INDUCTANCE SIMULATION FOR MICROELECTRONICS AND

INDUCTANCE SIMULATION FOR MICROELECTRONICS AND

TRANSISTORIZED LOY-FREQUENCY ACTIVE GIRATORS

by

KENNETH RAOUL MORIN

B.Sc., Queen's U n i v e r s i t y , Kingston, O n t a r i o , 1961

A THESIS SUBMITTED IN PARTIAL FULFILMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF APPLIED SCIENCE r > .

i n the Department

of

E l e c t r i c a l E n g i n e e r i n g

¥e accept t h i s t h e s i s as conforming to the

r e q u i r e d standard

THE UNIVERSITY OF BRITISH COLUMBIA

October, 1963

I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of

the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of

B r i t i s h C o l u m b i a , I agree that the L i b r a r y s h a l l make i t f r e e l y

a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree that p e r

m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s , t h e s i s f o r s c h o l a r l y

purposes may be granted by the Head of my Department or by

h i s r e p r e s e n t a t i v e s , , " i t i s unders tood that copying, or p u b l i

c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d

without my w r i t t e n p e r m i s s i o n .

Department of E l e c t r i c a l E n g i n e e r i n g

The U n i v e r s i t y of B r i t i s h Columbia , . Vancouver 8, Canada.

Date October 25, 1963

ABSTRACT

An inductance can be simulated f o r m i c r o e l e c t r o n i c s

a p p l i c a t i o n s u s i n g semiconductor elements (e.g., the "inductance

d i o d e " ) , u s i n g c i r c u i t s c o n t a i n i n g a m p l i f i e r s , or u s i n g g y r a t o r s .

The l a s t two methods are considered i n t h i s t h e s i s .

Several " a m p l i f i e r methods" have appeared i n the l i t e r a t u r e ;

these methods are c l a s s i f i e d i n t o i n t e g r a t i n g - or d i f f e r e n t i a t i i i g -

type c i r c u i t s , and a d i f f e r e n t i a t i n g - t y p e c i r c u i t i s proposed

which i s b e l i e v e d to be new.

Gyrator r e a l i z a t i o n methods are t a b u l a t e d and compared.

An " a c t i v e g y r a t o r " ("AG") i s proposed as a c i r c u i t element ( i t

has unequal g y r a t i o n r e s i s t a n c e s ) . The AG behaves much l i k e a

g y r a t o r ; i t can be used to simulate inductance, and an a n a l y s i s

shows that i t can be used to make i s o l a t o r s and c i r c u l a t o r s

with a power g a i n .

Methods of r e a l i z i n g an AG with a m p l i f i e r s are i n v e s t i g a t e d ,

and an a n a l y s i s leads to seven 2 - a m p l i f i e r c i r c u i t s . One of

these AG c i r c u i t s appears "best" f o r inductance s i m u l a t i o n , and

t h i s one i s i n v e s t i g a t e d e x p e r i m e n t a l l y u s i n g a t r a n s i s t o r

c i r c u i t .

An extensive b i b l i o g r a p h y of the inductance s i m u l a t i o n and

g y r a t o r l i t e r a t u r e i s presented.

ACKNOWLEDGEMENT

The author i s indebted to Dr. M.P. Beddoes, the s u p e r v i s i n g

p r o f e s s o r of t h i s p r o j e c t , f o r h i s help and guidance throughout

the course of the p r o j e c t . Thanks are also given to those p r o f

essors and graduate students who provided h e l p f u l d i s c u s s i o n , and

i n p a r t i c u l a r , to C R . James, f o r h i s thorough p r o o f - r e a d i n g .

Acknowledgement i s g r a t e f u l l y gîven to the N a t i o n a l Research

C o u n c i l of Canada f o r the a s s i s t a n c e r e c e i v e d through a Bursary

and a Studentship h e l d by the author during h i s s t u d i e s .

The work d e s c r i b e d i n t h i s t h e s i s was supported by the Nat-.

i o n a l Research C o u n c i l under Grant BT - 6 8 .

i x

TABLE OP CONTENTS

ACKNOWLEDGEMENT i x

1. INTRODUCTION .. . . . . . . c 1

2. INDUCTANCE SIMULATION 5

2.1 Methods f o r E l i m i n a t i n g , M i n i a t u r i z i n g , and Simulatin g Inductors . 5

2.2 C l a s s i f i c a t i o n of the Various " A m p l i f i e r Methods" of Inductance S i m u l a t i o n 7

.1 I n t e g r a t i n g c i r c u i t ..............»•«.<>.. 7

.2 D i f f e r e n t i a t i n g c i r c u i t 12

2.3 R e l a t i o n s h i p Between the Various " A m p l i f i e r Methods" of Inductance Simulation,and the AG ...... .......... . 15

3. GYRATOR AND ACTIVE GYRATOR THEORY AND APPLICATIONS ..... 17

3.1 S u r v e y of Gyrator L i t e r a t u r e .«.......<> . 17

3.2 Gyrator Symbols 20

3.3 D e f i n i t i o n of an Active Gyrator 24

3.4 The A c t i v i t y of the A c t i v e Gyrator 25

3.5 A p p l i c a t i o n s of the AG 26

.1 S i m u l a t i o n of inductance 26

.2 The AG used as an i s o l a t o r 27

.3 The AG used as a c i r c u l a t o r 29

a) I n t r o d u c t i o n 29

b) Conditions f o r c i r c u l a t i o n , 30

c) Matching 34

d) Power Gains 36

Page

4. CIRCUITS FOB REALIZING ACTIVE GTRATORS 39

4.1 I n t r o d u c t i o n .*•...........................«««•••..« 39

4.2 E f f e c t s of the 2 Types of C i r c u i t Components on the

I Matrix . 41

.1 Input conductance ............. 41

.2 Output connections 42

• 3 Negative r e s i s t a n c e . . . . . . . . . . . . . . 4 3

4.3 Design of a 3-Âmplifier AG 43

4.4 Clue f o r Designing 2-Amplifier AGs 46 4.5 C l a s s i f i c a t i o n and P r e l i m i n a r y Screening of

2-Amplifier AGs ...... 48

4.6 Results of the A n a l y s e s of the 16 C o n f i g u r a t i o n s ... 49

5. EXPERIMENTAL RESULTS 52

5.1 Allowance f o r the F i n i t e Output Impedance of the A m p l i f i e r s 52

5.2 C i r c u i t Diagrams f o r the Prototype AG 55

5.3 Experimental Results 57

.1 P r e l i m i n a r y adjustment of the AG ................. 57

a ) Adjustment f o r Y-^ & 0 ......................... 58

b) Adjustment f o r Y 2 2 ~ 0 59

.2 I n v e r s i o n of r e s i s t a n c e .....•...«......•••««•.«•« 60

.3 Si m u l a t i o n of inductance ........................ . 60

6. CONCLUSION 64

7. BIBLIOGRAPHY ..... 66

7.1 Subject Index to B i b l i o g r a p h y ............. ........ . 66

7.2 References 68

Page

APPENDIX 81

A . l The 4 P o s s i b l e output Connections f o r Each

C o n f i g u r a t i o n 81

k»2 A Systematic Method of A n a l y s i s ........ 82

.1 The f i r s t step 84

.2 The second step ................................... 87

LIST OF ILLUSTRATIONS

F i g u r e Page

1.1 - D e f i n i t i o n of an i d e a l g y r a t o r ................«... • 1

2.1 - (a) The simple R-C i n t e g r a t o r ; (b) the o p e r a t i o n a l a m p l i f i e r i n t e g r a t o r 8

2.2 - '•. Simulated inductance u s i n g a simple R-C i n t e g r a t i n g c i r c u i t and a pentode tube ( g r i d - l e a k r e s i s t o r omitted) .•.«.•.*..........................•«.•.«•»• 8

2.3 - Stern's c i r c u i t f o r a simulated inductance .««•««••. 9

2.4 - Basic c i r c u i t used by Holbrook and McKeown. (a)

a c t u a l c i r c u i t ; (b) approximate e q u i v a l e n t c i r c u i t 10

2.5 - Midgley and Stewart's simulated inductance c i r c u i t 12

2.6 - (a) The simple R-C d i f f e r e n t i a t o r , and (b) the o p e r a t i o n a l a m p l i f i e r d i f f e r e n t i a t o r , shown with a small r e s i s t o r R^ connected (to produce v^ ' from "the c u r r e n t x ) + o e o o < > e o o * * w * 0 * Q o o o o » o t > * * * » + 4 r i * < t - 6 - a « - 4 + 13

2.7 - Diagram of Towner's a r t i f i c i a l i n d u c t o r .......... 14

2.8 - An inductance s i m u l a t i o n c i r c u i t which uses the o p e r a t i o n a l a m p l i f i e r d i f f e r e n t i a t o r of Figure 2.6(b) 14

2.9 - C l a s s i f i c a t i o n of simulated inductances which use a m p l i f i e r me"fcho.cLs + o o c o * e * i o o o a 9 o t > » + Q o « o 9 9 0 4 i * * + «<> + <r*e- 15

3»1 —• Representative cross s e c t i o n of a f i e l d e f f e c t

t e t r o d e ^ ^ ^ ^ ^ « ^ ^0 « ^ a m o • ^ o ^ o o c « c « o o 0 a « « « » » 0 • * ^ ^ ^ ^ ^ ^ ^ 20

3.2 - Comparison of g y r a t o r r e a l i z a t i o n methods .«...„..«« 21

3.3 - T e l l e g e n ' s symbol f o r the i d e a l g y r a t o r . Sometimes

one of the s e m i c i r c l e s i s omitted (e.g., st57c) «... 22

3.4 - Gyrator symbol proposed by F e l d k e l l e r .............. 22

3.5 - Shekel's symbol f o r the 3-terminal g y r a t o r •»«.«.«.. 23

3.6 - Hogan's symbol f o r the (microwave) g y r a t o r ......... 24

3.7 — C i r c u i t symbol and equations f o r the AG ............ 25

3.8 - The AG as an i s o l a t o r : (a) p a r a l l e l connected^ and (b) s e r i e s v i

Figure Page

3.9 - E q u i v a l e n t c i r c u i t f o r the i s o l a t o r of Figure 3.8(a)

3.10 - A 3-port network made from a 3-terminal device ..... 29

3«11 - The two b a s i c c i r c u l a t o r c o n f i g u r a t i o n s : (a) voltage sources, and (b) current sources ..... » ........•• «.« 31

3 012 •=- The f i r s t step i n d e r i v i n g the matrix d e f i n e d i n

3.13 - (a) and (b) Input admittances to the c i r c u l a t o r shown i n Figure 3.11(b); (c) and (d) values of input admittance obtained when an AG i s used. The d i r e c t i o n of c i r c u l a t i o n i s i n d i c a t e d by the arrow.. 35

3.14 - An a p p l i c a t i o n of a c i r c u l a t o r . The c i r c l e represents a 3-port c i r c u l a t o r ........ <» o.......... . . 37

3.15 - The product Gm "GT ^, as a f u n c t i o n of G_/G_ ... 38 ia->c c—>b n m

4.1 - The two components which w i l l be used to b u i l d AGss (a) conductance, and (b) ideali'zed v o l t a g e a m p l i f i e r , which has A r e a l j p o s i t i v e or negative 39

4 = 2 •=- Skeleton of the. AG- s t a r t i n g p o i n t f o r each AG design 40

4.3 =• Successive steps taken i n the a n a l y s i s of AG

4.4 - I l l u s t r a t i n g the e f f e c t of output connections ...... 42

4.5 - Negative r e s i s t a n c e s u s i n g v o l t a g e a m p l i f i e r s .«»«•« 43

4.6 - The s t a r t of

4 o 7 D 3 ™*CHT1J) X i f X G I* AG!" • aa«««-«0*«*»*0e*a«0a««0o4>aA0 4»«a« i0*« 43

4.8 - Obtaining the v o l t a g e s +V +V"2 from the po r t v o l t a g e s 46

4.9 => Changing the s k e l e t o n of an AG c i r c u i t .......»oo.«. 47

4.10 - Summary of 2 - a m p l i f i e r AG c i r c u i t s ......a... 50

5.1 = The output connections f o r an a m p l i f i e r with f i n i t e output conductance, G : (a) a c t u a l c i r c u i t ; (b) approximate - e q u i v a l e n t when G ^ r v G ^ « G o ......... 52

5.2 - Design sheet f o r c o n f i g u r a t i o n number 8, using the

approximation given i n Figure 5.1(b) ............... 54

5.3 - C i r c u i t diagram of the Q a m p l i f i e r ................• 55 v i i

Figure Page

5.4 - C i r c u i t diagram of the P. a m p l i f i e r ...........»«•».. 56

5.5 — Schematic diagram of the AG prototype .............. 56

5.6 - Artwork f o r the p r i n t e d c i r c u i t board ( a c t u a l s i z e ) 57

5.7 - The AG used to i n v e r t an impedance Z- : experimental

apparatus 58

5.8 - Experimental r e s u l t s , i n v e r s i o n of r e s i s t a n c e ...... 61

5.9 - Experimental r e s u l t s , s i m u l a t i o n of inductance; (a) low frequency measurements, Z- ; (b) high frequency measurements, 62

5.10 - Approximate eq u i v a l e n t c i r c u i t f o r the simulated inductance ( l o s s e s are neglected) 63

A . l - Design sheet used f o r the i n v e s t i g a t i o n of AG

c i r c u i t s 83

A.2 - Design sheet f o r c o n f i g u r a t i o n number 8, D - l - ...... 85

A.3 - E q u i v a l e n t c i r c u i t f o r finding- 1;the " i n t r i n s i c terms" • f o r the c o n f i g u r a t i o n D - l - 86

A.4 - E q u i v a l e n t c i r c u i t s f o r f i n d i n g some of the " a m p l i f i e r terms" f o r the c o n f i g u r a t i o n D-1-. T matrices are given ................................. 87

A.5 - Design sheet f o r c o n f i g u r a t i o n number 5, D+1+ ...... 90

A.6 - Design sheet f o r c o n f i g u r a t i o n number 6, D+l- ...... 91

A.7 - Design sheet f o r c o n f i g u r a t i o n number 11, D-2+ ..... 92

v i i i

1. INTRODUCTION

There i s c u r r e n t l y much i n t e r e s t i n the making of t i n y

e l e c t r o n i c a s s e m b l i e s . Both r e s i s t o r s and c a p a c i t o r s have q u i t e

s u c c e s s f u l l y been m i n i a t u r i z e d ; however, i n d u c t o r s have not

been m i n i a t u r i z e d so s u c c e s s f u l l y . I n i t i a l l y , an attempt was made

to d e v e l o p m i n i a t u r e i n d u c t a n c e s by c i r c u i t t e c h n i q u e s ; but as t h e

work p r o g r e s s e d , emphasis s h i f t e d t o the s y n t h e s i z i n g of^ g y r a t o r

c i r c u i t s .

Techniques have been d e s c r i b e d which e l i m i n a t e the need f o r

i n d u c t o r s by r e d e s i g n of c i r c u i t s . Other approaches d e a l w i t h

making m i n i a t u r e i n d u c t o r s * f o r i n s t a n c e by w i n d i n g f i n e w i r e on

a t i n y f e r r i t e c o r e . S t i l l o t h e r approaches i n v o l v e the s i m u l a t i o n

of i n d u c t a n c e .

One approach f o r s i m u l a t i n g an in d u c t a n c e uses a g y r a t o r — a

4-^terminal, a n t i r e c i p r o c a l d e v i c e w hich obeys the e q u a t i o n s shown

i n F i g u r e 1.1. One p r o p e r t y of the g y r a t o r i s t h a t of impedance

L V 2 .

0

R

-R

0

h

- 1 2

F i g u r e 1*1 D e f i n i t i o n of an I d e a l G y r a t o r ,

i n v e r s i o n . I f an impedance Z T i s connected from c to d i n F i g u r e Li

1 . 1 , t h e n V_ = ™I_.ZT , and the i n p u t impedance between a and b i s

seen to be

i n ,11 -HI 2

z T

. . ( 1 * 1 )

In p a r t i c u l a r , i f = j^jj"» "then Z i n = ja> CR , and a simulated 2

inductance of value L - CR r e s u l t s .

However, gyrators are commercially available only for

microwave frequencies* although several experimental low-frequency

gyrators have been made, for instance by using the Hall effect, or

coupled electromechanical transducers. In view of the current

interest i n solid—state techniques, a gyrator made using trans

i s t o r s would be of value. One achievement reported i n this thesis

i s the design of a gyrator-like device (made with transistors)

which i s suitable for inductance simulation.

The impedance matrix of this device i s p r a c t i c a l l y i d e n t i c a l

with that shown i n Figure 1*1, except that the R's are unequal;

i . e . , i t obeys the equation

0

R L m

-R n

0 ( 1 . 2 )

This equation reduces to that of a gyrator i f = = R. This

device inverts an impedance Z T into R R / Z , , and has other * L m n L ?

* properties similar to a. gyrator; however, i t i s an active device,'

and on this account i t i s called an "active gyrator", or "AG" f o r

sho r t. * This i s shown i n Section.3*4. ,<,.

. 3 Although many papers on inductance s i m u l a t i o n and gyr a t o r s

have been publis h e d , as yet no comprehensive b i b l i o g r a p h y has

appeared f o r e i t h e r f i e l d . I t i s hoped that the extensive b i b l i o

graphy contained i n t h i s t h e s i s w i l l f i l l t h i s need.

Chapter 2, on inductance s i m u l a t i o n , contains a survey of

the l i t e r a t u r e i n t h i s f i e l d . A t t e n t i o n i s focussed on those

methods of inductance s i m u l a t i o n which use voltage a m p l i f i e r s ,

and these methods are c l a s s i f i e d according to t h e i r b a s i c p r i n c i p l e

of o p e r a t i o n . F u r t h e r , i t i s shown that they a l l behave l i k e AGs.

Chapter 3, on gyr a t o r s and a c t i v e g y r a t o r s , begins with a

survey of the gyrator l i t e r a t u r e ; the v a r i o u s gyrator symbols are.

then given. The r e s t of the chapter deals with the c i r c u i t

p r o p e r t i e s of a c t i v e g y r a t o r s : the AG i s d e f i n e d , and then some

of i t s p r o p e r t i e s and a p p l i c a t i o n s are given.

In Chapter 4, on c i r c u i t s f o r r e a l i z i n g AGs, a method i s

developed f o r o b t a i n i n g AG c i r c u i t s which c o n t a i n only v o l t a g e

a m p l i f i e r s and r e s i s t o r s . I t i s shown t h a t a 3 - a m p l i f i e r AG can

be designed i n t u i t i v e l y . The main problem i s to design a 2-

a m p l i f i e r AG, and a r a t h e r lengthy a n a l y s i s produces 7 such

designs. Of these 7 designs, 4 have been p u b l i s h e d p r e v i o u s l y ,

and 3 are b e l i e v e d to be new. In f a c t , one of the 3 new c i r c u i t s

had been designed i n the e a r l y stages of t h i s p r o j e c t , and was an

i n c e n t i v e f o r c a r r y i n g out the present a n a l y s i s . This p a r t i c u l a r

c i r c u i t a l so appears to be the best of the 7 f o r inductance

s i m u l a t i o n .

