Gain of an amplifier is defined as V_{OUT}/V_{IN}. For
the special case of a differential amplifier, the input V_{IN}
is the difference between its two input terminals, which is equal to (V_{1}-V_{2})
as shown in the following diagram.

So the gain of this differential amplifier is

Instead of applying superposition theorem with V_{1} and V_{2
}separately,
a better way is to first combined V_{1} and V_{2
}in a
different format, viz. (V_{1}-V_{2}). This is known as
the differential mode input - V_{d}. Associated
with this differential mode component will be the common mode input - V_{cm}.,
which is equal to the average value of V_{1} and V_{2}.

Common mode component : V_{cm} = (V_{1}+V_{2})/2

By using these alternate representation of the input components (V_{d
}and
V_{cm}) instead of the original components
(V_{1} and V_{2}), we can re-express eqn (2) in terms of
V_{d} and V_{cm}
as follows.

Since
V_{d} = V_{1 }- V_{2 }
---- (4)

Substitute eqns (5) & (6) into eqn (2) :

**
V _{OUT}** = 1/2[R

= V_{OUT}/V_{d}

= 1/2[R_{3}/(R_{1}+R_{3})]
[(R_{4 }+ R_{2})/R_{2 }+ R_{4}/R_{2}]

**
A _{d} = 1/2[R_{3}/(R_{1}+R_{3})]
[(R_{4 }+ R_{2})/R_{2 }+ R_{4}/R_{2}]**

As there is another component in V_{OUT} due to the common-mode
component V_{cm} of the input, we define another
gain for the differential amplifier, the **Common Mode Gain **(A_{cm}=V_{OUT}/
V_{cm}). From eqn (7), this is

To eliminate the effect of the common mode component, we can either

(i) make the input common mode component equal to zero,
i.e. make V_{2} = -V_{1}

such that the average value of the two input signals equal to zero

or

(ii) can be achieved __theoretically__ by making R_{1} =
R_{2} and R_{3} = R_{4}. However, this is not feasible
in practice due to the tolerance of the resistors used.

Because of this imperfection, a figure of merit used to describe differential
amplifier is the **Common Mode Rejection Ratio (CMRR)**, which is defined
as

**
CMRR = 20 log (A _{d}/A_{cm})**

For a __perfect__ differential amplifier, the CMRR is equal to ¥,
as A_{cm} is zero.

In practice, a CMRR in excess of 80dB to 100dB will be needed for high
accuracy measuring system (e.g. a microcomputer data acquisition system).
This is very difficult to achieve if the differential amplifier uses discrete
resistors for R_{1} to R_{4}.