In electronics, complex gain is the effect that 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.
Considering the general LTI system[1]
P(D)x=Q(D)f(t)
where
f(t)
P(D),Q(D)
P(s) ≠ 0
f(r)=F0\cos(\omegat)
xp(t)=\operatorname{Re} (F0
Q(i\omega) | |
P(i\omega) |
ei\omega).
Consider the following concepts used in physics and signal processing mainly.
\bullet
F0
\bullet
\omega
\bullet
A=F0|Q(i\omega)/P(i\omega)|
\bullet
g(\omega)=|Q(i\omega)/P(i\omega)|
\bullet
\phi=-\operatorname{Arg}(Q(i\omega)/P(i\omega))
\bullet
\phi/\omega
\bullet
Q(i\omega)/P(i\omega)
Suppose a circuit has an input voltage described by the equation
Vi(t)=1 V ⋅ \sin(\omega ⋅ t)
where ω equals 2π×100 Hz, i.e., the input signal is a 100 Hz 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
Vo(t)=2 V ⋅ \cos(\omega ⋅ t)
In complex notation, these signals can be described as, for this frequency, j·1 V and 2 V, respectively.
The complex gain G of this circuit is then computed by dividing output by input:
G=
2 V | |
j ⋅ 1 V |
=-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).