ip

iD = I s(e‘>v° lkT - 1) * I seqVDlkT, (1 0 .2 -3 )

w hich is a re a so n a b le a p p ro x im a tio n for m o st sem ico n d u cto r d io d es, then iD m ay be exp ressed in term s o f v¡(t) by th e e q u a tio n

k T iv¡ = iDR H In -p . (1 0 .2 -4 )

q h

E q u a tio n (1 0 .2 -4 ) m ay be a lso w ritten in th e n o rm alized form

z = y + In y - In W, (1 0 .2 5)

w here z = qv¡/kT, y = qiDR /k T , an d W = q I sR /k T . A p lo t o f y vs. z w ith — In VKas a p a ra m e ter is show n in F ig . 1 0 .2 -2 a lo n g w ith th e co rre sp o n d in g p lo t fo r a n ideal d iod e. A s — In W in creases , the e n tire z-y c h a ra c te r is tic sh ifts lin early to the right. H ere we see th a t th e p h y sica l d io d e d o es n o t tu rn on at v¡ = 0 (o r z = 0), b u t ra th er req u ires a m o re p o sitiv e v alu e o f v¡ b e fo re an y sig n ifican t d iod e cu rren t flow s. T h e effect o f this tu rn -o n req u irem en t is to p revent sm all inp u t sig n als from p ro d u cin g

4 70 A M P L IT U D E D EM O D U L A T O R S 10.2

a n o u tp u t. T h u s su ch a d e te c to r c a n n o t p o ssib ly d em o d u la te 1 0 0 % m o d u lated s ig n a ls w ith o u t d is to rtio n .

T h e tu rn -o n re q u ire m e n t b e co m e s m o re o b v io u s if th e a c tu a l half-w av e rec tifie r c h a ra c te r is tic is m o d eled b y th e tw o -seg m en t p iecew ise-lin ear c h a ra c te r is t ic show n in F ig . 1 0 .2 -2 an d given by

y =z + In W — 5,

0,

z > - I n W + 5,

z < — In W + 5,( 10.2-6)

In W = 5 In ^ = 2 0

z = y + l t \ y —\n W z=qVi/kT y=qi„R/kT

W = q lsR/kT [ ] Values in volts

if kT/q=26 mV

( z - 5 + ln fV, z > 5 -T n W z < 5 —In W

0 20 40 60 80 100 120 140 160 180 200 220 240 260[0.52] [2.6] [5.2]

Fig. 10.2-2 P lot of y [i0/?] vs. z [u j with —In W as a param eter,

o r e q u iv a le n tly , by

- v0Vi > V0 ,

Vi < Vn ,

z

[v,]

(10.2-7)

w h e re

1 0 . 2 PR A C T IC A L A V ER A G E EN V ELO PE D ET E C T O R S 4 7 1

It sh ou ld be a p p a ren t from E q . (1 0 .2 7) th a t c o n stru c tin g the p iecew ise-lin ear c h a ra c te r is tic is e q u iv a len t to rep la cin g th e p hy sica l d io d e in the c ircu it show n in Fig . 1 0 .2 -1 by a n ideal d io d e in series w ith a b a tte ry o f valu e K0 , as show n in F ig . 1 0 .2 -3 . F ro m th e eq u iv a le n t c ircu it it is o b v io u s th a t, unless i ,(f) exceed s V0 , n o o u tp u t exists.

W e shou ld n o te th a t th e tu rn -o n v o ltag e V0 is a fu n ctio n o f k T /q , R , and l s . F o r exam p le, w ith a g erm a n iu m d io d e fo r w hich /s % 2 x 1 0 ~ 7 A, R = \ k f i , and k T /q = 26 m V ,

- l n W = 4 .8 6 and V0 = 2 5 6 m V .

O n th e o th e r h an d , w ith a s ilico n d io d e fo r w hich Js x 2 x 1 0 “ 12 A , R = I kf2, and k T /q = 2 6 m V ,

- l n ( P y ) = 16.4 an d V0 = 5 5 6 m V .

It is a p p a ren t th at th e larger the valu es o f R and I s , the low er the valu e o f V0 and the sm aller th e d ep a rtu re o f th e p h y sica l half-w av e rectifier c h a ra c ter is tic . C learly , th en , a g erm an iu m d io d e w ith a larg e valu e o f R p rov id es th e b est a p p ro x im a tio n to the ideal half-w av e rectifier.

T o o b serv e the effect o f the tu rn -o n v o ltag e V0 on the o u tp u t o f the c ircu it o f F ig . 1 0 .2 -3 , c o n sid er i>,(i) to be eq u a l to Vl co s cot. T h e d io d e cu rren t th erefo re has the form o f a p e rio d ic tra in o f sin e-w av e tip s o f p eak value I DP = (Vi - V0)/R and co n d u ctio n an g le 2</> — 2 c o s - 1 {V0/V y). C o n se q u e n tly the av erage v alu e o f iD, w hich is ex tra c te d by the o u tp u t filter, ta k e s the fo rm (cf. the ap p en d ix to C h a p te r 4)

472 A M P L IT U D E D EM O D U LA TO R S 1 0 .2

w h ere*

f ( - = 1 \ k * n _ r

A p lo t o f F ( Vt IV 0) a p p e a rs in F ig . 10.2—4, fro m w hich it is o b serv ed th a t, fo r K, > 4K 0 , F (V i/V 0) m ay be a p p ro x im a te d by its a sy m p to tic valu e

» I k ,T h u s, fo r V, > 4 V 0 ,

Kil° izR

r o 2 R

(1 0 .2 -9 )

( 10.2- 10)

and , if the o u tp u t filte r o f th e a v era g e en v e lo p e d e te c to r show n in F ig . 1 0 .2 -3 rem ov es sig nal co m p o n e n ts a t a> an d its h a rm o n ics w hile b e in g w ide en o u g h to pass the v a ria tio n s in Vt , th en va is g iv en by

v0 = ctiDR 0 =a J^ R o *V 0R 0

n R 2 R(1 0 .2 -1 1 )

^ 3

g :

(ft) J jlV\

y,Yo1.0 0

1.5 0.088

2.0 0.218

2.5 0.3623.0 0.510

5.0 1.122

10.0 2.700

Fig. 10.2-4 Plot of F(VJV0) vs. VJV0 .

10.2 PRACTICAL AVERAGE ENVELOPE DETECTORS 4 7 3

I f V, = g(t), then

(102_12)n R 2 R

T h u s th e effect o f the tu rn -o n v o lta g e is to su b tra c t a c o n sta n t value, <xV0 R 0/2 R in th is ca se , fro m th e o u tp u t, p ro v id ed th a t g(r) > 4 V 0 . I f g(r) d ro p s b e lo w 4 V 0 . o u tp u t d is to rtio n o ccu rs , s in ce iD an d v jt ) a re n o n lin e a rly re la ted to g(i), as ca n be seen from

Fig . 1 0 .2 -4 .T h e re s tr ic tio n g(t) > 4 V 0 fo r a ll t severely lim its the m ax im u m m o d u la tio n index

o f the A M sig nal th a t m ay be lin early d em o d u lated by the d e te c to r o f F ig . 10.2 3.

C o n sid er, fo r exa m p le ,

g(t) = F , [ l + m f(t)}

w h ere | / ( f ) |max = 1 a n d th u s m is th e m o d u la tio n index . F o r th is case , g(r) > 4K0is equivalent to the condition

4 VV. > ------5 - , (1 0 .2 -1 3 )

1 — m

C o n seq u en tly , if K0 = 2 5 0 m V and m = 0 .8 , I/, m u st be g rea ter th an 5 V to ach iev e lin ea r d em o d u la tio n . As a p ra c tica l m a tter, fo r v o ice d em o d u la tio n , the d is to rtio n w ould b e w ith in to le ra b le lim its w ith K, as low as 1.5 V.

A co m p le te c irc u it w hich h as as its m o d el the c ircu it show n in F ig . 1 0 .2 -3 is illu stra ted in Fig . 10.2 5. In th is c ircu it th e v o ltag e supply Vc c sup plies the b ias to keep the c o lle c to r-b a s e ju n c t io n o f th e tra n s is to r rev erse b iased , w hile C is a co u p lin g c a p a c ito r w hich iso la tes t’,(f) fro m th e b ia s supply. T h e e m itte r-b a se ju n c t io n a c ts as the half-w ave rectifier, w hile the d io d e D p ro v id es a re tu rn p ath fo r th e c a p a c ito r c u rren t and thu s p rev en ts c la m p in g ; i.e., i f th e av erage valu e o f vt(t) is zero , vc m ust

ch arg e to Vc c to k eep iD = iE, w hich is n ecessary to keep th e av erage c a p a c ito r cu rren t eq u al to zero . W ith vc = Vc c , w ith th e e m itte r cu rren t d escrib ed by the exp ressio n

iE = I ESe ^ BlkT, (1 0 .2 -1 4 )

4 7 4 A M P L IT U D E D EM O D U LA TO R S 10.2

and w ith the a ssu m p tio n th a t C is a sh o rt to vh the e m itte r c u rren t m ay be re la ted to r ,(i) by th e exp ress io n

k T iv,{t) = R iE + — l n - ^ , (1 0 .2 -1 5 )

q I es

w hich is the sam e re la tio n sh ip as th a t o b ta in e d from the c ircu it o f F ig . 1 0 .2 - 1 ; thus we see th a t th e e m itte r -b a se ju n c t io n fu n ctio n s e x a c tly as the d io d e in th e p rev io u s p ro b lem , w hile th e c o lle c to r cu rren t p ro v id es th e cu rren t d riv e , a iE , fo r th e lo w -p ass filter. C o n se q u e n tly , E q s. (1 0 .2 -1 1 ) an d (1 0 .2 -1 2 ) p ro v id e exp ress io n s for th e o u tp u t o f the c ircu it o f F ig . 1 0 .2 -4 .

In ch o o sin g co m p o n e n ts fo r th e c ircu it o f F ig . 1 0 .2 -5 we u su ally c h o o se tw o id en tica l g erm an iu m tra n s is to rs an d c o n n e c t the b a se an d c o lle c to r o f o n e to g e th e r to o b ta in a d iod e. W e th en select a v alu e o f R som ew h ere in th e v ic in ity o f 10 k i ) and d eterm in e V0 . A sm a ller valu e o f R in cre a se s V0 , w hile a la rg e r v alu e o f R req u ires i\{t) to be to o large to a ch iev e a re a so n a b le o u tp u t cu rren t. F o r an y given m o d u la tio n index th e req u ired am p litu d e Vx o f

v,{t) = K ,[ l + m f(t)] co s (at

m ay be d eterm in ed to satisfy E q . (1 0 .2 -1 3 ) .W ith the a ssu m p tio n th a t Vc c is a lso sp ecified , R 0 is ch o se n to p ro d u ce a re a so n

a b ly large o u tp u t sw ing w ith o u t ca u sin g th e tra n s is to r to sa tu ra te . T h e c a p a c ito r

C 0 is ch o se n su ch th a t 1//?0Q> = w 3 — w h ere ®m *s th e m a x im u m m o d u la tio n freq u en cy an d a>3 is th e — 3 d B b an d w id th o f th e lo w -p ass filter. F in a lly aiC is ch o se n to b e a t lea st ten tim es as g re a t as 1 /R .

E x a m p le 1 0 .2 -1 F o r th e av erag e en v e lo p e d e te c to r show n in F ig . 1 0 .2 -6 , d eterm in e valu es fo r a ll u n specified p a ra m e ters such th a t lin ea r d em o d u la tio n is achieved . A lso d eterm in e a n e x p re ssio n fo r t?0( f ).

Solution. S in ce R = 10 k Q and Is = 2 x 1 0 ~ 7 A , the tu rn -o n v o ltag e K0 h as the v alu e V0 = 197 m V ; h en ce w ith a m o d u la tio n ind ex o f 0 .6 lin ea r d em o d u la tio n req u ires Vx > 1.97 V (cf. E q . 1 0 .2 -1 3 ). W e shall c h o o se Vx = 2 .5 V. W ith th is c h o ice ,

C R

Figure 10.2-6

1 0 .2 P R A C T IC A L A V ER A G E EN V E LO PE D ET EC TO R S 475

va(t) is given by (cf. E q . 1 0 .2 -1 2 )

v0 = - (2 5 0 / x A )( l + 0 -6 c o s 10 3i ) ^ + (9 .9 f iA )R 07t

with the assumptions that the transistor does not saturate, that a = 1, and that the o u tp u t filter p asses freq u en cies up to 1 0 3 rad/sec u n d istorted . T h e m inu s sign, o f co u rse , resu lts fro m e m p lo y in g a n N P N tra n s is to r in stead o f a P N P tra n sisto r.

Since the minimum value of v0 must be greater than - 10.2 V to avoid saturating th e g erm an iu m tra n s is to r , we o b ta in

R 0 > — 10.2 V ,

o r eq u iv a len tly , R 0 < 87 k i i T o leave so m e m arg in for p a ra m e ter v a ria tio n , we c h o o se R 0 = 68 k f i (a s tan d ard 5 % valu e) an d o b ta in finally

v0 = - ( 4 . 7 V ) - (3 .2 V ) c o s 1 0 3i.

B y ch o o sin g 1//?0C 0 = 104 rad/sec o r eq u iv a len tly C 0 ss 1 5 0 0 p F , we ensu re th a t the signal in fo rm a tio n is tra n sm itte d an d th a t freq u en cy c o m p o n e n ts at th e fu n d am en tal and h a rm o n ics o f the c a rr ie r freq u en cy a re rem oved fro m the o u tp u t. At i d = 108 rad/sec, l/ 108C o = 6 .8 0 , w hich is ind eed a sh o rt c ircu it co m p a red w ith

68 k fi. I f th e im p ed a n ce o f th e o u tp u t filter a t th e ca rr ie r freq u en cy is n o t sm all co m p a red w ith R 0 , rip p le a p p ea rs in the o u tp u t. A lth ou g h th is rip p le m ay be re m oved by su b seq u en t filterin g , c a re m u st b e ta k e n to ensu re th a t it d o es n o t sa tu ra te th e tra n s is to r w hen ad d ed to vB. F in a lly , to en su re th a t the co u p lin g c a p a c ito r is an a c sh o rt c ircu it, we need C > 10/co/? = 10 p F . A gain , to a llo w fo r a m arg in o f safety , we c h o o se C = 100 p F .

T h e n o n id ea l effect o f the p h y sica l d io d e m ay be g reatly red uced by c o n stru c tin g a half-w ave rec tifie r th a t in c o rp o ra te s the d io d e in th e feed b ack lo o p o f an o p e ra tio n a l

R2

Fig. 10 .2-7 Half-wave rectifier employing a diode in the feedback loop of an operational amplifier.


a m p lifie r , a s sh o w n in F ig . 1 0 .2 -7 . T h e half-w av e re c tifica tio n is a ch iev ed w ith d iod e D j , w hich c o n d u cts fo r n eg ativ e valu es o f t;, an d o p en s fo r p o sitiv e valu es o f i\. D io d e D 2 is in c o rp o ra te d in th e c irc u it to p rev en t o p e n -lo o p o p e ra tio n an d th e p o ssib ility o f d ev ice sa tu ra tio n fo r p o s itiv e v alu es o f u, fo r w h ich D x o p en s. S in ce w ith D j and D 2 in th e c irc u it th e m a x im u m valu e o f v'„ exceed s th e m a x im u m valu e o f va by o n ly th e sm all d io d e “ o n ” v o lta g e (a p p ro x im a te ly 2 2 0 m V fo r germ an iu m and 6 5 0 m V fo r s ilico n — cf. S e c tio n 5 .4 ), th e m a x im u m v alu e o f i;| is c lo se ly a p p ro x im a te d by vammJ A , w here A is the v o lta g e a m p lifica tio n o f th e o p e ra tio n a l am p lifier. In a d d itio n , i f th e p o sitiv e an d n eg ativ e p eak valu es o f vt a re o f th e sam e o rd e r o f m ag n itu d e, th e p resen ce o f D 2 en su res th a t the m in im u m v alu e o f i>j is o f th e o rd er o f — vamJ A . C o n se q u e n tly , fo r A > 1 0 0 (w hich is tru e even fo r th e p o o re s t-q u a lity o p e ra tio n a m p lifier), a p p ea rs as a “ v ir tu a l” grou n d re la tiv e to va, an d th u s va m ay b e very c lo se ly a p p ro x im a te d by

N o tin g th a t D 2 is o p en fo r iDi > 0 , we m ay re la te iDl to v t by the exp ressio n

In d eriv in g E q . (1 0 .2 -1 7 ) , w e a ssu m e th a t th e d io d e c u rren t an d v o lta g e a re ag a in re la ted by th e a p p ro x im a tio n o f E q . (1 0 .2 -3 ) , th a t the o p e ra tio n a l am p lifie r inp u t im p ed a n ce is sev era l o rd ers o f m ag n itu d e g re a te r th a n R t o r R 2 and thu s m ay be n eg lected , an d th a t th e o p e ra tio n a l a m p lifie r o u tp u t im p ed a n ce is sm all in c o m p a riso n w ith R 2 . T h e s e a ssu m p tio n s a re u su ally tru e fo r m o st in teg ra ted o p e ra tio n a l a m p lif ie r s ; an d ev en in th o se ca ses w here they a re n o t s tric tly tru e , it c a n b e read ily show n th a t th e resu lts we d eriv e a re still valid . T h e e q u a tio n s , how ever, b e co m e m o re cu m b erso m e.

E q u a tio n (1 0 .2 -1 7 ) m ay b e w ritten in th e n o rm a liz ed fo rm

It is ap p a ren t th a t, fo r A = 0 , E q . (1 0 .2 -1 8 ) red u ces to E q . (1 0 .2 -5 ) an d n o im p ro v em e n t o v er th e p rev io u s half-w av e re c tifie r is o b ta in e d . A s A is in crea sed fro m zero , th e effect o f th e n o n lin e a r te rm in y is red u ced an d th e z-y c h a ra c te r is t ic a p p ro a ch e s th a t o f a n id ea l d iod e. A p lo t o f y vs. z fo r — In W = 2 0 is sh o w n in F ig . 1 0 .2 -8 . T h e sca le in th is fig u re is g rea tly exp an d ed fro m th e sca le o f F ig . 1 0 .2 -2 , a s c a n b e seen fro m th e z-y c h a ra c te r is t ic fo r A = 0 . F r o m F ig . 1 0 .2 -7 w e o b se rv e th a t fo r A > 9 9 th e z-y c h a ra c te r is t ic is w ith in 7 m V (u n n o rm a liz ed ) o f th e id ea l half-w av e re c tifie r c h a ra c te r is t ic (A -» oo). In a d d itio n , fo r A > 9 9 9 th e c h a ra c te r is t ic is w ith in 0 .7 m V o f th e id ea l half-w av e re c tifie r c h a ra c te r is t ic an d fo r a ll p ra c tica l p u rp o ses m ay

(1 0 .2 -1 6 )

(1 0 .2 -.1 7 )

z = y + { | ^ (ln y - In W ), (1 0 .2 -1 8 )

w here

an d W = ^

10.2 PR A C T IC A L A V ER A G E EN V ELO PE D E T E C T O R S 477

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

[260] [520] [780] [v,]

Fig. 10 .2-8 P lo t o f y vs. z w ith — In W = 20 an d with A as a param eter.

be co n sid ered to b e id eal, th a t is,

or eq u iv a len tly ,

0,

z > 0,

z < 0,

- vf .R , + R 2/( 1 + A ) R , ’

( 0 ,

y, < 0,

(1 0 .2 -1 9 )

(1 0 .2 -2 0 )

C le a rly , th en , va is re la ted to v, by th e id eal half-w av e rec tifie r c h a ra c te r is t ic given by

~ Vi^2R x ’ Vi ’ (1 0 .2 -2 1 )

0, Vi > 0.

I f h o t-c a rr ie r d io d e s a re used fo r a n d D 2 , E q . (1 0 .2 -2 1 ) p ro v id es an a c cu ra te d escrip tio n o f th e tra n sfe r c h a ra c te r is t ic o f va vs. vt fo r c a rr ie r freq u en cies up to th e ten s o f m eg ah ertz , p ro v id ed th a t a t th ese freq u en cies the o p e n -lo o p a m p lifica tio n o f th e o p e ra tio n a l am p lifie r still exceed s 100.

A co m p le te av erag e en v elo p e d e te c to r em p lo y in g tw o o p e ra tio n a l am p lifiers

is show n in F ig . 1 0 .2 -9 . I f R 3 » R 2 , w hich sh o u ld b e th e c a se to en su re th a t th e

R2

478 AMPLITUDE DEMODULATORS 10.3

cu rre n t fro m D x flow s .p rim a rily th ro u g h R 2 , th en th e first a m p lifie r p erfo rm s as an id ea l h a lf-w av e rectified , w h ile th e seco n d a m p lifie r p erfo rm s a s a lo w -p ass filter w ith a tra n sfe r fu n ctio n

H (p ) =V0 (P) r j r 3Va(p) 1 + pR*C

(10.2-22)

E q u a tio n (1 0 .2 -2 2 ) is th e tra n sfe r fu n c tio n o f a lo w -p ass filte r w ith a tra n sm issio n a t lo w freq u en cies o f R J R 3 an d a — 3 d B b an d w id th o f co3 = 1/R4C. T h e re fo re , i f V / ( t ) = g(f) c o s cot, th en

vjt) = * hU),7lK j

(1 0 .2 -2 3 )

w here h(t) = lH(p). I f co3 is la rg e in c o m p a riso n w ith th e m a x im u m freq u en cy c o m p o n e n t o f g(i), th en

. . R 2R 4 . . VoU) = ^ g ( 0 - (1 0 .2 -2 4 )

B y c o rre c tly a d ju stin g R i , R 2 , R 3 , a n d R A, an y d esired a m p lific a tio n m ay b e ach iev ed p ro v id ed th a t th e o p e ra tio n a l am p lifie rs a re n o t d riven in to sa tu ra tio n . In a d d itio n , by ca sca d in g sev era l m o re jo p e ra tio n a l am p lifie rs , a h ig h e r-o rd er , lo w -p ass filter c a n read ily b e synth esized .

10.3 NARROW BAND PEAK EN V ELO PE D ETEC TO R

O n e o f th e m o s t w idely used d em o d u la to rs fo r n o rm a l A M sig n a ls is th e n a rro w b a n d p e a k e n v e lo p e d e te c to r sh o w n in F ig . 1 0 .3 -1 . A lm o st every su p erh etero d y n e A M rece iv er h as a d e te c to r o f th is fo rm , w h ich is co n stru c te d b y p la c in g th e series c o m b in a tio n o f a d io d e D a n d a p a ra lle l R 0- C 0 c irc u it a c ro ss th e o u tp u t-tu n e d c irc u it o f th e la s t I F a m p lifie r in th e I F strip . T h e p a ra lle l R L C c irc u it d riv en by th e cu rren t so u rce i,{t) p ro v id es the m o d el fo r th e fin al I F am p lifier. T h e n a rro w b a n d p e a k

10.3 NARROWBAND PEAK ENVELOPE DETECTOR 4 7 9

envelope detector is always designed so tha t the loaded Q T of the parallel R L C circuit is high and the time constan t o f the parallel R 0-C0 circuit is long in com parison with the period of a carrier cycle. The high value of QT ensures tha t the last IF stage is sufficiently selective and also prevents the nonlinear diode circuit from distorting the waveform of The long R 0C 0 time constan t enables v0(t) to rem ain essentially constant a t the peak value of over each carrier cycle. Thus ou tpu t ripple is minimized.

In analyzing the detector of Fig. 10.3-1 we assume that the diode is ideal. This assum ption greatly simplifies the analysis and, in addition, as we shall dem onstrate, provides results tha t are directly applicable to the identical circuit em ploying a physical diode.

W e begin the analysis by obtain ing an expression for i>,(i) and va(t) for the case where

i,(f) = / 1 cos a)0t.

In particular, we show that is indeed directly proportional to / , , as is required by a linear envelope detector. We then generalize the analysis to the dynam ic case, where

= b(t) cos oj0t = /( [ I + mf(t)] cos oj0t,

and show tha t a simple equivalent circuit exists for determ ining u0(t) as a function of b(t). Finally, we consider the problem of “ failure-to-follow” distortion which results when the detector is im properly designed.

Static Analysis

If the R 0C0 tim e constant is long in com parison with the duration T = 2n/a>0 of a carrier cycle (or, equivalently, C 0 is an ac short circuit at co0), then with i;(f) = 11 cos a>0t, the ou tpu t voltage va(t) stabilizes a t a dc value va = Vdc. W ith Kdc developed across the capacitor C 0, it is apparen t from Exam ple 5.5-3 that, if the loaded Q T of the parallel RLC circuit is high, then the voltage across the tuned circuit i\{t) m ust have the form

v,{t) = Vic co s u>0t. (1 0 .3 -1 )

4 8 0 AMPLITUDE DEMODULATORS 10.3

T o determ ine ■Klc we observe tha t the diode D is reverse biased except in the im m ediate vicinity o f the peak of r,(i); hence the diode current iD m ust flow in narrow pulses occurring a t the peak of each cycle o f v,-(t), as shown in Fig. 10.3-2. Consequently, iD may be expanded in a Fourier series of the form

iD{t) = / D0(l + 2 cos co0t + •••)» (10.3-2)

Fig. 10 .3-2 Current and voltage waveforms occurring in the circuits of Fig. 10.3-1.

where I D0 is the average value of iD(t). [The narrow pulse width causes the am plitudeof the fundam ental com ponent of iD(t) to be twice the average value.] Since theaverage diode current m ust flow through R 0,

V*c = I doR o- (10-3-3)

In addition, by applying K irchhoff’s current law to the fundam ental current com ponents flowing into the parallel R L C circuit, we obtain

/ i = ^ T - 2 / D0; (10.3-4)R +

10.3 NARROWBAND PEAK ENVELOPE DETECTOR 481

a n d b y c o m b in in g E qs. (10 .3 -3 ) a n d (10 .3 -4 ), w e fina lly o b ta in th e d e s ire d ex p re ss io nfor Vdc :

Vic = I lR T , (1 0 .3 -5 )

w here R T = R[|(R0/2). W e a lso o b serv e th a t the eq u iv a len t lin ea r lo ad in g on the tuned c ircu it p ro d u ced by the n o n lin e a r d io d e c ircu it is

Kic Ro

an d th a t Q T is given by

21 r

R yu)0 L

(1 0 .3 -6 )

= R t w 0C. (1 0 .3 -7 )

It is q u ite a p p a ren t fro m E q . (10 .3 -5 ) th a t K c is indeed lin early re lated to I j and , th ere fo re , th a t th e c ircu it o f F ig . 10.3-1 fu n ctio n s , a t least s ta tica lly , as an en v elo p e d etec to r. T o u n d erstan d th e o p e ra tio n o f th e c ircu it still b e tte r , we d eterm in e the fo rm o f iD(t). I f we assu m e th a t th e d io d e b e co m e s forw ard b iased at t , and b eco m es

reverse b iased a t t2 , w here the tim e in terv a l t0 = t2 — ?i is sh o rt in c o m p a riso n w ith T an d o ccu rs in th e v ic in ity o f th e p ea k o f y,(f), and if in a d d itio n we assu m e th a t

v0(t) rem ain s essen tia lly c o n s ta n t a t VAq d u rin g d io d e c o n d u ctio n , then for f j < t < t2

ii ~ * i , t

■ _ Kdc _ •I d — — Id •R

i t = I l U i ) + — — ■ (10.3 8)

S in ce all the n o n re a ctiv e sh u n t b ra n ch e s o f th e c ircu it o f F ig . 1 0 .3 -1 c o n ta in c o n sta n t c u rren ts fo r f , < t < t2 , the a c ra m p co m p o n e n t o f iL m u st d iv ide betw een C and C 0 ; how ever, the ra m p co m p o n e n t th ro u g h C 0 a lso flow s th ro u g h th e d iod e. T h u s the d iod e cu rren t co n sis ts o f a ram p w ith slop e — VdcC 0/ L ( C 0 + C), w hich rea ch es 0 fo r t = r2 (the d io d e b e co m e s rev erse b iased w hen its cu rren t reach es zero). A sk e tch o f iD(t) is sh o w n in F ig . 1 0 .3 -3 , w hich in d ica tes th a t i0(t) co n sis ts o f a p erio d ic tra in o f narro w tr ia n g u la r pulses.

T h e pulse w idth t0 is read ily d eterm in ed by n o tin g fro m F ig . 1 0 .3 -3 th a t the av erage value, I D0, o f i'D(f) m ay be exp ressed as

V t 2 CI = ------------------— (1 0 .3 -9 )

D0 2 L T ( C 0 + C )

t A sine wave rem ains essentially constant in the vicinity o f its peak.

while from Eq. (10.3-3) I D0 = VAJ R 0 ; hence elim inating l m from Eq. (10.3-12) yields


tg _ j 2L(C0 + C) ^ /oj0L(C0 + C) T T R 0C0 \ 7iR qC 0

C0 + C 1V n R 0a)0C 0 C V nQrs (10.3-10)

where Q TS = co0C sR 0 and C s = CC0/(C + C 0) is the series com bination of C and C 0 .In alm ost all narrow band detectors of interest, QTS is slightly greater than QT ;

hence a high value of Q T ensures a high value of QTS and, in tu rn , a narrow pulse width. F o r example, if C 0 = C, then

Qts = h ( 1 0 . 3 - h )

if C 0 » C, then

2 R

< 2rs= (2 + ^ W (10-3—12)

If Q t s — 200, then t J T = <t>/2n = 0.0399 and (j> = 14.4°, where (f> is the conduction angle. If Q TS = 100, <j) = 20.3°. F o r Q T as low as 50 the conduction angle is sufficiently small to justify the assum ptions that the am plitude of the fundam ental com ponent of iD is twice the average value and that may be approxim ated by its peak value during the interval tQ. In addition, with a conduction angle less than 30°, the small flat occurring on the peak of vt(t) due to the forward bias of the diode is hardly perceptible.

If the R 0C 0 tim e constant is not infinite, t>0 increases slightly during the time that the diode is forward biased and decreases during the time the diode is reverse biased, thus causing a small am ount of ripple on the output. It is readily shown that, even with a small am ount of ripple, Fig. 10.3-3 and Eq. (10.3-10) still provide very good approxim ations for the diode current waveform and the current pulse duration. In particular, if v0 increases during diode conduction, then the current pulse duration

Fig. 10.3-3 Sketch o f i^ t) vs. t.

1 0 . 3 NARROWBAND PEAK ENVELOPE DETECTOR 4 8 3

j slightly shorter than the dura tion given by Eq. (10.3 10); this gives even m ore validity to the previous assum ptions.

To get some idea of the size of the increase A Fin v0 during diode conduction, we observe that the net charge entering C 0 during a cycle must equal zero. W ith the assum ption that i 0 « T, the charge entering the capacitor is Q+ = C 0AK while the charge leaving is » Vdl.T/R0. (The approxim ation becomes exact as t 0 -> 0.) Consequently, the fractional ripple AV/Vdc appearing at the envelope detector output is closely approxim ated by

A V T T^ T = 1 T 7 ^ = - ' (10.3-13)►dc J ' o ' - 'O T 0

where r 0 = R 0C 0- For 1 % ripple, x = 1007] If the fractional ripple rem ains below0.1, all the previous results obtained provide excellent approxim ations (within 5%) to the actual results; however, with AF/Kdc = 0.1, some additional low-pass filtering must follow the detector to remove the excess ripple.

Example 10.3-1 For the circuit shown in Fig. 10.3-4, (a) determ ine the value of v0 and (b) determ ine the peak-to-peak ripple on v0.

Ideal diode

(2 mA) cos 108/

------------------ , >------------------ »----------------- 1 i----------------- • -

1 0 n H ; 50 k f i < > 10 p F ^ - l O O k n i > 10 p F "

-of

Figure 10.3-4

Solution. If we assum e QT > 10, we can replace the diode and the circuitry to its right by an equivalent 50 kQ resis to r; hence

vt{t) = (2 mA)(25 k ii) cos 108r = (50 V) cos 108r

and, in turn,

v0(t) = (50 V) + ripple.

From Eq. (10.3-14) it is apparent that the fractional ripple has the value

A V 2n 1Fdc 108 (10 pF) x (100 kfì)

= 0.0628;

or equivalently, the peak-to-peak ripple has the value A V = 3.14 V. It is also apparent that Qt = (25 kQ)/co0L = 25, which justifies the initial assum ption.


Dynamic AnalysisWe now let

i,(f) = / j [ l 4- m/(i)] cos <y0(i) = b(t) cos co0t, ( 1 0 . 3 - 1 4 )

where b(t) = /,[1 + mf(t)], and assume again tha t the loaded Q T is sufficiently high so tha t vt(t) has frequency com ponents only in the vicinity of a>0 and may therefore be written in the general form

v M = g(t) cos a)0t, ( 1 0 . 3 - 1 5 )

where g(t) is some low-frequency waveform. If we assume, in addition, tha t the ripple on v0(t) is small and tha t the diode D conducts every carrier cycle, then va{t) is given by

v0(t) = g ( f ) . ( 1 0 . 3 — 1 6 )

[N ote tha t v„(t) cannot be less than g(i), since this would require a voltage across a forward-biased ideal diode at the peak of vt{t) or a large am ount of ripple. O n the other hand, t>„(i) cannot be greater than g(f) o r the diode could not conduct every cycle; hence, with the above assum ptions, v0(t) = g(f) ] If the diode does not conduct every cycle, the result is failure-to-follow distortion, which we investigate later in this section.

Clearly, our objective is to evaluate g(t) in term s of b(t). T o achieve this objective we first observe tha t a high value of Q T ensures an even higher value of QTS and thus ensures tha t the diode current iD(t) flows in narrow pulses. Therefore, in this case, as in the static analysis, iD may be expanded in a series of the form

'd(0 = ' d o ( £ ) ( 1 + 2 cos a>0t + ■••), ( 1 0 . 3 - 1 7 )

where iDO( 0 ' s the slowly varying average value of iD(t). [The reader should convince himself tha t the expansion of Eq. ( 1 0 . 3 - 1 7 ) is a unique representation for iD(t) provided tha t iD(i) is band-lim ited to co0/2.] The slowly varying diode current flowing through the parallel R 0-C0 circuit gives rise to g(i); hence

g(t) = !DoW * ( 1 0 . 3 - 1 8 )

where z0(i) is the response of the parallel R 0-C0 circuit when driven by a unit impulse of current.

In addition, the input current /¡(t), less the fundam ental com ponent of the diode current, flowing through the parallel R L C circuit gives rise to vt(t); that is,

viU) = gW cos co0t = [b(t) cos a>0t — 2i'D0(i)cos co0i] * z(t), ( 1 0 . 3 - 1 9 )

where z(i) is the im pulse response of the parallel R L C circuit. By noting tha t z(t) is a symmetric narrow band filter with 0(coo) = 0 , and by em ploying Eq. ( 3 . 3 - 3 ) , we can simplify Eq. ( 1 0 . 3 - 1 9 ) to the form

g(t) cos w 0t = {[¿»(i) - 2 /D0(r)] * z,(t)} cos w 0t, (10 .3 -20 )


or equivalently, to the form

g(f) = 2 1d o ( 0 * 2z L(t), (10.3-21)

where : L(t) is the low-pass equivalent of z(f). The response zL(t) is the impulse response of the circuit shown in Fig. 3.1-3. The response 2zL(t) is the impulse response of a circuit with double the im pedance level of the circuit shown in Fig. 3.1 3.

Equations (10.3-18) and (10.3-21) com prise two sim ultaneous equations from which g(f) may be obtained. The m ethod for obtaining g(t) from these equations becomes apparent only when the two circuits described by these equations are draw n as in Fig. 10.3 5. C onnecting term inals a-a' to term inals b-b' yields an equivalent circuit from which g(t) may be obtained directly in term s of b(t).

'ooO) 'ooiOa b

Fig. 10.3 5 T w o circuits from w hich Eq. (10.3- 18) (right) and Eq. (10.3-21) (left) can be ob tained .

We note from the equivalent circuit of Fig. 10.3-5 that g(t) may be related to h(t) by the transfer function

G(p) R tZ T(p) = J ' ; = ' , (10.3-22)

B(p) 1 + p/(u3

where

, R o" 3 “ 2 R t(C0 4- C )’ R t ~ R 11 2 ’

and G(p) and B(p) are the Laplace transform s of g(f) and b(t) respectively. Here we see that the — 3 dB bandw idth, which m ust be chosen large enough to pass b(t), is a function of not only the parallel R 0-C0 circuit but also the parallel R L C circuit. Hence in designing a narrow band peak envelope detector one m ust consider the complete equivalent circuit of Fig. 10.3-5. However, if a>3 is wide enough to transm it the inform ation of b(t) undistorted, then

g(t) = b(t)RT = /j [1 + mf(t)]RT (10.3-23)


and the desired envelope inform ation is obtained. Clearly, Eq. (10.3-23) could have been obtained directly from Eq. (10.3-5) with the assum ption of slowly varying m odulation. For C 0 « C, Eq. (10.3-22) can be obtained by reflecting R J 2 across the parallel R L C circuit (as was done in the static case) and then determ ining the re lationship between g(f) and b{t) for the resultant parallel R L C circuit.

If additional circuitry is added to the peak detector ou tpu t for the purpose of developing an A G C voltage (Fig. 10.3-6a) o r for the purpose of ac-coupling the dem odulated signal to a subsequent stage (Fig. 10.3-6b), the previous analysis is still

»',-(/) ~ b ( t ) c o s <*>ol VVolume control

(b)

Fig. 10 .3-6 Additional circuits placed on detector output to (a) develop an A G C voltage and to (b) ac-couple the detector output.

valid provided tha t QT is high and the ou tpu t ripple is sm all; i.e., the bandpass circuit to the left of the diode may be replaced by the equivalent circuit shown in Fig. 10.3-5 (left). A com plete equivalent circuit for the envelope detector containing a coupling network (Fig. 10.3 6b) is shown in Fig. 10.3-7. In this figure the polarity o f the current source has been reversed, since the diode in the original circuit is positioned to obtain the negative envelope. Two examples now dem onstrate the usefulness of the equivalent circuits.

Example 10.3-2 For the narrow band peak envelope detector shown in Fig. 10.3-8, determ ine expressions for va(t) and vt(t). Repeat the problem for C 0 = 10/iF.


v„U)

v,(t)

b(i) cos 10’' 0

:1000pFft

(Ideal)

- O h

io jiH■R:io k ii

.Co ^ Ro " 1 0 0 0 p F < 2 0 k f i v„(0

b(t) = (2 mA)(l fco s 5 x 10 i)

Figure 10.3-8

Solution. For the circuit of Fig. 10.3-8,

(ÜQ 1 in ? a , a n ( R ^ R7= = 1 0 r a d / s e c a n d 0 T = ------------- = 50.ALC

In addition, the fractional ripple has the value

AK/Kdc = 2n/t»0R 0C 0 = 0.0314;

hence, with i' high value of QT, a small am ount of ripple, and the assum ption of no "failure-to-follow ” distortion, the detector of Fig. 10.3 8 may be replaced by the equivalent circuit shown in Fig. 10.3-9. From the equivalent circuit it is apparent that

v0(t) = (10 V) 1 H— cos I 5 x 104f — n )v/2 \ 4 I

In addition, since v„(t) is equal to the envelope of i>,(r),

1 / . . 7 t \m = d o v ) cos 107f.

We observe that, even though the input current has a m odulation index of unity, the voltage developed across the tuned circuit has a m odulation index of l / v / 2.


Figure 10.3-9

If C 0 is replaced by a 10 fiF capacitor, the capacitor in the equivalent circuit of Fig. 10.3-9 m ust be replaced by a 10 /iF + 1000 pF x 10 p F capacitor. Such a large capacitor effectively shorts the signal com ponent of ¿>(f)/2 to ground to yield v„(t) = 10 V, and in tu rn vt(t) = (10 V) cos 107t. Consequently, since all the envelope inform ation has been stripped from u,(f), the circuit functions as an am plitude limiter provided that no failure-to-follow distortion occurs.

Example 10.3-3 F o r the circuit shown in Fig. 10.3-10, m ake an accurate sketch o f v0(t), v'0(t), and ^(t).

Figure 10.3-10

Solution. Assuming that Q r is high, that the ripple is small, and that there is no failure- to-follow distortion, we replace the circuit of Fig. 10.3-10 by its equivalent circuit, shown in Fig. 10.3 -11. Since the 10 ¡iF capacitor appears as a short circuit to the ac com ponent of b(t)/2 and an open circuit to the dc com ponent of b(t)/2, we decompose b(t)/2 into its ac and dc com ponents as shown in Fig. 10.3-12 and solve for v j t ) and v'a(t) by superposition. The dc com ponent of v0(t) is equal to ( j mA) x 40 k il = 20 V ;


*(i)

400 |isec

mA

M2

b ( t )2 + m

Figure 10.3-12

the dc c o m p o n en t o f v0(t) is equal to zero. In addition , the ac co m p o n en ts of va(t) and v0(t) are equal, since the 10 ¡xF capac i to r is an effective ac short. A sketch of i''a(t), v„(t), and r,-(f) = va{t) cos 107i is show n in Fig. 10.3 13.

T he presence of the coupling capac ito r has significantly reduced the m odu la tion index of r,(/| from unity. This effect is best un d ers to o d by observing that the im pedance presented to the m o d u la t io n in fo rm ation [the ac co m ponen ts of hu)] is s ignificanth less than the im pedance presented to the carrie r [the dc co m p o n en ts o f h it

F or the circuit of Fig. 10.3-11, if h(t) = / , . then

i-oin = = (40 kcs)(/, 2)and, in turn,

r , ( / | = ( 2 0 k Q ) 7 , c o s 1 0 7 r.

4 9 0 AMPLITUDE DEMODULATORS 10 .3

Figure 103-13

Consequently, R T = 20kQ and QT = 200. In addition, the fractional ripple is W / V it. = 0.0157; hence two of our original assum ptions are justified. The m ethod for justifying the third assum ption, tha t of no “ failure-to-follow” distortion, is now considered.

Failure-to-Follow Distortion

In a narrow band peak envelope detector of the form shown in Fig. 10.3-1 or Fig.10.3-6 a form of distortion know n as failure-to-follow distortion or diagonal clipping may result if the circuit is im properly designed. Physically, this form of d istortion results when the input m odulation on i,(i) causes the envelope of vt(t) to decrease a t a rate greater than the natural decay of the parallel R 0-C0 c ircu it; hence the diode fails to conduct on each cycle and v0(t) decays tow ard a steady-state value which is independent of the input m odulation. Typical waveforms of v,{t) and v„(t) during failure- to-follow distortion are shown in Fig. 10.3-14.


Intuitively we can m ake a num ber of observations concerning failure-to-follow d istortion which are helpful in preventing its occurrence:

1. Failure-to-follow distortion always occurs on the negatively sloping envelopeof the input current.

2. Failure-to-follow distortion is m ost likely with high m odulation indices andhigh m odulation frequencies. Both of these conditions tend to increase the negative slope of the envelopes of i,(i) and in turn v\(t), and thus increase thechance that v j t ) will not be able to follow the envelope of t ,(i).

3. Failure-to-follow distortion cannot occur if the parallel R L C circuit contains no loss, that is if R -> x . If no loss is present and diode D conducts on the peak of a particular cycle of i\(f), then it must also conduct on the peak of each subsequent cycle, thus elim inating the possibility of diagonal clipping. This statem ent follows from the fact that the envelope of the lossless tuned circuit must increase or rem ain the same with D d isconnected ,t while with D open v j t ) decreases; hence conduction is ensured on the peak of every cycle. Thus, diagonal clipping results when the tuned circuit is sufficiently heavily loaded to perm it the envelopeof r,(f) to decrease m ore rapidly than v0(t) during the interval when D is back- biased. (It is apparent that diagonal clipping is impossible in the circuit of Example 10.3-3, since no loss exists in the tuned circuit.)

4. Failure-to-follow distortion cannot occur in the circuit of Fig. 10.3-1 if 2RC > R 0C 0. Assume that for an arb itrary / = i, the diode conducts and r,(i,) = vjt,-) = g(f;). If the diode now opens, v j t ) decays exponentially to zero with an initial slope equal to — g(tJ/R0C 0. In addition, if it is equal to zero, the envelope of t',(i) decays exponentially to zero with an initial slope equal to — g(i;)/2KC (cf. Section 3.3). If i, / 0, energy is supplied to the tuned circuit and the initial slope of the envelope of i>,(f) is greater than g(f,)/2R C ; hence, if

g(f.) < g(tj)IR C - R 0C 0'

or equivalent!v.2RC > R 0C 0, (10.3 24;

the envelope of c,(f) decays less rapidly than v j t ) and the diode conducts at t — t,- + 7. The same argum ent may be applied to subsequent cycles; thus the diode conducts on the peak of every cycle of v,-(t) and diagonal clipping is irupossible.It is apparent at this point that, when diagonal clipping occurs, the diode current

and in turn its average value drop to zero for a large num ber of carrier cycles; consequently, a sufficient condition to ensure that failure-to-follow distortion cannot occur is that the average diode current iD0 exceed zero for all time, that is,

iD0 > 0 (10.3-25)

t So long as h{t) cos <unt supplies energy to the lossless tuned circuit, its voltage amplitude increases ; if b(t) = 0, no additional energy is supplied and the voltage amplitude remains constant.


for all t. Thus by obtaining an expression for iD0 from the equivalent envelope detector circuits of Fig. 10.3-5 or Fig. 10.3-7 and requiring that it satisfy Eq. (10.3-25), we obtain an inequality involving the param eters of the envelope detector which ensures that diagonal clipping cannot occur. Therefore, in addition to solving for v0(t), we m ust solve for iD0{t) and check to see that it exceeds zero. If it does, the model being used is valid. Two examples should clarify the use of this method.

Example 10.3-4 F o r the narrow band peak envelope detector shown in Fig. 10.3-15, determ ine the relationship between m, R, C, R 0, and C0 which ensures no diagonal clipping for all values of com.

i, =b(t ) cosbjQ/

b{t) /,(! + m cos uimt)

Figure 10.3-15

Solution. Assuming that Q T is high and that the output ripple is small, we replace the envelope detector by its equivalent circuit, shown in Fig. 10.3-16. For the equivalent circuit we obtain

'o o (f) = I i R^ ( ojJ m i )2 + 1 ^

1 + m v7 f . = cos (wmt + <t>)2)2 + 1R 0 + 2 R

where

1 1 j - 1 01 m - (<y,“ l = and ^ = t a n ---------tan ~RoC0 (C0 + C )(2r? |j ftjj to

Clearly, fo ri:D0(t) to be greater than zero for all t, we require

s /^m / iO l)2 + T< 1, (10.3-26)

since cos(a>mi + </>) reaches a value of — 1.F or to i > w2 the coefficient of m in Eq. (10.3 26) has its maxim um value of unity

for vjm = 0 ; hence for to, > w 2, or equivalently for

R 0C 0 < 2RC,

no diagonal clipping occurs for any value of com provided that m < 1 (a condition which is always satisfied). This, of course, is an expected result. O n the o ther hand,


for a), < co2 , the coefficient of m in Eq. (10.3 -26) has its m axim um value of a)2/<oi = C 0(R0 + 2R)/2R(C0 + C) a t o»m -♦ oo ; hence for c«! < to 2 no diagonal clipping occurs for any value of com provided that

m <2R(C0 + C)

C o ( R 0 + 2 R )

(10.3-27)

For example, if 2R = R 0 and C 0 = 3C, then the maximum m odulation index which may be used w ithout failure-to-follow distortion is m = §. Also, for the case where the detector is used as a lim iter and C0 -> oo,

m <2 R

R 0 + 2 R(10.3-28)

(Note that diagonal clipping does occur in the second part o f Example 10.3 2.)

Example 10.3-5 W ith the assum ption tha t Cc-is an ac short circuit at the frequency com, determ ine the m axim um value of m odulation index m which does not cause diagonal clipping in the envelope detector of Fig. 10.3-17.

Solution. Again assum ing a high QT and a low output ripple, we can replace the envelope detector by its equivalent circuit, shown in Fig. 10.3-18. For this circuit,W O is 8iven by

• , * l i m l i'Do(f) = 6 + 4 COS (Umt

i i t ) b i t ) cosou(„'( J C

it { I) / ,( I i m cos cu,„f)

Figure 10.3 17

4 9 4 AMPLITUDE DEMODULATORS 10 .3

Cr

(we assume that Cc is an ac short circuit). Hence, for iDO(0 to be greater than zero for all i,

m < f .

