# Complex gain

In electronics, Complex gain is the effect circuitry has on the amplitude and phase of a sine wave signal. The term complex is used because mathematically this effect can be expressed as a complex number.

## Example

Suppose a circuit has an input voltage described by the equation

$V_{i}(t) = 1V \cdot \sin (\omega \cdot t)$

where ω equals 2π×100Hz, i.e., the input signal is a 100Hz sine wave with an amplitude of 1 Volt.

If the circuit is such that for this frequency it doubles the signal's amplitude and causes a 90 degrees forward phase shift, then its output signal can be described by

$V_{i}(t) = 2V \cdot \cos (\omega \cdot t)$

In complex notation, these signals can be described as, for this frequency, j·1V and 2V, respectively.

The complex gain G of this circuit is then computed by dividing output by input:

$G = \frac {2V}{j\cdot1V} = -2j.$

This (unitless) complex number incorporates both the magnitude of the change in amplitude (as the absolute value) and the phase change (as the argument).