The Effective or RMS Value of Sinusoids

The effective value of a time-varying current $i(t)$ or voltage is the constant value of current $I_{rms}$ or voltage $V_{rms}$ that in period $T$ would transfer the same amount of energy:

$$W=\int_0^T p(t) dt = R\int_0^T i^2(t) dt =R I^2_{rms}T$$

$$= \frac{1}{R}\int_0^T v^2(t) dt=\frac{1}{R}V^2_{rms}T$$

i.e.,

$$I_{rms}=\sqrt{\frac{1}{T}\int_0^T i^2(t) dt},\;\;\;\;\;\;\;\... ...{or}\;\;\;\;\;\; V_{rms}=\sqrt{\frac{1}{T}\int_0^T v^2(t) dt}$$

As $I_{rms}$ or $V_{rms}$ is the ``square root of the mean of the squared value'', it is also called the root-mean-square (rms) current or voltage.

For a sinusoidal variable $i(t)=\cos(\omega t)$, we have

$$I^2_{rms} = \frac{1}{T}\int_0^T i^2(t) dt = \frac{1}{T}\int_... ...dt = \frac{1}{2T}\int_0^T [1+\cos(4\pi t/T)]\; dt=\frac{1}{2}$$

(Recall trigonometric identity: $\cos^2\alpha=[1+\cos(2\alpha)]/2$.) Similarly, we also have

$$V_{rms}=\frac{1}{\sqrt{2}}=0.707$$