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Chapter 7
Frequency Response
Introduction
7.1 s-Domain analysis: poles,zeros and bode plots
7.2 the amplifier transfer function
7.3 Low-frequency response of the common-source
and common-emitter amplifier
7.4 High-frequency response of the CS and CE
amplifiers
7.5 The CB, CG and cascode configurations
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Introduction


Why shall we study the frequency response?
Actual transistors exhibit charge storage
phenomena that limit the speed and frequency of
their operation.
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.
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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)
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Part I:




s-Domain analysis
Zeros and poles
Bode plots
The amplifier transfer function
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7.1 s-Domain analysis–
Frequency Response

Transfer function: poles, zeros

Examples: high pass and low pass

Bode plots: Determining the 3-dB
frequency
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Transfer function: poles, zeros

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
am s m  am1s m1  ...  a0
T ( s)  Vo ( s) / Vi ( s) 
s n  bn 1s n 1  ...  b0
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Transfer function: poles, zeros
( s  Z1 )( s  Z 2 )......( s  Z m )
T ( s )  Vo ( s ) / Vi ( s )  am
( s  P1 )( s  P2 )......( s  Pn )

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.
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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
Low pass : T ( s ) 
equal to the inverse of the time
constant of circuit
network(STC).
a0
s  0
High pass : T ( s ) 
a1s
s  0
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a1s  a0
T ( s) 
s  0
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Example1: High pass circuit

Au 
f
fL
 f 
1   
 fL 
  90  arctan
2
f
fL
jRC
RCs
s
T ( s) 


jRC  1 RCs  1 s  1 / RC
RC is the time constant; ωL=1/RC
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Example2: Low pass circuit
1 / RC
T ( s) 
s  1 / RC
RC is the time constant; ωH=1/RC

Au 
1
 f 

1  
 fH 
   arctan
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2
f
fH
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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
20 log 10 a 2   2  20 log 10 1  ( / a) 2
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Bode Plots
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Bode Plots
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Example 7.1
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7.2 the amplifier transfer function
(a) a capacitively coupled amplifier
(b) a direct-coupled amplifier
Bandwidth : BW  H  L
Gain - bandwidth product : GB  AM BW  AM H
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The Gain Function
A(s)  AM FL (s) FH (s)

Gain function

Midband: No capacitors in effect
A(s)  AM

Low-frequency band: coupling and bypass
capacitors in effect
A(s)  AM FL (s)

High-frequency band: transistor internal capacitors
in effect
A(s)  AM FH (s)
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The low-Frequency Gain Function

Gain function
A( s)  AM FL ( s)

s  Z 1 s  Z 2 .....s  ZnL 
F ( s) 
L
s   s   .....s   
p1
p2
pnL

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.
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Determining the 3-dB Frequency

Definition
A(L )  AM  3dB

or
A( L ) 
AM
2
Assume ωP1< ωP2 < ….<ωPn and ωZ1 < ωZ2
< ….<ωZn
L  
2
P1
 P 2  ...  2(
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2
2
Z1
 Z 2  ...)
2
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Determining the 3-dB Frequency

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,
AM s
A( s ) 
s   P1
 L   P1
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The High-Frequency Gain Function

Gain function
A( s )  AM FH ( s )
(1  s Z 1 )(1  s Z 2 ) .....(1  s  Zn )
FH ( s ) 
(1  s  P1 ) (1  s  P 2 )..... (1  s  Pn )

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.
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Determining the 3-dB Frequency

Definition
or

A( H )  AM
A(H )  AM  3dB
2
Assume ωP1< ωP2 < ….<ωPn and ωZ1 < ωZ2
< ….<ωZn
H  1
(
1
P1
2
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
1
P 2
2
 ....)  2(
1
Z 1
2

1
Z 2
2
 ....)
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Determining the 3-dB Frequency

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,
AM
A( s ) 
1  s /  P1
 H   P1
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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.
1  a1s  a2 s 2  ...  anH s nH
FH ( s ) 
1  b1s  b2 s 2  ...  bnH s nH
b1 
1
1
1

