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