Many physical quantities vary sinusoidally with time. A general sinusoid can be written

$$x(t)=X_m\cos(\omega t+\phi),$$

where $X_m$ is the amplitude, $\omega$ the angular frequency and $\phi$ the phase.

Frequency and period

Angular frequency and ordinary frequency are related by

$$\omega=2\pi f,$$

with period

$$T=\frac1f=\frac{2\pi}{\omega}.$$

Phase

The phase $\phi$ shifts the oscillation in time. Two sinusoids with the same frequency but different phases reach corresponding points of their cycles at different times.

For example,

$$\cos\left(\omega t-\frac\pi2\right)=\sin\omega t.$$

So one sinusoid lags the other by a quarter cycle.

Same-frequency sinusoidal steady state

In many linear AC systems, all voltages and currents in steady state oscillate at one common angular frequency while differing only in amplitude and phase.

This makes it possible to replace repeated trigonometric calculations by a compact complex-number representation: the phasor.