EE 105 Fall 1998Lecture 11

(Saturated) MOSFET Small-Signal Model

■ Concept: find an equivalent circuit which interrelates the incremental changes 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 = µn Cox (W/2L) (vGS - VTn )2 (1 + λnvDS) = iD(vGS, vDS)

Solution: do a Taylor expansion around the DC operating point (also called the quiescent point or Q point) 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 vGS∂∂iD

Q

vgs( ) 12---

vGS2

2

∂

∂ iD

Q

vgs( )2 …+ + +=

iD ID vGS∂∂iD

Q

vgs( )+ ID gmvgs+= =

11

EE 105 Fall 1998Lecture 11

Transconductance

The small-signal drain current due to vgs is therefore given by

id = gm vgs.

D

S

G

+

_

B VDS = 4 V+

_

1 2 3 4 5

100

200

300

400

500

600

iD

(µA)

VDS (V)6

vGS = VGS = 3 V

VGS = 3 V

+

_vgs

iD = ID + id

vGS = VGS + vgs

Q

id

gm = id / vgs

EE 105 Fall 1998Lecture 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

(µA)

vGS (V)6

vGS = VGS = 3 V

VGS = 3 V

+

_vgs

iD = ID + id

vGS = VGS + vgs

Q

id gm = id / vgs

iD(vGS, VDS = 4 V)

EE 105 Fall 1998Lecture 11

Transconductance (cont.)

■ Evaluating the partial derivative:

■ In order to find a simple expression that highlights the dependence of gm on the DC drain current, we neglect the (usually) small error in writing:

For typical values (W/L) = 10, ID = 100 µA, and µnCox = 50 µAV-2 we find that

gm = 320 µAV-1 = 0.32 mS

gm µnCoxWL-----

VGS VTn–( ) 1 λnVDS+( )=

gm 2µnCoxWL-----

ID

2ID

VGS VTn–--------------------------= =

EE 105 Fall 1998Lecture 11

(Partial) Small-Signal Circuit Model

■ How do we make a circuit which expresses id = gm vgs ? 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 1998Lecture 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 ro added looks like:

go

iD∂vDS∂------------

Q

µnCoxW2L------

VGS VTn–( )2λn λnID≅= =

ro1go----- 1

λnID------------= =

gmvgs ro

gate

source

drain

+

_

vgs

id+

_

vds

id = gm vgs + (1/ro)vds

EE 105 Fall 1998Lecture 11

MOSFET Capacitances in Saturation

In saturation, the gate-source capacitance contains two terms, one due to the channel charge’s dependence on vGS [(2/3)WLCox] and one due to the overlap of gate and source (WCov, where Cov is the overlap capacitance in fF per µm of gate width)

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.

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� gatedrainsource

n+ n+qN(vGS)

overlap LD overlap LD

fringe electric field lines

Csb Cdbdepletionregion

Cgs23---WLCox WCov+=

EE 105 Fall 1998Lecture 11

Complete Small-Signal Model

■ All these capacitances are “patched” onto the small-signal circuit schematic containing gm and ro ... gmb is open-circuited for EECS 105 since vbs = 0 V.

gmvgs gmbvbs ro

gate

source__

vgs Cgs Cgb

Cgd

CsbCdb

vbs

drain

id

+

+

bulk

EE 105 Fall 1998Lecture 11

p-channel MOSFETs

■ Structure is complementary to 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 withp-channel MOSFETshorted to source

common bulk contact forall n-channel MOSFETs(to ground or to the − supply)

n well

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n-channel MOSFET

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(b)

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EE 105 Fall 1998Lecture 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:

µpCox is a measured parameter. Typical value: µpCox = 25 µAV-2

VTp = -0.7 to -1.0 V, which should be approximately -VTn for a well-controlled CMOS process

I– D 0 A V( SG V– T )≤=

I– D µpCox W L⁄( ) VSG VTp VSD 2⁄( )–+[ ] 1 λpVSD+( )VSD= V( SG V– Tp VSD VSG VTp )+≤,≥

I– D µpCox W 2L( )⁄( ) VSG VTp+( )2 1 λpVSD+( )= V( SG V– Tp VSD VSG VTp )+≥,≥

VTp VTOp γp V– SB 2φn+( ) 2φn–( )–=

λp0.1µmV

1–

L--------------------------≈

EE 105 Fall 1998Lecture 11

p-channel MOSFET small-signal model

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

gmvsg gmbvsbro

gate

drain

bulk

+

_

vsgCgs

CsbCdb

Cgd

Cgb

_

source

−id

vsb