6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-1

Lecture 23 - Frequency Resp onse of Amplifiers (I)

Common-Source Amplifier

May 6, 2003

Contents:

1. Intro duction

2. Intrinsic frequency resp onse of MOSFET

3. Frequency resp onse of common-source amplifier

4. Miller effect

Reading assignment:

Howe and So dini, Ch. 10, §§10.1-10.4

6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-2

Key questions

• How do es one assess the intrinsic frequency resp onse of a transistor?

• What limits the frequency resp onse of an amplifier?

• What is the ”Miller effect”?

6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-3

1. Intro duction

Frequency domain is a ma jor consideration in most ana­log circuits.

Data rates, bandwidths, carrier frequencies all pushing up.

Motivation:

• Pro cessor sp eeds ↑ • Traffic volume ↑ ⇒ data rates ↑ • More bandwidth available at higher frequencies in the

sp ectrum

[from Agilent Technologies]

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6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-4

2. Intrinsic frequency resp onse of MOSFET

How do es one assess the intrinsic frequency resp onse of a transistor?

f ≡ short-circuit current-gain cut-off frequency [GHz] t

Consider MOSFET biased in saturation regime with small-signal source applied to gate:

VDD

iD=ID+iout

iG=iin

vs

VGG

at input ⇒ i : transistor effect s out

⇒ i due to gate capacitance in

i out Frequency dep endence: f ↑⇒ i ↑⇒ | | ↓ in i in

i out f ≡ frequency at which | | = 1 t

iin

v

6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-5

Complete small-signal mo del in saturation:

S

D+

-

vgs Cgs

Cdb

Csb

gmvgs gmbvbs ro

+

vbs

-

+

-

iin G Cgd iout

vs

B

vbs=0

iin 1

Cgd 2 iout

+ +

-

vgs gmvgsCgsvs -

no de 1: i in − v gs jω Cgs − v gs jω Cgd = 0

⇒ i in = v gs jω (Cgs + Cgd )

no de 2: i out − g m v gs + v gs jω Cgd = 0

⇒ i out = v gs (g m − jω C gd )

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6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-6

Current gain:

i g − jω C out m gd h = = 21

i jω (C + C ) in gs gd

Magnitude of h :21

2 2 2 g + ω C gd m |h | = 21 ω (C + C ) gs gd

g m • For low frequency, ω , C gd

g m |h | � 21 ω (C + C ) gs gd

g m • For high frequency, ω , C gd

C gd |h | � < 1 21 C + C gs gd

6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-7

log |h21|

1 log ω

Cgd

Cgs+Cgd

ωT

-1

|h | b ecomes unity at: 21

g m ω = 2πf = T T

C + C gs gd

Then:

g m f =T

2π (C + C ) gs gd

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6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-8

� Physical interpretation of f : T

Consider:

1 C + C C gs gd gs =

2πf g g T m m

Plug in device physics expressions for C and g : gs m

2 1 C LW C L gs ox 3 = = W V −V 3 GS T 2πf g µC (V − V ) µ T m ox GS T L 2 L

or

1 L L = = τ t

2πf µ < E > < v > T chan chan

τ ≡ transit time from source to drain [s] t

Then:

1 f T

2πτ t

f gives an idea of the intrinsic delay of the transistor: T

go o d first-order figure of merit for frequency resp onse.

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6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-9

To reduce τ and increase f : t T

• L ↓: trade-off is cost

• (V − V ) ↑⇒ I ↑: trade-off is p ower GS T D

• µ ↑: hard to do

• note: f indep endent of WT

Impact of bias p oint on f : T

W W 2 µC I g µC (V − V ) ox D m ox GS T L L f = = = T 2π (C + C ) 2π (C + C ) 2π (C + C ) gs gd gs gd gs gd

fT fT

0 0 VT VGS

0 ID

In typical MOSFET at typical bias p oints: f ∼ 1 − 25 GH z T

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6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-10

3. Frequency resp onse of common-source amp

VDD

signal source

vs

VGG

RS

vOUT

iSUP

+

-

signal� load

RL

VSS

Small-signal equivalent circuit mo del (assuming current source has no parasitic capacitance):

RS Cgd

+

-

vgs gmvgsCgs rocroCdb

+ +

RL vout

-

vs -

Rout'

