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TRANSCRIPT
EE 435
Lecture 22
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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evie
w f
rom
last
lectu
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How linear is the amplifier ?
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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R
evie
w f
rom
last
lectu
re .•
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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
TF IT
td in t 2 Vm
hV
IT
Q1 Q
2
IC1 I
C2
V1
V2
VE
-0.1 -0.05 0 0.05 0.1
x% Deviation
I
Vd
Signal Swing and Linearity of Bipolar Differential Pair
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
Signal Swing and Linearity of Bipolar Differential Pair
IT
Q1 Q
2
IC1 I
C2
V1
V2
VE
How linear is the amplifier ?
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 Vln
I
1
d
t
d
t
V
V T C1
C1
V
V
C1 T
I -Ie
I
I I 1+e
How linear is the amplifier ?
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
How linear is the amplifier ?
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
4
d d d
t t t
V 2V V
V V V+e e 1+e
How linear is the amplifier ?
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 4
d 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
How linear is the amplifier ?
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 48V CI
3 1
34 4
3sin t sin t sin t
How linear is the amplifier ?
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 (single transistor) is often used to
program the transconductance of an OTA with the tail bias current IABC
MOS
m ABC OXW
g = I μCL
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 =
2V
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
Linearity of Common-Source Amplifier
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
Linearity of Common-Source Amplifier
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
Linearity of Common-Source Amplifier
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
Linearity of Common-Source Amplifier
VSS
VDD
ViS
VOUT
IB2
11 2
iS iSB B OS SS OQ 2 2
EB 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?
2
B EBI β V
but 11 0
BSS OQ 2
EB
I V -V
λ βV
Thus
Linearity of Common-Source Amplifier
VSS
VDD
ViS
VOUT
IB1 2
OS iS iSEB EB
2
λV 2V
V V V
Is this a linear or nonlinear relationship?
when ViS = -VEB (the minimum value of ViS to maintain saturation operation)
the error in VOS will be VEB/2 which is -50% !
Is this a linear or nonlinear relationship?
-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
Linearity of Common-Source Amplifier
VSS
VDD
ViS
VOUT
IB1 2
OS iS iSEB EB
2 -
λV 2V
V V V
Is this a linear or nonlinear relationship?
Over what output voltage range are we interested?
Note this is a high gain amplifier
VEB= 1V
λ=0.1
VOS
VIS
-40
-30
-20
-10
0
10
20
30
-1 -0.5 0 0.5 1
Linearity of Common-Source Amplifier
VSS
VDD
ViS
VOUT
IB1 2
OS iS iSEB EB
2 -
λV 2V
V V V
Is this a linear or nonlinear relationship?
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 VOS
VIS
Linearity of Common-Source Amplifier
VSS
VDD
ViS
VOUT
IB1 2
OS iS iSEB EB
2 -
λV 2V
V V V
Is this a linear or nonlinear relationship?
-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
VIS
VOS
-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
Linearity of Common-Source Amplifier
VSS
VDD
ViS
VOUT
IB1 2
OS iS iSEB EB
2 -
λV 2V
V V V
Is this a linear or nonlinear relationship?
VEB= 1V
λ=0.01
FIT oSε = -V V
1 2 iS2
EB
λV
V
FIT iSEB
2 -
λVV V
VOS
VIS
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
Linearity of Common-Source Amplifier
VSS
VDD
ViS
VOUT
IB1 2
OS iS iSEB EB
2 -
λV 2V
V V V
Is this a linear or nonlinear relationship?
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
Linearity of Common-Source Amplifier
VSS
VDD
ViS
VOUT
IB1 2
OS iS iSEB EB
2 -
λV 2V
V V V
Is this a linear or nonlinear relationship?
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
Linearity of Common-Source Amplifier
VSS
VDD
ViS
VOUT
IB1 2
OS iS iSEB EB
2 -
λV 2V
V V V
Is this a linear or nonlinear relationship?
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
Linearity of Common-Source Amplifier
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|
Linearity of Common-Source Amplifier
VSS
VDD
ViS
VOUT
IB
Is this a linear or nonlinear relationship?
VSS
VDD
ViS
VOUT
IB
CL
IOUT
High-Gain Amplifier Transconductance Amplifier
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
Linearity of Common-Source Amplifier
Is this a linear or nonlinear relationship?
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
Linearity of Common-Source Amplifier
VSS
VDD
ViS
VOUT
IB
Is this common-source amplifier linear?
VSS
VDD
ViS
VOUT
IB
CL
IOUT
M1M2
High-Gain Amplifier Transconductance Amplifier
• 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?
High-Gain Amplifier Transconductance Amplifier
• 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 22