Chapter 5 contains experimental r e s u l t s . In order to t e s t

the v a l i d i t y of the t h e o r e t i c a l design, a prototype model was made

of the AG chosen as most s u i t a b l e f o r inductance s i m u l a t i o n . The

sample measurements made on t h i s model confirmed the v a l i d i t y of

4

the design.

Chapter 6 contains the c o n c l u s i o n , and Gjiapter 7 contains

the b i b l i o g r a p h y . This b i b l i o g r a p h y i s a major p a r t of the

t h e s i s , and consequently i t i s i n c l u d e d i n the main body of the

t h e s i s . I t i s indexed according to subject f o r handy reference .

• 5

2, INDUCTANCE SIMULATION

The f i r s t s e c t i o n of t h i s chapter i s a survey of v a r i o u s

proposals f o r e l i m i n a t i n g , m i n i a t u r i z i n g , and s i m u l a t i n g i n d u c t o r s .

Then c e r t a i n of these s i m u l a t i o n methods are considered i n more

d e t a i l , and i t i s shown that they are a l l r e l a t e d to one

another, and a l s o to the A G .

2.1 Methods f o r E l i m i n a t i n g , M i n i a t u r i z i n g , and Simu l a t i n g Inductors.

Some approaches to the problems of avo i d i n g the use of

i n d u c t o r s , or a l t e r n a t i v e l y to m i n i a t u r i z e or simulate them, are

considered i n the general works on m i c r o e l e c t r o n i c s (e.g. du6l,

ho58, ho62., as l i s t e d i n the b i b l i o g r a p h y , Chapter 7 ) .

In some cases, i t may be p o s s i b l e to avoid the use of

in d u c t o r s a l t o g e t h e r , f o r example by s u b s t i t u t i n g R-C c i r c u i t s

i n place of i n d u c t i v e c i r c u i t s (co58), or by us i n g ceramic

transformers, e t c . (e!58, lu58, ma61a, ma61b). For power supply

smoothing c o i l a p p l i c a t i o n s , i t i s not necessary a c t u a l l y to have

an i n d u c t o r , f o r any device with a low d.c. impedance and a high

a.c. impedance can be used} a cur r e n t l i m i t e r f i l l s these

s p e c i f i c a t i o n s n i c e l y (du61, pp. 47-48, la62, wa59).

In other cases, i t i s p o s s i b l e to a l t e r conventional

techniques and produce t i n y i n d u c t o r s . The most s t r a i g h t - f o r w a r d

approach i n v o l v e s the development of s p e c i a l t i n y f e r r i t e cores,

on which very f i n e wire i s wound (du6l, pp. 201-03, and 248-52,

st57a, st57b? ). Inductances of the order of a m i l l i h e n r y , and

* Some of the re f e r e n c e s given i n the b i b l i o g r a p h y were una v a i l a b l e to the author, so t h e i r c l a s s i f i c a t i o n i s u n c e r t a i n .

6

small enough to mount on a t i n y "micromodule 1 1 substrate have been

produced by t h i s method* For very small values of inductance (of

the order of a microhenry) p r i n t e d c i r c u i t techniques can be used

to p r i n t a s p i r a l , two-dimensional i n d u c t o r (br55). Using t h i a -

f i l m techniques, s p i r a l i n d u c t o r s with 1 m i l l i n e - w i d t h can be

f a b r i c a t e d on a f e r r i t e s u b s t r a t e , y i e l d i n g inductances of s e v e r a l

hundred microhenries (du61, p* 202, to62)i F i n a l l y , there are s e v e r a l methods f o r a c t u a l l y s i m u l a t i n g

an inductance* An AG and a c a p a c i t o r can be used, as was d i s c u s s e d

i n Chapter 1, and thus any method of c o n s t r u c t i n g an AG i s a l s o a

method of s i m u l a t i n g an inductance. (Several c i r c u i t s f o r

r e a l i z i n g the AG are considered i n the next c h a p t e r ) .

Semiconductor elements have been used f o r inductance

s i m u l a t i o n ; these elements use one or s e v e r a l s o l i d - s t a t e e f f e c t s

to produce i n d u c t i v e behaviour* Examples are the "inductance

diode"* and the i n d u c t i v e t r a n s i s t o r * D i l l ( d i 6 l ) g i v e s a com

prehensive survey of these methods. He concludes that the /

i n d u c t i v e t r a n s i s t o r , with which i t "can be p o s s i b l e to get

inductances i n the henry r e g i o n with reasonably high Q", i s " v p r y

promising", although i t i s plagued with temperature and s t a b i l i t y

problems* and r e q u i r e s a h i g h power d i s s i p a t i o n . Refer t o the

b i b l i o g r a p h y , page 66 f f o r f u r t h e r r e f e r e n c e s *

There are s e v e r a l methods of inductance s i m u l a t i o n which

might c o l l e c t i v e l y be c a l l e d " a m p l i f i e r methods" (fu60, h o — y

mc63j, mi60, st58, to53* wa47), since they a l l employ some s o r t of

a m p l i f y i n g means. These c i r c u i t s a l l c o n t a i n a c a p a c i t o r as a

major component. These c i r c u i t s are considered i n much more d e t a i l

i n the r e s t of t h i s chapter, where i t i s shown that they can a l l

be d e r i v e d from e i t h e r an i n t e g r a t i n g or a d i f f e r e n t i a t i n g

c i r c u i t . In a d d i t i o n , i t i s shown t h a t they a l l behave as AG

only under the same c o n d i t i o n s that are necessary f o r inductance

s i m u l a t i o n .

2.2 C l a s s i f i c a t i o n of the Various " A m p l i f i e r Methods" of Inductance S i m u l a t i o n .

The v a r i o u s a m p l i f i e r methods of inductance s i m u l a t i o n a l l

employ the same b a s i c p r i n c i p l e : an a m p l i f i e r i s used i n con

j u n c t i o n with the -90° phase angle of a c a p a c i t i v e impedance to

produce the +90° phase angle of an i n d u c t o r . The c a p a c i t o r appears

i n e i t h e r an i n t e g r a t i n g or a d i f f e r e n t i a t i n g c i r c u i t .

2.2.1 I n t e g r a t i n g c i r c u i t .

The c i r c u i t equation f o r an i n d u c t o r may be w r i t t e n

Given an empty, two-terminal, b l a c k box with the voltage v^

a p p l i e d between the t e r m i n a l s , what can be put i n s i d e the box i n

order that the input c u r r e n t be given by (.2.1)?

If an i n t e g r a t i n g c i r c u i t i s connected to the two t e r m i n a l s ,

one can o b t a i n a v o l t a g e

c i r c u i t s (terminated i n the c a p a c i t o r mentioned above), a l b e i t

• • • (2.1)

8

I f i t now can be arranged f o r the input current to be p r o p o r t i o n a l

to t h i s v , then the b l a c k box w i l l behave l i k e an inductance, o Common ways of i n t e g r a t i n g are with e i t h e r a simple R-C • i f

i n t e g r a t o r or an o p e r a t i o n a l a m p l i f i e r i n t e g r a t o r , as i n

F i g u r e 2.1. 4 R i » f c — o — w 1 R 1 \L 2

C v

(a) (b)

F i g u r e 2.1 - (a) The simple R-C i n t e g r a t o r } a t i o n a l a m p l i f i e r i n t e g r a t o r .

(b) the oper-

For the simple i n t e g r a t o r of F i g u r e 2.1(a) some s o r t of

a m p l i f y i n g device w i l l be needed. A pentode tube may be used

(hu42, wa47), connected as shown i n F i g u r e 2.2.

O-11 a

b O

B l A / y

h1

+ A t c ap

F i g u r e 2.2 - Simulated inductance u s i n g a simple R-C i n t e g r a t i n g c i r c u i t and a pentode tube ( g r i d - l e a k r e s i s t o r omitted).

* A simple L-R i n t e g r a t o r might be used as w e l l , but t h i s would be d e f e a t i n g the purpose.

I f the i n t e g r a t i n g c i r c u i t draws n e g l i g i b l e c u r r e n t , and the

b l o c k i n g c a p a c i t o r C, i s l a r g e , then the input impedance to the R 1 C

c i r c u i t i s approximately t h a t of an inductance, L = » m

Fulenwider (fu60) gives an improved v e r s i o n of t h i s c i r c u i t * which

con t a i n s 3 pentodes and 4 t r i o d e s . Instead of u s i n g a simple

R-C i n t e g r a t i n g c i r c u i t , however, he uses a t r i o d e i s o l a t i o n

a m p l i f i e r and a pentode loaded with a c a p a c i t o r ; the other e x t r a

tubes cancel the remaining l o s s e s , to produce a Q l i m i t e d only by

s t a b i l i t y c o n s i d e r a t i o n s . The p a r t i c u l a r c i r c u i t which he gives

has "a frequency range of 60 to 2000 cps, with an e q u i v a l e n t magni

tude range of 28 to 2800 h e n r i e s . "

Stern (st58) completes the c i r c u i t of F i g u r e 2.1(a) with

a t r a n s i s t o r . The base of the t r a n s i s t o r i s connected to term

i n a l 2, and the e m i t t e r i s connected to t e r m i n a l 3 through a

r e l a t i v e l y high r e s i s t a n c e (so as to not load the c a p a c i t o r C too

h e a v i l y ) , thereby f o r c i n g the base cu r r e n t to be p r o p o r t i o n a l to

V q . F i n a l l y , the c o l l e c t o r i s connected to t e r m i n a l 1 (see

F i g u r e 2.3) so t h a t , i f the i n t e g r a t i n g c i r c u i t draws n e g l i g i b l e

c u r r e n t , then the input c u r r e n t i s the c o l l e c t o r c u r r e n t ,

^ pi - dt,

+ a (1)

b O-

R l

-vW ' V = : V o .2

(2)

(3)

-O O —

cap -O O— 1

R,

F i g u r e 2.3 - Stern * s . c i r c u i t f o r a simulated inductance.

10 and a simulated inductance r e s u l t s . In p r a c t i c e , a s i n g l e t r a n s

i s t o r seems i n s u f f i c i e n t to provide a reasonably high Q. A one-

t r a n s i s t o r c i r c u i t was assembled i n t h i s l a b , and produced a Q

of only about 2. D i l l ( d i 6 l ) r e p o r t s a c h i e v i n g inductances as

h i g h as 10 h e n r i e s with t h i s c i r c u i t , but always with a Q l e s s

than 10.

An i n t e r e s t i n g r e a l i z a t i o n of the i n t e g r a t i n g c i r c u i t of

F i g u r e 2.1(a) has been s t u d i e d from a p r a c t i c a l p o i n t of v.iew .

by Holbrook and McKeown (ho—> mc63). The b a s i c c i r c u i t which

they use (suggested by Bogert (bo55)) i s shown i n F i g u r e 2.4.

This c i r c u i t c o n s i s t s of a tube (or common-emitter t r a n s i s t o r )

with a c a p a c i t i v e l o a d (C^), and a r e s i s t i v e feedback path (R^)*

The R-C i n t e g r a t o r i s formed by r and C^, as shown i n F i g u r e 2.4(b).

I f I^yfl-^ ( t h i s r e q u i r e s g^p^l/R^t then the input impedance

to t h i s c i r c u i t i s given by

R (1+jtfCr ) Z. ^ 2_ , i n (u+l)+j«Cr P

+ V.

R(bias)

(a).

R, •AAr

p <-•AAr-:•. l i n +

•(b)- ^

F i g u r e 2.4 - B a s i c c i r c u i t used by Holbrook and McKeown: (a) a c t u a l c i r c u i t ; (b) approximate e q u i v a l e n t c i r c u i t .

and i f

1 « < o C r p « ([i+l) s

then

Z. ^ j t t j C ^ ) . \ gm/

I t i s seen t h a t the input s i g n a l i s a m p l i f i e d before i t i s

i n t e g r a t e d ; t h i s arrangement might, r e s u l t i n . a low noise l e v e l .

A gainst t h i s , the a m p l i f i e r must be capable .of handling, a large,

s i g n a l .

The s i n g l e - s t a g e t r i o d e and t r a n s i s t o r c i r c u i t s s t u d i e d by

Holbrook and McKeown e x h i b i t e d a maximum'Q of l e s s than 2, while

a pentode c i r c u i t had a maximum Q of about 5. Two and 3-trans-

i s t o r c i r c u i t s had maximum Q's of about 5 and 8 r e s p e c t i v e l y ,

while a 3 - t r a n s i s t o r c i r c u i t , employing 2 of the t r a n s i s t o r s to

produce a negative r e s i s t a n c e , i s repor t e d to produce "high values

of Q f a c t o r , up to the p o i n t of i n s t a b i l i t y and o s c i l l a t i o n . "

Most of the measurements were taken at 1000 c.p.s. 1, with values

of C^ ranging from 0.01 to 1.0 |xf.; the r e s u l t i n g inductances

were of the order of 1 henry.

For the o p e r a t i o n a l a m p l i f i e r i n t e g r a t o r of Fig u r e 2.1(b),

one need merely connect a r e s i s t a n c e from the output t e r m i n a l

2 to the. i n p u t t e r m i n a l 1, as shown i n Figure 2.5. The input

impedance w i l l be .predominantly- inductive, i f ( i ) : the ; i n t e g r a t i n g j .

c i r c u i t draws n e g l i g i b l e current- •(•I^vV^,^'":i'.^)V''&rid-:(dl.)vy^<^:yî;.*

This c i r c u i t was used by Midgley and Stewart (mi6.0), who. *

simulated inductances of the order of 100 h e n r i e s , at •power ••''.'

f r e q u e n c i e s . Even i f the o p e r a t i o n a l a m p l i f i e r has an i n f i n i t e

12

F i g u r e 2*5 - Midgley and Stewart's simulated inductance c i r c u i t .

g a i n , t h e s i m u l a t e d i n d u c t a n c e i s . s h u n t e d b y . a r e s i s t a n c e 5 a s e c o n d

o p e r a t i o n a l a m p l i f i e r c a n be u s e d t o ' c a n c e l : t h i s , r e s i s t a n c e ^ How

e v e r , i f t h e g a i n o f t h e f i r s t o p e r a t i o n a l a m p l i f i e r i s f i n i t e , a

s e r i e s r e s i s t a n c e a l s o a p p e a r s ; one c a n c o m p e n s a t e f o r t h i s b y

u s i n g p o s i t i v e f e e d b a c k w i t h i n . t h e o p e r a t i o n a l a m p l i f i e r : t o

i n c r e a s e i t s g a i n . '. •>.• •.•'•

2 . 2 . 2 D i f f e r e n t i a t i n g c i r c u i t . , ' , \

The c i r c u i t e q u a t i o n f o r a n i n d u c t o r may be w r i t t e n

di-j-'' , V-j^ = Ii — 4 .0 * ( 2 . 2 )

G i v e n a n e m p t y , t w o - t e r m i n a l , , b l a c k b o x w i t h t h e c u r r e n t i - ^

i n j e c t e d i n t o one t e r m i n a l a n d o u t t h e o t h e r , w h a t c a n be p u t

i n s i d e t h e b o x i n o r d e r t h a t t h e i n p u t v o l t a g e be g i v e n b y ( 2 i 2 ) ?

I f i - ^ f l o w s t h r o u g h a s m a l l s e r i e s r e s i s t o r c o n n e c t e d

13 i n s i d e the b l a c k box, a v o l t a g e v ' = Rî-j^ w i l l - appear across R^.

A d i f f e r e n t i a t i n g c i r c u i t can then be used to develop a v o l t a g e

dv, ' d i v = K — ^ - = KR, —

0 dt L dt

I f now K ' V q (where K ' v J ^ j ^ v ^ ' ) i s made to appear i n s e r i e s with

R^ across the input t e r m i n a l s of the box, the t e r m i n a l impedance

w i l l be i n d u c t i v e .

Common ways of d i f f e r e n t i a t i n g use e i t h e r a simple R=C

d i f f e r e n t i a t o r or an o p e r a t i o n a l a m p l i f i e r d i f f e r e n t i a t o r ,

(Figure 2.6). I t i s assumed t h a t the d i f f e r e n t i a t i n g c i r c u i t s

F i g u r e 2.6 - (a) The simple R-C d i f f e r e n t i a t o r j and (b) the o p e r a t i o n a l a m p l i f i e r d i f f e r e n t i a t o r , shown w i t h a

s m a l l r e s i s t o r R^ connected (to produce v^' from the current i-j ) .

provide l i t t l e l o a d i n g on R^.

/ The simple R-C d i f f e r e n t i a t o r was used by Towner (to63)

f o r h i s a r t i f i c i a l i n d u c t o r . ' He solved the problem of connecting

V q ( a m p l i f i e d ) i n s e r i e s with R^, by u s i n g what Millman (mi58)

c a l l s a "bootstrap" c i r c u i t , as shown i n Figure 2.7. The two

a m p l i f i e r s shown diagrammatically, and the R-C d i f f e r e n t i a t o r

between them, produce the a m p l i f i e d V q voltage ( JA-J | - -2! V o ^ '

F i g u r e 2.7 - Diagram of Towner's a r t i f i c i a l i n d u c t o r .

which i s the input to the b o o t s t r a p ( t r i o d e ) c i r c u i t . The out

put of the b o o t s t r a p c i r c u i t appears a c r o s s H ^ , and i s thus i n

s e r i e s with R^, as r e q u i r e d . Towner has " e a s i l y obtained* an

inductance of 10,000 henries with such a c i r c u i t , and although

the frequency range was l i m i t e d to below 500 cps by the p a r t i c u l a r

components used, he says the upper l i m i t can be r a i s e d

i n d e f i n i t e l y . I f a high—Q inductance i s d e s i r e d , a r e s i s t o r can

be used to produce p o s i t i v e feedback from the output of the

second a m p l i f i e r to the input of the f i r s t .

No p u b l i s h e d c i r c u i t based upon the o p e r a t i o n a l a m p l i f i e r

d i f f e r e n t i a t o r of F i g u r e 2«6(b) was.found, but the c i r c u i t shown

i n F i g u r e 2.8 might be used.

An - ^ ( C R l V -

F i g u r e 2.8 - An inductance s i m u l a t i o n c i r c u i t which uses the o p e r a t i o n a l a m p l i f i e r d i f f e r e n t i a t o r of Figure 2.6(b).

15 The r e s u l t s of t h i s s e c t i o n are summarized i n a t a b l e ,

Figure 2.9.

C l a s s i f i c a t i o n

I n t e g r a t i n g c i r c u i t D i f f e r e n t i a t i n g c i r c u i t C l a s s i f i c a t i o n

Simple R-C O p e r a t i o n a l A m p l i f i e r Simple R-C Operational

A m p l i f i e r

Proposed

by

-Hund, Figure 2,2, (hu42, wa47).

-Fulenwider, (fu6 0 ) .

-Stern, Figure 2.3, ( s t 5 8 ) .

-Midgley and Stewart, Fig u r e 2.5, (mi60).

-Towner, Fi g u r e 2.7, (to53).

-Morin, Fig u r e 2.8, ( t h i s t h e s i s ) .

F i g u r e 2.9 - C l a s s i f i c a t i o n of simulated inductances which use a m p l i f i e r methods.