 ... 
p1 p2
pnH
nH
1
b1   Ci Rio 
p1
i 1
H 
1
nH
C R
i 1
i
io
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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.
s nL  d1s nL1  d 2 s 2  ...
FL ( s)  nL
s  e1s nL1  e2 s 2  ...
e1  p1  p2  ...  pnL
nL
1
  p1 if dominant pole exists
e1  
i 1 Ci Rio
nL
1
L  
i 1 Ci Rio
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Example7.3
FL ( s) 
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s( s  10)
( s  100)( s  25)
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Example7.4
1  s / 10 5
FH ( s) 
(1  s / 10 4 )(1  s / 4 10 4 )
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summary (The Fouth Edition:P601)
(A) Poles and zeros are known or can be easily determined
Low-frequency
High-frequency
A( s)  AM FL ( s)
A( s )  AM FH ( s )
FL ( s) 
FH ( s ) 
s  Z1 s  Z 2 .....s  ZnL 
s   s   .....s   
p1
A( s ) 
(1  s Z 1 )(1  s Z 2 ) .....(1  s Zn )
(1  s  P1 ) (1  s  P 2 )..... (1  s  Pn )
AM
A( s ) 
1  s /  P1
p2
pnL
AM s
s   P1
 H   P1
 L   P1
L 

2
P1
 P 2  ...  2(
2
2
Z1
 Z 2  ...)
2
H  1
(
1
P1
2

1
P 2
2
 ....)  2(
1
Z 1
2

1
Z 2 2
 ....)
(B) Poles and zeros can not be easily determined
Low-frequency
s nL  d1s nL1  d 2 s 2  ...
FL ( s)  nL
s  e1s nL1  e2 s 2  ...
e1  p1  p2  ...  pnL
nL
e1  
i 1
1
  p1 if dominant pole exists
Ci Rio
nL
L  
i 1
1
Ci Rio
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High-frequency
FH ( s ) 
b1 
1  a1s  a2 s 2  ...  anH s nH
1  b1s  b2 s 2  ...  bnH s nH
1
1
1

 ... 
p1 p2
pnH
nH
b1   Ci Rio 
i 1
H 
1
if dominant pole exists
p1
1
nH
C R
i 1
i
io
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Homework

April 17th, 2008
7.1; 7.2; 7.7; 7.10
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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
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Internal Capacitances of the BJT and
High Frequency Model

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
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The Base-Charging or Diffusion
Capacitance


Diffusion capacitance almost entirely exists in
forward-biased pn junction
Expression of the small-signal diffusion
capacitance
IC
Cde   F g m   F
VT
 F : forward base - transit ti me, represents the
average time a charge carrier spends in crossing
the base10ps - 100ps

Proportional to the biased current
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Junction Capacitances

The Base-Emitter Junction Capacitanc
C je
C je0

 2C je0
VBE m
(1 
)
Voe
C je0 : is the value of Cje at zero voltage
Voe : is the build in voltage (tipically , 0.9v)

The collector-base junction capacitance
C 0
C 
V
(1  CB ) m
Voc
C  0 :is the value of C  at zero voltage
Voc : is the CBJ bulid in voltage (typically , 0.75v)
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The High-Frequency Hybrid- Model
C  Cde  C je

Two capacitances Cπ and Cμ , where

One resistance rx . Accurate value is obtained from high
frequency measurement.
rx  r
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The Cutoff and Unity-Gain
Frequency


IC
h fe ( s ) 
Circuit for deriving an expression for
IB
vCE  0
According to the definition, output port is short
circuit
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The Cutoff and Unity-Gain Frequency

Expression of the short-circuit current
transfer function
h fe ( s) 

0
1  s(C  C )r
Characteristic is similar to the one of firstorder low-pass filter
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The Cutoff and Unity-Gain Frequency
1
 
(C  C )r
gm
T   0 
C  C r
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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
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The Gate Capacitive Effect

