Analog and Mixed-Signal Design for SOC in Emerging Digital ...class.ece.iastate.edu/ee435/lectures/EE 435 Lect 8 Spring 2019.pdfSS This establishes a lower bound on V DD-V SS and it

1

EE 435

Lecture 8:

High-Gain Single-Stage Op Amps

2

Basic Op Amp Design

• Fundamental Amplifier Design Issues

• Single-Stage Low Gain Op Amps

• Single-Stage High Gain Op Amps

• Other Basic Gain Enhancement Approaches

• Two-Stage Op Amp

Review from Last Lecture

3

Determination of op amp characteristics

from quarter circuit characteristics

VDD

F

P

F

P

VSS

2

dv

2

dv

VBB VBB

VO+ VO

-

IBB

VDD

F

P

F

P

F

P

F

P

VSS

2

dv

2

dv

VBB VBB

VO+ VO

-

IBB

V1G M1V1 G 1

V1G M1 G 1

G2

CL

2

Vd

Vo++

21L

M1

d

OV

GGsC2

G

V

VA

L

M1

L

21

21

M1VO

2C

GGB

C

GGBW

GG2

GA

Small signal differential half-circuit

Recall from a previous lecture:

4

Single-Stage High Gain Op Amps

How can the gain of the op amp be increased?

21

M1VO

GG

GA

2

1

Recall from Quarter-Circuit Concept

A possible strategy :

Increase GM1 or Decrease G1 (and G2)

in Quarter Circuit or Both


5

Single-Stage High-Gain Op Amps

• If the output conductance can be

decreased without changing the

transconductance, the gain can be

enhanced

• Will concentrate on quarter-circuits and

extend to op amps


6

Determination of 2-port parameters

gO2

gM2

V1

V1 g

O1

gM1

V2

V2

General Two-Port Network

VTST

CL

O2L

M2

TST

2

TSTM2LO22

gsC

g

sV

sVsA

0VgsCgV

0L

0

bsC

asA

Background

Determination of {go1, go2,gM1,gM2 }

Method 2 Load Termination Approach

Since first-order express the gain A(s) in form

gO2

gM2

V1

V1 g

O1

gM1

V2

V2

General Two-Port Network

VTST

CL

observe

(must express with coefficient of s in den equal to CL)


7

o2OUTXm2INm1o2o1X

o2XXm2Lo2OUT

gVVgVgggV

gVVgsCgV

M1

VIN

M2

VOUT

CL

IN1

X2

o2OUT2m21m1o2o1X

o2X2m2Lo2OUT

VV

VV

gVVgVgggV

gVVgsCgV

Analysis of Cascode AmplifierBackground

m1 o2 m2OUT m1 m2

IN L o1 o2 m2 o1 o2 L m2 o1 o2

g g gV g g

V sC g g g g g sC g g g

m2

o2o1L

m1

IN

OUT

g

ggsC

g

V

V gMEQ

gOEQ


8

High output impedance quarter-circuits

VSS

CLM2

M1

IDB

VDD

VBB

VOUT

VIN

Cascode Amplifier

Output conductance appears to have been

decreased !1

21

2

1 20

1 2

1

( ) mv

oL o

m

m mV

o o

m

L

gA s

gsC g

g

g gA

g g

gGB

C

But must verify in the practical parameter

domain to be sure!

O2OEQ O1

m2

mEQ m1

gg g

g

g g


9


EB

V0λV

2A

Cascode AmplifierSubstantial increase in dc gain

No improvement in GB but also

no deterioration in GB !

How does this compare with previous amplifier

using single transistor as quarter circuit?

EBLDD V

1

CV

2PGB

m1

L

gGB

C

VSS

CLM2

M1

IDB

VDD

VBB

VOUT

VIN

1 20

1 2

m mV

o o

g gA

g g

0

1 EB1 2 EB2

DD L EB1

2 2•

λ V λ V

2P 1•

V C V

VA

GB

Cascode amplifier quarter circuit:

Single transistor quarter circuit:


10


Cascode Amplifier

(small-signal equiv)

Quarter Circuit

M3

M1

VOUT

VIN

11


Cascode Amplifier

Quarter Circuit

M5

M7

VDD

VB1

VB2

Counterpart Circuit

M3

M1

VOUT

VIN

VB3

12

Telescopic Cascode Op Amp

Needs CMFB Circuit for VB1 or VB5

Either single-ended or differential outputs

Can connect counterpart as current mirror to eliminate CMFB

M3

M1

VOUT

VIN

VDD

VSS

VB5 M

11

VB1

IT

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

VB2

VB3

VOUT

CL

13

Determination of op amp characteristics from quarter circuit

characteristics (for single-ended gain)

FVIN

IXX

VOUT

VSS

CL

Note: Factor of 4 reduction of single-ended gain

VDD

F

P

F

P

VSS

2

dv

2

dv

VBB VBB

IBB

VDD

F

P

F

P

F

P

F

P

VSS

2

dv

2

dv

VBB VBB

IBB

CL CL

VO-VO- VO

+VO+

1 2

3 4

Small signal Quarter Circuit Small signal differential amplifier

Recall:

1

0

1 3

1 3

1

1

1 3

2

2

2

M

V

L

M

L

M

L

G

AG G

G GBW

C

GGB

C

G

A ssC G G

0M

V QC

L

M

L

M

L

GA

G

GBW

C

GGB

C

GA s

sC G

14


Single-ended operation

VDD

VSS

VB5 M

11

VB1

IT

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

VB2

VB3

VOUT

CL

gOQC =

gOCC =

gmQC =

15


L

m

C

gGB

2

1


m1

oo3 o7

o1 o5m3 m7

g-

2A =g g

g + gg g

VDD

VSS

VB5 M

11

VB1

IT

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

VB2

VB3

VOUT

CL

m1

do3 o7

L o1 o5m3 m7

g-

2A s =g g

sC g + gg g

16


L

m

C

gGB

2

1


m1

oo3 o7

o1 o5m3 m7

g-

2A =g g

g + gg g

This circuit is widely used !!(CMFB circuit not shown)

VDD

VSS

VB5 M

11

VB1

IT

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

VB2

VB3

VOUT

CL

m1

do3 o7

L o1 o5m3 m7

g-

2A s =g g

sC g + gg g

• Large improvement in A0

• No change in GB

17


(CMFB circuit not shown)

VSS

VB5

VB1

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

VB2

VB3

VOUT

CL

M11M12

IB

IT

VDD

Current Mirror Bias

VDD

• One tail bias current generator shown

• IT often one of many outputs for current mirror

• IB and M12 often common to many blocks

• IB often generated from a reference generator circuit

• Tail voltage bias can also be used

19


• Current-Mirror p-channel Bias to Eliminate CMFB

• Only single-ended output available

• Doubles dc gain

VSS

VB5

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

VB3

VOUT

CL

M11M12

IB

IT

VDD

Current Mirror Bias

VDD

VSS

VB5

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

VB3

VOUT

CL

M11M12

IB

IT

VDD

Current Mirror Bias

VDD

VB2

Standard p-channel Cascode Mirror Wide-Swing p-channel Cascode Mirror

20


VSS

VB5

VB1

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

VB2

VB3

VOUT

CL

M11M12

IB

IT

VDD

Current Mirror Bias

VDD

m1

do3 o7

L o1 o5m3 m7

g-

2A s =g g

sC g + gg g

VSS

VB5

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

VB3

VOUT

CL

M11M12

IB

IT

VDD

Current Mirror Bias

VDD

m1d

o3 o7L o1 o5

m3 m7

-gA s =

g gsC g + g

g g

?