2.3 R e l a t i o n s h i p Between the Various " A m p l i f i e r Methods" of Inductance Simulation, and the AG.

A l l the c i r c u i t s d i s c u s s e d so f a r c o n t a i n a c a p a c i t o r (the

c a p a c i t o r used i n the i n t e g r a t i n g or d i f f e r e n t i a t i n g c i r c u i t )

p l u s some "e x t e r n a l c i r c u i t r y " . Since a c a p a c i t o r plus t h i s

e x t e r n a l c i r c u i t r y produces anc.inductance, and a c a p a c i t o r . p l u s an

AG produces an inductance, i t may w e l l be asked i f the e x t e r n a l

c i r c u i t r y i s not i n f a c t an AG. I t i s e a s i l y shown that the

answer to t h i s question i s a f f i r m a t i v e , but with some r e s e r

v a t i o n s , because' the c o n d i t i o n s which are necessary to r e a l i z e an

inductance are a l s o necessary i n order t h a t the " e x t e r n a l

c i r c u i t r y " behaves l i k e an AG.

For example, consider the c i r c u i t of F i g u r e 2.3. I f t h i s

c i r c u i t i s thought of as a 4-terminal device (terminals a,b,c,d)

terminated at the output side (terminals c,d) by the c a p a c i t o r

C, then the c u r r e n t i through the c a p a c i t o r i s j u s t the C Etp

current i0 at the output p o r t , ,

16

i ~ = i ^ - D~" (because. v 1 » v 0 ) . ...(2.3) z cap it^ JL <i

i , ^ i (because i - , » i ~ ) 1 c l ^ 2 ...(2.4)

Equations (2.3) and (2.4) are, taken together, i n the form of

equation (1.2). Stern's c i r c u i t , minus h i s c a p a c i t o r , t h e r e f o r e

acts l i k e an AG.

S i m i l a r r e s u l t s can be obtained f o r the other inductance

s i m u l a t i o n c i r c u i t s . In a l l cases, the c o n d i t i o n s f o r the

" e x t e r n a l c i r c u i t r y " to behave l i k e an AG are the same as the con

d i t i o n s f o r o b t a i n i n g an inductance.

Al s o ,

17 3. GYRATOR AND ACTIVE GYRATOR THEORY AND APPLICATIONS

The f i r s t s e c t i o n i s a survey of the gyrator l i t e r a t u r e .

In the 2nd s e c t i o n , the gyrator symbols i n common use are given.

Then, the more general " a c t i v e g y r a t o r " (AG) i s d e f i n e d , and some

of i t s p r o p e r t i e s are shown. F i n a l l y , s e v e r a l a p p l i c a t i o n s Of

the AG are given - inductance s i m u l a t i o n , use as an i s o l a t o r , and

use as a c i r c u l a t o r .

3.1 Survey of Gyrator L i t e r a t u r e .

In 1948, T e l l e g e n (te48a) proposed a new, a n t i - r e c i p r o c a l

network element, which he c a l l e d the "gyrator". The gy r a t o r

i s an i d e a l i z e d element, which has the impedance matrix

0 -R

R 0

as was shown i n Fig u r e 1*1. The gyrator produces zero phase s h i f t

f o r t r a n s m i s s i o n i n the forward d i r e c t i o n , and a 180° phase s h i f t

f o r t r a n s m i s s i o n i n the reverse d i r e c t i o n . Among other t h i n g s ,

i t can be used to simulate inductance, and to make i s o l a t o r s and

c i r c u l a t o r s .

Two years before, McMillan had publi s h e d a paper (mc46) i n

which he d e s c r i b e d a 4-terminal " a n t i - r e c i p r o c a l box". This

device was r e a l l y a non-ideal g y r a t o r , f o r i t d i f f e r e d from the

gyr a t o r only i n t h a t the diagonal elements i n i t s impedance matrix

were not zero.

18

The problem of r e a l i z i n g ajj'gyrator has r e c e i v e d c o n s i d e r

able a t t e n t i o n , s i n c e , u n l i k e the r e s i s t o r , c a p a c i t o r , i n d u c t o r ,

and transformer, the gyrator i s not something which can e a s i l y

be made i n the workshop. Concurrently, much t h e o r e t i c a l work

has been done, and much has been w r i t t e n on network sy n t h e s i s

u s i n g g y r a t o r s .

The problem of r e a l i z a t i o n has been approached i n many

d i f f e r e n t ways: e l e c t r o m e c h a n i c a l , H a l l e f f e c t , f e r r i t e , tube,

t r a n s i s t o r , f i e l d - e f f e c t , and parametric devices have a l l been

used.

McMillan's " a n t i - r e c i p r o c a l box" (mc46), c o n s i s t i n g of

2 coupled e l e c t r o m e c h a n i c a l t r a n s d u c e r s , i s an approximate r e a l i z

a t i o n of a g y r a t o r , although the two self-impedance terms are not

zero. I t has been represented (b!53) by an e q u i v a l e n t c i r c u i t

which contains a g y r a t o r . Several papers have been p u b l i s h e d

i n the l a s t few years on p i e z o e l e c t r i c - p i e z o m a g r i e t i c g y r a t o r s

(7 references ). These coupled transducers have also been used

to make i s o l a t o r s (9 r e f e r e n c e s ) and c i r c u l a t o r s ( s i 6 2 b ) . Another

ele c t r o m e c h a n i c a l device, the g e n e r a l i z e d machine, has a l s o been

used as a gyrator (pe 58) .

McMillan (mc47) suggested u s i n g the H a l l e f f e c t to o b t a i n

an " a n t i - r e c i p r o c a l box" (a non-ideal g y r a t o r ) , and s e v e r a l

papers have since appeared on H a l l e f f e c t g y r a t o r s , i s o l a t o r s , and

c i r c u l a t o r s (9 r e f e r e n c e s ) .

Gyrators have been r e a l i z e d most s u c c e s s f u l l y f o r the micro

wave range of f r e q u e n c i e s . A microwave gyrator c o n s i s t s of a

piece of f e r r i t e m a t e r i a l i n s e r t e d i n s i d e a waveguide i n a b i a s i n g

These re f e r e n c e s are l i s t e d under the appropriate heading i n the b i b l i o g r a p h y , chapter 7.

19

magnetic f i e l d . The f i r s t s u c c e s s f u l microwave g y r a t o r was

reported by Hogan (ho52), who has subsequently given an e x c e l l e n t

summary of microwave g y r a t o r s , e t c . (ho56). Many other papers on

microwave gyrators have been publis h e d , some of which are

i n c l u d e d i n the b i b l i o g r a p h y (about 20 r e f e r e n c e s ) . The s p e c i a l

f e r r i t e s issue of the Proceedings of the IRE, October, 1956,

contains many r e f e r e n c e s *

Comparatively few papers have been w r i t t e n oh tube and

t r a n s i s t o r g y r a t o r s . Shekel (sh53) has proposed a 3-terminal

gyrator (the 4th t e r m i n a l i s e l i m i n a t e d by connecting b and d

together i n Figure 1.1,, page 1 ), and shown that any non-re

c i p r o c a l 3-terminal element, such as a vacuum tube or t r a n s i s t o r ,

may be represented by a 3-terminal g y r a t o r i n p a r a l l e l with a

t r i a n g l e of admittances; these admittances may be " s t r i p p e d " by

admittances of the opposite s i g n to leave the gyrator ( t h i s

process w i l l i n v o l v e at l e a s t one negative r e s i s t a n c e i f the non-

r e c i p r o c a l element i s a c t i v e •( sh'54) )". As an example of t h i s

method, Shekel shows how a gyrator can be b u i l t u s ing a vacuum ..'

tube and 3 conductances, 2 of which are negative. Shekel's c i r

c u i t (sh53, F i g . 4) i s e s s e n t i a l l y the same as one given by

Bogert (bo55, F i g . 3 ( d ) ) , who uses an a m p l i f i e r to provide the

two negative conductances, although the equivalence i s not at a l l

obvious. In a l l , Bogert gives 4 gyrator c i r c u i t s , each of which

contains 2 low-gain a m p l i f i e r s . Nonnenmacher (no54) has proposed

a c i r c u i t c o n t a i n i n g 2 low-gain a m p l i f i e r s and 2 transformers,

which can be used, among other t h i n g s , to make a g y r a t o r . de P i a n

(pi62) mentions Shekel's method of s y n t h e s i z i n g a gyrator with a

vacuum tube.

20 A f i e l d e f f e c t tetrode i s a 4-terminal semiconductor device

(et62, st59a, st59b, s t 6 l ) , with the r e p r e s e n t a t i v e c r o s s - s e c t i o n

shown i n Figure 3.1. Under proper b i a s i n g c o n d i t i o n s , i t behaves

Figure 3.1 - Representative cross s e c t i o n of a f i e l d e f f e c t t e t r o d e .

l i k e a g y r a t o r . Prototype f i e l d - e f f e c t t e trodes have been made

with d i f f i c u l t y , but i t i s hoped that t h e i r f a b r i c a t i o n w i l l be

Parametric devices have been d e s c r i b e d (ka60, ko6l) which

can be adjusted to perform as g y r a t o r s .

The foreg o i n g methods of r e a l i z i n g g y r a t o r s are compared

i n the t a b l e of Fi g u r e 3.2. This t a b l e i s intettded only as a

rough guide.

3.2 Gyrator Symbols.

In t h i s s e c t i o n , the " i d e a l g y r a t o r " as proposed by T e l -

legen i s considered. I t s equations may be w r i t t e n :

ohmic contacts. a, b, c, and d are

e a s i e r using e p i t a x i a l growth techniques (et62).

0 -R ... (3.1a1)

V R 0 I

21 Me^^F thod o R e a l i z

eature ^Frequency * f \ ^ j Range ation>J

How Close to I deal?

Accessory Equipment Required

Other Features

E l e c

t r o

ns ch-

a n i c -

a l

Coup

l e d

Trans

ducers

<v400cps .-lMc. Passband i s <30$,. probably

10$.

As i s o l a t o r : about 20db. minimum i s o l a t i o n , with 2db. minimum f o r ward l o s s , over e^lQfo passband.

None; These devices, being resonant mechanical s t r u c t u r e s , e x h i b i t narrow passbands. The temperature dependence from -40 to +80°C i s shown i n s i 6 1 .

E l e c

t r o

ns ch-

a n i c -

a l Genera l i z e d Machine

d. c. only. S e l f - a d m i t tances t£.15c7° of t r a n s f e r admittances.

D r i v i n g motor.

Rotor s e l f i n ductance, brush n o i s e , and s e l f speed voltages< r e s t r i c t p e r f o r mance to d.c.

H a l l

E f f e c t

d.c. to lOKmc. (ma53). Upper frequency l i m i t e d by lead attachment (gr59a).

As i s o l a t o r : 44db. i s o l a t i o n , with 17db. f o r ward l o s s (gr59a).

Minimum t h e o r e t i c a l power l o s s = 7.66db. (wi54).

Magnetic b i a s i n g f i e l d ; , A/1 5,000 gauss f o r germanium

C a l l e d " r e s i s t a n c e -g y r a t o r " by Mason et a l (ma53) to denote that i t i s "a n o n - r e c i p r o c a l element of the r e s i s t i v e type."

F e r r i t e

Lower l i m i t of about 0.2-1.OKmc. Upper l i m i t = ? (>15Kmc. at l e a s t ) .

I n s e r t i o n l o s s e s fi^ldb.

Magnetic b i a s i n g f i e l d ; ~3,000 gauss?

Commercially a v a i l a b l e .

Tube and

T r a n s i s t o r

d . c . -s e v e r a l Kmc .

Idealness l i m i t e d only by s t a b i l i t y problems.

Power supply.

The c i r c u i t s u t i l i z e conventi o n a l components.

F i e l d

E f f e c t

1 K c -lOOMc. (et62).

Inductances can be simu l a t e d with Q l i m i t e d only by s t a b i l i t y (et62).

Power supply f o r b i a s i n g (2 v o l t ages r e quired) .

F a b r i c a t i o n d i f f i c u l t -E p i t a x i a l techniques look promising.

Parametric

C i r c u l a t ors r e ported at lOOKc., & at 500Mc.

Loss £=i l d b . I s o l a t o r s with 20-45 db. I s o l a t i o n ,

Quadrature pumping sources.

Bandwidths of ifo are reported, although 30$ i s hoped f o r .

F i g u r e 3.2 - Comparison of g y r a t o r r e a l i z a t i o n methods.

2 2

or

V " 0 1

G

_X2. _-G 0_ - V2.

..•(3.1b)

.where R = l/G* . a. (3.1c)

R i s c a l l e d the " g y r a t i o n r e s i s t a n c e " . The. symbol proposed by

T e l l e g e n f o r the i d e a l g y r a t o r i s given i n Figure 3.3.

+ A O -IV a

b O -

) c d - O

+

F i g u r e 3.3

s t 5 7 c ) .

T e l l e g e n t s symbol f o r the i d e a l g y r a t o r . Sometimes one of the s e m i c i r c l e s i s omitted (e

An a l t e r n a t e symbol, which c l e a r l y shows the a n t i -

r e c i p r o c a l nature of the gy r a t o r , has been proposed by F e l d -

k e l l e r ( f e 54). This symbol i s shown i n Figure 3.4. S i g n a l s

a O --5K

b O - j f— O d

Figure 3.4 - Gyrator symbol proposed by F e l d k e l l e r .

23 t r a v e l l i n g from r i g h t to l e f t s u f f e r a phase i n v e r s i o n , while

those t r a v e l l i n g i n the opposite d i r e c t i o n do not.

In p r a c t i c e , g yrators are o f t e n b u i l t and/or used w i t h t e r

minals b and d common (Figure 3.3), that i s , as 3-terminal dev

i c e s . For such a 3-terminal g y r a t o r , the i n d e f i n i t e admittance

matrix can be w r i t t e n ( l i 6 l ) from (3.1b):

0 G -G*

i n d e f i n i t e -G 0 G . *>« . (3.2)

G -G 0

Depending upon which t e r m i n a l i s grounded, c r o s s i n g out of the

corresponding row and column y i e l d s the usual 2X2 admittance

matrix. No matter which t e r m i n a l i s grounded, the c i r c u i t has the

same matrix (with a p o s s i b l e change of s i g n , corresponding to an

interchange of input and output t e r m i n a l s ) , as pointed out by

Shekel (sh53). In view of t h i s p roperty, Shekel proposed the

symbol shown i n Figure 3*5 f o r the 3-terminal g y r a t o r . The

Figu r e 3.5 - Shekel's symbol f o r the 3-terminal g y r a t o r .

d i r e c t i o n of the arrow i n d i c a t e s t h at each Y.. term i s e i t h e r

+G or -G, according to whether j—>k i s i n the same or opposite d i r e c t i o n as the arrow r e s p e c t i v e l y ( j , k , = 1,2, or 3, and j ^ k ) .

24 A f o u r t h symbol was proposed by Hogan (ho52) f o r h i s micro

wave g y r a t o r . T h i s symbol, shown i n Figure 3.6, emphasizes the

• % • %

F i g u r e 3.6 - Hogan's symbol f o r the (microwave) g y r a t o r .

1 8 0 ° phase s h i f t i n one d i r e c t i o n ; the i n t e r p r e t a t i o n i s that the

element produces a 1 8 0 ° phase s h i f t f o r t r a n s m i s s i o n i n the

d i r e c t i o n of the arrow, and a zero phase s h i f t i n the opposite

d i r e c t i o n .

3.3 D e f i n i t i o n of an A c t i v e Gyrator.

A " g e n e r a l i z e d g y r a t o r " has been considered by Nonnenmacher

(no54), who r e p l a c e d the g y r a t i o n r e s i s t a n c e s R by a r b i t r a r y

complex impedances, Zffl and Z .. The case where these Imgedances are

pure r e s i s t a n c e s , R m and R n, i s of p a r t i c u l a r i n t e r e s t f o r

inductance s i m u l a t i o n :

" v l ~ "o -R ~ n

J2_ R m

0 . I 2 .

. . . (3.3)

Because of i t s a c t i v e property (see s e c t i o n 3.4), t h i s device w i l l

be c a l l e d an " a c t i v e g y r a t o r " , or "AG" f o r short. S e t t i n g

G = l/R , and G = l/R , m • m' n n' ...(3.4)

25 the Y-matrix equation f o r the AG can be w r i t t e n ,

•..(3.5)

F e l d k e l l e r ' s g y r a t o r symbol (Figure 3.4) w i l l be used

f o r the AG, since i t g i v e s more i n f o r m a t i o n than Tellegen's

symbol, and since Shekel's symbol i s c l e a r l y u n s u i t e d to the AG.

See F i g u r e 3.7.

F i g u r e 3.7 - C i r c u i t symbol and equations f o r the AG.

3.4 The A c t i v i t y of the A c t i v e Gyrator.

With regard to such things as phase i n v e r s i o n , i n t e r c h a n g i n g

p o r t s , cascading 2 u n i t s , land impedance i n v e r s i o n , the AG has

p r o p e r t i e s c l o s e l y resembling those of the g y r a t o r . Consequently,

these t o p i c s are not considered f u r t h e r . In t h i s s e c t i o n , the

a c t i v i t y of the AG i s considered.

7 *

This f a c t i s e a s i l y seen by comparing the i n d e f i n i t e Y-matrix f o r the AG with t h a t f o r the g y r a t o r (equation ( 3 . 2 ) ) .

26 The f o l l o w i n g c o n d i t i o n s are used f o r the p a s s i v i t y of a

l i n e a r 2 - t e r m i n a l - p a i r network (see, f o r example, mc46 or bo57):

R l l > 0

R 2 2 ^ " 0 . . . (3 . 6a,b ,

4 R 1 1 B 2 2 r- ( Z 1 2 + Z 2 | ) ( Z 1 * + Z 2 1 ) > 0,

where the impedances r e f e r to the Z matrix.

Comparing (3.6) with (3.3) , i t i s seen that the c o n d i t i o n s

of (3.6a) and(3.6b) are s a t i s f i e d with the equal s i g n , and the

c o n d i t i o n of (3.6c) becomes

-(-R + R ) 2 > - 0 . n m ^

Since R ^ R , the AG i s not p a s s i v e ; i . e . i t i s a c t i v e .

3.5 A p p l i c a t i o n s of the AG.

Three a p p l i c a t i o n s are considered i n t h i s sections

1) s i m u l a t i o n of inductance,

2) use as an i s o l a t o r , and

3) . use as a c i r c u l a t o r .

3.5.1 S i m u l a t i o n of inductance.

I f an AG i s terminated by a capacitance, Z^ = l / ( j w C ) , then

the input impedance (see Chapter l) i s given by

Z. = R R /Z T = j«(R R C), m m n' L 0 m n ' '

; 27 which represents a simulated inductance,

L = R.R C« «•.(3»7)

m- n

This r e s u l t i s of. much p r a c t i c a l i n t e r e s t i n m i c r o e l e c t r o n i c s ,

f o r i f a t i n y AG can be b u i l t , then a t i n y inductance can be

obtained by merely adding a c a p a c i t o r . In a d d i t i o n to being very

s m a l l , such a simulated i n d u c t o r woul<J produce a much (smaller

magnetic f i e l d than a conventional i n d u c t o r , and consequently

s h i e l d i n g problems would be minimized.

