ee 435 - iowa state universityclass.ece.iastate.edu/ee435/lectures/ee 435 lect 19...1 m 2 i d1 i d2...

EE 435

Lecture 19

Linearity of Bipolar Differential Pair

Linearity of Common Source Amplifier

Offset Voltages

How linear is the amplifier ?

Vd

VEB1

IT

I

EB1V2

ID1

1% Linear =

0.3VEB1

IT

M1

M2

ID1 I

D2

V1

V2

VS

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IT

M1

M2

ID1 I

D2

V1

V2

VS 3

0 1 3sin sinD2I ωt ωta a a

3

1 3

3

2T

m m

T

θI θV V

2 4 2 Ia a

2

2

1

20logTHDk

k

a

a

2

log m

T

θVTHD 20

4I

OXμC Wθ =

2L

2OXT EB1

μC WI = V

L

2

log m

2

EB1

VTHD 20

8V

Substituting in we obtain

where

This expression gives little insight. But in terms of the practical parameters:

Thus to minimize THD, want VEB large and VM small

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Bipolar Differential Pair

12d VVV

IT

Q1 Q

2

IC1 I

C2

V1

V2

VE

t

E1

V

VV

E1SC1 eAJI

t

E2

V

VV

E2SC2 eAJI

TC2C1 III

E1S

C1tE1

AJ

IlnVVV

E2S

C2tE2

AJ

IlnVVV

C1

C2t

AA

E1S

C1

E2S

C2td

I

IlnV

AJ

Iln

AJ

IlnVV

E2E1

Bipolar Differential Pair

12d VVV

IT

Q1 Q

2

IC1 I

C2

V1

V2

VE

C1

C2t

AA

E1S

C1

E2S

C2td

I

IlnV

AJ

Iln

AJ

IlnVV

E2E1

C1

C1Ttd

I

I-IlnVV

C2T

C2td

I-I

IlnVV

As IC1 approaches 0, Vd approaches infinity

As IC1 approaches IT, Vd approaches minus infinity

At IC1=IC2=IT/2, Vd=0

Transition much steeper than for MOS case

-0.2 -0.1 0 0.1 0.2

IT

I

Vd

Differential input in Volts

2

IT

IC1

IC2

Transfer Characteristics of Bipolar Differential Pair

Transition much steeper than for MOS case

Asymptotic Convergence to 0 and IT

IT

Q1 Q

2

IC1 I

C2

V1

V2

VE

C1

C1Ttd

I

I-IlnVV

-0.2 -0.1 0 0.1 0.2

Vd

I

IC1

Signal Swing and Linearity of Bipolar Differential Pair

?m

hVdint

hmVI dFIT

2

Ih T

pointQd

C1

V

Im

2

IIC1TC1

Tt

pointQC1

d

T/C1

III

IV

I

V

T

t

pointQC1

d

I

4V

I

V

2

IV

4V

II T

d

t

TFIT

td in t 2Vm

hV

IT

Q1 Q

2

IC1 I

C2

V1

V2

VE

-0.1 -0.05 0 0.05 0.1

x% Deviation

I

Vd


for 1% deviation, Vd=.56Vt

for 0.1% deviation, Vd=.27Vt

IT

Q1 Q

2

IC1 I

C2

V1

V2

VE

-0.2 -0.1 0 0.1 0.2

Vd

I

IC1

1% linear = .56Vt

2Vt


IT

Q1 Q

2

IC1 I

C2

V1

V2

VE


Distortion in the differential pair is another useful metric for characterizing

linearity of IC1 and Ic2 with sinusoidal differential excitation

12d VVV

Consider again the differential pair and assume excited differentially with

d d2 1

V VV = V = -

2 2

Recall:

and assume Vd=Vmsin(ωt)

Thus can express as

IT

Q1 Q

2

IC1 I

C2

V1

V2

VE

T C1d t

C1

I -IV V ln

I

1

d

t

d

t

V

V T C1

C1

V

V

C1 T

I -Ie

I

I I 1+e


Vd=Vmsin(ωt) IT

Q1 Q

2

IC1 I

C2

V1

V2

VE1

d

t

V

V

C1 TI I 1+e

Consider a Taylor’s Series Expansion

2 32 31 1 1

1 1 2 30

0 0 0

1 1. .

2! 3!d

d d d

C C CC C d d dV

d d dV V V

I I II I V V V H OT

V V V


Vd=Vmsin(ωt) IT

Q1 Q

2

IC1 I

C2

V1

V2

VE

2 32 31 1 1

1 1 2 30

0 0 0

1 1. .

