Microelectronic Circuits SJTU Yang Hua Chapter 7 Frequency Response Introduction 7.1 s-Domain analysis: poles,zeros and bode plots 7.2 the amplifier transfer

Microelectronic Circuits SJTU Yang Hua

Chapter 7 Frequency Response

Introduction7.1 s-Domain analysis: poles,zeros and bode plots7.2 the amplifier transfer function 7.3 Low-frequency response of the common-source and common-emitter amplifier7.4 High-frequency response of the CS and CE amplifiers7.5 The CB, CG and cascode configurations

SJTU Yang HuaMicroelectronic Circuits

Introduction Why shall we study the frequency response? Actual transistors exhibit charge storage

phenomena that limit the speed and frequency of their operation.

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Aims: the emphasis in this chapter is on analysis. focusing attention on the mechanisms that limit frequency response and on methods for extending amplifier bandwidth.


Three parts:

s-Domain analysis and the amplifier transfer function (April 13,2008)

High frequency model of BJT and MOS; Low-frequency and High-frequency response of the common-source and common-emitter amplifier (April 15,2008)

Frequency response of cascode, Emitter and source followers and differential amplifier (April 22,2008)


Part I:

s-Domain analysisZeros and poles Bode plotsThe amplifier transfer function


7.1 s-Domain analysis– Frequency Response

Transfer function: poles, zeros

Examples: high pass and low pass

Bode plots: Determining the 3-dB frequency



Most of our work in this chapter will be concerned with finding amplifier voltage gain as a transfer function of the complex frequency s.

A capacitance C: is equivalent an impedance 1/SC

An inductance L: is equivalent an impedance SL

Voltage transfer function: by replacing S by jw, we can obtain its magnitude response and phase response

01

1

01

1

...

...)(/)()(

bsbs

asasasVsVsT

nn

n

mm

mm

io



Z1, Z2, … Zm are called the transfer-function zeros or transmission zeros.

P1, P2, … Pm are called the transfer-function poles or natural modes.

The poles and zeros can be either real or complex numbers, the complex poles(zeros) must occur in conjugate pairs.

))......()((

))......()(()(/)()(

21

21

n

mmio PsPsPs

ZsZsZsasVsVsT


First-order Functions

All the transfer functions encountered in this chapter have real poles and zeros and can be written as the product of first-order transfer functions.

ω0, called the pole frequency, is equal to the inverse of the time constant of circuit network(STC).

0

01)(

s

asasT

0

1

0

0

)( :passHigh

)( :pass Low

s

sasT

s

asT


Example1: High pass circuit

2

1

L

Lu

f

f

f

f

A

RCs

s

RCs

RCs

RCj

RCjsT

/111)(

Lf

farctan90

RC is the time constant; ωL=1/RC


Example2: Low pass circuit

RCs

RCsT

/1

/1)(

Hf

farctan

RC is the time constant; ωH=1/RC

2

1

1

H

u

ff

A


Bode Plots

A simple technique exists for obtaining an approximate plot of the magnitude and phase of a transfer function given its poles and zeros. The resulting diagram is called Bode plots

A transfer function consists of A product of factors of the form s+a

210

2210 )/(1log20log20 aa


Bode Plots


Bode Plots


Example 7.1

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7.2 the amplifier transfer function

(a) a capacitively coupled amplifier (b) a direct-coupled amplifier

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LH

ABWAGB

BW

:productbandwidth -Gain

:Bandwidth


The Gain Function

Gain function

Midband: No capacitors in effect

Low-frequency band: coupling and bypass capacitors in effect

High-frequency band: transistor internal capacitors in effect

)()( sFAsA HM

)()()( sFsFAsA HLM

)()( sFAsA LM

MAsA )(


The low-Frequency Gain Function

Gain function

ωP1 , ωP2 , ….ωPn are positive numbers representing the frequencies of the n real poles.

ωZ1 , ωZ2 , ….ωZn are positive, negative, or zero numbers representing the frequencies of the n real transmission zeros.

pnLpp

ZnLZZL

LM

sss

ssssF

sFAsA

.....