(CMFB circuit needed)(No CMFB circuit needed)

21


• Differential Output

• CMFB to establish VB1 or VB5 needed

• Tail current generally generated with current mirror

VB1

IT

VIN VIN

M1M2

M3 M4

M5

M7

M6

M8

VB2

VB3

CLCL

+OUTV

-OUT

V

VDD

VB5

VSS

M11

22


Signal Swing and Power Supply Limitations

Thus, there are a minimum of 5 VDSAT

drops between VDD and VSS

This establishes a lower bound on VDD-

VSS and it will be further reduced by the

p-p signal swing required on the output

VSS

VB1

VINVIN

M1 M2

M3 M4

M5

M7

M6

M8

VB2

VB3

VOUT

CL

M11

IT

VDD

VB5

VDSAT

VDSAT

VDSAT

VDSAT

VDSAT

There are a minimum of 2 VDSAT drops

between VOUT and VDD and a minimum of 3

VDSAT drops between VOUT and VSS

23


VB1

IT

VIN VIN

M1M2

M3 M4

M5

M7

M6

M8

VB2

VB3

CLCL

+OUTV

-OUT

V

VDD

VB5

VSS

M11

IT

VIN VINM1

M2

M3 M4

M5

M7

M6

M8

VB2

VB3

CLCL

+OUTV

-OUT

V

VDD

VSS

VB1

VB5

M11

n-channel inputs p-channel inputs

How important is it to develop good approximate analysis methods for an op amp with the complexity of the telescopic cascade structure?

And many useful op amp circuits will have more complexity !!

VSS

VB5

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

VB3

VOUT

CL

M11M12

IB

IT

VDD

Current Mirror Bias

VDD

Telescopic Cascode Op Amp with Mirror-connected Counterpart Circuit

• Previous analysis obtained almost by inspection• Gain expressions were real simple• Gain expressions provide major insight into operation• Some assumptions were made to simplify analysis

⁻ Vac=0 at “approximate axis of symmetry”⁻ Matched left and right side transistors⁻ Current mirror used to mirror left-side current to right side

How much more involved would an exact analysis be ?Would an exact analysis provide additional insight ?

Vgs1

gm1Vgs1

g01

Vgs3

gm3Vgs3

g03

V-INVgs2

gm2Vgs2

g02

Vgs4

gm4Vgs4

g04

V+

IN

CL

VOUT

VS8

Vgs11

gm11Vgs11

g011

VS1

Vgs7

gm7Vgs7

g07

Vgs5

gm5Vgs5

g05

Vgs8

gm8Vgs8

g08

Vgs6

gm6Vgs6

g06

VS7

VG7

VS4VS3

Small-Signal model of Telescopic Cascode Amplifier

A bit tedious to obtain but really straight forward

1 01 02 011 3 01 4 02 1 1 2 1

3 01 03 1 1 01 1 03 7 3 3

4 02 04 2 1 02 1 04 4 4

04 08 4 4 7 8 04 4 08 88

7 07 03

S S S m IN S m IN S

S m IN S S G m S

S m IN S S OUT m S

OUT L m S G S S Sm

G m

V g g g V g V g g V V g V V

V g g g V V g V g V g V


V sC g g g V g V V g V g V

V g g g

7 7 7 3 3 03 3 07 7

8 06 08 8 7 8 7 8 08

7 05 07 5 7 7 7 7 07 7

G S m S S S

S m S m G S OUT

S m S m G S G

V V g V g V g V

V g g g V g V V V g

V g g g V g V V g V

Analysis of Telescopic Cascode Amplifier

Apply KCL at 7 nodes to obtain a set of 7 independent linear equations

A bit tedious to obtain but really straight forward

Time required to obtain set of equations is quite small

1 01 02 011 3 01 4 02 1 1 2 1

3 01 03 1 1 01 1 03 7 3 3

4 02 04 2 1 02 1 04 4 4

04 08 4 4 7 8 04 4 08 88

7 07 03

S S S m IN S m IN S

S m IN S S G m S

S m IN S S OUT m S

OUT L m S G S S Sm

G m

V g g g V g V g g V V g V V



V sC g g g V g V V g V g V

V g g g

7 7 7 3 3 03 3 07 7

8 06 08 8 7 8 7 8 08

7 05 07 5 7 7 7 7 07 7

G S m S S S

S m S m G S OUT

S m S m G S G

V V g V g V g V

V g g g V g V V V g

V g g g V g V V g V

Analysis of Telescopic Cascode Amplifier

Apply KCL at 7 nodes to obtain a set of 7 independent linear equations

111 12 13 1 2

321 22 26 1

31 33 37 4 2

43 45 46 47 7

52 54 56 8

64 65 66 67 7

74 76

0

0

0 0

0 0

0 0

0 0

S m m

S m

S m

IN

S

INS

G

OUT

Va a a g g

Va a a g

a a a V gV

a a a a VV

a a a V

a a a a V

a a V

In matrix form, this looks like (0 in blank locations)

INA V G V

1

INV A G V

• The A matrix has many 0’s which dramatically reduces complexity of solution• But there are still a large number of off-diagonal terms

Analytical solution of these equations is not practical

%Complete Analysis of Telescopic Cascode Op Amp

diary('matlabdiary')

clear

clc

fprintf('Complete Analysis of Telescopic Cascode Op Amp')

syms Av s CL VOUT VS1 VS3 VS4 VS7 VS8 VG7 Vinn Vinp g01 g02 g03 g04 g05

g06 g07 g08 g011 gm1 gm2 gm3 gm4 gm5 gm6 gm7 gm8

eqn1 =VS1*(gm1+gm2+g01+g02+g011)==VS3*g01+VS4*g02+Vinp*gm1+gm2*Vinn;

eqn2 = VS3*(gm3+g01+g03)+gm1*Vinp==VS1*(g01+gm1)+VG7*g03;

eqn3 = VS4*(gm4+g02+g04)+Vinn*gm2==VS1*(gm2+g02)+VOUT*g04;

eqn4 = VOUT*(s*CL+g04+g08)+VG7*gm8==VS4*(gm4+g04)+VS8*(gm8+g08);

eqn5 = VG7*(gm7+g07+g03)==VS3*(gm3+g03)+VS7*(gm7+g07);

eqn6 = VS8*(gm8+g06+g08)+VS7*gm8==VG7*gm8+VOUT*g08;

eqn7 = VS7*(gm7+gm5+g05+g07)==VG7*(gm7+g07);

eqn8 = Av==VOUT/Vinp;

sol=solve([eqn1,eqn2,eqn3,eqn4,eqn5,eqn6,eqn7,eqn8],[VOUT,VS1,VS3,VS4,VS7,

VS8,VG7,Av]);