If the size of the ou tpu t resistor or the resistor loading the tuned circuit is increased, the value of m that the circuit can accom m odate increases.

Summary and Design

In the previous analyses a num ber of constrain ts on the interrelationship of p arameters were required for the correct operation of the peak envelope detector shown in Fig. 10.3-1. These constraints are sum m arized as follows:

I) 1 « QT = oj0R t C = QtoQ i

Q to + Q t u

2)R 0C 0

T T

3) ü)m <a>3 =

» 1,

12 r t(c + c 0y

4) R 0C0 < 2RC.

(selectivity)

(low ripple)

(bandw idth sufficient to pass m odulation inform ation)

(no failure-to-follow distortion for any m)

Here QTfJ = co0R C is the unloaded Qr o f the tuned circuit, QT0 = co0C R 0/ 2, and wm is the m axim um m odulation frequency. F o r circuits of the type shown in Fig.10.3-6, only condition 4 in the above list m ust be modified slightly. To observe how these constraints may be em ployed in the design of an envelope detector, we first rewrite condition 4 in the form

R 0C0 t 0 2RCai0 ^ Q t u

T ~ 2n h(10.3-29)

To satisfy condition 2 and minimize the ripple, we wish x J T to be as large as possib le;

10.3 NARROWBAND PEAK ENVELOPE DETECTOR 4 9 5

however, Eq. (10.3-29) limits its value to QTu/TC'-> hence we choose

^ = 0 ™ , (10.3-30)T 71

which is equivalent to 2RC = R0C0.W ith the value of t 0 given by Eq. (10.3-30), condition 3 may be rearranged in the

form

— < ( 1 0 . 3 - 3 1 )(o0 ¿{¿tu

F or proper operation we desire a small am ount of ripple and thus a large value of Qtu ; however, from Eq. (10.3-31) we see tha t the maxim um size of QTV is lim ited by the ratio of com/co0 (broadband signals require low-(?r circuits). Consequently the peak envelope detector may be employed only when a>0 exceeds com, usually by a factor o f 100 or more. U nder these conditions we choose QTU as large as possible within thelim itations of Eq. (10.3-31). A large value of QTV also ensures that condition 1 willbe satisfied.

C ondition 1 also requires a high value of Q T0 which we choose, once the required value of Q t is specified, from the relationship

Q t o = n Q T Q lb n ■ (10 .3 -32)sctv V£r

W ith the arbitrary specification of one desired param eter in the circuit of Fig. 10.3-1, the relationship a>0 = XjyjLC and Eqs. (10.3-30), (10.3-31), and (10.3-32) provide a set of equations for completely determ ining all o ther param eters in the envelope detector. The following example dem onstrates this fact.

Example 10.3-6 The circuit o f Fig. 10.3-1 has an ou tput resistor R 0 = 10ki2. Choose the rem aining param eters in the circuit to ensure QT = 25, m inim um ripple, no failure-to-follow distortion regardless o f the input m odulation index, and a sufficient bandw idth to accom m odate m odulation frequencies up to u>m = 106 rad/sec. The carrier frequency is cj0 = 108 rad/sec.

Solution. M inim um ripple, sufficient bandw idth, and no diagonal clipping are ensured by choosing

lo _ Qtu _ wo — 159 T n 2 n w m

Thus t 0 = R 0Co = 1 /¿sec, QTV = a>0R C = 50, and RC = 0.5 /xsec. Since R 0 = lO kfi, C 0 = 100 pF. By solving for QT0 from Eq. (10.3-32), we obtain

„ Q tQ tv _ ( 5 0 ) ( 2 5 ) g n

Q tu ~ Q t 25

496 AMPLITUDE DEMODULATORS 10 .4

But Qto = (o0R 0C/2; therefore, R 0C = 10“ 6sec ensures QT = 25. Again, with R 0 = lO k fi we obtain C = 100 pF, and with R C = 0.5 /¿sec we obtain R = 5 k i l Finally, 1/cuqC = 1 /iff = L.

10.4 PRACTICAL NARROW BAND PEAK EN V ELO PE D ETECTO R

If a physical diode is placed in the envelope detector circuit of Fig. 10.3-1, the detector no longer functions properly for all levels of the input current ¡¡(t). Specifically, unless the am plitude of (,(t) is sufficiently large to develop a voltage t’,(r) across the tuned circuit which exceeds the diode turn-on bias, then the diode rem ains reverse biased and v„ = 0. To focus atten tion on this problem we model the physical diode by an ideal diode in series with a battery of V0 as shown in Fig. 10.4-1. For silicon diodes V0 is closely approxim ated by 650 mV ; for germ anium diodes, V0 is closer to 220 mV. From Fig. 10.4-1 it is apparen t that the effect o f V0 is to keep the diode reverse biased for small values of /¡(i) or, equivalently, to induce failure-to-follow distortion for input m odulation indices close to unity.

Fig. 10.4-1 N arrow band peak envelope detector with physical diode replaced with its model.

However, if the input drive level is sufficiently high and the input m odulation index is sufficiently low so tha t failure-to-follow distortion does not occur, if Q T is high, and if the ou tp u t ripple is small, then v'0(t) follows the envelope of v,{t) and v„(t) = v'0(t) - V0. T hat is, if Ufa) — g(t) cos co0t, then v j t ) = g(t) and

v0(t) = g(t) - V0. (10.4—1)

In addition, the diode current flows in sufficiently narrow pulses so that the detector can be replaced by the equivalent circuit o f Fig. 10.4-2. W ith the aid of this circuit v’a(t) and va(t) may be determ ined as functions of b(t) and, in addition, iD0(t) may be obtained to check tha t F0 has not induced failure-to-follow distortion.

For example, if b(t) = /,[1 + m cos comt] cos w 0t, then

v’„(t) = 2RV° ■ R °R 1'2 R + R q 2 R + R 0

m cos (a>mt + 0)”|1 + —j — -- - ■ — (10.4-2)

V 1 + K . M ) 2-!

1 0 . 4

VoPRACTICAL NARROWBAND PEAK ENVELOPE DETECTOR 497

and

, fl — , R oR It 1m cos (wmt + 0)

V 1 + (wm/w 3) -2 R + Rg 2 R + R q

where 0 = —tan 1 { w j u i 3) and co3 = 1/(K0 ||2K)(C + C 0). In addition,

/ i R

(10.4-3)

ivoi11Vn

+2 R + R q 2 R + jRq

, , s j(a> jw i)2 + 1 , , ,1 + m ~ = = = cos (a>mt + 4>) (10.4-4)

where (j> = tan 1 (wm/w ,) — tan 1 (a>m/a>3) and uj1 = l /R 0C0. To ensure that no failure-to-follow distortion occurs, we require iD0(t) > 0 for all ?, or equivalently,

1 — n > my / ( < a > i )2 + i

j )2 + 1(10.4-5)

where n = V0/ I XR. If 2RC > R 0C 0 (the case which ensures no failure-to-follow distortion for V0 = 0), then co3 > a», and the coefficient of m in Eq. (10.4-5) achieves its maximum value of unity for com = 0. Thus to ensure no failure-to-follow distortion for all values of com, we require

1 n > m. (10.4 6)

F rom Eq. (10.4-6) we observe that n approaches zero as V0 is decreased, / , is increased, or R is increased. Consequently, for an envelope detector em ploying a physical diode to dem odulate AM signals with m odulation indices close to unity, we choose a germ anium diode to keep K0 as small as possible and we make the im pedance of the tuned circuit as high as possible. (In the last section we saw that we could choose one param eter of the envelope detector arbitrarily.) In addition, as a practical m atter, we choose a diode with a stray capacity much less than C or C0 . W ith this accomplished, the rem aining com ponents in the detector may be selected by the m ethod discussed in Section 10.3. Here again, if the tuned circuit contains no loss, diagonal clipping is impossible regardless of the size of V0.


10.5 BROADBAND PEAK ENVELOPE DETECTOR

An envelope detector which linearly dem odulates AM signals over a broad range of input carrier frequencies is shown in Fig. 10.5-1. F o r this circuit the diode conducts at the peak o f each carrier cycle, thus increasing the capacitor voltage v0(t) to the peak value of v,{t) less the tu rn-on voltage of the diode. As u,(f) decreases from its peak value, the diode is reverse biased and the capacitor holds the voltage until the diode again conducts a t the peak o f the next carrier cycle. The outpu t voltage v0(t), therefore, follows the envelope of vfa) shifted dow n by the turn-on voltage of the diode.

Fig. 10.5-1 Broadband peak envelope detector.

The dc current source in the netw ork serves two valuable purposes. First, it provides a discharge path for the capacitor when the diode is reverse biased, thus perm itting v0(t) to follow negatively sloping envelopes of v((t) w ithout failure-to- follow distortion. Secondly, it ensures tha t the diode turns on a t the peak of each input cycle even when the input am plitude drops below the nom inal turn-on voltage of the diode. Consequently, even with a silicon diode in the detector, the ou tpu t faithfully responds to variations in the input envelope dow n to very low levels. The slightly undesirable effect of / dc is the negative diode turn-on voltage which appears a t the ou tpu t when vt{t) = 0. This voltage can readily be com pensated for in circuits where its presence is undesirable.

To obtain a m ore quantitative expression for v0{t) in term s of i;f(i), let us consider the case where vt(t) = K, cos cot and where C is sufficiently large so tha t over a carrier cycle v0{t) rem ains constant a t Vdc. F o r this case the diode current may be w ritten as

iD(f) = l^toVilWcosrtg-qVic/kT

= j sexcixate - ‘iv<i<=ikT (10.5-1)

with an average value

= I sI 0(x )e -“y^ lkT, (10.5-2)

w h e re x = q V i/k T a n d / 0(x) is th e m o d if ie d B esse l fu n c t io n o f o rd e r ze ro . S in ce in

10.5 BROADBAND PEAK ENVELOPE DETECTOR 499

the steady state iD = / dc, we can rearrange Eq. (10.5-2) to yield

k T ]Vic = — I n / 0(x) - l n - ^

<J ¡s

k T , IqVA = 7 ° | l r ] “ v°' (10^ 3)

where in this case Vu = ln ( /dc/7s), which has a value in the range of 200 mV for germ anium diodes and 650 mV for silicon diodes. A plot of

, M /, k T I qV1f ( V i) = — ln / o 7 T q \ k T

is shown in Fig. 10.5--2 for kT/q = 26 mV. It should be observed from Fig. 10.5 2 that / ( K,) is rem arkably linear for values of V1 in excess of 50 mV and thus may be closely approxim ated by

/ (V i ) = 0.997K, - (54 mV) % V,. (10.5-4)

Vx.V

Fig. 10.5-2 Plot of /'( V\ ) vs. V, with kT/q — 26 mV.


Consequently, with /(V ,) % V,, the output voltage is given by

I'd c = Vx - V0 (10.5-5)

for values of K, as low as 50 mV regardless of whether a germ anium or silicon diode is being employed. E quation (10.5-5) m akes it obvious tha t for Vx > 50 mV the exponential diode in the circuit o f Fig. 10.5-1 may be m odeled by an ideal diode in series with a battery of value V0, as was assum ed in Section 10.4.

O n the o ther hand, if q V J k T « 26 mV, then

and we observe th a t the peak envelope detector functions as a square-law detector for Vt in the millivolt range. U nfortunately, V0 m ust be rem oved from K>c by subtraction or capacitive dc blocking before the square-law property can be fully utilized.

Output Ripple

Even if the ou tpu t capacitor is large, a certain am ount of ripple exists on va(t) because of the narrow current pulses which flow each time the diode conducts to recharge the capacitor. It is apparent from Eq. (10.5-1) that the current pulses have the waveform shown in Fig. 4.5-3 which is indeed narrow for x » 1 or Vy » 26 mV.

If we model these narrow pulses as impulses of strength 77dc, where T is the period of the input carrier, then bo th the im pulse model and the actual pulses supply the same charge to the capacitor when the diode conducts. W ith the impulse model, however, it is apparen t tha t each pulse increases the capacitor voltage by

(A slightly smaller value of A V is obtained if the actual current pulse shape is used.)As should be expected, the ripple varies directly with the current / dc and inversely

with the capacitance and carrier frequency. Unlike the ripple in the narrow band peak envelope detector, the ripple A V (Eq. 10.5-9) is independent of Kdc. This effect is a direct result of providing / dc to discharge C in lieu of a resistor.

Failure-to-Follow Distortion

(10.5-6)

and

q V iV q V \k T 4 k T

(10.5-7)

Therefore,

(10.5-8)

(10.5-9)

Now, if u,(t) = g(t) cos co0t and we assum e th a t g(t) rem ains greater than 50 mV m ost o f the time, then we may model the exponential diode of Fig. 10.5-1 by an ideal

10.5 BROADBAND PEAK ENVELOPE DETECTOR 501

diode in series with a battery of value V0. If, in addition, we assum e that the ou tput ripple is small and that the diode conducts on the peak of each carrier cycle, then theoutput voltage v0 is given by

v0(t) = g(r) - V0, (10.5-10)

which is the desired ou tput voltage.To ensure that the diode conducts on the peak of each carrier cycle it is sufficient

to require, with the diode in Fig. 10.5-1 reverse biased, that the ou tpu t voltage decay m ore rapidly than the envelope g(i). Specifically, diode conduction is ensured if

g(t) > -I dc

c(10.5-11)

for all t. If the inequality of Eq. (10.5-11) is not satisfied, failure-to-follow distortion of the form shown in Fig. (10.3-14) results.

The requirem ent o f a small am ount of ripple m eans tha t / dcT/C = I dJ f C m ust be small in com parison with the average value of g(f); on the o ther hand, the requirement of no failure-to-follow distortion necessitates I J C > - g ( i ) for all I. If g(f) = y4[l + cos ojmt], then these two conditions require that

/ dc A oj A cum < — «

C 2n

Clearly, Eq. (10.5-12) can be satisfied only if a> » con n a r ro w b a n d p eak en v e lo p e d e te c to r .

(10.5-12)

as was also the case for the

Developing / dc with Resistor Bias

If the current source in the detector of Fig. 10.5-1 is practically im plem ented by a larger resistor in series with a negative supply voltage Vcc the circuit shown in Fig.10.5-3 results. For this circuit, with i>,(i) = Vx cos u>t and with a large value of C,

lD ‘S*

Fig. 10 .5-3 Broadband peak envelope detector with resistor bias.

5 0 2 AMPLITUDE DEMODULATORS 1 0 . 5

10 = Vdc. Since in this case the average diode current [given by Eq. (10.5-2)] mustequal (Fdc + Vcc)/R , Vi{. is related to x = qVl/ k T by the equation

+ i j = M / o(x). (10.5-13)\ ycc I Vcc

For Vcc » kT/q, Eq. (10.5-13) may be closely approxim ated by (cf. footnote on page 173)

eqViclkT = M / q(x )5 (10.5-14)Vcc

from which we obtain

Kdc = l n / 0(x) - V0, (10.5-15)

where V0 = - I n (ISR/VCC). In particular, the value of Kdc obtained from Eq. (10.5-13) differs by less than 1 % from the corresponding value of Vdc obtained from Eq. (.10.5-14) if Vcc > 2.6 V (a condition which is alm ost always true in practice). Consequently, by com paring Eqs. (10.5-3) and (10.5-15) we see tha t the ou tput voltage is not affected by replacing the current source / dc by the series R-Vcc com bination.

O n the o ther hand, the ripple voltage and the condition for no failure-to-follow distortion are affected by the bias change. F or the circuit of Fig. 10.5-3 the ripple voltage AK (assuming narrow current pulses) is given by

V c c + V A TA V x I | c : (10.5-16)

the condition which ensures no failure-to-follow distortion is

g(f) > - Vcc + ^ } ~ Kj>, (10.5-17)

which m ust be satisfied for all t. The reader should convince himself of the validity of Eqs. (10.5-16) and (10.5-17).

Example 10.5-1 D eterm ine the m axim um value o f R for the circuit shown in Fig.10.5-4 which ensures no failure-to-follow distortion.

Solution. If failure-to-follow distortion occurs, it m ust occur on the negatively sloping portion of g(f) between 0 and 10 /isec. During this interval,

g(t) = (5 V)( 1 - — — I and -g (r ) = i V//isec.\ 1 0 /isec/

However, from Eq. (10.5-17), if

(15 V)[ 1 — f, , 30 usee/j V / u s e c < ----------------- - (10.5-18)2 RflOOO’pF)

1 0 .5 PROBLEMS 503

for 0 < t < 10 //sec, then no lailure-to-follow distortion occurs. Clearly, if Eq. (10.5-18) is satisfied for f = 10 /xsec, it is satisfied over the entire interval of in te rest; therefore, with t — 10 /isec Eq. (10.5-18) reduces to

R < 20 kfi.

Thus R mdx = 20 kQ.

PRO BLEM S

10.1 For the circuit shown in Fig. 10.P-1,

v2(r) - ^ p c o s 107i — ^ s i n 107r,

where g(r) = (1 V) cos 103r and v,(t) has the form shown in the figure. Find r„(f).

10.2 Modify the circuit of Fig. 8.4-2 so that it functions as a synchronous detector.

10.3 F or the circuit shown in Fig. 10.P-2, va is related to r, by the relationship va = Kvf. Find an expression for vj t ) and show that the circuit functions as a synchronous detector if B » A. Show also that with B = 0 the circuit functions as an envelope detector if m « 1.

10.4 The half-wave rectifier in Fig. 10.1-6 is replaced by a full-wave rectifier. Sketch the frequency spectrum of va for this case and show that no spectral com ponents overlap/the low-frequency spectrum so long as w 0 > tom. For this case, dem onstrate also that the output of the average detector is doubled.

504 AMPLITUDE DEMODULATORS

12 V

Figure 10.P-1

r ^ :■ 1 / H, (jut)Mt)

J V,

Low-pass filter

v2(f) Scoscuqt

Figure 10.P-2

PROBLEMS 505

10.5 The baseband signal (after frequency dem odulation) of a composite SSB frequency- dem odulated telephony signal has the form

<v(f) 1 ~Y~cos nwo[ — y ~ sin na)° + A cos w 0t,

where the individual voice channel g„(t) is band-lim ited between 200 Hz and 3 kH z and fo = 4 kHz. D raw the block diagram of a system which can be employed to extract each g„(i). Include the bandw idths and center frequencies of all filters and indicate how the reference signal for each SSB dem odulator is obtained.

10.6 For the circuit of Fig. 10.2-5, R = R 0 = 5 k il, C = 0.1 fiF, C0 = 2000 pF, Vcc = 12 V, and

v,{t) = g(t) cos 108t,

where g(t) is shown in Fig. 10.P-3. Evaluate t>0(r).

£('>50 usee

10 V

10 0 n se c

Figure 10. P 3

10.7 A "back” diode is a Zener diode which has a breakdown voltage within a few millivolts of zero. The vD — iD characteristic for such a diode is given by

iD = ¡¿e"’° lkT - 1), vB > 0,

<o = vo/r, vp < 0,

where r is o f the order of 10 i i A silicon “back” diode with r = 20 i i and Is = 1 0 ' 16 Ais placed in the circuit of Fig. 10.2-1 in which R = 1 8 0 il,a = 1 ,R 0 = 1 0 k fi,C o = 0.01 /iF,and

= V, cos 108i.

Evaluate and plot the t'0-vs.-V, transfer characteristic for 0 < Vx < 2 V. Over approximately what range of V, does the circuit function as a linear average envelope detector?

10.8 F or the circuit shown in Fig. 10.P—4,

Dj(t) = (0.25 V )(l + cos 104f)cos 107t.

Find vjt). How many decibels below the ou tpu t signal is the carrier ripple?

10.9 F o r the circuit shown in Fig. 10.2-9, R , = R 2 = 1 k ii, R 3 = 50 k il, Rt = 500 kfi,C = 200 pF, A = 2000, and

v,{t) = (1 V)(l -(- m cos 104t)cos 108i.

5 0 6 AMPLITUDE DEMODULATORS

Figure 10.P-4

Find v0(t). If A is now reduced to 9 and D, and D2 are hot carrier diodes with turn-on voltage V0 » 0.4 V, determ ine the maximum size of m for linear demodulation.

10.10 F or the circuit of Fig. 10.3-1, R = 20k fl, R 0 = 40kfi, L = 10/iH , C0 = C = 1000pF, and

/¡(f) = (10 mA) cos 107f.

Find t’,(r), v„(t), the conduction angle of iD(f), and the fractional ripple of ('„(f).

10.11 F or the circuit of Fig. 10.3-1, R = R0/2, C = C0, and i, = cos a>0t. Plot the total harmonic distortion of v,(t) vs. Qr for the values of Qr sufficiently large that iD(t) may be modeled as shown in Fig. 10.3-3.

10.12 F or each of the circuits shown in Fig. 10.P-5, let b(t) = I t = 5 mA and determine vjt), v'Jt), Vj(t), Q T, the fractional ripple, the diode conduction angle, and the T H D for ii,(i).

10.13 If b(t) has the form shown in Fig. 10.P-6, determine i\, v0, and v’„ for each of the circuits shown in Fig. 10.P- 5.

10.14 Repeat Problem 10.13 with b(t) = (5m A )(l + cos 2 x 104i).

10. i 5 W hat is the minimum value of R which may shunt L in the circuit of Fig. 10.P-5(a) without causing diagonal clipping?

10.16 W hat is the minim um value of R which may shunt L in the circuit of Fig. 10.P-5(b) without causing diagonal clipping if 6(i) = (5m A )(l + j cos 2 x 104f)?

10.17 For the circuit shown in Fig. 10.P-7, determine expressions, for r,U), i 2(i), and t-3(f) assuming that g(t) = (1 mV)(l + cos 104t).

PROBLEM S 507

v , (Q .Ideal diode

b ( t ) COS 107i© 10nH ?1000 pF

:25ki21000 pF KU)

(a)

(b)

Figure 10.P 5

b{t)

J5 mA

t

" 400 (isec 400 fisec

Figure 10.P-6

5 0 8 AMPLITUDE DEMODULATORS

Figure 10.P- 7

10.18 Repeat Problem 10.13, replacing the ideal diode with a silicon diode having V0 * 600 mV. How much resistance can be placed across the 10 /¿H inductor in this case before failure - to-follow distortion occurs?

10.19 For the idealized broadband envelope detector shown in Fig. 10.P-8, show that no failure - to-follow distortion occurs if

- & ) < — for a lii.

F or this circuit, determ ine the maximum value of m which may be used w ithout failure - to-follow distortion if

g(t) = 4(1 + racosw„i), w„ « to0.

10.20 For the circuit shown in Fig. 10.5-4, g(i) = (10 V)(l + m cosiomf) and R = lO kil. Find the maximum value of m (as a function of a>J before failure-to-follow distortion occurs.

Figure 10.P-8

C H A P T E R 11

GENERATION OF FM SIGNALS

In this chapter, after discussing the theoretical lim itations on the transm ission of FM signals through filters and nonlinear networks, we study a num ber of circuits for generating frequency-m odulated signals. This discussion starts from the direct m echanization o f the FM differential equation and proceeds to both analog and digital generation circuits.

In all cases we explore both the theoretical and the practical limits of the circuits.

11.1 FREQ U EN CY -M O D U LA TED SIG NA LS

If the instantaneous frequency or phase of a high-frequency sinusoid is varied in proportion to a desired low-frequency signal f( t ) , a constant am plitude-m odulated signal is generated which has its spectrum concentrated about the frequency of the unm odulated sinusoid. Thus a frequency- or phase-m odulated signal, like an AM signal, can be efficiently transm itted with reasonably small an tennas if the carrier frequency is sufficiently high. In addition such a signal can be frequency division multiplexed.

However, frequency- and phase-m odulated signals, unlike am plitude-m odulated signals, have the advantage that no inform ation is carried in their envelope; thus atm ospheric and receiver noise which introduces unwanted perturbations on the envelope of a received signal can be removed by lim iting the signal am plitude before the FM or PM signal is dem odulated. As a consequence, the signal-to-noise ratio at the ou tput of an FM or PM receiver is higher than that of a com parable AM receiver which receives a signal with the same average carrier power as the received FM or PM signal and which has been subjected to the same unwanted disturbances.

The price tha t m ust be paid for the increased output signal-to-noise ratio in FM or PM receivers is increased transm itted signal bandwidth. For FM signals, the ou tput signal-to-noise ra tio increases only when the ratio of the bandw idth of the FM signal to the bandw idth of f ( t ) is increased.

A typical phase-m odulated signal has the form

v(t) — A cos <t>(t) = A cos [<y0f + A</>/(f)], (11.1-1)

where <j>(t) = a>0t + A ^/(f) is the instantaneous phase, A is the carrier am plitude, oj0 is the carrier frequency, A</> is the phase deviation, a n d /( i) is a signal proportional to the m odulation inform ation with the properties

\ f ( t ) U = 1 and f ( t ) = 0.509

(1 1 .1 -2 )

510 GENERATION OF FM SIGNALS 11.1

Clearly the instahtaneous phase varies in direct proportion to / ( f ) with the constant of proportionality A(f>.

Any time-varying instantaneous phase (j>(t) has associated with it an instantaneous frequency which by definition is the derivative of the instantaneous phase ; that is,

oj,(f) =d<t>(t)

dt '(11.1-3)

If, instead of varying <f>(t) in proportion to /(f) , we vary a»,(f) in proportion to /(f) , that is, if

co,(f) = a)0 + Aco/(f),

then

, </>(r) = u>0t + Aa> J f(Q) d6 + d0

and v(t) takes the form of a typical frequency-modulated signal:

u(f) = A cos to0t + Aa> d6 -(- 6q

(11.1-4)

(11.1-5)

( 11.1-6)

where Aw, the frequency deviation, is the constant of proportionality relating /( f ) to the instantaneous frequency and 90 is an arb itrary constant phase which may be taken as zero w ithout loss of generality. A plot of v(t) given by Eq. (11.1-6) with a saw tooth signal for / ( f ) is shown in Fig. 11.1-1.

It is apparent from Eq. (11.1-3) that we can convert any PM signal into an equivalent F M signal; therefore, by developing the properties of the FM signal, we are sim ultaneously developing the properties of PM signals. Hence we shall restrict our attention to the FM signal o f the form given by Eq. (11.1-6).

Before continuing further we should note that there are three frequencies associated with an FM signal. O ne frequency is obviously the carrier frequency co0 . The

A t )

Fig. 11.1-1 Sawtooth-modulated FM signal.

11.1 FREQUENCY-MODULATED SIGNALS 511

second frequency is the frequency deviation Au>, which is a measure pf how far the instantaneous frequency departs from the carrier frequency as / ( f ) varies between plus and m inus one. The third frequency is the maxim um m odulation frequency tom [the frequency to which / ( f ) is band-limited]. This maximum m odulation frequency is a measure of how rapidly the instantaneous frequency varies about co0.

We may express these three frequencies in the form of two param eters by norm alizing oj0 and a>m to Aw with the relationships

/ ? = — and D = — , (11.1-7)

where ft is referred to as the m odulation index and D is referred to as the deviation ratio {¡] can have values in the thousands, whereas D is always required to be less than unity and in many practical systems may be 0.005 or even smaller).

Frequency Spectrum

To obtain the frequency spectrum of an FM signal we may, in principle, obtain the Fourier transform V(to) of v(t) given by Eq. (11.1-6). U nfortunately such an operation is not m athem atically tractable for an arb itrary f i t) . Therefore, we restrict our initial a ttention to the special case of obtaining the spectrum of i?(f) with /( f ) = cos tomt. Fortunately, as we shall dem onstrate, the bandw idth of v(t) obtained with /( f ) = cos o)mt provides a conservative bound for the bandw idth of v(t) which is m odulated in frequency by any /( f ) band-lim ited to mm.

W ith /( f ) = cos u)mt. Eq. (11.1-6) may be w ritten in the form

lit) = A cos (a)0t + sin w j ) .

= A cos io0t cos (/? sin comt) — A sin to0t sin (JJ sin oJmf), (11.1-8)

which is the superposition of two AM waves, the first m odulated by cos (fi sin u>mt) and the second m odulated by sin (/? sin comf)./Thus if we obtain the spectra of the two low-frequency, periodic m odulating functions, we may shift them up in frequency by to0 to obtain the spectrum of v(t).

Now cos (fi sin a)mt) and sin (/? sin comt) may both be expanded directly as Fourier series whose coefficients are ordinary Bessel functions of the first kind with argum entJ»t:

cocos (fi sin w mt) = J o(/0 + 2 £ J 2n{p)cos2nu>mt, (11.1-9)

n ~ 1

00sin (/? sin u)mt) = 2 £ J 2n+ ¡(/J) sin (2n + l)a>mt. (11.1-10)

n = 0

A plot oiJJiP) vs. p for several typical values of /? and n is given in Fig. 11.1-2.

t Equations 9.1 -42 and 9.1 43 on p. 361 o f Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, Eds., N ational Bureau of Standards, Applied M athem atics Series No. 55, G overnm ent Printing Office, W ashington, D.C. (1964). Revised by Dover, New York (1965).

5 1 2 GENERATION OF FM SIGNALS 11.1

Fig. 11.1-2 Plot of J„(0)vs. p.

If we now recom bine Eqs. (11.1-9) and (11.1-10) with Eq. (11.1-8) by m aking use of the trigonom etric identities

cos x cos y = j[cos (x + y) + cos (x — y)]

and

— sin x sin y = j[cos (x + y) — cos (x — y)],

we finally obtain

v(t) = A { J 0(P) cos w 0t + JJP ) [cos (w0 + ( o j t - cos (co0 - co„)t]

+ J 2(P) [cos (co0 -(- 2(om)t + cos (o j0 - 2a>m)t]

+ J 3(P) [cos (coq + 3 a )J t - cos (co0 - 3 w ji]

+ J 4(P) [cos ( o j0 + 4(om)t + cos (oj0 - 4a>m)t]

+ ■••}. (11.1- 11)

From Eq. (11.1-11) it becomes apparen t tha t the spectrum of a sinusoidally m odulated F M wave contains an infinite num ber of sidebands occurring at frequencies com, 2wm, 3(Dm, . . . on each side of the carrier frequency a>0. Fortunately, as can be seen from Fig. 11.1-2, for any given m odulation index ¡3, only a finite num ber of Bessel functions have values significantly different from zero and thus only the corresponding num ber of upper and lower sidebands play a significant role in determ ining the FM spectrum. F o r example, with /? = 1, J 0(l) = 0.7652, J J 1) = 0.4401, J 2(l) = 0.1149,. J 3(l) = 0.01956, J 4(l) = 0.002477, J 5(l) = 0.0002498, and J 6(l) = 0.00002094. Thus if we retain only those sidebands whose coefficient J„{ 1) is greater than 0.01 (i.e., those sidebands which are greater than 1 % of the unm odulated carrier), we observe

11.1 FREQUENCY-MODULATED SIGNALS 513

that the FM signal with p = 1 occupies a band of frequencies of 6com. A plot of themagnitude of the line spectrum of v(t) [given by Eq. (11.1-11)] with p = 1 is shown in Fig. 11.1—3(b). Also shown in the figure are sim ilar plots for p = 0.2, /? = 2.4048, P = 5, and/3 = 10. In each case only those sidebands greater than 1 % of the unm odulated carrier are retained as being significant.f As expected from Fig. 11.1-2, the num ber of significant sidebands increases with p.

1

P =0.2

1J > Cl

(a)

.............................................1

)0 u>—~~

p 1.0

1 , .

P - 2.4048

j l l l i i I i ■___________________(c) wo u -*~

P 5BW

I \—

. . i l l . ! i

Acu — 1

. I I I . ,(d) OJ0 (Jj—m-

BWP 10

. “ ”' - 1 I * " , 1 1 , 1 1 . 1. , . 1 I

1 ‘

, 1 ,Aui

1 . 1 1 , 1 I I I ,(e) % <jJ~►

Fig. 11.1 3 Plot of am plitude spectrum of an FM signal for several values of m odulation index.

F ro m Fig. 11.1-2 w e a lso o b se rv e th a t th e coeffic ien ts o f th e v a r io u s s id e b a n d s o sc illa te a s th e m o d u la t io n in d ex is in c rea sed . In p a r t ic u la r , a s p is in c re a se d from ze ro , J 0(P) a n d th u s th e c a r r ie r a m p litu d e d e c re a se fro m un ity . T h e c a r r ie r a m p litu d e re a c h e s z e ro fo r P = 2 . 4 0 4 8 ; it th e n b e c o m e s n e g a tiv e a n d re tu rn s to z e ro fo r p = 5.5200; a n d so on . T a b le 11.1-1 ta b u la te s th e v a lu e s o f P fo r w h ich J„(P) = 0 lo r n = 1, 2, 3, a n d 4.

R e tu rn in g to F ig . 1 1 . 1 - 3 , w e o b se rv e th a t w e can d e te rm in e th e b a n d w id th (B W ) o f a s in u so id a lly m o d u la te d F M c a r r ie r fo r an y m o d u la tio n in d ex p by d e te r m in in g th e n u m b e r o f B essel fu n c tio n s J n(P) w h ich exceed 0.01 a n d th e n m u ltip ly in g

t Figure 11.1 3 would remain essentially unchanged if we retained all sidebands for which JAP) 0.005 because ./„(/¡) decreases very rapidly with increasing n

Table 11.1-1 Values of p for which J„(P) = 0


n = 0 n = 1 n = 2 n = 3 n = 4

First zero 2.405 3.832 5.136 6.380 7.588Second zero 5.520 7.016 8.417 9.761 11.065Third zero 8.654 10.173 11.620 13.015 14.372Fourth zero 11.792 13.324 14.796 16.223 17.616Fifth zero 14.931 16.471 17.960 19.409 20.827

The fact that the carrier am plitude and sideband am plitude vanish for known values of /? perm its the rapid determ ination of the frequency deviation vs. m odulation am plitude for an FVI generator whose calibration is not known. Specifically, m odulation of the form V\ cos a>mt is applied to the m odulator with a known value of w m. The am plitude of V\ is then increased from zero until the carrier a t the m odulator output disappears. (The carrier is m onitored on a narrow band receiver, or on a wave or spectrum analyzer.) For this value of V, , Aw = 2.405ajm and a point on the Vt — Aco calibration curve is determined. O ther points may be determined from subsequent zero values of the carrier or other sidebands.

by 2com. We can present this bandw idth in a universal form by norm alizing BW to 2Aco and plotting it vs. j3. F o r example, with /? = 1, BW = 6cum and thus

BW = 3 ^ = 3 2 Aw Aco /?

O ther points obtained in a similar fashion are presented in Fig. 11.1-4, in which a sm ooth curve is draw n as a best fit through all the points.

F or very large values of ft the curve approaches an asym ptote of 1. This asym ptote is intuitively reasonable for any FM signal with any /(f), since for a finite frequency deviation a large value of ¡3 implies a small value of com or a very slowly vary ing /(f). Very slow variations in / (i) cause v(t) to vary alm ost statically between co0 — Aco

Fig. 11.1-4 Plot o f norm alized FM bandwidth (BW) vs. m odulation index fi = Aw/wm.

11.2 TRANSMISSION OF FM SIGNALS THROUGH NONLINEAR NETWORKS 5 1 5

and c«0 + Aw; thus a filter with a bandw idth 2 Aw centered about w0 has the m inimum bandw idth to pass the slowly varying carrier of v(t) undistorted. Consequentlythe spectrum of v(t) must have BW = 2 Aw or equivalently BW/2 Aw = 1.

As an example of the usefulness of the norm alized bandw idth curve, consider an FM signal with a m odulation frequency wm = 2n x 10 kHz and a frequency deviation of Aw = 2n x 50 kHz. For this signal , / i —. 5 and thus from Fig. 11. 14 BW/2 Aw = 1.6. Consequently a bandw idth of 2n x 160 kH z is required for this sinusoidally m odulated FM signal.

Similarly with wm = 2n x 5 kH z and Aw = 2n x 50 kHz, ft = 10, BW/2 Aw =I.4, and the bandw idth is 2n x 140 kHz. As expected, a smaller bandw idth is required for a fixed frequency deviation if the m odulation frequency is decreased. Consequently, for any arbitrary /( f ) band-lim ited to wm, the frequency com ponents o f /( f ) closest to wm have a greater effect on increasing the bandw idth of the FM signal than do those com ponents close to w = 0. Thus if we were to model /( f ) as cos wmr, a signal with ail of its energy concentrated at wm, the bandw idth of the FM signal would be greater than the corresponding bandw idth with any o th e r /( f ) whose energy is distributed in the frequency range between w = 0 and w = wm. As a result, the bandw idth curve of Fig. 11.1-4 provides a conservative bound for any FM signal whose m odulation is band-lim ited to wm. Actually, in many practical cases, the highest-frequency sine-wave approxim ation gives much too conservative an estimate of bandwidth.

For high-/j cases, where the peak deviation is high in com parison to the maximum frequency com ponent of the m odulating signal, there is a rule known as "W oodw ard’s Theorem .” t According to this rule, the shape of the envelope of the FM spectrum is approxim ately that of the am plitude probability density of the m odulating function. The scale factor of this spectrum along the frequency axis is, of course, proportional to Aw. Intuitively this means that a high-/f square-wave m odulation has two impulses in frequency at f 0 ± Af that a high-/? triangular-wave m odulation has a flat spectrum, and that a high-/? G aussian or noiselike function has a Gaussian- shaped spectrum.

Thus a com bination of a num ber of voice channels that will com bine to provide a noiselike signal produces an FM spectrum (provided that the [i is large) that concentrates its energy near the carrier rather than near the edge of the band. In such cases the actual bandw idths required of filters and other circuits are substantially smaller than might be expected from the sine-wave approxim ation.

II.2 TRANSMISSION OF FM SIGNALS THROUGH NONI.INKAR NETWORKS

In this section we shall dem onstrate that an FM signal of the form

t P. M. W oodward, “The Spectrum of Random Frequency M odulation," Telecommunications Research Establishment, G reat Malvern, Worcs., England, Memo 666, December 1952.


when passed through a nonlinear nonm em ory circuit, behaves as if it were a singlefrequency sinusoid. Specifically, we shall show that the ou tpu t v0(t) of the nonlinear circuit may be expanded in a “ Fourier-like” series of FM waves of the form

oov0(t) = a0 + £ a„ cos

n = 1

where the coefficients are exactly the ones that would be obtained if the circuit were driven by Vx co sto 0t. This “ Fourier-like” expansion m akes evident the fact tha t a signal proportional to the original F M signal with the same instantaneous frequency can be obtained by filtering the com ponents of t>0(i) which are centered in frequency about o j 0 . If the nonlinear circuit causes a x to be independent of V-t , the ou tpu t of the narrow band filter is independent of Vt and an effective FM limiter results.

O n the o ther hand, if a narrow band filter is employed to extract the com ponents of v0(t) centered in frequency about na>0, an F M signal is obtained with its instantaneous frequency proportional to / ( f ) bu t with n times the frequency deviation and n times the carrier frequency of v(t). This technique for increasing the deviation of an FM wave finds wide application when a specified frequency deviation is required and yet the maxim um deviation available from a particular m odulator produces less than the specified value. If m ultiplication of the carrier frequency along with the deviation is not desired, the carrier may be heterodyned back to the desired frequency by any of the techniques of C hapter 7 after the required deviation m ultiplication has been achieved.

nco0t + n Aco J7<»> dd (11.2- 2)

y U) o—

Nonlinearnonmemory

device

v„(0 —o

Fig. 11.2-1 Nonlinear nonmemory device characteristic.

To dem onstrate the validity o f Eq. (11.2-2), let us assum e tha t v(t) given by Eq. (11.2-1) is placed through a nonlinear nonm em ory device, as shown in Fig.11.2—1. Since v(t) is, in general, not periodic in time, v„ = F(v) is not periodic in time and a conventional Fourier series expansion of i?0(f) is not possible. However, v(t) is periodic with period T = 2n/oj0 in r ( f ) , where

T ( f ) = t + — C f m d O , (11.2-3)co0 J

since, as a function of x, v(t) = ^ ( t ) is given by

u ( i ) = ^ ( t ) = Fj c o s o ) 0 t . (11.2—4)

Therefore, v0{t) = vol(x) = F [ i, ' , ( t )] is a periodic function in x and may be expanded in a Fourier series in x of the form

oo

= ôi(t) = a0 + £ a„ cos n(o0x. (11.2-5)n = 1

If x, given by Eq. (11.2-3), is resubstituted in to Eq. (11.2-3), the Fourier-like series of Eq. (11.2-2) follows immediately.

N ote tha t the coefficients an in Eqs. (11.2-5) and (11.2-2) are the coefficients of the Fourier series of the ou tpu t of the nonlinear device when the input is driven by a cosinusoidal signal of frequency a »0 and am plitude Vt . Consequently all of the results of C hapters 1, 4, and 5 obtained for cosinusoidal inputs to nonlinear devices apply directly to FM signals.

If at the ou tp u t of the nonlinear device we wish to reclaim an FM signal p ro portional to the input FM signal, we m ust be able to separate the fundam ental FM com ponent of v„(t) from all the higher-order F M com ponents of v0(t). Figure 11.2-2

11 .2 TRANSMISSION OF FM SIGNALS THROUGH NONLINEAR NETWORKS 5 1 7

illustrates a typical frequency spectrum for v0(t) in which F0(co) is the Fourier transform of va(t). It is apparen t from the figure tha t the fundam ental FM com ponent of v0(t) may be extracted by filtering provided that

BW j BW ,- ^ + ^ < < 00, (11.2- 6)

where BW , is the to tal bandw idth of the fundam ental FM com ponent of v„(t) and BW 2 is the to tal bandw idth of the second-harm onic FM com ponent of vjt). By noting that A co2 = 2 A to, we may express Eq. (11.2-6) in the equivalent form

° = “ - RW 1 R W ~ ’ (1 L 2“ 7)o)0 BW, „ BW , + 2 -2 Aw 2 Aoj2

w here D is the d ev ia tio n ra tio o f the in p u t F M signal.


Nonsinusoidal FM Signals

We have observed that, when an FM signal is passed through a nonlinear device, the resultant signal contains the same instantaneous frequency inform ation as the original FM signal. Therefore, such a d istorted FM signal is indeed also an FM signal, since a signal of the form of Eq. (11.2-1) can be extracted from it by placing it through either a bandpass filter o r an inverse nonlinear operation.

v(i) o— K,(/) — o

v(t) V\ cos [oJoi f AouJf(Q) dff\ v„ (r) is an FM square wave

Fig. 11.2-5 Generation of an FM square wave.

For example, if an FM signal is passed through the hard limiter shown in Fig.11.2-5, an FM square wave of peak am plitude V2 results. In a similar fashion, if r(f) given by Eq. (11.2-1) is passed through a nonlinear device with the transfer characteristic given by

2V-i= sin • i v{t)

v r■Vi < v ( t ) < Vu ( 11.2-8)

an FM triangular wave of peak am plitude V2 results. O ther nonlinearities yield o ther waveforms for the FM signals.

N ote, however, tha t all the nonsinusoidal FM signals, as well as the sinusoidal FM signal, have one property in co m m o n : when they are plotted vs.

r(i) = t + — f m d o ,(D0 J

they are periodic with period 2n/a)0 [cf. the paragraph containing Eq. (11.2-5)].Figure 11.2-6 illustrates an FM triangular wave plotted vs. t(t) and vs. t. N ote

that only when t>r (0 is plotted vs. x does the waveform consist of a series of straight line segments.

The fact that a sinusoidal F M signal can be reclaim ed from nonsinusoidal FM signals suggests a widely used m ethod of FM generation. Both FM square waves and FM triangular waves are readily generated by standard digital operations. Therefore, FM forms of these waveforms are generated and then converted into a sinusoidal FM signal by filtering (for the FM square wave) o r by non-linear processing (for the FM triangular wave). F o r example, if the F M triangular wave is placed through a nonlinear device with the characteristic

v 0 - Fj sinnvT(t) ■2V, ’

11.3 TRANSMISSION OF FM SIGNALS THROUGH LINEAR FILTERS 521

*r(0 = )

(a)

(b)

Fig. 11.2-6 FM triangular wave plotted (a) vs. x and (b) vs. ¡.

a sinusoidal FM signal results. W hereas filtering to reclaim the sinusoidal FM signal places a lim it on the m odulation index and deviation ra tio tha t can be used w ithout causing undue distortion, the use of nonlinear processing to reclaim the sinusoidal FM signal imposes no such restrictions.

11.3 TRANSMISSION OF FM SIGNALS THROUGH LINEAR FILTERS

In this section we shall consider the problem of determ ining the ou tpu t of a linear filter which is driven by an FM signal o f the form

U nfortunately the solution to such a problem , unlike the corresponding problem in AM, which was studied in C hapter 3, is no t possible in closed form for an arbitrary f ( t ) o r an arb itrary filter transfer function H(p). However, two particular cases with wide application in the design of FM systems are tractable m athem atically, and we shall restrict our atten tion to them.

The first tractable situation is tha t in which / (f) varies slowly in com parison with the duration of the im pulse response of the linear filter. In this “quasi-static” case the filter ou tpu t is obtained by substituting a>,{t) = a>0 + Acof{t) for to in the filter transfer function H(ja>) to obtain the ou tpu t v j t ) given by

In addition to establishing the validity of this intuitively satisfying “quasi-static” response, we shall a ttem pt to obtain a bound on the m axim um value of wm [the frequency to which f ( t ) is band-lim ited] for which Eq. (11.3-2) may be employed.

( 1 1 . 3 - 1 )

(11.3-2)


The second tractable situation i s tha t in which H{ja>) has both a m agnitude and a phase which are straight line functions of a>. In this case, regardless of the rate at which / (r) is varied, v0(t) is given by

v„(t) = IH[ju)i(t - i0)]| cos co0t + A oj J f(d)de + 0(cyo) (11.3-3)

where 6(co0) = arg H(ja>0) and t0 is the constant slope of the arg H(jco)-vs.-a) curve. We observe from Eq. (11.3-3) that \H(jco)\ establishes the transfer function between the envelope of v0(t) and w,(i — i0), whereas in the “quasi-static” case \ H(jaj)\ establishes the transfer function between the envelope of v0(t) and to,(t). A lthough Eqs. (11.3-2) and (11.3-3) seem to be quite different in form, it is readily shown that Eq. (11.3-3) reduces to Eq. (11.3-2) as com decreases relative to l/'f0.

In either of the above cases it is clear that, if \H(ju>)\ is a straight line function of co, the filter ou tpu t v0(t) has an envelope which varies in p roportion to a>,(r); thus if we place an envelope detector a t the filter ou tpu t we obtain an obvious m eans of FM dem odulation. This technique will be discussed in more detail in C hapter 12.

Linear filter

h{i\=v(t) = AQie (expyaV)[exp /AajJ /(0) d%)

Fig. 11 .3-1 Linear filter driven by FM signal.

To dem onstrate the validity of Eq. (11.3-2) consider the filter driven by v(t) which is shown in Fig. 11.3-1. If h{t) is the impulse response of the filter and we represent the filter input v(t) in the form

i>(i) = A cos [u)0t + \p(t)] = (11.3-4)

where denotes the real part and i¡/(t) = A cop f (6) d6, then em ploying the convolution integral we may write v„(t) as

v0(t) = A. elw" J” h{T)e-jo,°zeJ*«“ T) rftj. (11.3-5)

If we now expand ip(t — i) in a M acLauren series in t, we obtain

m - t 2i l / ( t - t ) = iA (0 - y { t ) T +

2 !(11.3-6)

where the dots represent differentiation with respect to t (or — r). However, if h(x) is zero for t < 0 and decays to zero for x > t m (for example, xM m ight represent four time constants if h(t) were the impulse response of a single-pole filter), then ij/(t — x)

1 1 . 3 TRANSMISSION OF FM SIGNALS THROUGH LINEAR FILTERS 523

contributes to the integral of Eq. (11.3-5) only for 0 < r < xM. Thus if

(that is, if the rate of change of the ac portion of ihe instantaneous frequency, is small, i.e., if iom « 1 /tm), then Eq. (11.3-5) simplifies to

v0(t) = A 0 leê jl‘O0, + 'l'{,)i | h{x)e-J ,+li,i,)]x dx

= A3l6{eJlmot+m] //[ /« ,-(f)]}

= A\H[ja)i(t)]\ cos {aj0i + ij/(t) + arg H[jaj;(r)]}, (11.3-7)

which is the desired result. In obtaining Eq. (11.3-7) use was m ade of the fact that the defining equation for H(joj) is the Fourier transform of the impulse response given by

H(jco) = | ° h(x)e dx. (11.3-8)

To obtain a limit on the usefulness of Eq. (11.3-7) we shall follow an approach suggested by B aghdady.f We first employ the expansion of Eq. (11.3-6) to write

expji/Kt - r) = [expji/'U)] [exp - j\I/(t)x] exp j’4>(t)x2 ÿ(t)x2

2 ! 3!+ (11.3-9)

We then further expand the last term of Eq. (11.3-9) in a M acLauren series in r to obtain

exp/iA(r - t) = [exp7'i//(r)][exp - j\jj{t)x] . , .iAWt2 .iA(t)T3+ j ~ y \ 3 r +

(11.3-10)

Finally, if we assume that for small values of xM the j[$(t)x2/2 !]-term is the main term contributing to inaccuracies in Eq. (11.3-7) (which is usually the case in practice, even though some nonrealistic examples can be set up where ip(t)x2/ 3 ! is more significant than 0'(i)r2/ 2 !), then Eq. (11.3-9) may be w ritten in the form of the desired “quasi-static” ou tpu t plus an error te rm :

v0{t) = +F

h(x)e dx

+ j. m

F .(11.3 11)

t E. J. Baghdady, Lectures on Communication System Theory, M cGraw-Hill, New York (1961), pp. 472 483.


