In sinusoidal steady state, resistors, capacitors and inductors can all be described by one phasor relation:

$$\underline V=Z\underline I.$$

The complex quantity $Z$ is the impedance.

Resistance and reactance

Write

$$Z=R+iX,$$

where $R$ is resistance and $X$ is reactance.

For the basic ideal elements,

$$Z_R=R,$$

$$Z_C=\frac1{i\omega C},$$

$$Z_L=i\omega L.$$

Magnitude and phase

The impedance magnitude is

$$|Z|=\sqrt{R^2+X^2},$$

and its argument gives the phase by which voltage leads current:

$$\angle Z=\angle\underline V-\angle\underline I.$$

Positive reactance is inductive and negative reactance capacitive.

Combining impedances

Series impedances add directly:

$$Z_{\mathrm{eq}}=Z_1+Z_2+\cdots.$$

Parallel combinations obey

$$\frac1{Z_{\mathrm{eq}}}=\frac1{Z_1}+\frac1{Z_2}+\cdots.$$

Impedance turns differential element behavior into an algebraic generalization of resistance at one chosen frequency.