Low-frequency voltage gain:

A v,LF =v out

v s = −g m (r o out//r oc //RL ) = −g m R

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6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-11

RS 1 Cgd 2

+

-

vgs gmvgsCgs Cdb

++

Rout' vout

-

vs -

v s −v gs no de 1: − v gs jω Cgs − (v gs − v out )jω Cgd = 0 R S

v out no de 2: ( v gs − v out )jω Cgd − g m v gs − v out jω Cdb − = 0 Rout

Solve for v gs in 2:

1jω (Cgd + Cdb ) + Rout v = v gs out jω C − g gd m

Plug in 1 and solve for v /v : out s

−(g − jωC )R m gd out A = v DEN

with

1 DEN = 1 + jω{R C + R C [1 + R ( + g )] + R C } S gs S gd m db out out R S

2 −ω R R C (C + C ) S gs gd db out

[check that for ω = 0, A = −g R ] v,LF m out

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6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-12

Simplify:

g m 1. Op erate at ω ω = ⇒ T C +C gs gd

g ω (C + C ) > ω C , ω C m gs gd gs gd

2. Assume g high enough so that m

1 + g g m m

RS

2 3. Eliminate ω term in denominator of A v

⇒ worst-case estimation of bandwidth

Then:

−g R m out Av 1 + jω [R S Cgs out) + R outCdb ]+ RS Cgd (1 + g m R

This has the form:

A v A v,LF

(ω ) = 1 + j ω

ω H

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6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-13

gm

log |Av|

-1

Rout'

ωH log ω

At ω = ω : H

1 √ |A (ω )| = |A | v H v,LF 2

ω gives idea of frequency b eyond which |A | starts rolling H v

off quickly ⇒ bandwidth

For common-source amplifier:

1 ω = H

R C + R C (1 + g R ) + R C S gs S gd m db out out

Frequency resp onse of common-source amplifier limited by C and C shorting out the input, and C shorting gs gd db

out the output.

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6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-14

Can rewrite as:

1f = H

2π {R [C + C (1 + |A |)] + R C } S gs gd v,LF db out

Compare with:

g m f = T

2π (C + C ) gs gd

In general: f f due to H T

1 • typically: g m R S

• C enters f but not f db H T

• presence of |A | in denominator v,LF

To improve bandwidth,

• C , C , C ↓ ⇒ small transistor with low parasitics gs gd db

• |A | ↓⇒ don’t want more gain than really needed v,LF

but...

why is it that effect of C on f app ears to b eing am­gd H

plified by 1 + |A |??!! v,LF

6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-15

4. Miller effect

In common-source amplifier, C lo oks much bigger than gd

it really is.

Consider simple voltage-gain stage:

iin C

+

vin

+

+ -

-Avvin vout

-

What is the input imp edance?

i = (v − v )jω C in in out

But

v = −A v out v in

Then:

i = v (1 + A )C in in v

6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-16

or

v 1 in Z = = in

i jω (1 + A )C in v

From input, C , lo oks much bigger than it really is. This is called the Miller effect.

When a capacitor is lo cated across no des where there is voltage gain, its effect on bandwidth is amplified by the voltage gain ⇒ Miller capacitance:

C = C(1 + A ) M iller v

Why?

v ↑ ⇒ v = −A v ↓↓ ⇒ (v − v ) ↑↑ ⇒ i ↑↑ in out v in in out in

In amplifier stages with voltage gain, it is critical to have small capacitance across voltage gain no des.

As a result of the Miller effect, there is a fundamental gain-bandwidth tradeoff in amplifiers.

6.012 - Microelectronic Devices and Circuits - Spring 2003 Lecture 23-17

Key conclusions

• f (short-circuit current-gaincut-off frequency): fig-T

ure of merit to assess intrinsic frequency resp onse of transistors.

• In MOSFET, to first order,

1 f =t

2πτ t

where τ is transit time of electrons through channel. t

• In common-source amplifier, voltage gain rolls off at high frequency b ecause C and C short out input gs gd

and C shorts out output. db

• In common-source amplifier, effect of C on band­gd

width is magnified by amplifier voltage gain.

• Miller effect: effect of capacitance across voltage gain no des is magnified by voltage gain

⇒ trade-off between gain and bandwidth.