2! 3!d

d d d

C C CC C d d dV

d d dV V V

I I II I V V V H OT

V V V

2

2 32

2 32

3

12

2

d d

t t

d d d d d

t t t t t

d d d d

t t t t

V V

V VC1 T

d t

V V V V V

V V V V VC1 T

2

d t t t

V V 2V V

V V V VC1 T

2 2

d t

C1 T

3

d

I I1+e e

V V

I I 11+e e e 1+e e

V V V V

I I1+e e e 1+e

V V

I I

V V

2 3 4 3

2 33

2 6 2

2 6

d d d d d d d d d d

t t t t t t t t t t

d d d d d

t t t t t

V V V V V 2V V V 2V V

V V V V V V V V V V

2

t t t t t

V V 2V V 3V

V V V V VC1 T

3 3

d t

1 1 1 21+e e e 1+e e e 1+e e e 1+e

V V V V

I I1+e e e 1+e e 1

V V

4 3

4d d d

t t t

V 2V V

V V V+e e 1+e


Vd=Vmsin(ωt) IT

Q1 Q

2

IC1 I

C2

V1

V2

VE

2 32 31 1 1

1 1 2 30

0 0 0

1 1. .

2! 3!d

d d d

C C CC C d d dV

d d dV V V

I I II I V V V H OT

V V V

2

2

00

2 32

2 3

00

23

0

2 0

d d

t t

d d d d

t t t t

d d

t

V V

V VC1 T T T

d t t t

V V 2V V

V V V VC1 T T

2 2 2

d t t

V V

VC1 T

3 3

d t

I I I I1+e e 2

V V V 4V

I I I1+e e e 1+e 2 2 2

V V V

I I1+e e

V V

dd

d

d

d

VV

VV

V

3 4 3

2 3 4 3

0

2 6 4 2 6 4d d d d d d

t t t t t t t

2V V 3V V 2V V

V V V V V V V T T

3 3

t t

I Ie 1+e e 1+e e 1+e 2 2 2 2

V 8VdV

2 3

0 0 0

0C1 C1 C1T T

2 3 3

d t d d t

I I II I

V 4V V V 8Vd d dV V V


Vd=Vmsin(ωt) IT

Q1 Q

2

IC1 I

C2

V1

V2

VE

2 32 31 1 1

1 1 2 30

0 0 0

1 1. .

2! 3!d

d d d

C C CC C d d dV

d d dV V V

I I II I V V V H OT

V V V

2 3

0 0 0

0C1 C1 C1T T

2 3 3

d t d d t

I I II I

V 4V V V 8Vd d dV V V

1

3T T Td d3

t t

I I I- V + V

2 4V 48VCI

1

3 3T T Tm m3

t t

I I I- V sin t + V sin t

2 4V 48VCI

3 1

34 4

3sin t sin t sin t


Vd=Vmsin(ωt) IT

Q1 Q

2

IC1 I

C2

V1

V2

VE

2 32 31 1 1

1 1 2 30

0 0 0

1 1. .

2! 3!d

d d d

C C CC C d d dV

d d dV V V

I I II I V V V H OT

V V V

1

1

3 13

4 4

3

3T T Tm m3

t t

3 3T T T Tm m m3 3

t t t

I I I- V sin t + V sin t sin t

2 4V 48V

I 3I I IV - V sin t - V sin t

2 4 48V 4V 4 48V

C

C

I

I

2

log m

2 2

t m

VTHD 20

48V 3V

THD (dB)2.5 -13.4049

1 -33.0643

0.5 -45.5292

0.25 -57.6732

0.1 -73.6194

0.05 -85.6647

0.025 -97.7069

0.01 -113.625

m tV /VThus:

