4. MOS Small Signal Model

Reading: Sedra & Smith Sec. 5.5 (S&S 5th Ed: Sec. 4.6)

ECE 102, Fall 2011, F. Najmabadi

Circuit response to signal is different

Recall:

o Response of the circuit to small signal is different than response of the circuit to Bias + signal.

o iv characteristics of the circuit elements in response to the signal is different than iv characteristics of the circuit elements in response to the Bias + signal

• Circuit looks different when signal is considered!

f(∙) aAA xXx += aAA yYy += )( AA xfy =Signal + Bias

)( AA XfY =f(∙) AX AYBias (signal = 0)

g(∙)

ax aySignal=

(Signal + Bias) –(Bias) )( aa xgy =

“Signal-only” circuit is different!

Signal only = (Bias + Signal) - Bias

RD: vrd ird = id

MOS: vgs, id, vds

No signal here!

RD: VRD IRD = ID

MOS: VGS, ID, VDS

VDD: VDD Bias

RD: vRD = VRD + vrd iRD = iD = ID + id

MOS: vGS = VGS + vgs vDS = VDS + vds iD = ID + id

VDD: VDD

Bias & Signal

Signal Model for linear circuit elements Independent voltage source (e.g., VDD)

o No signal: effectively grounded

Independent current source o No signal: effectively open circuit (Careful about current mirrors as they

are NOT “ideal” current sources, channel width modulation was ignored!)

Resistors, capacitors, inductors o Remain the same:

Dependent sources o Remain the same with the control parameter related to the signal!

Non-linear Elements: o Different!

iR = IR + ir vR = VR + vr = RIR + vr vR = R iR = R (IR + ir ) = RIR + R ir vr = R ir

−

×=

−

×

=

−

+=−=

1exp1expexp

expexp :Signal

T

dD

T

d

T

Dsd

T

Ds

T

dDsDDd

nVvI

nVv

nVVIi

nVVI

nVvVIIii

Diodes: signal response is non-linear but can be linearized when signal is small

VD

ID

vd

id ?

vd

id

R = nVT/ID

=+

T

DsD nV

vIi exp :Signal Bias

=

T

DsD nV

VII exp :Bias

vD

iD

DT

D

T

dDd

T

d

T

d

T

d

T

d

T

d

T

d

vnVI

nVvIi

nVv

nVv

nVv

nVv

nVv

nVv

=

−

+×≈

+≈

<<

+

+

+=

11

1exp :1 If

.... !2

11exp :Exapnsion SeriesTaylor 2

Formal derivation of small signal model

( ) ( ) ...!2

)()()( 2)2(

)1( +−⋅+−⋅+= AAA

AAAA XxXfXxXfXf

...!2

)()()( 2)2(

)1( +⋅+⋅+= aA

aAA xXfxXfXf

aAAaAA xXfYxXfXf ⋅+=⋅+≈ )()()( )1()1(

2)2(

)1(

!2)()( a

AaA xXfxXf ⋅>>⋅

)()(2 )2(

)1(

A

Aa Xf

Xfx ⋅<<

Small signal means:

aAaa xXfxgy ⋅== )()( )1(

f(∙) aAA xXx += aAA yYy += )( AA xfy =Signal + Bias

)( AA XfY =f(∙) AX AYBias (signal = 0)

g(∙)

Ax AySignal=

(Signal + Bias) –(Bias) )( aa xgy =

)( AAaA xfyyY ==+(Taylor Series Expansion)

Derivation of diode small signal model

−⋅= 1T

DnVv

SD eIi

−⋅= 1)( TnV

x

S eIxf

−⋅== 1)( T

DnVV

SDD eIVfI

dT

SDd

T

nVV

Sd

VxT

nVx

SdDd v

nVIIv

nVeIv

nVeIvVfi

T

D

D

T

⋅

+=⋅

⋅

=⋅

⋅

=⋅=

=

)()1(

dT

Dd

T

SDd v

nVI

vnV

IIi ⋅

≈⋅

+=

D

dd r

vi =D

TD I

nVr ≈vd

id

R = nVT/ID

vD

iD

For Small Signals!