Since m ultiplication in the t- (or r-) dom ain corresponds to differentiation in the frequency dom ain

r2h(r) —d2H(j(u)

den2(11.3-12)

Equation (11.3-11) may therefore be rew ritten in the form

Va(t) = A & e <i

It is now apparent tha t if

yjliOOt +

d2H[jœ,{t)] -IM t) dcof (11.3-13)

d2H [ M t ) ] $(t) dcof <

max

i}{t)d2H[jMi(t)}

dcof2 2 max H[j(Oi(t)]

« 1, (11.3-14)

then Eq. (11.3-13) reduces to Eq. (11.3-7) and the “ quasi-static” approxim ation is valid.

As an example of applying Eq. (11.3-4), consider passing an FM signal through a high-Q r parallel R L C circuit for which Z u (yco) is closely approxim ated by (cf. Eq. 2.2-9)

Z n (jco) =R

1 + j.CO — (O o

co > 0,

a

where co0 = \j~J~LC and a = 1/2R C (the reciprocal o f the circuit time constant). It follows immediately that

d2Z n (joj)dco2

- 2 R

a 2 1 + j -(0 — (l)f

and

d2Z n ( j w )d o r

Z u(ja ))

(11.3-15)

(11.3-16)

In deriving Eq. (11.3-16) we have assum ed that co/t) is centered about co0 . If this were not the case, Eq. (11.3-16) would give a smaller value.If we now let f ( t ) = cos comi (the worst case), then

and\j/{t) = Aco co s comt, (¡/(t) = — Acocom sin comi,

I^WLax = Acocom;

1 1 . 3 TRANSMISSION OF FM SIGNALS THROUGH LINEAR FILTERS 5 2 5

therefore, the quasi-static approxim ation is valid provided that

If we interpret “ m uch less than un ity” as 1/20 (5% error or less), then for P = Aoj/ojm = 5,

However, since the derivation for the m axim um value of Aco/a involved consistently conservative approxim ations, in m ost cases the bounds may be relaxed by 50 % and the quasi-static approach will still yield reasonably accurate results.

N ote from Eq. (11.3-17) that the “ quasi-static” approxim ation is valid if either com « a or Aa> « a. Therefore, if the m axim um m odulation rate of f ( t ) [or a>,(0] is m uch less than the reciprocal of the circuit time constant, the deviation may be large w ithout destroying the validity of Eq. (11.3-7). O n the o ther hand, if 2 Aco « 2a, where 2a is the circuit 3 dB bandw idth, then / (i) may vary quite rapidly w ithout

A oj 1— < -

and for fi = 10,

A cu 1

o ■o

Slope — io

ui

Fig. 11 .3-2 Filter with linearly sloping m agnitude and phase characteristic.


D irect substitu tion of u, and v2 given by Eq. (11.4-3) into Eq. (11.4-4) o r Eq. (11.4-5) indicates that v , and v2 are indeed two independent solutions ; and since the differential equation is second-order and linear [as clearly indicated by Eq. (11.4-5)], any other solution may be obtained as a linear com bination of i>, and v2. Hence the m ost general solution may be written as

v(t) = A i cos J (0,(0) dO + A 2 sin j" co,(0) dO

= C cos J cu¡(0) dO + 0OJ , (11.4-6)

where A 1, A 2,C , and 60 are arb itrary constants of integration which may be uniquely specified by the initial conditions y(0) and ¿(0).

F o r example if r(0) = F, and ¿(0) = 0, then

v(t) = Vt cos J a>i(0) dO ;

and if co((f) = cj0 + Aa> f (i), then

u(t) = Vl cos to0t + Aa> f (8) dd , (11-4-7)

which is the desired FM signal. Any other set o f initial conditions merely changes the am plitude of u(r) and introduces a constant phase angle which is of little consequence, since it may be taken as zero with a different choice of time axis.

N ote tha t Eq. (11.4-4) and (11.4-5) are even functions in cof(t); hence negative values of co,(t) produce the same i?(t) as the corresponding positive values of co,(i). Thus to avoid this full-wave rectification effect of co,((), the instantaneous frequency m ust be greater than z e ro ; that is,

u>i(t) = <y0 + Aa>/(f) > 0 (11.4-8)

or equivalently

A w < a>0. (11.4-9)

This restriction is of little consequence, since no dem odulator exists which can recover /(£) if Eq. (11.4-9) is no t satisfied.

G eneration of an FM signal by sim ulating the FM differential equation is particularly well suited to an analog com puter. Figure 11.4-1 illustrates an analog com puter block diagram which em ploys tw o multipliers, tw o integrators, and aninverter to im plem ent Eq. (11.4—4). F o r the block diagram of Fig. 11.4-1 the equationdescribing v(t) takes the form

J' K MK,vtf)v(0) d9 + v{t) = 0, (11.4-10)

1 1 . 3 TRANSMISSION OF FM SIGNALS THROUGH LINEAR FILTERS 525

therefore, the quasi-static approxim ation is valid provided that

01.3-17)

If we interpret “ much less than un ity” as 1/20 (5% error or less), then for /? =Ato/iom = 5,

A to 1 a 2

and for ¡3 = 10,

Aco 1

a ~

However, since the derivation for the m axim um value of Aai/a involved consistently conservative approxim ations, in m ost cases the bounds may be relaxed by 50 % and the quasi-static approach will still yield reasonably accurate results.

N ote from Eq. (11.3-17) that the “ quasi-static” approxim ation is valid if either iom « a or Aw « a. Therefore, if the m axim um m odulation rate of / ( f ) [or a>,(f)] is much less than the reciprocal of the circuit time constant, the deviation may be large w ithout destroying the validity of Eq. (11.3-7). O n the o ther hand, if 2 Aa> « 2a, where 2a is the circuit 3dB bandw idth, then / ( f ) may vary quite rapidly w ithout

Fig. 11 .3-2 Filter with linearly sloping magnitude and phase characteristic.

5 2 6 GENERATION OF FM SIGNALS 1 1 . 4

destroying the validity of the “quasi-static” solution. A lthough we derived Eq. (11.3-17) for a specific filter, a sim ilar equation results for alm ost all narrow band filters; specifically,

K l 2 Aw\ 2« 1 , (11.3-18)

P \ BW

where K usually lies between 1 and 5. Several m ore specific examples are derived by Baghdady.t

To dem onstrate the validity of Eq. (11.3-3), let us consider the filter with the m agnitude and phase function shown in Fig. 11.3 -2. If v(t) is again given by

v(t) = ¡M4Ae1[a">, + U)]lthen

i>0(f) = « . [ / - ‘H Ijw JF H ], (11.3-19)

where V(a>) is the Fourier transform of /4ejI" 0' + *(,,). W ith H(jaj) given in Fig. 11.3-2 substituted into Eq. (11.3-19), v„(t) reduces to

~ja)V(co)e~jw,°C pjO(u>o)gj<t>otova(t) = St A ---------------

I a>.

f Aeje(‘00)ejm0'0= die

0 >,(f ~ fo)

J

de 10 0<( ~,o) + _ ,o)1j d t

- « j V(o))e

= 3tcAa)o + 1 t o °Ji » (>-««) + »(moilcoa

= A \ H [ j i O i ( t - i0)]| cos [a)0t + i¡/(t - t 0 ) + 0(coo)], (11.3-20)

which is the desired result. In the above derivation, use has been m ade of the fact that the operations of e~ja>,° and jco in the frequency dom ain are equivalent to a time delay of t0 and the differentiation operation respectively in the time domain.

11.4 FREQ U EN CY M O D U LA TIO N TEC H N IQ U ES— T H E FM D IFFEREN TIA L EQU ATIO N

In this section and the sections to follow we shall study the theoretical m ethods (and their practical im plem entation) by which we may vary the instantaneous frequency of a sinusoidal carrier in p roportion to the m odulation inform ation / ( f ) to obtain an FM signal of the form

v(t) = A cos co0t + Aa> f(6 )d0 (11.4-1)

which has the instantaneous frequency

a),it) = a >0 + Aa>/(f). (11.4-2)

t Op. cit.

11 .4 FREQUENCY MODULATING TECHNIQUES 5 2 7

In g en eral, the th e o re tica l m eth o d s o f p ro d u cin g a freq u en cy -m o d u la ted signal can be b ro k e n in to six b a s ic c a te g o r ie s :

1. T h e a n a lo g s im u la tio n o f the F M d ifferen tia l eq u atio n .~2. T h e q u a s i-s ta tic v a r ia tio n o f the o sc illa tio n freq u en cy o f an o sc illa to r .3. T h e g e n era tio n and n o n lin e a r w av esh ap in g o f an F M tria n g u la r wave.4. T h e g e n era tio n and filte rin g o f an F M sq u are w ave.5. T h e g e n era tio n an d freq u en cy m u ltip lica tio n o f n a rro w b a n d F M (the A rm stro n g

6. The generation of FM in specialized devices, e.g. the Phasitron etc.O f a ll the a b o v e m eth o d s o n ly the a n a lo g s im u la tio n o f the F M d ifferen tia l e q u a

tio n and the g e n era tio n and w av esh ap in g o f a tr ia n g u la r F M w ave p ro d u ce a th e o re tica lly p erfect F M signal w hich m ay have a freq u en cy d ev ia tio n Aw o r m ax im u m m o d u la tio n freq u en cy wm as h igh as the ca rr ie r freq u en cy w 0. B o th the g e n era tio n

and filterin g o f an F M sq u are w ave and the A rm stro n g m eth o d req u ire the filterin g o f a d is to rte d F M sig nal to p ro d u ce th e o u tp u t F M signal an d th u s have Aw an d w m lim ited w ith resp ect to w 0 by cu rv es s im ila r to th o se o f F ig . 1 1 .2 -3 . In the q u a s is ta tic m eth o d b o th Aw an d w m a re ev en m o re severely restricted . T h e m ax im u m m o d u la tio n freq u en cy m ust be kep t sm all in co m p a riso n w ith the re c ip ro ca l o f the en v elo p e tra n s ie n t tim e c o n sta n t o f th e o sc illa to r , w hile Aw m u st b e m u ch less th an w 0 to p revent n o n lin e a ritie s in the in s ta n ta n e o u s freq u en cy o f the F M signal.

In th e rem a in d er o f th is s ec tio n an d th e sec tio n s fo llo w in g we shall c o n sid er each o f the ab o v e te ch n iq u es in som ew h at g re a te r d eta il. S p ecifica lly , we shall a tte m p t to o b ta in a b lo ck d ia g ra m fo r a n F M g e n era to r co rre sp o n d in g to ea ch o f th e a b o v e m eth o d s, a lo n g w ith the m a x im u m valu es o f Aw and wm (re lativ e to w 0) fo r w hich the g e n era to r fu n ctio n s p rop erly . In e a ch ca se we shall th en d ev elop one o r m o re p hysica l im p lem en ta tio n s fo r th e g en era to r .

The FM Differential Equationt

A seco n d -o rd er , lin ear, h o m o g en eo u s d ifferen tia l e q u a tio n w hich has as its tw o lin early ind ep en d en t so lu tio n s the F M signals

m ethod).

and (1 1 .4 -3 )

is g iven in in te g ro -d iffe re n tia l fo rm by

o r. eq u iv a len tly , is given in d ifferen tia l form by

(1 1 .4 -5 )

t D. T. Hess, "F M D ifferential E q u a tio n ” (letter), P ro c . I E E E , 54, No. S, p. 1089 (Aug. 1966).

528 GENERATION OF FM SIGNALS 1 1 . 4

D irect substitu tion of v t and v2 given by Eq. (11.4-3) into Eq. (11.4-4) o r Eq. (11.4-5) indicates that v t and v2 are indeed two independent solutions ; and since the differential equation is second-order and linear [as clearly indicated by Eq. (11.4-5)], any other solution may be obtained as a linear com bination of Vi and v2. Hence the most general solution may be w ritten as

v(t) = A j cos J a>¡(9) dO + A 2 sin J co;(0) dO

= C cos J a>i(0) dO + 0OJ , (11.4-6)

where A 1, A 2,C , and 90 are arb itrary constants of integration which may be uniquely specified by the initial conditions i’(0) and i;(0).

F o r example if i’(0) = K, and ¿(0) = 0, then

v(t) — Vx cos J cj,(0) dO ;

and if a>i(t) = <y0 + A w f (t), then

v(t) = Vl cos co0( + Aco I f{0 )d 0fJo(11.4-7)

which is the desired FM signal. Any other set of initial conditions merely changes the am plitude of u(t) and introduces a constan t phase angle which is of little consequence, since it may be taken as zero with a different choice of time axis.

N ote tha t Eq. (11.4—4) and (11.4-5) are even functions in «¡(t); hence negative values o f a);(t) produce the same u(t) as the corresponding positive values of co;(t). Thus to avoid this full-wave rectification effect of co,(f), the instantaneous frequency m ust be greater than ze ro ; that is,

a>i(t) = co0 + Acof(t) > 0 (11.4-8)

or equivalently

Aoj < (o0. (11.4—9)

This restriction is of little consequence, since no dem odulator exists which can recover f i t ) if Eq. (11.4-9) is no t satisfied.

G eneration of an FM signal by sim ulating the FM differential equation is particularly well suited to an analog com puter. Figure 11.4-1 illustrates an analog com puter block diagram which employs two multipliers, two integrators, and aninverter to im plem ent Eq. (11.4-4). For the block diagram of Fig. 11.4-1 the equationdescribing v(t) takes the form

K MK Jvi( t ) j , K MK Jvi(0)v(O)d0 + v(t) = 0, (11.4-10)

1 1 .4 FREQUENCY MODULATION TECHNIQUES 5 2 9

Fig. 11.4-1 Analog com puter im plem entation of FM differential equation.

which is identical in form to Eq. (11.4-4). It is thus apparent that the instantaneous frequency of v(t) in the block diagram of Fig. 11.4—1 is

Hence, if v,(t) is chosen as

1 + — At)(J)q

(11.4-11)

and K iK m V1 is adjusted to equal a>0 , the analog com puter has the desired instan taneous frequency co,{t) = oj0 + Acof (t).

If, in addition the initial conditions (IC) of the integrators are set so that

v(0) = Vl and «5(0)K , K Mv m

= 0,

with the result that i)(0) = 0, then the analog com puter generates the FM signal given by Eq. (11.4—7). (A practical circuit to take care of the initial-condition problem autom atically will be presented shortly.)

The block diagram of Fig. 11.4-1 can be constructed not only on the analog com puter, but also as an independent FM generator. If good-quality integrated operational amplifiers are em ployed as the basic com ponent of the integrators and the inverter and if high-speed integrated m ultipliers (Section 8.3) are employed, then


the resultant FM generator should perform satisfactorily with carrier frequencies exceeding 10 M Hz. In addition, linear frequency deviations equal to the carrier frequency may be obtained regardless of the m odulation frequency, which may exceed the carrier frequency. The extreme range of A i d and com is a direct result of the fact that the solution of the FM differential equation is a theoretically perfect FM signal regardless of the values of Aco and (om, provided that Atu < oj0.

Because of the high deviation ratio (D = Aa>/a)0) possible with the analog m odulator of Fig. 11.4-2, an FM signal with specified values of Aco and a>m can be generated at a reasonably low carrier frequency to0 . This FM signal can then be heterodyned by a crystal oscillator with frequency t o c to any desired carrier frequency (d '0 = a ) 0 + o j c .

If u>c » a>0, as is the case when co0 was initially a low value, then the stability of the center frequency of the resultant FM signal is controlled prim arily by the stable crystal oscillator and is alm ost independent of the less stable frequency w 0. For example, if a>c = 9co0 and a>0 varies by 1 %, then coc varies by only 0.1 %.

The limit on the m axim um ratio of coc to w0 to obtain center frequency stability is set by the problem of separating the FM signal at the difference frequency wc — u)0 (heterodyning produces both the sum and the difference frequencies) from the desired FM signal. Specifically, a filter with BW « 2a >0 and a center frequency of loc + <x>0 is required to perform the separation. The QT of such a filter m ust satisfy the relationship

= + 1) ; (11.4-12)BW 2 \a »0 I

hence, as c d c/ ( d 0 increases, the required QT and thus the filter complexity greatly increase.

Amplitude Limiting and Initial ConditionsTo avoid the problem of having to set the initial- conditions to ensure an FM signal of a desired am plitude and, in turn, to extend the usefulness of the FM generator beyond the low-frequency analog com puter two additional feedback loops may be added to the block diagram of Fig. 11.4-1 as shown in Fig. 11.4-2. The two additional loops have the same effect on the FM generator as an am plitude-lim iting circuit has on a sinusoidal oscillator. As a m atter of fact, with vt(t) = V2 (a constant), the effects are identical. Specifically, for values of Kenv < Vy, the system poles lies in the right- half complex plane and a growing oscillation occurs. W hen Kenv = Vx, the poles lie on the im aginary axis and the oscillations stabilize. If, however, for som e reason Kenv increases beyond Vx, the complex conjugate poles move into the left-half plane and the envelope of the oscillation decays back tow ard Vx.

It is apparent that even if v,{t) is a function of time, the envelope of tit) should stabilize at Vl , since for this value no signal is fed into m ultipliers M 3 and M 4, the two additional loops are opened, and the block diagram of Fig. 11.4-2 reduces to the original diagram of Fig. 11.4-1.

To obtain a m ore quantitative picture of how the envelope of v(t) stabilizes, we may write the differential equation governing the behavior of v(t) from the block

1 1 . 4 FREQUENCY MODULATION TECHNIQUES 5 3 1

Inverter

O utput : v(/) o ----------■*-

V\o—

M j f v r

In tegrator

M,

i K J t )

Peak envelope detector *„=1

In tegrator

M2

Fig. 11.4-2 Analog im plem entation of FM differential equation with am plitude stabilization mechanism.

diagram of Fig. 11.4—2f :

where

, a (h: a a \ / 2 a tbA v(t)0 = t'(i)( 1 H j H j 2 I — ^ (0 1 —2 ----- 3 ^----- 2 ’o f cof cof J \ cof cof I ,cof

<o,(r) = K , K Mv,{t) and a(i) = K IK^[V l - KCTV(r)].

(11 .4 -13)

For this case, the m ost general solution is given [as the reader should verify by substituting in Eq. (11.4—13)] by

¡’(f) = C exp J a(0) dû cos J co,(0) dd + 90 (11.4-14)

^ e n v ( 0

where again C and 90 are arb itrary constants. It is apparent that from Eq. (11.4-14) that

ênv(f) = C exp J K ,K ^ [ V i - vmm de , (11 .4 -15)

t Equation (11.4-13) is not obvious by inspection. The interested reader should make an effort to derive it from the block diagram of Fig. 11.4-2; the more casual reader should accept its validity and proceed to the conclusion draw n from the equation.


or equivalently,

ln = f ~ v^ 0 ) ] d e .Ç K ,K U [V t - K J O ) ] dO. (11.4-16)

By differentiating Eq. (11.4-16) with respect to time, we obtain

d V UVAt(11.4-17)

which has as its solution

(11.4-18)

where B is an arb itrary constant of integration. As expected, we observe from Eq. (11.4-17) that Kenv(i) approaches Vl and does so with a time constant r = 1 /K'MVlK l . Therefore, if power is applied at t = 0, for t > 4 t, v(t) (for Fig. 11.4-2) is given by

In the above analysis we assum ed that the envelope detector extracted the instantaneous envelope of v(t). In general, however, the detector contains some filtering, so that the detector ou tput is a filtered version of the instantaneous envelope. If this equivalent filter contains a single pole at some low frequency (which m ost peak envelope detectors do), then the above results are still valid except that Venv(() rises tow ard K, m ore slowly and with a second-order response which is usually dom inated by the low-frequency detector pole. If, on the other hand, the peak envelope detector contains two or more poles, the possibility of squegging ex ists; hence higher-order envelope detector filters should be avoided.

In addition, the single pole of the peak envelope detector should lie well below the lowest value of the instantaneous frequency of prevent the possibility of having the envelope detector ou tpu t vary over a carrier cycle. Such a variation would significantly d istort the FM signal, since cc(t) in Eq. (11.4-13) would vary over a cycle in this case.

11.5 Q UA SI-STA TIC FREQ U EN CY M O D U LA TIO N

W hen the frequency deviation Aw and the maximum m odulation frequency wm of an FM signal are both small fractions of the carrier frequency w0 (which is the case in more than 95 % of the practical applications for which an FM signal is required), then the FM differential equation given by Eq. (11.4-5) can be closely approxim ated by either of the following quasi-static forms, which act as the defining equations for a broad class of FM g en era to rs :

( 1 1 .5 - la )

1 1 . 5 QUASI-STATIC FREQUENCY MODULATION 5 3 3

a n d^ 2tf(i)ajj(i) v(t) 1U

v ( t -----------3— + - 7-2 = 0. (11.5-lb)co f w ,(i)

The justification for this approxim ation is based on the fact that, when both Aw « w0 and wm « w0 , the term — v(t)<b,{t)/a)l{t)3 in Eq. (11.4-5) becomes vanishingly small in com parison with the term s v(t) and i>(f)/a>;(f)2 ; hence it may be added to (Eq. 11.5-lb) or subtracted from (Eq. 11.5-la) Eq. (11.4-5) with negligible effect on the solution i’(0 -t

For example, if we consider the case where w,(i) = co0 + Aw cos wmt (the mostrestrictive case for given values Aw, wm, and w0), then the solution to the FM differential equation is given by

(A co

w0r -l sin wmi + 0o


i’(t) = —A(cu0 + Aw cos wmf) sin I a>0t -I- ^ sin wmr + 0O) (11.5-2)\ U)m I

where A and 0o are arbitrary constants of integration. Therefore, from Eq. (11.5-2) and the fact that w,(i) = — Awwm sin wmt, it is apparent that

fib.- /4Aww_ sin w_f I Aw \y = ;------ — ------------- —y sin (w0r H sin comt + 0O| (11.5-3)

and thus

W; (w0 + Aw cos w„i)‘

Il’Lax (®0 - Aw)2 /?(w0/Aw - l )2(11.5-4)

forAoj/oj0 « 1 F o r ft = Ato/com = 5 a n d w 0 = 20 Aw, Eq. (11.5-4) has the numerical value of 5 x 10 4, for which the second term in the FM differential equation relative to the first term is truly negligible. In most practical m odulators, as we shall see, w0 is usually much greater than 20 A w ; thus Eq. (1 1 .5 -laan d b) is an excellent approxim ation to the F1.1 differential equation.

The advantage of being able to employ Eq. (11.5 la and b) as the basis for p ro ducing an F"M signal is that it is much m ore simply implemented by practical circuits than is the exact equation. Specifically, each of the LC circuit models shown in Fig.11.5-1 has as its defining differential equation one of the quasi-static FM differential

t If a negligibly small term is added to or subtracted from the FM differential equation having a set of initial conditions specified at / = 0, it will be a considerable length of time before the solutions to the original differential equation and the modified differential equations differ significantly. However, if the circuit which implements the modified differential equation contains an envelope-limiting mechanism which is fast-acting in comparison with a>m (all practical FM generators must contain such a mechanism), then the initial conditions of the modified differential equation are updated at a sufficiently rapid rate to ensure that its solution differs from the solution of the original differential equation by a negligible amount.


w,M^ c + c« ) ( ' + S q )

,<o I V(0w,(0 3 (uXO2

^ J W ¥ W )

Fig. 11.5-1 Three quasi-static FM m odulators.

equations. In each circuit a time-varying controlled source is em ployed to vary the instantaneous frequency. The im aginative reader should be able to construct several other L C circuits with time-varying controlled sources which also have as their defining equations one of the quasi-static FM equations.

To obtain the differential equation for any of the circuits o f Fig. 11.5-1, one m ust write a single-node equation. F o r example, for circuit (a) we have

jl + !co + [1 + A(t)]ic — 0

= i f V(d)d6 + C0i>(t) + C[ 1 + A (f)]i)(0 (1 1 .5 -5 )L j

11.5 QUASI-STATIC FREQUENCY MODULATION 535

or equivalently,

0 = (11.5-6)

where

1

If we differentiate Eq. (11.5-6) with respect to tim e we obtain Eq. (11.5-lb), which is the desired result.

As an alternative point of view, the time-varying controlled source in the circuits of Fig. 11.5-1 may be thought of as varying the capacitance or inductance of the circuit model as a function of time. For example, the last term in Eq. (11.5-5) could be regarded as the current of a time-varying capacitance C(r) = [1 + A(t)]C. With this point of view, we obtain an additional technique for im plem enting quasi-static FM generators which entails placing time-varying capacitors or inductors in shunt with parallel LC circuits.

In each of the circuits of Fig. 11.5-1 we note that a>,(f) is a nonlinear function of A(t) \ thus if A(t) varies in p roportion to the m odulation inform ation /(f) , that is, if

then / l i / ( f ) m ust be kep sufficiently small so tha t co,(f) also varies in proportion to /(f), as is usually desired. Specifically, if for circuit (a) of Fig. 11.5-1 we expand a),-(f) in a M acLauren expansion in A 1f ( t ) to obtain

A(t) = A 0 + A J ( t ) , (11.5-7)

(11.5-8)

where

1co o =

and

Aft» —C A X 1^ = 2(C0 + C) [1 + CAJOCo + C)]’

then we observe th a t on ly if


does the instantaneous frequency have the desired form cy,(i) = oj0 + Aajf(t). For example, if the maxim um am plitude of the nonlinear term of Eq. (11.5-8) is to be less than 1 % of the m axim um am plitude of the linear term, then Aw/u>0 < 0.0133 or equivalently a »0 > 75 Aa>. If A 0 « 1 (which is usually true in practice), then

Identical restrictions on Aa>/a>0 are obtained for the o ther circuits o f Fig. 11.5-1 ; hence, if = Aw/o)m > 1 (which is usually true for the m ajority of FM signals) then satisfying Eq. (11.5-9) is sufficient to ensure that Eq. (11.5-1) is also valid. Therefore, with fi > 1 and w 0/A w > 75, the ou tpu t signals of the quasi-static FM generators are of the desired form

The problem now rem ains of synthesizing the circuit m odels o f Fig. 11.5-1 with practical circuits. The first obstacle to be overcome is the synthesis of a lossless LC circuit. This is readily accom plished by placing a lossy LC circuit which has the form of the circuits of Fig. 11.5-1 (plus some equivalent shunt loss) in the feedback loop of an oscillator. At the point where the loop gain of the oscillator is unity, the oscillator provides exactly the required negative resistance to -cancel the equivalent shunt loss of the lossy LC circuit, f In addition, m ost oscillators contain the necessary am plitude stabilization m echanism to ensure the self-starting of the FM generator as well as a specified level for the envelope of the ou tput FM signal.

Although the oscillator is capable of cancelling the LC circuit loss, the Q of the LC circuit should be kept as high as possible so that, if the oscillator limits in a non linear fashion, the sinusoidal waveform of the ou tpu t FM signal is m aintained. In addition, the tim e constant o f the envelope-lim iting circuitry (cf. Section 6.8) should be fast in com parison with variations in f ( t ) to ensure that the ou tpu t envelope readjusts itself to a fixed level for each small variation in / ( f ) and thus is independent of f( t) . The fast time constant also provides the continuous updating of the initial conditions of y(f), thereby further ensuring close correspondence between the solutions of the exact and of the quasi-static FM differential equations.

A practical L C circuit em ploying a differential pair which im plem ents the circuit model of Fig. 11.5-l(a) is shown in Fig. 11.5-2. For this circuit the capacitor current

t Consider, for example, the oscillator of Fig. 6.1-6. Since nv, — vh the controlled current source gmy, is equal to ngmv(t). However, the voltage across the current source is v,; thus the current source may be replaced by an equivalent negative conductance — ngm. With A,(jo>0) = 1, — ngm = — (Gl + rt2Gin), which is exactly the negative of the total circuit loss, GL + n2Gin. Therefore, the circuit loss is canceled, leaving a pure LC network.

1 A (oand —

(D0C A X

(Ü0 X =y /L (C 0 + C) 2 (c0 + cy

= A cos (11.5-10)


- Oscillator Time - varying controlled source

Q V

h(t) — ho + W U )

Fig. 11.5-2 Q uasi-static FM generator employing a differential pair.

ic is sampled by the small resistor r f (which is not loaded by the input impedance of (?3) to develop a voltage v ] = icr a t the base of Q3. The am plification of the differential pair, which is directly controlled by I k(t), produces a collector current at Q3 which simulates the time-varying controlled source A(t)ic where, on a small-signal basis,

w 1™ , *qrikl- s j , ) r ~ W + - i F F m

Therefore, for the circuit of Fig. 11.5-2 (cf. Eq. 11.5-8),

1<«n

J L ( C 0 + C )(l + B)

(11.5-1 l ) î

(11.5-12)

t The value of r should be sufficiently small so that, when it is transformed into an equivalent parallel resistor at a»0, it is much greater than R loss and thus is a negligible com ponent o f the total tuned-circuit loss. If this were not the case, the total tuned-circuit loss would be a function of r: however, r, like C, varies with A(t). Therefore, the envelope level at which the FM wave would stabilize (which is directly related to the tuned-circuit load—see Section 6.5) would be a function of A(t)\ this is an undesired effect.t The collector current o f Q3 also contains low-frequency terms proportional to Ik(t) which are shunted to ground by L and thus do not contribute to the output.


and

A <w — B(Ikl/ I k0)OJn 2(1 + B) ’

where B = CaqrIk0/4kT (C 0 + C). As a result, v,(t) is given by

vtU) = K cos + Am J f ( 9 ) d 6 + 0OJ

and v„(t) is given by

vdt) = Vcc + Vo cos co0t + A co j ' m d e + do^ ,

(11.5-13)

(11.5-14)

(11.5-15)

provided tha t ¡3 = Aco/tom > 1, co0 > 75 Ato, and the ou tpu t-tuned circuit has sufficient bandw idth to pass the spectrum o f the FM signal. B oth V, and V0 are determ ined by the am plitude-lim iting m echanism o f the oscillator in the fashion described in Section 6.5.

T o choose values for the various param eters o f the tim e-varying controlled source, we first observe th a t B is a function o f tem perature and should be as small as possible in com parison w ith unity to keep to0 independent of T. The limit on the smallness of B is determ ined by the required ra tio of Agj/co0 and the ra tio I ki/ I k0 (cf. Eq. 11.5-13). F o r a specified ratio of Aco/a>0 the m inim um value of B results when ¡Kl/ I k0 is as large as possible. The lim it on the size of this ra tio is 1, since I k(t) > 0 ; therefore, a sensible choice would be to select I kl x I k0 and to select B with its m inim um value of

B =2 Aoj/m0

1 + 2 Ao)/u)02 Aft)

to o&)0 » Acu. (11.5-16)

W ith this value of B, to0 simplifies to

1IL(C0 + C )

1 J L ( C 0 + C)a>0 » Aa>. (11.5-17)

1 + 2Aco/co0

We also observe that 10 keep Q3 operating as a linear controlled source, Q3 must function as a snv’ll tgnai amplifier or, equivalently, It^l < 10 mV. Since r « l/o)0C and co0 >; A , the am plitude of ic may be approxim ated by V,oj0C ; thus the am plitude of»': n r hiiappiuxiinatedby V,rco0C. Com bining the requirem ent of V,roj0C < 10 mV w .. definition of B (and letting a * 1 and k T /q - 25 mV), we obtain

co0(C0 + C)BV, B V f a

IkO^loss< 0.1, (11.5-18)

where Q r x to0(C0 + C)Rloss is the Q of the circuit with f ( t ) = 0. Since B, V, , Q r , and Kloss are usually known or specified quantities, Eq. (11.5-18) determ ines the m inim um value of I k0.

1 1 . 5 QUASI-STATIC FREQUENCY MODULATION 539

As a numerical example, consider the design of an FM generator of the form shown in Fig. 11.5-2, for which w 0 = 108 rad/sec, Aw = 106 rad/sec, wm = 2 x 105 rad/sec, QT = 20, Vt = 5 V, Vcc = 10 V, and R luss = lO k il From Eq. (11.5-16) we obtain th e xoptim um value of B for a tem perature-insensitive center frequency, namely, B « 2 A<u/a)0 = 0.02 provided t h a t / kl % I k0. In addition, from Eq. (11.5-18) we find that ¡k0 m ust be greater than 2 mA. A choice of I kl = Ik0 of 2.5 mA would be reasonable here. In addition, since QT = a>0(C0 + C )î0ss» then C0 + C = 20 pF, a value which when substituted into Eq. (11.5-15) yields L = 5 ¿¿H.

If we choose C 0 = C — 10 pF (the ratio of C to C0 may be chosen arbitrarily), then with k T /q = 26 mV and a « 1, from the definition of B we obtain r = 1.6 Q. Clearly r is not loaded by Q 3.

The oscillator differential pair can now be designed to yield the correct value of V,. Usually ¡lie oscillator is designed to stabilize with a large value of x such that, even if small am plitude variations exist at v(t), v0(t) has a constant am plitude. This effect follows from the fact that with large values of x the output of 0 , is a limited version of the input to Q2 ■

To cause the oscillator to stabilize at x = 10 we choose M such that

Thus Gm( 10) m ust equal 1/500 Q and from Fig. 4.6-5 gm must equal 1/130 Q. However, since gm = q lJ A k T , I k » 800 fiA.

Now if R l = 6 kQ and the output-tuned circuit bandw idth is sufficient for the FM signal (BW/2 Aw > 1.6 with ft — 5, or BW > 3.2 x 106 rad/sec), then

If we choose BW = 5 x 106, with the result that Qr for the output-tuned circuit is 20, then L l = 3 ¡xH and CL = 33.3 pF.

An alternative circuit for obtaining the time-varying controlled current source

provided that r is sufficiently small so that the transistor is operated in a small-signal fashion and tha t the im pedance of C E a t <w0 is small in com parison with the m inim um value of gin(f) so that appears across the transistor junction. The im pedance of CE at <om should also be large in com parison with the m axim um value of gm(t) so

M = 0.25 //H

and I k such that

v0 = (10 V) - (6 V) cos

required to im plem ent the circuit model of Fig. 11.5—1(a) is shown in Fig. 11.5 3. For this circuit, which employs a single transistor, ^4(0 is given by

¿ ( 0 = gm(0 r =<xqrIEo *qr IEi

k T k T m ,


Fig. 11.5-3 Im plem entation of time-varying controlled source with a single transistor.

th a t I ^ t ) flow s e n tire ly th ro u g h th e tra n s isto r . T h is la tter re q u ire m e n t c a n b e im p lem en ted o n ly if co0 exceed s u>m by a fa c to r o f 1 0 0 0 o r m o re.

F o r exa m p le , i f I E1 = 0 .9 I E0 (w hich is d esira b le , as we h av e seen , to k eep co0 ind ep en d en t o f T ), then

>inmax .8inmin

th ere fo re , if

a)0 C E > 10ginm>x an d (omC E <

th en u>0lajm m u st b e g re a te r th a n 19 0 0 to p erfo rm c o m p a ra b ly w ith th e d ifferen tia l- p a ir im p le m e n ta tio n o f the tim e-v a ry in g c o n tro lle d sou rce. C o n se q u e n tly , o n ly w hen e x trem ely w ide se p a ra tio n s ex ist b etw een co0 an d com ca n th e sin gle tra n s is to r b e used in p la ce o f th e d ifferen tia l p a ir as the tim e -v a ry in g c o n tro lle d so u rce . H o w ev er, w hen the c ircu it o f F ig . 1 1 .5 -3 is em p lo yed , th e sam e d esign c r ite r ia a re used as w ere used fo r th e c ircu it o f F ig . 1 1 .5 -2 . S till o th e r v a r ia tio n s o n the im p le m e n ta tio n o f th e tim e- v ary in g c o n tro lle d c u rre n t so u rce req u ired fo r th e c irc u it o f F ig . 1 1 .5 - l ( a ) w ill be exp lo red in th e p ro b le m s a t th e end o f th e ch a p ter.


T h e c ircu it m o d el o f F ig . 11.5—1(b) ca n be im p lem en ted in e x a c tly the sam e fash io n a s th e m o d e l o f F ig . 11.5—1(a) ex c e p t th a t a n in d u c to r L re p la c e s th e c a p a c i to r C. W ith th e in d u cto r , h ow ev er, a n a d d itio n a l co u p lin g c a p a c ito r m u st be in serted in

series w ith L to p reserv e the b ia s c o n d itio n s o f the c ircu its o f F igs. 1 1 .5 -2 and 1 1 .5 -3 . T h is a d d itio n a l la rg e c a p a c ito r m ak es the in d u ctiv e case less a ttra c tiv e th an the ca p a c itiv e case . A s im ila r a rg u m en t ap p lies to the c ircu it m o d el o f F ig . 11.5 1(c).

A tim e-v ary in g c a p a c ita n c e ca n a lso be o b ta in e d from any d evice w hose stored c h a rg e q is a n o n lin e a r fu n ctio n o f the ap p lied v o ltag e vD. A ny rev erse-b iased P N ju n c t io n h as th is p ro p erty . F o r ex a m p le , if a d evice h as th e re la tio n sh ip q = g(vD), w here g(vD) rep resen ts a n o n lin e a r fu n ctio n o f vD, th en th e d ev ice cu rren t is given by

dq dg(vD). m , IQ.iD = ~r = —— vD = C(vD)vD, (11.5-19)d t (*V[)

w here C (r 0 ) = dg(vD)/dvD. I f a h ig h -freq u en cy d eveloped v o ltag e v(t) and a low -

freq u en cy b ia s an d c o n tro llin g v o ltag e Vcc + V dt) are su p erim p osed a c ro ss the d evice as show n in F ig . 11 .5—4, th en iD(t) ca n b e w ritten as

'o (i) = C [ - Vc c - Vc (t) + ¡Xi)] [i3(f) - V Jt)]

~ c [ - vc c - vc (t) + m m . ( 11-5 2 0 )

[ I t is a p p a re n t th a t Kc (r) is n eg lig ib le in c o m p a riso n w ith v(t) b e ca u se o f th e large freq u en cy se p a ra tio n b etw een th ese tw o v o ltag es.]

In a d d itio n , if it is assu m ed th a t Kc (i) « Vcc and o(i) « Kc c ,-th en E q . (1 1 .5 —18) ca n b e a p p ro x im a te d by its first tw o p o w er-series term s in th e form

C(f)Co ,----------------- «----------------- ,

iD = C(-Vccm + dC{,/Vcc) v(t)v(t) - 8Ci V(c) Vc(t)v(t). ( 1 1.5-21) dVc c dVc c

To oscillator

Fig. 11.5-4 Nonlinear capacitor employed in quasi-static FM generator.


S in ce th e p ro d u ct u(t)i5(t) p ro d u ces te rm s in th e v ic in ity o f d c and 2co0 w h ich are n o t p assed by the L C c irc u it o f F ig . 1 1 .5 -4 , the seco n d co m p o n e n t o f iD m ay be n eg lected in th e co m p le te d ifferen tia l e q u a tio n fo r v (t):

KO + - ^ ~C0;(I)3 £0,(i)2

0 = "W - 7 7 7 ^ * 0 + 7 7 7 7 ^ (1 1 .5 -2 2 )

w here co,(i) = \ /y JL [C 0 + C(i)]- C le a rly E q . (1 1 .5 -2 0 ) is id en tica l w ith E q . ( 1 1 .5 - lb ) an d th u s the c irc u it o f F ig . 1 1 .5 -4 m ay be em p lo yed as a q u a s i-s ta tic F M g en era to r . A sp ecific a p p lica tio n o f th is c irc u it is e x p lo red in th e p ro b le m s a t the end o f the ch ap ter. In p ra c tica l s itu a tio n s o n e in cre a se s b o th th e lin ea r ity and the d ev ia tio n sen sitiv ity by using tw o (diodes in a p u sh -p u ll a rra n g em en t.

S p e cia l h y p e r-a b ru p t ju n c t io n s a re a v a ila b le in w hich C (i) v aries as th e sq u are o f the sig n a l v o lta g e . W ith su ch u n its th e s ta tic re s tr ic tio n s o n m a x im u m d ev ia tio n b e fo re excessiv e d is to rtio n m ay b e so m ew h at relaxed .

In th e o ld e r lite ra tu re th e c irc u it id ea s d escrib ed in th is sec tio n w ere k n o w n as “ re a c ta n ce tu b e ” an d “ v a ra c to r tu n e d ” F M g en erato rs. B o th typ es o f c irc u its .h a v e b een in use fo r o v er 4 0 years. T h e s ta tic re s tr ic tio n s o n the v acu u m tu b e v ersio n s w ere w ell d escrib ed m o re th a n 2 5 years ago . T h e d y n a m ic re s tr ic tio n s an d th e re la tio n sh ip o f th ese g e n era to rs to the e x a c t F M e q u a tio n a re n o t k n o w n to h av e b een p resen ted p rev iously .

11.6 TRIANGULAR-WAVE FREQUENCY MODULATION

T h e g e n era tio n an d n o n m e m o ry w av esh ap in g o f a n F M tr ia n g u la r w ave is th e m o st p ra c tica l m e th o d o f g e n era tin g a s in u so id a l F M signal w hich h a s n o th e o re tica l re s tr ic tio n s o n e ith e r its freq u en cy d ev ia tio n Ao> o r its m a x im u m m o d u la tio n fre q u en cy com. (T h e re s tr ic tio n Aa> < a>0 , to a v o id fu ll-w ave re c tif ica tio n o f th e in s ta n ta n e o u s freq u en cy , d o es , o f co u rse , ap p ly .) In th is se c tio n we shall stud y the p ro b le m o f F M tr ia n g u la r w ave g e n e ra tio n by first p resen tin g a g en era l b lo c k d iag ram o f a tr ia n g u la r w ave g e n e ra to r an d th en im p lem en tin g the b lo c k d ia g ra m w ith p h y sica l c ircu itry . W e sh a ll a ls o co n sid er the im p le m e n ta tio n o f a w av esh ap in g c irc u it w hich c o n v erts th e F M tr ia n g u la r w ave in to a s in u so id a l F M signal.

A s w as p o in ted o u t in S e c tio n 11.2, a n F M tria n g u la r w ave vT(t) is a sy m m etric tr ia n g u la r v o lta g e w hich is p e rio d ic (o f p erio d T = 2n/a>0) w hen p lo tted vs.

t ( i ) = t + — f f ( d ) dd co0 J

(cf. F ig . 11.2-6). W e a lso re ca ll fro m S e c tio n 11.2 th a t su ch a w av efo rm h a s the in s ta n ta n e o u s freq u en cy

ajj(t) = co0 + A a>/(£). (11 .6 -1 )

F ig u re 11.6-1 i llu s tra te s the b lo c k d ia g ra m o f a system w hich p ro d u ces a p erio d ic tr ia n g u la r w ave in z(t) o f p e a k -to -p e a k a m p litu d e 2 V2 and thu s a n F M tr ia n g u la r w ave

11.6 TRIANGULAR-WAVE FREQUENCY MODULATION 543

Voltage-controlledswitch

Integrator constant: K,

FM triangular wave

vr(/)

Inverter \V..v,(i)

Thresholdsensor± V 2

V

VA Vr

m —0

TZV rv(t) = A sin 2-y

(Schmitt trigger)

Fig. 11.6-1 B lo c k d i a g r a m o f a n FM t r i a n g u l a r w a v e g e n e r a to r .

o f th e sam e p e a k -to -p e a k am p litu d e in t. T h e system g en erates a n a ltern a tin g seq u en ce o f p o sitiv ely an d n eg ativ e ly s lo p in g ra m p s in t ( t ) by in te g ra tin g + i,{t) and — i,(i) resp ectiv ely , w here

i|(0 = ^0 1 H /(OL ®o

T o o b ta in th e c o r re c t p e a k -to -p e a k a m p litu d e fo r th e a lte rn a tin g seq u en ce o f ram p s at th e in te g ra to r o u tp u t, a th resh o ld sen so r sw itch es th e in te g ra to r inp u t from i,{t) to — i,(i) w hen th e in te g ra to r o u tp u t re a ch e s V2 an d sw itch es th e in te g ra to r inpu t b a c k to ¡,(i) w hen the in te g ra to r o u tp u t re a ch e s - V2 .

M o re sp ecifica lly , w hen the in p u t v o ltag e to the th resh o ld sen so r rea ch es a value o f V2 , a step o f v o lta g e ap p ea rs a t th e o u tp u t, ca u sin g th e v o lta g e -co n tro lle d S(t) to sw itch from c o n ta c t a to c o n ta c t b. S im ila rly , w hen th e in p u t v o lta g e to th e sen so r rea ch es — V2 , a s tep o f v o lta g e o p p o site to th e o rig in a l step c a u ses S(r) to re tu rn from c o n ta c t b to c o n ta c t a. F ig u re 1 1 .6 -2 illu s tra te s th e in p u t-o u tp u t c h a ra c te r is t ic o f th e

th resh o ld sen so r req u ired to o p e ra te the v o lta g e -co n tro lle d sw itch , as w ell as the co rre sp o n d in g p lo t o f S(t) vs. vT(t).

T o o b ta in a m o re q u a n tita tiv e e x p re ss io n for vT(t) assu m e, a t f = i , , i r (f i) = V2 such th a t S(r) sw itch es fro m a to b. H en ce , fo r t > t( ,

- W ) = -/ o f 1 + — /(*)L %is ap p lied to th e in te g ra to r o f F ig . 1 1 .6 -1 an d Vjist) is given by

vr (t) = V2 - K , J' ifm d o

= V2 - K , 1 0 ( i - t¡) + — Ç m d e" o J ,,

( 11.6- 2 )

— V2 — K , I 0[z(t) 'f(i’i)], f, < t < t i + 1,

w h e re K , is th e in te g ra to r c o n s ta n t. C lea rly , fo r t > f i? vT(t) is a n e g a tiv e r a m p w ith


7 Threshold V

sensor

V,

Switch - control level

- Hysteresis loop

Contact bS

v7

Contact

11

(a)

(b)

Fig. 11.6-2 (a) Plot of vc vs. vT for the threshold sensor, (b) Plot of switch position vs. vT for the threshold sensor-controlled switch com bination.

Fig. 11.6-3 P lo t o f Vj(t) vs. z(t).

slo p e — K , I 0 in x, a s sh o w n in F ig . 1 1 .6 -3 . T h e ra m p re a c h e s — V2 a t t = ti + l , a t w hich tim e S(r) sw itch es to a. F o r t > t i + l , it{ t ) is ap p lied to the in te g ra to r an d vT(t) ta k e s th e fo rm

vT(t) = — V2 + K fI 0[x(t) — t(/,+ 1)], (1 1 .6 -3 )

w hich is a p o sitiv e ram p o f s lo p e K , I Q in x. W h e n v T(t) ag ain re a ch e s V2 a t t = ti + 2 , the cy cle rep eats.

I t is now q u ite a p p a ren t fro m F ig . 1 1 .6 -3 th a t r r (f) is ind eed a p e rio d ic tria n g u la r w ave in x w ith a p erio d

4K 2

K ,I o( 1 1 . 6 - 4 )

11.6 TRIANGULAR-WAVE FREQUENCY MODULATION 5 4 5

an d a p e a k -to -p e a k a m p litu d e 2 V 2 . T h u s i f vT(t) is p laced th ro u g h a n o n lin e a r d ev ice

w ith a tra n sfe r fu n ctio n giv en by

v = A sin - — V, < Vt < Vt , (1 1 .6 -5 )

th e n o n lin e a r d ev ice o u tp u t v(t) is the d esired F M signal

v(t) = A c o s <uot ( 0

= A co s

w here

a>0

w 0t + Aco J f ( d ) d 6 + 6 0

2 tz tzK - j I q

(1 1 .6 -6 )

2 V,

an d 9 0 is an a rb itra ry p h a se an g le w hich m ay b e tak en as zero w ith the c o rre c t c h o ic e o f tim e orig in .

