Complex gain

Complex gain is a term used in electronics to describe 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

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

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

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

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:

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