Signal Circuit and Transistor Small-Signal Modelaries.ucsd.edu/.../ECE65/12-W/Slides/ECE65_W12-SSM.pdf · Signal Circuit and Transistor Small-Signal Model Lecture notes: Sec. 5

Signal Circuit and Transistor Small-Signal Model

Lecture notes: Sec. 5

Sedra & Smith (6th Ed): Sec. 5.5 & 6.7 Sedra & Smith (5th Ed): Sec. 4.6 & 5.6

F. Najmabadi, ECE65, Winter 2012

Transistor Amplifier Development


Bias & Signal

.....

:

,...)( ,,, : MOS

rRR

rRRD

gsGSGS

DDSGS

iIivVvR

vVvivv

+=+=

+=

....., :

,,, : MOS

RRD

DDSGS

IVR

IVV

....., :

,,, : MOS

rrD

ddsgs

ivR

ivv

+

Bias Signal only = (Bias + Signal) - Bias

?

Finding signal circuit elements -- Resistor


)( RRRRRRr IiRRIRiVvv −=−=−=

Resistor Voltage Current iv Equation

Bias + Signal: vR iR vR = R iR

Bias: VR IR VR = R IR

Signal: vr = vR − VR ir = iR − IR ??

rr Riv =

A resistor remains as a resistor in the signal circuit.

Finding signal circuit elements -- Capacitor


dtVvdC

dtdVC

dtdvCIii CCCC

CCc)( −

=−=−=

Capacitor Voltage Current iv Equation

Bias + Signal: vC iC iC = C dvC /dt

Bias: VC IC IC = C dVC /dt

Signal: vc = vC − VC ic = iC − IC ??

dtdvCi c

c =

A capacitor remains as a capacitor in the signal circuit. o Since VC = const., IC = 0 , i.e., A capacitor acts as an open circuit for bias circuit.

Finding signal circuit elements – IVS & ICS


0=−=−= SSIVSIVSivs VVVvv

Independent voltage source

Voltage Current iv Equation

Bias + Signal: vIVS iIVS vIVS = VS = const

Bias: VIVS IIVS VIVS = VS = const

Signal: vivs = vIVS − VIVS iivs = iIVS − IIVS ??

0 ,0 ≠= ivsivs iv

An independent voltage source becomes a short circuit!

An independent current source becomes an open circuit!

Similarly:

Exercise: Show that dependent sources remain as dependent sources

Summary of signal circuit elements

Resistors& capacitors: The Same o Capacitor act as open circuit in the bias circuit.

Independent voltage source (e.g., VDD) : Effectively grounded

Independent current source: Effectively open circuit o Careful about current mirrors as they are NOT “ideal” current sources (early

effect and/or channel width modulation was ignored!)

Dependent sources: The Same

Non-linear Elements: Different! o Diodes & transistors ?


Diode Signal Response


−

×

=

−

+=−=

1expexp

expexp :Signal

T

d

T

Dsd

T

Ds

T

dDsDDd

nVv

nVVIi

nVVI

nVvVIIii

VD

ID

vd

id ?

=+

T

DsD nV

vIi exp :Signal Bias

=

T

DsD nV

VII exp :Bias

vD

iD

A different iv equation! iv equation is non-linear! Related to bias value, ID!

1exp

−

×=

T

dDd nV

vIi

Diode small-signal model:


1exp

−

×=

T

dDd nV

vIi

vd

id ?

