# 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

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

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

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

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:

${\displaystyle G={\frac {2V}{j\cdot 1V}}=-2j.}$

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