Mean or Average Value of AC

Because a symmetric alternating current spends exactly as much time (and reaches exactly the same peak magnitude) in its positive half-cycle as in its negative half-cycle, the SIMPLE average of current over one FULL cycle is always exactly zero -- the positive and negative contributions cancel completely -- making a full-cycle average useless as a way to characterise an AC waveform's typical size. The average (or mean) value of alternating current is therefore instead defined over just ONE HALF cycle (positive or negative), as the average of all the current values over that half-cycle.

Deriving this value: with $i=I_m\sin\theta$, the sum of all current values over the positive half-cycle is represented by the AREA under the curve from $\theta=0$ to $\theta=\pi$, found by integrating an elementary strip $i\,d\theta$: $\text{Area}=\int_0^{\pi}I_m\sin\theta\,d\theta = I_m[-\cos\theta]_0^{\pi}=I_m(-\cos\pi+\cos0)=2I_m$. Dividing this area by the half-cycle's base length $\pi$ gives the average value,

$I_{av} = \dfrac{2I_m}{\pi} \approx 0.637\,I_m \qquad (4.32)$ …