1
Differential Amplifiers
•Single Ended and Differential Operation•Basic Differential Pair•Common-Mode Response•Differential Pair with MOS loads
Hassan AboushadyUniversity of Paris VI
• B. Razavi, “Design of Analog CMOS Integrated Circuits”,McGraw-Hill, 2001.
References
H. Aboushady University of Paris VI
2
Differential Amplifiers
•Single Ended and Differential Operation•Basic Differential Pair•Common-Mode Response•Differential Pair with MOS loads
Hassan AboushadyUniversity of Paris VI
Single Ended and Differential Operation
H. Aboushady University of Paris VI
• Single Ended Signal:- Measured with respect to a fixed potential, usually ground.
• Differential Signal:- Measured between 2 nodes that have equal and oppositeexcursions around a fixed potential.- The center potential is called “Common Mode” (CM).
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Rejection of Common Mode Noise
H. Aboushady University of Paris VI
• Single Ended Signal:- Due to capacitive coupling,transitions on the clock linecorrupt the signal on L1 .
• Differential Signal:- If the clock line is placedmidway, the transitions disturbthe differential signals by equalamounts, leaving the differenceintact.
Rejection of Power Supply Noise
H. Aboushady University of Paris VI
• Maximum Output Swing: • Maximum Output Swing:
)(max THGSDDout VVVV −−= [ ])(2maxmax THGSDDYX VVVVV −−=−
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Differential Pair
H. Aboushady University of Paris VI
21
21
21
mm
DD
inin
ggIIVVif
≠⇒≠⇒≠
21 DDSS III +=
Differential pair minimaldependence on input CM level.
Differential circuit sensitiveto the input CM level.
221
21
SSDD
inin
III
VVif
==⇒
=
2SS
DDDIRVCMOutput −=
Differential Pair: Qualitative Analysis
H. Aboushady University of Paris VI
21 inin VV << M1 OFF, M2 ON
SSD II =2 DDout VV =1
DSSDDout RIVV −=2
21 inin VV = M1 ON, M2 ON
221SS
DDIII == D
SSDDoutout RIVVV
221 −==
21 inin VV >> M1 ON, M2 OFF
SSD II =1
DDout VV =2
DSSDDout RIVV −=1
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Differential Pair: Qualitative Analysis
H. Aboushady University of Paris VI
• Maximum and minimum levels are well-defined andindependent of the input CM: VDD and VDD - RD ISS
• The small signal gain (the slope of Vout1-Vout2 vs Vin1-Vin2)is maximum for Vin1=Vin2 (equilibrium).
Differential Pair: Common-Mode Behavior
H. Aboushady University of Paris VI
For proper operation:• M3 in saturation
• M1 & M2 in saturation
CMininin VVV ,21 ==To study Common-Mode
)( 331, THGSGSCMin VVVV −+≥
THSS
DDDCMin VIRVV +−≤2,
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Differential Pair: Output Voltage Swing
H. Aboushady University of Paris VI
For M1 & M2 in saturation:
THCMinout
THPCMinPout
VVVVVVVV
−≥−−≥−
,
,
THCMinoutDD VVVV −≥≥ ,
Output Voltage Swing:
To increase output swing, we choose a low CMinV ,
Differential Pair: Quantitative Analysis
H. Aboushady University of Paris VI
2211 GSinGSinP VVVVV −=−=
2121 GSGSinin VVVV −=− 1
LWC
IVVoxn
DTHGS
µ
2)( 2 =−
TH
oxn
DGS V
LWC
IV +=µ
22
Assuming M1 & M2 in saturation:
LWC
I
LWC
IVVoxn
D
oxn
Dinin
µµ21
2122 −=−
From eq. 1 & 2 :
)2(2)( 212
21 DDSS
oxn
inin III
LWC
VV −=−µ
Squaring the 2 sides, and since: 21 DDSS III +=
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Differential Pair: Quantitative Analysis
H. Aboushady University of Paris VI
212
21 2)(2 DDSSininoxn IIIVVLWC −=−−µ
221
2
221
22121
)(
)()(4
DDSS
DDDDDD
IIIIIIIII
−−=
−−+=Squaring the 2 sides, and since:
Previous equation can be written:
221
421
22
21 )()(41)( ininoxnSSininoxnDD VV
LWCIVV
LWCII −+−⎟
⎠⎞
⎜⎝⎛−=− µµ
2212121 )(4)(
2 inin
oxn
SSinin
oxnDD VV
LWC
IVVLWCII −−−=−
µ
µ
We arrive at:
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Differential Pair: Quantitative Analysis
H. Aboushady University of Paris VI
2
2
/4
2/
4
2in
oxn
SS
inoxn
SS
oxn
in
Dm
VLWC
I
VLWC
I
LWC
VIG
∆−
∆−=
∆∂∆∂=
µ
µµ
SSoxnmin ILWCGVFor µ==∆ ,0
Deriving eq. 3 with respect to
2121 andLet DDDininin IIIVVV −=∆−=∆
inV∆
221121 DDDDDDDDoutout RIVRIVVV −−−=−Since:
Dinmout RVGV ∆=∆DDout RIV ∆=∆
DSSoxnDmin
outv RI
LWCRG
VVA µ==
∆∆=The small signal
differential voltage gain:
4
8
Drain Currents and Overall Transconductance
H. Aboushady University of Paris VI
2
2
/4
2/
4
2in
oxn
SS
inoxn
SS
oxnm
VLWC
I
VLWC
I
LWCG
∆−
∆−=
µ
µµ
LWC
IVoxn
SSin
µ
21 =∆ when is 11 SSDin IIV =∆ 111 THGSin VVV −=∆
2212121 )(4)(
2 inin
oxn
SSinin
oxnDD VV
LWC
IVVLWCII −−−=−
µ
µ3
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∆ID vs ∆VD
H. Aboushady University of Paris VI
LWC
IVoxn
SSin
µ
21 =∆
↑↑SSI
↑↑LWPlot the input-output characteristics
of a differential pair as the devicewidth and the tail current vary:
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Differential Pair: Small Signal Gain
H. Aboushady University of Paris VI
DSSoxnDmin
outv RI
LWCRG
VVA µ==
∆∆=
221SS
DDIII ==
In equilibrium, we have
Dmv RgA = Where gm is thetransconductance of M1 & M2.