VOUTX=sol.VOUT

collect(VOUTX,Vinn)

Simple program and effort to write program is small

fprintf('Differential Analysis of Telescopic Cascode Op Amp')

syms Av s CL VOUT VS1 VS3 VS4 VS7 VS8 VG7 Vdiff g01 g02 g03 g04 g05 g06 g07 g08 g011 gm1 gm2 gm3

gm4 gm5 gm6 gm7 gm8

eqn1 =VS1*(gm1+gm2+g01+g02+g011)==VS3*g01+VS4*g02+Vdiff*gm1/2-gm2*Vdiff/2;

eqn2 = VS3*(gm3+g01+g03)+gm1*Vdiff/2==VS1*(g01+gm1)+VG7*g03;

eqn3 = VS4*(gm4+g02+g04)-Vdiff*gm2/2==VS1*(gm2+g02)+VOUT*g04;

eqn4 = VOUT*(s*CL+g04+g08)+VG7*gm8==VS4*(gm4+g04)+VS8*(gm8+g08);

eqn5 = VG7*(gm7+g07+g03)==VS3*(gm3+g03)+VS7*(gm7+g07);

eqn6 = VS8*(gm8+g06+g08)+VS7*gm8==VG7*gm8+VOUT*g08;

eqn7 = VS7*(gm7+gm5+g05+g07)==VG7*(gm7+g07);

eqn8 = Av==VOUT/Vdiff;

sol=solve([eqn1,eqn2,eqn3,eqn4,eqn5,eqn6,eqn7,eqn8],[VOUT,VS1,VS3,VS4,VS7,VS8,VG7,Av]);

VOUTX=sol.VOUT;

GainAv=sol.Av

collect(GainAv,s)

Simple program and effort to write program is small

VOUTX =

(-(g01*g03*g04*g07*gm2*gm8^2 + g01*g03*g04*gm2*gm7*gm8^2 + g01*g03*g07*gm2*gm4*gm8^2 +

g01*g04*g07*gm2*gm3*gm8^2 + g03*g04*g07*gm1*gm2*gm8^2 + g01*g03*gm2*gm4*gm7*gm8^2 +

g01*g04*gm2*gm3*gm7*gm8^2 + g01*g07*gm2*gm3*gm4*gm8^2 + g03*g04*gm1*gm2*gm7*gm8^2 +

g03*g07*gm1*gm2*gm4*gm8^2 + g04*g07*gm1*gm2*gm3*gm8^2 + g01*gm2*gm3*gm4*gm7*gm8^2 +

g03*gm1*gm2*gm4*gm7*gm8^2 + g04*gm1*gm2*gm3*gm7*gm8^2 + g07*gm1*gm2*gm3*gm4*gm8^2 +

gm1*gm2*gm3*gm4*gm7*gm8^2 + g01*g03*g04*g05*g06*g07*gm2 + g01*g03*g04*g05*g07*g08*gm2 +

g01*g03*g04*g05*g06*g011*gm2 + g01*g03*g04*g05*g08*g011*gm2 + g01*g03*g04*g06*g07*g011*gm2 +


g03*g04*g05*g06*g07*g011*gm2 + g03*g04*g05*g07*g08*g011*gm2 + g01*g03*g04*g05*g06*gm2*gm7 +

g01*g03*g04*g06*g07*gm2*gm5 + g01*g03*g05*g06*g07*gm2*gm4 + g01*g04*g05*g06*g07*gm2*gm3 +













g01*g03*g04*g06*gm2*gm5*gm7 + g01*g03*g05*g06*gm2*gm4*gm7 + g01*g03*g06*g07*gm2*gm4*gm5 +











g01*g03*g06*g011*gm2*g\\\r\nm4*gm5 + g01*g03*g04*g08*gm2*gm7*gm8 + g01*g03*g07*g08*gm2*gm4*gm8 +












g05*g07*g08*g011*gm2*gm3*gm4 + g01*g03*g06*gm2*gm4*gm5*gm7 + g01*g04*g06*gm2*gm3*gm5*gm7 +

g01*g05*g06*gm2*gm3*gm4*gm7 + g01*g06*g07*gm2*gm3*gm4*gm5 + g03*g04*g06*gm1*gm2*gm5*gm7 +


















MATLAB simulation results using Symbolic Math Toolbox

MATLAB Solution:

Analysis with Vin+ and Vin-





g07*g08*g011*gm2*gm3*gm4*gm5 + g01*g06*gm2*gm3*gm4*gm5*gm7 + g03*g06*gm1*gm2*gm4*gm5*gm7 +

g04*g06*gm1*gm2*gm3*gm5*gm7 + g05*g06*gm1*gm2*gm3*gm4*gm7 + g06*g07*gm1*gm2*gm3*gm4*gm5 +












g08*g011*gm2*gm3*gm4*gm5*gm7 + g06*gm1*gm2*gm3*gm4*gm5*gm7 + g06*gm1*gm2*gm3*gm4*gm5*gm8 +

g01*gm2*gm3*gm4*gm5*gm7*gm8 + g03*gm1*gm2*gm4*gm5*gm7*gm8 + g04*gm1*gm2*gm3*gm5*gm7*gm8 +



gm1*gm2*gm3*gm4*gm5*gm7*gm8)/(g01*g02*g03*g04*g07*gm8^2 + g01*g02*g03*g04*gm7*gm8^2 +

g01*g02*g04*g07*gm3*gm8^2 + g02*g03*g04*g07*gm1*gm8^2 + g01*g02*g04*gm3*gm7*gm8^2 +

g02*g03*g04*gm1*gm7*gm8^2 + g02*g04*g07*gm1*gm3*gm8^2 + g02*g04*gm1*gm3*gm7*gm8^2 +

g01*g02*g03*g04*g05*g06*g07 + g01*g02*g03*g04*g05*g06*g08 + g01*g02*g03*g04*g05*g07*g08 +















































MATLAB Solution Continued 1:






































g05*g06*g08*g011*gm3*gm4*gm7 + g06*g07*g08*g011*gm3*gm4*gm5 + g02*g04*g06*gm1*gm3*gm5*gm7 +








g06*g08*g011*gm3*gm4*gm5*gm7 + g02*g04*gm1*gm3*gm5*gm7*gm8 + g06*g08*gm1*gm3*gm4*gm5*gm7 +

g06*g08*gm2*gm3*gm4*gm5*gm7 + CL*g01*g02*g03*g04*g05*g06*s + CL*g01*g02*g03*g04*g05*g08*s +

CL*g01*g02*g03*g04*g06*g07*s + CL*g01*g02*g03*g05*g06*g07*s + CL*g01*g02*g03*g04*g07*g08*s +