It is o f in te re st to n o te th a t the v o lta g e a t th e o u tp u t o f th e th resh o ld sen so r sits a t o n e v o ltag e level d u rin g th e p o sitiv ely s lo p in g p o rtio n s o f vT(t) and sits at a n o th e r v o ltag e level d u rin g th e n eg ativ ely s lo p in g p o rtio n s o f vT(t). H ence, w hen p lo tted versus t(r), the th resh o ld sen so r o u tp u t is a p erio d ic sq u are w ave w ith the sam e p erio d as Vj{t). T h e re fo re , th e th re sh o ld sen so r o u tp u t vs. tim e is a n F M sq u are w ave. C o n se q u e n tly it is p o ssib le to o b ta in fro m a single n etw o rk an F M tria n g u la r w ave, an F M sq u a re w ave, and a n F M sin u so id a l signal. (N o te the 90° p h ase shift betw een th e sq u a re w ave an d th e tw o o th e r w av eform s.) H o w ever, i f o n ly th e F M sq u are w ave is d esired , a m u ch sim p ler te ch n iq u e for g e n era tin g it is av a ila b le . Su ch a te ch n iq u e is exp lo red in S e ctio n 11.7. T h e a b ility to g en erate several d ifferent typ es o f F M signals s im u lta n eo u sly m a k es the tria n g u la r F M g e n era to r q u ite v a lu ab le as a p iece o f la b o ra to ry test eq u ip m en t.

T h e b lo ck d ia g ra m o f F ig . 1 1 .6 -1 c a n be p h y sica lly im p lem en ted by selectin g c ircu its w hich p erfo rm the fu n ctio n o f ea ch b lo ck and in te rco n n e ctin g them . A lth o u gh there are m an y p o ssib le c ircu its for ea ch o f the b lo ck s, we shall select only one rep resen ta tiv e c ircu it in e a ch case.

Threshold Sensor (Large-Hysteresis Schmitt Trigger Circuit)

T h e d esired th resh o ld sen so r c h a ra c te r is t ic is given in F ig . 11.6 2. F o r h igh -freq u en cy o p e ra tio n the d is ta n ce betw een the tw o sw itch in g p oin ts should be 4 V o r m ore. S u ch an a m p litu d e w ill a llow the tria n g u la r w ave to drive the d iod e w avesh ap in g n etw o rk w ith o u t fu rth er a m p lifica tio n . T h is d irec t d rive is h igh ly d esira b le , s in ce to lin early am p lify a tr ia n g u la r w ave req u ires an ex trem ely w id eban d am p lifier.

T h u s the b a s ic p ro b le m in th e S ch m itt trigg er design in th is ca se is to m axim ize the hysteresis o f the c ircu it. (In the m o re n o rm a l use o f the c ircu it, one m in im izes the hysteresis.) A p o ssib le S ch m itt trigg er c ircu it is illu stra ted in Fig. 11.6 4.

I f Q 2 is a llo w ed to m ove on ly betw een the a ctiv e reg ion and cu to ff, then the o u tp u t is a lw ays iso la ted fro m th e rest o f the c ircu it. A lso , keep in g Q 2 a ctiv e w ill av o id the


Fig. 11.6-4 Schmitt trigger circuit.

u n d esira b le c irc u it “ d e la y ” th a t w ould be cau sed by the tim e n ecessa ry fo r Q 2 to c o m e o u t o f sa tu ra tio n .

T h e re g en e ra tio n feed b a ck p a th is sup plied by th e tw o tra n s is to rs w hen b o th are activ e . I f y,(f) is in itia lly larg e an d n eg ativ e , th en Q l w ill b e o ff an d Q 2 o n :

vE = [ K - (1 - a 2) R J k] - V

W h e n vf(t) in crea ses to the n e ig h b o rh o o d o f the b ra ck e te d term in uE, th en b o th tra n s is to rs b e co m e a c t iv e ; a n d i f th e lo o p g a in exceed s un ity , reg en era tiv e sw itch ing o ccu rs w hich leaves Q , o n and Q 2 off. In cre a sin g v jt ) fu rth er h as n o effect o n the o u tp u t v o ltage. I f o n e s ta r ts to d ecre a se t\{t), th en reverse sw itch in g w ill n o t o cc u r until Qi co m es o u t o f sa tu ra tio n far en o u g h so th a t a p o sitiv e b a se -e m itte r v o lta g e m ay be ap p lied to Q 2 . T h is req u ires v£t) % — <xl R aI k. A gain , reg en erativ e a c tio nand sudden sw itch in g to th e in itia l s ta te w ill o ccu r , f

I f th e tw o in p u t sw itch in g levels a re d en o ted by vit a n d viu fo r lo w er an d up per valu es resp ectiv ely , then

v„ x V x - c t iR Jb an d viu k V, - (1 - oc2) R J k.

H en ce th e p e a k -to -p e a k v alu e o f the tr ia n g u la r w ave is

VPP = - I k R *[<*i - (1 - a 2)] » 0 .9 5 /jR « .

f F or a m ore detailed description of the Schmitt trigger circuit, see J. M illman and H. Taub, Pulse, Digital, and Switching Waveforms, M cG raw -H ill, New York (1965), pp. 389-402.

T h e o u tp u t v o lta g e o f th e c ircu it sw itches betw een and h en ce th e o u tp u t sw itch in g a m p litu d e is a 2 R L/fc. In a d d itio n , to k eep Q 2 in the a c tiv e reg io n ,

Vcc - a2RLI k > K - (1 - «2)/**.or, as a co n se rv a tiv e a p p ro x im a tio n ,

Vcc > K + IkR L.

F ig u re 1 1 .6 -5 illu s tra te s th ese c h a ra c ter is tics .

11.6 TRIANGULAR-WAVE FREQUENCY MODULATION 547

V0

f f r >K + ltRL

ycc - a2 R, lk

V

H - -- v

;/

»,, « K - a x R J k VPP 0.9S/tR,0 a2)R J k

F:g. 11.6-5 Schmitt trigger transfer characteristic.

F o r high-free, uen cy o p e ra tio n o n e tries to k eep the res is tan ce valu es low (b elow

1 k Q ); h en ce fo r larg e o u tp u t a n d in p u t v o lta g es th e cu rren t levels a re a p p re c ia b le — p erh ap s 10 ;n A o r m ore. T h u s Q 2 m u st be ch eck ed fo r excessiv e d issip atio n .

V o lta g o C j p .rolled Sw itch, Inverter, and Integrator

T h e re n a m in g p o rtio n o f th e c ircu it m ay be im p lem en ted by the c ircu it o f F ig . 1 1 .6 -6 .

T h e v o lta g e -co n tro lle d sw itch in g is a cco m p lish e d by the d ifferen tia l p a ir (Q x and Q 2), and th e c a p a c ito r C is em p lo yed as th e in teg ra to r. T ra n s is to r Q } fu n ctio n s as the inverter.

It is a p p a re n t th a t w ith uc (i) less th a n VB b y sev era l tim es kT/q (cf. F ig . 4 .6 -2 ) , ¡¡(f) flow s en tire ly o u t o f Q x, w ith th e resu lt th a t the co lle c to r cu rren t o f Q t has the valu e ic l = ait(t) w hile iC2 = 0. T h e c o lle c to r cu rren t o f Q t , less (1 — x)iE3, flow s th ro u g h R 3 to d ev elo p a v o ltag e

VR3 — [«l 'i — (1 ~ 0t3)i£3]^3- (11.6-7)

W ith the a ssu m p tio n th a t the ju n c t io n v o lta g e a cro ss D e q u a ls the e m itte r-b a sev o ltag e o f Q 3 (w hich is true if D an d Q 3 a re in teg ra ted on the sam e ch ip and have thesa m e g eo m etry an d if iD % iE3), vR3. = vR3 = R'3iE 3; th u s

R'3 '1 — ct3 + ——

« 3Now iC3 = a3iEJ flows into the integrating capacitor.

(11.6-8)


Inverter •<< Ri

y . D

Voltage- controlled

switch ^ S(t)

'« I

R'l= (2a — 1 )R3 « R3

Qi

| *Vi

1*0x— Integrator

+

c - - M<?2

/ vCO?

Inputcurrent

« )

0 vc(i)

' .W = / o [n : 4 f / ( 0 ]

Fig. 11.6-6 Practical im plem entation of the voltage-controlled switch, the inverter, and the in tegrator o f Fig. 11.6-1.

O n the o th e r h an d , w ith vc(t) g re a te r th a n VB by several tim es k T /q , ¿¡(t) flow s e n

tire ly o u t o f Q 2 , iE3 = iC 3 = ¿ci = 0 , an d iC2 = a 2 'i flow s o u t o f th e in teg ra tin g c a p a c ito r . C o n se q u e n tly , i f th e a lp h a s a re eq u a l an d the ra tio R 3/ R 3 is a d ju sted to eq u a l 2a — 1 so th a t iE3 = i, in E q . (1 1 .6 -8 ) , th en as vc exp erien ces a step in crea se fro m sev eral tim es k T / q b elo w VB to sev era l tim es k T / q a b o v e VB, th e in te g ra to r cu rren t sw itch es fro m a i / i ) to — a i/r ); th is is th e d esired effect fo r th e v o lta g e -c o n tro lle d sw itch -in v erter c o m b in a tio n .

W e n o te th a t, w hen a i,(i) flow s in to th e in te g ra to r , th e o u tp u t v o lta g e vT(t) is given by

M t ) = h ^ ) d T ; (1 1 .6 -9 )

th ere fo re , th e in te g ra to r c o n s ta n t K , fo r th e c irc u it o f F ig . 1 1 .6 -6 h as th e value

K t = ~ (1 1 .6 -1 0 )

W e a lso n o te th a t, to k eep Q 2 fro m sa tu ra tin g vc(t)m3[X m u st be less th a n vT(t)nnn + K0 ; to keep Q 3 fro m sa tu ra tin g , Vc c — ¿¡(t)maxR 3 m u st b e g re a te r th a n i>r (t)max.

1 1 .6 TRIANGULAR-WAVE FREQUENCY MODULATION 5 4 9

Fig. 11.6-7 Complete F M triangular wave generator.

F ig u re 1 1 .6 -7 illu s tra te s th e co m p le te c irc u it for th e F M tr ia n g u la r w ave g e n era to r w hich c o m b in e s the th re sh o ld sen so r o f F ig . 1 1 .6 -4 w ith th e c ircu it o f F ig . 1 1 .6 -6 . T w o em itte r fo llo w ers w ith co m p e n sa tin g b a se -e m itte r ju n c t io n b iases are em p loyed to iso la te th e in te g ra tin g c a p a c ito r fro m th e in p u t to th e S ch m itt trigger. In a d d itio n , a d c-lev el sh ifter is in c o rp o ra te d a t th e S ch m itt tr ig g er o u tp u t, w ith the resu lt th a t

uc (0 = v o ( t ) — V 0 — / d c ^ E lO - (1 1 .6 - 1 1 )

T h is sh ift p erm its vc(t)max to b e k ep t less th a n vT(t)mi„ + V0 , as is req u ired to k eep Q 2 fro m sa tu ra tin g . F o r ex a m p le , i f Vj{t)min = — V2 = — 3 V an d Vc c = 10 V , th e n a


c h o ic e o f /dc/?£ io o f 13 V w ould p rev ent Q 2 from sa tu ra tin g . In a d d itio n , if Vc c — <x.RLl k fo r th e S c h m itt tr ig g er is 5 V , th e n vc(t) sw itch es b etw een — (3 V ) — V0 and — (8 V ) — V0 . F o r th is ca se VB sh o u ld b e c h o se n b etw een th e tw o lim its o f vc(t) ; VB = — 5 V w ould b e re a so n a b le . T h e 10 p F c a p a c ito r a c ro ss R E10 co u p le s th e fast rise an d fall tim es o f v0(t) to vc(t).

I f for the F M tria n g u la r w ave g e n e ra to r o f F ig . 1 1 .6 -7 th e S ch m itt trigg er sw itches sta tes w hen its in p u t re a ch e s + V2 an d if R'3 = (2 a — l)/?3 so th a t K , = a/C, then the c a rr ie r frequenc y is given by (cf. E q . 1 1 .6 -6 )

_ n K ,I 0 nccl0 M i ^~ ~ 2 k T ~ 2K 2C ’ U 1 .6 - 12)

w hich is inversely , i p o rtio n a l to C. H en ce , for an y given valu es o f 10 and V2 , C m ay be ch o se n to yield t '.e d esired cen te r freq u en cy fo r the F M tria n g u la r w ave. F o r exa m p le , if I 0 = 3 inA an d V2 = 3 V , C = 157 p F yields a ca rr ie r freq u en cy o f 1 0 7 rad/sec.

T o o b ta in a sp ecM td freq u en cy d ev ia tio n , we first n o te th a t ¡¡(f) m ay b e w ritten in the form

/;(f) = /0 + (1 1 .6 -1 3 )

w here /, = I 0 Aco/co0 . WV then c h o o se 1 1 to o b ta in the d esired Aw. F o r exa m p le , if/0 = 3 m A an d a>0 1 0 7 rad/sec, a freq u en cy d ev ia tio n eq u a l to o n e -h a lf th e ca rr ie rfreq u en cy m ay b e o b ta in e d by c h o o sin g /, = I 0/2 = 1 . 5 m A .

T h e cu rren ts I 0 an d /j a re d ev elop ed by Q l 3 . T h e d io d e D 2 in th e b a se c ircu it o f Q i 3 co m p e n sa te s fo r th e b a se -e m itte r tu rn -o n b ia s so th at

and

(1 1 .6 -1 5 )K 5

I t is a p p a ren t th a t Vx m u st b e selected b e lo w — VB — V0 to k e e p Q l3 fro m sa tu ra tin g . F o r VB = — 5 V , a re a so n a b le valu e fo r Vx w ould be 6 V.

N o w if \\ = 6 V , Vk = 6 V , VEE = 10 V , an d a » 1, and we d esire /0 = 3 m A and /j = 1.5 m A to a ch iev e a>0 = 1 0 7 rad/sec an d Aw = 1 0 7/2 rad/sec, th en fro m E q .

(1 1 .6 -1 5 )

ViR s x t 1 = 4k ii

and from Eq. (11.6-14)

VFF - K

Ra ~ VJR S + I0 = 890 a

1 1 . 6 TRIANGULAR-WAVE FREQUENCY MODULATION 55!

T o co n v ert an F M tr ia n g u la r w ave to a s in u so id a l F M sig n al, a n o n lin e a r n etw o rk w ith th e tra n sfer c h a ra c te r is t ic

v = A s i n ^ f (1 1 .6 -1 6 )

is requ ired , j F ig u re 1 1 .6 -8 illu stra te s th is c h a ra c te r is tic for th e ca se w here A = 1 V an d V2 = 2 V. In a d d itio n , F ig . 1 1 .6 -8 in clu d es th e b rea k p o in ts an d the s lo p e o f a p o ssib le sev en -segm en t p iecew ise -lin ea r a p p ro x im a tio n to th e n o n lin e a r c h a ra c te r is tic. M o re seg m en ts ca n b e ad ded if a c lo se r a p p ro x im a tio n is desired.

F ig u re 11.6—9 illu s tra te s tw o p o ssib le d io d e w av esh ap in g n etw o rk s w hich p h y sica lly rea lize the p iecew ise -lin ea r c h a ra c te r is tic show n in F ig . 1 1 .6 -8 . T h e v o ltag e so u rces in th e n e tw o rk a re c h o se n to co rre sp o n d to the b re a k p o in ts o f F ig . 11.6—8,

Fig. 11.6-8 Plot of v = (1 V) sin [nvT/(4 V)] vs. vT plus a seven-segment piecewise-linear approxim ation of the characteristic.

t See L. Strauss, Wave Generation and Shaping, M cGraw-Hill, New York (1960). Section 2-8 and Problem 2-22 discuss and illustrate a diode waveshaping network. See also G. Klein, "Accurate Triangle-Sine C onverter,” ISSCC Digest o f Technical Papers (Philadelphia), pp. 120-121 (Feb. 1967).


(a)

Rt = l.23R *2 = 0.53/?

0 »

Fig. 11 .6 -9 (a) D iode waveshaping netw ork to realize the piecewise-linear approxim ation of Fig. 11.6-8. (b) M ore practical version of a diode waveshaping network to realize the approxim ation of Fig. 11.6-8.

w hile th e re s is to rs Rl,R2, a n d R3 a re selected to give the c o r re c t s lo p e in e a ch reg ion . F o r exa m p le , R3/(R3 + R) m u st eq u a l 0 .7 8 5 to p ro d u ce th e in itia l s lo p e o f the p iece - w ise-lin ear c h a r a c te r is t ic ; th ere fo re , R3 = 3 .6 4 R . S im ila rly ,

R A R i 0 .4 8 an d „ = 0 . 2 5 ,(Rx\\R3) + R {R.WRAR,) + rw ith th e resu lt th a t Rt = 1.23/? an d R2 = 0 .5 3R. I f m o re seg m en ts a re em p lo yed in th e p iecew ise -lin ea r a p p ro x im a tio n o f th e n o n lin e a r c h a ra c te r is t ic , sev era l m o re d io d e -re s is to r -v o lta g e -s o u rc e b ra n ch e s w ould b e req u ired in th e c irc u it o f F ig .

11.7 PRACTICAL SQUARE-WAVE FREQUENCY MODULATION 553

1 1 .6 -9 . W h e n p h y sica l d io d es are used in p la ce o f th e id eal d io d es o f F ig . 1 1 .6 -9 ,

each dc source should be reduced in magnitude by V0 to account for the turn-on bias ofth e d iodes.

S in ce the key to low d is to rtio n in su ch a n e tw o rk is sym m etry it is a p p a ren t th at

the circuit of Fig. 11.6—9(b) is much to be preferred. This is true because it avoids the n ecessity o f m a tc h in g p a irs o f re s is to rs fo r R i an d R 2, e tc ., as is th e ca se fo r c ircu its o f th e typ e rep resen ted by F ig . 1 1 .6 -9 (a )

W ith w ell-m a tch ed co m m e rc ia l s ilico n d io d es ( l N 9 1 4 B ’s, fo r exam p le) an d co m m e rc ia l 1 % re s is to rs , a f iv e -b re a k p o in t (p er h a lf-cy cle) n e tw o rk c a n yield to ta l h a rm o n ic d is to rtio n s o f 50 d B b elo w th e fu n d am en ta l. W h en “ h o t c a r r ie r” d iod es (V0 ~ 0 .4 V ) a re used to m in im ize ch a rg e s to ra g e p ro b lem s, th en ca rr ie r freq u en cies o f up to 30 M H z o r m o re c a n b e h an dled .

11.7 PRACTICAL SQUARE-W AVE FREQ U EN CY M O D U LA TIO N

S q u are-w av e freq u en cy m o d u la tio n fo llo w ed by n a rro w b a n d filte rin g is em p loyed in m an y p ra c tica l s itu a tio n s . In p a rticu la r , it is a p p lica b le in th o se s itu a tio n s w here the d ev ia tio n ra tio D = Aco/<x>0 m u st be g re a te r th a n th e few p e rce n t p o ssib le w ith th e q u a s i-s ta tic F M m o d u la to rs an d yet n o t so g re a t th a t D d eterm in es a p o in t a b o v e th e to p cu rv e o f F ig . 1 1 .2 -3 an d thu s p rev en ts the e x tra c tio n o f th e fu n d am en ta l F M w ave by filtering , f In g en era l, th is ra n g e o f d ev ia tio n ra tio s usually arises w hen a large

number o f re a so n a b ly lo w -freq u en cy carriers are to b e m o d u la ted in freq u en cy by in d ep en d en t in fo rm a tio n sig n a ls, so th a t they c a n b e m u ltip lexed in freq u en cy , and th en em p lo yed c o lle ctiv e ly as th e m o d u la tio n fo r a h igh -freq u en cy F M o r A M signal. M a n y sy stem s w h ich m u st tra n sm it a la rg e q u a n tity o f in d ep en d en t d a ta o r a larg e n u m b e r o f v o ice ch a n n e ls d o so in th is fash ion .

A lth o u g h a n F M sq u a re w ave ca n be o b ta in e d as a b y -p ro d u ct o f g en era tin g an F M tr ia n g u la r w ave (as w e saw in S e c tio n 11.6), an F M sq u a re w ave c a n b e gen erated d irec tly an d m o re sim p ly by c o n tro llin g th e freq u en cy o f a sy m m etric a s ta b le m u ltiv ib ra to r h av in g th e b a s ic fo rm sh o w n in F ig . 1 1 .7 -1 . E a ch am p lifie r in th e m u ltiv ib ra to r o f F ig . 1 1 .7 -1 is assu m ed to h a v e th e in p u t, o u tp u t, and tra n sfer c h a ra c te r is tic s show n in F ig . 1 1 .7 -2 . T h e se c h a ra c te r is tic s c lo se ly a p p ro x im a te th o se o f in teg ra ted in v erters w hich in c o rp o ra te a n o u tp u t buffer am p lifier.

B e fo re d ete rm in in g h o w th e a s ta b le m u ltiv ib ra to r p erfo rm s as an F M g e n era to r , le t us review its b a s ic o p e ra tio n fo r th e ca se w here I(t ) = J 0 (a co n sta n t). J F o r th is case , the w av efo rm s o f vn , vi2 , v o l , a n d vg2 a re show n in F ig . 1 1 .7 -3 . R e fe rrin g to th e w av eform s, we o b serv e th a t, fo r t < 0 , vBl = 0 an d vi2 < 0 ; thu s th e inp u t to a m p lifier 2 a p p e a rs as an o p e n c irc u it (am p lifie r 2 is cu t off), w hile th e o u tp u t o f am p lifie r 1

t Since a square wave contains only odd harm onics of its fundam ental frequency, the filter following the FM square wave generator must only be capable o f separating the FM wave centered at o)g from the FM wave centered at 3<w0.t In this treatm ent we are assum ing that the reader has some familiarity with the basic operation of m ultivibrators. If this is no t the case, he should consult either J. M illman and H. Taub, Pulse, Digital, and Switching Waveforms, M cGraw-Hill, New Y ork (1965), C hapter 11, o r L. Strauss, Wave Generation and Shaping, 2nd edition, M cGraw-Hill, New York (1970), pp. 385-389.


C, = C2= C

/(/) = / « + / , / ( 0 = / o [l + \Fig. 11.7-1 ,.FM square-wave generator employing a stable multivibrator.

Output impedance « 0 Reverse transfer impedance ~ 0

1,

Inputcharacteristic

V,1 '

V i' Forward transfer characteristic

Fig. 11.7-2 Input and transfer characteristics for the amplifiers in Fig. 11.7-1.

a p p ea rs as a sh o rt c ircu it (am p lifier 1 is sa tu ra ted ). T h e re fo re , /0 flow s e n tire ly in to C j , ca u sin g vi2(t) to in crea se as a lin ea r ram p w ith slo p e I 0/C . H o w ever, a t t = 0 , w hen vi2(t) = V6, the in p u t to am p lifie r 2 a p p e a rs as an in cre m e n ta l sh o rt c ircu it (cf. F ig . 1 1 .7 -2 ) ca u sin g I 0 to flow en tire ly in to am p lifie r 2 in stead o f C , . I f I 0 > I c (w hich we assu m e it is), th en v02(t) d ro p s in sta n tly to zero , w ith the resu lt th a t vn (t) d ro p s an eq u a l a m o u n t V2 , s in ce vC2(f) c a n n o t ch an g e in s ta n ta n e o u sly . T h e d ecrease in Vn below Vb ca u ses in to eq u a l 0 an d vol to rise rap id ly to V2 . (E v e n th o u g h C ] is co n n e cte d d irec tly a cro ss vo l , th e sm all assu m ed o u tp u t im p ed a n ce p erm its C j to be ch arg ed a lm o st in s ta n ta n e o u sly .) A t th is p o in t I 0 flow s in to C 2 , ca u sin g Dn to in cre a se a s a ra m p w ith slo p e I J C w hile vol = V2 , vi2 = V{ , an d vo2 = 0. W h en , at t = T /2 , vn = Vs , th e c irc u it sw itch es to its o rig in a l s ta te an d th e p ro cess co n tin u es .

11.7 PRACTICAL SQUARE-WAVE FREQUENCY MODULATION 555

Hi

"’I2 CV2

/o

t"" T 2n

Wo

V2

F r o m the w av efo rm o f vn o r vi2 it is a p p a ren t th a t the s lo p e I J C tim es T/2 m u st eq u a l V2 ; h en ce

T =2 V2C 1

r


V,.C'

(1 1 .7 -1 )

(1 1 .7 -2 )

In th e a b o v e an a ly sis we assu m ed /0 > I c . I f th is w ere n o t th e ca se , v0, and vo2 w ouid n o t d ro p to zero v 'hen th e resp ectiv e am p lifiers w ere on , b u t ra th e r w ould rem ain a t som e v o lta g e w hich w ould p ro b a b ly b e stro n g ly d ep en d en t on the ind iv idu al a m p lifica tio n fa c to rs o f the co m p o n e n ts m a k in g up the am p lifiers. In a d d itio n , th is v o ltag e w hich w ould p ro b a b ly b e sen sitive to tem p eratu re and p ow er supply v o lta g e , w ould d eterm in e the size o f the n eg ativ e step in vn and vi2 , an d thu s w ould stro n g ly in flu ence th e p erio d T . C o n se q u e n tly , in an y ra tio n a l d esign I 0 is ch o sen to be g re a te r th a n I c .

N ow if th e c u rren t d rives in the c ircu it o f F ig . 1 1 .7 -1 are p erm itted to be fu n ctio n s o f tim e w ith th e re s tr ic tio n

/(f) > I c for a ll f, (11 .7-3)

then the basic w aveform s o f Fig. 11 .7 -3 rem ain the sam e except that the r, w aveform s


fo r Vi < 0 b e co m e freq u en cy -m o d u la ted a n d ta k e th e form

vtjit) = - (V2 - Vt) + ± 1 + r f ( e ) dd• 7 = 1 o r 2 , (1 1 .7 -4 )*00

w here /(f) = I x f ( t ) an d t; is th e tim e a t w hich th e c ircu it sw itched fro m its prev iou s state . W e m ay rew rite E q . (1 1 .7 -4 ) in th e fo rm

w here i(t) = t + (I J I 0) J ' f ( 9 ) d 6 ; h e n ce vi}(t) vs. t is a lin e a r ra m p w ith slo p e I o/C w hich s ta r ts a t — (K2 — Vs) an d te rm in a te s a t Va, a t w hich p o in t th e c irc u it sw itches states. C lea rly , th en , th e w av efo rm s vn , v o l , vi2 , an d vo2 vs. x(t) a re id en tica l w ith the co rre sp o n d in g w av efo rm s vs. t fo r th e c a se w here I(t) = I 0 . C o n se q u e n tly th e o u tp u t sq u a re w ave is p e rio d ic in t w ith p erio d T = 2V 2C / I 0(ot co0 = n l J V 2C ), w hich is th e req u ired fo rm fo r a n F M sq u a re w ave w ith in s ta n ta n e o u s freq u en cy

In th e ease o f a n F M sq u a re w ave g e n era to r , it is ev en m o re essen tia l to keep I(t) > I c . I f th is req u irem en t is n o t m et, the p e a k -to -p e a k am p litu d e o f v„(t) w ill vary w ith tim e as /(f) varies re la tiv e to I c . In a d d itio n , s in ce T i s a fu n ctio n o f b o th I{t) an d the p e a k -to -p e a k am p litu d e o f v„ , th e in s ta n ta n e o u s freq u en cy w ill n o t vary lin early w ith /(f).

E x a m p le 1 1 .7 -1 G iv e n tw o am p lifiers w ith V2 = 5 V , Vs = 0 V , and I c = 1 m A , c h o o se valu es f o r /(<) = I 0 + I t f ( t ) an d C su ch th a t th e c irc u it o f F ig . 1 1 .7 -1 p ro d u ces an F M sq u a re w ave w ith cy,(f) = 1 0 8 + 5 x 1 0 7/ (f) rad/sec.

Solution.- S in ce Aoj/wq = I ¡ /¡o — b

T h u s we m u st c h o o se I 0 > 2 m A to k eep a c o n s ta n t p e a k -to -p e a k am p litu d e o u tp u t sq u a re w ave. A c h o ic e o f 3 m A fo r I 0 w ould b e re a so n a b le . H en ce I x = 1.5 m A. In ad d itio n , s in ce co0 = 1 0 8 rad/sec an d sin ce a>0 = n I 0/V 2C , C = 19 p F .

T h e p ro b le m now rem a in s o f im p lem en tin g the am p lifiers an d c u rren t so u rces o f F ig . 1 1 .7 -1 w ith p ra c tica l c ircu its . O n e p o ssib le am p lifie r co n fig u ra tio n em p lo y in g d iscre te tra n s isto rs is show n in F ig . 1 1 .7 -4 . F o r th is c ircu it g , p ro v id es th e b a s ic a m p lifica tio n , w h ereas Q 2 , fu n ctio n in g as an e m itte r fo llo w er, p ro v id es th e low o u tp u t im p ed an ce n ecessa ry to en su re th a t v„(t), w h ich is lo ad ed by th e c a p a c ito r C ,

Vijit) = —(V 2 - Va) + ^ [ t ( i ) - T(f,)], (1 1 .7 -5 )

(1 1 .7 -6 )

I f I J I 0 is ad ju sted to eq u a l A o)/oj0 , th en

w,-(f) = ft»0 + Aft>/(f). (1 1 .7 -7 )

I(t)min = l 0 ~ h = U 2 .

11 . 7 PRACTICAL SQUARE-WAVE FREQUENCY MODULATION 557

Fig. 11.7-4 Discrete com ponent im plem entation of amplifiers in the circuit of Fig. 11.7-1.

rises rap id ly w hen Q ! is cu t off. S p ecifica lly , th e tim e c o n sta n t c o n tro llin g th e rise tim e is c lo se ly a p p ro x im a te d by

r = [R c ( 1 - a)||K*]C.

T h e d io d e v o lta g e-so u rce c o m b in a tio n (D 2-V2) p erfo rm s tw o fu n ctio n s . F ir s t , it c a tch e s th e o u tp u t v o ltag e as it rises to w a rd its s tea d y -s ta te valu e, th u s fu rth er im p ro v in g the r ise tim e o f th e o u tp u t sq u a re w ave. S e co n d , it e s ta b lish es th e p eak o u tp u t v o ltag e a t a p p ro x im a te ly V2 (V2 < Vc c ), w hich is in d ep en d en t o f supply v o ltag e v a ria tio n s if V2 is d evelop ed by a Z en er d iod e. H en ce co0 = n I 0/V 2C is less sen sitive to pow er supply v a ria tio n s .

T h e d io d e D i p ro v id es a co m p e n sa tin g ju n c t io n v o ltag e d ro p for th a t o f Q 2 w hen g j is sa tu ra te d , th ereb y en su rin g v0 « 0 w hen Q 1 is tu rn ed on . I f D t w ere n o t

inclu d ed , v0 w ould vary by V2 + V0 in stead o f V2 w hen Q t sw itches b etw een sa tu ra tio n and c u to f f ; h en ce V2 w ould b e rep laced by V2 + V0 in d eterm in in g co0 , an d a>0 w ould be s ig n ifican tly m o re d ep en d en t o n tem p eratu re .

I f fo r the c irc u it o f F ig . 1 1 .7 -4 we assu m e th a t th e b a se -e m itte r ju n c t io n s o f <2i and Q 2 and th e d io d es D x an d D 2 c a n be m o d eled as id eal d io d es in series w ith rev erse- b ia sin g b a tte r ies o f F0 » 0 .7 V (fo r s ilico n tra n s is to rs and d io d es), then we o b serv e th a t th e i c h a r a c t e r i s t i c o f th e am p lifie r is id en tica l to th a t o f F ig . 1 1 .7 -2 w ith Vg = 2 V0 . T h e fo rw ard tra n sfe r c h a ra c te r is t ic o f th e am p lifie r is a lso id en tica l to th a t o f F ig . 1 1 .7 -2 w ith I c = (Vc c — Fq)//?/^, w here /? = a/(l — a). N o te th a t I c is th e valu e o f th e b ase cu rre n t to Q t w h ich cau ses Q t to s a tu ra te ; th a t is, vCEl & 0. S in ce litt le o r no signal p asses b etw een th e c o lle c to r and the b a se o f Q , (even w ith o u t


the em itte r fo llo w er), the reverse tra n sfe r im p ed an ce is essen tia lly zero w hile the o u tp u t im p ed an ce, b e ca u se o f the e m itte r fo llo w er, is q u ite sm all.

T o keep the o u tp u t im p ed an ce sm all, the em itter fo llo w er m u st rem a in in its a c tiv e reg ion a t a ll tim es. H o w ev er, w hen the am p lifier is o p e ra tin g w ith in the a s ta b le m u ltiv ib ra to r , an d w hen Q i is sa tu ra te d (vB % 0), th e cu rrerit /(f) m u st flow in to the v0(t) te rm in a l from the c a p a c ito r w hich is bein g ch arg ed . T h u s to k eep Q 2 a c tiv e , th e cu rren t th ro u g h R E m u st e xceed i( t) o r.

^ > !/U)lma, (1 1 -7 -8 )

E q u a tio n (1 1 .7 -8 ) d eterm in es th e m a x im u m size o f R E. F o r exa m p le , if |/(f)|max =4 .5 m A (E x a m p le 1 1 .7 -1 ) an d VEE = 9 V , R E m u st be ch o se n to be less th an 2 kQ. T h e re s is to r R c is u sually ch o se n som ew h ere b etw een 2 0 0 Q an d 5 kC2; th is is a c o m p ro m ise b etw een fast sw itch in g tim es (sm all re s is to rs a re n o t affected as m u ch as large res is to rs by s tra y c a p a c ita n c e ) an d p ow er d issip atio n .

F ig u re 1 1 .7 -5 illu stra te s a co m p le te F M sq u a re w ave g e n e ra to r em p lo y in g the am p lifiers o f F ig . 1 1 .7 -4 . F o r th is g e n e ra to r Q s and Q b a c t as th e cu rren t so u rces /(f). T h e K3 an d VA v o lta g e so u rces p ro v id e th e b ias fo r Q s an d Q 6 , w hile D s (w hich is tu rn ed o n by R ) p ro v id es a co m p e n sa tin g ju n c t io n v o lta g e fo r th e b a se -e m itte r v o ltag es o f Q s an d Q 6 so th a t som e d egree o f th e rm a l s ta b ility is o b ta in ed .

A gain , fo r th e c irc u it o f F ig . 1 1 .7 -5 , i f th e b a se -e m itte r ju n c t io n s o f Q 5 and Q 6 as w ell as D s a re m o d eled by a n id eal d io d e -b a tte ry co m b in a tio n , then

10 11

J k - y3R 1 R 2J R 2

I{t) = - - + ~ / ( t ) . (1 1 .7 -9 )

S in ce b o th V3 a n d V4 p lay a ro le in d ete rm in in g /(f), b o th sh o u ld be Z e n e r c o n tro lle d to k eep I(t) an d in tu rn a>,(t) in d ep en d en t o f v a r ia tio n s in supply v o ltag e. In a d d itio n , the in eq u ality

V3 > V S (1 1 .7 -1 0 )

m u st be satisfied to keep Q 5 and Q 6 fro m sa tu ra tin g w hen e ith e r <2, o r Q 3 is on.T h e res is to rs R t an d R 2 a re c h o se n to yield th e d esired valu es o f /0 an d /( o n ce

Vx, K3 , and V4 a re selected . F o r exa m p le , if Vt = 3 V , V3 = 2 V , V4 = 10 V , and a = 1, w hile (from E x a m p le 1 1 .7 -1 ) /0 = 3 m A an d /, = 1.5 m A , then

R 2 = ^ = 2 k Q a n d R , = ^ - = 2 k Q .‘ 1 ' 3 / K - z ~r I Q

T h e c ircu it o f F ig . 1 1 .7 -5 a lso illu stra te s a sim p le b a n d p a ss filter for e x tra c tin g th e fu n d a m en ta l F M w ave fro m th e F M sq u a re w ave t sq(f). S p ecifica lly , if th e filter B W = 1 !R l C is su fficien tly w ide to pass the fu n d am en ta l F M co m p o n e n t o f r sq(f) w hile a tte n u a tin g all o th e r F M c o m p o n e n ts , then the filter o u tp u t t>(f) is given by

2 V 2v{i) = — co s

n ■ rco0 t + Aa> f ( 9 ) d d f 9 0 ( 1 1 . 7 - 1 1 )

l l . i MISCELLANEOUS FREQUENCY MODULATOR 559

w here

and

Fig. 11.7-5 Complete FM square-wave generator.

n l 0 n ¡V 4 ~ V3co n =

3 V 3 ).|1±----V2C V2C \ R i R

nil n ViA to =

V2C V2C R 2

(N o te th a t the o u tp u t filter sh o w n h as a tra n sfer fu n ctio n o f u n ity at co0 .)

11.8 M ISC E L L A N E O U S FR EQ U EN C Y M O D U LA T O R S— T H E A R M STR O N G M ET H O D

In th is sec tio n w e sh a ll c o n sid er sev era l a d d itio n a l m eth o d s o f p ro d u cin g F M signals. T h e se m eth o d s in clu d e th e Armstrong method and th e m eth o d o f v ary in g the e le c tr ica l tim e d elay o f a d elay line. N e ith e r o f th ese m eth o d s is so v ersa tile as the


p rev io u sly d iscu ssed m eth o d s, s in ce th ey a re q u ite re stric te d in th e m a x im u m size o f b o th Aco an d a>m re la tiv e to co0 . T h e A rm stro n g m e th o d (n am ed a fte r its in v en to r E . H . A rm stro n g , w h o w as a p io n e e r in th e field o f freq u en cy m o d u la tio n ) h a s the a d v a n ta g e o f c a rr ie r freq u en cy s ta b ility , s in ce its c a rr ie r m ay b e o b ta in e d fro m a c ry sta l o sc illa to r . In a d d itio n , it is o f in te re st h is to r ica lly , s in ce it w as o n e o f th e first m eth o d s o f F M g e n era tio n .

T h e v a ria tio n o f th e e le c tr ica l tim e d elay o f a d elay lin e is o f in te re st b eca u se th is te ch n iq u e a p p e a rs to b e th e m e th o d by w hich th e freq u en cy m o d u la tio n o f a laser b e a m c a n b e m o st ea sily a cco m p lish ed . S p ecifica lly , i f a s tro n g tra n sv erse e le c tr ic field is p laced a c ro ss sp ecia lly co n stru c te d tra n sp a re n t c ry sta ls , u sually cad m iu m su lp hid e, th e d ie le c tr ic c o n s ta n t an d , in tu rn , th e v e lo c ity o f w ave p ro p a g a tio n th ro u g h th e cry sta l v ary w ith ch a n g es in th e ap p lied e le c tr ic field. S in ce th e c ry s ta l h a s a fin ite len g th d ; v a r ia tio n s in th e p ro p a g a tio n v e lo c ity u resu lt in co rre sp o n d in g v a ria tio n s in th e tim e ta k e n by a n e le c tro m a g n e tic w ave, la ser lig h t in th is ca se , to pass th ro u g h the c r y s ta l ; i.e. th e tim e d elay th ro u g h th e cry sta l is g iven by

t o = - . (1 1 .8 -1 )v

H en ce v ary in g th e ap p lied field as a fu n ctio n o f tim e p ro d u ces co rre sp o n d in g v a ria tio n s in i0(i) w h ich , as we sh a ll see, a re su fficien t to p ro d u ce a freq u en cy -m o d u la ted signal.

T h e A rm stro n g m e th o d is a d irec t resu lt o f the fact th a t, fo r very sm all freq u en cy d ev ia tio n s, a n F M sig nal o f th e fo rm

co0t + Aco J f ( 0 ) , ( 11.8-2 )v(t) = A co s

w hich m ay b e exp an d ed as

v(t) = A c o s co0t c o s Aco J f ( 9 ) d d — A s in co0t sin Aco J” f (& )d 0 , (1 1 .8 -3 )

c a n b e c lo se ly a p p ro x im a te d by

v(t) x A c o s co0f — A Aco j * f { 0 ) dd s in co0t. (1 1 .8 —4)

E q u a tio n (1 1 .8 -4 ) fo llo w s fro m the fa c t th a t, fo r x 1 and sin </>*</> w ith in 2 % . H o w ever, E q . (1 1 .8 -4 ) m a y b e rea lized by th e b lo c k d ia g ra m o f F ig . 1 1 .8 -1 , w hich co m b in e s A c o s co0i w ith th e A M signal

A A to f f { 0 ) d 0 s in co0t.

Since co0 may be crystal controlled, the carrier frequency is quite stable.

11.8 MISCELLANEOUS FREQUENCY MODULATOR 561

A c o s u v Crystal oscillator

Fig. 11.8-1 Block diagram o f Armstrong system for generating FM .

T h e o n ly d ifficu lty is th a t Aw fo r th e system o f F ig . 1 1 .8 -1 o f n ecessity m u st beq u ite sm all. C o n sid e r , fo r exa m p le , f ( t ) = c o s wt. F o r th is case

Aw f f ( 6 ) dd = sin wt,J w

w hich h a s a p ea k v a lu e o f Aw /w . F o r th e A rm stro n g m o d u la to r to fu n ctio n co rre c tly in th is case , A w /a) m u st b e less th a n o r eq u a l to 0 .2 . F o r th e m o re g en era l ca se in w hich f ( t ) is b a n d -lim ite d b etw een w L an d w m, Aa> J ' f ( 9 ) d6 h as a p eak v alu e ran g in g from a m a x im u m o f Ajj} /w l [w hen a ll th e en erg y o f f ( l ) is c o n ce n tra te d a t tt>L] to a m in im u m o f A w /w m [w hen a ll th e en erg y o f f ( t ) is co n ce n tra te d a t com]. H ow ever, s in ce th e p e a k v a lu e o f A w §' f (9 ) dO m u st b e less th a n 0 .2 fo r a ll p o ssib le freq u en cy d is tr ib u tio n s o f f ( t ) , Aco m u st b e su fficien tly sm all th a t

Aa> < 0 .2 . (1 1 .8 -5 )(oL

In p ra ctice , E q . (1 1 .8 -5 ) is u sually re laxed to

Aft)— < 0 . 5 , (1 1 .8 -6 )

sin ce very rarely is a ll the en erg y o f f ( t ) c o n ce n tra te d a t w L. E v en E q . (1 1 .8 -6 ) req u ires ex tre m e ly sm all v a lu es o f Aw. I f f ( t ) is b a n d -lim ite d betw een 100 H z an d 10 k H z, Aw m u st b e less th a n 27i(50 H z). S in ce A w is d irectly d eterm in ed by th e v alu e o f wL , th e m in im u m v alu e o f w L sh o u ld b e n o sm a lle r th a n n ecessary w hen using th e A rm stro n g m o d u la to r.

T o o v erco m e th e p ro b le m o f th e sm all freq u en cy d ev ia tio n a p p e a rin g a t the o u tp u t o f th e A rm stro n g m o d u la to r , we m ay p la ce the o u tp u t sig nal th ro u g h a seq u en ce o f freq u en cy d o u b le rs an d trip lers| (o f th e fo rm d iscu ssed in S e c tio n 1.4),

t H igher-order m ultiplication is usually not attem pted because of the difficulty in extracting the higher harm onics at the m ultiplier output with conventional tuned circuits.


th ereb y in cre a sin g the freq u en cy d ev ia tio n , as w as p o in ted o u t in S e c tio n 11.2. F o r ex a m p le , if th e A rm stro n g m o d u la to r p ro d u ces an F M sig n al w ith a)0 = 27r(100 kH z) a n d Au> = 27t(50 H z), by p la c in g ten freq u en cy d o u b lers a fte r th e m o d u la to r we o b ta in a t th e o u tp u t o f th e la s t d o u b le r a freq u en cy d ev ia tio n o f 27t(50 H z) x 2 10 = 2 ti(5 1 .2 kH z). In a d d itio n , we o b ta in a c a rr ie r freq u en cy o f 2 7 t(1 0 0 k H z ) x 2 10 = 27t(102.4 M H z ). T h is c a rr ie r m ay b e h e tero d y n ed to any d esired freq u en cy b y any o f th e te ch n iq u e s o f C h a p te r 7 w ith n o effect on th e freq u en cy d ev ia tio n . A n o th e r c ry s ta l-c o n tro lle d o sc illa to r sh o u ld be em p lo yed in th e m ixer to m a in ta in th e ca rr ie r freq u en cy stab ility .

S in ce a ll th e b lo c k s o f th e d ia g ra m o f F ig . 1 1 .8 -1 ca n b e p h y sica lly im p lem en ted by c ircu its d iscu ssed elsew h ere in th is b o o k , we leave the p h y sica l c ircu it co n stru c tio n as an exerc ise for th e read er.

Variable T im e -D e la y M o d u la to r

I f the d elay e x p erien ced by a c o n sta n t-a m p iitu d e sin u so id A c o s co0r p assin g th ro u g h a n e tw o rk is a fu n ctio n o f tim e, i.e., i f t0 = f0(f), th en th e s ig n al a p p e a rin g a t th e n e tw o rk o u tp u t h a s th e fo rm

v(t) = K A c o s [co0t — (w0r0(i)], (1 1 .8 -7 )

w here K is th e a tte n u a tio n in tro d u ced by th e n etw o rk . N o w if f 0(i) c a n som eh o w be cau sed to vary in p ro p o r tio n to J '/ ( 0 ) dO, i.e., if

m = — ( m d o , ( n . 8 - 8 )ft}0 J

th en a n F M sig n al o f th e d esired fo rm is p ro d u ced a t the n e tw o rk o u tp u t. In th e ca se o f th e freq u en cy m o d u la tio n o f a la se r b eam , th e field a c ro ss th e cry sta l th ro u gh w hich th e b e a m is p assed m u st b e v aried in p ro p o rtio n to j ' f (8) d 8 to p ro d u ce a tim e d elay o f th e fo rm given by E q . (1 1 .8 -8 ) .

11.9 FREQ U EN CY STA B ILIZA TIO N O F FREQ U EN CY M O D U LA TO R S

In m an y a p p lica tio n s th e c a rrie r freq u en cy a t th e o u tp u t o f a n F M g e n e ra to r is req u ired to b e ex tre m e ly s ta b le . F o r e x a m p le , th e F e d e ra l C o m m u n ic a tio n s C o m m issio n req u ires th a t co m m e rc ia l F M s ta tio n s (88 M H z to 108 M H z ) m a in ta in a ca rr ie r freq u en cy s ta b ility o f + 2 k H z a b o u t th e ir assign ed s ta tio n freq u en cy . T h isco rre sp o n d s to m a x im u m c a rr ie r v a r ia tio n o f tw o p a rts in 1 0 5 fo r a s ta tio n a t100 M H z . S u ch s ta b ility is u su ally n o t d irec tly o b ta in a b le w ith a n y o f th e F M g e n era to rs p rev io u sly d iscu ssed , w ith th e p o ssib le e x c e p tio n o f th e c ry s ta l c o n tro lle d A rm stro n g g en era to r .

T h is s ta b ility m a y b e o b ta in e d b y c o m p a rin g th e o u tp u t s ig n a l o f a n F M g e n era to r w ith a cry sta l o s c illa to r in su ch a fa sh io n th a t a n e rro r s ig n a l resu lts i f th e F M c a rr ie r is n o t a t its sp ecified freq u en cy . T h is e rro r s ig n a l is th en am p lified an d fed b a c k to th e F M g e n e ra to r in p u t w ith th e c o r re c t p h a se to re tu rn th e F M c a rr ie r to its req u ired frequ ency .

11.9 FREQUENCY STABILIZATION OF FREQUENCY MODULATORS 563

Fig. 11.9-1 Block diagram of frequency stabilization system.