3.5.2 The AG used as an i s o l a t o r .

I t i s shown here t h a t the AG can be uarad to make an

i s o l a t o r , and t h i s i s o l a t o r can have a power gain g r e a t e r than

u n i t y i f G > G . J n ' m

The element i n the Y matrix of an i d e a l i s o l a t o r must be

zero (to provide i s o l a t i o n i n the reverse d i r e c t i o n ) , while the Y21 e l e m e n t m u s t be non-zero (to provide t r a n s m i s s i o n i n the

forward d i r e c t i o n ) . The Y ^ an^ ^22 e-'- e m e n"' > s a r e then the input

admittances at the two p o r t s , since

T i n l = Y l l " T 1 2 Y 2 1 / / ^ 2 2 + T L ^

= Y^^ since Y ^ — 0»

and s i m i l a r l y ^-^n2 ~ Y 2 2 ' ^ n u s these elements should provide a

match with the input and output l i n e s .

Two ways of making an i s o l a t o r from an AG and a r e s i s t o r

are shown i n Fig u r e 3.8.

28

„ , „ "X . — - — _ •

0 - R ' n

R 0 . m -J

I R

(b)

V

F i g u r e 3 . 8

I n F i p r e

G (G - G) m

(G+G ) G n

V ,

L V 2 J

R

R + R

R - R n

m R

( b ) s e r i e s c o n n e c t e d .

I n F i g u r e 3 . 8 ( a ) , t h e Y . ^ e l e m e n t w i l l be z e r o i f G m = G . The

r e s u l t a n t Y m a t r i x i s

Y =•

m

- ( G + G ) : n

0

G

F o r a p e r f e c t m a t c h , G, s h o u l d e q u a l t h e c h a r a c t e r i s t i c a d m i t -

" *

t a n c e o f t h e l i n e i n w h i c h t h e i s o l a t o r i s i n s e r t e d .

The e q u i v a l e n t c i r c u i t o f t h i s i s o l a t o r i s s h o w n i n F i g u r e

3 . 9 , a n d t h e p o w e r g a i n ( m a t c h e d ) i s

o u t P .

i n

(G + G ) '

4 G 2

. . . ( 3 . 8 )

T h i s p o w e r g a i n w i l l be g r e a t e r t h a n u n i t y i f G> i . e . , i f

G > G . n ^ m

I t i s a s s u m e d t h a t t h e i n p u t a n d o u t p u t l i n e s h a v e t h e same c h a r a c t e r i s t i c a d m i t t a n c e .

G.

1 r -I

O r

| V / G

G .= G m

P , = V J " G = out I 2' ,2 „ N 2 ( G + GJ'

4G

29

i n

F igure 3.9 - E q u i v a l e n t c i r c u i t f o r the i s o l a t o r of Figure 3..8(a).

For the c o n f i g u r a t i o n of Figure 3.8(b), R = R f o r p e r f e c t n i s o l a t i o n , and the (matched) power ga i n i s

P , (R + R ) out m' P.

m 4R£

...(3.9)

3.5.3 The AG used as a c i r c u l a t o r .

I t i s shown here that an AG can be used as a c i r c u l a t o r i f

the1 t e r m i n a t i n g impedances are p r o p e r l y chosen. Such a c i r c u l a t o r

e x h i b i t s a power gain , which v a r i e s with the r a t i o G M/G .

a ) I n t r o d u c t i o n

A 3-port network can be made from a 3-terminal d e v i c e , as

\ shown i n Figure 3.10. I f the 3-terminal device and th6 t e r m i n

a t i o n s at each p o r t are s u i t a b l y chosen, such a network may

30 Port b

Port a Port c

Fig u r e 3c10 - A 3-port network made from a 3-terminal d e v i c e .

behave l i k e a c i r c u l a t o r ( s i 6 2 b ) . Such a c i r c u l a t o r would be

u s e f u l , f o r i n s t a n c e , f o r simultaneously sending and r e c e i v i n g

audio-frequency s i g n a l s over a s i n g l e p a i r of wir e s , or f o r

r e p l a c i n g the h y b r i d c o i l s i n two-way, two-wire a m p l i f i e r s .

b) C o n d i t i o n s f o r c i r c u l a t i o n

The ports shown i n Figure 3..10 may be d r i v e n by e i t h e r

v o l t a g e or current sources, as shown i n Figure 3.11.

For each case, a given d i r e c t i o n of c i r c u l a t i o n i s achieved

i f c e r t a i n elements of the a s s o c i a t e d 3 x 3 matrix ( d e f i n e d

s e p a r a t e l y f o r each f i g u r e ) are set to zero. These matrix

elements can be made zero by s u i t a b l e choice of the te r m i n a t i n g

(source) immitbances. These immittances are of course independent

of whether vol t a g e or current sources are used, so the a n a l y s i s

need be c a r r i e d out f o r only one of F i g u r e s 3.11(a) and ( b ) .

Since the a s s o c i a t e d matrix of Figure 3.11(b) can be d e r i v e d more

simply, t h i s c o n f i g u r a t i o n w i l l be analysed.

a'

\ Y " 12 /

- T 2 1 22j .A/

V a

C / C (a)

3.1

a

L c j

aa Y , Y ab £ Y Y Y ba bb x l Y

J - ca Y . Y cb c

V V " a

1**

Z aa âb Z "

ac I a

V V b ^ba Z,,

bb Z b c Xb

V V L cJ

Z - ca cb Z I

L c j

F i g u r e 3.11 - The two b a s i c c i r c u l a t o r c o n f i g u r a t i o n s : (a) v o l t age sources, and (b) current sources.

F i r s t , the c o n d i t i o n s f o r c i r c u l a t i o n w i l l be s t a t e d . For

c i r c u l a t i o n i n the d i r e c t i o n a - H>b-^-c, the a s s o c i a t e d matrix of

F i g u r e 3.11(b) must have the form

" V ~ a

V b =

V L c _

Z aa Jba

0

0

Jbb

ac 0

Z cb CC-!

a ...(3.10)

For c i r c u l a t i o n i n the opposite d i r e c t i o n , c

a s s o c i a t e d matrix must have the form

•a, the

V a

Z aa ab 0 I

a V b 0 Z..

bb be h V _ c _

Z ca

0 Z cc J

I c

.. . (3.11)

The a s s o c i a t e d matrix d e f i n e d i n Figure 3.1l(b) w i l l now be

d e r i v e d , i n terms of the Y parameters d e f i n e d i n the same f i g u r e .

The c i r c u i t of Figure 3.11(b) can be formed by connecting two

32 c i r c u i t s i n p a r a l l e l , as shown i n Figure 3.12. The r e s u l t a n t

i n p a r a l l e l .j

with

11

•21

•12

'22.

a b b

-Y, Y, +Y b b c_

Y, , +Y +T. Y, 0-T. 11 a b 1 2 b

L Y 2 1 - Y b Y 2 2 + Y b + Y c

Figure 3.12 - The f i r s t step i n d e r i v i n g the matrix d e f i n e d i n Figure 3.11(b).

Y matrix can be i n v e r t e d .

AY

Y 0 0 + Y, + Y 22 b c

Y - Y b 21

Y - Y x b x12

Y,, + Y + Y, 11 a b

...(3.12)

where AY = ( Y ^ + J& + X b ) ( T 2 2 .+ Yfc + Y c ) - ( Y 2 1 - t b ) ( T 1 2 - T b ) .

Hence the i n d e f i n i t e Z matrix ( l i 6 l ) i s e a s i l y w r i t t e n ,

"v a v 1 Vb ~ AY V . c.

Y 2 2 + T b + T c - ( Y 1 2 + Y 2 2 + Y c ) - ( Y 2 1 + Y 2 2 + Y c ) Y n + Y 2 2 + Y 1 2 + Y 2 1 + Y a + Y c

x21 b

< T 1 1 + I 1 2 + Ia >

a

...(3.13)

33 and t h i s i s i n f a c t the matrix d e f i n e d i n Figure 3.11(b).

Now, comparing (3.10) with (3.13), the c o n d i t i o n s f o r

c i r c u l a t i o n i n the d i r e c t i o n a — » b — ^ c are

Y - Y Lb ~ x21

Y c = -< T12 +" T22>'

Conditions f o r c i r c u l a t i o n i n d i r e c t i o n a—*>b—>c.

. . .(3.14)

Comparing (3.11) with (3.13)» the c o n d i t i o n s f o r c i r c u l a t i o n i n

the d i r e c t i o n c —>b—>a are

T a = "(Til + V

Y - Y Xb _ x12

Y c = "<Y21 + : T 2 2 > - J

Conditions f o r c i r c u l a t i o n i n d i r e c t i o n c — * b — » a .

..(3.15)

For the AG, T - , = Y 0 0 = 0 , Y, 0 = G , and T _ N •= -G ; there-1 1 - 2 2 12 m 21 n

f o r e , (3.14) and (3.15) become

a -G m

Y, =. -G b n

Y = -G , c m'

Conditions f o r c i r c u l a t i o n i n d i r e c t i o n a — > b — ^ c with AG,

...(3.14a)

34 and

Y = G a n

Y. = G b m

c n

Conditions f o r c i r c u l a t i o n i n d i r e c t i o n c—»>b — ^ a with AG.

...(3.15a)

For c i r c u l a t i o n i n the d i r e c t i o n a - ^ b — > c with the AG,

negative source immittances are r e q u i r e d . This r e s u l t might be

u s e f u l when i t i s d e s i r e d to use a c i r c u l a t o r i n c o n j u n c t i o n with

an a m p l i f i e r having a negative input and/or output impedance.

For . c i r c u l a t i o n i n the d i r e c t i o n c—s»b—*-a, the r e q u i r e d

source immittances are a l l p o s i t i v e . Suppose such a c i r c u l a t o r

i s o p e r a t i n g with the.source immittances given by (3.15a).. Are

the 3 po r t s simultaneously matched? Can a net power g a i n be

achieved?

c) Matching

The s e l f terms i n (3.13) are the o p e n - c i r c u i t i nput

impedances ( i n c l u d i n g the source immittances) at the r e s p e c t i v e

p o r t s i n Fig u r e 3.11(b);. Therefore, the admittance seen by

each source i s gi v e n by the r e c i p r o c a l of the r e s p e c t i v e s e l f

term minus the source admittance. This r e s u l t i s summarized i n

F i g u r e s 3.13(a) and (b), i n •which the source admittances have

been chosen according to (3.14) and (3.15), thereby producing ; 1

c i r c u l a t i o n i n the d i r e c t i o n s a f » b - » c and c->-b —:>a

r e s p e c t i v e l y . The r e s u l t s obtained when an AG i s used are given

i n (c) and (d) of the same f i g u r e ; note t h a t , i n t h i s c

35

Figure 3.13 - (a) aad (b) Input admittances to the c i r c u l a t o r shown i n Figure 3*11 (b); (c) and (d») values of input adm

i t t a n c e obtained when an AG i s used. The d i r e c t i o n of c i r c u l a t i o n i s i n d i c a t e d by the arrow.

match cannot be obtained at any port unless Gffl - G n ( i . e . , the AG

becomes a g y r a t o r ) , i n which case a l l 3 ports are matched

simultaneously. ' ... •

In general, however, the 3 ports are a l l mismatched. I t

might be argued that r e f l e c t i o n s caused by these mismatches would

destroy the c i r c u l a t i o n p r o p e r t i e s ; however, unless the sources

are connected by very long l i n e s , or by delay l i n e s , r e f l e c t i o n s

should be n e g l i g i b l e at the low frequencies (long wavelengths)

of i n t e r e s t . Of course* matching transformers cannot be u 3 e d ^

f o r then the e f f e c t i v e t e r m i n a t i n g immittances would no longer

36 s a t i s f y the c o n d i t i o n s f o r c i r c u l a t i o n .

One e f f e c t of the mismatches would be to cause the power

t r a n s f e r s to be l e s s than maximum; the question of power gain

i s considered next.

d) Power gains

For c i r c u l a t i o n i n the d i r e c t i o n c—*»b—> a u s i n g an AG,

(3.13) becomes

a

G + G m n

-1

0

L - i

o

- l

0 1 .

a

I .

... (3.13a)

Hence, i f a c u r r e n t I i s a p p l i e d at p o r t a (while I, = I - 6 ) , a

the power d e l i v e r e d at p o r t c i s given by

2

P c _ U +\ t y m n/ G . n

...(3.16a)

S i m i l a r l y , d r i v i n g ports c and b, r e s p e c t i v e l y ,

1 G ™ > a n d P o G + 0 I m' a m n/

- I , G + G m n n* ...(3.16b,c

Now, the powers a v a i l a b l e from each of the sources (P ) av

are

B/V a 4G

n ; P

[bl 2

.•av. 4G m av I 2 c1

4G n

..(3.17)

37 Therefore, the transducer power gains (G^) i n the d i r e c t i o n of

c i r c u l a t i o n are

4G G G m = G m n

T c - > b T b _ > a (G + G ) 2

m n' . . . (3.18)

G 4G 2 . n

a — ^ c (G + G )' m n

Note t h a t

...(3.19)

G T = G T — 1, and c — ^ b b — a

> G n > Grp = 1 as Q = 1.

a—>-c * m ^

Thus.; a power ga i n can be r e a l i z e d from port a to po r t c i f G " ^ G m «

How much of t h i s power ga i n can be u t i l i z e d ? A t y p i c a l

a p p l i c a t i o n of the c i r c u l a t o r i s shown diagrammatically i n

Figure 3.14. A measure of the power ga i n produced by the

load AAAAA

Port b

Port a

Si g n a l i n / ~ZT , \ , • . •. - • to Port c. t r a n s m i s s i o n l i n e

F i g u r e 3.14 - An a p p l i c a t i o n of a c i r c u l a t o r . , The c i r c l e represents a 3-port c i r c u l a t o r .

c i r c u l a t o r i s the product G ^ . G ^ . This product i s a —:> c c --s»b

38

given by

. G-n b c 1 6 G G ' m n

'T * "T . ~ P P a—>-c c—>b av av a c ( G + G ) ' m n'

....(.3.20)

This f u n c t i o n v a r i e s with G / . G ^ . as. shown i n Figure 3 . 1 5 . The

f u n c t i o n i s gre a t e r than 1 f o r 1 «C G n / G m < ^ 1 1 . 5 , reaching a

maximum of about 1 . 6 9 when G / G ~ 3 . 0 . n m

Fi g u r e 3 . 1 5 - The product G^, » a—*-c • c—^-b

G / G . n m

1 0 1 1 . 5 15

* G™ as a f u n c t i o n of

Another measure of the c i r c u l a t o r ' s power ga i n i s given by

the product G ^ . v.^ . ^ ... G,j, . Grj, . T h i s product v a r i e s with a—>- c c —>b b —>a

with G / G i n a manner s i m i l a r to t h a t shown i n Figure 3 . 1 5 . I t n m 6

i s g r e a t e r than 1 f o r 1 < , G / G ^ 4 . 2 4 , and reaches a maximum

of 1 . 3 9 when G / G = 2 . n m

In summary, the a c t i v e property of the A G may be t^ken ad

vantage of to r e a l i z e a c i r c u l a t o r with a net gain g r e a t e r than

u n i t y .

39

4. CIRCUITS FOR REALIZING ACTIVE GYRATORS

In the l i t e r a t u r e , c i r c u i t s have been d e s c r i b e d (bo55)

which are e s s e n t i a l l y In AG form ( i . e . , which obey 3.5)). These

c i r c u i t s are made up of only two types of elements, the i d e a l

conductance, and the i d e a l i z e d v o l t a g e a m p l i f i e r , as shown i n

Fig u r e 4.1. An approach based on these two elements w i l l be used

Output

G

-AAA"

(a)

F i g u r e 4.1 - The two components which w i l l be used to b u i l d AGs: (a).conductance, and (b) i d e a l i z e d v o l t a g e ampli

f i e r , which has A r e a l , p o s i t i v e or ne g a t i v e .

t o d e s i g n s e v e r a l AG C i r c u i t s .

4.1 I n t r o d u c t i o n .

As was mentioned above, only the two types of elements,

shown i n Fig u r e 4.1 w i l l be used to r e a l i z e AGs. S t a r t i n g with

nothing but the e x t e r n a l AG te r m i n a l s a,b,c,d, as shown i n

Figu r e 4.2, a m p l i f i e r s and conductances w i l l be connected i n s i d e

An analogous i n v e s t i g a t i o n could be made using i d e a l i z e d current a m p l i f i e r s .

40

a

b O

"Black Box"

(empty)

-O c

d -O

Y 1

V 2

}2. . T 2 l ' . Y 22, 2 j

F i g u r e 4.2 - Skeleton of the AG - s t a r t i n g p o i n t f o r each AG design.

the b l a c k box, one by one: each such a d d i t i o n w i l l a l t e r the Y

matrix*, and the aim w i l l be to add components so t h a t the Y

matrix becomes that of an AG,

...(3.5) V ' o G "

m

-G n

0 V L 2J

The f o r e g o i n g i d e a u n d e r l i e s the whole of t h i s chapter. The

success i v e steps are shd|m. i n a b l o c k diagram, Figure 4.3.

4 .4 4. 5 Appendix 4. 6 S e c t i o n * • 4.2 4.3 The two How to Design compon use of a ents of these 3-ampFigure c i r c u i t l i f i e r 4.1 comp AG

onents

U s e f u l i n i t s own :

r i g h t

Clue f o r de signi n g a 2-ampl i f i e r AG

16 configurations Ifor 2-amp-lifier AGs

Analys i s of the 16 configu r a t ions

Be s u i t s of the analys i s , and d i s c u s s i o n

One c i r c u i t p a r t i c u l a r l y u s e f u l f o r inductance s i m u l a t i o n

F i g u r e 4.3 - Successive, steps taken i n the a n a l y s i s of AG c i r c u i t s *

* Whenever the Y matrix i s mentioned i n t h i s chapter, reference i s made to the Y matrix f o r the te r m i n a l s a,b,c,d, as d e f i n e d m Fi g u r e 4.2.

41 4*2 E f f e c t s of the 2 Types of C i r c u i t Components on the T M a t r i x .

The input of a p a r t i c u l a r a m p l i f i e r w i l l be connected b e t

ween 2 of the t e r m i n a l s a,b,c»dj separate conductors w i l l then be

connected between the output t e r m i n a l , and some of the 4 t e r m i n a l s

a,b,c,d, each such conductor forming what w i l l be r e f e r r e d to as

an "output connection". For each a m p l i f i e r , both the input con

ductance and these "output connections" w i l l have an e f f e c t upon

the X matrix, and these two e f f e c t s w i l l be considered s e p a r a t e l y *

4*2,1 Input conductance.

Suppose only the i n p u t t e r m i n a l s of s e v e r a l a m p l i f i e r s

have been connected to c e r t a i n p a i r s of the 4 ter m i n a l s a,b,c,d*

The Y matrix may then be expressed i n terms of the input conduct

ances of the a m p l i f i e r s . A l l the self-admittance elements w i l l

n e c e s s a r i l y be of p o s i t i v e s i g n and the t r a n s f e r admittance

elements w i l l n e c e s s a r i l y be of negative s i g n . Since both s e l f -

admittances must be zero (equation (3.5)), any such s e l f - a d m i t

tance element w i l l have to be c a n c e l l e d somehow by a negative

conductance (see the next paragraph). C o n s i d e r i n g the t r a n s f e r

admittance matrix elements now-, ^2 m u s t be p o s i t i v e , and ^21

negative; thus any (negative) t r a n s f e r admittances caused by

input conductances w i l l be of proper s i g n for,-Y^l* but not f o r Y.^;

consequently, might not have to be a l t e r e d , but w i l l .

