M.H. Perrott

Analysis and Design of Analog Integrated CircuitsLecture 3

Large Signal Modeling of CMOS Transistors

Michael H. PerrottJanuary 29, 2012

Copyright © 2012 by Michael H. PerrottAll rights reserved.

M.H. Perrott

Introducing CMOS Devices

CMOS: Complementary Metal Oxide Semiconductor- Current flow through channel between Drain and Source

is controlled by Gate- Complementary: both PMOS and NMOS are available

2

(Gate)

(Drain)

(Source)

(Bulk)

D

G

S

B (Gate)

(Source)

(Drain)

(Bulk)

S

G

D

B

(Source) (Drain)(Gate)

(Bulk)

n+ n+p- p+ p+

n-

p-

NMOS PMOS

(Source) (Drain)(Gate)

(Bulk)

p+

(Bulk)

n+

(Bulk)

M.H. Perrott

Simplified MOS Symbol for Typical Bulk Connections

Bulk silicon below the channel under the gate also has an impact on the channel current- We often tie the Bulk to Gnd/Vdd for NMOS/PMOS devices

In such case, the symbol does not include the bulk terminal3

(Gate)

(Drain)

(Source)

D

G

S

(Gate)

(Source)

(Drain)

S

G

D

(Source) (Drain)(Gate)

(Bulk)

n+ n+p- p+ p+

n-

p-

NMOS PMOS

(Source) (Drain)(Gate)

(Bulk)

p+

(Bulk)

n+

(Bulk)VddGnd

M.H. Perrott

Symbol Notation Often Includes Size

4

M1WL L

W

The designer is generally free to choose the width (W) and length (L) of the device- Wider width is often chosen to achieve higher channel

current for a given gate bias voltage- Longer length is often avoided since it lowers the channel

current and decreases the operating speed of the device The minimum length for the gate is often used to define the

process name (i.e., 0.18u CMOS or 0.13u CMOS) Longer length is used in cases where better matching or

high resistance is desired

M.H. Perrott

Channel Current as a Function of Gate Voltage

If Vgs < VTH, then current density Id/W is small- The device is in the subthreshold operating region

For Vgs > VTH, then Id/W is much larger- The device is in strong inversion- If Vds > V, then Id is relatively independent of Vds

The device is in the saturation operating region- If Vds < V, then Id is strongly dependent on Vds

The device is in the triode operating region5

Id

Vgs

Id_op

ΔV

VTH Vgs_op

Vds > ΔV

M1

Id

Vgs

NMOS

g

s

d Note that we designateV as the overdrive voltage

and that V = Vdsatin strong inversion

M.H. Perrott

PMOS Devices are Complementary to NMOS Devices

Same observations and definitions apply to PMOS- However, voltage and current signs are flipped

Note that Vsg = -Vgs, Vsd = -Vds

Note that Id as defined above for PMOS is in the opposite direction as for NMOS

Note that VTH becomes negative

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M2

Vsg

Id

Id

Vsg

Id_op

ΔV

-VTH Vsg_opPMOS

gd

sVsd > ΔV

M.H. Perrott

Examine MOS Behavior As Vds is Increased

S D

G

Cchannel = Cox(VGS-VTH)

VGS

VDS=0

S D

GVGS

VD=ΔV

S D

GVGS

VD>ΔV

Triode

Pinch-off

SaturationVDS

ID

ID

ID

ID

Triode

Pinch-offSaturation

ΔV

Overall I-V Characteristic

How does VGS influence Id in the above curve ? 7

M.H. Perrott

MOS Behavior Is A Function of Vgs and Vds

VDS

ID

Triode

Pinch-offSaturation

ΔV

Overall I-V Characteristic

Increasing Vgs

See page 15-23 of Razavi…

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M.H. Perrott

MOS Current Equations in Triode and Saturation Regions

S D

G

Cchannel = Cox(VGS-VTH)

VGS

VDS=0

S D

GVGS

VD=ΔV

S D

GVGS

VD>ΔV

Triode

Pinch-off

Saturation

ID

ID

ID

ID = μnCoxWL

(VGS - VTH - VDS/2)VDS

ID μnCoxWL

(VGS - VTH)VDS

for VDS << VGS - VTH

ID = μnCoxWL

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(VGS-VTH)2(1+λVDS)

(where λ corresponds tochannel length modulation)

ΔV = VGS-VTH

ΔV =μnCoxW

2IDL

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M.H. Perrott

The Issue of Velocity Saturation

When in saturation, the MOS current is calculated as

Which is really

- Here Vdsat,l is the saturation voltage at a given length It may be shown that

- If Vgs-VTH approaches LEsat in value, then We say that the device is in velocity saturation The current becomes linearly related to Vgs-VTH

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M.H. Perrott

Example: Current Versus Voltage for 0.18 Device

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

Vgs

(Volts)

I d (m

illiA

mps

)

Id versus V

gsM1

IdVgs

WL = 1.8μ

0.18μ

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M.H. Perrott

The Tricky Issue of Modeling MOS Devices

The device characteristics of modern CMOS devices lead to complicated analytical models- This creates challenges for achieving accurate hand

calculations with reasonable effort Hand calculations are essential in achieving deeper

understanding and intuition of circuit and device behavior- Simple hand calculations lack accuracy- Detailed hand calculations often do not yield the desired

insight and understanding to make them worthwhile A typical compromise

- Assume simple models for hand calculations- Use SPICE to get a more accurate picture of the actual

circuit and device characteristics and performance

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M.H. Perrott

What is the Key Role of Large Signal Calculations?

