EE 330 Lecture 25class.ece.iastate.edu/ee330/lectures/EE 330 Lect 25 Spring 2020.pdf · Lecture 25 • Small Signal Analysis • Small Signal Analysis of BJT Amplifier. Exam 2 Friday

EE 330

Lecture 25

• Small Signal Analysis

• Small Signal Analysis of BJT Amplifier

Exam 2 Friday Mar 13

Exam 3 Monday April 13

Review in Rm 2245 Coover !

Small-Signal Analysis

IOUTSS

VOUTSS

VINSS or IINSS

Biasing (voltage or current)

Nonlinear

Circuit

IOUTSS

VOUTSS

VINSS or IINSSLinear Small

Signal Circuit

• Will commit next several lectures to developing this approach

• Analysis will be MUCH simpler, faster, and provide significantly more insight

• Applicable to many fields of engineering

Map Nonlinear circuit to linear

small-signal circuit

Nonlinear

Analysis

Linear

Analysis

Review from Last Lecture

Small-Signal Principle y

x

yQ

xQ

xSS

ySS

Q-point y=f(x)

Changing coordinate systems:

ySS=y-yQ

xSS=x-xQ

Q

Q Q

x=x

fy - y x - x

x

Q

SS SS

x=x

fy x

x


Arbitrary Nonlinear One-Port

yi VQV=V

defn I

Vy

d f

SS

e

= i i

d

SS

ef

= v V

V

IQ

I

Q-point

VQ

iSS

vSS

Q

SS S

V=

S

V

I

Vi v

Linear model of the nonlinear

device at the Q-point

V Nonlinear

One-Port

I

I = f V

yV

iA Small Signal Equivalent Circuit:

I

V

2-Terminal

Nonlinear

Device

f(v)

Arbitrary Nonlinear One-Port

QV=V

Iy

V

yi V

• The small-signal model of this 2-terminal electrical network is a resistor of value 1/y

or a conductor of value y

• One small-signal parameter characterizes this one-port but it is dependent on Q-

point

• This applies to ANY nonlinear one-port that is differentiable at a Q-point (e.g. a diode)

V Nonlinear

One-Port

I

Linear 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)


Nonlinear

Device

Linearized

Small-signal

Device

Linearized nonlinear devices

This terminology will be used in THIS course to emphasize difference

between nonlinear model and linearized small signal model


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

Example: Obtain the small-signal equivalent circuit

VINSS

VDD

M1

VOUT

VSS

C1

R1

R2

R4 R5

R3 R6

RL

C2

C4

C3

R7

C1,C2, C3 large

C4 small

IDD

Q1

VIN

M1

VOUT

R1

R2

R4 R5

R3 R6

RL

C4

R7

Q1M1

VOUT

R1//R2

R4

R5

R6

RL

C4

R7

Q1

VIN


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 4-Terminal Network

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

V1

V2

4-Terminal

Linear Device

V3

i3

I1

I2

V1

V2

4-Terminal

Device

V3

I3

(Nonlinear)

32133

32122

32111

V,,VVfI

V,,VVfI

V,,VVfI

Mapping is unique (with same models)


Small-Signal Model of 4-Terminal Network

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

V1

V2

4-Terminal

Linear Device

V3

i3

I1

I2

V1

V2

4-Terminal

Device

V3

I3

(Nonlinear)

32133

32122

32111

V,,VVfI

V,,VVfI

V,,VVfI

Systematic procedure for developing a small-signal model for any nonlinear device

will now be developed

Will use 4-terminal device as an example and obtain results for 3-terminal and 2-

terminal devices by inspection

Based upon multi-variate Taylor’s series expansion truncated after first-order terms

Will then use this procedure to get small-signal model of Diode, MOSFET, BJT,

and JFET

Recall for a function of one variable

Taylor’s Series Expansion about the point x0

...!2

1)(

2

02

2

0

00

0

xxx

fxx

x

fxfxfy

xxxx

xx

)(xfy

If x-x0 is small

0

0

0xx

x

fxfy

xx

xx

00

0

xxx

fyy

xx

(x0 is termed the expansion point or the Q-point)


)(xfy If x-x0 is small

00

0

xxx

fyy

xx

00

0

xxx

fyy

xx

If we define the small signal variables as

0yy y

0xx x


)(xfy If x-x0 is small

00

0

xxx

fyy

xx

00

0

xxx

fyy

xx

If we define the small signal variables as

0yy y

0xx x

Then

Qx X

f

x

y x This relationship is linear !

