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4. Active Loads and IC MOS Amplifiers ECE 102, Winter 2011, F. Najmabadi Reading: Sedra & Smith: Chapter 7 (MOS portions)

MOSFET: Transfer Function & Biasing

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Page 1: MOSFET: Transfer Function & Biasing

4. Active Loads and IC MOS Amplifiers

ECE 102, Winter 2011, F. Najmabadi

Reading: Sedra & Smith: Chapter 7 (MOS portions)

Page 2: MOSFET: Transfer Function & Biasing

Progress towards an IC relevant amplifier

Resistors and capacitors take a lot of space on ICs:o Minimize (i.e., very few) R & C and small sizes (e.g., nF or smaller)

o Get rid of coupling capacitors by direct coupling between stages (makes biasing design complicated)

o Replace Rs with a current sourceo Still need to get rid of Cs!o What to do about RD?

Page 3: MOSFET: Transfer Function & Biasing

Current mirrors are the principle method for biasing in ICs

Identical MOS: Same k’n and Vt

Q1 is always in saturationVDS1 = VGS > VGS – Vt

Q2 has to be in saturation for current mirror to workVDS2 > VGS – Vt

refo ILWLWI

1

2

)/()/(

=

Page 4: MOSFET: Transfer Function & Biasing

Small signal model of Current Mirrors

Small signal response of an ideal current source is an open circuit!

However, Current mirrors are NOT ideal current sources as we used λvDS << 1 in our “bias” analysis

Bias Model Small Signal Model

Real Circuit

refo ILWLWI

1

2

)/()/(

=

Page 5: MOSFET: Transfer Function & Biasing

Small signal model of Current Mirrors

0

in flows :D1at KCL

111

11

=⇒−== gsogsmgsD

ogsm

vrvgvvrvg

Real Circuit Small Signal Model

2orR =circuitopen is sourcecurrent 0 2 gsmgs vgv ⇒=

Current sourcebecomes open circuit

Page 6: MOSFET: Transfer Function & Biasing

But what happens if we replace Irefcurrent source with “practical” elements?

Not a Practical Circuit Practical Circuit

Would practical elements that fix Iref change the small signal response?

Page 7: MOSFET: Transfer Function & Biasing

Small signal response of a practical current mirror

Practical Circuit

0 )||(

||in flows :D1at KCL

111

11

=⇒−== gsDogsmgsD

Dogsm

vRrvgvvRrvg

2orR =circuitopen is sourcecurrent 0 2 gsmgs vgv ⇒=

Page 8: MOSFET: Transfer Function & Biasing

Generalized Small signal Model ofCurrent Mirrors

Any circuit that “fixes Iref”

V ..

)1()( )/('21

saturationin Q1

.

G1

12

111

11

1

→⇒=⇒==⇒

+−=

⇒=

==

GGGSGS

DStnGSnD

GSDS

refD

vconstvconstvv

vVvLWki

vvconstii

λ

Small Signal Model

Page 9: MOSFET: Transfer Function & Biasing

Summary of Current Mirrors

Bias Model Small Signal Model

refo ILWLWI

1

2

)/()/(

=

Bias current goes through this leg

Signal current goes through this leg(∞ capacitor)

It is sufficient to consideronly Q2 in circuit calculations

Page 10: MOSFET: Transfer Function & Biasing

PMOS Version:

Summary of Current Mirrors

NMOS Version:

“Intuitive” Model

It is sufficient to consideronly Q2 in circuit calculations

Page 11: MOSFET: Transfer Function & Biasing

Biasing a CS Stage: Can we place a current mirror in the source circuit?

Typical bias of a discrete CS amplifier

Placing a current mirror in the source circuit will not Work!

Current mirror Bias works fine!

o A large R in the source circuit for small signal reducing the gain by (1+ gm1 rO2)

Bias Small Signal

Page 12: MOSFET: Transfer Function & Biasing

We need to Bias a CS Stage by placing a current mirror in the drain circuit!

Signal

Current mirror provides RD !

Current mirror sets ID = Io

However, a precise bias voltage should be applied to the Gate (corresponding to the ID set by the current source)

Several ways to do this

Bias

Page 13: MOSFET: Transfer Function & Biasing

Basic gain cell in IC

NMOS Version:

PMOS Version:)||( 211 oomv rrgA −=

Page 14: MOSFET: Transfer Function & Biasing

Bias point of CS amp with current mirror

||22

2

tpSGOV

GDDSG

VVVVVV−=

−=

222 2 )/('

21

OVVLWkI pD =

222

111

111

//

/2

DAo

DAo

OVDm

IVrIVr

VIg

===

2/1

1

11

21

)/('2

=

=

LWkI V

II

n

DOV

DD

Ignore Channel Width Modulation,o Fast and relatively accurate

method to find gm1 , ro1 and ro2o Cannot find VDS1 and VDS2

Page 15: MOSFET: Transfer Function & Biasing

Bias point of CS amp with current mirror

Include Channel Width Modulation,o Lengthy Analysiso Gives VDS1 and VDS2o See S&S Example 7.2 (pp500-504)o Can gain insight with load-line analysis

iD2

vDS2

Page 16: MOSFET: Transfer Function & Biasing

Bias point of CS amp with current mirror

12 DSSDDD vvV +=

Setting Vss = 0 (For simplicity), the load line (or load curve!) equation for Q1 is (note iD1 = iD2)

