ee 330 lecture 23 - class.ece.iastate.educlass.ece.iastate.edu/ee330/lectures/ee 330 lect 23 fall...

EE 330

Lecture 23

• Small Signal Analysis– SS Models for MOSFET

– SS Models for BJT

Exam Schedule

Exam 1 Monday Oct 5

Exam 2 Monday October 26

Exam 3 Friday November 20

Comparison of Gains for MOSFET and BJT Circuits

R

Q1

VIN(t)

VOUT

VCC

VEE

BJT MOSFET

R1

VIN(t)

VOUT

VDD

VSS

M1

DQ

VM

SS T

2I RA

V V

CQ 1

VB

t

I RA

V

If IDQR =ICQR1=2V, VSS+VT= -1V, Vt=25mV

0CQ 1

VB

t

I R 2VA - =-8

V 25mV

DQ

VM

SS T

2I R 4VA = - 4

V V -1V

Observe AVB>>AVM

Due to exponential-law rather than square-law model

Review from Last Lecture

Operation with Small-Signal Inputs

• Analysis procedure for these simple circuits was very tedious

• This approach will be unmanageable for even modestly more complicated

circuits

• Faster analysis method is needed !


Small-Signal Principle y

x

yQ

xQ

xSS

ySS

Q-point y=f(x)

Q

SS SS

x=x

fy x

x

• Linearized model for the nonlinear function y=f(x)

• Valid in the region of the Q-point

• Will show the small signal model is simply Taylor’s series expansion

of f(x) at the Q-point truncated after first-order terms

Small-Signal Model:


“Alternative” Approach to small-signal

analysis of nonlinear networks

Nonlinear

Network

dc Equivalent

Network

Q-point

Values for small-signal parameters

Small-signal (linear)

equivalent network

Small-signal output

Total output

(good approximation)


Small-signal and simplified dc equivalent elements

Element ss equivalentSimplified dc

equivalent

VDCVDC

IDC IDC

R RR

dc Voltage Source

dc Current Source

Resistor

VACVACac Voltage Source

ac Current Source IAC IAC


Small-signal and simplified dc equivalent elements

C

Large

C

Small

L

Large

L

Small

C

L

Simplified

Simplified

Capacitors

Inductors

MOS

transistors

Diodes

Simplified

Element ss equivalentSimplified dc

equivalent

(MOSFET (enhancement or

depletion), JFET)


Element ss equivalent

Dependent

Sources

(Linear)

Simplified

Simplified

Bipolar

Transistors

Simplified dc

equivalent

VO=AVVIN IO=AIIIN VO=RTIIN IO=GTVIN

Small-signal and simplified dc equivalent elementsReview from Last Lecture

Small-Signal Model of BJT and MOSFET

32133

32122

32111

V,,VVfI

V,,VVfI

V,,VVfII1

I2

I3 V1

V2

V3

4-Terminal

Device

Define

3Q33

2Q22

1Q11

II

II

II

i

i

i

3Q33

2Q22

1Q11

VV

VV

VV

v

v

v

Small signal model is that which represents the relationship between the

small signal voltages and the small signal currents

Consider 4-terminal network


32133

32122

32111

V,,VVfI

V,,VVfI

V,,VVfI

Nonlinear network characterized by 3 functions each

functions of 3 variables


I1

I2

I3 V1

V2

V3

4-Terminal

Device


32133

32122

32111

V,,VVfI

V,,VVfI

V,,VVfI

3132121111 vyyy vvi

Nonlinear Model Linear Model at

(alt. small signal model)

VQ

3232221212 vyyy vvi

3332321313 vyyy vvi

QVVj

321iij

V

V,,VVfy

where

Small Signal Model DevelopmentReview from Last Lecture

QVVj

321iij

V

V,,VVfy

3232221212 vyyy vvi

3332321313 vyyy vvi

3132121111 vyyy vvi

• This is a small-signal model of a 4-terminal network and it is linear

• 9 small-signal parameters characterize the linear 4-terminal network

• Small-signal model parameters dependent upon Q-point !

where

Small Signal Model


y11y12V2

V1

i1

y13V3

y22y21V1

V2

i2

y23V3

y33y31V1

V3

i3

y32V2

A small-signal equivalent circuit of a 4-terminal nonlinear network(equivalent circuit because has exactly the same port equations)

