EECS 105 Fall 2003, Lecture 20
Lecture 20:
Frequency Response: Miller Effect
Prof. Niknejad
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Lecture Outline
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Department of EECS
Frequency response of the CE as
voltage amp
The Miller approximation
Frequency Response of Voltage
Buffer
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Last Time: CE Amp with Current Input
Can be MOS
Calculate the short circuit current gain of device
(BJT or MOS)
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
CS Short-Circuit Current Gain
MOS Case
MOS
BJT
0 dB
T
Ai ( j ) 
g m 1  jC gd / g m 
j (Cgs  Cgd )
z
gm

j (Cgs  Cgd )
Note: Zero occurs when all of “gm” current flows into Cgd:
g m vgs  vgs jC gd
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Common-Emitter Voltage Amplifier
Small-signal model:
omit Ccs to avoid
complicated analysis
Can solve problem directly
by phasor analysis or using
2-port models of transistor
OK if circuit is “small”
(1-2 nodes)
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
CE Voltage Amp Small-Signal Model
Two Nodes! Easy
Same circuit works for CS with
Department of EECS
r  
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Frequency Response
KCL at input and output nodes; analysis is made
complicated due to Z branch  see H&S pp. 639-640.
 r 
ro || roc || RL 1  j /  z 
 g m 
r  RS 
Vout


1  j /  p1 1  j /  p 2 
Vin
Low-frequency gain:
Zero
Two Poles
 r

 g m     ro || roc || RL  1  j 0 
 r

Vout
 r  RS 

  g m     ro || roc || RL 
Vin
1  j 0 1  j 0 
 r  RS 
gm
Zero:  z  T 
C C 
Department of EECS

Note: Zero occurs due to feed-forward
current cancellation as before
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Exact Poles
1
 p1 
 C  Rout
 C
RS || r C  1  g m Rout
Usually >> 1
 p2
 / RS || r 
Rout

 C  Rout
 C
RS || r C  1  g m Rout
These poles are calculated after doing some
algebraic manipulations on the circuit. It’s hard
to get any intuition from the above expressions.
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Miller Approximation
Results of complete analysis: not exact and little insight
Look at how Z affects the transfer function: find Zin
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Input Impedance Zin(j)
I t  (Vt  Vout ) / Z 
At output node:
   g mVt Rout

Vout  ( g mVt  I t ) Rout
Why?
I t  (Vt  AvC Vt ) / Z 
Z in  Vt / I t 
Department of EECS
Z
1  AvC
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Miller Capacitance CM
Effective input capacitance:
 1
1
Z in 

jCM  1  AvC
 1

 jC


1

 j 1  A C
vC




Cx
+
AV,Cx
+
Vin
─
Department of EECS
AV,Cx
+
Vout
─
(1-Av,Cx)Cx
─
+
Vout
─
(1-1/Av,Cx)Cx
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Some Examples
Common emitter/source amplifier:
AvC 
Negative, large number (-100)
CM  (1  AV ,C )C
100C
Miller Multiplied Cap has Detrimental Impact on bandwidth
Common collector/drain amplifier:
AvC 
Slightly less than 1
CM  (1  AV ,C )C
0
“Bootstrapped” cap has negligible impact on bandwidth!
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
CE Amplifier using Miller Approx.
Use Miller to transform C
RS || r
Analysis is straightforward now … single pole!
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Comparison with “Exact Analysis”
Miller result (calculate RC time constant of input pole):

1
p1
  C 
  RS || r  C  1  g m Rout
Exact result:
 C  Rout
 C
 p1  RS || r C  1  g m Rout
1
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Method of Open Circuit Time Constants
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This is a technique to find the dominant pole of a
circuit (only valid if there really is a dominant
pole!)
For each capacitor in the circuit you calculate an
equivalent resistor “seen” by capacitor and form the
time constant τi=RiCi
The dominant pole then is the sum of these time
constants in the circuit
 p ,dom 
Department of EECS
1
1   2 
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Equivalent Resistance “Seen” by Capacitor

For each “small” capacitor in the circuit:
–
–
–
–
–
–

Open-circuit all other “small” capacitors
Short circuit all “big” capacitors
Turn off all independent sources
Replace cap under question with current or voltage
source
Find equivalent input impedance seen by cap
Form RC time constant
This procedure is best illustrated with an
example…
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Example Calculation
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Consider the input capacitance C1  C  CM
Open all other “small” caps (get rid of output cap)
Turn off all independent sources
Insert a current source in place of cap and find
impedance seen by source
RM  r || RS
  C 
 1   RS || r  C  1  g m Rout
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Common-Collector Amplifier
Procedure:
1. Small-signal twoport model
2. Add device (and
other) capacitors
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Two-Port CC Model with Capacitors
Gain ~ 1
Find Miller capacitor for C -- note that the base-emitter
capacitor is between the input and output
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Voltage Gain AvC Across C
AvC  Rout /( Rout  RL ) 1
gm RL  1
Rout
1

gm
Note: this voltage gain is neither the two-port gain nor the
“loaded” voltage gain
Cin  C  CM  C  (1  AvC )C
1
Cin  C 
C
1  g m RL
Cin  C
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 20
Prof. A. Niknejad
Bandwidth of CC Amplifier
Input low-pass filter’s –3 dB frequency:
p
1

C 

 RS || Rin  C 
1  g m RL 

Substitute favorable values of RS, RL:
RS  1 / g m


 p 1  1 / g m  C 
RL  1 / g m
C 
  C / g m
1  BIG 
Model not valid at
these high frequencies
 p  g m / C  T
Department of EECS
University of California, Berkeley