: _ . * • • • ;

Some i n t e r e s t i n g exceptions would a r i s e i f a m p l i f i e r s with nega t i v e i n p u t conductance were used*

42 4.2.2 Output connections*

Consider the a m p l i f i e r and conductance connected as i n

F i g u r e 4.4. I f P T were connected to P l , a current I' = V ^ ( l - A ) G

would flow through G, and i f (l-A) « ^ 0 , the e f f e c t would be that

F i g u r e 4.4 - I l l u s t r a t i n g the e f f e c t of output connections,

of a negative conductance connected from a to b. This type of

connection a f f e c t s a self-admittance matrix element (Y-^-for. the

connection shown i n Figure 4.4), and could be used to make a

se l f - a d m i t t a n c e zero, as mentioned i n s e c t i o n 4.2.1 above. On

the other hand, i f P' were connected to P2, a current

I 1 = V^G - VÂG would flow through G. The f i r s t term on the r.h.s

(a s e l f term) i s j u s t the current which would flow i f G were

connected from c to d, and t h e r e f o r e i t s . e f f e c t i s s i m i l a r to that

of an a m p l i f i e r input admittance, as d i s c u s s e d i n s e c t i o n 4.2.1

above. The second term (a t r a n s f e r term) can be v a r i e d both i n

magnitude and s i g n by choice of A, and thus, a connection of t h i s

type a f f o r d s c o n t r o l of a t r a n s f e r admittance matrix element by

choice of A.

43

4.2.3 Negative r e s i s t a n c e

The p o i n t regarding negative conductance (or r e s i s t a n c e )

w i l l be considered i n more d e t a i l . Negative r e s i s t a n c e s (e.g.,

st60; see b i b l i o g r a p h y , page 68) are of two b a s i c types, which

are u s u a l l y c a l l e d the shunt, or s h o r t - c i r c u i t s t a b l e type,

and the s e r i e s , or o p e n - c i r c u i t s t a b l e type. A shunt—type

negative r e s i s t a n c e of value -G- w i l l remain s t a b l e f o r any passive

t e r m i n a t i o n G + jB which has G > G v A s e r i e s - t y p e negative

r e s i s t a n c e of value ''-B w i l l remain s t a b l e f o r any passive

t e r m i n a t i o n R + jX, f o r which R R q . The a m p l i f i e r of Figure

4.1(b) can be used to produce e i t h e r a shunt or a s e r i e s — t y p e

negative r e s i s t a n c e , and c i r c u i t s f o r both types are shown i n

Figu r e 4.5.

R

G G, + U - A ) G 1 0

B]_+ ( l - A ) j i

a) Shunt or s h o r t -c i r c u i t s t a b l e type.

b) S e r i e s , or open-c i r c u i t s t a b l e type.

F i g u r e 4.5 - Negative r e s i s t a n c e s u s i n g v o l t a g e a m p l i f i e r s .

4.3 Design of a 3- A m p l i f i e r AG.

The ideas d i s c u s s e d i n s e c t i o n 4.2.1 w i l l be used to design

an AG. Two a m p l i f i e r s may be connected as shdwn i n Figure 4.6 to

44

a Y 11 12

Y Y 21 22

Fi g u r e 4.6 - The s t a r t of an AG c i r c u i t .

provide c o n t r o l over the t r a n s f e r admittances. As the c i r c u i t

stands, the Y matrix i s

G l + G 5

" A 1 G 4

-A 2G 5

G 2 + G 4

Since t h i s Y matrix must have the form

0

-G _ n

G m

0

A 2 must be negative, and A^ must be p o s i t i v e . Next, the two s e l f -

admittance terms must be made zero, by p r o v i d i n g a negative

conductance (shunt-type) from a to b (to cancel G^ + G,-), and

another from c to d (to cancel G 2 + G^). A m p l i f i e r A^ can be

used f o r the former (since a p o s i t i v e — g a i n a m p l i f i e r i s needed),

45 and a 3rd a m p l i f i e r connected from c to d (using the c o n f i g u r a t i o n

of F i g u r e 4.5(a)) f o r the l a t t e r . The f i n a l c i r c u i t i s shown i n

Fig u r e 4.7.

Fi g u r e 4.7 - 3 - a m p l i f i e r AG.

This AG has s e v e r a l u s e f u l f e a t u r e s . One important f e a t u r e

i s t h a t a l l 3 a m p l i f i e r s have a common ground p o i n t , so only one

power supply i s needed. In a d d i t i o n , t h i s common ground p o i n t

i s connected to the common negative input-output t e r m i n a l of the

AG. (Several 2 - a m p l i f i e r AGs w i l l be designed l a t e r , but none

of them w i l l have t h i s property.) G m and G n can be independently

c o n t r o l l e d f o r t h i s AG, so G ^ G can be obtained. ' m < n

In the development above, i t was assumed that the ampli :'-

fiers a l l had zero output impedance, In a p r a c t i c a l c i r c u i t ,

t h i s w i l l not be t r u e . In p a r t i c u l a r , i f a m p l i f i e r 1 has a f i n i t e

output impedance, the a n a l y s i s w i l l be complicated .

The problem of des i g n i n g a 2 - a m p l i f i e r AG i s Considered

next.

This p o i n t i s considered f u r t h e r i n s e c t i o n 5.1.

46 4.4 Clue f o r Designing 2-Amplifier AGs.

In t h i s s e c t i o n * j u s t i f i c a t i o n i s given f o r c o n s i d e r i n g

only c i r c u i t s i n which b i s connected d i r e c t l y to d; t h e r e f o r e

a l l the c i r c u i t s w i l l have the same "sk e l e t o n " , c o n s i s t i n g of the

3 t e r m i n a l s a,c, and b-d.

In the 3 - a m p l i f i e r c i r c u i t of Figure 4.7 > a m p l i f i e r A-

a m p l i f i e s , and the other two amplify V^. G a n o n e °^ these

a m p l i f i e r s be eliminated? For i n s t a n c e , i f an a m p l i f i e r were

connected to amplify V^-V^, perhaps i t could perform the f u n c t i o n s

of both A^, and one of the other two. I t i s easy to show t h a t

t h i s i s i n f a c t the case, and both A^ and A 2 can be r e p l a c e d

by a s i n g l e +'ve ga i n a m p l i f i e r a m p l i f y i n g - V. ( t h i s c i r c u i t

i s analysed i n Fig u r e 4.16), Here, then, i s a c l u e : use one

a m p l i f i e r to amplify +V^+V2. W i l l t h i s clue l e a d to s t i l l more

2 - a m p l i f i e r AG c i r c u i t s ?

The v o l t a g e s iv'-j.+v^ can e a s i l y be obtained d i r e c t l y from the

4 te r m i n a l s a,b,c,d, as shown i n Figure 4.8. In each of the 4

cases shown, two t e r m i n a l s * one from each p o r t , are connected

+ y V -V, 1 '2

(a)

b o» O d

v. a o =0 c

b O'

(b)

\> d

a a

F i g u r e 4.8 - Obtaining the v o l t a g e s +Vr^±\1^ from the port voltages.

47

t o g e t h e r , and t h e r e q u i r e d v o l t a g e i s t a k e n b e t w e e n t h e o t h e r

two t e r m i n a l s . H o w e v e r , t h e 4 m e t h o d s a r e r e a l l y e q u i v a l e n t f o r ,

g i v e n a n AG b u i l t on one o f t h e s k e l e t o n s o f F i g u r e s 4 . 8 ( b ) t o

( d ) , t h e p o s i t i o n o f . i t s t e r m i n a l s c a n e a s i l y be c h a n g e d so t h e

s k e l e t o n becomes t h a t o f F i g u r e 4 . 8 ( a ) , w h i l e t h e f o r m o f t h e Y

m a t r i x r e m a i n s u n c h a n g e d . A n e x a m p l e i s g i v e n i n F i g u r e 4 . 9 .

o

- G n

G m

0

1 >

> v i

d O

c O

0

a • 2 b

- O

G n

-G 0 am

F i g u r e 4 . 9 - C h a n g i n g t h e s k e l e t o n o f an AG c i r c u i t .

C o n s e q u e n t l y , t h e c o m p l e t e s e t o f AG c o m b i n a t i o n s p o s s i b l e w i t h

t h e s k e l e t o n o f F i g u r e 4 . 8 ( a ) ;> w i l l i n c l u d e a l l t h e p o s s i b l e AG

c i r c u i t s f o r a n y o f t h e o t h e r s k e l e t o n s . T h e r e f o r e , o n l y one

The m a n i p u l a t i o n p r o c e e d s i n 2 s t e p s . F i r s t , r e v e r s e t h e p o l a r i t y a t one p o r t , whence t h e Y m a t r i x 0 G

m - G 0

n

becomes '0

G

- G m

n 0

S e c o n d ,

e i t h e r (a) r e v e r s e t h e p o l a r i t y a t t h e o t h e r p o r t , o r e l s e (b) i n t e r c h a n g e p o r t s , whence t h e Y m a t r i x becomes e i t h e r 0

- G n

G

0

o r

'0 G n

•G 0 m

r e s p e c t i v e l y , and i n e i t h e r c a s e i s o f t h e o r i g i n a l f o r m .

48 sk e l e t o n w i l l be considered, say that of Figure 4.8(a). This i s

# the consequence of our clue .

4.5 C l a s s i f i c a t i o n and P r e l i m i n a r y Screening of . 2-Am p l i f i e r AGs.

In t h i s . s e c t i o n , a l l the p o s s i b l e c i r c u i t s which can

be b u i l t on the s k e l e t o n of Fig u r e 4.8(a) using 2 a m p l i f i e r s

are c l a s s i f i e d , and i t i s shown that only 16 of the 24 p o s s i b l e

types need be considered.

The c i r c u i t s w i l l be c l a s s i f i e d by the way the a m p l i f i e r

inputs are connected. How many u s e f u l ways can the inputs of

two a m p l i f i e r s be connected to the 3 t e r m i n a l s of Figure 4.8(a)?

D i s r e g a r d i n g the p o l a r i t y of the inputs f o r the moment, a con

n e c t i o n between a and c w i l l be designated as "D" ( f o r d i f f e r e n c e ) ,

between a and b-d as "1", and between c and b-d as "2". Then

there are 6 p o s s i b l e ways of connecting the 2 a m p l i f i e r s : DD,

D l , D2, 12, 11, and .22. The l a t t e r 2 a,re e l i m i n a t e d , f o r they

allow only (one of) Y-^ or 22 ^° e s e ^ ^° z e r o •

In order to take p o l a r i t y i n t o account, a, a, and c w i l l

be considered as the p o s i t i v e t e r m i n a l s f o r the D, 1, and 2

connections r e s p e c t i v e l y . Then the input connections, of the two

a m p l i f i e r s may be completely d e s c r i b e d by 4 c h a r a c t e r s , f o r

example "D+1-". .There are th e r e f o r e 4 X 4 = 16 p o s s i b l e ways

of connecting the a m p l i f i e r inputs which might y i e l d an AG.

The problem of s y s t e m a t i c a l l y a n a l y s i n g these. 16 c o n f i g u r a t i o n s

i s considered i n d e t a i l i n the Appendix, and the r e s u l t s of t h i s

a n a l y s i s are contained i n the next s e c t i o n .

The s k e l e t o n was obtained on the b a s i s of using a d i f f e r e n c e a m p l i f i e r ; c i r c u i t s w i l l a l s o be considered which do not have ;

t h i s property (the 1+2+ c o n f i g u r a t i o n s ) . : :.. ,

49 4.6 Results of the Analyses of the 16 C o n f i g u r a t i o n s .

The r e s u l t s of the analyses c a r r i e d out i n the Appendix are

summarized i n a t a b l e , Figure 4.10. From the 1st column of

e n t r i e s , note that 7 of the 16 c o n f i g u r a t i o n s y i e l d an AG, numbers

5, 7, 8, 11, 12, 14, and 16. I t i s easy to show that the l a s t 4

are e q u i v a l e n t to c i r c u i t s g i v e n by Bogert (bo55).

From the 2nd column, note that only one of the 7 AGs,

c o n f i g u r a t i o n riuniber 8, has a common ground f o r both a m p l i f i e r s .

This i s a d e s i r a b l e f e a t u r e i n a p r a c t i c a l c i r c u i t , f o r otherwise

2 separate power s u p p l i e s would be needed.

From the 3rd column of e n t r i e s , note that c o n f i g u r a t i o n num

ber 8 r e q u i r e s one + 've and one —'ve gai n a m p l i f i e r , while the.

other 6 c o n f i g u r a t i o n s r e q u i r e both a m p l i f i e r s to have a +'ve gain.

In p r a c t i c e , a -t-'ve gain a m p l i f i e r may r e q u i r e 2 stages of amp

l i f i c a t i o n , while a -'ve gai n a m p l i f i e r r e q u i r e s only 1, and

thus only 3 stages of a m p l i f i c a t i o n are r e q u i r e d f o r c o n f i g u r a t i o n

number 8, while the others r e q u i r e 4. i

From the 4th column of e n t r i e s , note that most of the f e a s i b l e

c o n f i g u r a t i o n s r e q u i r e only one output connection per a m p l i f i e r ,

except f o r number 8, which r e q u i r e s 2 f o r the +'ve g a i n a m p l i f i e r .

In a p r a c t i c a l c i r c u i t , the a m p l i f i e r s w i l l have a f i n i t e output

r e s i s t a n c e . I f only one output connection i s r e q u i r e d , t h i s f i n i t e

output r e s i s t a n c e may be used as (p a r t of) the connection r e s i s

tance. For c o n f i g u r a t i o n number 8, though, the equations w i l l be

complicated i f the + 've g a i n a m p l i f i e r has a f i n i t e output

r e s i s t a n c e ; and i f t h i s output r e s i s t a n c e i s too high, perhaps the

c o n f i g u r a t i o n w i l l not be f e a s i b l e . This problem i s considered i n

50

N u m b e r

A m p l i f i e r I n * p u t C o n f i g u r a t i o n

I s t h e C o n f i g u r a t i o n F e a s i b l e ?

C o m m o n G r o u n d

f o r b o t h

A m p s . ?

How m a n y + ' v e A m p s . R e q d , ?

I s o n l y o n e R e s i s t o r p e r A p i p . R e q d . ?

C a n t h e G y r a t i o n C o n d u c t a n c e s b e m a d e =?

C o m m e n t s

1 D + D + NO

2 D + D - NO

3 D - D + \

4 D - D - NO

5 D+1 + Y E S NO 2 Y E S Y E S une D a i a n c e c o n d i t i o n i s t h a t G ~ = Gr-

6 D + l - NO

7 D - 1 + Y E S NO 2 Y E S Y E S Same a s f o r n u m b e r 5

8 D - l - Y E S Y E S 1 NO NO T h i s MJ i s best, f o r inductance s i m u l a t i o n

9 D + 2 + NO

1 0 D + 2 - NO

11 D - 2 + Y E S NO 2 Y E S Y E S B o g e r t g i v e s t h i a _ c i r c u i t (bo55,Fig.3(c)) . ;

12 D - 2 - Y E S NO 2 Y E S Y E S Same as f o r n u m b e r 11

13 1+2+ NO

1 4 1 + 2 - Y E S NO 2 Y E S Y E S B o g e r t g i v e s t h i s c i r c u i t Cbo5;5,Eig.Md))

1 5 1 - 2 + NO

16 1 - 2 - Y E S NO • 2 Y E S Y E S Same a s f o r n u m b e r 14

F i g u r e 4 . 1 0 - S u m m a r y o f 2 - a m p l i f i e r A G c i r c u i t s .

51

s e c t i o n 5.1.

From the 5th column of e n t r i e s , note that with a l l the

c o n f i g u r a t i o n s but number 8j the g y r a t i o n conductances can be

made equal. C o n f i g u r a t i o n number 8, however, w i l l not y i e l d a

g y r a t o r .

Which of the 7 c i r c u i t s i s best f o r inductance simulation?

C o n f i g u r a t i o n number 8 has the advantages of r e q u i r i n g only 1

power supply, and only one + rve gain a m p l i f i e r . The only d i s

advantage of c o n f i g u r a t i o n number 8 i s that one of the ampli

f i e r s has 2 output connections; i t i s shown i n s e c t i o n 5.1 th a t ,

f o r the case where the output r e s i s t a n c e of t h i s a m p l i f i e r i s

small, the c o n f i g u r a t i o n s t i l l y i e l d s an AG. Therefore, number

8 seems to be the best of the 7 c o n f i g u r a t i o n s . In f a c t , on

the b a s i s of these c o n s i d e r a t i o n s , number 8 i s j u s t as good as the

3 - a m p l i f i e r AG of s e c t i o n 4.3.

The c i r c u i t s of c o n f i g u r a t i o n numbers 5, 7, and 8 are a l l

b e l i e v e d to be new. However, since the a p p l i c a t i o n of primary

i n t e r e s t i s inductance s i m u l a t i o n , and since number 8 seems the

most advantageous f o r t h i s purpose, numbers 5 and 7 w i l l not be

pursued any f u r t h e r . In Chapter 5, c o n f i g u r a t i o n number 8 i s

considered i n more d e t a i l , and a few experimental measurements

which were obtained with t h i s c i r c u i t are given.

Questions of high Q, s t a b i l i t y , and noise performance (the f i r s t two are interdependent) have not entered i n t o t h i s d e c i s i o n . I t i s i n t e r e s t i n g to note t h a t the c i r c u i t of c o n f i g u r a t i o n number 8 was designed before the present a n a l y s i s was c a r r i e d out.

5, EXPERIMENTAL RESULTS 52

In t h i s chapter* a p r a c t i c a l c i r c u i t f o r AG c o n f i g u r a t i o n

number 8 (Figure A.£ ) i s i n v e s t i g a t e d . In the f i r s t s e c t i o n , the

a n a l y s i s of t h i s c o n f i g u r a t i o n i s repeated, t h i s time f o r the

case where the a m p l i f i e r s have a f i n i t e output r e s i s t a n c e . The

second s e c t i o n contains the: a c t u a l c i r c u i t diagram f o r the

prototype which was b u i l t * In the t h i r d s e c t i o n , the r e s u l t s of

some experimental measurements are g i v e n . . These r e s u l t s are

s u f f i c i e n t to confirm the theory.

5.1 Allowance f o r the F i n i t e Output Impedance of the A m p l i f i e r s .

In the a n a l y s i s of c o n f i g u r a t i o n number 8 presented i n the

Appendix, i t was assumed t h a t the a m p l i f i e r s had zero output

impedance. In p r a c t i c e , t h i s i s not t r u e ; t h i s p o i n t was d i s

cussed i n s e c t i o n 4*6*

Let the P a m p l i f i e r . ( s e e Appendix) have a f i n i t e output

conductance, G q . Then the output connections f o r the a m p l i f i e r

are as shown i n F i g u r e 5*1(a). The a p p r o x i m a t e V - e q u i v a l e n t of

t h i s network i s shown i n F i g u r e 5.1(b), f o r the case where

G , Go p c M v -A/V-o

G ' ^ G - G . / G PI ° , 4 0

-A/V P2

4.