log

2

t

m

VTHD 20 48 3

V

or, equivalently

Comparison of Distortion in BJT and MOSFET Pairs

Vd=Vmsin(ωt) IT

Q1 Q

2

IC1 I

C2

V1

V2

VE

log

2

t

m

VTHD 20 48 3

V

2

log EB1

m

VTHD 20 32 3

V

IT

M1

M2

ID1 I

D2

V1

V2

VS

THD (dB) THD (dB)2.5 -13.4049 2.5 -6.52672

1 -33.0643 1 -29.248

0.5 -45.5292 0.5 -41.9382

0.25 -57.6732 0.25 -54.1344

0.1 -73.6194 0.1 -70.0949

0.05 -85.6647 0.05 -82.1422

0.025 -97.7069 0.025 -94.1849

0.01 -113.625 0.01 -110.103

m tV /V m EB1V /V

-0.2 -0.1 0 0.1 0.2

Vd

I

IC1

1% linear = .56Vt

2Vt

Linearity and Signal Swing Comparison of

Bipolar/MOS Differential Pair

IT

Q1 Q

2

IC1 I

C2

V1

V2

VE

Vd

VEB1

IT

I

EB1V2

ID1

1% Linear =

0.3VEB1

IT

M1

M2

ID1 I

D2

V1

V2

VS

Applications as a programmable OTA

gm

IABC

The current-dependence of the gm of the differential pair is often used to

program the transconductance of an OTA with the tail bias current IABC

MOS

m ABC OXW

g = I 2uCL

m OX EBW

g =uC VL

Two decade change in current for every

decade change in gm

One decade decrease in signal swing for

every decade decrease in gm

BJT

ABCm

t

Ig =

V

One decade change in current for every

decade change in gm

No change in signal swing when gm

Is changed

Limited gm adjustment possibility Large gm adjustment possible

Signal Swing and Linearity Summary

• Signal swing of MOSFET can be rather large if VEB is large but this limits gain

• Signal swing of MOSFET degrades significantly if VEB is changed for fixed W/L

• Bipolar swing is very small but independent of gm

• Multiple-decade adjustment of bipolar gm is practical

• Even though bipolar input swing is small, since gain is often very large, this small swing does usually not limit performance in feedback applications

Linearity of Common-Source Amplifier

VSS

VDD

ViS

VOUT

IB

For convenience, will consider situation where current source biasing

Is ideal


VSS

VDD

ViS

VOUT

IB

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5

VDS

ID(m

A)

VGS=1V

VGS=1.5V

VGS=2V

VGS=2.5V

0 1 2 3 4 5

VDS

ID(m

A)

VGS=1V

VGS=1.01V

VGS=0.99VIB


VSS

VDD

ViS

VOUT

IB

2OX B iS SS T OUT SS

μC WI = -V -V 1+λ V -V

2LV

2 B iS EB OS OQ SSI = β -V 1+λ +V -V V V

21

1

B

2 iSEB

EB OS SS OQ

I

βVV

= V -V -

V

V


VSS

VDD

ViS

VOUT

IB

21

1

B

2 iSEB

EB OS SS OQ

I

βVV

= V -V

V

V

2

1

1

iSB

EB2EB

OS SS OQ

IV

βV

V -V

V

V

21

1 2 iS iSB OS SS OQ 2

EB EBEB

I V -V

V VλβV

V VV

21

1 2 iS iSB B OS SS OQ 2 2

EB EBEB EB

I I V -V

λ V VβV λβV

V VV


VSS

VDD

ViS

VOUT

IB2

11 2 iS iSB B

OS SS OQ 2 2EB EBEB EB

I I V -V

λ V VβV λβV

V VV

21

2 iS iS OS

EB EB

-λV λ V

V VV

1 2 OS iS iS

EB EB

2 -

λV 2V

V V V

Is this a linear or nonlinear relationship?


VSS

VDD

ViS

VOUT

IB1 2

OS iS iSEB EB

2

λV 2V

V V V


when ViS = -VEB (the minimum value of ViS to maintain saturation operation)

the error in VOS will be VEB/2 which is -50% !


-40

-30

-20

-10

0

10

20

30

-1 -0.5 0 0.5 1

VEB= 1V

λ=0.1

Fit Line

-40

-30

-20

-10

0

10

20

30

-1 -0.5 0 0.5 1


VSS

VDD

ViS

VOUT

IB1 2

OS iS iSEB EB

2 -

λV 2V

V V V


Over what output voltage range are we interested?

