Complex Impedance

The circuitous impedance ZL (in ohms) of an inductor with inductance L (in henries) is

Z_L \ = \ Ls

The circuitous abundance s is a circuitous number,

s \ = \ \sigma + j \omega

where

j represents the abstract unit:

j2 = − 1

\sigma \ is the exponential adulteration connected (in radians per second), and

\omega \ is the angular abundance (in radians per second).

edit Eigenfunctions

The complex-valued eigenfunctions of ANY beeline time-invariant (LTI) arrangement are of the afterward forms:

V(t) \ = \ \mathbf{A}e^{st} \ = \ \mathbf{A}e^{(\sigma + j \omega) t} \ , or absolution \mathbf{A} \ = \ A e^{j \phi} and rewriting; \ = \ A e^{j \phi}e^{(\sigma + j \omega) t} , and accession agreement is \ = \ A e^{\sigma t}e^{j ( \omega t + \phi )}

From Euler's formula, the real-part of these eigenfunctions are exponentially-decaying sinusoids:

v(t) \ = \ \mathrm{Re} \left \{ V(t) \right \} \ = \ A e^{\sigma t} \cos(\omega t + \phi )

edit Sinusoidal Abiding State

Sinusoidal abiding accompaniment is a appropriate case in which the ascribe voltage consists of a authentic sinusoid (with no exponential decay). As a result,

\sigma \ = \ 0

and the appraisal of s becomes

s \ = \ j \omega

edit Series circuit