P 6 F i g u r e 5.1 - The output connections f o r an a m p l i f i e r with f i n i t e

output conductance, G 0 : (a) a c t u a l c i r c u i t ; (b) approximate ^ — e q u i v a l e n t when G.j'V G ^ « G 0 .

G 3 ^ G4«V 53

..(5.1)

The e f f e c t of the f i n i t e output . impedance, then, i s to

introduce a conductance G ' ^ G^G./G between P l and P2 (c and d).

o 3 4 / o

In Figure 5.2, c o n f i g u r a t i o n number 8 i s analysed f o r the

case where the P a m p l i f i e r has a f i n i t e output conductance G q . The

approximation of Figure 5.1(b) i s used. I t i s seen t h a t , with

G q i n c l u d e d , the c o n f i g u r a t i o n s t i l l y i e l d s an AG. I t i s easy

to show that

f o r t h i s AG ; i n f a c t , equation (2 ) of Figure 5.2 can be w r i t t e n

T12 = I T 2 l | + ( G 2 + G 4 + G o r ) * •••(5.2)

This s e c t i o n i s concluded with a sample design, based on

the a n a l y s i s given i n F i g u r e 5.2. Suppose

G = G, = G~ = 200 umho, o 1 2 '

A 1 = 6,

and A 2 - " 1 0 -

Then equations ( l a ) and (2 ) i n Figure 5.2 become, r e s p e c t i v e l y ,

(5000)G 3G 4 +0.0002 + G & - 5G 3 = 0 ,

and ...(5.3)

0.0002 - (5000)G 3G 4 + 10G 6 - 5G 4 = 0 .

Since there are 2 equations i n 3 unknowns, one of the unknowns

Note t h a t the AG t e r m i n a l s can e a s i l y be re-named so that t h i s i n e q u a l i t y i s reversed. See footnote i n s e c t i o n 4,4, page 47.

54

I n t r — i n s i c term, i n t r

Terms Produced by Amps. AG admittance term

Subtotals I n t r — i n s i c term, i n t r

P A m p l i f i e r f i f i r

AG admittance term

Subtotals I n t r — i n s i c term, i n t r PI P2 Ql v 02

AG admittance term

Subtotals

Affec-v i a i n = o 0 Affec-v i a v a Ml o Y 1 2 = +

A f f e c t I ? 2

v i a v, - S i %M 0 T 2 1 = . -A f f e c t I ? 2 v i a * X 0 0 T22= 0 o 0 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) ( l l )

+ a 'P-Pl •V

P2.QI

P-P2

Q - Q i Q-Q2

Vifl-AiVfl&Ai

(0

(3)

^ t'sth-z output <*</H/Trance o-f ^ck4**p//z^r^v\

tt**t t>*t/> Pt«UPZ u>*'ll U us<d, (fee.

For- y^2T0^ & 6* a?€c/.

T^-e ce>* J rife* Ti^paSeJ /"f -hd^C/lî)^, Cl)

T~/4.e sn£>~tot<*/s <ar<z, <^faeisi A i c<o/u, 3 -

Tk<z rey«/t ,<QW€h i h c<*/nh,h /D, re prefers on 4<y,

F i g u r e 5.2 - Design sheet f o r c o n f i g u r a t i o n number 8, usi n g the approximation given i n Figu r e 5.1(b).

55 may be chosen at w i l l , say

G^ = 50 u-mho,

i n order to comply with ( 5 . l ) . S o l u t i o n of equations (5.3) y i e l d s

G^ = 91 jumho,

Gg = 27*. 4 p,mho .

In the next s e c t i o n a prototype i s d e s c r i b e d which was b u i l t

u s i n g these v a l u e s as a design c e n t r e .

5.2 C i r c u i t Diagrams f o r the Prototype AG.

A o n e — t r a n s i s t o r , common-emitter a m p l i f i e r was designed to

produce a negative g a i n of about 10. The c i r c u i t diagram i s

shown i n Fig u r e 5*3*

^ 4 . 7 K

F i g u r e 5»3 - C i r c u i t diagram of the Q a m p l i f i e r .

A t w o — t r a n s i s t o r , common-emitter, R-C coupled a m p l i f i e r

was designed to produce a p o s i t i v e g a i n of about 6. The c i r c u i t

56

~ +6, at

OUT f•= 1000c.p.s. O p

Z o u t - 5 K

F i g u r e 5*4 - C i r c u i t diagram of the P a m p l i f i e r .

diagram i s shown i n Fig u r e 5.4. The emitter r e s i s t o r s i n both

c i r c u i t s provide negative current feedback, and lower the output

r e s i s t a n c e .

These 2 a m p l i f i e r s were connected i n the D - l - c o n f i g u r a t i o n

(number 8.) as shown i n Fig u r e 5.5. Design centre values c a l c u l a t e d

200K

A H c a p a c i t o r s a r e 10 u-f t a n t a l u m e l e c t r o l y t i c .

0 4 E 3

F i g u r e 5.5 - Schematic diagram of the AG pr o t o t y p e .

i n the sample design of s e c t i o n 5.1 were used, and v a r i a b l e

r e s i s t o r s were used f o r G- and G^ to permit adjustment* The

c i r c u i t was b u i l t on a p r i n t e d c i r c u i t board, Figure 5*6*

57

Fi g u r e 5.6 - Artwork f o r the p r i n t e d c i r c u i t board ( a c t u a l s i z e ) .

For convenience* i d e n t i c a l c o u p l i n g c a p a c i t o r s were used through

out.

5.3 Experimental R e s u l t s .

Two sets of measurements were, taken with the prototype AG

d e s c r i b e d i n the preceeding s e c t i o n . In the f i r s t s e t , the AG

was used to i n v e r t a r e s i s t a n c e , f o r comparison w i t h s i m i l a r

r e s u l t s g i v e n by Bogert (bo55). In the second s e t , i t was used to

simulate an inductance.

The experimental apparatus i s shown i n Fi g u r e 5.7. Z^ n was

measured w i t h a General Radio X-Y b r i d g e , type 1603A.

5.3.1 P r e l i m i n a r y adjustment of the AG.

R e c a l l from s e c t i o n 5*2 t h a t G_ and G, were made v a r i a b l e

D e t e c t o r : T e k t r o n i x Type 502A O s c i l l o s c o p e

Heathkit Audio O s c i l l a t o r Model AG.-8

58

General Radi o X-Y Bridge Type 1603A O

Figure 5.7 - The AG used to i n v e r t an impedance Z^: experimental apparatus.

to permit adjustment. These two v a r i a b l e conductances can be

adjusted to set Y ^ — 0 and ^ 0 *

a) Adjustment f o r — 0

Terminals c and d are s h o r t - c i r c u i t e d . Under t h i s c o n d i t i o n ,

Y. i n

'v 2.=.o

so Y-j^ can be measured d i r e c t l y . Y ^ can be set a r b i t r a r i l y close

to zero by a d j u s t i n g G^ (see equation (2 ), Figure 5.2). Since

the c i r c u i t would be p o t e n t i a l l y unstable i f Y ^ were -'ve, Gg i s

adjusted to make Y ^ s l i g h t l y +'ve, say

Y. - Y,, = 5 umho i n 11

This value of Y,, w i l l provide an ( a r b i t r a r y ) margin of s t a b i l i t y .

59 The adjustment was c a r r i e d out by s e t t i n g the dials of "the X-T bridge

to 5 + JO u.mhos, and then a d j u s t i n g G^ to balance the b r i d g e .

b) Adjustment f o r Y 2 2 > ' 0

Terminals c and d a r e . o p e n r c i r c u i t e d . Under t h i s c o n d i t i o n ,

z. •• • 2 2

J i n T 1 1 Y 2 2 + J Y12|| Y2l|

But Y^^ ~ 0 because of the adjustment made i n ,a) above, and t h e r e

f o r e Y

2 ^ 22 i n l I i 2 | P a i l

Y 2 2 can be set a r b i t r a r i l y c l o s e to zero by a d j u s t i n g G-j (see

equation ( l a ) i n Figure 5.2). Again, a margin of s t a b i l i t y can

be achieved by making Y 2 2 s l i g h t l y p o s i t i v e ; i n p r a c t i c e , t h i s i s

e a s i l y done by a d j u s t i n g G^ to make Z^ n s l i g h t l y p o s i t i v e , say

Z. : = 5 J l . i n

This adjustment was c a r r i e d out by s e t t i n g the d i a l s of the X-Y

bridge to 5 + JO ohms, and then a d j u s t i n g G^ to balance the b r i d g e .

In p r a c t i c e , a s l i g h t s l i d i n g balance i s observed between

adjustments a) and b ) , so the p r e l i m i n a r y adjustment i s completed

by r e p e a t i n g a) and b ) . This s l i d i n g balance i s not evident i n

equations ( l a ) and (2 ) of Figure 5.2, because of the approximation

(Figure 5.1(b)) which was made-.

60 5.3.2 I n v e r s i o n of r e s i s t a n c e .

A c a l i b r a t e d Heathkit r e s i s t a n c e s u b s t i t u t i o n box was used

f o r Z T i n Figu r e 5.7. The r e s i s t a n c e was v a r i e d from 22 ohms to Li

10 megohms i n 18 steps, and Z^ n was measured f o r each s e t t i n g .

Three such sets of readings were taken, at frequencies of 200

c.p.s., 1 Kc., and 5 Kc. The measured values were converted to

p o l a r form, and are p l o t t e d i n Figure 5.8 (the curve given by

Bogert (bo55, taken at 400 c.p.s.) i s i n c l u d e d i n the f i g u r e f o r

comparison).. At 1000 c.p.s., the i n v e r s i o n i s e x c e l l e n t f o r

values of ranging from about 500 ohms to 100 kilohms (about

2\ decades), w i t h a phase angle of l e s s than 1° over t h i s e n t i r e

range. At both 200 c.p.s. and 5 K c , the phase response i s much

poorer.

5.3.3 S i m u l a t i o n of inductance.

A 0.1 ufd., 75v., ceramic d i s k c a p a c i t o r was used f o r Z-

i n Figure 5.7. was measured as the frequency was v a r i e d from

20 c.p.s. to 20 Kc. The measured v a l u e s , p l o t t e d i n Figure 5.9,

i n d i c a t e t h at the simulated inductance has the approximate e q u i

valent c i r c u i t shown i n Fig u r e 5.10. The s e r i e s capacitance i n

t h i s e q u i v a l e n t c i r c u i t represents the coup l i n g c a p a c i t o r at the

input to the Q a m p l i f i e r (C i n Figure 5.5), while the p a r a l l e l s

capacitance i s probably the r e s u l t of s t r a y capacitances and phase

s h i f t e f f e c t s w i t h i n the AG c i r c u i t .

At 1000 c.p.s., the Q of the ind u c t o r was about 16. Note,

however, that t h i s Q can be in c r e a s e d by merely a d j u s t i n g G^ and

G A, I f the Q i s made too l a r g e , however, d r i f t i n the a m p l i f i e r s

-100K

t - l O K -

1 l Z i nl

b i K -

-ft-

- X r - l ° ( O ) (phase a n g l e o f Z ± n )

10 o _>>N

+ 5 ° (A) ^ - 1 0 ° ( + )

- 5 ° •( + )

, - 5 ° (A)

C u r v e g i v e n by B o g e r t (bo55) ( s h i f t e d 1 d e c a d e t o the r i g h t )

f = 0 . 2 K c .

f "= 1 . 0 K c .

f = 5 . 0 K c .

f = 0 . 4 K c .

h i o o

••(+>•••-.•• s^--io° (A)

i o v

A o +

( B o g e r t 1 s c u r v e )

+5° ( O )

100 1 O > I I I M ' t l

IK » I i t i n>

R ,_n_. ' : I0K 100K I i I i \

> IM > t i i > J i l + i o ° Id).

10M "Q i i i i i > i l

F i g u r e 5 . 8 - E x p e r i m e n t a l r e s u l t s , i n v e r s i o n o f r e s i s t a n c e .

20 c . p . s . •300

(a)

62

I.. u.mho,

0 . 1 K c .

F i g u r e 5 . 9 - E x p e r i m e n t a l r e s u l t s , s i m u l a t i o n o f i n d u c t a n c e : (a) l o w f r e q u e n c y m e a s u r e m e n t s , Z±N', (b ) h i g h f r e q u e n c y

m e a s u r e m e n t s , Y ; i n

63

~1".C- ' — 15u.f. S e r i e s resonance. s at f ^ 37 c.p.s.

600pf. P a r a l l e l resonance at f 6Kc .

O— J

Figure 5.10 - Approximate e q u i v a l e n t c i r c u i t f o r the simulated inductance (losses are n e g l e c t e d ) .

may cause the Q to become -'ve, thereby producing p o t e n t i a l

i n s t a b i l i t y . Although no e x p l i c i t d r i f t measurements were made,

the i n i t i a l and f i n a l readings i n one set of measurements i n d

i c a t e d t h at had1 d r i f t e d from about +0.5 umho to -l.0u.mho,

over a p e r i o d of a few hours (Y-^ a n £ l | ^ 2 l | ^ 300 urriho).

On the b a s i s of the r e s u l t s given above i n s e c t i o n s 5.3.2

and 5.3.3, one can conclude that the prototype AG performs as

expected from the t h e o r e t i c a l design.

http://-l.0u.mho

64

6. CONCLUSION

The t o p i c s of inductance s i m u l a t i o n and g y r a t o r r e a l i z a t i o n

have been d e a l t with i n t h i s t h e s i s *

An extensive survey has been given of the inductance

s i m u l a t i o n l i t e r a t u r e . I t has been shown that s e v e r a l seemingly

u n r e l a t e d " a m p l i f i e r methods" of inductance s i m u l a t i o n are r e a l l y

c l o s e l y r e l a t e d .

An extensive survey Of the gy r a t o r l i t e r a t u r e has been given

The " a c t i v e g y r a t o r " (A|J) has been d e f i n e d , and s e v e r a l of i t s

p r o p e r t i e s and a p p l i c a t i o n s given* One pro p e r t y of the AG i s

that of inductance s i m u l a t i o n . An a c t i v e gyrator can a l s o be

used to r e a l i z e an i s o l a t o r or a c i r c u l a t o r with a power g a i n .

An a n a l y s i s has been presented which r e s u l t s i n 7 c i r c u i t

c o n f i g u r a t i o n s f o r the r e a l i z a t i o n of the AG. One of these c i r

c u i t s i s the best f o r inductance s i m u l a t i o n . Of the other c i r

c u i t s ^ 4 have been p u b l i s h e d p r e v i o u s l y , arid 2 are b e l i e v e d to

be new. The " b e s t " c i r c u i t i s also b e l i e v e d to be new.

Experimental r e s u l t s obtained with the best AG c i r c u i t f o r

inductance s i m u l a t i o n have been g i v e n . These r e s u l t s confirm

the v a l i d i t y of the d e s i g n .

The main c o n t r i b u t i o n s of t h i s t h e s i s are:

1) c l a s s i f i c a t i o n of the " a m p l i f i e r methods" of i n d

uctance s i m u l a t i o n (Chapter 2),

2) proposal of ari o p e r a t i o n a l a m p l i f i e r method of

inductance s i m u l a t i o n which i s b e l i e v e d to be new

(Figure 2.8),

3) comparison of gyrator r e a l i z a t i o n methods (Figure 3

65 proposal of the AG ( s e c t i o n 3 . 3 ) ,

a n a l y s i s of i s o l a t o r s and c i r c u l a t o r s made wit h

AGs, showing the power gains a v a i l a b l e ( s e c t i o n 3*5) 9

design of an AG c i r c u i t which turns out to be "best"

f o r inductance s i m u l a t i o n ( s e c t i o n 4.6),

p r e s e n t a t i o n of an a n a l y s i s which r e s u l t s i n several.

c i r c u i t s f o r the r e a l i z a t i o n of the AG (Chapter 4, and

the Appendix),

experimental t e s t i n g of one of these AG c i r c u i t s

(Chapter 5), and

c o m p i l a t i o n of an extensive b i b l i o g r a p h y of the

inductance s i m u l a t i o n and gyrator l i t e r a t u r e

(Chapter 7)«

7. BIBLIOGRAPHY 66

The b i b l i o g r a p h y which was compiled d u r i n g the course of

t h i s p r o j e c t contains w e l l over 100 r e f e r e n c e s . I t i s b e l i e v e d

t h a t a b i b l i o g r a p h y of t h i s extent has not appeared elsewhere.

Each refer e n c e i s indexed with a 4-character code; f o r

example, "do63". The f i r s t two characters are the f i r s t two

l e t t e r s of the surname of the p r i n c i p a l author, while the l a s t

two c h a r a c t e r s i n d i c a t e d i r e c t l y the year of p u b l i c a t i o n . For

in s t a n c e , a paper w r i t t e n by J . Doe, and p u b l i s h e d i n 1963 would

be indexed under "do63"«. I f s e v e r a l papers f a l l under the same

index code, a 5th character i s added to d i f f e r e n t i a t e them; f o r

example, do63a, do63b, e t c .

I t i s hoped that t h i s b i b l i o g r a p h y w i l l serve as a u s e f u l

guide to anyone seeking i n f o r m a t i o n regarding inductance sim

u l a t i o n and g y r a t o r s .

7.1 Subject Index to B i b l i o g r a p h y .

INDUCTANCE, e t c *

Conventional i n d u c t o r s : ho49, hu25, po37.

Microminiature i n d u c t o r s : br55, du61, ho58, ho62, st57a, st57b, to62.

Simulated inductance:

Semiconductor devices : di61, di62a, di62b, du63 , ei52, f i 5 9 , ga61, go57, gu56, hu60 r ka55« ko54, ko55, la60, l a 6 l , ni58, ni60, no63, on56, sc60, sc62, sp58.

67 Other: bo55, bo56, co58, et62, fu60, h o — ,

hu42(?-), mc63, mi55, mi60, st58, to53, wa47.

Current l i m i t e r s : du61, la62, wa59.

Ceramic IF transformers, e t c . : el58, lu58, ma61a, ma61b.

Incr e a s i n g Q f a c t o r : s a 6 l .

GYRATORS, e t c .

T h e o r e t i c a l :

Network theory: au55, ca55a, ca55b, ca56, f r 5 8 , on6la, oo54, pr57, te48b, te4'9a, te49b, te50, te51, te55, tw55, wa51.

Others: ca45, fa58, na6l, pi62, sh53, sh54, te48a r

( o r i g i n a l paper), te52, te56.

P r a c t i c a l ( p h y s i c a l r e a l i z a t i o n ) :

E l e c t r o m e c h a n i c a l : Coupled t r a n s d u c e r s : bl53, br44, c l 6 1 , c u — ,

ga52, ge61, ha54, mc46, mi47, on61b, on62, s i 6 l , s i62a.

" " as i s o l a t o r : ga52, ga54, ga59, mc46, mc47, on62, s i 6 1 , ' s i 6 2 a .

" " as c i r c u l a t o r : si62b. Generalized' machine: pe58.

H a l l E f f e c t : ar60, be63, ch58, gr58, gr59a,•gr59b, hu61, ma53, mc47, se52, si62a, wi54.