.....)(

)()(

21

21


Determining the 3-dB Frequency

Definition

or

Assume ωP1< ωP2 < ….<ωPn and ωZ1 < ωZ2 < ….<ωZn

2)( M

LAA dBAA ML 3)(

...)(2... 22

21

22

21 ZZPPL



Dominant pole

If the lowest-frequency pole is at least two octaves (a factor of 4) away from the nearest pole or zero, it is called dominant pole. Thus the 3-dB frequency is determined by the dominant pole.

Single pole system,

1

1

)(

PL

P

M

s

sAsA


The High-Frequency Gain Function

Gain function

ωP1 , ωP2 , ….ωPn are positive numbers representing the frequencies of the n real poles.

ωZ1 , ωZ2 , ….ωZn are positive, negative, or infinite numbers representing the frequencies of the n real transmission zeros.

)1().....1()1(

)1.....()1)(1()(

)()(

21

21

PnPP

ZnZZH

HM

sss

ssssF

sFAsA



Definition

or

Assume ωP1< ωP2 < ….<ωPn and ωZ1 < ωZ2 < ….<ωZn

2)( M

HAA dBAA MH 3)(

....)11

(2....)11

(1 22

21

22

21

ZZPP

H



Dominant pole

If the lowest-frequency pole is at least two octaves (a factor of 4) away from the nearest pole or zero, it is called dominant pole. Thus the 3-dB frequency is determined by the dominant pole.

Single pole system,

1

1/1)(

PH

P

M

s

AsA


approx. determination of corner frequency

Using open-circuit time constants for computing high-frequency 3-dB Frequency: reduce all other C to zero; reduce the input source to zero.

nH

iioi

H

nH

iioi

nH

nHnH

nHnH

H

RC

pRCb

pppb

sbsbsb

sasasasF

1

11

21

221

221

1

1

1...

11

...1

...1)(

1

1


approx. determination of corner frequency

Using short-circuit time constants for computing low-frequency 3-dB Frequency: replace all other C with short circuit; reduce the input source to zero.

nL

i ioiL

p

nL

i ioi

nL

nLnL

nLnL

L

RC

RCe

pppe

seses

sdsdssF

1

11

21

22

11

22

11

1

exists poledominant if 1

...

...

...)(

1

1


Example7.3

)25)(100(

)10()(

ss

sssFL


Example7.4

)104/1)(10/1(

10/1)(

44

5

ss

ssFH


summary (The Fouth Edition:P601)

(A) Poles and zeros are known or can be easily determined

Low-frequency High-frequency

(B) Poles and zeros can not be easily determined

Low-frequency High-frequency

...)(2... 22

21

22

21 ZZPPL

1

1

)(

PL

P

M

s

sAsA

nH

iioi

H

nH

iioi

nH

nHnH

nHnH

H

RC

pRCb

pppb

sbsbsb

sasasasF

1

11

21

221

221

1


1...

11

...1

...1)(

1

1

nL

i ioiL

p

nL

i ioi

nL

nLnL

nLnL

L

RC

RCe

pppe

seses

sdsdssF

1

11

21

22

11

22

11

1


...

...

...)(

1

1

pnLpp

ZnLZZL

LM

sss

ssssF

sFAsA

.....

.....)(

)()(

21

21

)1().....1()1(

)1.....()1)(1()(

)()(

21

21

PnPP

ZnZZH

HM

sss

ssssF

sFAsA

1

1/1)(

PH

P

M

s

AsA

....)11

(2....)11

(1 22

21

22

21

ZZPP

H


Homework

April 17th, 2008

7.1; 7.2; 7.7; 7.10


Part II: Internal Capacitances of the BJT BJT High Frequency Model Internal Capacitances of the MOS MOS High Frequency Model Low-frequency of CS and CS amplifiers


Internal capacitance The base-charging or diffusion capacitance Junction capacitances

The base-emitter junction capacitance The collector-base junction capacitance

High frequency small signal model Cutoff frequency and unity-gain frequency

Internal Capacitances of the BJT and High Frequency Model


Diffusion capacitance almost entirely exists in forward-biased pn junction

Expression of the small-signal diffusion capacitance

Proportional to the biased current100ps-base10ps the

crossingin spendscarrier charge a timeaverage

therepresents me,transit ti-base forward:F

T

CFmFde V

IgC

The Base-Charging or Diffusion Capacitance


The Base-Emitter Junction Capacitanc

The collector-base junction capacitance

0.9v) ,(tipicallyin voltage build theis :

voltagezeroat C of value theis :

