EE 330 Lecture 28 Small-Signal Model Extension Applications of the Small-signal Model

EE 330Lecture 28

Small-Signal Model Extension

Applications of the Small-signal Model

32133

32122

32111

V,,VVfI

V,,VVfI

V,,VVfI

Nonlinear network characterized by 3 functions each functions of 3 variables

Consider 4-terminal networkI1

I2

I3 V1

V2

V3

4-TerminalDevice

Review from Last Lecture

Consider now 3 functions, each a function of 3 variables

1 11 1 12 2 13 3y y y i v v v

Extending this approach to the two nonlinear functions I2 and I3

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 linear9 small-signal parameters characterize the linear 4-terminal networkSmall-signal model parameters dependent upon Q-point !

where


y11y12V2

V1

i1

y13V3

y22y21V1

V2

i2

y23V3

y33y31V1

V3

i3

y32V2

A small-signal equivalent circuit of a 4-terminal nonlinear network

QVVj

321iij V

V,,VVfy

Equivalent circuit is not uniqueEquivalent circuit is a three-port network


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-TerminalDevice

3232221212 Vyyy VVi

3332321313 Vyyy VVi

3132121111 Vyyy VVi

QVVj

321iij V

V,,VVfy

3

Consider 3-terminal networkReview from Last Lecture

Small-Signal Model

2122

2111

,

,

VVi

VVi

g

g

I1

I2

V1V2

3-TerminalDevice

2221212 VVi yy 2121111 VVi yy

QVVj

21iij V

,VVfy

y11 y22

y12V2

y21V1

V1 V2

i1 i2A Small Signal Equivalent Circuit

2Q

1Q

V

VV

Consider 3-terminal network

4 small-signal parameters characterize this 3-terminal (two-port) linear networkSmall signal parameters dependent upon Q-point


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-TerminalDevice

3232221212 Vyyy VVi

3332321313 Vyyy VVi

3132121111 Vyyy VVi

QVVj

321iij V

V,,VVfy

2

Consider 2-terminal networkReview from Last Lecture

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

Consider 2-terminal network

I1

V1

2-TerminalDevice


Small Signal Model of MOSFET

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


Small Signal Model of MOSFET

DQI

Og

OX GSQ T

WμC V V

Lmg

gOgmVGS

VGS

G D

S

Alternate equivalent expressions:

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

Lmg

OX DQ

W2μC I

Lmg

2DQ

m

GSQ T

Ig

V V


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:


Small Signal Model of BJT

CQ

t

I

βVg

CQ

t

I

Vmg CQ

AF

I

VOg

21 22C BE CEy y i V V11 12B BE CEy y i V V

C m BE O CEg g i V V

B BEg

i V

gOgmVBE

VBEgπ

B

E

C


Active Device Model Summary

Simplified

Simplified

MOStransistors

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 2OXGSQ Tn

μC WV -V

2L

G D

S

VGSQ 2OXGSQ Tp

μC WV -V

2L

B C

E

BQβI0.6V

IBQ

B C

E

BQβI0.6V

IBQ


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

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 `1

VM

SS T

2I RA

V V CQ

VB

t

I RA

V

Verify the gain expression obtained for the BJT using a small signal analysis

R

Q1

VIN(t)

VOUT

VCC

VEE

R

VIN

VOUT

RVIN

VOUT

Vbe gmVbegπ

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 λ effects to be consistent with earlier analysis

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 VV

Example:


VGS1 gm1VGS1VIN

VOUT

VGS2 gm2VGS2

Analysis:

1

2

OUT m

V

IN m

gA

g VV

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 VV

1 2

2 1

V

W LA

W LRecall:

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

R1

VIN(t)

VOUT

VDD

VSS

M1

OUT DD DV =V -I R

2OX

DQ SS T

μ C WI V V

2L 2

OX

D IN SS T

μC WI V -V -V

2L

2TSSOX

DQ VV2L

WC μI

Graphical Analysis and Interpretation Consider Again

ID

VDS

VGS1

VGS6

VGS5

VGS4

VGS3

VGS2

R1

VIN(t)

VOUT

VDD

VSS

M1

OUT DD DV =V -I R

2OX

DQ SS T

μ C WI V V

2L

Graphical Analysis and Interpretation Device Model (family of curves) 2

1 OX

DQ GS T DS

μ C WI V -V V

2L

Load Line

Device Model at Operating Point

ID

VDS

VGS1

VGS6

VGS5

VGS4

VGS3

VGS2

Load LineQ-Point

R1

VIN(t)

VOUT

VDD

VSS

M1

OUT DD DV =V -I R

2OX

DQ SS T

μ C WI V V

2L

2OX

D IN SS T

μC WI V -V -V

2L

2TSSOX

DQ VV2L

WC μI

Graphical Analysis and Interpretation

VGSQ=-VSS

Device Model (family of curves) 2

1 OX

DQ GS T DS

μ C WI V -V V

2L

ID

VDS

VGS1

VGS6

VGS5

VGS4

VGS3

VGS2

Load LineQ-Point

R1

VIN(t)

VOUT

VDD

VSS

M1


VGSQ=-VSS


1 OX

DQ GS T DS

μ C WI V -V V

2L

Saturation region

ID

VDS

VGS1

VGS6

VGS5

VGS4

VGS3

VGS2

Load LineQ-Point

R1

VIN(t)