CL*g03*g04*g05*g06*g07*g011*s + CL*g03*g04*g05*g07*g08*g011*s + CL*g01*g02*g03*g04*g06*gm5*s +

CL*g01*g02*g03*g05*g06*gm4*s + CL*g01*g03*g04*g05*g06*gm2*s + CL*g01*g02*g03*g04*g05*gm8*s +






CL*g01*g02*g05*g06*g07*gm4*s + CL*g01*g03*ng04*g07*g08*gm2*s + CL*g01*g04*g05*g06*g07*gm2*s +



























CL*g04*g05*g07*g08*g011*gm3*s + CL*g01*g02*g03*g06*gm4*gm5*s + CL*g01*g03*g04*g06*gm2*gm5*s +

CL*g01*g03*g05*g06*gm2*gm4*s + CL*g01*g02*g03*g04*gm5*gm8*s + CL*g01*g02*g03*g05*gm4*gm8*s +
















































CL*g04*g07*g08*g011*gm3*gm5*s + CL*g05*g07*g08*g011*gm3*gm4*s + CL*g01*g03*g06*gm2*gm4*gm5*s +

CL*g01*g02*g03*gm4*gm5*gm8*s + CL*g01*g03*g04*gm2*gm5*gm8*s + CL*g01*g03*g05*gm2*gm4*gm8*s +










































CL*g07*g08*g011*gm3*gm4*gm5*s + CL*g01*g03*gm2*gm4*gm5*gm8*s + CL*g02*g06*gm1*gm3*gm5*gm7*s +

CL*g01*g03*gm2*gm4*gm7*gm8*s + CL*g01*g06*gm2*gm4*gm5*gm7*s + CL*g01*g02*gm3*gm5*gm7*gm8*s +




















CL*g02*gm1*gm3*gm5*gm7*gm8*s + CL*g06*gm1*gm3*gm4*gm5*gm7*s + CL*g01*gm2*gm4*gm5*gm7*gm8*s +





CL*g08*gm2*gm3*gm4*gm5*gm7*s + CL*g011*gm3*gm4*gm5*gm7*gm8*s + CL*gm1*gm3*gm4*gm5*gm7*gm8*s +

CL*gm2*gm3*gm4*gm5*gm7*gm8*s))*Vinn + (Vinp*g02*g03*g04*g07*gm1*gm8^2 +

Vinp*g02*g03*g07*g011*gm1*gm8^2 + Vinp*g03*g04*g07*g011*gm1*gm8^2 +

Vinp*g02*g03*g04*gm1*gm7*gm8^2 + Vinp*g02*g03*g07*gm1*gm4*gm8^2 + Vinp*g02*g04*g07*gm1*gm3*gm8^2

+ Vinp*g03*g04*g07*gm1*gm2*gm8^2 + Vinp*g02*g03*g011*gm1*gm7*gm8^2 +

Vinp*g02*g07*g011*gm1*gm3*gm8^2 + Vinp*g03*g04*g011*gm1*gm7*gm8^2 +

Vinp*g03*g07*g011*gm1*gm4*gm8^2 + Vinp*g04*g07*g011*gm1*gm3*gm8^2 +

Vinp*g02*g03*gm1*gm4*gm7*gm8^2 + Vinp*g02*g04*gm1*gm3*gm7*gm8^2 + Vinp*g02*g07*gm1*gm3*gm4*gm8^2


+ Vinp*g03*g04*gm1*gm2*gm7*gm8^2 + Vinp*g03*g07*gm1*gm2*gm4*gm8^2 +

Vinp*g04*g07*gm1*gm2*gm3*gm8^2 + Vinp*g02*g011*gm1*gm3*gm7*gm8^2 +

Vinp*g03*g011*gm1*gm4*gm7*gm8^2 + Vinp*g04*g011*gm1*gm3*gm7*gm8^2 +

Vinp*g07*g011*gm1*gm3*gm4*gm8^2 + Vinp*g02*gm1*gm3*gm4*gm7*gm8^2 +

Vinp*g03*gm1*gm2*gm4*gm7*gm8^2 + Vinp*g04*gm1*gm2*gm3*gm7*gm8^2 + Vinp*g07*gm1*gm2*gm3*gm4*gm8^2

+ Vinp*g011*gm1*gm3*gm4*gm7*gm8^2 + Vinp*gm1*gm2*gm3*gm4*gm7*gm8^2 +

Vinp*g02*g03*g04*g05*g06*g07*gm1 + Vinp*g02*g03*g04*g05*g07*g08*gm1 +

Vinp*g02*g03*g04*g05*g06*gm1*gm7 + Vinp*g02*g03*g04*g06*g07*gm1*gm5 +










Vinp*g02*g03*g04*g06*gm1*gm5*gm7 + Vinp*g02*g03*g05*g06*gm1*gm4*gm7 +


























Vinp*g02*g03*g06*gm1*gm4*gm5*gm7 + Vinp*g02*g04*g06*gm1*gm3*gm5*gm7 +































Vinp*g02*g06*gm1*gm3*gm4*gm5*gm7 + Vinp*g03*g06*gm1*gm2*gm4*gm5*gm7 +

















Vinp*g08*g011*gm1*gm3*gm4*gm7*gm8 + Vinp*g06*gm1*gm2*gm3*gm4*gm5*gm7 +

Vinp*g06*gm1*gm2*gm3*gm4*gm5*gm8 + Vinp*g02*gm1*gm3*gm4*gm5*gm7*gm8 +




Vinp*g08*gm1*gm2*gm3*gm4*gm7*gm8 + Vinp*gm1*gm2*gm3*gm4*gm5*gm7*gm8)

/

(g01*g02*g03*g04*g07*gm8^2 + g01*g02*g03*g04*gm7*gm8^2 + g01*g02*g04*g07*gm3*gm8^2 +

g02*g03*g04*g07*gm1*gm8^2 + g01*g02*g04*gm3*gm7*gm8^2 + g02*g03*g04*gm1*gm7*gm8^2 +

g02*g04*g07*gm1*gm3*gm8^2 + g02*g04*gm1*gm3*gm7*gm8^2 + g01*g02*g03*g04*g05*g06*g07 +








g02*g03*g05*g06*g07*g08*g011 + g03*g04*g05*g06*g07*g08*g011 + g01*g02*g03*g04*g05*g06*gm7 +
























































g04*g06*g07*g08*g011*gm3*gm5 + g05*g06*g07*g08*g011*gm3*gm4 + g01*g02*g04*g06*gm3*gm5*gm7 +






















g06*g07*g08*g011*gm3*gm4*gm5 + g02*g04*g06*gm1*gm3*gm5*gm7 + g02*g04*g06*gm1*gm3*gm5*gm8 +


















CL*g03*g04*g05*g07*g08*g011*s + CL*g01*g02*g03*g04*g06*gm5*s + CL*g01*g02*g03*g05*g06*gm4*s +




















































































CL*g05*g07*g08*g011*gm3*gm4*s + CL*g01*g03*g06*gm2*gm4*gm5*s + CL*g01*g02*g03*gm4*gm5*gm8*s +






























































CL*g07*g011*gm3*gm4*gm5*gm8*s + CL*g08*g011*gm3*gm4*gm5*gm7*s + CL*g02*gm1*gm3*gm5*gm7*gm8*s +






CL*g011*gm3*gm4*gm5*gm7*gm8*s + CL*gm1*gm3*gm4*gm5*gm7*gm8*s + CL*gm2*gm3*gm4*gm5*gm7*gm8*s)


We now (finally) have a complete analytic expression !




