Page 17: MOSFET: Transfer Function & Biasing

Biasing CS amp with current mirror allows a very large RD without increasing VDD

Q2 in triodeQ2 in saturation

Load line for a discrete resistor of value ro2

Needed VDD

Page 18: MOSFET: Transfer Function & Biasing

Biasing a Source Follower in ICs

Current mirror Bias works fine! Small signal OK

Common-Drain (Source Follower) stages are biased with current mirror in the source circuit (as above)

Common-Gate stages are biased with current mirror in the drain circuit similar to CS amplifier.

Page 19: MOSFET: Transfer Function & Biasing

PMOS version of Basic gain cell in IC

NMOS CS Amp PMOS CS Amp NMOS CD Amp PMOS CD Amp

Page 20: MOSFET: Transfer Function & Biasing

Implementation of CS and Follower configurations on IC

Page 21: MOSFET: Transfer Function & Biasing

Cascode Amplifiers and Current Mirrors

Page 22: MOSFET: Transfer Function & Biasing

Cascode amplifier is a two-stage, CS-CG configuration

Cascode Configuration

CG stage

CS stage

signalbias

Page 23: MOSFET: Transfer Function & Biasing

Cascode amplifier is a two-stage, CS-CG configuration

Cascode Configuration

CG stage

CS stage

Small Signal Model

Small Signal Model configured as two-stage amplifier

Page 24: MOSFET: Transfer Function & Biasing

Open-Loop gain of a Cascode amplifier

122 )1( vrgv omo ⋅+=⇒

22112211 )1( omomomomi

ovo rgrgrgrg

vvA ⋅⋅⋅−≈⋅+⋅⋅−==

Node Voltage Method:

0122

1 =⋅−− vg

rvv

mo

oNode vo:

0011

1 =+⋅+ imo

vgrv

Node v1: iom vrgv 111 ⋅−=⇒

By KCL around Q2

Open Loop Gain (RL →∞, io = 0):

Page 25: MOSFET: Transfer Function & Biasing

Output Resistance of a Cascode amplifier

1122 )1( oomoo rrgrR +⋅+=

RRgr mo ++ )1(

21221 oomooo rrgrrR ++=

Exercise: Compute Ro from the small signal circuit of the previous slide(Attach a voltage source vx to the output and compute ix, see S&S pp 509-510)

R

Page 26: MOSFET: Transfer Function & Biasing

Amplifier models*Voltage Amplifier

Current Amplifier

Transconductance Amplifier

0for == oivoo ivAv

Avo : Open-Loop voltage gain

0for == oimo vvGi

Gm : Short-circuit transconductance

0for == oiiso viAi

Ais : Short-circuit current gain

Ideal Amplifier

Ri →∞Ro = 0

Ri →∞Ro →∞

Ri = 0 Ro →∞

* See S&S pp 26-27

Page 27: MOSFET: Transfer Function & Biasing

Amplifier modelsVoltage Amplifier

Current Amplifier

Transconductance Amplifier

isiovo

imis

omvo

ARRARGARGA

)/(===

From the point of view of circuit analysis: amplifier models are identical (The output stages are

Thevenin/Norton Equivalent)

omvo

oimivoo

RGARvGvAvi

====

0For 0

Avo , Ais , and Gm are related to each other

Page 28: MOSFET: Transfer Function & Biasing

Amplifier models*Voltage Amplifier

Transconductance Amplifier

0for == oivoo ivAv

0for == oimo vvGi

Alternate method to compute Avo:1. Set RL = 0 (short output)2. Compute Gm = io/vi3. Avo = Gm Ro

omvo RGA =

Page 29: MOSFET: Transfer Function & Biasing

Example: CS amplifier gain from Gm

gsmo vgi ⋅−=

oo rR =

omomvo rgRGA −==

gsmimo vGvGi ==

gsmgsm vgvG −=

mm gG −=

CS Amplifier

Alternate method to compute Avo:1. Set RL = 0 (short output)2. Compute Gm = io/vi3. Avo = Gm Ro

Short circuit

Transconductance Amplifier

Gm = − gm because current source is flipped

0

Page 30: MOSFET: Transfer Function & Biasing

Cascode amplifier gain from Gm

1

11

1

111

oim

ogsmo r

vvgrvvgi −⋅−=−⋅−=

21221

2211 ) 1(

oomoo

omom

i

om rrgrr

rgrgviG

+++

−==

21221 oomooo rrgrrR ++=22112211 )1( omomomomomvo rgrgrgrgRGA −≈+−==

121 and vvvv gsigs −==

Node equation at v1

imoooo

moo

immoo

ogsmgsm

o

vgrrrr

grrv

vgvgrr

rvvgvg

rv

22121

1211

11221

2

12211

1

1

11

0

++−=

−=

++

=+−+

KCL at D1

Page 31: MOSFET: Transfer Function & Biasing

Insight into operation of Cascode amplifier

For simplicity assume ro1 = ro2 = ro and gm1 = gm2 = gm

CG section has kept Gm the same (i.e., since Gm = io/vi , same current is passed through) but has increased Ro substantially resulting in a large increase in overall open-loop voltage gain!

