EECS 105 Fall 2003, Lecture 19
Lecture 19:
Frequency Response
Prof. Niknejad
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Lecture Outline
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Department of EECS
Finish: Emitter Degeneration
Frequency response of the CE and
CS current amplifiers
Unity-gain frequency T
Frequency response of the CE as
voltage amp
The Miller approximation
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Typical “Discrete” Biasing
VCC

RC
R1

R2
A good biasing scheme must
be relatively insensitive to
transistor parameters (vary
with process and temperature)
In this scheme, the base
current is given by:
VB  VCC
RE

R1
R1  R2
The emitter current:


R1
I E  VCC
 VBE ,on  / RE
R1  R2


Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Gain for “Discrete” Design
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Rin  (   1)(r  RE )
 (   1) RE
vs
RC
R1
~ vs RC / RE
R2
vs / RE
~ vs
RE
Rin 2  (   1) RE || R1 || R2
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Let’s derive it by
intuition
Input impedance can
be made large
enough by design
Device acts like
follower,
emitter=base
This signal generates
a collector current
Can be made large to couple
All of source to input (even with RS)
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
CE Amplifier with Current Input
Find intrinsic current gain by driving with infinite source impedance and
Zero load impedance…
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Short-Circuit Current Gain
Pure input current
(RS = 0 )
Small-signal
short circuit
(could be a DC
voltage source)
Substitute equivalent circuit model of transistor and do the “math”
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Small-Signal Model: Ai
Note that ro, Ccs play no role (shorted out)
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Phasor Analysis: Find Ai
KCL at the output node:
I out  g mV  0  V  / Z 
KCL at the input node:
I in  V / Z  V  0 / Z 
Solve for V :
V  (1/ Z  1/ Z  )1 Iin
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Phasor Analysis for Ai (cont.)
I out  ( g m  jC )V
Substituting for V
Ai ( j ) 
( g m  jC )
(1 / Z )  jC
Substituting for Z = r || (1/jC)
Z 
 0 (1  jC / g m )
Ai ( j ) 
1  j r (C  C )
Department of EECS
r
1  j r C
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Short-Circuit Current Gain Transfer Function
Transfer function has one pole and one zero:
 o (1  j[C / g m ])
Ai ( j ) 
1  j[r (C  C )]
 o (1  j /  z )
Ai ( j ) 
(1  j /  p )
Note: Zero Frequency much larger than pole:
gm  1
1
z 


  p
C r C
r (C  C )
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Magnitude Bode Plot
pole
o
1
Ai ( j ) 

j /  p j / T
o
Unity current gain
0 dB
zero
p
Department of EECS
T
z
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Transition Frequency T
gm
T   o p 
C  C
Dependence on DC collector current:
C  C je  Cdiff
Base Region
Transit Time!
Cdiff  g m F   F qI / kT
Limiting case: f  T  1
T
2
2 F
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Common Source Amplifer: Ai(j)
DC Bias is problematic: what sets VGS?
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
CS Short-Circuit Current Gain
Transfer function: Ai ( j ) 
Department of EECS
g m 1  jC gd / g m 
j (C gs  C gd )
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
MOS Unity Gain Frequency
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Since the zero occurs at a higher frequency than
pole, assume it has negligible effect:
gm
Ai 
1
j (Cgs  Cgd )
T 
gm
(Cgs  Cgd )
W
g m Cox L (VGS  VT ) 3  (VGS  VT )
T 


2
2
Cgs
2
L
WLCox
3
Performance improves like L^2 for long channel devices!
For short channel devices the dependence is like ~ L^1
3  (VGS  VT )
T 
~
2
2
L
Department of EECS

VGS  VT
 Eeff v
L

 L
L
L
L
Time to
cross
channel
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Miller Impedance
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Consider the current flowing through an impedance
Z hooked up to a “black-box” where the voltage
gain from terminal to the other is fixed (as you can
see, it depends on Z)
Av 
v2
v1
I
v1
Z
v2
1  Av
v1  v2 v1  Av v1
I

 v1
Z
Z
Z
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Miller Impedance

Notice that the current flowing into Z from terminal
1 looks like an equivalent current to ground where
Z is transformed down by the Miller factor:
1  Av
Z
I  v1
 Z M ,1 
Z
1  Av
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From terminal 2, the situation is reciprocal
1  Av1
v2  v1 v2  Av1v2
I 

 v2
Z
Z
Z
Z M ,2
Department of EECS
Z

1  Av1
University of California, Berkeley
EECS 105 Fall 2003, Lecture 19
Prof. A. Niknejad
Miller Equivalent Circuit
Note:
Z M ,1  Z M ,2  Z
Z
Z M ,1 
1  Av
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
Z
Z M ,1 
1  Av1
We can “de-couple” these terminals if we can
calculate the gain Av across the impedance Z
Often the gain Av is weakly depedendent on Z
The approximation is to ignore Z, calculate A, and
then use the decoupled miller caps
Department of EECS
University of California, Berkeley