Vinp*g08*g011*gm1*gm3*gm4*gm7*gm8 + Vinp*g06*gm1*gm2*gm3*gm4*gm5*gm7 +






/

(g01*g02*g03*g04*g07*gm8^2 + g01*g02*g03*g04*gm7*gm8^2 + g01*g02*g04*g07*gm3*gm8^2 +


g02*g04*g07*gm1*gm3*gm8^2 + g02*g04*gm1*gm3*gm7*gm8^2 + g01*g02*g03*g04*g05*g06*g07 +








g02*g03*g05*g06*g07*g08*g011 + g03*g04*g05*g06*g07*g08*g011 + g01*g02*g03*g04*g05*g06*gm7 +

































Zoom in to look at a few terms



CL*g05*g07*g08*g011*gm3*gm4*s + CL*g01*g03*g06*gm2*gm4*gm5*s + CL*g01*g02*g03*gm4*gm5*gm8*s +






























































CL*g07*g011*gm3*gm4*gm5*gm8*s + CL*g08*g011*gm3*gm4*gm5*gm7*s + CL*g02*gm1*gm3*gm5*gm7*gm8*s +






CL*g011*gm3*gm4*gm5*gm7*gm8*s + CL*gm1*gm3*gm4*gm5*gm7*gm8*s + CL*gm2*gm3*gm4*gm5*gm7*gm8*s)

Vinp*g08*gm1*gm2*gm3*gm4*gm5*gm7+ Vinp*g06*gm1*gm2*gm3*gm4*gm7*gm8 +


/

(g01*g02*g03*g04*g07*gm8^2 + g01*g02*g03*g04*gm7*gm8^2 +

g01*g02*g04*g07*gm3*gm8^2 + g02*g03*g04*g07*gm1*gm8^2 +

g01*g02*g04*gm3*gm7*gm8^2 + g02*g03*g04*gm1*gm7*gm8^2 +

Individual product terms have product of 7 small-signal parameters

CL*g02*gm3*gm4*gm5*gm7*gm8*s + CL*g03*gm2*gm4*gm5*gm7*gm8*s +



CL*g011*gm3*gm4*gm5*gm7*gm8*s + CL*gm1*gm3*gm4*gm5*gm7*gm8*s +

CL*gm2*gm3*gm4*gm5*gm7*gm8*s)

• Approximately 2000 factors in output characteristics• Approximately 14,000 small-signal parameter appearances• GB expression much longer

MATLAB simulation results using Symbolic Math Toolbox

MATLAB Solution:

Differential Analysis with Vd = Vin+ - Vin-

GainAv = (g01*g03*g04*g07*gm2*gm8^2 + g02*g03*g04*g07*gm1*gm8^2 + g02*g03*g07*g011*gm1*gm8^2 +


g01*g04*g07*gm2*gm3*gm8^2 + g02*g03*g04*gm1*gm7*gm8^2 + g02*g03*g07*gm1*gm4*gm8^2 +

g02*g04*g07*gm1*gm3*gm8^2 + 2*g03*g04*g07*gm1*gm2*gm8^2 + g02*g03*g011*gm1*gm7*gm8^2 +

g02*g07*g011*gm1*gm3*gm8^2 + g03*g04*g011*gm1*gm7*gm8^2 + g03*g07*g011*gm1*gm4*gm8^2 +

g04*g07*g011*gm1*gm3*gm8^2 + g01*g03*gm2*gm4*gm7*gm8^2 + g01*g04*gm2*gm3*gm7*gm8^2 +

g01*g07*gm2*gm3*gm4*gm8^2 + g02*g03*gm1*gm4*gm7*gm8^2 + g02*g04*gm1*gm3*gm7*gm8^2 +

g02*g07*gm1*gm3*gm4*gm8^2 + 2*g03*g04*gm1*gm2*gm7*gm8^2 + 2*g03*g07*gm1*gm2*gm4*gm8^2 +

2*g04*g07*gm1*gm2*gm3*gm8^2 + g02*g011*gm1*gm3*gm7*gm8^2 + g03*g011*gm1*gm4*gm7*gm8^2 +

g04*g011*gm1*gm3*gm7*gm8^2 + g07*g011*gm1*gm3*gm4*gm8^2 + g01*gm2*gm3*gm4*gm7*gm8^2 +

g02*gm1*gm3*gm4*gm7*gm8^2 + 2*g03*gm1*gm2*gm4*gm7*gm8^2 + 2*g04*gm1*gm2*gm3*gm7*gm8^2 +

2*g07*gm1*gm2*gm3*gm4*gm8^2 + g011*gm1*gm3*gm4*gm7*m8^2 + 2*gm1*gm2*gm3*gm4*gm7*gm8^2 +







g02*g03*g05*g06*g07*gm1*gm4 + g02*g04*g05*g06*g07*gm1*gm3 + 2*g03*g04*g05*g06*g07*gm1*gm2 +





2*g03*g04*g05*g07*g08*gm1*gm2 + g01*g03*g04*g06*g07*gm2*gm8 + g02*g03*g04*g06*g07*gm1*gm8 +












g03*g04*g07*g08*g011*gm1*gm8 + g01*g03*g04*g06*gm2*gm5*gm7 + g01*g03*g05*g06*gm2*gm4*gm7 +




g02*g05*g06*g07*gm1*gm3*gm4 + 2*g03*g04*g05*g06*gm1*gm2*gm7 + 2*g03*g04*g06*g07*gm1*gm2*gm5 +

2*g03*g05*g06*g07*gm1*gm2*gm4 + 2*g04*g05*g06*g07*gm1*gm2*gm3 + g01*g03*g04*g06*gm2*gm5*gm8 +


g02*g03*g05*g06*gm1*gm4*gm8 + g02*g04*g05*g06*gm1*gm3*gm8 + 2*g03*g04*g05*g06*gm1*gm2*gm8 +







g02*g04*g07*g08*gm1*gm3*gm5 + g02*g05*g07*g08*gm1*gm3*gm4 + 2*g03*g04*g05*g07*gm1*gm2*gm8 +

2*g03*g04*g05*g08*gm1*gm2*gm7 + 2*g03*g04*g07*g08*gm1*gm2*gm5 + 2*g03*g05*g07*g08*gm1*gm2*gm4 +