QVVj

321iij

V

V,,VVfy

Equivalent circuit is not unique

Equivalent circuit is a three-port network

32133

32122

32111

V,,VVfI

V,,VVfI

V,,VVfI

4-terminal small-signal network summary

I1

I2

I3 V1

V2

V3

4-Terminal

Device

3232221212 vyyy vvi

3332321313 vyyy vvi

3132121111 vyyy vvi y11y12V2

V1

i1

y13V3

y22y21V1

V2

i2

y23V3

y33y31V1

V3

i3

y32V2

QVVj

321iij

V

V,,VVfy

Small signal model:

Small-Signal Model

2122

2111

,VVfI

,VVfI

I1

I2

V1

V2

3-Terminal

Device

Define

1 1 1Q

2 2 2Q

I I

I I

i

i 2Q22

1Q11

VV

VV

v

v




Small-Signal Model

1 1 1 2 3

2 2 1 2 3

3 3 1 2 3

, ,

, ,

, ,

g

g

g

i V V V

i V V V

i V V V

I1

I2

I3 V1

V2

V3

4-Terminal

Device

3232221212 Vyyy VVi

3332321313 Vyyy VVi

3132121111 Vyyy VVi

QVVj

321iij

V

V,,VVfy

3


Small-Signal Model

2122

2111

,

,

VVi

VVi

g

g

I1

I2

V1V2

3-Terminal

Device

2221212 VVi yy 2121111 VVi yy

QVVj

21iij

V

,VVfy

y11 y22

y12V2

y21V1

V1 V2

i1i2

A Small Signal Equivalent Circuit

2Q

1Q

V

VV


4 small-signal parameters characterize this 3-terminal (two-port) linear network

Small signal parameters dependent upon Q-point

3-terminal small-signal network summary

Small signal model:

y11 y22

y12V2

y21V1

V1 V2

i1i2

QVVj

21iij

V

,VVfy

I1

I2

V1V2

3-Terminal

Device

2221212 VVi yy 2121111 VVi yy

2122

2111

,VVfI

,VVfI

Small-Signal Model

1 1 1I f V

I1

V1

2-Terminal

Device

Define

1 1 1QI I i

1 1 1QV V v




Small-Signal Model

1 1 1 2 3

2 2 1 2 3

3 3 1 2 3

, ,

, ,

, ,

g

g

g

i V V V

i V V V

i V V V

I1

I2

I3 V1

V2

V3

4-Terminal

Device

3232221212 Vyyy VVi

3332321313 Vyyy VVi

3132121111 Vyyy VVi

QVVj

321iij

V

V,,VVfy

2


Small-Signal Model

1 11 1yi V

Q

1 1

11

1 V V

f Vy

V

y11V1

i1

A Small Signal Equivalent Circuit

1QV V


I1

V1

2-Terminal

Device

This was actually developed earlier !

Nonlinear

Device

Linearized

Small-signal

Device

Linearized nonlinear devices

How is the small-signal equivalent circuit

obtained from the nonlinear circuit?

What is the small-signal equivalent of the

MOSFET, BJT, and diode ?

Small Signal Model of MOSFET

3-terminal device 4-terminal device

MOSFET is actually a 4-terminal device but for many applications

acceptable predictions of performance can be obtained by treating it as

a 3-terminal device by neglecting the bulk terminal

When treated as a 4-terminal device, the bulk voltage introduces one additional

term to the small signal model which is often either negligibly small or has no

effect on circuit performance (will develop 4-terminal ss model later)

In this course, we have been treating it as a 3-terminal device and in this

lecture will develop the small-signal model by treating it as a 3-terminal device


1

GS T

DS

D OX GS T DS GS DS GS T

2

OX GS T DS GS T DS GS T

0 V V

VWI μC V V V V V V V V

L 2

WμC V V V V V V V V

2L

T

GI 0

3-terminal device

Large Signal Model

MOSFET is usually operated in saturation region in linear applications

where a small-signal model is needed so will develop the small-signal

model in the saturation region

ID

VDS

VGS1

VGS6

VGS5

VGS4

VGS3

VGS2

SaturationRegion

CutoffRegion

TriodeRegion


Q

G

11

GS V V

Iy

V

GI 0

2 2 1 2I f V,V

Q

i 1 2

ij

j V V

f V ,Vy

V

1 1 1 2I f V,V

12

D OX GS T DS

WI =μC V V V

2L

Small-signal model:

Q

G

12

DS V V

Iy

V

Q

D

21

GS V V

Iy

V

Q

D

22

DS V V

Iy

V

G 1 GS DSI f V ,V

D 2 GS DSI f V ,V


Q

G

11

GS V V

Iy ?

