EE 330 Lecture 27 - Iowa State University

EE 330 Lecture 27

Small-Signal Modelsss

models of BJT

Quiz 20 Obtain the small signal model of the following circuit. Assume MOSFET is operating in the saturation region

And the number is ….

6

31

2

45

7

8

9

And the number is ….

6

31

2

4

5

7

8

9

Quiz 20 Obtain the small signal model of the following circuit. Assume MOSFET is operating in the saturation region

Solution:

( )0mV g g I+ =

0

1 1EQ

m mR

g g g=

+

Small

Signal Model of BJT

3-terminal device

Usually operated in Forward Active Region when small-signal model is needed

1BE

t

VV CE

C S E

AF

VI J A eV

⎛ ⎞= +⎜ ⎟

⎝ ⎠t

BEV

VES

B eβAJI =

Forward Active Model:

Review from Last Time

Small

Signal Model of BJT

CQ

t

IβV

gπ= CQ

t

IVm

g = CQ

AF

IVO

g =

21 22C BE CEy y= +i V V

11 12B BE CEy y= +i V V

C m BE O CEg g= +i V V

B BEg

π=i V


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?

dc equivalent

( )2OXGSQ Tn

μC W V -V2L

G D

S

VGSQ ( )2OXGSQ Tp

μC W V -V2L

B C

E

BQβI0.6V

IBQ

B C

E

BQβI0.6V

IBQ


Example: Determine the small signal voltage gain AV

=VOUT

/VIN

.

Assume M1 and M2are operating in the saturation region and that λ=0

Small-signal circuit

Small-signal circuit


Example:

Small-signal circuit Analysis:

1

2

OUT mV

IN m

gAg

= = −VV

1 2

2 1

V

W LAW L

= −Recall:

If L1 =L2 , obtain

1 2 1

2 1 2

V

W L WAW L W

= − = −

The width and length ratios can be accurately set when designed in a standard CMOS processBut care must be taken in the layout to obtain this accuracy!


1

VIN

OUT

DD

SS

1

OUT DD DV =V -I R

( )2OX

DQ SS T

μ C WI V V2L

= + ( )2OX

D IN SS T

μC WI V -V -V2L

=

( )2TSSOX

DQ VV2L

WC μI +=

Graphical Analysis and Interpretation Consider Again

OUT DD DV =V -I R

( )2OX

DQ SS T

μ C WI V V2L

= +

Graphical Analysis and Interpretation Device Model (family of curves) ( ) ( )2 1 λ= +OX

DQ GS T DS

μ C WI V -V V2L

Load Line

Device Model at Operating Point

OUT DD DV =V -I R

( )2OX

DQ SS T

μ C WI V V2L

= +( )2

OXD IN SS T

μC WI V -V -V2L

=

( )2TSSOX

DQ VV2L

WC μI +=

Graphical Analysis and Interpretation

VGSQ

=-VSS

Device Model (family of curves) ( ) ( )2 1 λ= +OXDQ GS T DS

μ C WI V -V V2L


VGSQ

=-VSS


μ C WI V -V V2L

Saturation region


VGSQ

=-VSS


μ C WI V -V V2L

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


VGSQ

=-VSS


μ C WI V -V V2L

Saturation region

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


VGSQ

=-VSS


μ C WI V -V V2L

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 ExtensionExisting 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)