VOUT

VDD

VSS

M1


VGSQ=-VSS


1 OX

DQ GS T DS

μ C WI V -V V

2L

Saturation region

•Linear signal swing region smaller than saturation region

•Modest nonlinear distortion provided saturation region operation maintained

•Symmetric swing about Q-point

•Signal swing can be maximized by judicious location of Q-point

ID

VDS

VGS1

VGS6

VGS5

VGS4

VGS3

VGS2

Load LineQ-Point

R1

VIN(t)

VOUT

VDD

VSS

M1


VGSQ=-VSS


1 OX

DQ GS T DS

μ C WI V -V V

2L

Saturation region

Very limited signal swing with non-optimal Q-point location

ID

VDS

VGS1

VGS6

VGS5

VGS4

VGS3

VGS2

Load LineQ-Point

R1

VIN(t)

VOUT

VDD

VSS

M1


VGSQ=-VSS


1 OX

DQ GS T DS

μ C WI V -V V

2L

Saturation region

•Signal swing can be maximized by judicious location of Q-point

•Often selected to be at middle of load line in saturation region

Small-Signal MOSFET Model Extension

Existing model does not depend upon the bulk voltage !

Observe that changing the bulk voltage will change the electric field in the channel region !

VBS

VGS

VDS

IDIG

IB

(VBS small)

E

Further Model ExtensionsExisting model does not depend upon the bulk voltage !

Observe that changing the bulk voltage will change the electric field in the channel region !

VBS

VGS

VDS

IDIG

IB

(VBS small)

E

Changing the bulk voltage will change the thickness of the inversion layer

Changing the bulk voltage will change the threshold voltage of the device

BST0T VVV

VT

VT0

VBS

~ -5V

BST0T VVV

Typical Effects of Bulk on Threshold Voltage for n-channel Device

1-20.4V 0.6V

Bulk-Diffusion Generally Reverse Biased (VBS< 0 or at least less than 0.3V) for n-channel Shift in threshold voltage with bulk voltage can be substantialOften VBS=0

T T0 BSV V V

Typical Effects of Bulk on Threshold Voltage for p-channel Device

1-20.4V 0.6V

Bulk-Diffusion Generally Reverse Biased (VBS > 0 or at least greater than -0.3V) for n-channel

VT

VT0

VBS

Same functional form as for n-channel devices but VT0 is now negative and the magnitude of VT still increases with the magnitude of the reverse bias

Model Extension Summary

1

GS T

DS


2


0 V V


L 2


2L

T

BST0T VVV

Model Parameters : {μ,COX,VT0,φ,γ,λ}

Design Parameters : {W,L} but only one degree of freedom W/L

0I

0I

B

G

Small-Signal Model Extension

1

GS T

DS


2


0 V V


L 2


2L

T

BST0T VVV

000

QQQ VVGS

G13

VVDS

G12

VVGS

G11 V

Iy

V

Iy

V

Iy

000

QQQ VVGS

B33

VVDS

B32

VVGS

B31 V

Iy

V

Iy

V

Iy

Q Q Q

D D D

21 12 13

GS DS GSV V V V V V

I I Iy y y

V V Vm o mbg g g

GI =0

BI =0

DS2

TGSOXD VVV2L

WμCI 1

BST0T VVV

11

OX GS T DS OX EBQ

W WμC 2 V V V μC V

2L LQQ

D

m

V VGS V V

Ig

V

2

OX GS T DQ

WμC 2 V V λ λI

2LQQ

D

o

V VDS V V

Ig

V

1

OX GS T DS

WμC 2 V V 1+λV

2LQ Q

D T

mb

BS BSV V V V

I Vg

V V

1

21

1 12 Q

Q Q

D T

mb BSV V

BS BSV V V V

I Vg V

V V

OX EBQ OX EBQ

W WμC V μC V

L L

2m

BSQ

g-V

mbg

Small Signal Model Summary

OX

m EBQ

μC Wg V

L

DQo λIg

BSQ

mmbV

gg

2

G

S

Vgs

Vbs

Vds

id

ig

ibB

D

Small Signal

Network

dsobsmbgsmd

b

g

vgvgvgi

i

i

0

0

Relative Magnitude of Small Signal MOS Parameters

OX

m EBQ

μC Wg V

L

o DQg λI 5E-7

2mb m m

BSQ

g g .26gV

d m gs mb bs o dsi g v g v g v

DQOX

m IL

WC2μg

DQ

m

EBQ

2Ig

V

Consider:

3 alternate equivalent expressions for gm

If μCOX=100μA/V2 , λ=.01V-1, γ = 0.4V0.5, VEBQ=1V, W/L=1, VBSQ=0V

-4

22OX

DQ EBQ

μC W 10 WI V 1V =5E-5

2L 2L

OX

m EBQ

μC Wg V 1E-4

L

0 m mbg <<g ,g

mb mg < g

In this example

This relationship is common

In many circuits, VBS=0 as well

Large and Small Signal Model Summary

dsobsmbgsmd

b

g

vgvgvgi

i

i

0

0

1

GS T

DS


2


0 V V


L 2


2L

T

BST0T VVV

Large Signal Model Small Signal Model

OX

m EBQ

μC Wg V

L

DQo λIg

BSQ

mmbV

gg

2

where

End of Lecture 28

Documents

EE 330 Lecture 28 Small-Signal Model Extension Applications of the Small-signal Model