Note this is a high gain amplifier

VEB= 1V

λ=0.1

-40

-30

-20

-10

0

10

20

30

-1 -0.5 0 0.5 1


VSS

VDD

ViS

VOUT

IB1 2

OS iS iSEB EB

2 -

λV 2V

V V V


Linearity is reasonably good over practical input range

Practical input range is much less than VEB

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-0.1 -0.05 0 0.05 0.1

VEB= 1V

λ=0.1

Fit Line


VSS

VDD

ViS

VOUT

IB1 2

OS iS iSEB EB

2 -

λV 2V

V V V


-5

-4

-3

-2

-1

0

1

2

3

4

5

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025

VEB= 1V

λ=0.01Fit Line

Can’t see nonlinearity in this plot

-5

-4

-3

-2

-1

0

1

2

3

4

5

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025


VSS

VDD

ViS

VOUT

IB1 2

OS iS iSEB EB

2 -

λV 2V

V V V


VEB= 1V

λ=0.01

FIT oSε = -V V

1 2 iS2

EB

λV

V

FIT iSEB

2 -

λVV V

ε

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025

VIN


VSS

VDD

ViS

VOUT

IB1 2

OS iS iSEB EB

2 -

λV 2V

V V V


VEB= 1V

λ=0.01

1 2 iS2

EB

λV

V

ε

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025

VIN


VSS

VDD

ViS

VOUT

IB1 2

OS iS iSEB EB

2 -

λV 2V

V V V


VEB= 1V

λ=0.01

1

100% 100%

2 iS2

EB PCT iS

FIT EB iS

EB

λVε 100%

2 2V

λV

V

VV V

PCT OS

λ 100%

4

V

-1.5

-1

-0.5

0

0.5

1

1.5

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025

εPCT

VIN


VSS

VDD

ViS

VOUT

IB1 2

OS iS iSEB EB

2 -

λV 2V

V V V


VEB= 1V

λ=0.01

100% PCT iS

EB

2V

V

PCT OS

λ 100%

4

V

or, in terms of VOS,

1% deviation for this example occurs at OS

4 0.01 4V

λ V


VSS

VDD

ViS

VOUT

IB

Is this common-source amplifier linear or nonlinear?

VSS

VDD

ViS

VOUT

IB

CL

IOUT

High-Gain Amplifier Transconductance Amplifier

The transconductance amplifier driving a load CL is performing as an integrator

gm

CL

ViS VOS

Integrators often used in filters where |VOS| is comparable to |ViS|


VSS

VDD

ViS

VOUT

IB


VSS

VDD

ViS

VOUT

IB

CL

IOUT


OUT B DI = I -I

2OUT B iS EB OS OQ SSI = I β +V 1+λ V +V -V V

2OUT EB OS

2B EB OQ S OQ SS iS iSSI = I β V 1+λ V β +2 V 1+λ V V-V +V -

V V

2OUT EB iS iSI β +2 V V V

2BOUT iS iS

EB EB

2I 1I -

V 2V

V V

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1

VIN

As an OTA



At VIS=-VEB, the error in IOUT will be -50% !

VSS

VDD

ViS

VOUT

IB

CL

IOUT

VEB= 1V

λ=0.01

2BOUT iS iS

EB EB

2I 1I -

V 2V

V V

OUT

m

I

g

B m

EB

2Ig =

V


VSS

VDD

ViS

VOUT

IB

Is this common-source amplifier linear?

VSS

VDD

ViS

VOUT

IB

CL

IOUT

M1M2


• Reasonably linear if used in high-gain applications and

VEB is large (e.g. if AV=gm/go=2/((λVEB)=100 and Vo=1V, Vin=10mV)

• Highly nonlinear when used in low-gain applications

Linearity of Common-Emitter Amplifier

Is this common-emitter amplifier linear?


• Very linear if used in high-gain applications

(e.g. if AV=gm/g0=VAF/Vt=4000 and Vo=1V, Vin=250uV)

• Highly nonlinear when used in low-gain applications

VSS

ViS

VOUT

IB

CL

IOUT

Q1

VSS

ViS

VOUT

IB

CL

IOUT

Q1 RL

• Systematic Offset Voltage

• Random Offset Voltage

Offset Voltage

VICQ

VOUT

Definition: The output offset voltage is the difference between the desired

output and the actual output when Vid=0 and Vic is the quiescent common-

mode input voltage.

OUTOFF OUT OUTDESV = V - V

Note: VOUTOFF is dependent upon VICQ although this dependence is

usually quite weak and often not specified

Two types of offset voltage:

Offset Voltage

Definition: The input-referred offset voltage is the differential dc input voltage

that must be applied to obtain the desired output when Vic is the quiescent

common-mode input voltage.

VICQ

VOUT

VOFF

Note: VOFF is usually related to the output offset voltage by the expression

OUTOFFOFF

C

VV =

ANote: VOFF is dependent upon VICQ although this dependence is

usually quite weak and often not specified

• Systematic Offset Voltage

• Random Offset Voltage

Offset Voltage

VICQ

VOUT

Two types of offset voltage:

After fabrication it is impossible (difficult) to distinguish between the

systematic offset and the random offset in any individual op amp

Measurements of offset voltages for a large number of devices will

provide mechanism for identifying systematic offset and statistical

characteristics of the random offset voltage

Systematic Offset Voltage

Offset voltage that is present if all device and model parameters

assume their nominal value

Easy to simulate the systematic offset voltage

Almost always the designer’s responsibility to make systematic

offset voltage very small

Generally easy to make the systematic offset voltage small

Random Offset Voltage

Due to random variations in process parameters and device dimensions

Random offset is actually a random variable at the design level but

deterministic after fabrication in any specific device

Distribution nearly Gaussian

Has zero mean

Characterized by its standard deviation or variance

Often strongly layout dependent

Due to both local random variations and correlated gradient effects

Will consider both effects separately

Gradient effects usually dominate if not managed

Good methods exist for driving gradient effects to small levels and will be

discussed later

In what follows it will be assumed that gradient effects have been managed

End of Lecture 19

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