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

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

( )φφγ −−+= BST0T VVV

T

T0

BS


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

1-20.4Vγ 0.6Vφ

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 Vγ φ φ= − + −

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

1-20.4Vγ 0.6Vφ

Bulk-Diffusion Generally Reverse Biased (VBS

> 0 or at least greater than -0.3V) for n-channel

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

DSD OX GS T DS GS DS GS T

2

OX GS T DS GS T DS GS T

0 V VVWI μC V V V V V V V V

L 2WμC V V V V V V V V2L

T

λ

⎧⎪ ≤⎪⎪ ⎛ ⎞= − − ≥ < −⎨ ⎜ ⎟

⎝ ⎠⎪⎪

− • + ≥ ≥ −⎪⎩


Model Parameters : {μ,COX

,VT0

,φ,γ,λ}

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

0I0I

B

G

==

Small-Signal Model Extension

( ) ( )1

GS T


2




T

λ

⎧⎪ ≤⎪⎪ ⎛ ⎞= − − ≥ < −⎨ ⎜ ⎟

⎝ ⎠⎪⎪

− • + ≥ ≥ −⎪⎩


000 =∂∂

==∂∂

==∂∂

==== QQQ VVGS

G13

VVDS

G12

VVGS

G11 V

IyVIy

VIy

000 =∂∂

==∂∂

==∂∂

==== QQQ VVGS

B33

VVDS

B32

VVGS

B31 V

IyVIy

VIy

Q Q Q

D D D21 12 13

GS DS GSV V V V V V

I I Iy y yV V Vm o mb

g g g= = =

∂ ∂ ∂= = = = = =

∂ ∂ ∂

GI =0

BI =0

( ) ( )DS2

TGSOXD VVV2LWμCI λ+•−= 1


( ) ( )11

OX GS T DS OX EBQ

W WμC 2 V V V μC V2L L

QQ

Dm

V VGS V V

IgV

λ==

∂= = − • + ≅∂

( )2

OX GS T DQ

WμC 2 V V λ λI2L

QQ

Do

V VDS V V

IgV ==

∂= = − • ≅∂

( ) ( )1

OX GS T DS

WμC 2 V V 1+λV2L

Q Q

D Tmb

BS BSV V V V

I VgV V

= =

⎛ ⎞∂ ∂= = − • − •⎜ ⎟∂ ∂⎝ ⎠

( ) ( ) ( )12

11 12 Q

Q Q

D Tmb BS

V VBS BSV V V V

I Vg VV V

γ φ −

== =

∂ ∂ ⎛ ⎞= ≅ • = − − −⎜ ⎟∂ ∂ ⎝ ⎠OX EBQ OX EBQ

W WμC V μC VL L

2m

BSQ

g-Vmb

g γφ

≅

Small Signal Model Summary

OXm EBQ

μC Wg VL

=

DQo λIg =

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

BSQmmb V

ggφγ

2

G

S

gs

bs

ds

d

g

bB

D

dsobsmbgsmd

b

g

vgvgvgii

i

++==

=

0

0

Relative Magnitude of Small Signal MOS Parameters

OXm EBQ

μC Wg VL

=

o DQg λI 5E-7= =

2mb m m

BSQ

g g .26gV

γφ

⎛ ⎞= =⎜ ⎟⎜ ⎟−⎝ ⎠

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

DQOX

m IL

WC2μg =DQ

m

EBQ

2IgV

=

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

22OXDQ EBQ

μC W 10 WI V 1V =5E-52L 2L

≅ =

OXm EBQ

μC Wg V 1E-4L

= =0 m mb

g <<g ,g

mb mg < g

In this example

This relationship is commonIn many circuits, VBS

=0 as well


dsobsmbgsmd

b

g

vgvgvgii

i

++==

=

0

0

( ) ( )1

GS T


2




T

λ

⎧⎪ ≤⎪⎪ ⎛ ⎞= − − ≥ < −⎨ ⎜ ⎟

⎝ ⎠⎪⎪

− • + ≥ ≥ −⎪⎩


Large Signal Model Small Signal Model

OXm EBQ

μC Wg VL

=

DQo λIg =

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

BSQmmb V

ggφγ

2

where

How does gm

vary with IDQ

?

DQOX

m IL

WC2μg =

( )TGSQOX

m VVL

WμCg −=

EBQ

DQ

TGSQ

DQm V

2IVV

2Ig =−

=

Varies with the square root of IDQ

Varies linearly with IDQ

Doesn’t vary with IDQ

How does gm

vary with IDQ

?

All of the above are true –

but with qualification

gm

is a function of more than one variable (IDQ

) and how it varies depends upon how the remaining variables are constrained


( )TGSQOX

m VVL

WμCg −=

DQo λIg =

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

BSQmmb V

ggφγ

2

An

equivalent circuit

This contains absolutely no more information than the previous model


More convenient representation


Simplification that is often adequate


Even further simplification that is often adequate


Alternate equivalent representations for gm

DQOX

m IL

WC2μg =

( )TGSQOX

m VVL

WμCg −=

EBQ

DQ

TGSQ

DQm V

2IVV

2Ig =−

=

( )2TGSOXD VV2LWμCI −≅from

Alternate Equivalent Small Signal Model of the BJT

Small Signal BJT Model

t

CQm V

Ig =t

CQ

βVIg =π

AF

CQ

VIg ≅o

bbe iv =πg

π

mm g

gg bbe iv =

β

βVIVI

gg

t

Q

t

Q

π

m =

⎥⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡

=

bbe iv βgm =

Observe :

Small Signal BJT Model

t

CQm V

Ig =t

CQ

βVIg =π

AF

CQ

VIg ≅o

t

CQ

βVIg =π

AF

CQ

VIg ≅o

B

E

C

Vbe

ic

o Vce

ib

π biβ

Alternate equivalent small signal model

Relative Magnitude of Small Signal BJT Parameters

t

CQm V

Ig =t

CQ

βVIg =π

AF

CQ

VIg ≅o

β

βVIVI

gg

t

Q

t

Q

π

m =

⎥⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡

=

7726mV100

200VVβ

V

VIVβ

I

gg

t

AF

AF

Q

t

Q

o

π =•

≈=

⎥⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡

=

oπm ggg >>>>

Often the go term can be neglected in the small signal model because it is so small