F e r r i t e (microwave f r e q u e n c i e s ) : al57, be50, bl56, bl57, ca54, ca56, fo55, go53, he56, ho52, ho53, ho56, jo59, ka53, ow56,

^ re54, sc61, so60. " as i s o l a t o r : be57, jo59. " as c i r c u l a t o r : ho52, ho56.

Tube and t r a n s i s t o r : bo55, bo56, no54, pi62, sh53.

F i e l d e f f e c t t e t r o d e : et62, st59a, st59b, s t 6 l .

Parametric: ka60, ko61.

68

OTHER Network p r o p e r t i e s ! bo57, sh54.

C i r c u l a t o r s : pe62, st57c, t r 5 6 .

Negative r e s i s t a n c e : bo45, cr31, fa52, gi45, he35, ro54, st60, uz63.

7.2 References,

al57 A l l i n , P.E.V., F e r r i t e s at Microwaves, E l e c t r o n i c Eng., V o l . 29, pp. 292-96; June, 1957.

ar60 A r l t , G., H a l l e f f e k t - V i e r p o l e mit Hohem ¥irkungsgrad, S o l i d - S t a t e E l e c t r o n i c s , V o l . 1, pp. 75-84; March, 1960.

au5 5 A u r e l l , C.G., Representation of General L i n e a r 4-Terminal Network and Some of i t s P r o p e r t i e s , E r i c s s o n Technics, V o l . 11, pp. 155-79; 1955.

be50 B e l j e r s , H.G., and Snoek, J.L. , Gyromagnetic Phenomena Occurring W i t h i n F e r r i t e s , P h i l i p s Tech. Rev., V o l . 11, pp. 315-22; May, 1950.

be 57 ' B e l j e r s , H.G., A p p l i c a t i o n of Ferroxcube to U n i d i r e c t i o n a l Waveguides, P h i l i p s Tech. Rev., V o l . 18, p. 158; 1957..

be63 Beckman Instruments, Inc., F u l l e r t o n , C a l i f . , H a l l E f f e c t Manual, (pamphlet, with extensive b i b l i o g r a p h y ) ; 1963.

bl53 Black, L . J . and Sco t t , H.J.,-Measurements on N o n r e c i p r o c i t y i n Ele c t r o m e c h a n i c a l Systems, Journ. A c o u s t i c a l Soc. Am., V o l . 25, pp. 1137-40; Nov., 1953.

bl56 Bloembergen, N., Magnetic Resonance i n F e r r i t e s , PIRE, V o l . 44, pp. 1259-69; Oct., 1956, (B i b l i o g r a p h y of 63 a r t . , see #»s 6, 8, 10 ( b i b l i o ) , 43)

bl57 Blackman, L.C.F., Low Loss Magnesium Manganese F e r r i t e s , J . E l e c t r o n i c s , V o l . 2, pp. 451-56; March, 1957.

69 Bode, H.W., Network A n a l y s i s and Feedback A m p l i f i e r Design (book), Van Nostrand Co., New York, pp. 185-88; 1945V

Bogert, B.P., Some Gyrator and Impedance I n v e r t e r C i r c u i t s , PIRE, V o l . 43, pp. 793-96; J u l y , 1955.

Bogert, B.P., Impedance I n v e r t e r s , U.S.Pat. 2,757,345, J u l y 31, 1956.

B o l i n d e r , E.F., Survey of Some P r o p e r t i e s of L i n e a r Networksj IRE Trans., V o l . CT-4, pp. 70-18; Sept., 1957.

Braun, K., T i t l e u n a v a i l a b l e ,

T e l e g r . Fernspr. Technik, V o l . 33, pp. 85- $ 1944.

Bryan, H.E., P r i n t e d Inductors and C a p a c i t o r s , Tele-Tech and E l e c t r o n i c I n d u s t r i e s , V o l . 14, p. 68; D e c , 1955. Casmir, H.B.G., On Onsager's P r i n c i p l e of Mi c r o s c o p i c R e v e r s i b i l i t y , Rev. Mod'. Phys., V o l . 17, pp. 343-50; 1945.:

C a r l i n , H.J., P r i n c i p l e s of Gyrator Networks, Proc. Symp. Modern Advances i n Microwave Techniques, P o l y t e c h n i c I n s t i t u t e of Brooklyn, Nov. 8-10, 1954, pp. 175-204.

C a r l i n , H. J.,, Synthesis of ,Non-Reciprocal Networks, Presented at Symposium on Modern Network Synt h e s i s , P o l y t e c h n i c I n s t i t u t e of Brooklyn, A p r i l 13—15$ 1955.

C a r l i n , H.J., On P h y s i c a l R e a l i z a b i l i t y of L i n e a r Non-Reciproeal Networks,

PIRE, V o l . 43, pp. 608-16; May, 1955.

C a r l i n , H.J., Non-Reciprocal Network Theory A p p l i e d to F e r r i t e Microwave Devices, (Convention on F e r r i t e s , Oct.-Nov., 1956), IEE P r o c , V o l . 104, pa r t B, supp. #6, pp. 316-19; 1957. Champness, C.H., H a l l E f f e c t and Some of i t s P o s s i b l e A p p l i c a t i o n s , IEE Can. Convention B e e , 1958, pp. 265-69.

70

c l 6 l C l e v i t e Corp., Low Frequency Gyrator Development,, ( r e p o r t ) , Nov., 1961, 44 pages.

co58 Cooperman, J . I . , and F r a n k l i n , P.J., Some C i r c u i t Techniques to E l i m i n a t e Large-Volume Components; a L i t e r a t u r e Survey, Micro m i n i a t u r i z a t i o n of E l e c t r o n i c Assemblies (book), Hayden Book Co., Inc., N.Y., 1958; pp. 193-212. Also see E l e c t r o n i c Design, v o l . 7, pp. 57-61; March, 1959.

cr31 C r i s s o n , G., Negative Impedances and the Twin 21-Type Repeater, BSTJ, Vol.. 10, pp. 485-513; J u l y , 1931.

c u — Curran, D.R., Germano, CP . , Silverman, J.H., and D i s h o t e l s , ¥.J., Low-Frequency Gyrator Development, Armed Ser v i c e s T e c h n i c a l Information Agency AD 269-374.

di61 D i l l , H.G., Inductive Semiconductor Elements and Their A p p l i c a t i o n i n Bandpass A m p l i f i e r s , IRE Trans., V o l . MIL-5, pp. 239-50; J u l y , 1961.

di62a D i l l , H.G., Avalanche Pulse Generators, Semiconductor Products, V o l . 5, Feb., 1962, pp. 23-30.

di62b D i l l , E . G . , Semiconductor Inductive Elements, Semiconductor Products, V o l . 5, A p r i l , 1962, pp. 30-33; and May, 1962, pp. 28-31.

do36 Doherty, W.H., A New High E f f i c i e n c y Power A m p l i f i e r f o r Modulated Waves, PIRE, V o l . 24, pp. 1163-82; Sept., 1936 .(Also seem51,BP. 399-404.)

du6l Dummer, G.W.A., and G r a n v i l l e , J.W., M i n i a t u r e and Microminiature E l e c t r o n i c s (book), John Wiley & Sons, New York, 1961, 310 pages, ($7.50).

du63 Dutta Roy, S.C., The Inductive T r a n s i s t o r , IRE Trans., V o l . CT-10, pp. 113-15; March, 1963.

ei52 E i n s e l e , T., Uber die Traegheit des F l u s s l e i t w a r t s von Germaniumdioden, Z. Angew, Phys., V o l . 4, pp. 183-87; May, 1952.

el58 E l d e r s , D., and Gikow, E., Ceramic IF F i l t e r s Match T r a n s i s t o r s , E l e c t r o n i c s , Apr. 25, 1958, pp. 59-61. Also see 1957 E l e c t r o n i c Components Conference, Chicago, May 2, 1957.

71 et62 E t t e r , P.J., and Wilson, B.L.H.,

Inductance Prom a F i e l d - E f f e c t Tetrode, PIRE, V o l . 50, pp. 1828-29; Aug., 1962.

fa52 F a r l e y , B.G., Dynamics of T r a n s i s t o r Negative-Resistance C i r c u i t s , PIRE, V o l . 40, pp. 1497-1508; Nov., 1952.

fa58 Fabrikov, V.A., Vozmozhnost U s i l e n i y a Giromagnitnoi Snedoi Moshchnosti Slabogo Moduliruyushchego S i g n a l a , Radio-Tekhnika i E l e c t r o n i k a , V o l . 3, pp. 190-97; Feb.,.1958.

Pu b l i s h e d i n E n g l i s h : P o s s i b i l i t y of Power A m p l i f i c a t i o n of Weakly Modulated S i g n a l s by Gyromagnetic Medium, Radio Eng. and E l e c t r o n i c s , V o l . 3, pp. 265-75; 1958.

f i 5 9 F i r l e , T.E., and Hayes, O.E., Some Reactive E f f e c t s i n Forward Biased J u n c t i o n s , IRE Trans., V o l . ED-6, pp. 330-34; J u l y , 1959.

fo55 Fox, A.G., M i l l e r , S.E., and Weiss, M.T., Behaviour and A p p l i c a t i o n s of F e r r i t e s i n Microwave Region, BSTJ, V o l . 34, pp. 5-103; Jan., 1955.

fr58 Frank, V., and Hojgaard-Jensen, H., Note on R e c i p r o c i t y Theorem f o r E l e c t r i c a l Systems, A p p l i e d Science Research, Sec. 8, V o l . 7, pp. 145-49; 1958.

fu60 Fulenwider, J.E., High Q Inductance Simulation, PIRE, V o l . 48, pp. 954-55; May, 1960.

ga52 Gamo, H., Four Terminal Networks V i o l a t i n g R e c i p r o c a l Theorem and One-Way Systems, Proc. 26th J o i n t Conf. of E l e c . Eng'g, S o c , 1952, Paper # 9-1.

ga54 Gamo, H., Ele c t r o m e c h a n i c a l One-Way Systems, J . Acoust. Soc. (Japan), V o l . 10, pp. 65-76; June, 1954.

ga59 Gamo, H.^ On Passive One-Way Systems, Trans. I n t . Symp. on Cct. & I n f . Theory, 1959, pp. 283-98.

ga6l Gartner, WW., and S c h u l l e r , M., • Three-Layer Negative Resistance and Inductive Semiconductor Diodes, PIRE, V o l . 49, pp. 754-67; Apr., 1961.

ge6l Germano, C P . , and Curran, D.R., Low Frequency Gyrators, Proc. E l e c t r n c . Compts. Conf., May, 1961, pp. 24-1 to 24-13.

72

gi45 Ginzton, E.L., S t a b i l i z e d Negative Impedances, E l e c t r o n i c s , V o l . 18, J u l y , Aug., and Sept., 1944, pp. 140-50, 138-48, and 140-44 r e s p e c t i v e l y .

go53 G o l d s t e i n , L., and Lampert, M.A., A New L i n e a r Passive Non-Reciprocal Microwave C c t . Component, PIRE, V o l . 41, pp. 295-6; 1953.

go57 Good, E.F., A fwo-Phase Low Frequency O s c i l l a t o r , E l e c t r o n i c E n g i n e e r i n g , V o l . 29, pp. 164-69 and,210-l3; A p r i l & May, 1957.

gr58 Grubbs, ¥.J., The H a l l - E f f e c t C i r c u l a t o r - a Passive ... Device, IRE Wescon, 1958, p t . 3, pp. 83-93.

gr59a Grubbs, W.J., H a l l E f f e c t Devices, BSTJ, V o l . 38, pp. 853-76; May, 1959.

gr59b Grubbs, W.J., H a l l E f f e c t C i r c u l a t o r , PIRE, Vol.. 47, p. 528; 1959.

gu56 Guggenbuehl, W., Theoretische Ueberlegungen zuj|, P h y s i k a l i s c h e n Begruendung des E r s a t z s c h a l t b i l d e s von H a l b l e i t e r d i o d e n b e i Hohen Stromdichten, Arch. E l e k t r o t e c h . Ubertragung, V o l . 10, pp. 483—85; Nov., 1956.

.ha!4 Haus, H.A., Eq u i v a l e n t C i r c u i t f o r Passive Non-Reciprocal Network, J . Appl. Phys., V o l . 25, pp. 1500-02; D e c , 1954.

he35 Herold, F.W., Negative Resistance and Devices f o r Obtaining i t , PIRE, V o l . 23, pp. 1201-23; Oct., 1935.

he56 H e l l e r , G.S., F e r r i t e s as Microwave C i r c u i t Elements, PIRE, V o l . 44, pp. 1386-93; Oct., 1956.

ho49 Howe, G.W.O., The Q F a c t o r of Sin g l e - L a y e r C o i l s , W i r e l e s s Eng., V o l . 26, pp. 179- ; June, 1949.

ho52 Hogan, C.L., Ferromagnetic Faraday E f f e c t at Microwave Frequencies and: i t s A p p l i c a t i o n s - Microwave Gyrator, BSTJ, V o l . 31, pp. 1-31; Jan., 1952., See a l s o W i r e l e s s Engr., V o l . 29, pp. 171-73; J u l y , 1952.

7 3

ho53 Hogan, C L . , The Ferromagnetic Faraday E f f e c t at Microwave Frequencies and i t s A p p l i c a t i o n s , Rev. Mod. Phys., V o l . 25, pp. 253-263; Jan., 1953.

ho56 Hogan, C L . , The Elements of N o n r e c i p r o c a l Microwave Devices, PIRE, V o l . 44, pp. 1345-68; Oct., 1956. Paper presented at Symposium on Microwave Props. &' Appl s . of F e r r i t e s , Harvard Univ., Cambridge, Mass., A p r i l 2-4. 1956. (See p. 1345, r h , top, re impedance i n v e r s i o n )

ho58 Horsey, E.F., and S h e r g a l i s , L.D. ( e d i t o r s ) , M i c r o m i n i a t u r i z a t i o n c o f E l e c t r o n i c Assemblies, Hayden Book Co., Inc., N.Y., 1958.

ho62 Horsey, E.F. and F r a n k l i n , , Status of M i c r o m i n i a t u r i z a t i o n , IRE Trans., V o l . CP-9, pp. 10- ; Mch., 1962.

h o — Holbrook, G.W., and McKeown, D.L., S i m u l a t i o n of Inductance by A c t i v e C i r c u i t Elements, To be p u b l i s h e d .

hu25 Hund, A., and DeGroot, H.B., Radio Frequency Resistance and Inductance of C o i l s Used i n Broadcast Reception, Technologic Papers of the Bureau of Standards, Washington, D.C., No. 298; Oct. 22, 1925.

hu4 2 Hund, A., Frequency Modulation (book), McGraw-Hill, 1942, pp. 15 5-74.

hu60 Hughes A i r c r a f t Co., Semiconductor Division., Newport Beach, C a l i f o r n i a , M olecular Bandpass A m p l i f i e r , I nterim S c i . Rept., Nos. I F and 2, Contract AF33(616) -7252; J u l y 15 and Oct. 15, I960, r e s p e c t i v e l y .

hu61 Hubbard, C.H., LoSasso, L.A., and Rousso, E., Microwave I s o l a t o r Combines H a l l E f f e c t and Tunnel Diodes, E l e c t r o n i c s , June 16, 1961, pp. 56-57.

jo59 Jones, EwM.T., Matthaei, G.L., and Cohn, S..B,, Non-Reciprocal, TEM-Mode St r u c t u r e s f o r "Wide-Band Gyrator and I s o l a t o r A p p l i c a t i o n s , IRE Trans., V o l . MTT-7, pp. 453-60; Oct., 1959.

ka53 Kales, M.L., C h a i t , H.N., and S a k i o t i s , N.G., A Non-Reciprocal Microwave Component, J . A p p l . Phys., V o l . 24, p. 816; 1953.

74

Kanai, Y., On the Inductive Pa r t i n the a . c C h a r a c t e r i s t i c s of the Semiconductor Diodes,

J . Phys. Soc. Japan, V o l . 10, pp. 719-20; 1955.

Kamal, A.K., A Parametric Device as a Non-Reciprocal Element, PIRE, V o l . 48, pp. 1424-30; Aug., 1960. Kohn, G., and Nonnenmacher, V . , Inductives V e r h a l t e n von p-n Uebergaengen i n F l u s s r i c h t u n g Arch. E l e k t r o t e c h . Ubertragung, V o l . 8, pp. 561-64; D e c , 1954. * Kohn, G., Die Beruecksichtigung des Uebergangsgebietes Zwischen Pluss und Sperrgebiet im E r s a t z s c h a l t b i l d f u e r Traege' Germaniumdioden, Arch. E l e k t r o t e c h . Ubertragung, V o l . 9, pp. 241-45; May, 1955.

K o r p e l , A., and Desmares, P., Experiments "with Non-Reciprocal Parametric Devices, PIRE, V o l . 49, p. 1582; Oct., 1961.

Ladany, I., An A n a l y s i s of I n e r t i a l Inductance i n a J u n c t i o n Diode IRE Trans., V o l . ED-7, pp. 303-10; Oct., I960.

Ladany, I . , and Kearney, R.J., A High-Q Tuned C i r c u i t Using a S o l i d - S t a t e Inductance, J . of E l e c t r o n i c s and C o n t r o l , V o l . 10, pp. 241-43; Mch., 1961.

Lawrence, H., A D i f f u s e d F i e l d - E f f e c t Current L i m i t e r , IRE Trans., V o l . ED-9, pp. 82-87; Jan., 1962.

L i n v i l l , J.G., and Gibbons, J.F., T r a n s i s t o r s and A c t i v e C i r c u i t s (book), McGraw-Hill, 1961, pp.. 21-5-18.

Lungo, A., and Henderson, K., A p p l i c a t i o n of - P i e z o e l e c t r i c Resonators to Modern Bandpass A m p l i f i e r s , IRE N a t l . Conv. R e c , 1958, p a r t 6, pp. 235-42.

Mason, W.P., Hewitt, W.H., and Wick, R.F., H a l l E f f e c t Modulators and. "Gyrators" Employing Magnetic F i e l d Independent O r i e n t a t i o n s i n G e r m a n i u m , J . Appl. Phys., V o l . 24 , p p . 166-75; Feb., 1953..

Macario, R.C.V., Design Data f o r Bandpass Ladder F i l t e r s Employing Ceramic Resonators, E l e c t r o n i c E n g i n e e r i n g , V o l . 33, pp. 171-77; Mch., 1961.

75

ma6lb Macario, R . C J . , Ceramic IP Transformers, W i r e l e s s World, V o l . 67, pp. 253-56; May, 1961..

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76 nb63

on56

on61a

on61b

on62

oo54

ow56

pe58

pe62

pi62

po37

pr57

ra63

Nordman, J.E., and Grei n e r , R.A., The S m a l l - S i g n a l Inductive E f f e c t i n a Long P-I-N Diode, IEEE Trans., V o l . ED-10, pp. 171-77; May, 1963.

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77 re54 R e d f i e l d , A.G.,

Electrodynamic P e r t u r b a t i o n Theorem, with A p p l i c a t i o n to Non-Reciprocal Systems, J . Appl. Phys., V o l . 25, pp. 1021-24; Aug., 1954.

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78

Soohoo, R.F., Theory and A p p l i c a t i o n of F e r r i t e s (book), P r e n t i c e H a l l , 1960, pp. 2, 9, and 159-61.