In analog circuits, we are often focused on amplifiers in which the small signal behavior is of high importance- Large signal calculations lead to the operating point

information of the circuit which is used to determine the small signal model of the device

Example amplifier circuit:

RS

RG

RD

vinvout

Vbias

ID 1) Solve for bias current Id2) Calculate small signal parameters (such as gm, ro)3) Solve for small signal response using transistor hybrid-π small signal model

Small Signal Analysis Steps

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M.H. Perrott

A Key Small Signal Parameter: Transconductance

Transconductance from input gate voltage, Vgs, to channel current, Id, is very important for amplifier circuits- Assuming device is in saturation:

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Id

Vgs

Id_op

ΔV

VTH Vgs_op

Vds > ΔV

M1

Id

Vgs

NMOS

g

s

d

gm = ΔVgs

ΔId

Vgs_op

M.H. Perrott

A Key Small-Signal Nonideality: Output Resistance

Ideally, Id would not change with Vds when the device is in saturation- Practical CMOS transistors exhibit Id dependence on Vds

due to channel length modulation- The parameter is often used to characterize this effect

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Vds

ID

Triode

Pinch-offSaturation

ΔV

gds =ΔVds

ΔId

Vds_op

Vds_op

M.H. Perrott

Another Non-Ideality: Back-Gate Effect

The threshold voltage of the device, VTH, is dependent on the potential between the source and bulk

- This implies that changes in the source node voltage, Vs, lead to changes in the channel current, Id

We model this effect as backgate transconductance, gmb

- MIC503 will provide details (also see pages 34-36 of Razavi) 16

(Source) (Drain)(Gate)

(Bulk)

n+ n+p- p+ p+

n-

p-

NMOS PMOS

(Source) (Drain)(Gate)

(Bulk)

p+

(Bulk)

n+

(Bulk)VddGnd

M.H. Perrott

MOS DC Small Signal Model

Assuming transistor is in saturation:- Note that designers often determine gmb impact from SPICE

RS

RG

RDRD

RS

RG

-gmbvsvgs

vs

rogmvgs

gm = μnCox(W/L)(VGS - VTH)(1 + λVDS)

= 2μnCox(W/L)ID (assuming λVDS << 1)

Cox

2qεsNA

2 2|ΦF| + VSB

γgm where γ =gmb =

In practice: gmb = gm/5 to gm/3

λID1ro =

ID

See Chapter 2 of Razavifor more discussion of

these formulas

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M.H. Perrott

MOS DC Small Signal Model

Assuming transistor is in triode region:- The channel of the device can be approximated as a

resistor whose value depends on the DC operating point of Vgs

RS

RG

RDRD

RS

RG

vgs

vs

rds

μnCox(W/L)(VGS - VTH)

ID

rds = 1

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M.H. Perrott

Example: Determine V and Operating Region (NMOS)

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1V0.2V

0.2V1V

1V0.7V

1V0.2V

0.4V

1V1V

0.7V0.2V

0.4V0.4V

ΔV =

Region =

ΔV =

Region =

ΔV =

Region =

ΔV =

Region =

ΔV =

Region =

ΔV =

Region =

Assume VTHn = 0.5V

M.H. Perrott

Example: Determine V and Operating Region (PMOS)

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1.3V

0.5V

0.7V

1.3V

0.9V

0.7V

1.3V

0.5V

1.1V

1.2V

0.5V

1.3V

0.8V

0.9V

0.7V

1.3V

1.1V

ΔV =

Region =

ΔV =

Region =

ΔV =

Region =

ΔV =

Region =

ΔV =

Region =

ΔV =

Region =

Assume VTHp = -0.5V

M.H. Perrott

Determine operating region for M1 and M2 assuming:- Vbias = 1.2- Vbias = 0.2- Vbias = 0.65

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Example: Determine Operating Region of M1 and M2

M2

M1

Vbias

vout

1.3V

Assume VTHn = 0.5V, VTHp = -0.5V, nCox = 50A/V2, pCox = 20A/V2, = 0, and M1 and M2 have the same value of W and L

M.H. Perrott

Determine Vbiasn such that Vout = 0.5V- Note that with = 0, a variety of Vout solutions will exist for

the same Vbiasn – I’m just trying to keep calculations simple Determine the resulting operating region of M1 and M2

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Example: Determine V and Operating Region

M2

M1Vbiasn

vout

1.3V

Vbiasp

1.3u0.13u

3.25u0.13u

Assume Vbiasp = 0.7V, VTHn = 0.5V, VTHp = -0.5V, nCox = 50A/V2, pCox = 20A/V2, = 0