2)1(

je0

00

oe

je

jem

oe

BE

jeje

V

C

C

VV

CC

0.75v) ,(typicallyin voltage bulid CBJ theis :

voltagezeroat of value theis:

)1(

0

0

oc

m

oc

CB

V

CC

VV

CC

Junction Capacitances


Two capacitances Cπ and Cμ , where

One resistance rx . Accurate value is obtained from high frequency measurement.

jede CCC

The High-Frequency Hybrid- Model

rrx


Circuit for deriving an expression for

According to the definition, output port is short circuit0

)(

CEvB

Cfe I

Ish

The Cutoff and Unity-Gain Frequency


Expression of the short-circuit current transfer function

Characteristic is similar to the one of first-order low-pass filter

rCCs

sh fe )(1)( 0



rCC )(

1

rCC

gmT

0



The MOSFET Internal Capacitance and High-Frequency Model

Internal capacitances The gate capacitive effect

Triode region Saturation region Cutoff region Overlap capacitance

The junction capacitances Source-body depletion-layer capacitance drain-body depletion-layer capacitance

High-frequency model


The Gate Capacitive Effect

MOSFET operates at triode region

MOSFET operates at saturation region

MOSFET operates at cutoff region

oxgdgs WLCCC 21

032

gd

oxgs

C

WLCC

oxgb

gdgs

WLCC

CC 0


Overlap Capacitance

Overlap capacitance results from the fact that the source and drain diffusions extend slightly under the gate oxide.

The expression for overlap capacitance

Typical value

This additional component should be added to Cgs and Cgd in all preceding formulas.

oxovov CWLC

LLov 1.005.0


The Junction Capacitances

• Source-body depletion-layer capacitance

• drain-body depletion-layer capacitanceo

SB

sbsb

V

V

CC

＋1

0

o

DB

dbdb

VV

CC

＋1

0


High-Frequency MOSFET Model


High-Frequency Model

(b) The equivalent circuit for the case in which the source is connected to the substrate (body).

(c) The equivalent circuit model of (b) with Cdb neglected (to simplify analysis).


The MOSFET Unity-Gain Frequency

Current gain

Unity-gain frequency

)( gdgs

m

i

o

CCs

g

I

I

)(2 gdgs

mT CC

gf


7.3 Low-frequency response of the CS and CE amplifiers


Low-frequency response of the CS


The procedure to find quickly the time constant:1. Reduce Vsig to zero2. Consider each capacitor separately; that is , assume that the other Capacitors are as perfect short circuits3. For each capacitor, find the total resistance seen between its terminals


Analysis of Common-emitter amplifier

Low frequency small signal analysis:1) Eliminate the DC source2) Ignore Cπ and Cμ and ro3) Ignore rx, which is much smaller than rπ


(b) the effect of CC1 is determined with CE and CC2 assumed to be acting as perfect short circuits;


(c) the effect of CE is determined with CC1 and CC2 assumed to be acting as perfect short circuits


(d) the effect of CC2 is determined with CC1 and CE assumed to be acting as perfect short circuits;


(e) sketch of the low-frequency gain under the assumptions that CC1, CE, and CC2 do not interact and that their break (or pole) frequencies are widely separated.


Homework:

April 15th, 2008

7.29; 7.35; 7.39


7.4 High-frequency response of the CS and CE amplifiers

• Miller’s theorem.

• Analysis of the high frequency response.

Using Miller’s theorem.

Using open-circuit time constants.


High-Frequency Equivalent-Circuit Model of the CS Amplifier

Fig. 7.16 (a) Equivalent circuit for analyzing the high-frequency response of the amplifier circuit of Fig. 7.15(a). Note that the MOSFET is replaced with its high-frequency equivalent-circuit. (b) A slightly simplified version of (a) by combining RL and ro into a single resistance R’L = RL//ro.


High-Frequency Equivalent-Circuit Model of the CE Amplifier

Fig. . 7.17 (a) Equivalent circuit for the analysis of the high-frequency response of the common-emitter amplifier of Fig. 7.15(b). Note that the BJT is replaced with its hybrid- high-frequency equivalent circuit. (b) An equivalent but simpler version of the circuit in (a),

oLL

xss

ssss

rRR

rrRR

rrR

rVV

//

// '

'

'


Miller’s Theorem

Impedance Z can be replaced by two impedances:

Z1 connected between node 1 and ground

Z2 connected between node 2 and ground


Miller’s Theorem

The miller equivalent circuit is valid as long as the conditions that existed in the network when K was determined are not changed.