Relative Magnitude of Small Signal Parameters

t

CQm V

Ig =t

CQ

βVIg =π

AF

CQ

VIg ≅o

β

βVIVI

gg

t

Q

t

Q

π

m =

⎥⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡

=

7726mV100

200VVβ

V

VIVβ

I

gg

t

AF

AF

Q

t

Q

o

π =•

≈=

⎥⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡

=

oπm ggg >>>>

Often the go term can be neglected in the small signal model because it is so small

Simplified small signal model

Vbe

ic

mVbe Vce

ib

π

Comparison of BJT and MOSFET

Comparison of MOSFET and BJT

t

CQm V

Ig =

EBVL

WμCg OXm =

DQOX

m IL

WC2μg =

EBQ

DQm V

2Ig =

BJT MOSFET

⎪⎪⎩

⎪⎪⎨

⎧

==>

==>===

2VVif50mV

2V50mV

V

100mVV if250mV

100mV50mV

V

2VV

V2IVI

gg

EBEB

EBEB

t

EB

EB

DQ

t

CQ

mMOS

mBJT

40

The transconductance

of the BJT is typically much larger than that of the MOSFET (and larger is better!) This is due to the exponential rather than quadratic output/input relationship


t

CQm V

Ig =

EBVL

WμCg OXm =

DQOX

m IL

WC2μg =

EBQ

DQm V

2Ig =

BJT MOSFET

⎪⎪⎩

⎪⎪⎨

⎧

==>

==>===

2VVif50mV

2V50mV

V

100mVV if250mV

100mV50mV

V

2VV

V2IVI

gg

EBEB

EBEB

t

EB

EB

DQ

t

CQ

mMOS

mBJT

40

The transconductance

of the BJT is typically much larger than that of the MOSFET (and larger is better)This is due to the exponential rather than quadratic output/input relationship


AF

CQ

VIg ≅oDQIλ=og

BJT MOSFET

5.020001.

11 =≈== − VVAFDQ

AF

CQ

oMOS

oBJT

V1

IVI

gg

λλ

The output conductances

are comparable but that of the BJT is usually modestly smaller (and smaller is better!)


t

CQ

βVIg =π

BJT MOSFET

0=πg

gπ

of a MOSFET is much smaller than that of a BJT (and smaller is better!)

gπ

is the reciprocal of the input impedance

Standard Approach to small-signal analysis of nonlinear networks

NonlinearNetwork

dc EquivalentNetwork

Q-point

Values for small-signal parameters

Small-signalequivalent network

Small-signal output

Total output(good approximation)

Systematic Approach to Small-Signal Circuit Analysis

•

Obtain dc equivalent circuit by replacing all elements with large-signal (dc) equivalent circuits

•

Obtain dc operating points (Q-point)

•

Obtain ac equivalent circuit by replacing all elements with small-signal equivalent circuits

•

Analyze linear small-signal equivalent circuit

Dc and small-signal equivalent elementsRecall

Square-Law Model

( ) ( )1

GS T


2




T

λ

⎧⎪ ≤⎪⎪ ⎛ ⎞= − − ≥ < −⎨ ⎜ ⎟

⎝ ⎠⎪⎪

− • + ≥ ≥ −⎪⎩


0I0I

B

G

==

Simplified MOS Model for Q-point Analysis

( )2

D OX GS T

WI μC V V2L

= −

0I0I

B

G

==

( )2OX

GS T

μC W V -V2L

⎛ ⎞⎜ ⎟⎝ ⎠

Simplified dc equivalent circuit

dc BJT model

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

AF

CEBC V

V1βII

t

BE

VV

ESB e

βAJI =

qkTVt =

VBE

=0.7VVCE

=0.2V

IC

=IB

=0

Forward Active

Saturation

Cutoff

VBE

>0.4V

VBC

<0

IC

<βIB

VBE

<0

VBC

<0

A small portion of the operating region is missed with this model but seldom operate in the missing region

Simplified dc BJT model for Q- point Analysis

C BI βI=

t

BE

VV

ESB e

βAJI =

C BI βI=

BEV 0.6V=

Simplified dc equivalent circuit

Examples

Not convenient to have multiple dc power suppliesVOUTQ

very sensitive to VEE

Examples

Not convenient to have multiple dc power suppliesVOUTQ

very sensitive to VEE

B

E

C

CC

Vin

1

Vout

1

B

1

ExamplesDetermine VOUTQ

, AV

, RIN

B

E

C

CC

Vin

1

Vout

1

B

1

ExamplesDetermine VOUT

and VOUT

(t) if VIN

=.002sin(400t)

B

E

C

CC

Vin

1

Vout

1

B

1

End of Lecture 27

EE 330 Lecture 27 - Iowa State University

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Transcript of EE 330 Lecture 27 - Iowa State University