Editing Differential amplifier (section)

== Mathematics of the amplifier ==
: <math>V_\text{out} = A_\text{d}(V_\text{in}^+ - V_\text{in}^-),</math>

where <math>V_\text{in}^+</math> and <math>V_\text{in}^-</math> are the input voltages, and <math>A_\text{d}</math> is the differential gain.

In practice, however, the gain is not quite equal for the two inputs. This means, for instance, that if <math>V_\text{in}^+</math> and <math>V_\text{in}^-</math> are equal, the output will not be zero, as it would be in the ideal case. A more realistic expression for the output of a differential amplifier thus includes a second term:

: <math>V_\text{out} = A_\text{d}(V_\text{in}^+ - V_\text{in}^-) + A_\text{c} \frac{V_\text{in}^+ + V_\text{in}^-}{2},</math>

where <math>A_\text{c}</math> is called the common-mode gain of the amplifier.

As differential amplifiers are often used to null out noise or bias voltages that appear at both inputs, a low common-mode gain is usually desired.

The [[common-mode rejection ratio]] (CMRR), usually defined as the ratio between differential-mode gain and common-mode gain, indicates the ability of the amplifier to accurately cancel voltages that are common to both inputs. The common-mode rejection ratio is defined as

: <math>\text{CMRR} = 10 \log_{10} \left(\frac{A_\text{d}}{A_\text{c}}\right)^2 = 20 \log_{10} \frac{A_\text{d}}{|A_\text{c}|}.</math>

In a perfectly symmetric differential amplifier, <math>A_\text{c}</math> is zero, and the CMRR is infinite. Note that a differential amplifier is a more general form of amplifier than one with a single input; by grounding one input of a differential amplifier, a single-ended amplifier results.