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Differential and Common-

Mode GainRecall that in a previous handout, we analyed thiscircuit!

and "ound that the output is related to the inputs as!

#2 22 1

1 3 # 1

1out

RR Rv v v

R R R R

= + +

R2

R1

$

%v

out

ideal

v1

v2

R3

R#

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1 2+ , and + ,v t v t i" you now + ,dv t and + ,c(v t , you now 1+ ,v t and

2+ ,v t .

oreover, we can e'press the ehavior o" our differential

amplifierin ter(s o" + ,dv t and + ,c(v t . nsertin) these

"unctions into the e'pression o" the a(pli"ier output + ,ov t , we

"ind!

#2 22 1

1 3 # 1

#2 2

1 3 # 1

#2 2

1 3 # 1

#2 2

1 3 #

1

+ , + ,1 + , + ,2 2

11 + ,

2

1

out

d dc( c(

d

RR Rv v v

R R R R

v t v t RR Rv t v t R R R R

RR Rv t

R R R R

RR R

R R R R

= + +

= + + +

= + + +

+ + +

( ) ( )

1

1 # 2 3 2 # 1 # 2 3

1 3 # 1 3 #

+ ,

2 + , + ,2

c(

c(d

v t

R R R R R R R R R R v t v t R R R R R R

+ + = ++ +

&hus, we now have an e'pression "or the open-circuitoutput in

the "or(!

+ , + , + ,out c( c( d d

v t * v t * v t = +

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where!

di""erential )ain

co((on%(ode )ain

d

c(

*

*

B

B

ote that each o" these )ains are open-circuit voltage)ains.

* *n idealdi""erential a(pli"ier has zeroco((on%(ode

)ain +i.e., *c( 40

*n other words, the output o" an ideal di""erential

a(pli"ier is independento" the common-mode+i.e.,

avera)e o" the two input si)nals.

*e re"er to this characteristic as common-mode

suppression.

&ypically, real di""erential a(pli"iers e'hiit small, ut non%

ero co((on (ode )ain.

&he Common-Mode Rejection Ratio+RR is there"ore used

to indicate the qualityo" a di""erential a(pli"ier!2

10 2RR 10 lo) + d

c(

*d8

*=

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ote the RR o" a gooddi""erential a(pli"ier is very large

+e.)., #0 d.

:or our eample circuit, we "ind that the di""erential and

co((on%(ode )ain are!

( )

( )

1 2 3 2# #

1 3 #

1 2 3#

1 3 #

2

2d

c(

RR R R R R *

R R R

RR R R *

R R R

+ +=

+

=

+

&he ratio o" these two )ains is thus!

( ) ( )

1

1 2 3 2 1 2 3# # #

1 3 1 3# #

1 2 3 2# #

1 2 3#

2

2

21

2

d

c(

* RR R R R R RR R R

* R R R R R R

RR R R R R

RR R R

+ +

= + +

+ +=

and there"ore RR is!

2

1 2 3 2# #

101 2 3#

21RR 10 lo) +

2

RR R R R R d8

RR R R

+ +=

t is evident that "or thise'a(ple, the co((on%(ode )ain *c(is

minimized, and thus the RR is maimized, when!

1 2 3#RR R R =

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so that 1 2 3# 0RR R R = .