1

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28

Lecture 28: Single Stage Frequency response

Prof J. S. Smith

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Context

In today’s lecture, we will continue to look at the frequency response of single stage amplifiers, starting with a more complete discussion of the CS amplifier, and then looking at the frequency response of CG and the CD connections.

Next: Multi-state amplifiers

2

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Reading

Reading: We are discussing the frequency response of single stage amplifiers, which isn’t treated in the text until after multi-state amplifiers (beginning of chapter 10). I feel that it is important to get warmed back up on linear circuit analysis for simple circuits before jumping into multi-stage amplifiers.

We will be starting on chapter 9, multi-state amplifiers, later this week.

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Lecture Outline

Frequency response of the CS as voltage ampThe Miller approximationFrequency Response of a Voltage BufferFrequency Response of Current Buffer

3

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Last Time: CS Amp with Current Input

Calculate the short circuit current gain of device (BJT or MOS)

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

CS Short-Circuit Current Gain

( )1 /( )

( ) ( )m gd m m

igs gd gs gd

g j C g gA jj 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=

4

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Input impedance

Look at how Zgd affects the transfer function: find Zin

gdC

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Input Impedance Zin(jω)

At output node:

gdouttt ZVVI /)( −=

outtmoutttmout RVgRIVgV ′−≈′−−= )( Why?

gdtvCtt ZVAVIgd

/)( −=

vCgd

gdttin A

ZIVZ

−==

1/

5

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Miller Capacitance CM

Effective input capacitance:

( )[ ]gdvCgdgdvCMin CAjCjACj

Zgd

−=⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−==

111

111

ωωω

AV,Cx

+

─

+

─

VinVout

Cx

AV,Cx

+

─

+

─Vout

(1-Av,Cx)Cx

(1-1/Av,Cx)Cx

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Some Examples

Common source (emitter) amplifier:

=gdvCA Negative, large number (-100)

Common drain (collector) amplifier:

=gsvCA Slightly less than 1

→Miller Multiplied Cap has Detrimental Impact on bandwidth

“Bootstrapped” cap has negligible impact on bandwidth!

( ) gdgdCVM CCACgd

1001 , ≈−=

( ) gsgsCVM CCACgs

01 , ≈−=

6

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

CE Amplifier using Miller Approx.

Use Miller to transform Cgd

Analysis is straightforward now … single pole!

−

+

gsv gsC gsmvg

)1( outRgCC mgdM ′+=

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Comparison

Miller result (calculate RC time constant of input pole):

If we hadn’t made the Miller approximation, the result would have been:

{ }gdmbssp CRgCR )1( out11 ′++=−ω

{ } gdgdmbssp CRCRgCR outout11 )1( ′+′++=−ω

7

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Method of Open Circuit Time Constants

Here 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 a time constant τi=RiCi

The dominant pole then is the sum of these time constants in the circuit

,1 2

1p domω

τ τ=

+ +L

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

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…

8

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. SmithExample Calculation: CE input impedance

Consider the input capacitanceOpen all other “small” caps (get rid of output cap)Turn off all independent sourcesInsert 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 Spring 2004, Lecture 28 Prof. J. S. Smith

Common-Drain Amplifier

21 ( )2DS ox GS T

WI C V VL

µ= −

2 DSGS T

ox

IV V WCL

µ= +

Weak IDS dependence

9

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

CD Voltage Gain

1out m

in mb m

v gv g g

≈ ≈+

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

CD Output Resistance

Sum currents at output (source) node:

|| || tout o oc

t

vR r ri

= t m t mb ti g v g v= +

1out

m mb

Rg g

≈+

10

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

CD Output Resistance (Cont.)

ro || roc is much larger than the inverses of the transconductances ignore

1out

m mb

Rg g

≈+

Function: a voltage buffer• High Input Impedance• Low Output Impedance

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Add capacitors

Procedure:Start with small-signal two-port modelAdd device (and other) capacitors

gdC

gsC

−+

inout vv ≈

11

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Common-Collector Amplifier

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

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

12

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Voltage Gain AvCπ Across Cπ

Note: this voltage gain is neither the two-port gain nor the “loaded” voltage gain

πµµ πCACCCC vCMin )1( −+=+=

11in

m L

C C Cg Rµ π= +

+

inC Cµ≈

1out

m

Rg

=

1m Lg R >>

( ) 1/ ≈+≈ LoutoutC RRRAπν

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Bandwidth of CC Amplifier

Input low-pass filter’s –3 dB frequency:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

+=−

LminSp Rg

CCRR1

||1 πµω

Substitute favorable values of RS, RL:

mS gR /1≈ mL gR /1>>

( ) mmp gCBIG

CCg /1

/11µ

πµω ≈⎟

⎠⎞

⎜⎝⎛

++≈−

/p m Tg Cµω ω≈ >

Model not valid at these high frequencies

13

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Common Gate Amplifier

DC bias:

SUP BIAS DSI I I= =

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

CG→Current buffer

out d ti i i= = −

1iA = −

14

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

CG Input Resistance

gs tv v= −

mbmin gg

R+

≈1

We found the approximation:

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

CG Output Resistance

)]1([||][|| SmoocSomoocout RgrrRrgrrR +=+≈

15

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

CG Two-Port Model

The function of the CG amp was a current buffer:•Low input impedance•High output impedance

The only parasitic capacitances are directly across theInput and output: frequency response can be directlydetermined

( )SmOC Rrgrr 00|| +

gsC gdC

No Miller-transformed capacitor!

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

CB Current Buffer Bandwidth

Same procedure: startwith two-port model andcapacitors

16

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. Smith

Two-Port CB Model with Capacitors

Unity-gain frequency is on the order of ωT for small RL

No Miller-transformed capacitor!

Department of EECS University of California, Berkeley

EECS 105 Spring 2004, Lecture 28 Prof. J. S. SmithSummation of Single-Stage Amp Frequency Response

CS, CE: suffer from Miller-magnified capacitor for high-gain caseCD, CC: Miller transformation nulledcapacitor “wideband stage”CG, CB: no Millerized capacitor wideband stage (for low load resistance)