A w idely used freq u en cy s ta b iliz a tio n fee d b a ck system is sh o w n in b lo ck d ia g ra m

fo rm in F ig . 1 1 .9 -1 . In th is figure th e o u tp u t o f th e F M g e n e ra to r is assu m ed to be o f th e form

y(t) = A cos w 0t + Et + yt + Aco j* f ( 9 ) d d

w here w 0 is th e d esired c a rr ie r freq u en cy , y is th e freq u en cy sh ift d ue to th e fe ed b ack lo o p , an d e is the freq u en cy d ifferen ce b etw een the a c tu a l c a rr ie r freq u en cy w ith n o feed b a ck an d th e d esired c a rr ie r freq u en cy . T h e F M g e n era to r o u tp u t is m ixed w ith a h igh ly s ta b le c ry s ta l-c o n tro lle d o sc illa to r an d th e d ifference te rm is ex tra c te d . I f th e c ry s ta l-c o n tro lle d o sc illa to r o p e ra te s a t the freq u en cy co0 — cod, th en th e m ixer o u tp u t h as th e fo rm

v t(t) = C c o s + y)t + Aco Jf (G )d 6J. (1 1 .9 -2 )

T h e m ixer o u tp u t is th en p la ced th ro u g h a freq u en cy d em o d u la to r (to b e d iscu ssed in C h a p te r 12) w h ich h a s th e tra n sfe r c h a ra c te r is t ic b etw een in s ta n ta n e o u s freq u en cy and o u tp u t v o lta g e sh o w n in F ig . 1 1 .9 -1 , w ith th e resu lt th a t th e d e m o d u la to r o u tp u t is g iven by

v2(t) - K d[e + y - öa) + Acof{t)]. (11.9-3)


(In reality , th e c e n te r freq u en cy o f the d iscr im in a to r sh ou ld b e a>d. T h e term Soj rep resen ts th e e rro r in o b ta in in g th e d esired frequ ency . T h is term c o m b in e s the in a ccu ra c ie s o f th e d em o d u la to r and the c ry s ta l o sc illa to r .)

T h e d iscr im in a to r o u tp u t is p laced th ro u g h a lo w -p ass filter w hich is su fficien tly n a rro w to rem o v e th e / (i)- te rm fro m v2(t). T h e filter o u tp u t is th en am p lified by— A 0 an d ap p lied to th e in p u t o f th e F M g en era to r . T o p ro d u ce a te rm yt in the

p h ase o f the F M g e n e ra to r o u tp u t, th e filter o u tp u t m ust have the valu e y /K g a s w ell as - A 0K d[e + y — da) ] ; thu s

7 = ~ A 0K gK d[e + y - <5«], (1 1 .9 -4 )

w here K g is th e g e n e ra to r c o n s ta n t w hich re la te s inp u t v o ltag e to freq u en cy d ev iatio n . E q u a tio n (1 1 .9 -4 ) m ay b e re a rra n g ed in the form

- A 0K gK d(e - S o ) - e + S oj n 1 9 -5 17 1 + A 0K gK d 1 + l / A 0K gK d K ‘

F r o m E q . (1 1 .9 -5 ) it is a p p a ren t th a t, as A 0K gK d a p p ro a ch e s in fin ity , y a p p ro a ch e s— e + da) and th e resu lta n t n et c a rr ie r freq u en cy a p p ro a ch e s a>0 + Sw. T h u s for a h ig h -g a in fe ed b ack system th e fin a l s ta b ility is a fu n ctio n o f o n ly the cry sta l o sc illa to r s ta b ility and th e d em o d u la to r s tab ility .

O n c e o n e h as a h ig h -g a in (n o n o sc illa tin g ) system , o n e h as th e p ro b le m o f sp rea d ing th e a llo w a b le v a ria tio n s b etw een th e cry sta l o sc illa to r an d the freq u en cy d em o d u lator. I f the re feren ce freq u en cy is very c lo se to the d esired o u tp u t freq u en cy , then cod w ill b e m u ch lo w er th a n oi0 and a 1 % d etu n in g o f the d e m o d u la to r w ill ca u se a m u ch sm a ller p e rcen ta g e ch a n g e in to0 .

A s we shall see, freq u en cy d em o d u la to rs ca n be c o n stru c te d o f q u a rtz cry sta l tu n in g e le m e n ts ; h en ce , w ith p ro p er te m p e ra tu re c o n tro l, s ta b ilitie s o f sev eral p arts p er m illio n a r t a v a ila b le b o th fro m th e in itia l cry sta l o sc illa to r and fro m th e d isc r im in a to r . A s an exa m p le , co n sid er a ca se w here A 0K gK d = 100 an d w here an F M g e n era to r a t 2 0 M H z is used to g e n era te a final ca rr ie r a t 6 0 M H z . T h e s ta b iliz in g cry sta l o sc illa to r o p e ra te s a t 5 .9 M H z , an d th is freq u en cy is m u ltip lied by a fa c to r o f 10 to o b ta in th e re feren ce frequ en cy . (In a p ra c tica l c irc u it , m u ltip lica tio n by 3 x 3 = 9 o r 2 x 2 x 3 = 12 w ould b e m o re re a so n a b le .) T h e freq u en cy d e m o d u la to r o p e ra te s a t 1 M H z .

F r o m a m od ified v ersio n o f E q . (1 1 .9 -4 ) ,

y = — 100(3£ + 3y — ¿disci0 — 10 3 ^ 0 ) )or

- 3 0 0 1 0 0 , 1 0 0 0 ,y - - j Q j - e + 3 0 i 5 ducO> + 3 0 1 3OSCOJ

H en ce, i f th e o r ig in a l 2 0 M H z o s c illa to r h a s a m a x im u m d rift o f 1 % o r e = 2 n (2 0 0 k H z), th en th e fin a l 6 0 M H z sig n al h a s th e fo llo w in g “ d r ift” t e r m s :

6 0 0 1 ,« , 3 0 0 , /• 3 0 0 0 ^301 301 301 osc^

PROBLEMS 565

I f th e d isc r im in a to r is a c cu ra te and s ta b le to w ith in \ % an d th e 5.9 M H z o sc illa to r is s ta b le to w ith in 15 p a rts per m illio n , th en th e fin a l w o rst-ca se o u tp u t v a ria tio n from 6 0 M H z shou ld n o t exceed a p p ro x im a te ly

2 k H z + 2.5 k H z + 900 H z = 5.4 kH z,

or less th a n o n e p a rt in 104. In th is ca se a b e tte r cry sta l o sc illa to r w ould n o t b e o f m u ch help until o n e had raised the fe ed b ack g a in an d im p roved the d iscr im in a to r s tab ility .

P R O B L E M S

11.1 For the sinusoidally m odulated FM signal

v(t) = V, cos (a>0r + fi sin ojmt ),

a) show that the rms value o f lit) is given by

ym = J y + 2 £ J i m

= A

sfi-

(Hint : £ Jl(fi) = 1) andn = - x

b) determine the am plitude of the significant sidebands for the case where p = 5 andK, = 5 V.

11.2 Determine the bandw idth occupied by the phase-m odulated signal

v{t) = A cos [co0i + A 0/(i)]

for the case where Aij> = 5 rad and / ( / ) = sin 105f. Retain only frequency com ponents which are greater than 1 °/0 of the unm odulated carrier.

11.3 A 10 V oeak-to-peak square wave is generated with a carrier frequency of 14 M Hz and m odulated with a baseband signal which is band-limited to 5 MHz. The FM square wave is then placed through a rectangular bandpass filter centered at 70 M H z in order to obtain a 70 M H z FM test signal. The filter BW passes all sidebands greater than 1 % ol the unm odulated carrier at 70 MHz. D eterm ine the maximum frequency deviation which may be applied to the FM square wave w ithout introducing excessive distortion into the 70 M H z test signal.

11.4 F or the circuit shown in Fig. ll .P -1 , determ ine an expression for i j t) . Assume that the FET has a square-law characteristic in its saturation region.

11.5 The FM signal

^ , I ■ \v(t) = A cos w 0t H sin wmt\ w- I

is passed through a high-Q single-tuned filter with bandw idth BW = 2a centred at o»0.

566 GENERATION OF FM SIGNALS

Figure l l .P - 1

Assuming the “quasi-static” approxim ation to be valid, show that the instantaneous frequency at the filter output is given by

_ ,A © c o s comt\tt).(i) = m0 + Aw cos wmt — — tan --------------- 1.

dt\ « I

For the case where Aco/com = [! = 5 and Aco/a = | (a set of param eters for which the quasi-static approxim ation is valid), determ ine the percentage of third-harm onic distortion (relative to Aw) introduced into the instantaneous frequency by the nonlinear phase characteristic of the filter. (H int: Expand tan - 1 6 in its M acLauren series and keep the first two terms.)

11.6 Verify that v(t) = C cos [ f co,(0) d6 + 0O] is indeed a solution of Eq. (11.4—4).

11.7 For the circuit shown in Fig. 1 l.P -2 , the t = 0 value o f i’c , is 100 mV while the i = 0 value of vC2 is 0. Determ ine an expression for v0(t) for t > 0 (cf. Section 8.3). (In practice, the circuit of Fig. 1 l.P -2 has a sufficient num ber of leakage paths to be self-starting. In addition, the oscillation am plitude is limited by the nonlinearity o f the drain-source resistance of the FET at about 200 mVJ

PROBLEMS 567

O perational amplifier g a in = —3000

Figure l l .P - 2

1 l.o For the circuit shown in Fig. 1 l.P -3 , Q , has a collector-base capacitance given by C(vCB) = Ai'ca' l/2. where

A = 15.7 pF y / vo lts ..

a) Shew that the instantaneous frequency of the oscillator is given by

1a )i( t) = — = = = = = ,j l [c 0 + c m

where

Q = § ^ + c o Co = AFcc- " 2, and C(t) = ^ Vcc' V ( t ) .

b) Determ ine a numerical expression for v„(t). (In a practical FM oscillator with small values of C , and C 2, m ost o f C 2 would be supplied by the em itter-base capacitance o f the transistor and thus the physical capacitor C2 would be omitted.)

11.9 Verify that Eq. (11.4-14) is the solution to Eq. (11.4—13).


silicon transistors

Figure l l .P - 3

11.10 Determ ine expressions for v,(t) and v0(t) in the circuit o f Fig. 11.5-2 with the following param eter values:

Ve e = V c c = 15 V, /*„ = /*, = 1mA, /* = 2m A , C0 = 0,

f ( t ) = cos 5 x 102t, Rloss = Rl = 5 kft, r = -n, ii,

C = CL = lOOOpF, L = L l = 1 0 / jH , and M = 0.5/iH .

All transistors are silicon and identical

11.11 F or the triangular-wave FM generator shown in Fig. 1 l.P -4 , determ ine the instantaneous frequency. The Schmitt trigger in the circuit has the following characteristics:1) W hen v„(t) increases upw ard through + 5 V, the Schmitt trigger ou tpu t switches from

- l O V t o . + lOV.2) W hen v„(t) decreases dow nward through — 5 V, the Schmitt trigger switches from

+ 1 0 V to - 1 0 V.11.12 Design an eight-diode wave-shaping netw ork 'sim ilar to the one shown in Fig. 11.6—9(b)

which converts the ou tpu t o f the circuit o f Fig. 1 l.P -4 into a 4 V peak-to-peak sinusoidal FM wave. The time intervals between break points on the ou tpu t waveform should be equal.

PROBLEMS 569

i k no— v W — f+

ß >100

ß ,> 5 :

5kÜ■ v w -

Schmitttrigger

v2( 02gmv2<0

i

©c <b v„(i)

gm= /100 f i , C =1000 pF, v,=(5 V)[l+0.1 /(f)]

Figure 1 l .P 4

Figure ll.P -5


11.13 Show that the-FET circuit in Fig. l l .P -5 may be modeled as a controlled current source A(t)ic where

provided that

a) r « 1 /a>0C,b) p,(t) « \VF\,c) the FET operates in its square-law region, andd) the low-frequency com ponents of iD are shunted to ground by L.If C0 = C = 500 pF, L = 10/xH, r = l f t VP = —4 V, Vss = 2 V, Vt = 1 V, VDD = 10 V, l DSS = 4 mA, and the oscillator limits the output waveform to 4 V peak-to-peak, find an expression for the FM voltage across the tuned circuit for the case where / ( f ) = cos 103f.

11.14 Design an FE T oscillator which can be used in conjunction with the circuit shown in Fig. l l .P -5 which has the numerical values given in Problem 11.13.

11.15 In the circuit shown in Fig. 11.7-5, Vcc = K££ = F4 = 12 V, V2 = 5 V, = 1.5 V, V3 =

p > 100, and /( f ) = cos 104f. Sketch csq(r) and find an expression for the instantaneous frequency.

Ao AifU)

2 lo s s lK fit ) 'VP \ VP

3 V, R , = 4.5 kii, R 2 = 3 kQ, RE = 4 kil, Rc = 2 kfi, r = 47 il, C t = C2 = 1000 pF,

C H A P T E R 12

FM DEMODULATORS

In b ro a d o u tlin e a n F M rece iv er is s im ila r to an A M receiver. B o th a re n o rm ally su p erh etero d y n e typ es w ith R F am p lifiers, R F filtering , m ixers, and I F a m p lifica tio n and filtering . In th e A M rece iv er the d esired o u tp u t in fo rm a tio n is ca rried in the c a rrie r en v elop e f lu c tu a tio n s ; h en ce en v elo p e lin ea rity m ust be m a in ta in ed n o t on ly in the fro n t end but th ro u g h o u t th e receiver. In th e F M rece iv er the d esired in fo rm a tio n is co n ta in e d in th e v a ria tio n s in the in s ta n ta n e o u s fre q u e n cy ; hen ce, in c rd e r to d em o d u la te th e F M sig n al, o n e m u st co n v ert these freq u en cy v a ria tio n s in to b a seb a n d am p litu d e v aria tio n s .

U n fo rtu n a te ly , m o st F M d em o d u la to rs are a t least to som e ex ten t a lso ca p a b le o f d em o d u la tin g A M signals. H en ce if, in the co u rse o f p assage th ro u g h the .rans- m issio n ch a n n el o r the fro n t end o f the receiv er, the en v elop e o f th e F M i >nal is co rru p ted by n o ise , fad ing o r filter-in d u ced F M -to -A M co n v ersio n , then v ’ch en v elo p e c o rru p tio n s w ill b e d etected an d w ill a p p ea r a t th e o u tp u t as b ase 'an d n o ise o r d isto rtio n .

As we shall see, som e F M d e te c to rs a re m o re su scep tib le th an o th ers to th is type o f d is to rtio n . H o w ever a lm o st a ll F M rece iv ers o f re a so n a b le q u a lity in clu d e an en v elo p e am p litu d e v a ria tio n rem o v a l c ircu it, co m m o n ly ca lled a lim iter, betw een the last I F am p lifie r and the d em o d u la to r . In som e p ra c tica l c ircu its , the ra tio d e te c to r for exa m p le , the lim itin g and th e d em o d u la tio n fu n ctio n s b eco m e som ew h at en tang led . F o r the sak e o f in itia l u n d ersta n d in g we shall first stud y d em o d u la to rs for w hich these o p e ra tio n s a re co m p le te ly sep a ra te an d ind ep endent.

A fter a d iscu ssio n o f lim iters , th is c h a p te r o u tlin es the th e o re tica l l im ita tio n s on variou s typ es o f F M d em o d u la to rs . I t th en an a lyzes all the co m m o n F M d em o d u la tio n c ircu its as w ell as sev eral useful but n o t yet co m m o n c ircu its . T h e c h a p te r end s w ith a sec tio n o n the m o re co m p le x d ete c to rs th a t a re used in “ th resh o ld e x te n s io n ” F M receivers.

12.1 LIM ITER S

T h e lim iter c ircu it is ideally a z e ro -m e m o ry n o n lin e a r c ircu it th a t p ro d u ces a c o n sta n t- a m p litu d e, c o n sta n t-w a v e sh a p e typ e o f o u tp u t fo r a ll its p erm issib le inp u t levels o r w avesh ape types. F ig u re 1 2 .1 -1 show s a n ideal lim iter c h a ra c te r is t ic ; F ig . 1 2 .1 -2 show s tw o p ra c tica l ch a ra c te r is tics .

A ny n o ise -co rru p te d o r am p litu d e-m o d u la ted F M signal ap p lied to th e c ircu it o f F ig . 1 2 .1 -1 w ill be co n v erted in to a c o n sta n t-a m p litu d e F M sq u are w ave. T h is

571

572 FM DEMODULATORS 12.1

+ V2v„(r)

v,(0

- v 2

Fig. 12.1-1 Ideal lim iter characteristic.

(a)Fig. 12.1-2 “ Practical” lim iter charactistics.

sq u a re w ave m ay be ap p lied d irec tly to c er ta in typ es o f F M d ete c to rs . In o th e r cases it m ay b e filte red — th e filter m u st rem o v e the c a r r ie r ’s th ird an d h ig h er od d h a r m o n ics and th e ir sid eb an d s w ith o u t d is tu rb in g the sid eb an d s a ro u n d th e c a fr ie r— to p ro d u ce a co n sta n t-a m p litu d e s in u so id a l F M signal. In a c tu a l c ircu its , such filters a t the o u tp u ts o f lim iters tend to be d esigned p rim a rily to sa tisfy th e re q u ire m en ts o f the d isc r im in a to r ra th e r th a n th o se o f the lim iter. N o rm a lly th e ir b a n d - w idths a re ju s t la rg e en o u g h to p rev en t sig n a l-in d u ced A M w hich ca u ses d is to rtio n a t th e d em o d u la to r outp u t.

F ig u re 1 2 .1 -3 show s a c irc u it th a t m ig h t be m od eled by th e c h a ra c te r is t ic o f F ig . 1 2 .1 -2 (a ). T o a cco m p lish su ch a m o d elin g , o n e re p la ce s th e real d io d es by a c o m b in a tio n o f an id eal d io d e , a b a tte ry , an d a series resisto r. T h e tw o o u tp u t v o ltag e b re a k p o in ts o cc u r w hen

v.{t)R0

± K '

Fig. 12.1-3 Diode limiter.

12.1 LIMITERS 573

w here V0 is th e d io d e tu rn -o n v o ltag e. T h e m id sectio n slop e is R 0/(R q + ^¡)» w hile th e o u tsid e s ec tio n s lo p es in clu d es r diode in p a ra lle l w ith R 0 . I t is o b v io u s th a t, as r diode a p p ro a ch e s zero an d as th e in p u t d riv in g a m p litu d e b e co m e s very large w ith resp ect to V0 , th e o u tp u t w av esh ap e a p p ro a c h e s a sq u a re w ave. O f co u rse , even if rdiode is zero , th e re w ill b e n o lim itin g a t a ll if th e in p u t am p litu d e is less th a n V0 . W h e re a s the rise an d fall tim es o f th e o u tp u t sig nal from th e c h a ra c te r is tic o f F ig .1 2 .1 -1 are a lw ays in d ep en d en t o f th e d riv in g sig n al, th e o u tp u t s lo p es for th e c h a ra c te r is tic o f F ig . 12 .1—2(a) a re a lw ays a fu n ctio n o f th e drive.

T h o u g h th e o u tp u t s ig n al fro m F ig . 1 2 .1 -2 is n o t a sq u are w ave, its fu n d am en ta l co m p o n e n t c a n be ca lcu la te d d irectly th ro u g h the use o f th e sin e-w ave tip c h a ra c te ristics o f C h a p te r 4. T h u s th e q u a s i-s ta tic o u tp u t o f such a lim ite r fo llow ed by a b a n d p a ss filter is k n o w n exactly .

W ith th e sy m m e trica l c h a ra c te r is t ic s o f F ig . 1 2 .1 -2 g o th e trem en d o u s a d v a n tages o f h av in g n o d c o r even h a rm o n ics . H ow ever, v ario u s sin g le-en d ed c h a ra c te r istics ca n a lso b e used as lim iters. F o r exa m p le , th e I t( x ) / I 0(x ) cu rv e o f F ig . 4 .5 -5

in d ica tes th e in p u t-filte re d -o u tp u t re la tio n sh ip fo r a co n sta n t-c u rre n t b ias-d riv en ex p o n en tia l ju n c tio n . S u ch lim iters a re re stric te d in th e ir m o d u la tin g freq u en cy c a p a b ilit ie s ; as w as p o in ted o u t in S e c tio n 5.6 , the in p u t co u p lin g c a p a c ito r req u ires th a t o j0 be o f th e o rd e r o f 1 0 4 com i f p ro p e r lim itin g a c tio n is to occu r.

T h e p resen t-d ay p ra c tica l c ircu it th a t co m e s c lo se st to a p p ro x im a tin g the ideal c h a ra c te r is tic is a d iffe re n tia l-p a ir lim ite r p reced ed b y o n e o r m o re stag es o f d ifferen tia l-p a ir d rivers. F ig u re 4 .6 - 2 in d ica tes th e b a s ic ch a ra c te r is tic , w hile F ig . 4 .6 -4 in d ica tes the o u tp u t o f a b a n d p a ss filter fo llo w in g a d ifferen tia l-p a ir lim iter. As in d ica ted in T a b le 4 .6 -1 , an in p u t v o ltag e sw ing o f + 7 8 m V sw ings the o u tp u t cu rren t in e ith er tra n s is to r fro m 5 % to 9 5 % (o r vice versa) o f its p eak value. T h u s if the lim iter stage is preced ed by a n o th e r sy m m etrica l stage w ith a v o lta g e gain o f 78 , then a + 1 m V in p u t v o lta g e sw ing w ill p ro d u ce th is nearly co m p le te o u tp u t swing.

S u ch a c h a ra c te r is t ic d o es indeed a p p ro a c h the id ea l cu rv e o f F ig . 1 2 .1 -1 q u ite closely .C o m m e rc ia l in te g ra te d -c ircu it d iffe re n tia l-p a ir lim iters w ith th ree ca scad ed p airs

are read ily a v a ila b le . T h e se u n its hav e v o ltag e g a in s o f 70 d B o r m o re (g rea ter th a n 3 0 0 0 tim es) in to a 1 k Q load . F ig u re 1 2 .1 -4 illu stra te s a sim plified v ersio n o f such a limiter. This unit has a level-shifting emitter-follower output to a llo w it to supply a “ sq u a re -w a v e ” o u tp u t w h ich is cen tered a b o u t 0 V. If the c irc u it w ere used in c o n ju n c tio n w ith a b a n d p a ss filter, th e filter w ould rep la ce R 2 in th e c o lle c to r o f Q 4 .

I f a s in u so id a l F M sig n al is req u ired a t th e lim ite r o u tp u t, a b a n d p a ss filter

cen tered a t th e c a rr ie r freq u en cy m u st fo llo w th e n o n lin e a r lim ite r ch a ra c te r is tic . T h is filte r m u st h a v e su ffic ien t b an d w id th (u su ally B W > 8 Aa>) so th a t its am p litu d e c h a ra c te r is t ic re m a in s su fficien tly fla t in th e v ic in ity o f w 0 to p rev en t d ev ia tio n - ind u ced a m p litu d e v a ria tio n s (cf. E q . 1 1 .3 -2 ). O n th e o th e r h an d , th e filter b a n d w idth m u st be su fficien tly n a rro w so th a t th e h igh er h a rm o n ic co m p o n e n ts o f the n o n lin e a r lim ite r o u tp u t d o n o t e n te r its p assb an d . S u ch c o n flic tin g req u irem en ts a re p o ssib le o n ly i f D an d j8 fo r a g iven F M sig nal d eterm in e a p o in t w ell below the a p p ro p ria te cu rv e in F ig . 1 1 .2 -3 . P ro b le m s 12.4 and 12.5 co n sid er th is p o in t in m o re

d etail.

.u

level shiftFig. 12 .1 -4 Differential-pair limiter with “square-wave” output.

FM D

EMO

DU

LATO

RS

12.1

12.1 LIMITERS 575

D y nam ic L im itin g

W h en a peak en v elo p e d ete c to r is p laced a cro ss a tuned c ircu it d riven by an am p litu d e- m o d u la ted cu rren t as show n in F ig . 1 2 .1 -5 , th e m o d u la tio n o f the tu n ed -c ircu it vo ltag e d ep ends n o t on ly on th e b an d w id th o f the p ara lle l R L C c ircu it, but a lso on the R 0C 0 tim e c o n s ta n t o f the en v elo p e d ete c to r c ircu it. In p a rticu la r, as w as show n in E x a m p le (1 0 .3 -2 ) , if th e en v elo p e d e te c to r tim e c o n sta n t b e co m e s su fficien tly large,

all a m p litu d e m o d u la tio n is strip p ed fro m the tu n e d -c ircu it v o ltag e, p rov id ed that fa ilu re -to -fo llo w d is to rtio n d o es n o t o ccu r. In a d d itio n , if ¡,(f) is an am p litu d e-

m o d u la ted F M sig n al, th e a m p litu d e v a ria tio n s m ay be rem ov ed fro m the tuned- c ircu it v o ltag e w hile its in s ta n ta n e o u s freq u en cy rem ain s id en tica l w ith the in s ta n ta n eo u s freq u en cy o f i',(t).

As we shall now d em o n stra te , even if i,(t) is an am p litu d e-m o d u la ted F M signal,the envelope of the tuned-circuit voltage i)(i) and the point at which diagonal clipping o ccu rs m ay be d eterm in ed from th e e q u iv a len t c ircu it for the en v elop e d ete c to r d evelop ed in S e ctio n 10.3. At th e sam e tim e we sh a ll show th a t the in s ta n ta n e o u s p hase o f ¡¡(f) and v(t) a re id en tica l, p rov id ed th a t the ban d w id th o f the u n load ed tuned c ircu it is su fficien tly large so th a t the c ircu it a p p ears resistiv e (o f value R) over the b an d o f freq u en cies o ccu p ied by the F M signal.

T o d em o n stra te th a t th e resu lts o f S e c tio n 10.3 rem ain valid w hen /,(f) is an a m p litu d e-m o d u la ted F M sig nal o f th e form

i,(f) = b(t) co s io0t + Au> f ( 0 ) d d

- b(t) cos to0r(i), (12.1-1)

we re tra ce th e a n a ly sis o f S e c tio n 10.3. W e first assu m e that Q T for th e p ara lle l R L C c ircu it is su fficien tly high so th a t v(t) c o n ta in s freq u en cy co m p o n e n ts o n ly in the v icin ity o f co0 . T h e re fo re , v(t) m ay b e w ritten in th e gen eral form

u(f) = g (t )c o s (OqT ! (i).

« , ( < ) = M O C O S Î L ü < ) l ( 0 ]

Fig. 12.1-5 Dynamic limiter.

5 7 6 FM DEMODULATORS 12.1

N ow if th e R 0C 0 tim e c o n s ta n t is su fficien tly lo n g so th a t r„(i) co n ta in s little o r no ripp le and if no fa ilu re -to -fo llo w d is to rtio n o ccu rs, then t;0(r) m ust eq u al g(f). A lso , the d io d e cu rren t m u st c o n sis t o f a tra in o f n a rro w pulses o ccu rrin g a t the p eak s o f v(t) : thu s i'D(f) m ay be exp an d ed in a F o u r ie r - lik e series o f the form

¿d(0 = *do(0 + 2/D0(t)cos co0r,(f) + ■ • •. (12 .1 -2 )

S in ce iD0(t) flow in g th ro u g h an R 0C 0 c ircu it m u st d ev elo p v0(t) = g(f), g (i) m ay be exp ressed as

g(t) = iDo(t) * zo(0 , (1 2 .1 -3 )

w here z0(t) is th e im p u lse resp o n se o f the R 0C 0 netw ork . In a d d itio n , the eq u a tin go f th e fu n d am en ta l F M sig nal co m p o n e n ts flow in g in to th e tuned c ircu it to zeroresu lts in

¿>(i) co s o »0 tg(i) , » .

+ 2 iD0(i) c o sw q T ^ î ). (1 2 .1—4 )

[H ere it is assu m ed th a t th e tuned c ircu it a p p ea rs resistiv e to i?(f).] F o r E q . (1 2 .1 -4 )to be satisfied , i^ r ) m u st be eq u al r(f) and a lso

b(t) g(r) _ . . . .2 2 R ~ 1d o U )• (1 2 .1 -5 )

A s in S e c tio n 10.3 th e s im u lta n e o u s e q u a tio n s , E q s. (12.1 3) an d (1 2 .1 -5 ) , m ay be re lated to the eq u iv a len t c irc u it show n in F ig . 1 2 .1 -6 . W e n o te th a t this c ircu it is id en tica l w ith th e co rre sp o n d in g c irc u it o f F ig . 1 0 .3 -5 [excep t th a t C = 0 , w hich is a d irect resu lt o f assu m in g the tun ed c ircu it to b e resistiv e to u(f)]; th erefo re , the en v elo p e g(t) and th e p o in t w here fa ilu re -to -fo llo w d is to rtio n o ccu rs m ay be d ete rm in ed in e x a ctly th e sam e fash io n as they w ere in C h a p te r 10.

It is now c le a r from F ig . 1 2 .1 -6 th a t, i f C 0 is ch o se n su fficien tly large so th a t it ap p ears as an a c sh o rt c irc u it o v er th e ban d o f freq u en cies o ccu p ied by the a c c o m p o n en t o f b(t), th en g(t) rises slow ly to th e d c level

gW = £ w ( K | | y ) , (1 2 .1 -6 )

Fig. 12.1-6 Equivalent circuit for determining g(t) and i^ t).

12.2 FREQUENCY DEMODULATION TECHNIQUES 577

w here b(t) is th e av erage en v elo p e level o f i',(f). C o n seq u en tly , s in ce t ^ f ) = t(f), v(t) is a lim ited F M sig nal g iven by

r / _ . . K nIit) = b(t) co s a>0t + Aco f { 0 ) d d

f '(1 2 .1 -7 )

A lth ou g h the en v elo p e o f i^f) is in sen sitiv e to rap id v a ria tio n s in b(t), it d o es resp ond to slow ch a n g es in the av erage value o f b{t) ; h en ce the c ircu it lim its o n ly d yn am ically .

T h e lim itin g a c tio n o f the c ircu it o f F ig . 1 2 .1 -5 assu m es th a t fa ilu re -to -fo llo w d is to rtio n d o es n o t o ccu r. It w as d eterm in ed in E q . (1 0 .3 -4 ) th a t the c o n d itio n w hich ensured no fa ilu re -to -fo llo w d is to rtio n for th e d y n a m ic lim iter w as given by

m <2 R

R 0 + 2 R

C o n seq u en tly , if large en v elo p e v a ria tio n s in b(t) a re exp ected , R 0 m u st be k ep t sm all in c o m p a riso n w ith R, a t the exp en se o f a g reatly reduced o u tp u t am p litu d e , to p revent fa ilu re -to -fo llo w d isto rtio n .

12.2 FR EQ U EN C Y D EM O D U LA TIO N T E C H N IQ U E S

In th is sec tio n we sh a ll e x p lo re som e o f the m eth o d s by w hich a sig nal p ro p o rtio n a l to / (f) ca n be o b ta in e d from a lim ited F M signal h av in g e ith er o f th e form s

or

r(f) = A co s

v j t ) = B F

a>0t + Aco J f ( 0 ) d 6

a>0t + Aw J f ( 8 ) d d

= A co s o j0T(f)

B F [ co0t(î )],

( 1 2 .2 - la )

( 1 2 .2 - lb )

w here t(r) = t + (Aco/co0 ) p f ( 6 ) d 6 and F (6 ) is the p erio d ic sq u are w ave show n in F ig . 1 2 .2 -1 . T h e v o ltag e i<r) rep resen ts the o u tp u t o f a lim iter fo llow ed by a b a n d pass filter to e x tra c t the fu n d am en ta l F M sig n al, w hereas vsq(t) rep resen ts the d irect o u tp u t o f an id eal lim iter.

^(0)

0

2 n

Fig. 12.2-1 Plot of F(0) vs. 9.


T h e re a re o n ly two b a s ic te ch n iq u es fo r d em o d u la tin g [o r e x tra c tin g f ( t ) from ] an F M signal. T h e first te ch n iq u e p la ces an F M m o d u la to r in th e re tu rn b ra n ch o f a feed b ack am p lifier. T h e resu ltan t d em o d u la to r , referred to as the p h a se-lo ck ed lo o p (P L L ) , m ak es use o f th e fact th a t a fe ed b ack am p lifier w ith sufficien t lo o p g ain p erfo rm s in its forw ard b ra n ch the inverse o p e ra tio n o f th a t w hich is p erfo rm ed in its retu rn b ra n ch . S in ce the P L L is so m ew h at sp ecia liz ed in its o p e ra tio n , we shall defer its e x p la n a tio n to S e ctio n 12.7.

T h e second an d m o re gen era l te ch n iq u e fo r a cco m p lish in g F M d em o d u la tio n is th at o f first p lacin g the a m p litu d e-lim ited F M signal th ro u g h a d ifferen tia tin g n e tw ork w hich p ro d u ces an en v elo p e m o d u la tio n p ro p o rtio n a l to th e in s ta n ta n e o u s freq u en cy o f the F M sig n al, and th en p la cin g the resu ltan t am p litu d e-v ary in g , freq u en cy -m o d u la ted sig n a l th ro u g h an am p litu d e d em o d u la to r w hich e x tra c ts a signal p ro p o rtio n a l to w^t). S in ce cojt) = w 0 + A w f(t), o n ce is ex tra c te d , f i t ) m ay be o b ta in e d e ith er by su b tra c tin g th e co0-term o r by rem o v in g it by p lacin g a>,(f) th ro u g h a h ig h -p ass (ca p a citiv e ) co u p lin g n etw ork . A b lo ck d ia g ra m o f th is b a s ic freq u en cy d e m o d u la to r (o r d iscr im in a to r , as it is a lso ca lled ) is show n in F ig .1 2 .2 -2 . M o re th a n 9 9 .9 % o f a ll the freq u en cy d em o d u la to rs in ex iste n ce to d ay em p lo y som e v a r ia tio n o f th e d em o d u la tio n tech n iq u e illu stra ted by the b lo ck d iagram .

Differentiation Envelopenetwork demodulator

Fig. 12.2 2 Block diagram of the basic frequency dem odulator.

T o see m o re q u a n tita tiv e ly how the d em o d u la to r o f F ig . 1 2 .2 -2 p erfo rm s, let us d eterm in e a n exp ress io n fo r its o u tp u t w hen e a ch o f the sig n als given by E q . (1 2 .2 -1 ) is ap p lied a t its inpu t. W ith u(r) ap p lied , the o u tp u t o f th e d ifferen tia tio n n e tw o rk is given by

envelope proportional tocoi(f)

( 12.2- 2)vb(t) = - K DA [w 0 + A w f(t)] sin

w ith th e resu lt th a t th e o u tp u t o f the en v elo p e d em o d u la to r has the d esired form

oj0t + A w f ( 6 ) d df '

v0(t) — A K dK m [w 0 + A w f(t)} — A K DK Mw i(t), (1 2 .2 -3 )

w here K D is the c o n s ta n t o f th e d iffe re n tia tio n n etw o rk an d K M is the c o n sta n t o f the am p litu d e d em o d u la to r . O n the o th e r h an d , w ith r sq(r) ap p lied , th e o u tp u t o f the d ifferen tia tio n n e tw o rk is given by

vb(t) = K dB [w 0 + A w f(t ) ]F ’ w 0t + A co | / ( f l ) d ö j , (1 2 .2 -4 )+

+ This form is obtained by applying the chain rule when differentiating u5q(i).


w here F ’{ 6 ) = dF(6)/dO . A p lo t o f F '(9 ) vs. 6 is show n in F ig . 12.2 -3 . H ere we o b serve th a t B K d[co0 + Ato/(f)] en v e lo p e -m o d u la te s a h igh -frecm ency tra in o f p o sitiv e and

n egativ e im p u lses o f a re a 2 ; h en ce , if the en v elo p e d em o d u la to r e x tra c ts the en v elo p e, i 0(f) is a g a in given by

v0(t) = + A o)f(t)]. (1 2 .2 -5 )

T h e valu e o f K M in th is ca se is in g en era l, b u t n o t a lw ays, d ifferent from K M given inE q . (1 1 .2 -3 ) , s in ce m o st a m p litu d e d em o d u la to rs are w aveform d ep endent.

I t is n o w a p p a re n t th a t to im p le m e n t th e freq u en cy d e m o d u la to r o f F ig . 1 2 .2 -2

we req u ire a d iffe re n tia tio n n e tw o rk an d an am p litu d e d e m o d u la to r ; h en ce in the rem a in d er o f th is s ec tio n an d in th e n e x t few sec tio n s th e phy sica l im p lem en ta tio n o f these b lo ck s w ill b e con sid ered . S p ecifica lly , in th e rem a in d er o f »his sec tio n we sh a ll

co n sid er th e th e o re tic a l re s tr ic tio n s im p o sed o n th e in p u t F M sig n al w hen e a ch o f th e th ree am p litu d e d em o d u la to rs d iscu ssed in C h a p te r 10 is em p loyed in th e fre q u en cy d em o d u la to r o f F ig . 1 2 .2 -2 . In p a rticu la r , we shall o b ta in a cu rv e o f the m a x im u m p erm issib le d ev ia tio n ra tio D = A<w/a>0 as a fu n ctio n o f m o d u la tio n ind ex /? = Aco/com fo r w hich th e o re tica lly u n d isto rted d em o d u la tio n o f the in p u t F M signal is p o ssib le .

W e sh a ll th en , in su b seq u en t sec tio n s , c o n sid er th e th ree d ifferen t w ays in w hich th e d iffe re n tia tio n n e tw o rk c a n b e rea lized o r c lo se ly a p p ro x im a ted . F o r e a ch o f th e th re e ca ses we sh a ll first o b ta in th e th e o re tica l lim its o n th e o p e ra tio n o f the

d ifferen tia to r and th en p ro ceed to im p lem en t it w ith a p hy sica l n etw ork . W e shall then c o m b in e the d iffe re n tia to r w ith o n e o f th e am p litu d e d em o d u la to rs d iscussed in C h a p te r 10 to o b ta in a co m p le te freq u en cy d em o d u la to r .

N o te th a t, s in ce th ere a re th ree b a s ic m e th o d s fo r en v elo p e d em o d u la tio n and

th ree b a s ic w ays to m a k e a d ifferen tia tin g c ircu it, th ere are n in e d ifferent freq u en cy d em o d u la to rs o f th e fo rm sh o w n in F ig . 1 2 .2 -2 . W e shall e x p lo re o n ly the m o st p ra c tica l o f these an d leave the stu d y o f the rem a in d er to the in terested read er.


Synchronous Detection

T o a c co m p lish th e sy n ch ro n o u s d e te c tio n o f a d ifferen tia ted F M sig n al w e m u ltip ly vb(t) g iv en b y E q . (1 2 .2 -2 ) o r E q . (1 2 .2 -4 ) by th e u n d ifferen tia ted F M re feren ce signal — F , sin [co0t + Ao ) j ' f ( 6 ) d 6 ] , a s sh o w n in F ig . 1 2 .2 -4 , a n d p ass th e re su lta n t s ig n a l

Fig. 12.2-4 Synchronous detector for an FM signal.

th ro u g h a lo w -p ass filter. W h e re vb(t) is given by E q . (1 2 .2 -2 ) a n d th e in p u t F M sig nal is s in u so id a l, th e m u ltip lier o u tp u t o f th e sy n ch ro n o u s d e te c to r h a s th e fo rm

v j t ) = + A K dK m [o)0 + A cof(t)]

— A K dK m[co0 + A co/ (f)]cos J 2ct>0f + 2Aco J f ( 9 ) d 0 (12.2-6)

w here K u = K V J 2 . T h e m u ltip lier o u tp u t c o n ta in s a lo w -freq u en cy co m p o n e n t vm j( f ) an d a h ig h -freq u en cy co m p o n e n t vm2{t), a s is sh o w n in F ig . 1 2 .2 -5 . I f th e s p e ctra o f th ese tw o term s d o n o t o v erla p , th en vmi(t) m ay b e e x tra c te d b y the lo w -p ass filte r show n in F ig . 1 2 .2 -4 to yield

v„(t) = A K dK m [u)0 + Aco/(i)]* M < ), (1 2 .2 -7 )

K .M |

/IK,, M |

A

BW2+ 2 u im K.2 (W) |

2ui0

K.(ai)=J[M*)l, l i .M -iF IM l)]

Fig. 12.2-5 Spectrum of ujt) plotted vs. w.

ui


w here hL(t) is the im p u lse resp o n se o f the lo w -p ass filter. I f th e filter is su fficiently fla t a t H l(0) = 1 o v er the b a n d o f freq u en cies 0 < w < w m, th en E q . (1 2 .2 -7 ) s im p li

fies to th e d esired form

v0(t) = A K DK M[w 0 + Aa>f(t)]. (1 2 .2 -8 )

T h e a b ility to sep a ra te vml(t) fro m vm2(t) by filterin g p laces a th e o re tica l lim it on

the m ax im u m v alu es o f Aa> an d wm re la tiv e to a>0 w hich ca n b e d em o d u lated by fo llo w in g th e d iffe re n tia tio n n e tw o rk by a sy n ch ro n o u s d etec to r. A s we sh a ll n ow show , these lim its ca n be exp ressed as a p lo t o f th e m a x im u m p erm issib le d ev ia tio n ra tio

D = A w /w 0 versu s ft = A w /w m. W e sh a ll o b ta in a s im ila r p lo t fo r th e av erag e en v elo p e d e te c to r an d th e p eak en v elo p e d e te c to r w hich will in d ica te th a t sy n ch ro n o u s d ete c tio n p erm its th e larg est p o ssib le d ev ia tio n ra tio fo r any given valu e o f p.

T o d eterm in e un der w h at c o n d itio n s the sp e ctra l co m p o n e n ts o f V J oj) d o n o t o v erlap , we n o te th a t the u p p er sp e ctra l co m p o n e n t Vm2(w ) is the F o u r ie r tra n sfo rm o f th e p ro d u ct o f w 0 + Aw f(t) (w hich is b a n d -lim ite d to wm) an d a n F M sig n al c e n tered a t 2w 0 w ith a freq u en cy d ev ia tio n Au)2 = 2 Aw , a m o d u la tio n in d ex p 2 = 2fi, an d a b an d w id th B W 2. S in ce m u ltip lica tio n in th e tim e d o m a in c o rre sp o n d s to c o n v o lu tio n in th e freq u en cy d o m a in , th e sp ectru m o f Vm2(w ) h a s a b an d w id th o f B W 2 + 2ft)m. f T h e re fo re , sep a ra tio n o f vml(t) fro m vm2(t) by filterin g is p o ssib le p rov id ed th at

B W ,— + 2a>m < 2<o0 . (1 2 .2 -9 )

o r eq u iv a len tly .

Aw 1D = — ^ --------— = Z)max. (1 2 .2 -1 0 )

w0 1 B W 2

P + 2 A w 2

F o r an y v alu e o f P a co n se rv a tiv e e stim a te o f B W 2/2 Aa>2 c a n be o b ta in e d fro m F ig .1 1 .1 -4 ; thus D max, th e m ax im u m d ev ia tio n ra tio for w hich sy n ch ro n o u s d ete c tio n (or an y fo rm o f d e te c tio n , fo r th a t m a tte r) is p o ssib le , m ay b e p lo tted as a fu n ctio n o f p. F o r exam p le, i f p = 5, then p 2 = = 10 an d B W 2/Aco2 = 1.4. T h e re fo re ,D max = 0 .6 2 5 .

A p lo t o f £>max vs. P fo r th e sy n ch ro n o u s d e te c to r is show n in F ig . 1 2 .2 -6 . As P - * oo, D max -» 1 ; h en ce th e cu rv e h a s a kn o w n lim itin g value. I f th e in p u t F M sig n al h a s a m o d u la tio n in d ex P an d a d ev ia tio n ra tio D w hich d eterm in e a p o in t lying u nder th e cu rv e o f F ig . 1 2 .2 -6 , th en it is p o ssib le to d em o d u la te it w ith a freq u en cy d em o d u la to r e m p lo y in g a sy n ch ro n o u s d ete c to r. H o w ever, i f th e p o in t lies c lo se to th e cu rv e , th e co m p le x ity o f the filter a t th e o u tp u t o f the sy n ch ro n o u s d e te c to r will in crea se s ig n ifican tly .

t Convolution of two band-lim ited spectra produces a spectrum which is also band-limited, with a bandw idth equal to the sum of the two original bandwidths.


Fig. 12.2-6 P lot of Dmax vs. (i for which synchronous dem odulation of an FM signal is possible.

F o r th e ca se w here vb(t) g iven b y E q . (1 2 .2 -4 ) is ap p lied to th e sy n ch ro n o u s d ete c to r , th e e ffect o f m u ltip ly in g F '[w 0T(t)] by — sin (o0x{t) is th e co n v e rs io n o f the tra in o f a lte rn a tin g p o sitiv e an d n eg ativ e im p u lses in to a tra in o f a ll p o sitiv e im pu lses. H o w ever, s in ce th is fu ll-w ave re c tif ica tio n c a n b e p erfo rm ed in a m o re s tra ig h tforw ard fash io n w ith d io d e n e tw o rk s, sy n ch ro n o u s d ete c tio n is n o t em p lo y ed in c o n ju n c tio n w ith h a rd -lim ite d F M signals.

T o d ev elo p th e re feren ce s ig n a l fo r th e sy n ch ro n o u s d ete c to r , th e in p u t F M signal m ay b e p la ced th ro u g h a n e tw o rk w h ich sh ifts its p h ase by n /2 . A n o th e r m eth o d ,

- the o n e m o st o ften used in p ra c tice , is to sh ift the o u tp u t o f th e d iffe re n tia tio n n etw o rk by — n /2 an d use th e u n d ifferen tia ted in p u t F M sig n al d irec tly as th e reference.

A verage E nvelope D etection

W h e n vb(t) g iven by E q . (1 2 .2 -2 ) is p laced th ro u g h a n av erage en v elo p e d ete c to r o f th e fo rm show n in F ig . 1 0 .1 -6 , va(t) m ay b e w ritten a s th e p ro d u ct o f vb(t) and a sw itch in g fu n ctio n S FM(f) w h ich is a u n it am p litu d e sq u a re w ave s im ila r to the o n e show n in F ig . 1 0 .1 -7 e x ce p t th a t it is p e rio d ic in r (t) r a th e r th a n t. T h u s S FM(i) m ay b e exp an d ed in a F o u r ie r series, w ith th e resu lt th a t va(t) m a y b e w ritten as fo llo w s (cf. E q . 1 0 .1 -1 1 ) :

*>a j(0 »a 2 ( 0

' ¡ ( a * ' K A * 'v„(t) = — — [a>o + Aa>/(f)] + — f - [ c o 0 + Aa>/(r)]sin cuot ( 0

K

+ h ig h er h a rm o n ic A M - F M signals. (1 2 .2 -1 1 )

I f uo l(i) c a n be e x tra c te d fro m va2(t) by the o u tp u t filter o f F ig . 1 0 .1 -6 , then

v0(t) = ~ K + Acof(t)]* hL(t). (1 2 .2 -1 2 )

In a d d itio n , i f th e lo w -p ass filter p asses / (f) in a n u n d istorted fash io n , v0(t) sim p lifies to th e d esired form

v0(t) = K MK DA[a>0 + A a>/(f)], (12.2-13)where K M = HL(0)/n.

1 2 . 2 FREQUENCY DEMODULATION TECHNIQUES 583

In th is ca se , in o rd er to sep a ra te val(t) fro m va2(t) by filterin g , th e sp ectru m o f the am p litu d e-m o d u la ted F M w ave cen tered a t a>0 w ith a b an d w id th B W t + 2tom (w here B W , is th e b an d w id th o f th e inp u t F M sig n al) m u st n o t o v erla p w ith the low - freq u en cy sp ectru m o f / (t) , w h ich e x ten d s to com. T o en su re th is sep a ra tio n , wereq u ire that

B W ,2wm + - y r ^ "o>


jL DW j

+ 2 & oj

W ith the aid o f F ig . 1 1 .1 -4 , Z)max m ay b e p lo tted as a fu n ctio n o f /?, as show n in F ig .12.2 7. B y c o m p a rin g F ig . 1 2 .2 -7 w ith F ig . 1 2 .2 -6 (w hich is re p lo tted w ith d ash ed

lin es in F ig . 1 2 .2 -7 ) , w e o b serv e th a t, s in ce th e sy n ch ro n o u s d ete c to r “ rip p le ” term w as cen tered a t 2a>0 w hile th e av erag e d ete c to r “ r ip p le ” term is cen tered a t o n ly a>0 , th e sy n ch ro n o u s d e te c to r is less res tr ic tiv e o n the p a ra m e ters o f th e in p u t F M sig nal th a n is the av erag e en v elo p e d ete c to r . H o w ever, i f th e p a ra m eters o f th e in p u t F M sig nal d eterm in e a p o in t w hich lies u n d er th e cu rv e o f F ig . 1 2 .2 -6 , av erage en v elo p e d ete c tio n m ay be em p lo y ed in stead o f th e m o re c o m p le x sy n ch ro n o u s d etectio n .