V

GI 0

12

D OX GS T DS

WI =μC V V V

2L

Small-signal model:

Q

G

12

DS V V

Iy ?

V

Q

D

21

GS V V

Iy ?

V

Q

D

22

DS V V

Iy ?

V


0

Q

G

11

GS V V

Iy

V

GI 0

2 2 1 2I f V,V

1 1 1 2I f V,V

12

D OX GS T DS

WI =μC V V V

2L

Small-signal model:

0

Q

G

12

DS V V

Iy

V

2 1 1Q

Q

1D

21 OX GS T DS OX GSQ T DSQ

V VGS V V

I W Wy μC V V V μC V V V

V 2L L

Q

Q

2D

22 OX GS T DQ

V VDS V V

I Wy μC V V I

V 2L

21 OX GSQ T

Wy μC V V

L


011

y GI 0

12

D OX GS T DS

WI =μC V V V

2L

012

y

22 DQy I

21 OX GSQ T

Wy μC V V

L

21 22D GS DSy y i V V

11 12G GS DSy y i V V

y22y21VGS

VGS

G D

S

An equivalent circuit


22 DQy I

Og

21 OX GSQ T

Wy μC V V

Lm

g

y22y21VGS

VGS

G D

S

by convention, y21=gm, y22=g0

gOgmVGS

VGS

G D

S

D m GS O DSg g i V V

0Gi

Note: go vanishes when λ=0


DQI

Og

OX GSQ T

WμC V V

Lm

g

gOgmVGS

VGS

G D

S

Alternate equivalent expressions for gm:

12 2

DQ OX GSQ T DSQ OX GSQ T

W WI =μC V V V μC V V

2L 2L

OX GSQ T

WμC V V

Lm

g

OX DQ

W2μC I

Lm

g

2DQ

m

GSQ T

Ig

V V

Small signal analysis example

VDD

R

M1

VIN

VOUT

VSS

VIN=VMsinωt

DQ

v

SS T

2I RA

V V

Consider again:

RM1

VIN

VOUT

2

D OX GS T

WI =μC V V

2L

Derived for λ=0

gOgmVGS

VGSVIN

VOUT

R

Recall the derivation was very tedious and time consuming!

ss circuit


Consider again:

RM1

VIN

VOUT

For λ=0, gO= λIDQ = 0

1OUT m

V

IN O

V gA

V g R

gOgmVGS

VGSVIN

VOUT

R

OUT

V m

IN

A g R V

V

2DQ

m

GSQ T

Ig

V V

but

thus

DQ

v

SS T

2I RA

V V

VGSQ = -VSS

VDD

R

M1

VIN

VOUT

VSS


Consider again:

RM1

VIN

VOUT

For λ=0, gO= λIDQ = 0

1OUT m

V

IN O

V gA

V g R

gOgmVGS

VGSVIN

VOUT

R

DQ

v

SS T

2I RA

V V

Same expression as derived before

More accurate gain can be obtained if

λ effects are included and does not significantly

increase complexity of small signal analysis

Small Signal Model of BJT

3-terminal device

VCE

IC

Forward Active

Cutoff

Saturation

Usually operated in Forward Active Region

when small-signal model is needed

1BE

t

V

V CE

C S E

AF

VI J A e

V

t

BE

V

V

ESB e

β

AJI

Forward Active Model:


Q

B

11

BE V V

Iy

Vg

2 2 1 2I f V,V

Q

i 1 2

ij

j V V

f V ,Vy

V

1 1 1 2I f V,V

Small-signal model:

Q

B

12

CE V V

Iy

V

Q

C

22

CE V V

Iy

VO

g

t

BE

V

V

ESB e

β

AJI

1BE

t

V

V CE

C S E

AF

VI J A e

V

11 12B BE CEy y i V V

21 22C BE CEy y i V V

Nonlinear model:

Q

C

21

BE V V

Iy

Vm

g

Note: gm, gπ and go used for notational consistency with legacy terminology


Q

B

11

BE V V

Iy ?

Vg

Q

i 1 2

ij

j V V

f V ,Vy

V

Small-signal model:

Q

B

12

CE V V

Iy ?

V

Q

C

22

CE V V

Iy ?

VO

g



t

BE

V

V

ESB e

β

AJI

Nonlinear model:

1BE

t

V

V CE

C S E

AF

VI J A e

V

Q

C

21

BE V V

Iy ?

