Once every sinusoidal voltage and current is represented by a phasor and every linear element by an impedance, an AC circuit can be analyzed with the same network laws used for resistive circuits.

Ohm's law in phasor form

For each element or equivalent network,

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

Because $Z$ is complex, this one equation contains both amplitude and phase information.

Kirchhoff's laws

At each node,

$$\sum\underline I=0,$$

and around each loop,

$$\sum\underline V=0.$$

These are complex equations, but their physical meaning is still charge and energy consistency in the sinusoidal steady state.

Example of a series circuit

For series impedances $Z_1$ and $Z_2$,

$$Z_{\mathrm{eq}}=Z_1+Z_2,$$

so a source phasor $\underline V_s$ produces

$$\underline I=\frac{\underline V_s}{Z_{\mathrm{eq}}}.$$

Individual element voltages then follow from $\underline V_k=Z_k\underline I$.

Return to time domain

After solving the phasor problem, magnitude and phase are converted back to a sinusoid at the common angular frequency.

Phasor analysis is valid for linear sinusoidal steady state; it does not by itself describe switching transients or mixtures of unrelated frequencies.