Consider now a function of n variables

1( ,... ) ( )ny f x x f x

If we consider an arbitrary expansion point

The multivariate Taylor’s series expansion around the point is given by

0 10 20 0{ , ,... }nX x x x

0X

0

0

0

1

2,

11

( )

..

n

x x

k x x

n n

kj x x

fy f x f

f

k k0

k

j j0 k k0

j k

x x -xx

1x -x x -x (H.O.T.)

x x 2!

Truncating after first-order terms, we obtain the approximation

0

0

1

k k0

k

x -xx

n

k x x

fy y

where 00 x xy f x

Multivariate Taylors Series Expansion

1( ,... ) ( )ny f x x f x

0

1

0 k k0

k

y-y x -xx

n

k x x

f

Linearized approximation

1

ky a x

k

n

ss ss

k

This can be expressed as

where

0y y yss

k k0x x -xkss

0k

ax

k

x x

f

1

ka

k

n

k

y x

0y y y

k k0x -xkx

1 http://www.chem.mtu.edu/~tbco/cm416/taylor.html

In the more general form1, the multivariate Taylor’s series

expansion can be expressed as

32133

32122

32111

V,,VVfI

V,,VVfI

V,,VVfI

Nonlinear network characterized by 3 functions each

functions of 3 variables

Consider 4-terminal network

I1

I2

I3 V1

V2

V3

4-Terminal

Device

32133

32122

32111

V,,VVfI

V,,VVfI

V,,VVfI

Consider now 3 functions each functions of 3 variables

1Q

Q 2Q

3Q

V

V V

V

Define

In what follows, we will use as an expansion point in a Taylor’s

series expansion. QV

32133

32122

32111

V,,VVfI

V,,VVfI

V,,VVfI

Consider now 3 functions each functions of 3 variables

1Q

Q 2Q

3Q

V

V V

V

Define

3Q3

VV3

32112Q2

VV2

32111Q1

VV1

3211

3Q2Q1Q132111

VVV

V,,VVfVV

V

V,,VVfVV

V

V,,VVf

V,,VVfV,,VVfI

QQQ

3Q3

VV3

32112Q2

VV2

32111Q1

VV1

32111Q1 VV

V

V,,VVfVV

V

V,,VVfVV

V

V,,VVfII

QQQ

The multivariate Taylors Series expansion of I1, around the operating point . when

truncated after first-order terms, can be expressed as: QV

Consider first the function I1

or equivalently as:

Make the following definitions

3Q3

VV3

32112Q2

VV2

32111Q1

VV1

32111Q1 VV

V

V,,VVfVV

V

V,,VVfVV

V

V,,VVfII

QQQ

QVV1

321111

V

V,,VVfy

QVV2

321112

V

V,,VVfy

QVV3

321113

V

V,,VVfy

3Q33

2Q22

1Q11

II

II

II

i

i

i

3Q33

2Q22

1Q11

VV

VV

VV

v

v

v

repeating from previous slide:

It thus follows that

3132121111 vyyy vviThis is a linear relationship between the small signal

electrical variables !

Extending this approach to the two nonlinear functions I2 and I3

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 Development

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 Development

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 !

• Termed the y-parameter model or “admittance” –parameter model

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 is that which represents the relationship between the

small signal voltages and the small signal currents


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 (not unique)

2Q

1Q

V

VV


• Small-signal model is a “two-port”

• 4 small-signal parameters characterize this 3-terminal 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 is that which represents the relationship between the

small signal voltages and the small signal currents


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

y11V 1

i1A Small Signal Equivalent Circuit

1QV V


I1

V1

2-Terminal

Device

This was actually developed earlier !

Small-signal model is a one-port

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

Recall: termed the y-parameter model


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

(y-parameter model)

Small-Signal Model of MOSFET

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

(y-parameter model)

still y-parameter model

but use “g” parameter notation

Small-Signal Model of MOSFET

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 (equivalently g0=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

This gain is expressed in terms of small-signal model parameters


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

• Will develop small-signal model in Forward Active Region

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

V

g

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

V

O

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

V

m

g

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

y-parameter model


Q

B

11

BE V V

Iy ?

V

g

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 ?

V

O

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 ?

V

m

g


1 BE

t

QQ

V

V BQ CQS EB

11

V VBE t tV V

I IJ AIy e

V β V βV

t

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 V

m

g

BE

t

Q

Q

V

V

CQC S E

22

CE AF AFV VV V

II J A ey

V V V

O

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

y-parameter model using “g” parameter notation

Small signal analysis example

VIN=VMsinωt

Consider again:

Derived for VAF=0 (equivalently go=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

End of Lecture 25

Documents

EE 330 Lecture 25class.ece.iastate.edu/ee330/lectures/EE 330 Lect 25 Spring 2020.pdf · Lecture 25 • Small Signal Analysis • Small Signal Analysis of BJT Amplifier. Exam 2 Friday