Vm

g


1 BE

t

QQ

V

V BQ CQS EB

11

V VBE t tV V

I IJ AIy e

V β V βVt

gV

Small-signal model:

0

Q

B

12

CE V V

Iy

V

1BE

t

Q Q

V

V CQC CE

21 S E

BE t AF tV V V V

II V1y J A e

V V V Vm

g

BE

t

Q

Q

V

V

CQC S E

22

CE AF AFV VV V

II J A ey

V V VO

g

BE

t

V

VS E

B

J AI e

β

1BE

t

V

V CE

C S E

AF

VI J A e

V

Nonlinear model:



Note: usually prefer to express in terms of ICQ


CQ

t

I

βVg

CQ

t

I

Vm

g CQ

AF

I

VO

g



C m BE O CEg g i V V

B BEg

i V

gOgmVBE

VBEgπ

B

E

C

An equivalent circuit


VIN=VMsinωt

Consider again:

Derived for VAF=0

Recall the derivation was very tedious and time consuming!

ss circuit

R

Q1

VIN(t)

VOUT

VCC

VEE

CQ

VB

t

I RA

V

R

VIN

VOUT

RVIN

VOUT

VbegmVbe

gπ g0

R

VIN

VOUT

RVIN

VOUT

VbegmVbe

gπ

OUT m BE

IN BE

= -g R

=

V V

V V

OUTV m

IN

A = -g RV=V

CQIm

t

g =V

ICQV

t

RA -

V=

Note this is identical to what was obtained with the direct nonlinear analysis

Neglect VAF effects (i.e. VAF=∞) to be consistent with earlier analysis

0CQ

AF

I

V

AFO V

g

Small Signal BJT Model – alternate

representation

t

CQm

V

Ig

t

CQ

βV

Ig

AF

CQ

V

Ig o

B

E

C

Vbe

ic

gmVbego Vce

ib

gπ

bbe iv πg

π

mm

g

gg bbe iv

β

βV

I

V

I

g

g

t

Q

t

Q

π

m

bbe iv βgm

Observe :

Can replace the voltage dependent current source

with a current dependent current source

B

E

C

Vbe

ic

gmVbego Vce

ib

gπ

t

CQm

V

Ig

t

CQ

βV

Ig

AF

CQ

V

Ig o

t

CQ

βV

Ig

AF

CQ

V

Ig o

B

E

C

Vbe

ic

go Vce

ib

gπbiβ

Alternate equivalent small signal model

Small Signal BJT Model – alternate

representation

Bi-Linear Relationship between

i1 ,i2, v1, v2

i1

Linear Two Port

v1

i2

v2

Small-Signal Model Representations

Have developed small-signal models for the MOSFET and BJT

Models have been based upon arbitrary assumption that v1, v2 are independent

variables

Have already seen some alternatives for parameter definitions in these models

(3-terminal network – also relevant with 4-terminal networks)

Alternative representations are sometimes used


i1 ,i2, v1, v2

i1

Linear Two Port

v1

i2

v2


The good, the bad, and the unnecessary !!

i1

Linear Two Port

v1

i2

v2

y-parameters

1 11 1 12 2y y i V V

2 21 1 22 2y y i V V

i1

Linear Two Port

v1

i2

v2

y-parameters

1 1 2mrg g i V V

2 1 2m og g i V V

i1

Linear Two Port

v1

i2

v2

H-parameters

(Hybrid Parameters)

1 11 1 12 2h h V i v

2 21 1 22 2h h i i v

i1

Linear Two Port

v1

i2

v2

H-parameters

(Hybrid Parameters)

1 1 2ie reh h V i v

2 1 2fe oeh h i i v

what we have developed:

The hybrid parameters:


i1 ,i2, v1, v2

i1

Linear Two Port

v1

i2

v2


The z-parameters

The ABCD parameters:i1

Linear Two Port

v1

i2

v2

ABCD-parameters

1 2 2A Bv V i

1 2 2C D i v i

i1

Linear Two Port

v1

i2

v2

z-parameters

1 11 1 12 2z z V i i

2 21 1 22 2z z V i i


i1 ,i2, v1, v2

i1

Linear Two Port

v1

i2

v2


The S-parameters

The T parameters:

i1

Linear Two Port

i2

v2

S-parameters

v1

i1

Linear Two Port

i2

v2

T-parameters

v1

(embedded with source and load impedances)

