ece 2c, notes set 3: small-signal models of active … notes, m. rodwell, copyrighted 2009 ece 2c,...

class notes, M. Rodwell, copyrighted 2009

ECE 2C, notes set 3: Small-Signal Models of Active Devices

Mark Rodwell

University of California, Santa Barbara

[email protected] 805-893-3244, 805-893-3262 fax


Goals of this note set:

devicesnonlinear of models signal-smalllinear make toHow

elementscircuit nonlinear Linear vs.

?network output) (input,port - twoa isWhat


2-Port Descriptions ( 3-Wire Network or Device)

subcircuita element, passivea or,a transist :containmight Box

s.t variableindependen theof functions as writtenbe thencan

,variables*dependent * the, variables tworemaining The

.*st variableindependen* theasset be can variablesAny two

freedom. of degrees 2 are There

. and , , , variables therelate sticscharacteri terminalThe 2121 IIVV


Two-Port Parameters: Represent Device or Network

devices. active modelingon is focusour now But,

network.any represent port to- twoa usecan We


Transistors are Nonlinear Elements

*signals small* assume weifeasy made is Analysis

difficult. very theforeis analysiscircuit sistor Exact tran

apply.not doesion superposit of principle The

. voltagesapplied theof functionsnonlinear are currents Transistor

ID

VDS

increasingVGS

)1())(2/( 2

, DSthgsggoxD VVVLWcI


A Nonlinear Two-Port

currents. signal AC and voltagessignal AC

plus

currents bias DC and voltagesbias DC

of sumsoften are currents and voltages thecircuits,tor In transis

voltages. theof functionsnolinear are currents The

always).not (but small often very are signals AC the...and


Nonlinear Two-Port Characteristics

...

22

...22

21

21

2

22

2

2

2

2

22

1

1

2

2

2

2

2

21

1

22

21

21

1

22

2

2

2

1

22

1

1

2

1

2

2

2

11

1

11

VVVV

IV

V

IV

V

IV

V

IV

V

II

VVVV

IV

V

IV

V

IV

V

IV

V

II

series...Taylor a Write

seriesTaylor in the sorder term-first but the allneglect can wecase, In this

smaller.even are and , , then small, are and ifBut

d.complicate very still is This

21

2

2

2

121 VVVVVV


Assuming Small Signals (Derivatives)

...

...

2

2

21

1

22

2

2

11

1

11

VV

IV

V

II

VV

IV

V

II

instead. and writesmall very are and If 2121 VVVV

small. are and if sorder term-higher eneglect thcan We 21 VV


Small-Signal 2-Port Admittance Parameters

2221212

2121111

VYVYI

VYVYI

shown)not (..areimplicit as taken are currents & voltagesDC

Where

2

222

1

221

2

112

1

111

V

IY

V

IY

V

IY

V

IY


Admittance Parameters

)(

)(

)()(

)()(

)(

)(

2

1

2221

1211

2

1

jV

jV

jYjY

jYjY

jI

jI

etc. , Re)( , Re)(

:ndescriptio domain-Frequency

1111

tjtj eItieVtv

implicit. as takenis bias DC

voltages.of functions as writtenare Currents


Side Point: Impedance Parameters

currents. of functions as writtenare Voltages

2

1

2221

1211

2

1

I

I

ZZ

ZZ

V

V

matrix. Y theof inverse theismatrix ZThe


General Comment About 2-Port Parameters

models. signal-smallansistor explain tr way tonatural a

provide they because parametersport -2 introduced have We

years.Senior or Junior theuntilleft becan this:My view

.parametersport -2about discuss tomoremuch is There


Small-Signal Equivalent-Circuit

2

1

2221

1211

2

1

V

V

YY

YY

I

I

)()( )( :partsimaginary and real havemight and

frequency with might vary )( assumed have wefar, So

.generatorscurrent controlled twoand sadmittance twoas

parameters-Yfour thengrepresenti now are We

jjBjGjY

jY


Small-Signal Equivalent-Circuit

.negligibleusually is ports),-2other not (but storsFor transi

uctance transcondreverse ctancetranscondu

econductancoutput econductancinput

:notation thechanged also have that weNote

)( parts.imaginary no have and

frequency,ith not vary w do parameters-Y that thesuppose Now

1221

2211

r

out

inr

in

outm

out

outout

in

inin

g

V

IgG

V

IgG

V

IGG

V

IGG

GjY


Small-Signal Equivalent-Circuit: Neglecting Gr.

ctancetranscondu

econductancoutput econductancinput

in

outm

out

outout

in

inin

V

Ig

V

IG

V

IG

model.circuit equivalentfrequency -lowr transisto typicala is This


Recall MOSFET Characteristics: Mobility-Limited Case

:current limitedmobility

Id

VgsVth

mobility-limited

velocity-limited

voltage,knee thenlarger tha voltagesdrainfor Applies

ID

VDS

increasingVGS

)1())(2/( 2

, DSthgsggoxD VVVLWcI


Recall MOSFET Characteristics: Velocity-Limited Case

))(1(, VVVVvWcI thgsDSsatgoxvD

Id

VgsVth

mobility-limited

velocity-limited

ID

VDS

increasingVGS

voltage,knee thenlarger tha voltagesdrainfor Applies

current limitedvelocity

/gsatLvV


FET Small-Signal Model: Mobility-Limited

.1 asroughly varies that Note

using are wemodels theofaccuracy the within to

1

1

eConductancOutput

))(/()1)()(/(

ctanceTranscondu

)1())(2/(

Current Drain

2

,

Dds

D

DS

D

DS

D

ds

ds

thgsggoxDSthgsggox

GS

Dm

DSthgsggoxD

/IR

I

V

I

V

I

RG

VVLWcVVVLWcV

Ig

VVVLWcI

RDS

gmVGS

VGS

D

S

G


FET Small-Signal Model: Velocity-Limited

1

1

eConductancOutput

)1(

ctanceTranscondu

))(1(

Current Drain

,

D

DS

D

DS

D

ds

ds

gsatoxDSgsatox

GS

Dm

thgsDSgsatoxvD

I

V

I

V

I

RG

WvcVWvcV

Ig

VVVVWvcI

RDS

gmVGS

VGS

D

S

G


Transconductance vs Vgs

gthgsgox

GS

Dm

gthgsgoxD

LVVWcV

Ig

LVVWcI

/)(

2/)(

limitedmobility

2

,

satgox

GS

Dm

thgssatgoxvD

vWcV

Ig

VVVvWcI

)(

limitedvelocity

,

Id

VgsVth

V

gm

VgsVth

V

mobility-limited velocity-limited


Rough Estimate: 250 nm NMOS Characteristics


Rough Estimate: 250 nm PMOS Characteristics


Rough Estimate: 45 nm NMOS Characteristics


Rough Estimate: 45 nm PMOS Characteristics

ece 2c, notes set 3: small-signal models of active … notes, m. rodwell, copyrighted 2009 ece 2c,...

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