CG section acted as a current buffer!

Cascode amplifier needs a large load!

mom

omom

oomoo

omomm g

rgrgrg

rrgrrrgrgG −=−≈

+++

−= 221221

2211 )() 1.(

221221 )2( omomooomooo rgrgrrrgrrR ≈+=++=Cascode amplifier:

oomm rRgG =−= and 1CS amplifier (1st

stage of cascode):

Page 32: MOSFET: Transfer Function & Biasing

Distribution of the gain in a Cascode Amp.

)||( 222 Lomv RrgA =

22222

221 1

1 om

L

mom

LoiL rg

Rgrg

RrRR +≈+

+==

)||( 2111 iomv RrgA −=

21 vvv AAA =

CG stages “reduces” the load seen by the CS stage by gm2ro2

CG StageCS Stage

Page 33: MOSFET: Transfer Function & Biasing

Cascode Amplifier needs a large load

)||( 222 Lomv RrgA =22222

221 1

1 om

L

mom

LoiL rg

Rgrg

RrRR +≈+

+==)||( 2111 iomv RrgA −=

For simplicity assume ro1 = ro2 = ro and gm1 = gm2 = gm

∞ ∞ − gmro gmro − (gmro)2

(gmro) ro = Ro ro − 0.5gmro gmro − 0.5(gmro)2

ro 2/gm − 2 0.5 gmro − gmro

RL Ri2 = RL1 Av1(CS) Av2 (CG) Av= Av1 Av2

Max. Gain

Same gainas a CS Amp.

PracticalGain

Page 34: MOSFET: Transfer Function & Biasing

Cascode amplifier is a two-stage, CS-CG configuration

Cascode Configuration

CG stage

CS stage

Small Signal Model

Small Signal Model configured as two-stage amplifier

Page 35: MOSFET: Transfer Function & Biasing

Cascode Amplifier needs a large load

)||( 222 Lomv RrgA =22222

221 1

1 om

L

mom

LoiL rg

Rgrg

RrRR +≈+

+==)||( 2111 iomv RrgA −=

For simplicity assume ro1 = ro2 = ro and gm1 = gm2 = gm

∞ ∞ − gmro gmro − (gmro)2

(gmro) ro = Ro ro − 0.5gmro gmro − 0.5(gmro)2

ro 2/gm − 2 0.5 gmro − gmro

RL Ri2 = RL1 Av1(CS) Av2 (CG) Av= Av1 Av2

Max. Gain

Same gainas a CS Amp.

PracticalGain

Page 36: MOSFET: Transfer Function & Biasing

Cascode amplifier needs a large load to get a high gain

RL = ro3

Av ≈ − gmro

Gain did not increase compared to a CS amplifier.

This is still a useful circuit because of its high gain-bandwidth (we see this later).

To get a high gain, Av = − 0.5(gmro)2 , we need to increase the small-signal resistance of the current mirror to ≈ (gmro) roo Cascode current mirror

Current Mirror

Cascode

Page 37: MOSFET: Transfer Function & Biasing

Cascode Current mirror Identical MOS: Same k’n and Vt ,

and

Usually: (W/L)1 = (W/L)3 and (W/L)2 = (W/L)4

1

2

3

4

)/()/(

)/()/(

LWLW

LWLW

=

Q1 and Q3 are always in saturation Q2 and Q4 have to be in saturation

for current mirror to work VDS2 > VGS – Vt

VDS4 > VGS – Vt

Straight forward to show

refo ILWLWI

1

2

)/()/(

=

Exercise: For VSS = 0, show that a single current mirror (no cascoding) works only if VD2 > VOV and a cascode current mirror requires VD4 > Vt + 2VOV

Page 38: MOSFET: Transfer Function & Biasing

Small signal resistance of a cascode current mirror is quite large

Equivalentcircuit

Small-signalcircuit

2244 )1( oomoo rrgrR +⋅+=

Page 39: MOSFET: Transfer Function & Biasing

Cascode amplifier with a cascode current mirror

Cascodeamplifier

Cascodecurrent mirror

Signal

Bias

Page 40: MOSFET: Transfer Function & Biasing

Cascode amplifier with a cascode current mirror

A high gain, Av ≈ − 0.5(gmro)2 , high gain-bandwidth circuit.

Draw-back: Low voltage headroom because 4 MOS should be in active for a given VDD

Cascodeamplifier

Cascodecurrent mirror

4433 )1( oomo rrgr +⋅+

Page 41: MOSFET: Transfer Function & Biasing

Folded Cascode increases voltage overhead

PMOS CG stageNMOS CS stageBiased with I1 – I2

Exercise: Draw the small-signal circuit of a folded cascode and show that it is exactly the same as a regular cascode.