(embedded with source and load impedances)


i1 ,i2, v1, v2

i1

Linear Two Port

v1

i2

v2



i1

Linear Two Port

v1

i2

v2

y-parameters

1 11 1 12 2y y i V V

2 21 1 22 2y y i V V

i1

Linear Two Port

v1

i2

v2

y-parameters

1 1 2mrg g i V V

2 1 2m og g i V V

i1

Linear Two Port

v1

i2

v2

H-parameters

(Hybrid Parameters)

1 11 1 12 2h h V i v

2 21 1 22 2h h i i v

i1

Linear Two Port

v1

i2

v2

H-parameters

(Hybrid Parameters)

1 1 2ie reh h V i v

2 1 2fe oeh h i i v

i1

Linear Two Port

v1

i2

v2

z-parameters

1 11 1 12 2z z V i i

2 21 1 22 2z z V i i

i1

Linear Two Port

v1

i2

v2

ABCD-parameters

1 2 2A Bv V i

1 2 2C D i v i

i1

Linear Two Port

i2

v2

S-parameters

v1

i1

Linear Two Port

i2

v2

T-parameters

v1

• Equivalent circuits often given for each representation

• All provide identical characterization

• Easy to move from any one to another


i1 ,i2, v1, v2

i1

Linear Two Port

v1

i2

v2



As of Oct 16, 2015


i1 ,i2, v1, v2

i1

Linear Two Port

v1

i2

v2



As of Oct 28, 2013

Active Device Model Summary

Simplified

Simplified

MOS

transistors

Diodes

Simplified

Simplified

Simplified

Bipolar

Transistors

Element ss equivalent dc equivalnet

What are the simplified dc equivalent models?

Active Device Model SummaryWhat are the simplified dc equivalent models?

Simplified

Simplified

Simplified

Simplified

Simplified

0.6V

dc equivalent

G D

S

VGSQ 2OX

GSQ TnμC W

V -V2L

G D

S

VGSQ 2OX

GSQ TpμC W

V -V2L

B C

E

BQβI0.6V

IBQ

B C

E

BQβI0.6V

IBQ

Small-signal Operation of Nonlinear Circuits

• Small-signal principles

• Example Circuit

• Small-Signal Models

• Small-Signal Analysis of Nonlinear Circuits

Recall:

1. Linearize nonlinear devices(have small-signal model for key devices!)

2. Replace all devices with small-signal

equivalent

3. Solve linear small-signal network

Alternative Approach to small-signal

analysis of nonlinear networks

Remember that the small-signal model is operating point dependent!

Thus need Q-point to obtain values for small signal parameters

M1

M2

VIN

VOUT

VDD

VSS

Example:

Determine the small signal voltage gain AV=VOUT/VIN. Assume M1 and M2

are operating in the saturation region and that λ=0

M1

M2

VIN

VOUT

VDD

VSS

Example: Determine the small signal voltage gain AV=VOUT/VIN. Assume M1 and M2


VIN

VOUT

M1 M2

Small-signal circuit

M1

M2

VIN

VOUT

VDD

VSS



VIN

VOUT

M1 M2


gmVGSVGS

G

S

D

Small-signal MOSFET model for λ=0



VIN

VOUT

M1 M2



VGS1 gm1VGS1VIN

VOUT

VGS2 gm2VGS2

Example:


VGS1 gm1VGS1VIN

VOUT

VGS2 gm2VGS2

Analysis:

By KCL

1 1 2 2m GS m GSg gV V

but

1GS INV V

2GS OUT-V V

thus:

1

2

OUT m

V

IN m

gA

g V

V

Example:


VGS1 gm1VGS1VIN

VOUT

VGS2 gm2VGS2

Analysis:

1

2

OUT m

V

IN m

gA

g V

V

1

1 1 2

2 12

2

2

2

D OX

V

D OX

WI C

L W LA

W LWI C

L

Recall: 1

1

2m D OX

Wg I C

L

Example:


VGS1 gm1VGS1VIN

VOUT

VGS2 gm2VGS2

Analysis:1

2

OUT m

V

IN m

gA

g V

V

1 2

2 1

V

W LA

W L

Recall:

If L1=L2, obtain

1 2 1

2 1 2

V

W L WA

W L W

The width and length ratios can be accurately set when designed in a standard

CMOS process

Example:Obtain the small signal model of the following circuit. Assume

MOSFET is operating in the saturation region

ExampleObtain the small signal model of the following circuit. Assume

MOSFET is operating in the saturation region

Solution:

gmvgsvgsg0

G D

V

I

0mV g g I

0

1 1EQ

m m

Rg g g

End of Lecture 23

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