2*g04*g05*g07*g08*gm1*gm2*gm3 + g01*g03*g04*g06*gm2*gm7*gm8 + g01*g03*g06*g07*gm2*gm4*gm8 +


g02*g04*g06*g07*gm1*gm3*gm8 + 2*g03*g04*g06*g07*gm1*gm2*gm8 + g01*g03*g06*g011*gm2*gm4*gm5 +



2*g03*g04*g07*g08*gm1*gm2*gm8 + g01*g03*g04*g011*gm2*gm5*gm8 + g01*g03*g05*g011*gm2*gm4*gm8 +











MATLAB Solution Continued 1:g03*g04*g06*g011*gm1*gm7*gm8 + g03*g04*g07*g011*gm2*gm5*gm8 + g03*g04*g08*g011*gm2*gm5*gm7 +







g02*g05*g06*gm1*gm3*gm4*gm7 + g02*g06*g07*gm1*gm3*gm4*gm5 + 2*g03*g04*g06*gm1*gm2*gm5*gm7 +

2*g03*g05*g06*gm1*gm2*gm4*gm7 + 2*g03*g06*g07*gm1*gm2*gm4*gm5 + 2*g04*g05*g06*gm1*gm2*gm3*gm7 +

2*g04*g06*g07*gm1*gm2*gm3*gm5 + 2*g05*g06*g07*gm1*gm2*gm3*gm4 + g01*g03*g06*gm2*gm4*gm5*gm8 +










g02*g07*g08*gm1*gm3*gm4*gm5 + 2*g03*g04*g05*gm1*gm2*gm7*gm8 + 2*g03*g04*g07*gm1*gm2*gm5*gm8 +




















g04*g08*g011*gm1*gm3*gm7*gm8 + g07*g08*g011*gm1*gm3*gm4*gm8 + g01*g06*gm2*gm3*gm4*gm5*gm7 +

g02*g06*gm1*gm3*gm4*gm5*gm7 + 2*g03*g06*gm1*gm2*gm4*gm5*gm7 + 2*g04*g06*gm1*gm2*gm3*gm5*gm7 +

2*g05*g06*gm1*gm2*gm3*gm4*gm7 + 2*g06*g07*gm1*gm2*gm3*gm4*gm5 + g01*g06*gm2*gm3*gm4*gm5*gm8 +


2*g05*g06*gm1*gm2*gm3*gm4*gm8 + g01*g03*gm2*gm4*gm5*gm7*gm8 + g01*g04*gm2*gm3*gm5*gm7*gm8 +



g02*g07*gm1*gm3*gm4*gm5*gm8 + g02*g08*gm1*gm3*gm4*gm5*gm7 + 2*g03*g04*gm1*gm2*gm5*gm7*gm8 +

2*g03*g05*gm1*gm2*gm4*gm7*gm8 + 2*g03*g07*gm1*gm2*gm4*gm5*gm8 + 2*g03*g08*gm1*gm2*gm4*gm5*gm7 +



g01*g06*gm2*gm3*gm4*gm7*gm8 + g02*g06*gm1*gm3*gm4*gm7*gm8 + 2*g03*g06*gm1*gm2*gm4*gm7*gm8 +

2*g04*g06*gm1*gm2*gm3*gm7*gm8 + 2*g06*g07*gm1*gm2*gm3*gm4*gm8 + g01*g08*gm2*gm3*gm4*gm7*gm8 +


2*g07*g08*gm1*gm2*gm3*gm4*gm8 + g01*g011*gm2*gm4*gm5*gm7*gm8 + g06*g011*gm1*gm3*gm4*gm5*gm8 +



g08*g011*gm2*gm3*gm4*gm5*gm7 + g08*g011*gm1*gm3*gm4*gm7*gm8 + 2*g06*gm1*gm2*gm3*gm4*gm5*gm7 +

2*g06*gm1*gm2*gm3*gm4*gm5*gm8 + g01*gm2*gm3*gm4*gm5*gm7*gm8 + g02*gm1*gm3*gm4*gm5*gm7*gm8 +

2*g03*gm1*gm2*gm4*gm5*gm7*gm8 + 2*g04*gm1*gm2*gm3*gm5*gm7*gm8 + 2*g05*gm1*gm2*gm3*gm4*gm7*gm8 +

2*g07*gm1*gm2*gm3*gm4*gm5*gm8 + 2*g08*gm1*gm2*gm3*gm4*gm5*gm7 + 2*g06*gm1*gm2*gm3*gm4*gm7*gm8 +

2*g08*gm1*gm2*gm3*gm4*gm7*gm8 + g011*gm2*gm3*gm4*gm5*gm7*gm8 + 2*gm1*gm2*gm3*gm4*gm5*gm7*gm8)

/

((2*CL*g01*g02*g03*g04*g05*g06 + 2*CL*g01*g02*g03*g04*g05*g08 + 2*CL*g01*g02*g03*g04*g06*g07 +

2*CL*g01*g02*g03*g05*g06*g07 + 2*CL*g01*g02*g03*g04*g07*g08 + 2*CL*g01*g02*g04*g05*g06*g07 +






MATLAB Solution Continued 2:2*CL*g01*g03*g04*g07*g08*g011 + 2*CL*g01*g04*g05*g06*g07*g011 + 2*CL*g02*g03*g05*g06*g07*g011 +


2*CL*g03*g04*g05*g07*g08*g011 + 2*CL*g01*g02*g03*g04*g06*gm5 + 2*CL*g01*g02*g03*g05*g06*gm4 +

2*CL*g01*g03*g04*g05*g06*gm2 + 2*CL*g01*g02*g03*g04*g05*gm8 + 2*CL*g01*g02*g03*g04*g06*gm7 +
































2*CL*g01*g02*g03*g06*gm4*gm5 + 2*CL*g01*g03*g04*g06*gm2*gm5 + 2*CL*g01*g03*g05*g06*gm2*gm4 +




































MATLAB Solution Continued 3:2*CL*g01*g02*g05*g011*gm7*gm8 + 2*CL*g01*g02*g07*g011*gm5*gm8 + 2*CL*g01*g02*g08*g011*gm5*gm7 +













2*CL*g05*g07*g08*g011*gm3*gm4 + 2*CL*g01*g03*g06*gm2*gm4*gm5 + 2*CL*g01*g02*g03*gm4*gm5*gm8 +

2*CL*g01*g03*g04*gm2*gm5*gm8 + 2*CL*g01*g03*g05*gm2*gm4*gm8 + 2*CL*g01*g03*g06*gm2*gm4*gm7 +









































2*CL*g01*g03*gm2*gm4*gm5*gm8 + 2*CL*g02*g06*gm1*gm3*gm5*gm7 + 2*CL*g01*g03*gm2*gm4*gm7*gm8 +
















MATLAB Solution Continued 4:2*CL*g04*g05*gm2*gm3*gm7*gm8 + 2*CL*g04*g07*gm2*gm3*gm5*gm8 + 2*CL*g04*g08*gm2*gm3*gm5*gm7 +