MOSFET operates at triode region
C gs  C gd  12 WLCox

MOSFET operates at saturation region
C gs  23 WLC ox

C gd  0

MOSFET operates at cutoff region
C gs  C gd  0

C gb  WLCox
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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
Cov  WLovCox

Typical value

This additional component should be added to Cgs and Cgd in
all preceding formulas.
Lov  0.05  0.1L
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The Junction Capacitances
•
Source-body depletion-layer capacitance
C sb 
•
C sb0
VSB
1＋
Vo
drain-body depletion-layer capacitance
C db 
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C db 0
VDB
1＋
Vo
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High-Frequency MOSFET Model
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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).
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The MOSFET Unity-Gain Frequency

Current gain
Io
gm

I i s(Cgs  Cgd )

Unity-gain frequency
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gm
fT 
2 (Cgs  Cgd )
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7.3 Low-frequency response of
the CS and CE amplifiers
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Low-frequency response of the CS
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The procedure to find quickly the time constant:
1. Reduce Vsig to zero
2. Consider each capacitor separately; that is , assume that the other
Capacitors are as perfect short circuits
3. For each capacitor, find the total resistance seen between its terminals
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Analysis of Common-emitter amplifier
Low frequency small
signal analysis:
1) Eliminate the DC
source
2) Ignore Cπ and Cμ
and ro
3) Ignore rx, which is
much smaller than
rπ
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(b) the effect of CC1 is determined with CE and CC2 assumed to be acting as perfect short circuits;
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(c) the effect of CE is determined with CC1 and CC2 assumed to be acting as perfect short circuits
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(d) the effect of CC2 is determined with CC1 and CE assumed to be acting as perfect short circuits;
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(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.
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Homework:

April 15th, 2008

7.29; 7.35; 7.39
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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.
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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.
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High-Frequency Equivalent-Circuit Model
of the CE Amplifier
Vs'  Vs
r
Rs  rs  r
Rs'  Rs  rx  // r
RL'  RL // ro
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),
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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
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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.
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Analysis Using Miller’s Theorem
Vo   g mVgs RL'
Neglecting the current through Cgd
Ceq  C gd (1  g m RL' )
CT  C gs  C gd (1  g m RL' )
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.
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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.
CT  C gs  C gd (1  g m RL )
'
1
fH 
2CT Rsig
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Analysis Using Open-Circuit Time Constants
Rgs  Rsig
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Rgd  Rsig (1  g m RL )  RL
'
'
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Analysis Using Open-Circuit Time
Constants
RCL  RL
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'
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High-Frequency Equivalent Circuit
of the CE Amplifier
Thevenin theorem:
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Equivalent Circuit with Thévenin
Theorem Employed
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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.
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The Situation When Rsig Is Low
fT, which is equal to the gain-bandwidth product
Bode plot for the gain of the circuit in (a).
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The Situation When Rsig Is Low

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.
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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.
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1

1

I e  V   sC   g mV  Ve   g m  sC 
 r

 r

Ie 1
1
  g m  sC   sC
Ve r
re
1
1
 p1 
for CG amplifier :  p1 
CB amplifier has a
C re // Rs 
C gs 1 / g m // Rs 
1
1
 p2 
for CG amplifier :  p 2 
C RL
C gd RL
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much higher uppercutoff frequency than
that of CE amplifier
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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
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Frequency Response of the BJT
Cascode
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Frequency Response of the BJT
Cascode

The cascode configuration combines the
advantages of the CE and CB circuits
1
1  '
 H
RS C 1  2C 1 
2 
1
C 2 re 2
1
3 
C  2 RL
Vo
r
AM 
  g m RL
VS
r  rx  RS
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The Source (Emitter) Follower
• Self-study
• Read the textbook from pp626-629
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Homework:
May 6th,2008
7.44; 7.45; 7.46; 7.57
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Test:
（1）the reason for gain decreasing of high-frequency
response ， 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˚
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