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EE 105 Fall 1998

Lecture 11

(Saturated) MOSFET Small-Signal Model

Concept: find an equivalent circuit which interrelates the incrementalchanges

in iD, vGS, vDS, etc. for the MOSFET in saturation

vGS= VGS+ vgs, iD=ID+ id-- we want to find id= (?) vgs

We have the functional dependence of the total drain current in saturation:

iD= nCox(W/2L) (vGS- VTn)2(1 + nvDS) = iD(vGS, vDS)

Solution: do a Taylor expansion around the DC operating point (also called the

quiescent point or Qpoint) defined by the DC voltages Q(VGS, VDS):

If the small-signal voltage is really small, then we can neglect all everything past

the linear term --

where the partial derivative is defined as the transconductance, gm.

iD

ID v

GSiD

Q

vgs( ) 12

---

vGS

2

2

iD

Q

vgs( )

2 + + +=

iD

ID v

GS

iD

Q

vgs

( )+ ID

gm

vgs

+= =

11

EE 105 Fall 1998

Lecture 11

Transconductance

The small-signal drain current due to vgsis therefore given by

id= gmvgs.

D

S

G

+

_

B VDS = 4 V+

_

1 2 3 4 5

100

200

300

400

500

600

iD

(A)

VDS

(V)6

vGS= VGS= 3 V

VGS= 3 V

+

_vgs

iD =ID+ id

vGS= VGS+ vgs

Q

id

gm= id/ vgs

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EE 105 Fall 1998

Lecture 11

Another View of gm

* Plot the drain current as a function of the gate-source voltage, so that

the slope can be identified with the transconductance:

D

S

G

+

_

B VDS = 4 V+

_

1 2 3 4 5

100

200

300

400

500

600

iD

(A)

vGS (V)6

vGS= VGS= 3 V

VGS= 3 V

+

_vgs

iD =ID+ id

vGS= VGS+ vgs

Q

idgm= id/ vgs

iD(vGS, VDS= 4 V)

EE 105 Fall 1998

Lecture 11

Transconductance (cont.)

Evaluating the partial derivative:

In order to find a simple expression that highlights the dependence of gmon the

DC drain current, we neglect the (usually) small error in writing:

For typical values (W/L) = 10,ID= 100 A, and nCox= 50 AV-2

we find that

gm= 320 AV-1= 0.32 mS

gm

nC

ox

W

L-----

VGS

VTn

( ) 1 nV

DS+( )=

gm

2nC

ox

W

L-----

ID

2ID

VGS VTn--------------------------= =

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EE 105 Fall 1998

Lecture 11

(Partial) Small-Signal Circuit Model

How do we make a circuit which expresses id= gmvgs? Since the current is not

across its controlling voltage, we need a voltage-controlled current source:

gmvgs

gate

source

drain+

_

vgs

_

id

EE 105 Fall 1998

Lecture 11

Output Conductance/Resistance

We can also find the change in drain current due to an increment in the drain-

source voltage:

The output resistance is the inverse of the output conductance

The (partial) small-signal circuit model with roadded looks like:

go

iD

vDS

------------

Q

nC

ox

W

2L------

VGS

VTn

( )2n

nID

= =

ro

1

go

-----

1

nID

------------= =

gmvgs ro

gate

source

drain

+

_

vgs

id+

_

vds

id= gmvgs+ (1/ro)vds

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EE 105 Fall 1998

Lecture 11

MOSFET Capacitances in Saturation

In saturation, the gate-source capacitance contains two terms, one due to the

channel charges dependence on vGS[(2/3)WLCox] and one due to the overlap of

gate and source (WCov, where Covis the overlap capacitancein fF per m of gatewidth)

In addition, there is the small but very important gate-drain capacitance (just the

overlap capacitance Cgd= Cov)

There are depletion capacitances between the drain and bulk (Cdb

) and between

source and bulk (Csb). Finally, the extension of the gate over the field oxide leads to

a small gate-bulk capacitance Cgb.

, ,

,

,

,

,

,

,

,

,

,

gatedrainsource

n+ n+qN(vGS)

overlap LD overlap LD

fringe electric

field lines

Csb Cdbdepletion

region

Cgs2

3---WLCox WCov+=

EE 105 Fall 1998

Lecture 11

Complete Small-Signal Model

All these capacitances are patched onto the small-signal circuit schematic

containing gmand ro... gmbis open-circuited for EECS 105 since vbs= 0 V.

gmvgs gmbvbs ro

gate

source__

vgs Cgs Cgb

Cgd

CsbCdb

vbs

drain

id

+

+

bulk

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EE 105 Fall 1998

Lecture 11

p-channel MOSFETs

Structure is complementaryto the n-channel MOSFET

In a CMOS technology, one or the other type of MOSFET is built into a well-- a

deep diffused region -- so that there are electrically isolated bulk regions in the

same substrate

n+source p+sourcen+drain p+drainp+ n+ ,

p-type substrate

isolated bulk contact with

p-channel MOSFET

shorted to source

common bulk contact for

all n-channel MOSFETs

(to ground or to the supply)

n well

,

,

,

,

,

,

,

,

,,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

n-channel

MOSFETp-channelMOSFET

(a)

(b)

,

,

,

,

,

,

,

,

,

, ,

,

,

,

,

,

,

,

A A,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

EE 105 Fall 1998

Lecture 11

p-channel MOSFET Models

DC drain current in the three operating regions: -ID> 0

The threshold voltage with backgate effect is given by:

Numerical values:

pCoxis a measured parameter. Typical value: pCox= 25 AV-2

VTp = -0.7 to -1.0 V, which should be approximately -VTnfor a well-controlled

CMOS process

I D 0 A V( SG V T)=

I D pCox W L( ) VSG VTp VSD 2( )+[ ] 1 pVSD+( )VSD= V( SG V Tp VSD VSG VTp )+,

I D pCox W 2L( )( ) VSG VTp+( )2

1 pVSD+( )= V( SG V Tp VSD VSG VTp )+,

VTp

VTOp

p

VSB

2n

+( ) 2n

( )=

p

0.1mV1

L--------------------------

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EE 105 Fall 1998

Lecture 11

p-channel MOSFET small-signal model

the source is the highest potential and is located at the top of the schematic

gmvsg gmbvsb ro

gate

drain

bulk

+

_

vsg Cgs

CsbCdb

Cgd

Cgb

_

source

id

vsb