Fig. 12 .2-7 Riot of Dmax vs. /> for which demodulation of an FM signal is possible with an average envelope detector.

W h en vb{t) g iv en by E q . (1 2 .2 -4 ) is p laced th ro u g h the av erag e en v elo p e d ete c to r o f F ig . 1 0 .1 -6 , Da(i) h a s th e fo rm

vJLt) = K ^ ojo + A (of(t)]F 'p[oj0T (t)l (1 2 .2 -1 6 )

w here F'p(d) is show n in F ig . 1 2 .2 -8 . E q u a tio n (1 2 .2 -1 6 ) is a d irec t resu lt o f th e fact th a t the h alf-w av e re c tifie r in the av erag e en v elo p e d e te c to r rem ov es the n eg ativ e im p u lses from F'(6).

(1 2 .2 -1 4 )

(1 2 .2 -1 5 )


I t is a p p a ren t th a t F p[oj0x(t)\ m ay be exp an d ed in a F o u r ie r series in oj0z(r), w here th e av erag e valu e o f th e series is g iven b y 1/jt; th u s th e lo w -freq u en cy c o m p o n en t o f v jt ) has .the fo rm (K DB /n)[a>0 + Aco/(f)]. I f th is co m p o n e n t o f v jt ) ca n be sep a ra ted fro m th e fu n d a m en ta l A M - F M sig n al co m p o n e n t o f va(t) by th e o u tp u t low -p ass filter, then

v0(t) = ^ K > + A co/(f)]* M O (1 2 .2 -1 7 )n

o r

v jt ) = K dK mB [0) o + A cof ( 0 1 (1 2 .2 -1 8 )

w here K u = H L(0 )/n fo r th e c a se w here h L(t) p asses/ (/) u n d isto rted . I t is in terestin g to n o te th a t K M fo r the av erag e en v e lo p e d ete c to r is the sam e fo r b o th im p u lse like and sin u so id a l inputs.

A gain in th is ca se , to en su re th e v alid ity o f E q . (1 2 .2 -1 7 ) o r E q . (1 2 .2 -1 8 ) , th e sp ectru m o f f ( t ) m u st n o t o v er la p w ith the sp ectru m o f the c o m p o n e n t o f v j t ) cen tered a t co0 . T h is re q u ire m e n t is, o f co u rse , id e n tica l w ith th a t w hich en su res th e sep a ra tio n o f ra l(f) from va(t) g iven in E q . ( 1 2 .2 - 1 1 ) ; h en ce fo r a n u n filtered F M sig nal th e cu rv eo f F ig . 1 2 .2 -7 a g a in d eterm in es th e m a x im u m valu e o f th e d ev ia tio n ra tio fo r anygiven p for w hich av erag e en v e lo p e d ete c tio n m ay be em p lo y ed in th e freq u en cy d em o d u lato r .

I f a fu ll-w ave re c tifie r rep la ces th e half-w av e rec tifie r o f F ig . 1 0 .1 -7 , th e o u tp u t am p litu d e is, o f c o u rse , in crea sed by a fa c to r o f 2. M o re im p o rta n t, how ever, as the in terested rea d er sh o u ld verify , the Z)max-vs.-/? -curve b e co m e s id en tica l to th e o n e o b ta in e d w ith a sy n ch ro n o u s d ete c to r . T h e b a s ic reaso n fo r th is is th a t w ith fu ll-w ave rec tifica tio n the co m p o n e n t o f v jt ) cen tered a b o u t a>0 is red u ced to zero.

P e a k Envelope D etectio n

A p eak en v elo p e d e te c to r e ith e r g rea tly d is to rts a tra in o f n a rro w pulses o r fu n ctio n s essen tia lly as an a v era g e d e te c to r w hen su ch p tilses a re ap p lied a t its in p u t ; h en ce it is seld o m em p lo y ed in th e freq u en cy d em o d u la tio n o f un filtered sq u are-w av e F M signals.

1 2 . 2 FREQUENCY DEMODULATION TECHNIQUES 5 8 5

O n the o th e r h an d , w hen th e lim ited F M sig n al is b a n d -lim ited and h a s th e form given b y E q . ( 1 2 .2 - la ) , th e p eak en v e lo p e d e te c to r m ay b e em p lo yed . H o w ever, w hen it is em p lo y ed , as we re ca ll fro m S e c tio n s 10.3 and 10.5 , th e c a rr ie r freq u en cy m u st b e m u ch g re a te r th a n u>m, u su ally by a fa c to r o f 100 o r m o re, for its c o rre c t o p e ra tio n (i.e., to k e e p th e rip p le sm all an d to p rev ent fa ilu re -to -fo llo w d isto rtio n ). F o r th e c a se o f F M sig n als it is th e in s ta n ta n e o u s freq u en cy o>,(f) ra th e r th a n th e c a rrie r freq u en cy w hich g ives rise to rip p le a t th e d ete c to r o u tp u t ; h en ce , w hen th e p eak e n v elop e d ete c to r is em p lo yed in th e freq u en cy d em o d u la to r , we m u st req u ire

K ( f )L n = w 0 - Aw > 100o ;m,


° < 100 + p'

(1 2 .2 -1 9 )

( 12.2-20)

A p lo t o f D max = /?/(100 + /?) is show n in F ig . 1 2 .2 -9 a lo n g w ith the p lo t o f F ig . 1 2 .2 -6 (d ashed lines) on the sam e set o f co o rd in a te s . A gain we o b serv e th a t th e sy n ch ro n o u s d ete c to r is fa r less re s tric tiv e o n th e p a ra m e ters o f th e in p u t F M sig n a l th a n is th e p eak

en v elo p e d ete c to r. S in ce D mi% -* 1 a s ft -* oo fo r a ll th ree am p litu d e d em o d u la to rs , it m a k e s little o r n o d ifferen ce th e o re tica lly w h ich en v elo p e d e te c to r is used w here th e m o d u la tio n in d ex is h igh . S im p lic ity o f c o n s tru c tio n in p ra c tica l c irc u its usually d ic ta te s the use o f th e p eak o r av erag e en v elo p e d e te c to r in p la ce o f th e sy n ch ro n o u s d etec to r.

1.0

0.8

0.6

0.4

0.2

0

D„

S y n ch ro n o u s d etecto r

//

<• Peak envelope detector

1 2 3 4 5 6 7 8 9 10 P

Fig. 12.2 9 Plot of Dmax vs. for the peak envelope detector.

N o te th a t, if th e F M sig nal h as b een d isto rted som ew h ere in th e system p rio r to the d em o d u la to r , fo r exa m p le by a lim iter, an d then filtered to rec la im r(f), D max m u st b e low er th a n the to p cu rv e o f F ig . 1 1 .2 -3 . H ow ever, s in ce th is filterin g cu rv e is


m o re res tr ic tiv e th a n th e a m p litu d e d e m o d u la to r lim itin g cu rv es o n £>max given in F ig s . 1 2 .2 -6 a n d 1 2 .2 -7 , D is n o rm a lly lim ited b y th e b a n d p a ss filte rs in th e system a n d n o t by th e a m p litu d e d e m o d u la to r in th e d iscr im in a to r .

T o p u t th in g s in to p ro p e r p ersp ectiv e , o n e sh o u ld re c a ll th a t o rd in a ry e n te r ta in m e n t-ty p e F M b ro a d ca s tin g h a s A f = 7 5 k H z a n d f m = 15 k H z a n d u ses an I F o f10 .7 M H z . H e n ce D max = 0 .0 0 7 0 an d ft = 5. T h u s a w ell-d esign ed p eak d ete c to r w ill nev er p o se a p ro b le m fo r su ch a sy stem , s in ce D max c a n nev er o cc u r w ith /? less th a n 5 ; h ow ev er, E q . (1 2 .2 -2 0 ) is sa tisfied d ow n to P = 0 .7 .

O n ly in sp ecia l w id eb an d system s w ill th e re s tr ic tio n s o f th is se c tio n b e co m e im p o rta n t.

123 DIRECT DIFFERENTIATION—THE CLARKE-HESS FREQUENCY DEMODULATOR

T h re e m e th o d s e x ist b y w hich th e d iffe re n tia tio n o f a s ig n a l m ay b e a c c o m p lis h e d :

1. D ire c t d iffe ren tia tio n .2. F re q u e n c y -d o m a in d ifferen tia tio n .3. T im e -d e la y d ifferen tia tio n .

In th is s ec tio n a n d in su b seq u en t se c tio n s e a ch o f th ese d iffe re n tia tio n m e th o d s is in v estig ated th e o re tica lly , im p lem en ted p ra c tica lly , and co m b in e d w ith a p ra c tica l e n v elo p e d e te c to r to yield a freq u en cy d e m o d u la to r (o r d iscr im in a to r) o f th e b a s ic fo rm show n in F ig . 1 2 .2 -2 . In th is se c tio n w e c o n sid er th e d ire c t d iffe ren tia to r.

A d irec t d iffe re n tia to r is a d ev ice w h ich p erfo rm s th e o p e ra tio n o f d ifferen tia tio n e x a c tly an d th u s p la ce s n o th e o re tica l l im ita tio n s o n th e sig nal a t its inp u t. W h e n such a ,d ev ice is em p lo y ed in a freq u en cy d em o d u la to r , a ll the l im ita tio n s o n th e in co m in g F M s ig n al a r ise fro m th e a m p litu d e d em o d u la to r . A n ex a m p le o f a d irec t d iffe re n tia to r w ould b e a c irc u it , em p lo y in g a s in g le c a p a c ito r C , fo r w h ich th e in p u t is th e c a p a c ito r v o lta g e and th e o u tp u t is th e c a p a c ito r cu rren t.

A freq u en cy d e m o d u la to r e m p lo y in g a c a p a c ito r as a d ire c t d iffe re n tia to r an d em p lo y in g a n a v e ra g e en v e lo p e d e te c to r o f th e fo rm show n in F ig . 1 0 .2 -5 is illu stra ted in F ig . 1 2 .3 -1 . A m o d el fo r th e c irc u it o f F ig . 1 2 .3 -1 in w h ich b o th D an d th e em itter- b a se ju n c t io n o f Q a re m o d eled by id ea l d io d es in series w ith b a tte r ie s o f v a lu e V0 is sh o w n in F ig . 1 2 .3 -2 . F o r p ro p e r c irc u it o p e ra tio n , F0 m u st b e sm a ll in co m p a riso n w ith th e m a x im u m valu e o f vfa) (as w e saw in S e c tio n 1 0 .2 ); i.e ., g erm a n iu m d io d es and tra n s isto rs sh o u ld b e em p lo yed . H e n ce in th e fo llo w in g a n a ly ses we a ssu m e V0 * 0.

W ith V0 x 0 , th e tw o id ea l d io d es in p a ra lle l a re in d is tin g u ish a b le fro m a sh o rt c ir c u i t ; thu s th e c a p a c ito r v o lta g e is g iven by

M 0 = ”f(0 - Vcc,

w ith th e resu lt th a t th e c a p a c ito r c u rre n t h a s th e fo rm

(1 2 .3 -1 )

(12.3-2)

1 2 . 3 THE CLARKE-HESS FREQUENCY DEMODULATOR 5 8 7

- Differentiator Half-wave ' Low-pass filter

Fig. 123-1 Frequency dem odulator employing a direct differentiator and an average envelope detector.

T h e c a p a c ito r c u rren t is a d irec t d eriv a tiv e o f v,{t) w ith th e d ifferen tia tio n co n sta n t K d = C. (In p ra c tice , a fin ite g e n e ra to r im p ed a n ce term and a fin ite d io d e and tra n s is to r lo ss term c o m b in e as r L to m a k e th e d ifferen tia tio n a p p ro x im a te instead o f e x a ct. I f co0C r T < 0 .1 , .then the o p e ra tio n is w ithin 1 % o f pu re d ifferen tia tio n . If r x - 5 0 0 and ca0 = 10 7 rps, then C m u st be less th a n 2 0 0 p F .)

I f i' ¡ ( t ) = i it), w here v(t) is given by E q . ( 1 2 .2 - la ) , then

i,-(f) = - AC[u>i(t)] sin co0t ■+ Aco f (0 ) dO (12.3 3)

O n ly th e p o sitiv e h a lf o f i,(t) flow s in to D 2 ; h en ce i^ t) is a half-w av e rectified v ersio n o f i,(i). T h e av erage v alu e o f th is rectified w av efo rm is ex tra c te d b y p lacin g aiE(t) th ro u g h th e R 0 C 0 lo w -p a ss filter. C o n se q u e n tly , i f D = A co/co0 and fi = Aco/com fo r t;(i) d eterm in e a p o in t b e lo w th e cu rv e o f F ig . 1 2 .2 -7 and the lo w -p ass R 0C 0 filter e x tra c ts the lo w -freq u en cy co m p o n e n t o f a i£(i), th en i>0(i) m ay be w ritten as fo llow s (cf. Eq.12.2- 12):

xACaiiU ) * hL(t) v„(t) = ------------------------ ( 1 2 . 3 - 4 )


where hL{t) is the impulse response of the low-pass filter. For the case where the filter passes a>t{t) = a>0 + A<of(t) in an undistorted fashion (the filter is flat for 0 < (o < a)m), Eq. (12.3-4) simplifies to the desired form

ocACRq vj(t) = co,(t)

aACR0[<u0 + Aco/(t)]. (12.3-5)

A plot of v0 versus a>f is shown in Fig. 12.3-3 for this case. From the figure we observe that the output voltage is linearly related to aj^t) up to the frequency for which v„ = Vc c + V0 « Vc c and Q saturates. For any given cofa), the slope of the characteristic shown in Fig. 12.3-3 must be chosen sufficiently small so that Q does not saturate. For example, if A = 10 V, ifw,(t) = 108[1 + ^/(t)] rad/sec, if C = 50 pF, and if Vcc = 15 V, then R 0 must be chosen less than

nV ,cc (3.14) (15)aACu>i_ (50 x 10“J2) (1.5 x 108)(10)

= 628 Q.

Fig. 123-3 Plot o f the discriminator characteristic for the frequency demodulator o f Fig. 12.3-1.

Clark e-Hess Frequency Demodulator!

The circuit shown in Fig. 12.3-2 functions best when Acw is a reasonable fraction of co0 . However, when Aw « g)0, significant advantages over the circuit of Fig. 12.3-2 result when a balancing branch, which reduces v„ to zero when a)t = co0 , is added to the basic frequency demodulator. A circuit containing such a branch is shown in Fig. 12.3-4. The balancing branch in the circuit subtracts a term proportional to the input envelope from v„(t). In particular, if V0 * 0, a term of value otARJnR (cf. Eq. 10.2-11) is subtracted from the output, with the result that v„(t) takes the form

. , <xACR0 f 1 1v0(t) = U 0 + A (of {t) - — . (12.3-6)

f K . K . Clarke and D. T. Hess, “Demodulator for Frequency Modulated Signals,” U.S. Patent 3292093, December 13,1966.

H o w ever, if 1 / R C is a d ju sted to e q u a l a)0 , th en va(t) s im p lifies to

t,0(t ) = ( ^ ^ ) A « / (i). ( 1 2 .3 -7 )

12.3 THE CLARKE-HESS FREQUENCY DEMODULATOR 589

c

T h e tra n sfe r fu n ctio n o f v„(t) g iven by E q . (1 2 .3 -7 ) vs. Aco/(f) is sh o w n in F ig . 1 2 .3 -5 .

H e re we o b serv e th a t v0 is lin ea r ly re la ted to A a )f(t) p rov ided th a t n e ith e r Q i n o r Q 2 sa tu ra tes .

T h e a d v a n ta g es o f b a la n c in g o u t th e c a rr ie r c o m p o n e n t fro m th e o u tp u t by c h o o s in g R C = l/co0 a re n u m ero u s. F ir s t , th e b a la n c in g su p p resses sm all am p litu d e


v a ria tio n s w h ich m ig h t sn e a k b y th e lim iter. F o r exa m p le , if

A = ^4o[l + £(0]>

w here |e(f)| « 1 is a sm all d is tu rb a n ce o n th e en v elo p e A , th en a t th e o u tp u t o f the

c irc u it o f F ig . 1 2 .3 -4

» .(*) = + « • (1 2 .3 -8 )71

H ere w e see th a t a co rre sp o n d in g sm a ll d is tu rb a n ce a p p e a rs a t th e o u tp u t w hich d ecrea ses in d irec t p ro p o rtio n to / (f). O n the o th e r h an d , w ith A = A 0[l + e(i)], th e o u tp u t o f th e sin g le-en d ed c irc u it o f F ig . 1 2 .3 -1 is g iven by

V0(t) = a '4 ° C R o [co0 + A « / ( f ) + a>0e(t) + Au>f(t)e(t)]

(xA qC R qCOq1 + — f { t ) + e(t)

w 0(1 2 .3 -9 )

In th is c a se th e o u tp u t d is tu rb a n ce is in d ep en d en t o f f ( t ) a n d , i f Aco/a>0 is sm a ll, the d is tu rb a n ce m ig h t easily d o m in a te / (f). F o r th is re a so n a lo n e , a lm o s t a ll h ig h -q u a lity F M d em o d u la to rs a re b a la n ced in so m e fa sh io n o r a n o th e r . [N o te th a t su b tra c tin g a c o n s ta n t w hich is in d ep en d en t o f A fro m th e o u tp u t d o es n o t p ro d u ce th e sam e effect as b a la n c in g , even th o u g h vg(t) = 0 fo r co,(r) = o 0 .]

T h e b a la n c in g b ra n ch a lso p ro v id es a m e a n s o f d ev elo p in g a n e rro r s ig n a l fo r the a u to m a tic freq u en cy c o n tro l (A F C ) in an F M rece iv er o r the freq u en cy sta b iliz in g c irc u it o f an F M g en era to r . I f 1 /R C is a d ju sted to eq u al a>0 (w hich is u su ally the ce n te r freq u en cy o f the I F strip in a n F M rece iv er) an d the F M sig n al p assin g th ro u g h th e I F s tr ip d rifts in freq u en cy b y Sto, th e n th e o u tp u t o f th e c irc u it o f F ig . 1 2 .3 -4 is given by

17,(0 = ~ ^ ^ -[A<of{t) + ¿to]. (1 2 .3 -1 0 )71

I f v„(t) is p laced th ro u g h a su fficien tly lo w -p ass filter to rem o v e th e f ( t ) co m p o n e n t, i.e., th e d em o d u la ted sig n a l, th en th e filte r o u tp u t c o n ta in s a d c s ig n a l d irec tly p ro p o rtio n a l to th e freq u en cy d rift. T h is s ig n al c a n th en b e am p lified and ap p lied as a freq u en cy c o n tro l to th e lo ca l o sc illa to r (w hich in th is c a se is a n F M g e n era to r) to re tu rn th e m ix e r o u tp u t sig nal to th e ce n te r o f th e I F strip .

T h ird , b a la n c in g p erm its us to in cre a se g rea tly th e sen sitiv ity o f t?„(r) to / (f). A s c a n b e seen fro m F ig . 1 2 .3 -5 , th e s lo p e o f th e Acof(t)-v0(t) c h a ra c te r is t ic ca n be in crea sed to th e p o in t w here v a r ia tio n s in / (t) b etw een + 1 an d — 1 ca u se c o r resp o n d in g v a r ia tio n s in v0{t) b etw een + Vc c an d — Vc c ; th a t is, a s lo p e o f Vc c /Aco is p o ssib le . O n th e o th e r h an d , w ith o u t b a la n c in g , th e c h a ra c te r is t ic o f F ig . 12 .3 3

1 2 . 3 THE CLARKE-HESS FREQUENCY DEMODULATOR 591

p revails . T h is c h a ra c te r is t ic ca n hav e a m a x im u m slo p e (w ith o u t sa tu ra tin g the tra n s isto r) o f

J c c _ _ K c

«¡m ax W 0 +

T h u s w ith b a la n c in g th e sen s itiv ity m a y b e in c re a se d by a fa c to r o f

co0 + Aa> _ c«o

A oj Aci>’

w hich is q u ite s ig n ifica n t fo r m o st p ra c tica l c irc u its in w hich a>0 » Acu.

Example 12.3-1 C h o o s e v alu es fo r th e v a rio u s co m p o n e n ts in th e c irc u it o f F ig .

1 2 .3 -4 such th a t th e c irc u it p ro d u ces a 2 V p e a k -to -p e a k o u tp u t w hen d riven by an F M sig n al o f th e fo rm

r,(f) = (1 0 V ) c o s [1 0 7i + 1 0 5 J co s 2 x 10 4r],

T h e a v a ila b le p o w er su p p lies are + Vc c = ± 12 V.

Solution. W e first n o te th a t a>0 = 1 0 7 rad/sec, Acu = 1 0 s rad/sec, an d a>m = 2 x 104rad / sec; thu s /? = 5 an d D = 1/100. T h e re fo re , s in ce D and ft d eterm in e a p o in t w ellb elow th e cu rv e o f F ig . 1 2 .2 -6 , we o b serv e th a t d em o d u la tio n is p o ssib le w ith o u t req u irin g an e la b o ra te o u tp u t lo w -p ass filter.

W e n ext se lect tw o g erm a n iu m tra n s isto rs an d tw o g erm a n iu m d io d es w ith freq u en cy lim its w ell a b o v e 10 M H z. In a d d itio n , w e select R = 5 k£2 as a c o m p ro m ise betw een a larg e o u tp u t s ig n al an d an in crea sed v a lu e o f F0. A s we saw in S e ctio n 10.2, a sm all v alu e fo r R in cre a se s th e e ffectiv e size o f F0 . O n th e o th e r h an d , a large value o f R d ecrea ses th e sen sitiv ity o f th e d em o d u la to r .

W ith / ? = 5 k f la n d a > 0 = 10 7 rad/sec we o b ta in C = 1/co0R = 2 0 p F . T o o b ta in a 2 V p e a k -to -p e a k o u tp u t sig n al, th e s lo p e o f th e c h a ra c te r is tic o f F ig . 12.3 5 m u st be ad ju sted to e q u a l (1 V)/10? rad / sec; thu s R 0 m u st h av e the valu e

R 0 = . 1 V — = 157 kQ.10 rad/sec a A C

In ad d itio n , to p ass th e m o d u la tio n in fo rm a tio n , R 0C 0 m ust b e less th a n o r eq u a l to l/com. I f we c h o o se R 0C 0 = l/com, th en C 0 = 318 p F . (A ctu a lly , fo r th is valu e o f R 0C 0 th e o u tp u t b a seb a n d sig n al suffers a 3 d B a tte n u a tio n . T o av o id th is a tte n u a tio n C 0 m u st be re d u c e d ; th is leads to an in cre a se in o u tp u t rip p le.)

F in a lly , a c h o ic e o f C c = 2 0 0 0 p F en su res th a t th e co u p lin g c a p a c ito r is an effective sh o rt c ircu it in co m p a riso n w ith R a t th e c a rr ie r freq u en cy (,YC- = .R/100).

I f an F M sq u a re w ave is ap p lied to the freq u en cy d em o d u la to r show n in F ig .1 2 .3 -4 , the c ircu it fu n ctio n s in e x a c tly the sam e fa sh io n as it d id fo r a s in u so id a l F M


sig n al, provid ed th a t R C is re a d ju sted so th a t

R C = 7t/2ft>0 . ( 1 2 .3 -1 1 )

A s w as p o in ted o u t in S e c tio n 12.2 , w hen a n F M sq u are w ave is p la ced th ro u g h a d iffe re n tia to r an d a n a v era g e en v e lo p e d ete c to r , th e o u tp u t is id en tica l w ith th a t p ro d u ced by a s in u so id a l F M s ig n a l w ith th e s a m e am p litu d e. H e n ce th e c o n tr ib u tio n to v0(t) fro m th e up p er b ra n ch o f th e c irc u it o f F ig . 1 2 .3 -4 is u n ch an g ed w hen an F M sq u a re w ave is ap p lied . O n th e o th e r h an d , w hen a n F M sq u a re w ave o f a m p litu d e B is ap p lied to th e b a la n c in g b ra n c h o f th e d e m o d u la to r , a c o m p o n e n t o f v alu e B R 0/2 R is su b tra c te d fro m v0, a s th e re a d e r sh o u ld verify. T h u s , w ith a n F M sq u a re w ave ap p lied to th e freq u en cy d e m o d u la to r o f F ig . 1 2 .3 -4 , v0(t) is given by

V„(t) = | \ o 0 + Acof(t) - , ( 1 2 .3 -1 2 )

w hich red u ces to

ocBCRnv„(t) = -A co/(t) (1 2 .3 -1 3 )

71

p ro v id ed th a t R C = n/2co0. C le a rly , v0(t) given by E q . (1 2 .3 -1 3 ) is id en tica l w ith v„(t) g iven by E q . (1 2 -3 -7 ) fo r th e ca se w here th e F M sq u are w ave an d th e s in u so id a l F M sig nal h a v e th e sam e am p litu d e.

W ith a sq u a re -w a v e in p u t to th e freq u en cy d em o d u la to r o f F ig . 1 2 .3 -4 , a m u ch la rg er rip p le co m p o n e n t ap p ea rs a t th e o u tp u t th a n w ith a c o rre sp o n d in g sin u so id a l inp u t. T h is is a d ire c t resu lt o f th e n a rro w cu rren t p u lses o f a re a 2 B C w h ich flow th ro u g h th e up p er b ra n ch o f th e freq u en cy d em o d u la to r in to th e o u tp u t c a p a c ito r C 0. In p a rticu la r , th e rip p le o n v„(t) fro m th is c o m p o n e n t a lo n e h a s a v alu e o f 2 B C /C 0 . O f co u rse , th is r ip p le c a n b e rem ov ed by su b seq u en t stages o f f i lte r in g ; h ow ev er, it still ad d s d irec tly to t7„(f)- T h is d ire c t a d d itio n lim its th e m a x im u m a m p litu d e o f the s ig n al c o m p o n e n t o f v„(t), s in ce th e s ig n a l p lu s th e ripp le m u st n o t sa tu ra te th e tra n s is to rs o f th e d em o d u la to r . In p a rticu la r , th e m a x im u m sig n al co m p o n e n t o f v j t ) m u st b e k e p t less th a n Vc c + V0 — 2 B C /C 0 .

In th e p ra c tic a l c irc u it th e fin ite v alu e o f r x ca u sed b y th e g e n e ra to r an d tra n s is to r- d io d e im p e d a n ces w ill ca u se th e “ d iffe re n tia to r” b ra n ch cu rre n t p u lses to ta k e th e sh a p e o f d eca y in g e x p o n e n tia l p u lses o f in itia l am p litu d e 2 B /r t ra th e r th a n im p u lses o f in fin ite am p litu d e . In e ith e r c a se th e p u lse a re a w ill be 2 B C . In th e p ra c tic a l ca se o n e m u st m a k e su re th a t 5 rxC is sm a lle r th a n a h a lf-p e rio d o f th e h ig h est in s ta n ta n e o u s freq u en cy f 0 + A/ in o rd e r th a t th e p u lse ca u sed b y o n e s tep is c o m p le te d b e fo re th e n ex t s tep o ccu rs. T h is re q u ire s th a t

C - l(K/o + Af ) r z (12.3-14)

1 2 .4 FREQUENCY-DOMAIN DIFFERENTIATION 5 9 3

12.4 FR EQ U EN C Y -D O M A IN D IFFEREN TIA TIO N — T H E S L O PE D EM O D U LA TO R

A freq u en cy -d o m a in d iffe re n tia to r is a lin ea r n e tw o rk w h o se tra n sfer fu n ctio n H (jo i) h a s a lin ea r ly s lo p in g m a g n itu d e o v er th e b a n d o f freq u en cies o ccu p ie d by the in p u t F M signal. S u c h a n e tw o rk , as w as sh o w n in S e c tio n 11.3 , fu n ctio n s as a d iffe re n tia to r an d c o n v erts v a r ia tio n s in th e in s ta n ta n e o u s freq u en cy a t its in p u t to en v elo p e v a ria tio n s a t its o u tp u t. F o r exa m p le , i f th e n e tw o rk h as the tra n sfe r fu n ctio n show n in F ig . 1 1 .3 -2 an d if v{t) given by E q . ( 1 2 .2 - la ) is ap p lied a t its in p u t, th e n e tw ork o u tp u t is g iven by (cf. E q . 1 1 .3 -2 0 )

,,, io) ® ivb(t) = A ----------------------------co s co0t + Aco f { B ) d 6 + 9(o)0) (1 2 .4 -1 )

H e n ce a t the o u tp u t o f the a m p litu d e d em o d u la to r fo llow ing th e freq u en cy -d o m a in d ifferen tia to r we o b ta in

v0(t) = A K dK m [(o0 - o jj + Aco/(f - i 0)], (1 2 .4 -2 )

w here K D = \/u>a is the slo p e o f the |/i(jo>)|-vs.-a> c h a ra c te r is tic and K M is th e c o n sta n t o f th e en v elo p e d em o d u la to r . I t is a p p a ren t th a t a freq u en cy -d o m a in d ifferen tia to r fo llow ed by an a m p litu d e d e m o d u la to r d o es ind eed p ro d u ce a sig nal p ro p o rtio n a l to / (f) [to be m o re p recise , / (f — f0)].

H ow ever, b e ca u se m o st p ra c tica l filters hav e lin ea r c h a ra c te r is tic s o n ly o v er a narrow ban d o f freq u en cies, a freq u en cy d e m o d u la to r e m p lo y in g a freq u en cy -d o m a in d ifferen tia to r is b est su ited fo r F M sig n als for w hich the d ev ia tio n ra tio is sm all. T h is freq u en cy re s tr ic tio n a lso p rev en ts th e d em o d u la tio n o f a n F M sq u a re w ave (w hich h as freq u en cy co m p o n e n ts in th e v ic in ity o f w 0 , 3w 0 , 5a>0 , e tc .) unless the sq u a re w ave is first filtered to e x tra c t its fu n d a m en ta l F M co m p o n e n t. T h is filtering , in so m e cases, c a n be a cco m p lish e d by th e d iffe re n tia tio n n e tw o rk itself.

T h e sim p lest p ra c tica l im p le m e n ta tio n o f th e freq u en cy -d o m a in d ifferen tia to r is a p a ra lle l R L C c irc u it w hich is tu n ed so th a t th e in p u t F M sig n al is cen tered on the s lo p in g p o rtio n o f th e tra n sfe r fu n ctio n . F ig u re 1 2 .4 -1 illu s tra te s su ch a c ircu it

fo llow ed by a p eak en v elo p e d e te c to r to fo rm a “ s lo p e ” d em o d u lato r .

Fig. 12.4-1 Single-ended slope frequency demodulator.


I f w e assu m e in itia lly th a t R 0 » R an d C 0 « C so th a t th e en v e lo p e d e te c to r d o es n o t lo a d th e tun ed c irc u it , th a t A co an d com h a v e a p p ro p ria te v alu es, in c o m p a riso n w ith th e p a ra m e ters o f th e tuned c irc u it , fo r th e “ q u a s i-s ta tic ” a p p ro x im a tio n o f E q . (1 1 .3 -3 ) to b e v alid (we sh a ll c h e ck th ese assu m p tio n s), th a t (¡(f) = /t c o s [co0t + Aco f ' f ( 0 ) d O ] , a n d th a t D an d /? fo r th e in p u t F M sig nal d ete rm in e a p o in t b e lo w th e cu rv e o f F ig . 1 2 .2 -8 , th en th e en v e lo p e v a r ia tio n in v(t), w hich a p p ea rs a s v0(t), is given

by

vJLt) = h\Zn[jcom= / ,| Z n [ ; « 0 + y A « / (i)]| , (1 2 .4 -3 )

w here Z u (p) is th e in p u t im p ed a n ce o f th e p a ra lle l R L C c ircu it. F ig u re 1 2 .4 -2 d ep ic ts th is tra n sfe r c h a ra c te r is t ic w h ich re la te s co,(t) and v jt).

Fig. 12.4-2 Transfer characteristic relating to ¡(f) and t^ i).

T o d esign a s lo p e d em o d u la to r w e m u st select valu es fo r th e c e n te r freq u en cy coc an d the b an d w id th B W o f th e tu n ed c irc u it o f F ig . 1 2 .4 -1 . In g en era l, we ch o o se th ese p a ra m e ters to a ch iev e th e la rg est p o ssib le o u tp u t s ig n a l w h ich is lin early re la ted to a>,(f)- T o see h ow th is c h o ic e c a n b est be a cco m p lish e d , we exp an d |Z, in a T a y lo r series in w a b o u t a>0 :

|ZuO'co)| = \ Z i i ( j o j 0 )\ + IZ nO iU oM ® ~ " o )

, 1-7 - f f l o ) 2 , 1-7 / • u J " - w o ) 3 , t 1 A ^+ l ^ n 0 ^ o ) l -------2\----------l^ n O ^ o ) ' ------------- (-•••• (1 2 .4 -4 )

I t is a p p a re n t th a t m a x im u m lin ea r ity is a ch iev ed i f coc is c h o se n su ch th a t \ Z u ( j o ) 0 ) \ "

and as m an y su b seq u en t d eriv ativ es o f |ZX 1(y'a>)||lu=cuo as p o ssib le are zero . T h e b an d w id th is th en c h o se n as n a rro w as p o ssib le to m axim ize I Z ^ i j a j Q ^ ' . [A sm all b an d w id th steep en s th e sk irts o f |Zu (;'a))| and thu s in creases its s lop e.] H ow ever, a sm all b an d w id th a lsa in c re a se s th e size o f th e first n o n z ero d eriv ativ e o f| Z ( , ( jo>)\ |C) = a)0,

12 .4 FREQUENCY-DOM AIN DIFFERENTIATION 595

th ereb y in ci e asin g the n o n lin e a r d is to rtio n fo r a fixed Aco. T h e p erm issib le d is to rtio n puts a lim it o n the m in im u m valu e o f the ban d w id th .

If Q T for the p a ra lle l R L C c irc u it is g re a te r th a n 10, then |Z, , 0 « ) ! ca n be c lo se ly a p p ro x im a te d by (cf. E q . 2 .2 -9 )

\Z(jco)\ = ^ _ £ ==== , (1 2 .4 -5 )t + t < o -

a

w here a = 1/2R C = BW /2 an d foc = f/v /L C . F o r th is case

iZu(M ,)r = Rn0(1 + Q 2)31 / 2 ’

R ( m 2 - 1)a 2( l + n 2 | 5 / 2 ’

- 3 R (2 i2 3 - 3C1)

Z\ lO ^ o )!” — ~2T, , r*2\5/2' (12.4—6)

I Zlt(ja>or = a (1 + fi 2 \ 7 / 2 ’

w here Q = (w 0 — coc)/a. I t is a p p a ren t th a t |Z, i(ju>0 )\" = 0 provid ed th at Q 2 = j o r, eq u iv a len tly , th a t

wt = w 0 ± oi/y/2. (12 .4- 7)

T h e plus term o f E q . (1 2 .4 -7 ) co rre sp o n d s to o p e ra tio n on the low er sk irt o f |Z, ¡(jio)\, w hile the m in u s term co rre sp o n d s to o p e ra tio n on the up per sk irt. W ith Q 2 = j , n o n e o f the h igh er d eriv ativ es o f\ Z l i0'a))|)| = V i « , \Z = +

3 ^ 3 « ’

_ 6 R /2\3 | Z n ( jw 0)l — +

h en ce , va(t) m ay be w ritten (on the a ssu m p tio n th a t the q u a s i-s ta tic a p p ro x im a tio n is valid ) as

v0(t) = I A Z n [ j o ) 0 + j Aco/(f)]|

r2 l l ‘ A ojf(t)± •••!>. (1 2 .4 -8 )

N ow , if 2A «j/3a < 0 .0 4 o r, eq u iv a len tly , B W > 100 Acu/3, then th e th ird term o f E q . (1 2 .4 —8) is less th a n 1 " „ o f t h e seco n d term fo r a l l/ ( f ) , an d thu s v„(t) v aries re a so n a b ly lin early w ith / (f). S p ecifica lly , if the b an d w id th -is ch o se n a t its m in im u m valu e


B W = 100 Aw/3, th en v0(t) is g iven by

v0(t) = | l + - ~ = . (1 2 .4 -9 )2 5 ^ 2 j

A s w e saw in S e c tio n 11.3 th e q u a s i-s ta tic a p p ro x im a tio n is v alid fo r th e s in g le tun ed c irc u it p ro v id ed th a t (cf. E q . 1 1 .3 -1 7 )

1 1 Aco\2

fi\ « ' <<L

F o r th e c irc u it ju s t co n sid ere d , (Aoj/a)2 = 0 .0 0 3 6 ; h en ce , fo r a n y v alu e o f /? g re a te r th a n 0 .3 6 ,

1 /Aw\2 _

/»( « ) < ' ’

th e v alid ity o f q u a s i-s ta tic a p p ro x im a tio n is ensu red , an d th u s E q . (1 2 .4 -9 ) p rov id es an a c cu ra te e x p re ss io n fo r v0(t).

O n c e w 0, Aw, an d com fo r th e in p u t F M sig n al are k n o w n , th e p a ra m e ters o f the tu n ed c irc u it c a n b e selected . F o r ex a m p le , i f w 0 = 27t(10.7 M H z ), Aw = 27c(75 kH z), an d w m = 27t(15 k H z ) (c o m m e rc ia l F M ) , th e n B W = 100 Aw/3 = M H z ) and

a B Ww c = w 0 H----7= = w 0 H--------- 7= = 2 ^ (1 1 .2 3 M H z ),

y / 2 2 y J 2

w ith th e resu lt th a t Q T = a>JB W = 7 .5 . If, in ad d itio n , R = lO k Q an d I x = 1 m A , th en C = 1/(BW)/? = 6 7 p F , L = l/wc2C = 12 //H, and

vJLt) = (8 .1 5 V ) + (0 .2 3 V )/ (i)

(P = Aw/wm = 5 en su res th e v a lid ity o f th e q u a s i-s ta tic a p p ro x im a tio n ).

Balanced Slope Demodulator

A s w as p o in ted o u t in S e c tio n 12 .3 , sin g le-en d ed d em o d u la to rs a re v irtu a lly never used i f Aw/w0 is sm a ll, b e ca u se o f th e ir h igh sen sitiv ity to en v e lo p e v a r ia tio n s o n the in p u t signal. C o n se q u e n tly , to m a k e th e s lo p e d e m o d u la to r less sen sitive to en v elo p e v a ria tio n s a t its in p u t, a b a la n c in g p a th m u st b e in tro d u ced . A b a la n c e d s lo p e d em o d u la to r w h ich n o t o n ly p ro d u ces z e ro o u tp u t fo r w, = w 0 b u t a lso h a s a n in creased lin ea r o p e ra tin g ra n g e is sh o w n in F ig . 1 2 .4 -3 . F o r th is c irc u it , b a la n c in g is ach iev ed by ta k in g th e d ifferen ce o f th e o u tp u t v o lta g es o f tw o s in g le-en d ed slo p e d em o d u la to rs , o n e tu n ed a b o v e w 0 an d th e o th e r tu n ed b e lo w w 0 by a s im ila r am o u n t.

I f a g a in w e a ssu m e th a t th e en v e lo p e d e te c to rs d o n o t lo a d th e tu n ed c ircu its , th a t th e “ q u a s i-s ta t ic ” a p p ro x im a tio n is .valid fo r d ete rm in in g th e v o lta g e a cro ss e a ch tu n ed c irc u it , an d th a t i,(t) is g iven by

/',(/) = / , cos w0i + Aw J f(0) dd , (12,4-10)

12 .4 FREQUENCY-DOMAIN DIFFERENTIATION 597

Fig. 12.4-3 Balanced slope dem odulator.

then fo r the d e m o d u la to r show n in F ig . 1 2 .4 -3

v„(t) = i„ i( t ) - vo2(t)

= M l z u [ M > + j Aco/(f)]|) - I Z u i j w + j Acof{t)]\2 )

= ll\ZT[jw0+ jAtiifm, (12.4 11)

w here \ZT(jio)\ = \Zl l (jw)\1 — \Zl l (jw)\2 an d Z i i (p )l an d Z l i (p)2 a re the input im p ed an ces o f the p ara lle l and R 2L 2C 2 c ircu its resp ectiv ely . T h e transferc h a ra c te r is tic now re la tin g a>,(f) and v0(t) is show n in F ig . 1 2 .4 -4 .

F r o m Fig . 12.4 4 th e re a so n fo r the in creased lin earity b e co m e s ap p aren t. If the tw o ind iv idu al c h a ra c te r is t ic s show n in F ig . 1 2 .4 -4 a re c o rre c tly a d ju sted , they c o m b in e in such a w ay as to form a ^„(O-vs.-wî) c h a ra c te r is tic w hich has oud

Fig. 12.4-4 T ransfer ch aracte ris tic re la ting to ¡(f) an d v0(t) for the circuit o f Fig. 12.4-3.


sym m etry a b o u t c j0 . T h u s i f |Z^j'co)! is exp an d ed in a T a y lo r series in a> a b o u t a>0 , th e even term s v a n is h :

T h e re fo re , ojc1 an d coc2 c a n n o w b e c h o se n su ch th a t |Z j { j o j 0)\'" = 0 [s in ce IZ jijw o)]" is a lre a d y z e ro ], w ith th e resu lt th a t th e first te rm in tro d u c in g a n o n lin e a r ity i n \ZT(ja >)\, an d in tu rn va(t), in v o lv es th e fifth d eriv ativ e o f \ZT(jto)\ |(U=(Uo.

I f Q t fo r b o th th e tu n ed c irc u its o f F ig . 1 2 .4 -3 is g re a te r th a n 10, th en \ZT(juj)\ m a y b e c lo se ly a p p ro x im a te d by

w here = l / 2 R 1C 1 = B W ^ an d a 2 = 1/2R 2C 2 = B W 2/2. F o r th is ca se , od d sy m m etry in \ZT(jaj)\ a b o u t co0 is o b ta in e d by c h o o sin g R , = R 2 = R , tx1 = a 2 = a = BW /2, ft)cl = co0 + ¿co, an d coc2 = co0 — Sa>. In a d d itio n (as th e rea d er sh ou ld verify), \Z j{jta0)\"' = 0 if 5a> is c h o se n as

F o r th is ca se , if Aco/a. < 0 .5 8 5 , th en th e seco n d te rm o f E q . (1 2 .4 -1 6 ) is less th a n 1 % o f th e first te rm fo r a ll valu es o f / ( i ) ; th u s v0(t) v aries in d irec t p ro p o r tio n to f ( t ) .

H o w ev er, a v alu e o f Aco/a a s la rg e a s 0 .5 8 5 p u ts a s tra in o n th e q u a s i-s ta tic a p p ro x im a tio n fo r v alu es o f fi le ss th a n 7 (cf. E q . 1 1 .3 -1 7 ) . T h e re fo re , Aco/a is u sually ch o se n to b e j , a v a lu e w hich en su res th e v a lid ity o f E q . (1 1 .3 -1 7 ) w ith P a s lo w a s 5.

W ith Aco/a = j (o r B W = 4 Aco), v„(t) g iven b y E q . (1 2 .4 -1 6 ) sim p lifies to

\ZT( M = \ZT(jco0)\'(co - co0) + \ZT(jo j0)\J œ 3!t° o)3

(1 2 .4 -1 2 )

\ZT{j(o) ( 1 2 .4 -1 3 )

(1 2 .4 -1 4 )

W ith th is c h o ic e fo r Sa>, th e T a y lo r series e x p a n sio n fo r \ZT(jai)\ red u ces to

co — co0

a 6 2 5 \ a(1 2 .4 -1 5 )

and in tu rn

, , 4 I tR 3T A co f(t ) 54J a 6 2 5) 5 + (1 2 .4 -1 6 )

(12 .4 -17 )

12.4 FREQUENCY-DOMAIN DIFFERENTIATION 599

H en ce , i f /, = 1 m A an d R = lO k i i , v0(t) = (3.1 V )/ ( i ) ; th u s th e b a la n ced slo p e d em o d u la to r , in a d d itio n to its o th e r a d v a n ta g es, p ro v id es a co e ffic ie n t fo r t h e / ( i ) - te rm in its o u tp u t w hich is m o re th a n 13 tim es as larg e as th e co rre sp o n d in g co e ffic ie n t fo r th e sin g le-en d ed slo p e d em o d u la to r .

H e re a g a in , p a ra m e ters fo r th e tun ed c ircu its o f the b a la n ced slo p e d em o d u la to r m ay b e c h o se n o n c e co0 , Aco, an d com a re sp ecified . U sin g th e c o m m e rc ia l F M vaiues, we o b ta in B W = 4A a> = 2 7 c(300k H z), ¿a> = 27r(l84 kH z), cocl = 27t(10.884 M H z ), and cuc2 = 2 tt(1 0 .5 1 6 M H z ); h en ce Q Tl = 3 6 .3 , Q T2 = 3 5 .2 , C x = C 2 = 53 p F , L j = 4 .0 4 /¿H, a n d L 2 = 4 .5 5 ¿iH.

A p o ssib le p h y sica l c irc u it fo r rea liz in g th e b a la n ced slo p e d em o d u la to r is sh o w n in F ig . 1 2 .4 -5 . F o r th is c irc u it th e first d ifferen tia l p a ir n o t o n ly p ro v id es th e

tw o id en tica l cu rre n t drivesi,-(i), b u t a lso a c ts as a lim iter. (A ctu a lly th e cu rren ts a t th e c o lle c to rs o f Q x an d Q 2 a re 180° o u t o f p h ase w ith e a ch o th er. T h e en v elo p e d ete c to rs , how ever, a re n o t sen sitive to th is p h ase in v ersio n .) T h e d io d es in th e d ete c to rs are

Fig. 12.4-5 Practical embodiment of balanced slope demodulator.


m ad e o f g erm a n iu m to m in im ize th e ir e ffectiv e V0 ; how ever, i f th e d io d es a re m a tch ed , th e F0-term is b a la n ced o u t a t th e o u tp u t. F in a lly , th e seco n d d ifferen tia l p a ir a cce p ts th e d ifferen tia l in p u t an d p ro v id es a sin g le-en d ed o u tp u t. T h e re s is to rs a re ad ded in series w ith th e e m itte rs o f th is d ifferen tia l p a ir to in cre a se th e lin ea r o p e ra tin g ran ge and red u ce th e lo a d in g o f th e d ifferen tia l p a ir o n th e en v elo p e d ete c to rs . T h e m o d u lated in p u t to th is c irc u it is th e tra n sfo rm er-co u p le d v o lta g e v{t) a p p e a rin g a t th e b ase o f Q , .

Envelope Detector Loading

In m an y a p p lica tio n s it is n o t d es ira b le to m a k e th e o u tp u t re s is ta n ce R 0 o f the en v e lo p e d e te c to r in th e slo p e d e m o d u la to r so la rg e th a t it d o es n o t lo a d th e tun ed

c ircu it. H o w ever, p ro v id ed th a t C 0 lo o k s lik e a n o p en c irc u it in co m p a riso n w ith R 0 o v er the b a n d o f freq u en cies o ccu p ied by / (f), th a t is, 0 < com, th e lo a d in g o f th e en v elo p e d e te c to r is read ily a cco u n te d for. In p a rticu la r , w ith C 0 a n o p en c ircu it to / (f), th e en v e lo p e d e te c to r fu n ctio n s in a s ta tic fash io n an d its p rim a ry effect is to low er th e Q T o f th e tuned c irc u it by re fle ctin g a re s is to r o f valu e R 0/2 in p a ra lle l w ith the p ara lle l R L C c ircu it. F o r exa m p le , i f th e tun ed c irc u it h a s a p a ra lle l re s is ta n ce o f 20 k Q an d R 0 = 4 0 k f i , th e lo a d in g o f th e en v elo p e d ete c to r m ay b e a cco u n te d fo r by d ecrea sin g the to ta l p a ra lle l re s is ta n ce R T = R||(R0/2) to l O k i l T h is v alu e o f R T is th en used in a ll th e e x p ressio n s p rev io u sly derived.

Series-Tuned Balanced Slope Demodulator

Series re so n a n t c irc u its ca n a lso b e used to o b ta in a slo p e d em o d u la to r . F o r th is d u al a rra n g e m en t a c o m m o n v o lta g e m u st d rive th e tw o series-tu n ed c ircu its , and th e d ifferen ce o f th e en v e lo p es o f th e ir c u rren ts m u st be used to p rov id e an o u tp u t. A c ircu it fo r a cco m p lish in g th is is illu s tra te d in F ig . 1 2 .4 -6 . F o r th is d em o d u la to r a n av erag e en v elo p e d e te c to r is used in stead o f th e p eak en v e lo p e d e te c to r ; thu s th e re s tr ic tio n s o n D an d /? o f th e in p u t F M sig n al a re co n sid e ra b ly re laxed . S p ecifica lly , D an d P m u st d ete rm in e a p o in t u n d er th e cu rv e o f F ig . 1 2 .2 -7 in stead o f d eterm in in g a p o in t u nder the m o re re s tric tiv e cu rv e o f F ig . 1 2 .2 -9 .