Miller theorem can be used to determining the input impedance and the gain of an amplifier; it cannot be applied to determine the output impedance.


Analysis Using Miller’s Theorem

Approximate equivalent circuit obtained by applying Miller’s theorem.

This model works reasonably well when Rsig is large.

The high-frequency response is dominated by the pole formed by Rsig and CT.

)1(

)1('

'

'

LmgdgsT

Lmgdeq

Lgsmo

RgCCC

RgCC

RVgV

Neglecting the current through Cgd


Analysis Using Miller’s Theorem

Using miller’s theorem the bridge capacitance Cgd can be replaced by two capacitances which connected between node G and ground, node D and ground.

The upper 3dB frequency is only determined by this pole.

sigTH

LmgdgsT

RCf

RgCCC

2

1

)1( '


Analysis Using Open-Circuit Time Constants

siggs RR '')1( LLmsiggd RRgRR


Analysis Using Open-Circuit Time Constants

'LC RR

L


High-Frequency Equivalent Circuit of the CE Amplifier

Thevenin theorem:


Equivalent Circuit with Thévenin Theorem Employed


The Situation When Rsig Is Low

High-frequency equivalent circuit of a CS amplifier fed with a signal source having a very low (effectively zero) resistance.



Bode plot for the gain of the circuit in (a).

fT, which is equal to the gain-bandwidth product



The high frequency gain will no longer be limited by the interaction of the source resistance and the input capacitance.

The high frequency limitation happens at the amplifier output.

To improve the 3-dB frequency, we shall reduce the equivalent resistance seen through G(B) and D(C) terminals.


7.5 Frequency Response of the CG and CB Amplifier

High-frequency response of the CS and CE Amplifiers is limited by the Miller effect Introduced by feedback Ceq.To extend the upper frequency limit of a transistor amplifier stage one has to reduce or eliminate the Miller C multiplication.


sCr

sCgrV

I

sCgr

VVgsCr

VI

em

e

e

meme

11

11

Lgdp

Lp

smgsp

sep

RCRC

RgCRrC

1:amplifierCG for

1

///1

1 :amplifierCG for

//

1

22

11

CB amplifier has a much higher upper-cutoff frequency than that of CE amplifier


Comparisons between CG(CB) and CS(CE)

Open-circuit voltage gain for CG(CB) almost equals to the one for CS(CE)

Much smaller input resistance and much larger output resistance

CG(CB) amplifier is not desirable in voltage amplifier but suitable as current buffer.

Superior high frequency response because of the absence of Miller’s effects

Cascode amplifier is the significant application for CG(CB) circuit


Frequency Response of the BJT Cascode


Frequency Response of the BJT Cascode

The cascode configuration combines the advantages of the CE and CB circuits

SxLm

S

oM

L

e

HS

Rrr

rRg

V

VA

RC

rC

CCR

23

222

11'1

1

1

2

1


The Source (Emitter) Follower

• Self-study• Read the textbook from pp626-629


Homework:

May 6th,2008

7.44; 7.45; 7.46; 7.57


Test: （ 1 ） the reason for gain decreasing of high-frequency respons

e ， the reason for gain decreasing of low-frequency response 。

A. coupling capacitors and bypass capacitors B. diffusion capacitors and junction capacitors C. linear characteristics of semiconductors D. the quiescent point is not proper （ 2 ） when signal frequency is equal to fL or fH ， the gain of of

amplifier decreases compare to that of midband frequency 。 A.0.5times B.0.7times C.0.9times that is decreasing 。 A.3dB B.4dB C.5dB （ 3 ） The CE amplifier circuit ， when f ＝ fL ， the phase is 。 A. ＋ 45˚ B. － 90˚ C. － 135˚ when f ＝ fH ， the phase is 。 A. － 45˚ B. － 135˚ C. － 225˚

Documents

Microelectronic Circuits SJTU Yang Hua Chapter 7 Frequency Response Introduction 7.1 s-Domain analysis: poles,zeros and bode plots 7.2 the amplifier transfer