This post assumes a working knowledge of elementary circuit theory as well as Fourier Analysis.

It seems to me that often in an introductory circuits course complex impedance is a major concept but its mathematical basis is never taught. Thus I'd like to take a moment to discuss some of the mathematics surrounding this concept.

The resistor is a purely resistive elementary circuit element possessing a time independent current voltage characteristic described by Ohm's Law.

\[ V(t) = I(t) R \]The capacitor is a purely reactive elementary circuit element that stores energy in its electric field.

\[ I(t) = C \frac{dV(t)}{dt} \]The inductor is a purely reactive elementary circuit element that stores energy in its magnetic field.

\[ V(t) = L \frac{dI(t)}{dt} \]What we aim to do is to derive a linear current voltage characteristic for the two reactive circuit elements by reducing the differential equations to algebraic equations. Using the Fourier Transform we can compute a complex quantity in the frequency domain that is analogous to resistance in the time domain.