Complex Impedance

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.