Calculating Small Signal Gain by Superposition
H. Aboushady University of Paris VI
Set Vin2 =0M1 forms a common sourcestage with a degenerationresistance
2/1 mS gR =
21
1
1 /1 mm
Dm
in
X
ggRg
VV
+−=
Sm
Dm
in
Xv Rg
RgVVA
1
1
1 1+−==
Neglecting channel lengthmodulation and body effect
21
111mm
D
in
X
gg
RVV
+−=
5
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Calculating Small Signal Gain by Superposition
H. Aboushady University of Paris VI
Replacing M1 by its Thévenin equivalent:
21
111mm
D
in
Y
gg
RVV
+=
1inT VV = 1/1 mT gR = 22 /1 min gR =
1
21
112)(
1in
mm
DVtodueYX V
gg
RVVin
+
−=−
6
From eq. 5 & 6, we get:
Calculating Small Signal Gain by Superposition
H. Aboushady University of Paris VI
1
21
112)(
1in
mm
DVtodueYX V
gg
RVVin
+
−=−
mmm ggg == 21
11
)( inDmVtodueYX VRgVVin
−=−
22
)( inDmVtodueYX VRgVVin
=−
Dminin
totalYX RgVVVV
−=−
−
21
)(
Similarly we can say that:
Since:
The small signaldifferential voltage gain:
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The concept of Half Circuit
H. Aboushady University of Paris VI
If a fully symmetric differential pair senses differentialinputs then the concept of half circuit can be applied.
• A differential change in the inputs Vin1 and Vin2 isabsorbed by V1 and V2 leaving VP constant
Application of The Half Circuit Concept
H. Aboushady University of Paris VI
Since VP experiences no change, node P can be considered“ac ground” and the circuit can be decomposed into twoseparate halves
Dmin
X RgVV −=
1Dm
in
Y RgVV −=
2
Two common source amplifiers:
Dminin
YX RgVVVV −=
−−
21
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The Half Circuit Concept : Example
H. Aboushady University of Paris VI
Taking into account the output resistance(channel length modulation)
( )11
// ODmin
X rRgVV −= ( )2
2
// ODmin
Y rRgVV −=
Two common source amplifiers:
( )ODminin
YX rRgVVVV //
21
−=−−
Arbitrary Inputs to a Differential Pair
H. Aboushady University of Paris VI
Conversion of arbitrary inputs to differential and common-modecomponents:
DifferentialCommon
Mode
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Arbitrary Inputs to a Differential Pair: Example
H. Aboushady University of Paris VI
Calculate VX and VY if Vin1 =Vin2 and λ=0
( ) ( )2
// 211
ininODmX
VVrRgV −−=For differential mode operation:
( ) ( )21// ininODmYX VVrRgVV −−=−
( ) ( )2
// 122
ininODmY
VVrRgV −−=
For common mode operation:2/21 SSDD III ==
Assuming fully symmetric circuitand Ideal Current Source:•ID1 and ID2 independent of•VX and VY independent of
inCMV ,
inCMV ,
The Differential pair circuit:Amplifies , Eliminates the effect of inCMV ,21 inin VV −
Common Mode Response: Non-Ideal Current Source
H. Aboushady University of Paris VI
Assuming fully symmetric circuitwith finite output impedancecurrent source, RSS :
SSm
D
CMin
outCMv Rg
RVVA
+−==
)2/(12/
,,
Equivalent circuit:• Degenerated Common Source
gm the transconductance of 1 transistor.
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Common Mode Response: RD Mismatch Effect
H. Aboushady University of Paris VI
SSm
DCMinX Rg
RVV+
∆−=∆)2/(1,
Assuming M1 & M2 identical:
Common mode to differential conversion
SSm
DDCMinY Rg
RRVV+
∆+∆−=∆)2/(1,
Effect of CM noise in thepresence of resistor noise
Common Mode Response: M1-M2 Mismatch Effect
H. Aboushady University of Paris VI
)( ,11 PCMinmD VVgI −=
)( ,22 PCMinmD VVgI −=
( ) SSPCMinmmP RVVggV )( ,22 −+=( )
( ) CMinSSmm
SSmmP V
RggRggV ,
22
22
1+++=
21 mm gg ≠
( ) DPCMinmX RVVgV −−= ,1 ( ) CMinDSSmm
mX VR
RgggV ,
21
1
1++−=
( ) DPCMinmY RVVgV −−= ,2 ( ) CMinDSSmm
mY VR
RgggV ,
21
2
1++−=
( )( ) 121
,21
++−
=−SSmm
CMinDmmYX Rgg
VRggVV
( ) 121, ++∆=−=−
SSmm
Dm
CMin
YXDMCM Rgg
RgV
VVACM to DM Conversion Gain:
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Common Mode Rejection Ratio
H. Aboushady University of Paris VI
DMCM
DM
AACMRR
−
=: Differential Mode GainDMA
: Common-Mode to Differential Mode Gain
DMCMA −
Cascode Differential Pair
H. Aboushady University of Paris VI
)//( OPONmNv rrgA =
Low gain 10 to 20.
To increase the gain:Cascode Differential Pair
)//( 7551331 OOmOOmmv rrgrrggA =
Current Source Load: