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Department of EECS University of California, Berkeley
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
� Frequency response of the CE as voltage amp
� The Miller approximation
� Frequency Response of Voltage Buffer
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 20 Prof. A. Niknejad
Last Time: CE Amp with Current Input
Calculate the short circuit current gain of device (BJT or MOS)
Can be MOS
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 20 Prof. A. Niknejad
CS Short-Circuit Current Gain
( )1 /( )
( ) ( )m gd m m
igs gd gs gd
g j C g gA j
j C C j C C
ωω
ω ω−
= ≈+ +
MOS Case
0 dB
MOS
BJT
Tω zω
Note: Zero occurs when all of “gm” current flows into Cgd:
m gs gs gdg v v j Cω=
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 directlyby phasor analysis or using2-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 rπ → ∞
Department of EECS 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 Zµ branch � see H&S pp. 639-640.
[ ]( )
( )( )21 /1/1
/1||||
pp
zLocoS
m
in
out
jj
jRrrRr
rg
V
V
ωωωω
ωωπ
π
++
−
+−
=
Low-frequency gain:
Zero:µπ
ωωCC
gmTz +
=>
[ ]( )
( )( ) [ ]|| || 1 0
|| ||1 0 1 0
m o oc LSout
m o oc Lin S
rg r r R j
r RV rg r r R
V j j r R
π
π π
π
− − + = → − + + +
�
Two Poles
Zero
Note: Zero occurs due to feed-forwardcurrent cancellation as before
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 20 Prof. A. Niknejad
Exact Poles
( ) ( ){ } µµππ
ωCRCRgCrR outoutmS
p ′+′++≈
1||
11
( )( ) ( ){ } µµππ
πωCRCRgCrR
rRR
outoutmS
Soutp ′+′++
′≈
1||
||/2
These poles are calculated after doing some algebraic manipulations on the circuit. It’s hard to get any intuition from the above expressions.
Usually >> 1
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 Zµ 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(jωωωω)
At output node:
µZVVI outtt /)( −=
outtmoutttmout RVgRIVgV ′−≈′−−= )( Why?
µµZVAVI tvCtt /)( −=
µ
µ
vCttin A
ZIVZ
−==
1/
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 20 Prof. A. Niknejad
Miller Capacitance CM
Effective input capacitance:
( )[ ]µµµ ωωωµ
CAjCjACjZ
vCvCMin −
=
−==
1
11
1
11
AV,Cx
+
─
+
─
Vin
Vout
Cx
AV,Cx
+
─
+
─
Vout(1-Av,Cx)Cx
(1-1/Av,Cx)Cx
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 20 Prof. A. Niknejad
Some Examples
Common emitter/source amplifier:
=µvCA Negative, large number (-100)
Common collector/drain amplifier:
=πvCA Slightly less than 1
,(1 ) 100M V CC A C Cµ µ µ= − �
,(1 ) 0M V CC A Cπ π= − �
Miller Multiplied Cap has Detrimental Impact on bandwidth
“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 Cµ
Analysis is straightforward now … single pole!
||SR rπ
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):
Exact result:
( ) ( ){ } µµππω CRCRgCrR outoutmSp ′+′++=− 1||11
( ) ( ){ }11 || 1p S m outR r C g R Cπ π µω − ′= + +
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 20 Prof. A. Niknejad
Method of Open Circuit Time Constants
� 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
,1 2
1p domω
τ τ=
+ +�
Department of EECS 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
� Consider the input capacitance
� 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
1 MC C Cπ= +
||M SR r Rπ=
( ) ( ){ }1 || 1S m outR r C g R Cπ π µτ ′= + +
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 20 Prof. A. Niknejad
Common-Collector Amplifier
Procedure:
1. Small-signal two-port model
2. Add device (andother) capacitors
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 20 Prof. A. Niknejad
Two-Port CC Model with Capacitors
Find Miller capacitor for Cπ -- note that the base-emitter capacitor is between the input and output
Gain ~ 1
Department of EECS University of California, Berkeley
EECS 105 Fall 2003, Lecture 20 Prof. A. Niknejad
Voltage Gain AvCππππ Across Cππππ
/( ) 1vC out out LA R R Rπ
≈ + ∼
Note: this voltage gain is neither the two-port gain nor the “loaded” voltage gain
πµµ πCACCCC vCMin )1( −+=+=
1
1inm L
C C Cg Rµ π= +
+
inC Cµ≈
1out
m
Rg
=1m Lg R >>
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:
( )
++=−
LminSp Rg
CCRR
1||1 π
µω
Substitute favorable values of RS, RL:
mS gR /1≈ mL gR /1>>
( ) mmp gCBIG
CCg /
1/11
µπ
µω ≈
++≈−
/p m Tg Cµω ω≈ >
Model not valid at these high frequencies
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