2*CL*g07*g011*gm3*gm4*gm5*gm8 + 2*CL*g08*g011*gm3*gm4*gm5*gm7 + 2*CL*g02*gm1*gm3*gm5*gm7*gm8 +

2*CL*g06*gm1*gm3*gm4*gm5*gm7 + 2*CL*g01*gm2*gm4*gm5*gm7*gm8 + 2*CL*g06*gm2*gm3*gm4*gm5*gm7 +





2*CL*g011*gm3*gm4*gm5*gm7*gm8 + 2*CL*gm1*gm3*gm4*gm5*gm7*gm8 + 2*CL*gm2*gm3*gm4*gm5*gm7*gm8)*s +

(2*g01*g02*g03*g04*g07*gm8^2 + 2*g01*g02*g03*g04*gm7*gm8^2 + 2*g01*g02*g04*g07*gm3*gm8^2 +

2*g02*g03*g04*g07*gm1*gm8^2 + 2*g01*g02*g04*gm3*gm7*gm8^2 + 2*g02*g03*g04*gm1*gm7*gm8^2 +

2*g02*g04*g07*gm1*gm3*gm8^2 + 2*g02*g04*gm1*gm3*gm7*gm8^2 + 2*g01*g02*g03*g04*g05*g06*g07 +

2*g01*g02*g03*g04*g05*g06*g08 + 2*g01*g02*g03*g04*g05*g07*g08 + 2*g01*g02*g03*g04*g06*g07*g08 +

2*g01*g02*g03*g04*g05*g06*g011 + 2*g01*g02*g03*g05*g06*g07*g08 + 2*g01*g02*g04*g05*g06*g07*g08 +

2*g01*g02*g03*g04*g05*g08*g011 + 2*g01*g02*g03*g04*g06*g07*g011 + 2*g01*g03*g04*g05*g06*g07*g08

+ 2*g02*g03*g04*g05*g06*g07*g08 + 2*g01*g02*g03*g04*g07*g08*g011 +


+ 2*g01*g02*g04*g05*g07*g08*g011 + 2*g01*g03*g04*g05*g06*g08*g011 +


+ 2*g02*g03*g04*g05*g07*g08*g011 + 2*g01*g04*g05*g06*g07*g08*g011 +

2*g02*g03*g05*g06*g07*g08*g011 + 2*g03*g04*g05*g06*g07*g08*g011 + 2*g01*g02*g03*g04*g05*g06*gm7

+ 2*g01*g02*g03*g04*g06*g07*gm5 + 2*g01*g02*g04*g05*g06*g07*gm3 + 2*g02*g03*g04*g05*g06*g07*gm1









+ 2*g01*g02*g03*g04*g06*g011*gm7 + 2*g01*g02*g03*g04*g08*g011*gm5 +

2*g01*g03*g04*g05*g06*g08*gm7 + 2*g01*g03*g04*g06*g07*g08*gm5 + 2*g01*g03*g05*g06*g07*g08*gm4 +



2*g03*g04*g05*g06*g07*g08*gm2 + 2*g01*g02*g03*g04*g07*g011*gm8 + 2*g01*g02*g03*g04*g08*g011*gm7













+ 2*g01*g02*g03*g04*g06*gm5*gm7 + 2*g01*g02*g04*g05*g06*gm3*gm7 + 2*g01*g02*g04*g06*g07*gm3*gm5





















+ 2*g04*g05*g06*g07*g08*gm2*gm3 + 2*g01*g02*g03*g04*g011*gm7*gm8 +

MATLAB Solution Continued 5:2*g01*g02*g04*g06*g011*gm5*gm7 + 2*g01*g02*g04*g05*g011*gm7*gm8 + 2*g01*g02*g04*g07*g011*gm5*gm8


2*g02*g03*g04*g06*g011*gm5*gm7 + 2*g02*g04*g05*g06*g011*gm3*gm7 + 2*g02*g04*g06*g07*g011*gm3*gm5









+ 2*g01*g02*g04*g06*gm3*gm5*gm7 + 2*g02*g03*g04*g06*gm1*gm5*gm7 + 2*g02*g04*g05*g06*gm1*gm3*gm7


















+ 2*g02*g04*g06*g011*gm3*gm5*gm7 + 2*g02*g03*g04*g011*gm5*gm7*gm8 +

2*g02*g04*g05*g011*gm3*gm7*gm8 + 2*g02*g04*g07*g011*gm3*gm5*gm8 + 2*g02*g04*g08*g011*gm3*gm5*gm7

+ 2*g01*g06*g08*g011*gm4*gm5*gm7 + 2*g02*g06*g08*g011*gm3*gm5*gm7 +

2*g03*g06*g08*g011*gm4*gm5*gm7 + 2*g04*g06*g08*g011*gm3*gm5*gm7 + 2*g05*g06*g08*g011*gm3*gm4*gm7

+ 2*g06*g07*g08*g011*gm3*gm4*gm5 + 2*g02*g04*g06*gm1*gm3*gm5*gm7 + 2*g02*g04*g06*gm1*gm3*gm5*gm8

+ 2*g01*g02*g04*gm3*gm5*gm7*gm8 + 2*g02*g03*g04*gm1*gm5*gm7*gm8 + 2*g02*g04*g05*gm1*gm3*gm7*gm8






+ 2*g06*g07*g08*gm2*gm3*gm4*gm5 + 2*g02*g04*g011*gm3*gm5*gm7*gm8 +

2*g06*g08*g011*gm3*gm4*gm5*gm7 + 2*g02*g04*gm1*gm3*gm5*gm7*gm8 + 2*g06*g08*gm1*gm3*gm4*gm5*gm7 +

2*g06*g08*gm2*gm3*gm4*gm5*gm7))

• Difference Mode Gain has only approximately 1100 product terms• Difference Mode Gain has approximately 7700 small-signal parameters in expression

• This does not even include the common-mode gain expression !

• Extremely difficult to get insight into how this relatively simple circuit performs

from this solution

VSS

VB5

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

VB3

VOUT

CL

M11M12

IB

IT

VDD

Current Mirror Bias

VDD


• Previous analysis obtained almost by inspection• Gain expressions were real simple• Gain expressions provide major insight into operation• Some assumptions were made to simplify analysis

⁻ Vac=0 at “approximate axis of symmetry”⁻ Matched left and right side transistors⁻ Current mirror used to mirror left-side current to right side

How much more involved would an exact analysis be ?Would an exact analysis provide additional insight ?

Where this started

VSS

VB5

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

VB3

VOUT

CL

M11M12

IB

IT

VDD

Current Mirror Bias

VDD


• Some assumptions were made to simplify analysis⁻ Vac=0 at “approximate axis of symmetry”⁻ Matched left and right side transistors⁻ Current mirror used to mirror left-side current to right side

Where this started

• Simplified Difference Mode Gain has 4 product terms• Simplified Difference Mode Gain has 12 small-signal parameters in expression

m1d

o3 o7L o1 o5

m3 m7

-gA s =

g gsC g + g

g g

How many product terms are present in the simplified analysis?