DT

D

T

dDd

T

d

T

d

T

d

T

d

T

d

T

d

vnVI

nVvIi

nVv

nVv

nVv

nVv

nVv

nVv

=

−

+×≈

+≈

<<

+

+

+=

11

1exp :1 If

.... !2

11exp :Exapnsion SeriesTaylor 2

dddD

Td iri

InVv ==

Formal derivation of small signal model


2)2(

)1(

!2)()( a

AaA vVfvVf ⋅>>⋅

)()(2 )2(

)1(

A

Aa Vf

Vfv ⋅<<

Small signal means:

aAaa vVfvgi ⋅== )()( )1(

Signal + Bias for element A (iA, vA) : iA = f (vA) Bias for element A (IA, VA) : IA = f (VA) Signal for element A (ia, va) : ia = g (va)

( ) ( )

aAA

aA

aAA

AAA

AAAA

AA

vVfVf

vVfvVfVf

VvVfVvVfVf

vfi

⋅+≈

+⋅+⋅+=

+−⋅+−⋅+=

=

)()(

...!2

)()()(

...!2

)()()(

)(

)1(

2)2(

)1(

2)2(

)1( (Taylor Series Expansion)

aAAAaA vVfIIii ⋅+=+= )()1(

Derivation of diode small signal model


)(1 DnVv

SD vfeIi T

D

=

−⋅= TT nV

v

ST

nVv

S eInV

vfeIvf 1)( 1)( )1( ×=

−=

1)( SnVV

SnVV

SDD IeIeIVfI T

D

T

D

−=

−⋅==

dT

SDd

T

nVV

Sd

VvT

nVv

SdDd v

nVIIv

nVeIv

nVeIvVfi

T

D

D

T

×

+=×

⋅

=×

⋅

=×=

=

)()1(

dT

Dd

T

SDd v

nVIv

nVIIi ×

≈×

+=

d

dd r

vi =D

Td I

nVr ≈ vd

id rd = nVT/ID

vD

iD

Diode can be replaced with a resistor in the signal circuit!

Small signal model vs iv characteristics


Small signal model is equivalent to approximating the non-liner iv characteristics curve by a line tangent to the iv curve at the bias point

D

T

Dd

dDd

InV

Vfr

vVfi

≈=

×=

)(1

)(

)1(

)1(

Derivation of MOS small signal model (1)


Signal + Bias for MOS (iD, vGS , vDS) : iD = f (vGS, vDS), iG = 0 Bias for MOS (ID, VGS , VDS) : ID = f (VGS, VDS), IG = 0 Signal for MOS (id, vgs , vds) : id = g (vgs , vds), ig = 0

MOS iv equations: iD = f (vGS, vDS) iG = 0

dsVVDS

gsVVGS

D

DSDSVVDS

GSGSVVGS

DSGS

DSGSDdD

vvfv

vfI

VvvfVv

vfVVf

vvfiiI

DSGSDSGS

DSGSDSGS

×∂∂

+×∂∂

+≈

+−⋅∂∂

+−⋅∂∂

+=

==+

,,

,,

...)()(),(

),(

,,

dsVVDS

gsVVGS

d vvfv

vfi

DSGSDSGS

×∂∂

+×∂∂

≈

(Taylor Series Expansion in 2 variables)

Derivation of MOS small signal model (2)


dsVVDS

gsVVGS

d vvfv

vfi

DSGSDSGS

⋅∂∂

+⋅∂∂

=,,

),()1()( 5.0 2DSGSDStGSoxnD vvfvVv

LWCi =+−= λµ

mOV

D

tGS

DStGSoxn

VVDStGSoxnVVGS

gV

IVV

VVVL

WC

vVvL

WCvf

DSGS

DSGS

≡=−

+−×=

+−×=∂∂

2)(

)1()( 5.02

)1)(( 5.02

2

,,

λµ

λµ

oD

DS

D

DS

DStGSoxn

VVtGSoxn

VVDS

rI

VI

V

VVVL

WC

VvL

WCvf

DSGSDSGS

1)1()1(

)1()( 5.0

)( 5.0

2

,

2

,

≡≈+

=+

+−×=

−×=∂∂

λλ

λλ

λµλ

µλ

0 =+⋅= go

dsgsmd i

rvvgi

MOS small signal “circuit” model


and 0 o

dsgsmdg r

vvgii +⋅==

Do I

r⋅

≈λ

1

OV

Dm V

Ig ⋅=

2 122>>==

OV

A

OVom V

VV

rgλ

Statement of KCL Two elements in parallel Input open circuit

PMOS “circuit” small signal model is identical to NMOS


=

PMOS small-signal circuit model is identical to NMOS o We will use NMOS circuit model for both! o For both NMOS and PMOS, while iD ≥ 0 and ID ≥ 0, signal quantities: id,

vgs, and vds , can be negative!