F o r th e c irc u it o f F ig . 1 2 .4 -6 , i f V0 % 0 fo r th e d io d es an d tra n s isto rs , then v(t) ap p ea rs d irec tly a c ro ss th e series-tu n ed c ircu its , w ith the resu lt th a t th e en v elo p e o f i',(r) is given by

V iW ^ U too + jAcof(t)]\l

an d th e en v elo p e o f i2(t) is given by

V ilY nU co o + j Aoj/(f)]|2 ,

w here Y n { p ) { a n d Yl l (p )2 a re th e a d m itta n ce s o f th e series r , L , C , an d r 2L 2C 2 re so n a n t c ircu its resp ectiv ely . F o r e a ch tuned c irc u it , th e series re s is ta n ce inclu d es an y .in p u t re s is ta n ce o f th e tra n s is to r-d io d e c o m b in a tio n s . C o n se q u e n tly , i f th e b a n d -

12.4 FREQUENCY-DOMAIN DIFFERENTIATION 601

width of the R 0~C0 low-pass filter is greater than com, then v0(t) is given by

atj-Fj.Ro71

otj-V^Ro (12.4-18)

where | YT(jo>)\ = | y, i(joj)\ = \ l (yoj)|, — I Yt l(jco'fi2. (A subscript T is used to differentiate the transistor ctT from the a of the tuned circuit.)

It is apparent tha t Eq. (12.4-18) is of the same form as Eq. (12.4-11); hence the same considerations employed in choosing param eter values for the tuned circuits of Fig. 12.4—3 apply to the tuned circuits of Fig. 12.4—6. In particular, if QT > 10 for both of the tuned circuits of Fig. 12.4-6, then

v„(t) =

where

<*t K R o 1 1

1 +j~oj,(Q - coc,

L “ i1 +

CO,(0 - w c(12.4-19)


Hence, if we again choose a! = a 2 = a, r , = r 2 = r,aicl = co0 + Sa>, a>c2 = a>0 — Soj, and doj = ^ / f a , then provided that Aco/a is kept less than v0(t) reduces to

, „ 4ar F,/?0 ¡3 Acof(t)v„(t) = 1 ° / - — (12. 4-20)

5nr \ 5 a

As with the C lark e-Hess dem odulator discussed in Section 12.3, R 0 should be chosen sufficiently large to maximize »„(t),'-but no t so large tha t either transistor in the circuit of Fig. 12.4-6 saturates on the peaks of /( t) .

A lthough we have assum ed F0 % 0 in this analysis, the expression for v0(t) given by Eq. (12.4-20) is not appreciably affected if V0 is a few tenths of a volt. This insensitivity is a direct result of the balanced nature of the c ircu it; thus, to aid in this balancing, every effort should be m ade to m atch the input characteristics of the tran sistors and diodes of Fig. 12.4-6. If this is not done, some variations in center frequency with input am plitude will result.

Crystal Slope Demodulator

In place of the two tuned circuits of Fig. 12.4-6, two crystal resonators may be inserted, one with its series resonance slightly above <o0 and the o ther with its series resonance slightly below co0. The result is a slope dem odulator with an extremely stable center frequency and an ou tpu t voltage which is extremely sensitive to small deviations about co0.

Sm ithf has reported such a discrim inator in which, instead of separate crystals, three sections of a single quartz crystal are mechanically coupled. He presents results for a detector with a center frequency of 15 M H z and a linear bandw idth of ± 3 kHz. His device is reported to be generally useful in the 5-40 M H z range and to have possible useful bandw idths of up to 0.2 % of the center frequency.

A single crystal used with a balancing branch is often employed as the discrim inator in the frequency stabilizing situation described in Section 11.9. In such situations it is norm ally m ore satisfactory to operate the crystal in the vicinity of the zero in its im pedance function rather than near its parallel resonant frequency. (Section 6.7 discusses the crystal’s im pedance function in some detail.)

12.5 TIME-DELAY DIFFERENTIATOR, TIME-DELAYDEMODULATOR, FOSTER-SEELEY DEMODULATOR, AND RATIO DETECTOR

A time-delay differentiator is a linear netw ork which im plem ents the fundam ental definition of a derivative,

dv(t) v ( t ) - v ( t - t 0) „ - . e n—:— = h m --------------------- . (12.5-1)

dt <o-o t o

The block diagram for such a netw ork is shown in Fig. 12.5-1. In this diagram we

t Warren L. Smith, “The Monolithic FM Discriminator: A New Piezoelectric Device,” IEEE Trans, on Communication Technology, Coro-16, No. 3, pp. 460-463 (June 1968).

12.5 TIME-DELAY DIFFERENTIATOR AND DEMODULATOR 603

v(f)o + *0

Amplifierv« - 0

Delay line to

l0 small compared with variations in v(t)

Fig. 12.5-1 Block diagram of a time-delay differentiator.

observe that a delayed version of v(t) is subtracted from i>(r) and the resultant signal is amplified by l / r 0. In general, t0 m ust be chosen equal to zero to realize the exact derivative of v(t); however, as a practical m atter, so long as i0 is small in com parison with the time interval during which variations in v(t) occur, the derivative of v(t) is approxim ated quite accurately.

I t is apparent tha t t 0 should not be chosen any smaller than necessary, since such a choice requires an additional amplifier gain (1 j t 0) to bring the signal up to a useful level. Therefore, we shall now determ ine the maxim um value of t0 which perm its the network to function as a differentiator (i.e., to convert variations in instantaneous frequency into envelope variations) for the case where i’(i) has the form

In the next section we shall again determ ine the maximum permissible size of f0 where v{t) is a square-wave FM signal given by Eq. (12.2-lb). Once the m axim um size of f0 is determ ined, we shall implement the block diagram of Fig. 12.5 -1 with practical circuits and then interconnect them with the various am plitude dem odulators to form frequency dem odulators.

For the purpose of analysis we shall consider the m ore general time-delay differentiator, shown in Fig. 12.5-2, from which the amplifier has been om itted and the

(12.5-2)

v(/) v„(/)o- + ■o

— Delay network

(0 - time delay0 0 = carrier phase shift

Fig. 12.5-2 General time-delay differentiator.


Fig. 12.5-3 Magnitude and phase characteristic of the delay network of Fig. 12.5-2.

provision for a phase shift of 0o a t co0 in the delay network has been included. Figure12.5- 3 illustrates the m agnitude and phase characteristic of the delay netw ork of Fig.12.5-2. N ote that, if 60 = - (o0t0, then the phase characteristic passes through the origin and the norm al delay line shown in Fig. 12.5-1 results.

If we apply v(t) given by Eq. (12.5-2) to the time-delay differentiator shown in Fig. 12.5-2, then with the aid of Eq. (11.3-3) we can write

vb(t) = A cos [ o v + Aco J f (0 ) dd~ — A cos + Acu J f(Q) d6 + 60

Envelope te rm : g(!)

= — 2 A sinin i^ A o > £ m d O - 9 oj

x sin |co0f + Ao) J" f(6)d6 — jjÂco f(8) d6 — 0o(12.5-3)

H ig h -frequency te rm

The second expression in Eq. (12.5-3) is ob tained with the aid o f the trigonom etric identity

x + y . x - y cos x — cos y = —2 sin — ~— sin —-— .

2 2

Since vb(t) will be placed through an envelope detector, only the envelope g(r) of vb(t) need be considered in determ ining the m axim um permissible value for t0. M ore specifically, the m axim um value of tQ is tha t value which still perm its g(f) given by

g(i) = 2 A sin ^ £aco J

to vary in direct p roportion to / ( f ) .

f m d o - e ,•] (12 .5 -4 )


The envelope g(t) may be arranged in a form in which its dependence on f„ is m ore explicit by defining

Aa>k(t) =Aco~Y

’ f mde• 'i-ro

r A«? r toJ m d o I2 J f(0 )d0 . (12.5-5)

Now by taking the Fourier transform of .Eq. (12.5-5) we obtain

^ Aa>F(a>) 1K(w) = r .-(1

2 jo)

= F(co)e 2 Aa}to sin (cofp/2)wto/2

= F(a>)H D(ja>), (12.5-6)

where K(ctj) and F(a>) are the Fourier transform s of k(t) and f i t ) respectively and

H D(ja>) = e — jaito/2Acyt0 sin {(ot0/2) 2 o)t0/2

A plot of \HD(ja>)\ and arg H D(jcj) vs. a> is shown in Fig. 12.5-4. from which we observe that \HD(jio)\ is essentially constant a t Acoi0/2 for a> < 2 /i0 ; hence, if a>m < 2/t0 , where com is the limit on the spectrum of F(a>), then

K M = F(«»Hd(oj) ^ e - J“ o /2 F (4

or equivalently,

m

(12.5-7)

(12.5-8)

Fig. 12.5-4 Plot of the magnitude and phase characteristic o f H^joS).


Thus we see that k(t) is p roportional to a delayed versionf of /( f ) provided that

t0 < — .

W ith k(t) given by Eq. (12.5-8), g(f) simplifies to

Acofg(t) = 2A sin Atofp , / fp\ Q0

2 J \ 2 1 2 _

(12.5-9)

(12.5-10)

The transfer characteristic relating g(f) and (Acof0/2) f ( t - f0/2) is shown in Fig.12.5-5, from which we observe tha t only if

Fig. 12.5-5 Transfer characteristic relating g(t) and (Atot0/2)/(f — t j 2).

Acw t0for all f

does g(f) vary in p roportion to f ( t — t0/2) as desired. M ore specifically, if we expand g(f) given by Eq. (12.5-10) in the form

git) = 2A sin fc (I)c o s^ — 2 A cosfc(t)sin (12.5-11)

we observe that, if k(t) is kept less than 0.2 for all f, then sin k(t) « k(t) and cos k(t) % 1 (within 2 %), with the result that

g(f) = 2Ak(t) cos — 2A sin ^

= A Ao)t0/ | f - c o s y - 2A

The requirem ent that k(t) < 0.2 is equivalent to

Acot0 < 0.4,

as is readily seen from Eq. (12.5-8).

sin 0o (12.5-12)

(1 2 .5 -1 3 )

t A slight delay is unimportant in most communication applications.

12.5 TIM E-DELA'. r' I f C I :r . n H '"» « jR AND DEMODULATOR 607

The desired proportionality between g(f) an-i f i t as expressed in Eq.(12.5-12), requires that both Eqs. (12.5-9) and (12.5-13) be satisfied. When P > 0.4/2 = 0.2 (which is usually the case in practice), Eq. (12.5-13) is m ore restrictive than Eq. (12.5- 9) hence for this case the m axim um size of t0 is restricted to 0.4/Aw.

In general, a value for 60 m ust also be selected. It is apparent that, if 60 is chosen as 0, 2n, 4n, etc., the com ponent of g(i) p roportional to f ( t 0/2 ) is maximized. The difficulty with this choice is that g(i) is not positive (or negative) for all t and thus peak o r average envelope detection is impossible. To overcom e this difficulty, 90 must be chosen such that (cf. Eq. 12.5-12)

or equivalently,

0O „ • 0o Acui0 cos — < 2 sin — , 2 2

O n A C O tnt a n ^ > — A

2 2(12.5-14)

Since Aa>t0 < 0.4, Eq. (12.2-33) is satisfied provided that

0o > 0.4(22.7°). (12.5-15)

In many practical circuits a phase shift at the carrier frequency of 0O = ± n /2 is readily ob ta ined ; thus this value is usually employed, with the result that Eq. (12.5-12) simplifies to

g(i) = + A y / 2 (12.5-16)

Consequently, if the time-delay differentiator (with 0o = ± n /2 ) is followed by an envelope detector as shown in Fig. 12.5-6 and if t0 is chosen at its m axim um value of 0.4/Aco, then the resultant frequency dem odulator ou tpu t is given by

vjtt) = A K U^ 2 [ 1 + 0.2f { t - t0/2)], (12.5-17)

where K M is the am plitude dem odulato r constant and the — and + correspond to 0O = n/2 and 0Q — — n/2 respectively.

Fig. 12.5-6 Single-ended tim e-delay frequency dem odulator.


The single-ended time-delay dem odulator finds wide application in dem odulating FM microwave signals. Two paths are provided for v(t) through waveguides or transm ission lines of differing lengths. The difference in length is adjusted so that one signal experiences a tim e delay i 0 and a phase shift 90 = — oj0t 0 = —2>n/2 = — n/2 — n (that is, a phase shift o f —n/2 plus an inversion). The ou tp u t o f the two waveguides (or transm ission lines) is then com bined and envelope-detected w ith a crystal detector inserted in the guide. The detector ou tpu t is the dem odulated FM signal. A sim ilar scheme should apply to the dem odulation of frequency-m odulated laser beams.

A balanced frequency dem odulator employing a time-delay differentiator is shown in block diagram form in Fig. 12.5-7. In this diagram tw o single-ended time- delay frequency dem odulators, one with 60 = —n/2 and the o ther with 90 = n/2 ,

Fig. 12.5-7 Balanced time-delay frequency dem odulator.

are arranged in such a way th a t their ou tpu ts subtract, with the result th a t va{t) is given by

f0(t) = y / 2 A K M Aa>t0f ( t - t0/2). (12.5-18)

F or this case, not only are all the advantages of balancing obtained, but also the coefficient of the f ( t — f0/2)-term is increased by a factor of 2 over the single-ended dem odulator.

By noting tha t a phase shift of n/2 can be obtained through a phase shift of— ti/2 plus a signal inversion (a phase shift of —n), we may rearrange the balanced time-delay dem odulator of Fig. 12.5-7 in a form which requires only phase shifts of— n/2 at the carrier frequency. (A phase shift of — n/2 is m ore readily realized in a physical circuit than is a phase shift of + 7i/2.) T w o versions of the resultant circuit are shown in Fig. 12.5-8.

W hen an actual piece o f delay line is used to implement the circuit, the singledelay-line circuit is the obvious choice. For this case, 0 O = — to 0l 0 ; hence, for 0o = ~ n / 2 ,

t 0 = To/4,

12.5 TIM E-DELAY DIFFERENTIATOR AND DEMODULATOR 6 0 9

Fig. 12.5-8 A lternative fo rm s for the b a lanced tim e-delay dem o d u la to r.

where T0 is the carrier period. Thus to construct a delay-line dem odulator at T0 = lOOnsec (/„ = 10 MHz), one requires a m inim um overall delay of 25 nsec. Since an ordinary coaxial line has a delay of the order of 5 nsec/m, such a detector will be ra ther cum bersom e if built v/ith this type of line. Fortunately helically wound delay lines existf with delays of up to 30nsec/ar> o r 3 ¿¿sec/10 m. Thus, with (he use of distributed lines, delay-line dem odulator,' ire a practical possibility for values of / 0 as low as 800 kHz.

The Foster Seeley Demodulator

M any existing FM dem odulators use the nearly constant phase characteristic of a tuned circuit near resonance to approxim ate the delay line and thus to realize a delay-line type of dem odulator. In the analysis of such circuits, one should rem em ber that two layers of approxim ations are being used. In the first layer, one is approxim ating an ideal differentiator with a delay line; in the second layer, one is approxim ating the delay line by the tuned circuit. In the second approxim ation, one norm ally further neglects the circuit-induced am plitude variations.

t See J. M illm an an d H. T au b , P u ls e a n d D ig it a l C ir c u it s , M cG raw -H ill, N ew Y ork (1956). C h ap te r 10, Section 1 describes a n u m b er o f lines w ith delays from 140 nsec/m to 30,000 nsec/m .


Figure 12.5 9 illustrates the widely used Foster-Seeley FM dem odulator.! As we shall show, in this circuit the tuned transform er introduces a time delay t0 and a phase shift of — n/2 between v(t) and vb(t). The voltage vb(t) is then com bined with v(t) in a fashion identical with tha t of the block diagram of Fig. 12.5-8 to yield va(t) given by Eq. (12.5-18). Since peak envelope detection is employed, K M = 1.

Ideal

Fig. 12.5-9 Foster-Seeley dem odulator which physically implements the block diagram of Fig. 12.5-8 (top).

To obtain a suitable expression for t0 in term s of the param eters of the tuned circuit o f Fig. 12.5-9, let us assume that the envelope detectors do not load the tuned transform er circuit. (If they do, as was pointed out in the previous section, a resistor R 0 m ust be placed in parallel with R.) Neglecting detector loading, we can determ ine vb(t) from the m odel of the tuned transform er illustrated in Fig. 12.5-10 (cf. Fig. 2,A-3c).

W ith this m odel, vb(t) is related to v(t) by the transfer function

H(p) = MV(p)

k a i l 2p2 + 2a p + a»o’

(1

(12.5-19)

—........ . i» . rvnrx, a >------ c

+

•k v { t )

►—--------------------- -

> R C -

\ — —<

^ 2i

\------ cIdeal

Fig. 12.5-10 M odel for the tuned transform er of Fig. 12.5-9.

t D. F. Foster and S. H. Seeley, P r o c . I R E , 25, p. 289 (1937).


where co0 = l / , / (T ~ - k 2)LC and a = 1/2RC. The pole-zero diagram o f H(p) is shown in Fig. 12.5-11. N ote, however, th a t this pole-zero diagram is identical with that of Fig. 3.1-5 (except for the scale factor), for which 90 was found to equal —n/2. In addition, for QT = co0R C > 10 (cf. Eq. 3.1-9), H(jco) may be closely approxim ated as

F o r |(&) - a>o)/a| < 0.2, tan 1 [(co — w 0)/a] may be approxim ated by (o> - a»0)/a within 2 % and ^ / l + [(oj — a>0)/a]2 may be approxim ated by unity within 2 %. Thus, if we operate sufficiently close to co0 , H (ja>) simplifies to

which is the transfer function of a delay network with a delay of i0 = 1/a, with a carrier phase shift of —n/2, and with a scale factor of kQ T/2. By choosing

we obtain the desired scale factor for the delay network.For proper operation, the bandw idth BW of the FM signal r(f) must lie within

the range over which H(ju>) appears as a delay n e tw o rk ; that is,

(12.5 20)2-v/l + [(cu - <y0)/a]2

H ( p )

x

Fig. 12.5-11 Pole-zero diagram of H (p ) .

H(jU)) = Ì ^ Z g _J[(ra_“o)/a + it/2 ] (12.5-21)

kQr _ j (12.5-22)

BW (BW)t0 ^ 0 22a 2

(12.5-23)


However, since BW /2 is greater than Aw, Eq. (12.5-23) is m ore restrictive than Eq. (12.5-13); hence the m axim um size of f0 [and in turn «„(f)] is lim ited in this case by the physical im plem entation of the delay network.

At this point we are in a position to determ ine the param eters of the tuned transform er once w 0, Aw, and wm are specified. Consider, for example, w 0 = 27r( 10.7 MHz), A w = 27i(75kHz), and wm = 27t(15kHz). F o r this case p — 5 and, from Fig. 11.1-4, BW = 3.2 A w , thus from Eq. (12.5-23) we see tha t the m axim um value of t0 is

1 0.4 0.4BW " 3.2 Aio

tn — = 266 nsec,

or a =/3.76 M rad/sec. Since Q T = w j l c . = 9, the coefficient o f coupling m ust be adjusted to equal 2/QT = 0.222.

Now if/? = 1 0 k ii, then

C = —- — = 13.3 pF 2a R F

and L = - T - -1

=17.6 (iVl-wlC{ 1 - k 2,

In addition, if A = 5 V, from Eq. (12.5-18) we obtain

r0(r) = (0.886 V )/(r - 133 nsec) ^ (0.886 V ) /( ,).

In writing the expression for v0(t) we implicitly assum ed that the diodes in the envelope detector were ideal. Even if they are not, so long as their characteristics are m atched, the F0-term they introduce cancels in the output.

A final version for the circuit o f Fig. 12.5-9 is shown in Fig. 12.5-11. In this circuit, v(t) is developed from ji current source drive by tuning the input of the transformer to cj0 ; tha t is, 1 / yJ L C l = oj0 . In addition, the “ floating” voltage source u(i) is developed across the R F choke by placing a coupling capacitor from the top o f the

12.5 TIME-DELAY DIFFERENTIATOR AND DEMODULATOR 6 1 3

input-tuned circuit to one end o f the R F choke. The o ther end is grounded to high frequencies th rough C 0.

If the prim ary tuned circuit for the circuit of Fig. 12.5-10 has approxim ately the same QT and thus the same bandw idth as the secondary tuned circuit, the bandw idth of i,(t) is concentrated over the range of frequencies where the input im pedance ofthe tuned transform er appears resistive. Hence

A =

where R in is the input resistance of the tuned transform er evaluated at co0 and A is the am plitude of v(t). Consequently, once R in is known, v(t) can be determ ined directly from i,{t) and the previous analysis may be employed to determ ine v0(t). The value for R in is readily shown to be given by

*.n = * l l £ t

Thus if R = 10 kO, R , = 2.5 kQ, and

‘M = (4 mA) cos â)0f + Aoj J f (0 ) dO ,

then

v(t) = (5 V )cos a)0t + Aa) J f ( 0 )d 0 .

The Ratio Detector

If one of the diodes of the Foster-Seeley dem odulator is reversed and a large capacito! is placed in the circuit as shown in Fig. 12.5-13, the ra tio detector results. J The ratio detector performs the functions of both a dynam ic limiter and a frequency dem odulator and is thus used in m any entertainm ent-type FM receivers. The lim iting operation is accom plished a t the expense of balancing, and thus the signal com ponent ^ f

t T o show that R in = R , || (K/4) let (, = / , cos a»0i with the result that v ( t ) — A c o c ù J . II' envelope detector loading is neglected, then 2ub(i) -- 2 A cos (a»0i - n /2 ) . Now the input power to the tuned transform er is given by P in = A 2/ 2 R in . This power is dissipated in R t and R ; thus P in = P K1 + P R . However, P R = 4 A 2/ 2 R and P RJ = A 2/ 2 K ; ; therefore,

If envelope detector loading is no t neglected, then

R.„ = R ,« * ll^ -

i See S. W. Seeley and J. Avins, “The R atio D etector,” R C A R e v i e w , 8, p. 289 (1947). See also B. D. Loughlin, “The Theory o f A m plitude M odulation Rejection in the Ratio D etector,” Proc I R E , 40, pp 289 296 (M arch 1952).


i-Ji) has half the am plitude of the ouput of a corresponding Foster-Seeley dem odulator in addition to a dc com ponent.

To get some idea of the operation of the ra tio detector, consider the circuit of Fig. 12.5-13 with C 2 removed. For this case, if we again assum e that the loading of the envelope detectors can be modeled by placing R 0 in parallel with R, that operation is such that the tuned transform er transfer function is given by Eq. (12.5-21), and thatkQr/2 = 1, then the upper and lower branches of the ratio detector function assingle-ended time-deJay dem odulators, with the result tha t (cf. Eq. 12.5-16)

v'0(t) = A j 2[1 - (Aojt0/ 2 ) f ( t - to/2)],

v0(t) = A y /2 [ l + (A(ot0/ 2 ) f (t - f0/2)], (12.5-24)

and

v2(t) = V0(t) + v0(t) = 2A S[2.

Di

Fig. 12.5-13 Ratio detector.

N ow if C 2 is returned to the circuit, it charges up to a po in t where the voltage across it has the value

c2(f) = lA y J l , (12.5-25)

at which level no additional current flows through C 2 and the circuit functions as if C 2 were not there; that is, v0(t) is still given by Eq. (12.5-24). However, in a fashion similar to that discussed in Section 12.1, C 2 placed across the secondary of the tuned transform er in series with diodes D x and D2 performs the function of a dynam ic limiter and keeps the am plitude of v(t) and vb(t) constant at A even if /¡(t) contains a small am ount of am plitude m odulation. However, the m odulation index (m ) cannot be too large; otherwise, failure-to-follow distortion will result.

12.5 TIME-DELAY DIFFERENTIATOR AND DEMODULATOR 6 15

In the case of the ratio detector it should be noted that the am plitude A of v(t) is determ ined by the average am plitude of i;(f) flowing into the tuned transform er with C2 removed. Thus, once the average am plitude of /¡(f) and the param eters of the tuned circuit are known, A and in turn v0(t) can be calculated in a straightforw ard fash ion :

A =

In many practical ra tio detector circuits, small resistors (small in com parison with R 0) are added in series with the diodes. These resistors serve two purposes. First, they provide a way of correcting for slight imbalances in the diodes and the center tapped transfo rm er; second, they som ewhat reduce the tendency of the circuit tow ard diagonal clipping.

Time-Delay Demodulator with a Synchronous DetectorThe time-delay differentiator may be followed by a synchronous detector to yield an FM dem odulator. Such an arrangem ent is shown in Fig. 12.5-14. F o r this dem odulator the top branch is superfluous, since v(t) m ultiplied by — Vx sin a > 0 T ( f ) produces

Fig. 12 .5-14 F requency d e m o d u la to r com prising a tim e-delay d ifferen tia to r follow ed by a synchronous detector.

a term centered abou t 2co0 which is no t passed by the low-pass filter. In addition, if a phase shift o f — te/2 at w 0 is incorporated into the delay netw ork and v(t) is used as the reference signal, the ou tpu t v0{t) rem ains unchanged ; thus the simplified circuit of Fig. 12.5-15 produces the same outpu t as the circuit of Fig. 12.5-14.

It is apparen t for the circuit of Fig. 12.5-15 tha t the low-frequency com ponent at the filter input is given by

K A 2s in 2k(t),


which simplifies to

KA2 4 I t0\ — &(Dt0f \ t — y ! (12.5-26)

provided that t0 < 0.2/Aco and t0 < 2/com. If the low-pass filter extracts this low- frequency term (which is possible only if D and determ ine a point below the curve of Fig. 12.2-6), then

v„(t) = - K mA Acot0/ | i - | | , (12.5-27)

where K M = KA/2. E quation (12.5-27) could have been obtained by letting 0o = 0 in Eq. (12.5-12) and placing the resultant signal through a synchronous detector.

R eferen ce: A cos [co0r(i)] = — A sin I <o0t(r) — ^

Fig. 12 .5-15 Sim plified tim e-delay frequency d em o d u la to r.

To arrive a t Eq. (12.5-27) we require t 0 < 0.2/Aca instead of t0 < 0.4/Aw as given by Eq. (12.5-13). This is a direct result of em ploying - Vx sin uj0n t) as the reference signal for the synchronous detector in place of a m ore complex signal p roportional to the high-frequency term of Eq. (12.5-3). Fortunately, the sim plification of the reference signal has only this small effect.

N ote that v„(t) given by Eq. (12.5-27) contains no dc com ponent dependent on A. Thus the frequency dem odulator of Fig. 12.5-15 has the same insensitivity to am plitude variations as any of the balanced frequency dem odulators previously discussed. In addition, it is easier to im plem ent physically.

Figure 12.5-16 illustrates one possible physical im plem entation of the block diagram of Fig. 12.5-15. In this circuit, as in the circuit o f Fig. 12.5-9, the transform er with a tuned secondary is employed to achieve a time delay of t0 = 1/a as well as a phase shift of —n / l a t the carrier frequency. Again the bandw idth of the input FM signal

\it) = Vl cos co0i -t- Aw i f(0)d0


must be less than 0.4a (cf. Eq. 12.5-23) and

10 = - <0.2

a A toThe differential pair is used as a m ultiplier, with the result that the low-frequency com ponent of the collector current of Q2 is given by

■ cCjkQ-j'Vxa x(x) 2R Ea

12a

Ad)/1 t - r1-) + CCjIk0, (12.5-28)

f ( 6 ) d ( )


where a T is the transistor alpha, aj(x) is the function of x given by Fig. 4.6-4, x = qV jkT , and I k0 is the quiescent em itter current o f Q3 with v(t) = = 0. If theR 0C0 low-pass filter extracts the low-frequency com ponent of the collector current of Q2 while elim inating the com ponents centered about co0 and its harm onics, then v„(t) is given by

Acu/(i - 1/2«)v o ( 0 — K :c ~~ a r ^ o ^ * o + 2 R m

(12.5-29)

O f course, Eq. (12.5-29) is valid only if the bandw idth of the input FM signal is less than 0.4a and t0 = 1/a < 0.2/Ato. The stricter of the two conditions determ ines the maxim um value of t0 and, in turn, the largest am plitude for v„(t).

B ilottif has reported an integrated 10.7 M H z differential-pair detector based on this approach. He also obtained an experim ental ou tpu t characteristic for his circuit.

12.6 PULSE-COUNT FREQUENCY DEMODULATOR

A pulse-count frequency dem odulator is a time-delay frequency dem odulator for which the time-delay netw ork is a physical delay line, the am plitude dem odulator is an average envelope detector, and the input signal is a hard-lim ited FM square wave vsq(t) as described by Eq. (12.2-lb). The basic block diagram of a pulse-count dem odulator is shown in Fig. 12.6-1. Figure 12.6-2 illustrates the waveforms of the signals appearing at the various points of the block diagram of Fig. 12.6-1. F rom Fig.12.6-2 we observe that a pulse of dura tion t0 is generated in v j t ) a t each positive zero crossing of vsq(t) and then averaged (or counted) by the low-pass filter to obtain V0 (0-

Intuitively the operation of ,the pulse-count dem odulator m akes good sense. W hen the instantaneous frequency is high, the pulses of va(t) are closely spaced and the average value (the low-frequency com ponent) of the pulses a t the low-pass filter ou tpu t is high. Similarly, as the instantaneous frequency decreases, the pulses of v jt) become m ore widely spaced and the low-pass filter ou tpu t decreases. Thus the low-pass filter provides an ou tpu t which is proportional to instantaneous frequency.

t A. Bilotti, “FM Detection Using a Product Detector,” Proc. IEEE, 56, pp. 755-757 (April 1968).

12.6 PULSE-COUNT FREQUENCY DEMODULATOR 619

»«,(0 / *«,(/-*o)~l

1T- 1“ n

I1 1

1 11 | 1 1 1 1 '1 I 1 J 1 j

[«-*0

i n “ nU- t u '

— 'o **1(0 — — r t0 'o(t)

i [average

______n ___________

i

Fig. 12.6-2 Waveforms for the block diagram of Fig. 12.6-1.

F or slowly varying co,(i) or f( t ) , the constant o f proportionality between va(t) and a)i(t) can readily be determ ined by letting <y,(i) = o> (a constant). F o r this case, va(t) is a periodic train of pulses with an am plitude 2B, with a duration f0 , with a period 2n/w, and thus with an average value B t0co/n. Therefore,

v„(t) = ^ H l( 0), (12.6-1)n

where H L(0) is the dc transfer function of the low-pass filter. A plot of v„(t) vs. co is shown in Fig. 12.6-3. Hqwever, the voltage v0(t) is proportional to co only for T > 2t0. As T is decreased beyond this level, the pulses of vb(t) and va(t) begin to decrease in duration with decreasing T, with the result that v0{t) decreases with increasing co (for co > n /t0) as show n in Fig. 12.6-3. [The reader should convince himself of this


fact by subtracting v ^ t — t0) from v ^ t ) for the case where T/2 < i0.] Because of the discontinuity in the v0(t)-vs.-(ot{t) curve, t0 must be chosen sufficiently small so that

f 2* " lW m a x = <W0 + A(U, ( 1 2 .6 - 2 )r0

or equivalently,

to ^ —I a - (12.6-3)a)0 + A tofor any given value of to0 + A to.

The transfer characteristic of Fig. 12.6-3 was obtained on the assumption that / (f) was a constant. We must now determine how rapidly f ( t ) may vary relative to l/r0 without altering the static transfer characteristic (or equivalently, how small t0 must be relative to a given com in order not to alter the static transfer characteristic). To accomplish this we observe that va(t) and thus v0(t) for the pulse-count demodulator and the circuit shown in Fig. 12.6-4 are identical, provided that the linear pulse- forming filter in Fig. 12.6-4 has an impulse response h(t) given by

! 0, t < 0,

1, 0 < t < t0 , (12.6-4)

0, t > t0 ,

or equivalently a transfer function H(jco) = ¿F[h(t)] of the form

H(jco) = toe- ^ sm 2\ (12.6-5)« » o /2

Exact Half-wave Linear pulse-differentiator rectifier forming filter

H (ju ))~ ~ h (t)

‘o IFig. 12.6-4 Block diagram of a circuit whose output is identical with that of the circuit of Fig. 12.6-1.

The equivalence is apparent when we note that the exact differentiator produces an alternating train of positive and negative impulses of strength 2B at the respective positive and negative zero crossings of usq(i). The positive impulses, which get past the half-wave rectifier of Fig. 12.6-4, are converted into rectangular pulses of height 2B and duration t0 by the linear pulse-forming filter.

12.6 PULSE-COUNT FREQUENCY DEMODULATOR 621

The pulse-forming filter has a m agnitude and phase response identical with that shown in Fig. 12.5—4 except that |//(0)| equals t0 instead of Aa>t0/2. However, this filter is in cascade with the ou tp u t low-pass filter which cuts off im m ediately beyond wm [the m axim um frequency com ponent of / (t)]; hence, if

so that \H(ju>)\ rem ains essentially constant at t0 over the passband of the output low-pass filter, the linear pulse-form ing filter may be replaced by an attenuato r of value t0 w ithout affecting v„(t). This substitu tion reduces the circuit of Fig. 12.6—4 to an exact differentiator followed by an average detector and an attenuato r of i0 ; hence with the aid of Eq. (12.2-17) (K D = 1) we can write v„(t) in the form

provided that the ou tpu t filter passes v„(t) w ithout distortion.Clearly, then, if t0 is chosen sufficiently small so that both Eqs. (12.6-6) and

(12.6-3) are satisfied, the constant of proportionality relating v0(t) and co,(t) is BtoH L(0)/n both dynam ically and statically. Thus, once a>0, Aa>, and <um are specified for the input FM signal and /? and D are found to be below the curve o f Fig. 12.2-7 [so that the low-frequency com ponent of v0(t) can be extracted by the ou tpu t low-pass filter], then t0 m ay be chosen sufficiently small so tha t v j t ) is linearly related to «,(()■ For the usual case where cum « co0, Eq. (12.6-3) is m ore restrictive than Eq. (12.6-6) and thus we need only concern ourselves with choosing t0 sufficiently small so that the static dem odulator transfer characteristic rem ains linear. O f course, t0 should be chosen no smaller than necessary, since the am plitude of v0(t) is directly p ro portional to f0.

A practical system which performs the same operation as the time-delay differen tia tor followed by the average envelope detector is the pulse-count frequency dem odulator shown in Fig. 12.6-5. F o r this dem odulator the m onostable m ultivibrator is triggered on the positively sloping edge of vsq(t) and produces a pulse of am plitude V2 and duration f0. The low-pass filter then extracts the low-frequency com ponent of the pulse train.

The static transfer characteristic o f v0(t) vs. <y,(i) (which is valid provided that t0 < 2/o»m) is shown in Fig. 12.6-6. F o r this case no reduction in pulse du ra tion occurs until T = Info) decreases to the po in t where the m onostable m ultiv ibrator cannot recover between the end o f the pulse o f du ra tion f0 and the beginning o f the next pulse. If the tim e for com plete recovery of the m onostable m ultiv ibrator is tR, then the m inim um perm issible size of T is T = i0 + tR. Consequently, in o rder to obtain

2 ( 12.6-6)

v„(t) = — [«o + A(U/ ( f)] * M O (12.6-7)

or


H L(0)I H l ( j o j )\

Vsq(<) Monostable v ,(i) Low-passfilter

Hdj(o)

>'„(multivibrator

Pulse length: i„Pulse am plitude: V2

Fig. 12.6-5 Pulse-count demodulator.

Fig. 12.6-6 Plot of the transfer characteristic for the pulse-count frequency dem odulator shown in Fig. 12.6-5.

the desired linear relationship between v0(t) and (o,{t),

K2toi i L(0)v„(t) 2n

t0 m ust be chojen sufficiently small so that

2nto + tR <

Ui(t),

2n a)0 + Aa>

and

( 12.6- 8)

(12.6-9)

in < ( 12.6- 10)

The validity of Eq. (12.6-8) depends, o f course, also on the ability of the low-pass ou tput filter to extract the low-frequency com ponent of the pulse train in an undistorted fashion.

Com m ercial discrim inators of this type are available as separate units. Center frequencies from 1 Hz to 10 M H z are available. Above 10 M H z the pulse duration . m ust be less than lOOnsec and the recovery time must be of the order of lO nsec; hence practical circuit considerations impose limitations. W hen operated within the previously described lim itation, such circuits are capable of achieving significantly better than 0.1 % linearities with frequency deviations approaching the carrier frequency.

12.7 MORE EXOTIC FM DETECTORS 6 2 3

Com m ercial divide-by-two and divide-by-ten networks are now available which operate with input frequencies as high as 100 MHz. With such counting networks the useful frequency range of the pulse-count discrim inator can be extended considerably in frequency.

F or cases where D liés reasonably close to the lower curve of Fig. 12.2-7 the m odulation cannot easily be separated from the carrier ripple w ithout a reasonably complex low-pass filter. To remedy this situation somewhat, the m onostable m ultiv ibrator of Fig. 12.6-5 may be triggered on every zero crossing of vsq(t) rather than only on positive zero crossings of vsq(t). W ith such triggering the closest carrier frequency com ponents to the desired m odulation at the input to the low-pass filter result from the FM spectrum centered at 2œ0 ra ther than at io0 .

Therefore, the upper curve of Fig. 12.2-7 provides a bound for D. If the m onostable m ultiv ibrator is triggered on every zero crossing, the ou tput voltage of the pulse-count dem odulator is given by

, 12.6- 11)71

from which we observe that the ou tpu t level has apparently been increased by a factor of 2. This increase cannot norm ally be realized, since for this case

t0 + tR < ------— -—, (12.6- 12)co0 + Au>

which constrains the maxim um permissible value of t0 to be less than one-half the previous value.

12.7 M O R E EX O TIC FM D ET E C T O R S—T H E PH A SE-LO CK ED L O O P , T H E FREQUENCY-LOCKED LOOP, AND THE FREQUENCY DEMODULATOR WITH FEEDBACK

As the final section of this chapter and of the book we present the block diagram s for and brief resumes of the operation of three “ threshold-extending” FM dem odulators. All of these circuits are m ore complex than the dem odulators discussed previously ; however, since all the individual elements within each block diagram have been considered in this book, the interested reader should have an adequate basis for understanding their operation.

because this section makes no pretense of offering a complete explanation of these useful but not yet widely used circuits, we have included a fairly substantial list of references a t the end of the chapter which will supply further inform ation. Even m ore recent inform ation will probably be available from an inspection of recent indexes to papers in the IE E E Transactions on Communication Technology or Aerospace and Electronic Systems.

Figures 12.7-1, 12.7-2, and 12.7-3 show all three circuits in block diagram form. All three circuits contain feedback loops, all contain multipliers within the loop, and all three take their outputs after a low-pass filter that is within the loop. If preceded by a limiter, the a(t) am plitude term s shown would become constan ts;


Low-pass loop filter

Fig. 12.7 1 Block diagram of frequency-locked loop.

and though the circuits would detect FM signals uncorrupted by noise, they would lose their ability to remove or reduce the effects of noise-induced “spikes” and hence their ability to extend the FM noise threshold. It is just this ability to extend the threshold that justifies the added complexity of these circuits in various weak-signal FM situations— satellite links, for example.

The frequency-locked loop com bines an envelope detector and a feedback loop as a circuit to process the ou tpu t from a conventional lim iter-discrim inator circuit. So long as the input signal is large, the envelope detector ou tput keens the loop gain high and hence the loop bandw idth wide and the discrim inator ou tput reaches

a(l) cos [ a v + '/'(')]

Input ° “

Multiplier a(i)B

& sin [i/z — <t>] +second-harmonicterms

B sin [iu0t +ho(0

veow0

Low-pass loop niter

♦ s“1 M Ö —$(*)]} *Ao(0 Voltage-controlledHD

? Output

oscillator

Fig. 12.7-2 Block diagram of phase-locked loop.

12.7 MORE EXOTIC FM DETECTORS 625

Fig. 12.7-3 Block diagram of frequency dem odulator with feedback.

the circuit ou tpu t unchanged. W hen the input am plitude drops briefly to a small am plitude (it can be shown that sudden drops to a small am plitude are often caused by the same noise mechanism that introduces “spikes” or impulselike disturbances in the FM signal), the loop gain drops and the circuit output is uncoupled from the discrim inator ou tput until the disturbance passes. Some of the references listed at the end of the chapter contain experim ental results as well as much more complete discussions of the circuit operation.

As was pointed out previously, one approach to the phase-locked loop is to consider the voltage-controlled oscillator as an FM generator in the return path of a feedback loop. If a(t) is large, then the loop gain is high and the forward path must try to perform the inverse of FM generation or it must approxim ate an FM detector. When a(t) falls again during a noise d isturbance— the loop tends to open and the output tends to be unaffected by any noise spikes present in the input FM signal. As pointed out below, the phase-locked loop suffers from problem s of initial locking on a signal, from VCO drift, and, if poorly designed, from unwanted “ loss-of-lock” conditions.

The frequency dem odulator with feedback includes both a voltage-controlled oscillator and a conventional discrim inator within a feedback loop. Also included within the same loop are a low-pass filter and a single-pole IF filter. H essf has shown that as the IF filter bandw idth goes to zero the equations for this circuit reduce to those for the phase-locked loop, whereas when the IF filter becomes wide and the low-pass filter becomes narrow the equations approach those for the frequency- locked loop.

Again an intuitive argum ent for the circuit operation is that, when the am plitude is high, the loop gain is high and the feedback widens the circuit bandw idth and

t D. T. Hess. "E qu ivalence of F M T hresho ld E xtension R eceivers," I E E E Trans, o n C o m m u n i c a t io n T e ch n o logy . Com-16. No. 5, pp. 946 748 (Oct. 1968).


presents the discrim inator ou tpu t unchanged. W hen a(t) falls, then the loop gain and circuit bandw idth fall rapidly and the ou tpu t is uncoupled from the unw anted d istortions in the input sine wave.

Phase-Locked Loop O peration

To see m athem atically how the simple phase-locked loop initially becomes “ locked” to the incom ing signal, consider the simplified case where an unm odulated carrier a t co0 + £ is tuned slowly tow ard the free-running VCO frequency co0. Then in Fig. 12 .7 -21l/(t) is et and a t t = 0 one m ay assum e the VCO o u tp u t to be cosinusoidal. O ne also assumes that the low-pass loop filter rejects additive frequency term s and passes difference term s with an attenuation independent of frequency. U nder these conditions,

<p(t) = cOj, cos [et - $(i)]> (12.7-1)

where cox contains a(i) (assumed constant for the moment), B, the filter constant, the j term from the m ultiplication, and the m ultiplier c o n s ta n t; <oa is know n as the “loop bandw idth” or the loop “hold-in range.”

N ow Eq. (12.7-1) is a first-order nonlinear differential equation which can besolved, on the assum ption tha t <f> = 0 a t t — 0, to yield

fat) = ---------- , - e, M < a»., (12.7-2)e — cox cosh y/coZ — £ t

or£2 — ft)2

</>«) = --------------------7 T = t - £’ l£l > <117- 3>£ — (O,, COS y /E — ( « ; t

These tw o equations are p lo tted in Figs. 12.7-4 and 12.7-5 for positive values ofc/ctv

W hen a>x > |e|, then as time passes <j> approaches e and the loop is “ locked.” If e approaches zero, then also approaches zero. F rom Fig. 12.7-4 it is apparen t tha t it takes abou t five “ tim e constan ts” equal to 1 ¡ J w \ — e2 for this locking to occur.

N ow , since in the steady state (j>(t) = e, if e is varied slowly in com parison with s/col — e2, then the ou tp u t o f the phase-locked loop produces an o u tp u t directly proportional to the frequency difference between the input and the V CO frequency. Consequently, if

m = a co j ‘m d d ,

if the m axim um frequency com ponent of / ( f ) is com, and if

com « y / c o \ - Aw2, (12.7-4)

then the ou tp u t o f the phase-locked loop is directly p roportional to f ( t ) and frequency dem odulation is accomplished.

12 .7 MORE EXOTIC FM DETECTORS 6 2 7

Fig. 12.7-4 Plot of (t)/wx vs. tJ w2 — e2t for |s/wj < 1. The “lock-in” range for a first-order phase-locked loop.


The inequality o f Eq. (12.7-4) implies th a t m, > Aco and th a t co* » (om. In addition, it can be shown that, in order to avoid d istortion in the detected signal, D and /? m ust fall below the curve of Fig. 12.2-6; hence the phase-locked loop detector suffers from the same type of lim itations concerning carrier frequency co0, m odulation frequency u)m, and frequency deviation Aw as do the o ther dem odulators discussed in this chapter.

W hen |fi| > w a, then </>(r) goes on oscillating with time at a beat frequency dependent on ^ J s2 — w \. In this case locking never occurs unless one either decreases e or increases the “ loop bandw idth” a>x.

Real phase-locked loops have m ore complex filters and have “pull-in” and “hold-in” ranges that are different. (The “hold-in” range exceeds the “pull-in” or “ lock-in” range.) However, in all cases these ranges are a function of the input signal am plitude. Thus, if the am plitude suddenly dips, the circuit may “ lose lock” ; if this happens, the ou tpu t will be unaffected by the input frequency disturbances that accom pany such suddenly dipping am plitudes.

REFERENCES

L. C alandrino and G . Immovilli, “Coincidence o f Pulses in A m plitude and Frequency D eviations,” A l t a F r e q u e n z a , 36, English Issue N o. 3 (Aug. 1967).

F. Cassara, “ FM D em odulator w ith Feedback,” M.S. Project R eport, Polytechnic Institute o f Brooklyn (June 1968). ■

L. H. Enloe, “ Decreasing the Threshold in FM by Frequency Feedback,” P r o c . I R E . 50, pp. 18-30 (Jan. 1962).

P. Frutiger, “ Noise in FM Receivers with Negative Frequency Feedback,” P r o c . I E E E ,

54, pp. 1506-1520 (Nov. 1966).D . T . Hess, “Cycle Slipping in First O rder Phase Locked Loops,” I E E E T r a n s , o n C o m m u n i

c a t io n T e c h n o l o g y , Com-16, N o. 2, pp. 255-260 (April 1968).

D. T . Hess, “ Equivalence o f FM Threshold Extension Receivers,” I E E E T r a n s , o n C o m m u n i

c a t io n T e c h n o l o g y , Com-16, N o. 5, pp. 746-748 (Oct. 1968).

D . T. Hess and K. K . C larke, “O ptim ization o f the Frequency Locked Loop,” P r o c . N a t i o n a l

E l e c t r o n i c s C o n f e r e n c e , pp. 528-533 (Dec. 1969).

S. O. Rice, “ Noise in FM Receivers,” in T i m e S e r i e s A n a l y s i s , M. R osenblatt, Ed., Wiley, New Y ork (1963), C hapter 25.

M. Schwartz, W. R . Bennett, and S. Stein, C o m m u n i c a t i o n S y s t e m s a n d T e c h n i q u e s , M cGraw-Hill, New Y ork (1966), C hapter 3.

M . U nkauf, “Q uantized Frequency Locked L oop FM D em odulator,” Ph.D . D issertation, Polytechnic Institute o f Brooklyn (June 1969).

PROBLEMS

12.1 F or the circuit shown in Fig. 12.P-1, plot the ou tpu t voltage am plitude V2 vs. the input voltage am plitude V , f o r 0 < F, < 50 mV. At what value of V t is V2 within 95 % of its

PROBLEMS 6 2 9

a sy m p to tic lim itin g v a lu e? I f

t , = ( 1.6 m V)( 1 + 0 .5 «'os o )mt ) cos co0(, u>m « <d„ -

find the m odulation index of vjt).

M 1— = 0 . 0 5 , - = =

y/L^Cat0 , m0R C = 20, Vcc = 10 V, R = !0 k ii

Figure 12.P 1

! >r the circuit show n in Fig. 12.P-2, p lo t the transfer characteris tic o f V { vs. / , for 0 < / , < > mA. A ssum e th a t the loaded Q rem ains sufficiently high to justify the a ssu m p tio n o f a lin u so id a ’ o u tp u t voltage.