MOS iv equations: iD = f(vGS, vDS)

),( AAA yxfz =Signal + Bias f(∙, ∙) aAA xXx +=aAA zZz +=

aAA yYy +=

Bias (signal = 0) f(∙, ∙) AX

AZAY

),( AAA YXfZ =

Signal= (Signal + Bias) –(Bias) g(∙, ∙)

axaz

ay),( aaa yxgz =

),( AAAaA yxfzzZ ==+

...)(),()(),(),(,,

+−⋅∂

∂+−⋅

∂∂

+= AAYX

AAYX

AA Yyy

yxfXxx

yxfYXfAAAA

aYX

aYX

A yy

yxfxx

yxfZAAAA

⋅∂

∂+⋅

∂∂

+≈,,

),(),(

aYX

aYX

a yyfx

xfz

AAAA

⋅∂∂

+⋅∂∂

≈,,

Derivation of MOS small signal model

dsVVDS

gsVVGS

d vvfv

vfi

DSGSDSGS

⋅∂∂

+⋅∂∂

=,,

yvxvyxfi

vvfvVvL

WCi

i

DSGSD

DSGSDStGSoxnD

G

↔↔=

=+−=

=

, with ),(

),()1()( 5.0

0

2 λµ

mOV

D

tGS

DStGSoxn

VVDStGSoxnVVGS

gV

IVV

VVVL

WC

vVvL

WCvf

DSGS

DSGS

≡=−

+−×=

+−×=∂∂

2)(

)1()( 5.02

)1)(( 5.02

2

,,

λµ

λµ

oDS

D

DS

DStGSoxn

VVtGSoxn

VVDS

rVI

V

VVVL

WC

VvL

WCvf

DSGSDSGS

1)1()1(

)1()( 5.0

)( 5.0

2

,

2

,

≡+

=+

+−×=

−×=∂∂

λλ

λ

λµλ

µλ

1 *

2

0

Do

OV

Dm

g

o

dsgsmd

Ir

VIg

irvvgi

⋅=

=

=

+⋅=

λ(* For λ vDS << 1)

)1()( 5.0

0

2DStGSoxnD

G

VVVL

WCI

I

λµ +−=

=

MOS small signal “circuit” model

and 0 o

dsgsmdg r

vvgii +⋅==

Do I

r⋅

≈λ

1

OV

Dm V

Ig ⋅=

2 122>>==

OV

A

OVom V

VV

rgλ

id Statement of KCL Two elements in parallel Input open circuit

PMOS “circuit” small signal model is identical to NMOS

Do I

r⋅

=λ

1

OV

Dm V

Ig ⋅=

2

id

vsg gmvsg

id

=

PMOS small-signal circuit model is identical to NMOS o For PMOS small signal model, id flows into the drain (while iD and ID

flows out of the drain). o For both NMOS and PMOS, while iD ≥ 0 and ID ≥ 0, signal quantities: id,

vgs, and vds , can be negative!

and 0 o

sdsgmdg r

vvgii +⋅==

Body Effect

It has been found that with a few unusual exceptions, body effects can be ignored in the initial design of MOS amplifiers (and we ignore it here).

When body effect included:

We can use Taylor Series Expansion in three variables to get MOS small-signal model:

( )),,()1()( 5.0

|2||2|

2

0,

SBDSGSDStGSoxnD

FSBFtt

vvvfvVvL

WCi

vVV

=+−=

−++=

λµ

φφγ

o Additional element. o Note body is connected to

ground for small signal (because it is connected to the negative terminal of the power supply).

Example 1: Draw the small-signal equivalent of the circuit below (assume capacitors are short for small signal).

IVS → 0 R remains

Ground at the bottom

Example 2: Draw the small-signal equivalent of the circuit below (assume capacitors are short for small signal).

Flip PMOS

IVS → 0 Ground at the bottom (100k || 33k = 24.8 k)

Example 3: Draw the small-signal equivalent of the circuit below (assume capacitors are short for small signal).

ICS → 0 (This makes ICS an open circuit)

IVS → 0