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1959 IRE N a t l . Conv. R e c , p t . 3, pp. 3-8.

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79

te49b T e l l e g e n , B.D.H., Complementary Note on the Synthesis of Passive Resistance-l e s s 4-Poles, , P h i l i p s Res. Rept., V o l . 4, pp. 366-69; Oct., 1949.

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to53 Towner, G.H., . A r t i f i c i a l Inductor, U.S. Patent # 2,800,586, f i l e d J u l y 31, 1953; granted. J u l y 23, 1957.

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80 wa51 Wallace, B,L. J r . , and Baisbeck, G. ,

D u a l i t y as a Guide i n T r a n s i s t o r C i r c u i t Design, BSTJ, V o l . 30, pp. 381-417; Apr., 1951 (pp. 394-95 i n p a r t i c u l a r ) .

va59 Warner, R.M. J r . , Jackson, W.H., Doucette, E . I . , and Stone, H.A. J r . , A Semiconductor Current L i m i t e r , PIRE, V o l . 47, pp. 44-56; Jan., 1959.

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A P P E N D I X •<

A N A L Y S I S OF T H E 1 6 C O N F I G U R A T I O N S OP) S E C T I O N 4.5

31

In the search f o r 2 - a m p l i f i e r AG c i r c u i t s , i t vas decided

( s e c t i o n 4.4) to consider only c i r c u i t s which have the s k e l e t o n

of F i g u r e 4.8(a), ( t e r m i n a l s b and d are connected). A pre

l i m i n a r y screening ( s e c t i o n 4.5) narrowed the multitude of poss

i b l e c i r c u i t c o n f i g u r a t i o n s to 1 6 * Each such c o n f i g u r a t i o n

represents a d i f f e r e n t connection of the input t e r m i n a l s of the

two a m p l i f i e r s . The problem at hand i s to determine which

c o n f i g u r a t i o n s y i e l d AG c i r c u i t s *

I t i s f i r s t shown that only 2 output connections need be

considered f o r each a m p l i f i e r . Thus, since each c o n f i g u r a t i o n

contains 2 a m p l i f i e r s , the problem reduces to choosing some or

a l l of the 4 p o s s i b l e output connections. A systematic method

i s given f o r making t h i s c h o i c e .

A . l The 4 P o s s i b l e Output Connections f o r Each C o n f i g u r a t i o n .

For each given c o n f i g u r a t i o n , the input t e r m i n a l connections

are f i x e d f o r the 2 a m p l i f i e r s . I t remains to connect the output

t e r m i n a l of each a m p l i f i e r , through separate conductances, to

1 or more po i n t s i n the c i r c u i t . Only the 3 p o i n t s a, c, and b-d

w i l l be considered as candidates f o r such connections. The ground

of each a m p l i f i e r w i l l a lready be connected to one of these 3

p o i n t s , and connection of the output to t h i s p o i n t as w e l l would

produce a loop current which would merely load the a m p l i f i e r .

82 Thus, only the remaining 2 p o i n t s need be considered as candid

ates f o r output connections. One of these 2 w i l l already be

connected to the input of the a m p l i f i e r ; t h i s p o i n t w i l l be

c a l l e d the "1" p o i n t , and the other the "2" p o i n t , f o r the amp

l i f i e r i n q u e s t i o n . The output terminals of the 2 a m p l i f i e r s

w i l l be l a b e l l e d P and Q, and a c c o r d i n g l y the 1 and 2 p o i n t s

f o r the P a m p l i f i e r w i l l be c a l l e d the P i and P2 p o i n t s , e t c .

(see Figure 4.4). Therefore, f o r each of the 16 given con

f i g u r a t i o n s , an attempt w i l l be made to design an AG by choice

of some or a l l of the 4 output t e r m i n a l connections P to PI, P to

P2, Q to Ql, and Q to Q2.

A.2 A Systematic Method of A n a l y s i s .

The systematic a n a l y s i s of these c i r c u i t s i s f a c i l i t a t e d

by the use of a t a b l e , as shown i n Figure A . l . The t a b l e cont

ains 4 rows ( l a b e l l e d 1 to 4 along the r i g h t hand side) and 11

columns ( l a b e l l e d 1 to 11 along the bottom), of which the f i r s t

2 and the 8th one are already f i l l e d i n . Each row r e l a t e s to a

p a r t i c u l a r Y matrix element, and both the matrix element concerned,

and i t s d e s i r e d value (equation 3.5)) are i n d i c a t e d i n the 8th

column.

For each given c o n f i g u r a t i o n , the a n a l y s i s i s done i n 2

steps. The f i r s t step i s to f i l l i n the diagram and v o l t a g e s

shown above the t a b l e , and columns 3 to 7 of the t a b l e . This

i s a s t r a i g h t f o r w a r d process. The second step i s to t r y and

8 3

a o O c k +

b o- •o d

'P-Pl V P-P2

rQ-Qi rQ-Q2

I n t r i n s i c term, Y i n t r

Terms Produced by Amps, AG admittance term

i Subtotals I n t r i n s i c term, Y i n t r

P A m o l i f i e r Q Ampl] f i fir

AG admittance term

i Subtotals I n t r i n s i c term, Y i n t r P l P2 : oi ••••' 02

AG admittance term

i Subtotals

Affec" I i 7

v i a ' u - 0 Affec"

I i 7 v i a Va

A f f ec-t I ?

v i a v, T21= " A f f ec-t I ? v i a

I22= 0

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) ( l l )

(I)

(3)

Fi g u r e A . l - Design sheet used f o r the i n v e s t i g a t i o n of AG c i r c u i t s .

84 design an AG, with the help of columns 3 to 7 of the t a b l e

(columns 9 to 11 are used to keep t r a c k of the progress made).

The procedure i s i l l u s t r a t e d by an example.

A.2.1 The f i r s t step.

The f i r s t step i s c a r r i e d out as f o l l o w s ( r e f e r to Figure

A.2) :

1) Draw i n the a m p l i f i e r s on the " s k e l e t o n " provided

(terminals a, c, and b-d), with t h e i r inputs connected

according to the p a r t i c u l a r c o n f i g u r a t i o n being analysed.

2) Label the v a r i o u s P and Q p o i n t s on t h i s diagram, as

d i s c u s s e d i n s e c t i o n A.1 above.

3) Write down the expressions f o r the 4 v o l t a g e s

Vp_p-^, e t c . , i n the space provided.

4) F i l l i n the blanks i n columns 3 to 7 of the t a b l e .

This process i s explained i n d e t a i l i n the next few

paragraphs.

Columns 3 to 7 of the t a b l e are designed to f a c i l i t a t e the

s e l e c t i o n of some or a l l of the 4 output connections P to PI,

e t c . , i n an attempt to o b t a i n a Y matrix having the form of

(3.5). They are completed i n such a manner that the matrix

element Y - ^ J f ° r example, can be found by merely adding together

( s u p e r p o s i t i o n theorem) the e n t r i e s i n the Y ^ 2 r o'w which c o r

respond to the p a r t i c u l a r output connections used.

The terms i n column 3 represent the c o n t r i b u t i o n to the Y

matrix made by the input conductances, G- and G 2 > and are t h e r e

i n : :—: ~~ !

• Whenever the Y matrix i s mentioned i n t h i s appendix,: •. r e f e r ence i s intended to the Y matrix f o r the terminals a,b,c,d, as d e f i n e d i n Figure 4.2.

85

I n t r i n s i c term, x i n t r


Subtotals I n t r i n s i c term, x i n t r

P Amu'lifier Q Ampl] f i e r

AG admittance term

Subtotals I n t r i n s i c term, x i n t r PI P2 01 * 02

AG admittance term

Subtotals

Affec-I ? 1

v i a (aw T n = ° o Affec-

I ? 1

v i a v2 0 i 1 2 = + o 0 Affeci I ?

v i a v, -6, 0 0 Affeci I ? v i a

Vz

0 T 2 2 = 0 o 0 0

+ a i I

b 6 P 2 , Ql O d

P-Pl

P-P2

Q-Qi Q-Q2

V l 0 - A , H V a ( A l - l )

V.O-A,) +V 2A,

(0

(3)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

for V 2 2 = 0^ PI must b*. uSZ<J$

*7""/?€ <s6ôt<*Is art, ^./Vei /Vi C<J/<***** 3»

f o r V 2y - V e j <#2 mus^ ^ je« / ,

*ôiS^ —* V e .

/ ^ r Yn~Oj p2. must £e US€</,

refn^y '*•» co/u*,* /Ij r e p u t e a h A<J-»

Figure A.2 -. Design sheet f o r c o n f i g u r a t i o n number. 8, D-1-.

86

f o r e c a l l e d " i n t r i n s i c t e r m s " . T h e y a r e f o u n d b y r e m o v i n g

e v e r y t h i n g b u t G a n d G^ f r o m t h e c i r c u i t s k e l e t o n , as shown

f o r t h e D - l - c o n f i g u r a t i o n i n F i g u r e A . 3 .

I 2 j

G l + G2 ~G1

-G, G,

F i g u r e A . 3 - E q u i v a l e n t c i r c u i t f o r f i n d i n g t h e " i n t r i n s i c t e r m s " f o r t h e c o n f i g u r a t i o n D - l - .

The t e r m s i n c o l u m n s 4 t o 7 , t h e " a m p l i f i e r t e r m s " , r e p r e s

e n t t h e c o n t r i b u t i o n s t o t h e Y m a t r i x w h i c h w i l l be made b y e a c h

o u t p u t c o n n e c t i o n ( i f i t i s u s e d ) . T h e s e t e r m s c a n be d e t e r

m i n e d , f o r e a c h o u t p u t c o n n e c t i o n , b y d r a w i n g a n e q u i v a l e n t c i r

c u i t s i m i l a r t o t h a t o f F i g u r e A . 3 . The e q u i v a l e n t c i r c u i t s f o r

t h e two P c o n n e c t i o n s i n t h e D - l - c o n f i g u r a t i o n shown i n F i g u r e

A . 2 a r e g i v e n b e l o w i n F i g u r e A . 4 , a l o n g w i t h t h e t e r m s t h e y

p r o d u c e . The t e r m s c a n a l s o be o b t a i n e d d i r e c t l y f r o m t h e

v o l t a g e s Vp_p^> e t c . , w i t h t h e h e l p o f c o l u m n s 1 a n d 2 o f t h e

t a b l e . F o r i n s t a n c e , t h e P2 c o n n e c t i o n w o u l d c a u s e a c u r r e n t t o

f l o w f r o m a t o b - d , a n d t h u s a f f e c t o n l y 1^ ( see c o l u m n 1 o f t h e

t a b l e ) . T h i s c u r r e n t w o u l d be e q u a l t o

G 4 V P - P 2 = W l - V + V 2 G 4 A X ,

The i n t r i n s i c t e r m s w i l l be t h e same, o f c o u r s e , f o r a l l o f t h e 4 c o n f i g u r a t i o n s D+1+.

87

O c

r+ VP-P2

i - — O d

"G^l-A^ Vl 0 0

Figure A.4 - E q u i v a l e n t c i r c u i t s f o r f i n d i n g some of the "ampl i f i e r terms" f o r the c o n f i g u r a t i o n D-1-. Y mat

r i c e s are given.

i n the same d i r e c t i o n as 1^, and thus i t s e f f e c t on 1^ v i a

(see column 2 of the t a b l e ) would y i e l d the admittance term

G^(l-A^), and i t s e f f e c t on 1^ v i a would y i e l d the term GÂ^.

These admittance terms are entered i n the proper spaces i n the

ta b l e (column 5 i n t h i s case). When columns 3 to 7 have been

f i l l e d , the f i r s t step i s completed.

A. 2.2 The second step.

The second step i s i n t e l l i g e n t l y to s e l e c t which output

connections to use. Refer to the t a b l e of Figure A.2. In the

row f o r Y - , the i n t r i n s i c term i s (always) +'ve, and t h e r e f o r e

a -'ve a m p l i f i e r term i s needed to make Y ^ = 0; any of the 4

output connections could be used f o r t h i s purpose. In the row

f o r Y^2> the i n t r i n s i c term i s (always) -'ve, and t h e r e f o r e a

+'ve a m p l i f i e r term i s needed to make Y, - = +G : e i t h e r con-1 12 m' n e c t i o n P l or P2 could be used f o r t h i s purpose. In the row f o r

CKjU-A^-- G 3(A 1-1)

G 3(A X-1) G 3 ( l - A 1 )

88 ^2±9 ^ e i n t r i n s i c term i s (always) -'ve, and thus no change i s

needed to have Y_, = -G (=-G.. here) , so t h i s row need not be 21 n 1 '

considered f o r the time being. F i n a l l y , f o r the Y 2 2 r o w > ^ n e

i n t r i n s i c term i s (always) +'ve, and t h e r e f o r e a -'ve a m p l i f i e r

term i s needed to make ^ 2 = ^' o n l y "the PI connection can provide

such a term. Now, a f t e r l o o k i n g at a l l 4 rows, what can be

concluded? Obviously, the PI connection must be used, f o r i t

i s the only one which can be used to set 22= 0. The elements

of the Y matrix when G^ i s connected from P to PI are found by

adding the terms i n columns 3 and 4. Of course, the c o n d i t i o n

Y 2 2 = G x + G 3 ( l - A 1 ) = 0 . . . ( l )

must be imposed; t h i s i s equation ( l ) i n Figure A.2. A^ must be

+'ve to s a t i s f y t h i s c o n d i t i o n . The elements of the r e s u l t a n t

Y matrix are w r i t t e n i n column 9 (columns 9 to 12 are provided

f o r t h i s purpose).

Now the e n t r i e s i n column 9 must be compared with those i n

the columns f o r the s t i l l unused output connections P2, Ql, and

Q2. This comparison i s s i m i l a r to that made i n the paragraph

above, except that t h i s time the row f o r n e e a - n o t be c o n s i d

ered. I t i s seen that connection Q2 must be used, f o r i t i s

the only one which can be used to make Y^^ -'ve. Note that

t h i s connection w i l l a lso a f f e c t Y 2 2 , so the c o n d i t i o n of equation

( l ) must be a l t e r e d to read

Y 2 2 = G l + G 6 + G 3 ( l - A l ) = 0. ...... ( l a )

89 The c o n d i t i o n f o r Y 21 to be negative i s

Y 21 = -G 1 + G 3(A 1-1) + G 6 ( A 2 - l ) • 9 • (2)

n

A 2 must be -'ve to s a t i s f y t h i s c o n d i t i o n . The, r e s u l t a n t Y

matrix i s w r i t t e n i n column 10..

Another comparison i s made, t h i s time between column 10,

and the columns f o r the unused output connections, P2 and Q l .

I t i s seen that connection P2 must be used, both to, set - 0,

and to make Y _ +'ve. The c o n d i t i o n imposed i s

The r e s u l t a n t Y matrix i s shown i n column 11. . Since t h i s matrix

has the form of (3.5), the a n a l y s i s stops here. Con c l u s i o n :

the given c o n f i g u r a t i o n y i e l d s an AG.

below the ta b l e i n Figure A.2. The c o n f i g u r a t i o n j u s t analysed

a c t u a l l y y i e l d e d an AG; however, some of the c o n f i g u r a t i o n s w i l l

not (e.g., the D+l- c o n f i g u r a t i o n , analysed i n Figure A.6). The

a n a l y s i s sheets f o r s e v e r a l other c o n f i g u r a t i o n s are given i n

Fi g u r e s A.5 to A.7.

T l l = G2 + G n + G 4 ( l - A x ) ;= 0. (3)

The f o r e g o i n g a n a l y s i s i s d e s c r i b e d c o n c i s e l y i n the space

I n t r i n s i c term, y i n t r


Su b t o t a l s I n t r i n s i c term, y i n t r

P A m p l i f i e r Q Amp!] f i p r

AG admittance term

Su b t o t a l s I n t r i n s i c term, y i n t r

P l P2 Ql • 02

AG admittance term

Su b t o t a l s

Affec-v i a = Vi

0 i n = o 0 Affec-v i a v 2

0 o 0. Affec-t I ?

v i a v, -<?i o Affec-t

I ? v i a v 2

&. 0 I 2 2 - o o O

_9_0_

V.

Pi,ai a 0 —

P2 o d

p P 1 =M(Ari)+v2(i-A,)

V2-Q2 =V ,A 2 - V 2

V

V

V

(0

(3)

/ô-r - V e ; <P2 A^JT* i*, u s e * / , wrid flz 7^\« |

For- y / f ~ o , ^

F i g u r e A, 5 -. Design sheet f o r c o n f i g u r a t i o n number 5, D+1-+.

91

b 6

Pi Q2.'

P2,QI -O d

'P-Pl V P-P2

Q-Qi Q-Q2

I n t r i n s i c term, i n t r


Subtotals I n t r i n s i c term, i n t r

P A m u l i f i e r 0 A.mpl I f i e r

AG admittance term

Subtotals I n t r i n s i c term, i n t r PI P2 •. Ql - 02

AG admittance term

Subtotals

A f f e c -v i a

&,+<*, 0 G5(l-A2\ A f f e c -v i a v 2

0 0 T i . ; -A *

A f f e c t I ?

v i a v, 64 A, 0 Y21= " A f f e c t I ? v i a 6 y H ) 0 ^ 6 *22= 0

(0

(3)

(1) (2) (3) (4) ( 5 ) ( 6 ) (7) (8) (9) (10) (11)

For- y a , P/ J>€ OfS<zd.

fee Q2. t^^st he. otS^d.

'* 9\2. Must oe. — V e . 7 ^ e so,{> to~tci/s are. <pVe/i /V? ô/oirny\ /O.

For- Yzz:~Oj 5 e £

7 ^ € r e ^ r « < /-e cahf^rci"Hoy\ does /\/bT~ yfe/d Qh f\G0

F i g u r e A.6 - Design sheet f o r c o n f i g u r a t i o n number 6, D+1-,

92

I n t r i n s i c term, y m t r


Subtotals I n t r i n s i c term, y m t r

P A m p l i f i e r Q A.mpl -i f i P r

AG admittance term

Subtotals I n t r i n s i c term, y m t r P l P2 Ql # 02

AG admittance term

Subtotals

Affec-I ? 1

v i a >V. T l l = 0

& < 0 Affec-

I ? 1

v i a Va 0 T12= +

Affec-t I ? 2

v i a v, 0 T 2 1 = " -Cr, Affec-t I ? 2

v i a v 2 0 0 i 2 2 = o 0 o

QZ iP a Q

PI, Ql ,

b o- P2 ^

r P - P i

P-P2

rQ-Q2

(0

(3)

0 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

/ S r y,/-0, e/"Her- PI cr PZ m«S"6 6e use,*/. If PI rSdSeJ} 67 e re^/retj cawJ^r^ Kmfas VÔar u/^ll. •/laar« r/*ice d crkhvc c^nec^ 1V1C affect yZf y*~P£with used* 7fi9. coh-c/rfian imposed TS <r( -J-<j-y. 0~f\,) = O . '

/hrs c/rctf/t? /y e^t/Vdle*?^' 6* i f ^ e -oLive* &y

Figure A.7 - Design sheet f o r c o n f i g u r a t i o n number 11, D-2+.

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INDUCTANCE SIMULATION FOR MICROELECTRONICS AND