An alternating voltage or current can have an ordinary time average of zero even though it transfers energy continuously. The root-mean-square (RMS) value gives an effective magnitude that treats positive and negative portions of the waveform equally.

The RMS idea

Over one complete cycle, RMS is obtained conceptually in three steps:

1. square the instantaneous value;
2. average those squared values over the cycle;
3. take the square root of that average.

Squaring prevents positive and negative half-cycles from cancelling.

Sinusoidal RMS values

For a sinusoid

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

the RMS value is

$$X_{\mathrm{rms}}=\frac{X_m}{\sqrt2}.$$

Thus

$$V_{\mathrm{rms}}=\frac{V_m}{\sqrt2},\qquad I_{\mathrm{rms}}=\frac{I_m}{\sqrt2}.$$

The phase does not affect the RMS magnitude.

Why RMS is useful

A sinusoidal current with RMS value $I_{\mathrm{rms}}$ produces the same average heating in a resistor as a DC current of magnitude $I_{\mathrm{rms}}$.

For a resistor,

$$P_{\mathrm{avg}}=I_{\mathrm{rms}}^2R.$$

RMS is therefore an effective magnitude for comparing alternating and steady electrical quantities without confusing it with the ordinary average of the waveform.