PMOS* NMOS

Exercise: Derive PMOS small signal model (follow derivation of NMOS small-signal model)

Derivation of BJT small signal model (1)


Signal + Bias for BJT (iB, iC, vBE , vCE) : iB = f1 (vBE), iC = f2 (vBE, vCE)

Bias for BJT (IB, IC, VBE , VCE) : IB = f1 (VBE), IC = f2 (VBE, VCE)

Signal for BJT (ib, ic, vbe , vce) : ib = g1 (vbe), ic = g2 (vbe, vce)

BJT iv equations: iB = f1 (vBE) iC = f2 (vBE, vCE)

+=

=

A

CEVv

sC

Vv

sB

VveIi

eIi

T

BE

T

BE

1

)/( β

We need to perform Taylor Series Expansion in 2 variables for both iB and iC.

,

1be

VVBEB v

dvdfi

CEBE

×≈ ,

2

,

2ce

VVDCEbe

VVBEC v

vfv

vfi

CEBECEBE

×∂

∂+×

∂∂

≈

Derivation of BJT small signal model (2)


o

cebemc

beb r

vvgirvi +==

π

)( )/( 1 BEVv

sB vfeIi T

BE

== β

bebeVVBE

B vr

vdvdfi

CEBE

×=×≈π

1 ,

1

π

βrV

IeIVdv

df

T

B

V

Vv

sTVVBE

BE

T

BE

CEBE

1 )/(1

,

1 ≡==

T

BEVV

sB eII )/( β=

+=

A

CEVv

sC VveIi T

BE

1

A

CEAVV

sA

CEVV

sC VVVeI

VVeII T

BE

T

BE +×=

+= 1

mT

C

VA

CEVv

T

s

VVBE

gVI

Vve

VI

dvdf

CEVBE

T

BE

CEBE

≡=

+= 1

,,

2

oCEA

C

V

Vv

A

s

VVCE rVVIe

VI

dvdf

CEVBE

T

BE

CEBE

1

,

,

2 ≡+

==

BJT small signal “circuit” model


C

A

C

CEAo I

VI

VVr ≈+

=

T

Cm V

Ig =

Statement of KCL Two elements in parallel A resistor, rπ ,

between B & E

o

cebemc

beb r

vvgirvi +==

π

B

T

IVr =π

We follow S&S: vbe is denoted as vπ

Similar to NMOS/PMOS, the small circuit model for a PNP BJT is the same as that of a NPN.

Alternative BJT small signal “circuit” model


πππ

π

π

ββ

β

ivr

vg

rVI

II

VIg

m

T

B

B

C

T

Cm

=×=

=×==

gm Model

β ib Model

Summary of transistor small signal models


Do

OV

Dm I

rV

Ig⋅

≈⋅

=λ

1 2

Comparison of MOS and BJT small-signal circuit models: 1. MOS has an infinite resistor in the input (vgs) while BJT has a finite resistor, rπ

(typically several kΩ). 2. BJT gm is substantially larger than that of a MOS (BJT has a much higher gain). 3. ro values are typically similar (10s of kΩ). gm ro >> 1 for both.

NMOS/PMOS

C

Ao

T

Cm

B

T

IVr

rVIg

IVr ≈===

ππ

β

NPN/PNP BJT

Documents

Signal Circuit and Transistor Small-Signal Modelaries.ucsd.edu/.../ECE65/12-W/Slides/ECE65_W12-SSM.pdf · Signal Circuit and Transistor Small-Signal Model Lecture notes: Sec. 5