How many small-signal parameters are in simplified expression?

m7 m1 m3d

L m7 m3 m7 o3 o1 m3 o5 o7

-g g gA s =

sC g g g g g + g g g

• Difference Mode Gain has only approximately 1100 product terms• Difference Mode Gain has approximately 7700 small-signal parameters in expression

How important is it to develop good approximate analysis methods for an op amp of this complexity?

And many useful op amp circuits will have much more complexity !!

Will you impress your boss if you use an “exact” analysis?

52

Are there other high

output impedance

circuits that can be

used as quarter

circuits?

L

M1

L

21

21

M1VO

2C

GGB

C

GGBW

GG2

GA

53

Are there other high output impedance circuits that can be used

as quarter circuits ?

Regulated cascode

circuits have the high

output impedance

property

54

M1

M3

-A

VIN

VOUT


Regulated Cascode Amplifier

or “Gain Boosted Cascode”

Quarter Circuit

• A is usually a simple amplifier, often the reference op amp with + terminal

connected to the desired quiescent voltage

• Assume biased with a dc current source (not shown) at drain of M3

55

go1gM1V1V1

go3gM3V2V2

VIN

VOUT

CL

-A VX

-A

M1

VIN

M3

VOUT

CL

Analysis of Regulated Cascode Amplifier

VX and V2 can be easily eliminated from these 3 equations

Background

OUT o3 L m3 2 X o3

X o1 o3 m1 IN m3 2 OUT o3

2 X X

V g +sC +g V =V g

V g +g +g V -g V =V g

V +V =-AV

56

-A

M1

VIN

M2

VOUT

CL

Analysis of Regulated Cascode AmplifierBackground

m1 o3 m3 m1 m3OUT m1

o1 o3IN L m3 o1 o3L o1 o3 m3 o1 o3L

m3

g g g 1 A g g 1 AV g

g gV sC g 1 A g gsC g g g 1 A g g sCg 1 A

for A large: gMEQ

gOEQ

OUT o3 L m3 2 X o3

X o1 o3 m1 IN m3 2 OUT o3

2 X X

V g +sC +g V =V g

V g +g +g V -g V =V g

V +V =-AV

OUT o3 L m3 X X o3

X o1 o3 m1 IN m3 X OUT o3

V g +sC -g V 1 =V g

V g +g +g V +g V 1 =V g

A

A

1

OUT m1

IN o3L o1

m3

V g

V gsC g

g A

57


Regulated Cascode Amplifier

or “Gain Boosted Cascode”

Based upon small-signal analysis it appears that the

output conductance has been decreased even more!

Same GB as for previous two circuits

IDB1

M1

M3

-A

VIN

VOUT

CL

VSS

1

1o3o1

m3

gg

g A

OEQ

mEQ m

g

g g

m1V

o3L o1

m3

gA

gsC g

g A

m3m1O

o1 o3

g AgA

g g

m1

L

gGB

C

Must verify improvement in gain in

practical parameter domain !!

Verification will show predicted improvements

58

Gain-Boosted Telescopic Cascode Op Amp

M1

M3

-A

VIN

VOUT

Needs CMFB Circuit for Vb1

Either single-ended or differential outputs

Can connect counterpart as current mirror to eliminate CMFB

Use differential op amp to facilitate biasing of cascode device

VDD

V OU

T

CL

VB2

VB3

V S

S

VB5 M 1

1

VB1

A1A2

A3A4

IT

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

59

Gain-Boosted Telescopic Cascode Op AmpV

DD

VOUT

CL

VB2

VB3

VSS

VB5 M

11

VB1

A1

A2

A3

A4

IT

VIN

VINM

1M

2

M3 M

4

M5

M7

M6

M8


gOQC =

gOCC =

gmQC =

60


m1

oo3 o7

o1 o5

1 m3 3 m7

g-

2A =g g

g + gA g A g

This is modestly less efficient at

generating GB because now power is

consumed in both the cascode

devices and the boosting amplifier

VDD

V OU

T

CL

VB2

VB3

V S

S

VB5 M 1

1

VB1

A1A2

A3A4

IT

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

m1

L

gGB

2C

61


m1

L

gGB =

C

m1o

o3 o7o1 o5

1 m3 3 m7

-gA =

g gg + g

A g A g

This is modestly less efficient at

generating GB because now power is

consumed in both the cascode

devices and the boosting amplifier

VDD

V OU

T

CL

VB2

VB3

V S

S

VB5 M 1

1

A1A2

A3A4

IT

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

Elimination of need for CMFB Circuit

62


VDD

CL

VB2

VB3

V S

S

VB5 M

11

A1A2

A3A4

IT

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

VDSAT

VOUT

VDSAT

VDSAT

VDSAT

VDSAT

A minimum of 5 VDSAT drops

between VDD and VSS

Signal Swing and Power Supply Limitations

This establishes a lower bound

on VDD-VSS and it will be reduced

by the p-p signal swing on the

output

63


VDD

V OU

T

CL

VB2

VB3

V S

S

VB5 M 1

1

A1A2

A3A4

IT

VINVIN

M1M2

M3 M4

M5

M7

M6

M8

Advantages:

Significant increase in dc gain

Limitations:

:• Signal swing (4VDSAT+VT between VDD and VSS)

• Reduction in GB power efficiency

- some current required to bias “A” amplifiers

• -additional pole in “A” amplifier -may add requirements for some compensation

• Area Overhead for 4 transistors and 4 amplifiers-actually minor concern since performance will usually

justify these resources

(with or w/o current mirror counterpart circuits)

64

Are there other useful high output impedance

circuits that can be used for the quarter circuit?

V1G M1V1 G 1

V1G M1 G 1

G2

CL

2

Vd

Vo++

1

0

1 2

1 2

1

2

2

MV

L

M

L

GA

G G

G GBW

C

GGB

C

65

What circuit is this?

VSS

CL

M3

M1

IDB

VDD

VBB

VOUT

VIN

Cascode Amplifier

• Cascode Amplifier– Often termed a “Folded Cascode Amplifier”

– Same small-signal performance as other

– VOUT swing VDSAt1-VDSAT2 could be small or negative

– But a biasing problem !!

IB1

M1

M3

VOUT

VBB

VIN

VSS

CL

66

Biased Folded Cascode Amplifier

IB1

M1

M3

VOUT

VBB

VIN

VSS

Folded Cascode Amplifier

VDD

VIN

M1

M3V

B1IB1

IB2 V

OUT

VSS

Biased Folded Cascode

67

Implementation of Biased Folded

Cascode Amplifier?

VDD

VIN

M1

M3 M

5

VB1 V

B3

Biased Folded CascodeImplementation of Biased

Folded Cascode

VDD

VIN

M1

M3V

B1IB1

IB2 V

OUT

VSS

68

End of Lecture 9

Documents

Analog and Mixed-Signal Design for SOC in Emerging Digital ...class.ece.iastate.edu/ee435/lectures/EE 435 Lect 8 Spring 2019.pdfSS This establishes a lower bound on V DD-V SS and it