V j cos U)0l

Figure 12.P-2

630 FM DEMODULATORS

12.3 F or the circuit shown in Fig. 12.P-3, show that if the loaded Q is high, the am plitude of the output voltage V , is related to the am plitude of the input current by the expression

f 1 = 2/,(x),‘s

w here / ¡ ( x ) is the modified Bessel function o f order one and x = q V J k T . Using the values for / ¡ ( x ) and the asym ptotic expression for I ¡ ( x ) given in the Appendix at the back o f the

• book, plot V t vs. / , for 0 < / , < 5 mA. Does this circuit function as a limiter? For what range of / ,? W hat value of C ensures Q r = 20 if I x — 5 mA and o>„ = 10s rad/sec?

V \ cos (jJ0 t

Figure 12.P3

12.4 A high-Q single-tuned circuit follows an ideal nonlinear limiter (Fig. 12.1-1) to extract the fundamental com ponent o f the FM signal. Assuming that the filter functions in a quasistatic fashion (Eq. 11.3-2), determ ine the m inim um filter bandw idth (in terms of Aoj) which ensures no m ore than 1 % am plitude m odulation on the ou tpu t signal (m = 0.01). W ith P = 5, determ ine the maximum deviation ratio which is possible if the tuned circuit must suppress the third harm onic com ponent of the limited FM signal to less than 1 % of the fundam ental while keeping the induced AM below 1 %.

12.5 Repeat Problem 12.4 for the case where the bandpass filter is a maximally flat doubletuned circuit.

12.6 F or the bandpass limiter shown in Fig. 12.P-4, r ,( i) = V , cos co0 t , which results in r j r ) =

V2 cos co0t. Using the sine-wave tip analysis in Sections 4.2 and 4.3, plot the limiter characteristic of V2 vs. V , . If

Vi(0 — (5 V)[l + a i / ( i ) ] c o s a i0r

and the m odulation is passed by the bandpass filter, determine the AM m odulation index of

12.7 F or the dynam ic limiter shown in Fig. 12.1-5, L = 10/iH, C = 1000 pF, ft = R g = 10kf2, C 0 — 10 f j F, and

i,(r) = (1 mA)(l + 0.5 cos 103f)cos | l 0 7t + 104J cos lO30di)J.

Find i’(i)- W hat is the AM m odulation index of y(f)? W hat is the output AM m odulation index if C „ is reduced to 1 //F? Does failure-to-follow distortion occur?

PROBLEMS 631

v0 ' v%2Bandpass

filter tuned to o,

0 > 2 O

Figure 12.P 4

12.8 For the circuit shown in Fig. 12.P-5,r,(r) = (0.5 V)(l + 0.5 cos 2.5 x 10s) cos 107f.

Find (■„(()• W hat is its AM m odulation index? If

i,(r) = (0.5 V) cos 1107f + A w> J cos 1O40 dO

6 3 2 FM DEMODUL ATORS

12.9

12.10

is applied to the circuit of Fig. 12. P 5, find the maximum value of Aw which ensures less than 1 am plitude m odulation (m = 0.01,’ for r0(/).

The Clarke-Hess dem odulator shown in Fig. 12.3 A is to be used to dem odulate an FM triangular wave. Show that in order to balance the dem odulator for this case one must choose R C = jr/4w0. Show also that when the dem odulator is balanced, the output is given by Eq. (12.3-7), where A is now the peak amplitude of the FM triangular wave.

For the Clarke-Hess dem odulator shown in Fig. 12.3-4, R = lOOOfi. C' = lOOpF, R 0 =15.7 kQ, C0 = 6300 pF, r cc = 10 V and

= (10 V) cos 1107/ + 106 f cos 10*0d0

12.11Find i j t ) - How much carrier ripple appears at the output?

A 10 V peak-to-peak FM square wave with instantaneous frequency co,(r) is applied toth tc ircu it o f Fig. 12.3-4 with C== lOOpF, R0 = 15.7 kil. C0 = 6300 pF, and K v = 1«V.a) Find the value of R V'hi.-h balances the dem odulator at ro0 = 107 rad/'sec.b) If r l = 100 i i and w,(i) = (o c = 107 rad/sec. make an accurate sketch of the current

flowing into the ou tpu t R 0 - C 0 filter. Determine the output carrie1 ripple.c) Plot the [„-vs.-a»//) characteristic as w,(t) is varied between 0 and ¿ w 0 .

d) If ajj(t) = (107 + 5 x 104 cos 10 'il rad/sec, determine ti0(i).

12.12 For the circuit shown in Fig. 12.P-6, i’¡(t) is given by

i/U ) = (2 V)cos

/ Crystal

u)nt + A oj

PROBLEMS 633

where w0 = (107 + 176) rad/sec. The crystal param eters are given by C0 = 4 pF, C = 0.04 pF, L = 250 m H, and r = 125 i l (cf. Section 6.7). Plot the static dem odulator characteristic v„ vs. Aco f ( t ) for — 500 rad/sec < Aoj/(r) < 500 rad/sec. ( H i n t : Since operation is in the vicinity of the crystal’s series resonance, the effect of C0 can be neglected.) W hat is the m aximum permissible value of Aco which ensures that the nonlinearity of the transfer characteristic does not exceed 1 %? W ith this value o f Aa>, with p — 5, and with /( f ) = cos w mt , find v 0(t) . If R increases by 10%, w hat is the change in the center frequency of the dem odulator? Would this dem odulator be useful in the system shown in Fig. 11.9-1? Why?

12.13 Verify the condition given in Eq. (12.4-14) and the expansion of Eq. (12.4-15).

12.14 For the balanced dem odulator shown in Fig. 12.4-3,R , = R 2 = 5 0 k C ian d R o = 100 kQ. Choose optim um values for the rem aining unspecified param eters if the circuit is to demodulate an input FM signal limited to 10 mA peak-to-peak having a carrier frequency co0 = 108 rad/sec, a frequency deviation of Atu = 106 rad/sec, and a maximum m odulation frequency of 105 rad/'sec. Find v„(t) for this choice of parameters.

12.15 A ruby laser beam is m odulated in frequency with a deviation of A/ = 1 G H z and a m aximum m odulation frequency of f m = 6 MHz. Draw a diagram of an optim um frequency dem odulator for such a beam. The dem odulator should employ only beam splitters (prisms), polished mirrors, and a photodiode (or phototransistor). Indicate all effective path lengths, assuming that the carrier beam propagates at the speed of light through the dem odulator medium.

If the received beam has an intensity of 30 foot-candles and if the photodiode has a linear sensitivity of 100 /iA/foot-candle and delivers its output current to a 600 fi impedance, determine the output voltage for the case where /'(f) = cos(2rt x 6 x 106f).

12.16 In the bottom circuit of Fig. 12.5-8 the delay network is a piece of helically wound delay line with a delay of20nsec/cm and a length of 10cm. If ((f) = (5 V)cosa>f and K M = I, plot the r (,-vs, - w transfer characteristic for 0 < w < 5 x 107. Over what ranges of w is this characteristic linear? D o these ranges of w agree with the results of Section 12.5?

12.17 If the envelope detector in the dem odulator shown in Fig. 12.P-7 is assumed not to load 75 ii, show that i - J t ) is given by

<„U) = i[ t’i(f) - I ' iU - 2f0)].

where t 0 = 400 nsec. If, in addition.

r,(f) = (5 V) cos | w0f + Ao) J cos oj„ 0 J o j ,

where u>0 = 3.93 x 10b rad/sec, A w = 106 rad/sec, and p = 10, find c0(f). Is the dem odulation nonlinearity less than 1 %?

12.18 A 10 V peak-to-peak square wave with frequency w is applied to the dem odulator of Fig. 12.P-7.a) Sketch v a( t) for w < n / 2 t 0 and for a> > n / 2 t 0 .

b) Plot the („-vs.-oj characteristic for 0 < w < n / t 0 . C om pare this characteristic with that shown in Fig. 12.6-3.

12.19 A 2 V peak-to-peak FM square wave i ^ r ) triggers the m onostable m ultivibrator shown in Fig. 12.P-8.a) If the instantaneous frequency u),(f) = w (a constant), plot the v a- v s . - w characteristic

for 0 < co < 106 rad/sec. At what point does the characteristic cease to be linear?

6 3 4 FM DEMODULATORS

75 i lV W -

10cm -

Delay line

Delay: 20nsec/cm

Characteristic impedance: 75 i l

z aOutputshort

*.(»>Envelopedetector v„(0

*« = !

Figure 12.P-7

Figure 12.P-8

PROBLEMS 635

b) If a>,(r) = (105 + 104 cos 103t) rad/sec, determine an expression for v„(f). W hat is the maximum carrier ripple which appears at v j t ) ?

12 20 Repeat Problem 12.19 for the case where the trigger circuit at the base of Q 2 in Fig. 12.P- 8 is replaced by the trigger circuit shown in Fig. 12.P-9. W hat is the advantage of this more complicated circuit?

Figure 12.P-9

A P P E N D I X

MODIFIED BESSEL FUNCTIONS

The modified Bessel function of order n and argum ent x may be defined as the integral function given by

A functional relationship between /„(x) and x may be achieved by integrating Eq. (A—1) num erically (on a com puter or by hand) for various values of x. Table A -l presents values of I„(x) for n = 0, 1, 2, 3 and values of x between 0 and 10. Table A -2 presents values of I„(x) norm alized to ex for n = 0, 1, 2, 3,4, 5 and values of x between 0 and 20; In addition, Table A -3 presents values of 2/„(x)//0(x) for the same values of n and x. It is quite clear th a t /„(x) is a m onotonically increasing function of x.

F o r small values of x we may expand e*cos9 in a M acLauren expansion and keep only the first few term s in the series. W ith this approxim ation inserted in Eq. (A -l) we obtain

In a similar fashion we can show that, as x -» 0,

By keeping all the term s in the expansion for é^cosS and perform ing a term -by-term integration, we can reduce Eq. (A -l) to an equivalent M acLauren series in x for I„(x). This series expansion is found in all the literature on modified Bessel functions.

F o r large values o f x, e*cose has a very sharp dom inant peak in the vicinity of 0 = 0 and may therefore be approxim ated by

(A -l)

and

^xcasO ^ x( I ~d2/2).

636

MODIFIED BESSEL FUNCTIONS 637

Table A -l Tabulation of l„ ( x ) vs. x for n = 0, 1, 2, 3

X /,(x ) 1 2W /.,(*)

0.0 1.0000 0.00000 0.00000 0.000000.5 1.0635 0.25789 0.03191 0.002651.0 1.2661 0.56516 0.13575 0.022171.5 1.6467 0.98167 0.33783 0.08077

2.0 2.2796 1.59060 0.68895 0.212742.5 3.2898 2.51670 1.27650 0.47437

3.0 4.8808 3.9534 . 2.2452 0.959753.5 7.3782 6.2058 3.8320 1.826404.0 11.3020 9.7595 6.4222 3.337304.5 17.4810 15.3890 10.6420 5.93010

5.0 27.2400 24.3360 17.50o0 10.33100

5.5 42.695 38.588 28.663 17.7436.0 67.234 61.342 46.787 30.1506.5 106.290 97.735 76.220 50.8307.0 168.590 156.040 124.010 • 85 1757.5 268.160 249.580 201.610 142.060

8.0 427.56 399.87 327.59 236.078.5 683.16 641.62 532.19 391.179.0 1093.6 1030.9 864.49 646.699.5 1753.5 1658.4 1404.30 1067.20

10.0 2815.7 2671.0 2281.50 1758.40

with this approxim ation, Eq. (A -l) reduces to

/„(x) ^ f e~x82/2 cos nd dd as I* e~182/2 cos nt) dfl. (A—3)2 n J - n 2 7 tJ_ OD

Equation (A—3) is directly integrable and yields

Eor .V > 10, Eq. (A-4) is accurate within 1% for I0(x).

638 APPENDIX

Table A -2 Tabulation of vs. x for n = 1,2, 3 ,4 ,5

X I u( x ) / e x h i * ) / ? 1 l 3( x ) /e *

0.0 1.0 0.0 0.0 0.0 0.0 0.0-0.5 0.64503 0.15642 0.01935 0.00160 0.00010 0.000051.0 0.46576 0.20791 0.04994 0.00816 0.00101 0.000101.5 0.36743 0.21904 0.07538 0.01802 0.00329 0.000482.0 0.30851 0.21527 0.09324 0.02879 0.00687 0.001332.5 0.27005 0.20658 0.10478 0.03894 0.01133 0.002703.0 0.24300 0.19683 0.11178 0.04778 0.01622 0.004543.5 0.22280 0.18740 0.11572 0.05515 0.02117 0.006764.0 0.20700 0.17875 0.11763 0.06112 0.02594 0.009244.5 0.19420 0.17096 0.11822 0.06588 0.03038 0.01187

5.0 0.18354 0.16397 0.11795 0.06961 0.03442 0.014545.5 0.17448 0.15770 0.11714 0.07251 0.03804 0.017196.0 0.16666 0.15205 0.11597 0.07474 0.04124 0.019756.5 ,*0.15980 0.14694 0.11459 0.07642 0.04405 0.022207.0 0.15374 0.14229 0.11308 0.07767 0.04651 0.024527.5 - 0.14832 0.13804 0.11150 0.07857 0.04865 0.026688.0 0.14343 0.13414 0.10990 0.07919 0.05050 0.028698.5 0.13900 0.13055 0.10828 0.07959 0.05210 0.030569.0 0.13496 0.12722 0.10669 0.07981 0.05348 0.032279.5 0.13125 0.12414 0.10512 0.07988 0.05467 0.03384

10.0 0.12783 0.12126 0.10358 0.07983 0.05568 0.0352810.5 0.12467 0.11858 0.10208 0.07969 0.05655 0.0366011.0 0.12173 0.11606 0.10063 0.07947 0.05728 0.0378011.5 0.11899 0.11369 0.09922 0.07918 0.05790 0.0389012.0 0.11643 0.11146. 0.09785 0.07885 0.05842 0.0399012.5 0.11402 0.10936 0.09652 0.07847 0.05886 0.0408113.0 0.11176 0.10737 0.09524 0.07807 0.05921 0.0416313.5 0.10963 0.10549 0.09400 0.07764 0.05950 0.0423814.0 0.10761 0.10370 0.09280 0.07718 0.05972 0.0430614.5 0.10571 0.10200 0.09164 0.07672 0.05990 0.04367

15.0 0.10390 0.10037 0.09051 0.07624 0.06002 0.0442215.5 0.1021# 0.09883 0.08943 0.07575 0.06011 0.0447316.0 0.10054 0.09735 0.08838 0.07526 0.06015 0.0451816.5 0.09898 0.09594 0.08736 0.07476 0.06017 0.0455917.0 0.09749 0.09458 0.08637 0.07426 0.06016 0.0459517.5 0.09607 0.09328 0.08541 0.07376 0.06012 0.0462818.0 0.09470 0.09204 0.08448 0.07326 0.06006 0.0465718.5 0.09340 0.09084 0.08358 0.07277 0.05998 0.0468319.0 0.09214 0.08969 0.08270 0.07227 0.05988 0.0470615.« 009094 0.08858 0.08185 0.07179 0.05977 0.0472720.0 0.08978 0.08750 0.08103 0.07130 0.05964 0.04744


Table A -3 Tabulation of 2/„(x)//0(x) vs. x for n = 1, 2, 3,4, 5

X 2 /,(x ) //0(x) 2 1 2( x ) / 1 0( x ) 2/ 3(x)//0(x) 2 U ( x ) / I 0( x ) 2 / 5(x)//0(x)

0.0 0.0 0.0 0.0 0.0 0.00.5 0.4850 0.0600 0.0050 0.0003 0.00001.0 0.8928 0.2144 0.0350 0.0043 0.00041.5 1.1923 0.4103 0.0981 0.0179 0.00262.0 1.3955 0.6045 0.1866 0.0445 0.00862.5 1.5300 0.7760 0.2884 0.0839 0.02003.0 1.6200 0.9200 0.3933 0.1335 0.03743.5 1.6822 1.0387 0.4951 0.1900 0.06074.0 1.7270 1.1365 0.5906 0.2506 0.08934.5 1.7607 1.2175 0.6785 0.3129 0.1222

5.0 1. 768 1.2853 0.7585 0.3751 0.15845.5 1.8076 1.3427 0.8311 0.4360 0.19706 .0 1.8247 1.3918 0.8969 0.4949 0.23706.5 1.8390 1.4342 0.9564 0.5513 0.27797.0 1.8511 1.4711 1.0104 0.6050 0.31897.5 1.8615 1.5036 1.0595 0.6560 0.35988 .0 1.8705 1.5324 1.1043 0.7042 0.40018.5 1.8784 1.5580 1.1452 0.7497 0.43969.0 1.8854 1.5810 1.1827 0.7926 0.47829.5 1.8916 1.6018 1.2172 0.8330 0.5157

1 0 .0 1.8972 1.6206 1.2490 0.8712 0.552010.5 1.9022 1.6377 1.2784 0.9072 0.587211.0 1.9068 1.6533 1.3056 0.9412 0.621111.5 1.9110 1.6677 1.3309 0.9733 0.653812.0 1.9148 1.6809 1.3545 1.0036 0.685412.5 1.9183 1.6931 1.3765 1.0324 0.715713.0 1.9215 1.7044 1.3970 1.0596 0.745013.5 1.9244 1.7149 1.4163 1.0854 0.773114.0 1.9272 1.7247 1.4344 1.1099 0.800214.5 1.9298 1.7338 1.4515 1.1332 0.8262

15.0 1.9321 1.7424 1.4675 1.1554 0.851315.5 1.9344 1.7504 1.4827 1.1765 0.875416.0 1.9365 1.7579 1.4970 1.1966 0.898716.5 1.9384 1.7650 1.5105 1.2158 0.921117.0 1.9403 1.7717 1.5234 1.2341 0.942617.5 1.9420 1.7781 1.5356 1.2516 0.963418.0 1.9436 1.7840 1.5472 1.2683 0.983518.5 1.9452 1.7897 1.5582 1.2843 1.002819.0 1.9466 1.7951 1.5687 1.2997 1.021519.5 1.9480 1.8002 1.5788 1.3144 1.039520.0 1.9493 1.8051 1.5883 1.3286 1.0569

640 APPENDIX

To obtain derivatives of /„(x), we differentiate both sides of Eq. (A -l) with respect to x to obtain

dIJtx) 1 rdx

= — f cos 0 i^ cos# cos nO dO 2 n J - n

= *xco,®cos(n + 1)0 ¿0 + ~ f e cos0 cos (n — 1)0 d92j_27t J - n 2 n J . „

(A—5)

(x) + i n_!(x)

E quation (A-5) is valid for n > 1. F o r n = 0 we obtain in an identical fashion

d l 0(x)dx

(A -6)

From Eq. (A—1) it is apparen t that

00

**«“ • = I 0(X) + 2 £ I„(x) cos n9. (A-7)

If we wish to evaluate the F ourier series o f exsme, we may define (j> = 0 — n /2 and observe that

= = + 2 £ /„(c) cos ntj)n = 1

^ I H7t= I 0(x) + 2 £ I„(x) COS I «0 - y

00

= / 0(x) + 2 X l)nI 2n(x)cos2n9

+ I ( - l)n+1/ 2B- 1(x )s in (2n - 1)0.n = 1

In a sim ilar fashion we can show that

e~ xcose = 10(x) + 2 £ ( - ir /„ (x )co s nOn = 1

by defining <j> = 9 — n.

(A—8)

(A—9)


W e m a y o b ta in a re c u rs io n fo rm u la fo r th e m o d ified Bessel fu n c tio n s by d iffe ren tia tin g b o th sides o f E q . (A -7 ) w ith re sp e c t to 9 :

x;

2 X n l „ ( x ) s in n 0 = x s in 9 e x c o s e

= x s in 9 7oW + 2 Z In(x ) cos n0n= 1 _oo

= x sin 910{x) + x Z IJ.X) sin (n + 1 )9n = 1

oo- x Z /„(x)sin(n - 1)0. (A -10)

By e q u a tin g th e co effic ien ts o f th e c o r r e s p o n d in g (sin n0 )-te rm s in th é first a n d last te rm s o f E q. ( A - 10), w e o b ta in th e d e s ire d re su lt,

2 n l „ { x ) = x [ /„ _ ,( x ) - /„+ ,(x )] . ( A l l )

Additional relationships involving Bessel functions can be found in the following references:

1. F. E. Relton, A p p l i e d B e s s e l F u n c t io n s , Blackie and Son, Limited (1946), C hapter 3.

2. G. N. W atson. T h e o r y o f B e s s e l F u n c t i o n s , Cam bridge University Press, Cambridge (1922).

3. E. T. W hittaker and G. N. Watson, A C o u r s e o f M o d e r n A n a l y s i s , Cambridge University Press, Cam bridge (1927), Section 17.7.

Additional tabulations and relationships of modified Bessel functions may be found in the following references:

1. M. Abramowitz and I. A. Stegun, Eds., H a n d b o o k o f M a t h e m a t i c a l F u n c t io n s , N ational Bureau of Standards, Applied M athem atical Series No. 55, Governm ent Printing Office, W ashington D.C. (1964). Reissued by Dover, New York (1965). Section 9.6 and Tables 9.8 through 9.11 deal with modified Bessel functions. Tabulated values as high as / ,oo(100) are given.

2. E. Jahnke and F. Emde, T a b le s o f F u n c t io n s , fourth edition, Dover, New York (1945). This standard reference work tabulates / 0(x) and /,(x ) for values of x up to 9.99. It also contains other functions up to /n (6 ), I 0( x ) = J 0{ j x ) , I ^ x ) = — j J , ( j x ) , etc.

3. M. Onoe, M o d i f i e d Q u o t i e n t s o f B e s s e l F u n c t io n s , Colum bia University Press, New York (1958). This work tabulates x l 0( x ) / l i ( x ) for values of x up to 20. The reciprocal of this function is within a factor of two of 2 /,(x ) /x /0(x), which is the ratio of Gm to g m :

ANSWERS TO SELEC T^) PROBLEMS

ANSWERS TO SELECTED PROBLEMS

C H A PTER 1

1.1. P , = 4.15 mW, P 2 = 1.65 mW, P } = 0.55 mW1.6. t>„(r) * —0.28 sin 107i + 7.3 cos 2 x 107f + 0.33 sin 3 x 107f1.7. v„ % - 5 .9 V, VEBi % 815 mV1.8. v0(t) * 118 mV cos 106i

CHAPTER 2

2.1. n = 19.1 ; ,/High = 25.8 k H z ,/Low = 266 H z; V2/ Vl = 1/20.9:»/ = 0.9142.7. Bandwidth is 242 kH z 2.8. P load = 4.6 W2.10. Loaded Q is 16.2

CHAPTER 3

This may be represented physically as the series combination of a conductance G and an inductor 2L .

b) 100 Ve - 106, 3 sin 107r u ( t)

c) 1500V[1 - e ,o6,/3]co s 107ru(r)The voltages appear large since unit drives were assum ed: i.e., an impulse of one coulom b strength and a step with a one-am pere level.

3.6. Vo22 = 200 / 2/v /5 (cf. Ex. 2.4-3); I 2 = 2.67 mA

3.7. v j t ) = v/2 V cos 12.5 x 105i - cos 107i

~ P + (1/2GL)

71

3.3. a) lO1* V e~ 106" 3 cos 107t u(r)

CHAPTER 4

4.2. Fundamental is 0.39 mA; third harmonic is 0.145 mA.4.6. G m = 400 ß A / V when Vdc = 3V4.7. / 2 = 44 ß A

4 .1 1 . K ank = 9 .3 V

4.8. / , = 8.45 mA, 1 2 = 7.8 mA4.12. /> = 27 mWbattery

645

646 ANSWERS TO SELECTED PROBLEMS

C H A PTER 5

5.2. P lk = 12.5 mW 5.5 Vc * 690 mV

5.3. Vc * 540 mV5.6. C > 63 / iF ; conduction angle ^ 120°.

CH A PTER 6

6.1. a) A min = C = 1370p F ; b) |K3M | = |K<co)| « 1 V6.2. b) |K3(a))| = 0.535 V with no change in frequency.6.3. w 0 = 107 rad/sec, | Vx(a>)\ = 4 V, |/,(w)| = 200 n A

6.5. All three poles originate at 1/2R C for f i = 0 and move outw ard along radial paths separatedby 120° as n is increased. O ne path lies on the negative real axis. F o r // = 4 ^ 8 , two of thepoles cross the imaginary axis a t a>0 = ^ / i / l R C .

6-7. oj0 = 104 rad/sec, = 1.66 V. (N ote that the attenuation through each cathodefollower is 0.815.)

6.11. r = 25 0(1 - 2/A) = 24.75 ft, -0 .2 6 8 rad/sec 6 .11 2.36 mA, 1.34 mA6.13. v0( t ) = 12.5 V cos 4.5 x 106r; T H D = 0.37%

C H A PTER 7

'd iffe re n ce 450 nA, 's ig n al 905 nA, ôscillator 1.62 mA^•3- 'd iffe re n ce = 6 6 f l A , 'o scilla to r = 1 - 7 2 ttlA7.5. G c 420 mhos7.9. v0{ t) = g(f)[860] cos (« , - w 2)t

C H A PTER 8

8.1. b) 5 V at a»0 ; 1.5 V at (w0 + w j and a t (w0 - c o j8.2. b) 125 m W ; 22.5 mW

6.9. A min = 9 , w 0 = 1 / y f i R C 6.10. A min = +93 , S F = 20

8.4. a) ¿(f) = (5 V) sin cu,f + (10 V) sin w 2t

8.8. 0.4958.9. v j t ) = (750 m V )[l + 0.5/(r)] cos 107f

C H A PTER 9

9.2. 48°C with no ac, 29°C with ac9.5. t] =; 0.30, t] = 0.60, q = 0.81 9 7 P * 7 Wy . t . l i r a n s i s t o r 9.10. ri v 0.77

ANSWERS TO SELECTED PROBLEMS 6 4 7

CHAPTER 10

10.1. Mr) = 7.5 V - (45 mV) cos 103r

10.3. t>„(r) = Y [1 + m f ( t ) ] 2 + A B [ 1 t m f ( t ) ] + y - * h L(t)

10.8. (40 m V)cos 104i

1 + 0.707m cos 104/10.9. (3.18 V)

10.10. u ,-(0 = (100 V)cos 107t; v„(t) * 100 V, 0 = 14.4°; F R = 0.015710.11. For large Q r , T H D * 0.47/Qr10.12. a) v„ = 62.5 V, l\ = (62.5 V) cos lO 't.ß ,- = 125, F R = 0.025, <p = 18.2", T H D = 0.0075

b) v'o = 125 V, v0 % 0,u, = (125 V) cos 107i,0 T = 250, FR = 0 0125,<p = 13.2°, T H D = 0.0038

10.14. v„(t) = (44.2 V) cos 2 x 104f -

cos 107r

cos 107r

Vo(t) = (125 V) + (44.2 V) cos 12 x 104i -

!-f(i) = (125 V)

10.15. R = 12.5 k n

1 + 0.353 cos 2 x 104 -

1 + 0.471 cos 104r -10.17. i;,(r) = -(3 .75 V)

v 2( t) = (3.75 V) + (1.77 V) cos 104i -

v 3(t) = (0.885 V) cos 104f -

CHAPTER 11

11.2. B W = 1.6 x 106 rad/sec 11.3. A / = 625 kHz

11.4. u„(t) = (11 V) — (5 V) cos 12 x 108r + 2 x 105 j " cos 1040d6

11.5. -5 7 .7 dB

11.7. v a = (100 mV) sin 1107i + 5 x 106 J f ( 9 ) d 6

11.10. üJq = 107 rad/sec, Aco « 104 rad/sec11.11. a>,(t) = (2.5 x 107 rad/sec) [1 + 0.1 /( i) ]

11.13. v ,( t) = 2 V c o s | l 0 7t + 1.25 x 10~3 J coslO 3# ^

11.15. 1 0 s H z[l + i / ( t ) ]

C H A PTER 12

12.1. V , * 0.8 mV, m 0 * 0.02 12.4. B W > 14Aw. D < 0.002812.5. B W > 5.3 Aco, D < 0.027 12.7. 0.0075,0.075. no

' 2.10. i;0(f) = (5 /7 2 ) V cos j l 0 4t -

12 11. a) 1570 i2, d) »„(0 = (250 mV) cos 103r12.12. / ,.(£) - 0.55 V cos ojJ

12.17 v 0( t) = j - ~ J ( l + 0 .2 cos 105 1)\ \ “/

12.19. b) v„ (t) - (0.58 V) + (0.041 V )cos 1103f -

648 ANSWERS TO SELECTED PROBLEMS

INDEX

INDEX

Abramowitz, M., 641 AFC, 590AGC, 308, 331-336, 486, 530 Alignability, 324-327 AM generators: balanced, 382-384

differential amplifier, 374-375 diode bridge, 376-379, 382-384 FET, 380-381 integrated circuit, 395-396 square law, 384-387 tuned circuit, 387-392 See also Diode bridge, FET, Chopper

modulation, Multiplier, and/or Modulation

AM methods: analog, 354-355 chopper, 355-358 direct, 361-362 high level, 447-450 nonlinear, 359-360

AM on FM signals, 589-590 AM receiver, superheterodyne, 478 AM signals: in capacitively coupled

circuits, 168-169, 195-199 in narrowband filters, 72, 76, 88-89 normal, 347single sideband, 350-352, 457-461,

464—466 suppressed carrier, 349, 357

Amplification, AM, 405-406 Amplifiers: cascode, 324-326

class A, 401-405 class AB, 401class B, broadband, 401, 421-425 class B “ linear,” 92, 405-408 class C “ linear,” 408-410 class C-RF, 401, 410-415 class D, broadband, 426-432

class D, narrowband, 401, 415-421 differential pair, 114-119, 324-326 feedback, 206, 475-478 FET, 98-104, 131-136 gain controlled, 214, 335-336 maximum signal without distortion, 4,

100, 110narrowband-tuned, 6- 8, 314-331, 405-421 power, 401-455small signal range, 4-5, 110, 118 transistor, 1-8, 104-113 tuned, 6- 8, 314-331, 405-421

Amplitude control, 212-215, 265-266,308, 331-336

Amplitude limiting, 7-8, 110, 188, 571-577 nonlinear, 207

Amplitude modulation; see specific headings unair AM

Amplitude sensitivity, function, 230-231, 264-265

Amplitude stabilization: FM, 530-532 oscillators, 212-215, 262-266

Analog computer, 528 Angelo, E. J., Jr., 13 Armstrong, E. H., 560 Astable multivibrator, 553-556 Attenuator, nonlinear, '<.'13, 265-266 Auto transformer, 49-i Automatic frequency co. ol (AFC), 590 Automatic gain control; see AGC AVC; see AGCAverage envelope detection, 461-466

in FM detectors, 582-584, 587 Average envelope detector, 468-478

dual transistor, 473-475 operational amplifier, 475-478

Avins, J., 6 i 3

652 INDEX

Back diode, 505-506 Baghdady, E. J., 523, 526 Bandwidth: FM, 513-515, 565

parallel RLC circuit, 28 peak envelope detector, 485, 494-495 pi network, 441-443 power amplifier coupling network,

433-437 transformer, 21

Barkhausen criterion, 11, 208 Bennett, W. R., 628 Bessel functions: modified

in amplifiers, 108-111 approximations, 113, 636 in mixers, 293-300 tabulations, 109, 637-639 theory, 636-641 uses, 3-9, 108-111, 293-300

ordinary, 427, 511-514 plot of, 512 zeros of, 514

Bias shifts, 4-5 in transistor circuits, 149, 169-181

Biasing: clamp circuit, 131-134 constant current, 1-2, 162-169 resistive, 169-181 triode grid, 432-433

Bilotti, A., 618 Binary AM, 410 Boltzmann’s constant, 2

Cadmium sulphide, 560 Calandrino, L., 628 Calibration: FM generator, 514

high-Q filter circuits, 88-89 Capacitive coupling, 162-171

to nonlinear devices, 149-180 Capacitive step-down circuits, 38-48 Carrier, pilot, 353"Carrier cancellation, 355, 376-383 Carrier filtering, 459-461 Cascode amplifier, 324-327, 345-346 Cassara, F„ 628 Chopper modulation, 355-358 Chudobiak, W. J., 421 Clamping circuits, 131-134, 153-180, 188

current-driven, 153-155 inductively biased, 154-156 peak FET, 131-134, 301-302, 304-305

Clarke, K. K„ 401, 588, 628 Clarke-Hess FM detector, 587-592, 632 Collector saturation in oscillators, 234-236 Colpitts oscillator, 223, 226, 231, 253 Complementary pair transistors, 423 Conduction angle, 93-103, 107

peak envelope detector, 482 Conversion transconductance; see

T ransconductance Converters, transistor, 311-314 Convolution, 74, 581 Coupling networks, 16-64, 433-447 Cross modulation, 123, 337-338 Crystal equivalent circuit, 243-247 Crystal filter, 459Crystal FM detector, 564, 602, 632 Crystal oscillator, 243-255 Current-driven junction, 249 Current inverter, 547-549 Current pulse, peak envelope detector,

480-482Current-switching circuit, 417-420,

431-432

Dammers, B. G., 260 Delay line, 603, 634

helical, 609 Delay network: ideal, 604

RLC, 610 Demodulation: amplitude, 457-508

frequency, 571-635 Derivative, definition, 602 Deviation ratio, 511Diagonal dipping; see Failure-to-follow

distortion Differential amplifier; see Differential

pairDifferential pair: amplifiers, 114-119

amplitude modulator, 365-375 distortion in, 114-119 mixers, 309-310 multiplier, 365-375 oscillator, 236-240, 251-252, 288, 291 R F amplifiers, 324-326

Differentiating network: direct, 586-587 frequency domain, 522, 586, 593 time delay, 586, 602-603

Diode bridge, 376-379 modulator, 376-383

INDEX 653

Diode clamping circuit, 153-162 Diode detector, 468-503 Diode shaping network, 551-553 Diodes: back, 505-506

germanium, 471, 586 hot carrier, 553 silicon, 2, 471, 496, 499 variable capacitance, 541-542 Zener, 557

Distortion: in AM, 361-362 failure-to-follow; see Failure-to-follow

distortion in FM detectors, 595-623 phase, 390total harmonic; see Total harmonic

distortion Double-tuned circuits, 328-331 Double-tuned transformers, 433-439 Drain resistance (in FET’s), 134-136

Edson, W. A., 244 Efficiency power amplifiers, 402ff Efficiency transformer, 19, 402 Emde, F., 641 Enloe, L. H„ 628Envelope detector, 468; see also Average

envelope detector, Peak envelope detector

Envelope detector loading, 489 in FM demodulators, 600, 610

Equivalent circuits: low pass, 65-81 peak envelope detector, 485 transformer, 51, 56, 62-64

Equivalent linear resistance, 181-194 envelope detector, 481

Exponential characteristic, 104-113

Failure-to-follow distortion, 361, 467, 389-390, 577

in amplitude modulators, 389-390 in broadband peak detector, 500 in peak narrowband detector, 490-494

Federal Communications Commission, 562 Feedback: amplifier, 206, 475-478

detector, 475-478 FM detector with. 623-628 FM generator stabilization, 562-565

FET: AGC circuit, 266 amplifiers, 98-104, 131-136 clamp-biased circuits, 131-134

distortion in, 98-104 drain resistance, 134-136 mixer, 295-297, 301-305 multiplier, 363-365 oscillator, 241-243, 253-255 switch, 380-381

Filaments, tungsten, 414 Filter: crystal, 459

mechanical, 315-316 pulse-forming, 620

FLL, 623-625 FM demodulation, 571ff FM demodulator: balanced slope, 596-600

balanced time delay, 608-613 balancing in, 589-591 Clarke-Hess, 587-592, 632 crystal, 564, 602, 632 direct differentiation, 586-588 FMFB, 623-626 Foster-Seeley, 609-613 frequency locked loop (FLL), 623-626 ideal, 578 laser, 608, 633 microwave, 608phase locked loop (PLL), 623-628 pulse count, 634, 635, 618-623 ratio detector, 613-615 series tuned slope, 600-602 simplified time delay, 615-618 slope, 593-602 time delay, 602-623

FM detectors; see FM demodulators FM differential equation, 527-528

quasi-static, 532-533 FM generators: analog, 528-529

Armstrong, 527, 559-562 crystal-controlled, 560 differential pair, 537 FET, 569-570 laser, 560, 562 quasi-static, 534-542 single transistor, 540 square wave, 553-559 time delay, 562 triangular wave, 568, 549 variable capacitance, 541-542

FM noise threshold, 624 FM signals: impulse train, 579, 584

nonsinusoidal, 520-521

654 INDEX

square, 577, 520-521 triangular, 520-521, 542-545, 632 sawtooth modulated, 510 sidebands of, 513

FM spectrum, 511-515 FMFB, 623-625 Foster, D. F., 610 Foster-Seeley detector, 609-613 “Fourier-like” FM expansion, 516 Fourier series: half-sine wave train, 92

impulse train, 80 narrow-pulse train, 480 periodic signal, 79 rectangular pulse train, 408 sine-wave tips, 93-95, 144-148 square wave, 92switching function, 355-356, 462

Fourier transform, 72 of AM signal, 72-73, 347-349 of Hilbert transformer, 351 of nonlinear FM, 517 of unit step, 71

Frequency demodulator with feedback (FMFB), 623-625

Frequency depression, 279, 282 Frequency deviation, definition, 510 Frequency division multiplexing (FDM),

73,509 Frequency doubler, 561 Frequency locked loop, 623-625 Frequency modulation, 509-570\ for

breakdown, see specific topics under FM

Frequency stability, 210, 216-222 direct, 216-217 FM generators, 562-565 indirect, 217-222

Frequency tripler, 561 Frutiger, P., 628 Full-wave rectifier, 503

Gain control, 265-266, 308, 331-336;see also AVC

Gerber, E. A., 243 Gdhnanium diode, 471, 496, 499 Gilbert, B., 365 Gray, P. E., 13 Groszkowski, J., 244, 279

Haantjej, J., 260 Hafner, E„ 243 Half-power frequency, 27 Half-wave rectifier, 462, 468-478, 618, 620

operational amplifier, 475-478 Hall effect multiplier, 362-363 Hayashi, C., 273 Hess, D. T., 527, 588, 625, 628 Hilbert transform, 351-353, 457 “Hold in range,” 626, 628 Hot carrier diodes, 553 Hysteresis, 544-545

IF amplifiers, 314-331 Immovilli, G., 628Impedance matching, 16-19, 326-327 Impulse response, narrowband filter, 70 Input impedance, variation with

frequency: in single-tuned circuits, 321-325 in double-tuned circuits, 330-331

Instantaneous frequency, definition, 510 Instantaneous phase, definition, 509 Intermodulation, 242

Jahnke, E., 641 Joyce, M. V., 401 Junction voltage, 2, 4

Kirchhoff’s current law, 480 Klein, G., 551

Large signal transconductance; see Transconductance, large signal

Lienard method, 273 Light bulb, 212, 213, 261 Limiters, 6-8, 110, 188, 571-577

differential pair, 573-574, 599, 629-631 diode, 572dynamic, 575-577, 613-614 tuned circuit, 629-630

Linearization: exponential characteristic, 119, 123-129

mixers, 302-305 Lissajous pattern, 88-89, 362 Load line, 183-190; see also Operating

pathLocal oscillator, 294

frequency modulated, 590

INDEX 655

Loop gain, 206Loudspeaker, 421Low pass equivalents, 65—69

IF strip, 69

MacLean, W. R., 260 Matching, impedance, 16-19 Maximum modulation frequency, 511 Meacham bridge oscillator, 212, 261-267 Measurements, high Q circuits, 88-89 Mechanical filters, 315-316 Midband range transformer, 17, 21 Miller capacitance, 52 Miller effect, 241 Millman, J., 546, 553, 609 Mixers: differential-pair, 309

FET, 295-297, 310-311 general problems, 294-295, 300-302,

305-306 series resistance in, 302-305 transistor, 8-10, 297-300, 307

Model transformer, 17, 51, 63-64 Modified Bessel functions: in amplifiers,

108-111 approximations, 113, 636 in mixers, 293-300 tabulations, 109, 637-639 theory, 636-641 uses, 3-9, 108-111, 293-300

Modulation, amplitude, 347-399 ;forbreakdown, see specific topics under AM

Modulation, frequency, 509-570; for breakdown, see specific topics under FM

Modulation index: AM, 8, 77 FM, 511

Monostable multivibrator, 622, 634-635 Multiplication of frequency deviation, 516 Multiplier: FET, 363-365, 379-381,

384-387 four quadrant, 368-373 frequency, 8integrated circuit, 365-373 single quadrant, 367

.square law, 354 transconductance, 395 two quadrant, 368-369

Multivibrator: astable, 553-556 monostable, 622, 634-635

Narrowband amplifiers, 6-7, 65-80,112-119, 314-331

Narrowband networks, 65-81 Narrowband power amplifiers, 405-420,

432-447Negative resistance, 205, 270-272 Networks: capacitive step-down, 38-48

coupling, 433-439 low pass equivalent, 65-69 narrowband, general, 65-81 pi, 439-447 “ transformer-like,” 51 transient response, 21-23, 70-72

Neutralization, 321, 327 Noise, 509Noise figure, 336-337Noise immunity in FM receivers, 571, 590Noise threshold in FM systems, 624-629

Onoe, M., 641 Operational amplifier, 262

half-wave rectifier, 475-478 low-pass filter, 478

Operating paths: class A amplifier, 407 class B amplifier, 407 class C amplifier, 410 effect of harmonics on, 438

Oscillators: Colpitts, 223, 226, 231-232 crystal, 243-255, 262

alignment, 251 tuning, 246

differential pair, 236-240, 251-252, 288, 291

Hartley, 223 junction FET, 241-243 large signal model: differential pair, 237ff

single transistor, 225ff local, 10, 233, 305-307 Meacham bridge, 212, 261-267 Miller, 253, 255negative-resistance, 268-272, 292 one-port, 268-272 pentode, 289 phase shift, 212, 283-284 Pierce, 253-254

656 INDEX

series mode crystal, 248-254 sine wave, 10-12, 205ff single transistor, 222-236, 286-287,

289-290variable frequency (VCO), 233-234,

624-628 Wien bridge, 212, 261, 267-268

Otte, J., 260 Overtone, crystal, 245

Page, D. F., 421 Parallel resonant circuits, 25-38 Peak envelope detection, 467-468

in FM detectors, 584-586 Peak envelope detector, 532

broadband, 498-503 ideal, 467narrowband, 478-497

Pentodes: characteristics, 120-123 mixers, 300-301 oscillators, 253-255 transconductance, 121-123

Periodic drives: effect of narrowband filters, 78-85

effect of nonlinear elements, 91-134 Phase deviation, definition, 509 Phase distortion, 390 Phase locked loop (PLL), 459-460,

623-628 Phase modulation, 509-510 Phase plane, 273-279 Phasitron, 527 Pi network, 439-447 Piecewise linear models, 91, 124-126,

151-153 Pierce oscillator, 253-254 PLL, 459-460, 623-628 •Pole zero diagram: delay network, 611

double-tuned transformer, 435 oscillator networks, 207, 208, 210, 212 quartz crystal, 245

Power in AM signals, 349 Power amplifiers: class A, 401-405

class AB, 401class B, broadband, 401, 421-425 class C, linear, 408-410 class C, RF, 401, 410-415, 442-447 class D, broadband, 415-421

class D, narrowband, 426-432 coupling networks, 442-447 modulation of, 387-391, 447-451 switched, broadband, 415-421 switched, narrowband, 426-432

Power gain, 326-327 Power loss in transformers, 17-19 “Pull in range,” 628 Pulse count FM detector, 618-623,

634-635

Q: to limit output distortion, 82, 85 parallel circuits, 27-28, 31

Quartz crystal, 210, 217 model, 244 oscillators, 243-255 parameters, 245

Quasi-static approximation, 521-526, 596,. 598

Quasi-static frequency modulation, 532-542

Ratio detector, 613-615 Raysistor, 266 Reactance tube, 542 Receiver design, 293-295 Relton, F. E., 641 Resonant frequency, 26-27, 33 Rice, S. O., 628 Ripple, 467

in broadband envelope detector, 500 in FM detectors, 592 in peak envelope detector, 483

Root locus, 207-212, 434-437 Rowe, H. E., 426

Saturating collector, AM modulator, 388-392

Saturation current, transistor or diode, 2, 104, 171

Sawtooth generator, 428 Schmitt trigger, 343-547 Schwartz, M., 628 Seeley, S. W., 610, 613 Semiconductor resistor, 212 Series resistance: effect on clamping

operation, 156-159 effect on exponential characteristic,

123-130

INDEX 657

effect on mixer characteristic, 302-305 Signal generator, laboratory, 213 Signals: telephony, 505

s e e a l s o AM signals a n d FM signals Silicon diode, 2, 471, 496, 499 Sine wave tips: with piecewise

discontinuities, 92-98 in power amplifiers, 414-415, 454-455 with square law characteristic, 102-104

Single sideband (SSB): amplification, 405 demodulation, 457-461, 464-466 generation, 349-353, 368 spectrum, 350

Small signal operation, 3, 101, 110, 118 Smith, W. L., 602 Smith Chart, 444—446 Source impedance, effect on clamping

operation, 156-159 Spectrum: FM signal, 513

SSB, 350 video signal, 465

SpikesSquare law characteristics, 98-104, 131-134,

146-148 Square law detector, 500 Square law modulation, 359-360 Squegging, 255-261, 532 Stegun, I. A., 641 Stein, S., 628Step response: narrowband networks,

70-75 transformer, 21-23

Strauss, L., 551, 553 Superheterodyne, 10, 293-295 Superposition, 90, 149, 153 Suppressed carrier AM, 349, 357 Switch, voltage-controlled, 547-549 Switching function, 355-356, 462 Switching power amplifiers: broadband,

415-421 narrowband, 426-432

Sykes, R. A., 243 Symmetric networks, 66 Synchronous detection, 457-461

in FM detectors, 580-582 Synchronous detector, 615

Taub, H„ 546, 553, 609

T aylor Series o f R L C circuit m agnitude,594-595, 598

Thermal runaway, 400 Thermal time constant, 213 Thermistor, 212-213Threshold extension FM receivers, 623-628 Threshold sensor, 543-544 Time-delay FM detector, 602-623 Time-varying controlled source, 534 Total harmonic distortion (THD), 82-85,

192-193in differential pair oscillator, 236, 240 in transistor oscillator, 227-230

Transconductance: differential pair, 116-119 effect of series resistance, 123-130 FET, 100-101, 133-134 pentodes, 121small signal, 100, 105, 116, 121 transistor, 105-112, 128-129

Transconductance, conversion, 9, 297-314 bipolar transistor, 9, 297-301 differential-pair, 309-310 FET, 295-297peak-clamped FET, 304-305 pentode, 301-302, 304-305

Transconductance, large signal: differential pair, 119

exponential characteristic, 7, 11, 110 with series resistance, 129-130

FET’s, 100, 122, 134 pentodes, 122-123 resistively coupled transistor circuit,

177-179square law characteristic, 100, 134

Transconductance multiplier, 395 Transfer function, 65 Transformer: broadband, 16-24

three-winding, 54-59 Transformer-like networks, 38-48

summary, 51 Transient buildup: clamping circuit,

159-162 transistor bias circuit, 165-169

Transient response: narrowband circuits,71-76

Transistor: as amplifier, 104-113 average detector, 473, 586-592 biasing, 1-2,162-180

658 INDEX

collector modulation of, 387-392 limiters, 573 mixers, 297-300 simple oscillator, 222-227 see also individual topics such as FET,

Oscillators, Amplifiers, Biasing, etc.Trapezoidal pattern, 88, 362Triangular wave frequency modulation,

542-553Triangular wave pulse train, harmonics in,

454-456Triode class C amplifier, 411-415Tuned circuit, nonlinear loading, 181-194Turn on bias, 425Turns ratio, transformer, 18-19

Unkauf, M., 628

Vacuum tube: AGC circuits, 332-334 characteristics, 120-123, 334, 413 mixers, 300-301modulated power amplifiers, 450-451

oscillators, 253-255 power amplifiers, 410-415, 432-439 transconductance, 120-123

Van Suchtelen, H., 260 Varactor, 542 Varicap, 541-542 VCO, 541-542, 624-628 Vestigial sideband, 464-466 “Virtual” ground, 363, 476 Voltage-controlled oscillator (VCO),

541-542, 624-628

Watson, G. N., 641Waveform function, 165Waveshaping network, sinusoidal, 551-553Whittaker, E. T., 641Woodward, P. M., 515Woodward’s theorem, 515

Y parameters, 315-346

Zener diode, 557

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