Average Value of Sinusoids

The average voltage is defined as:

$$V_{av}=\frac{1}{T}\int_0^T v(t) dt$$

If the current/voltage is periodic, i.e., it repeats itself every time cycle $T$, then $i(t)=i(t+T)$ and $v(t)=v(t+T)$. The reciprocal of $T$ is called the fundamental frequency. In particular, for a sinusoidal current

$$i(t)=I_p \sin(\omega t)=I_p \sin(2\pi ft)=I_P \sin(2\pi t/T)$$

the average over the complete cycle $T=1/f$ is always zero (the charge transferred during the first half is the opposite to that transferred in the second). However, we can consider the half-cycle average over $T/2$:

$$I_{av} = \frac{1}{T/2}\int_0^{T/2} i(t)\; dt =\frac{2}{T}\int_0^{T/2} \;I... ...\pi t/T)\;dt =-\frac{2}{T}\frac{TI_P}{2\pi} \cos(2\pi t/T)\bigg\vert _0^{T/2}$$

$$= \frac{1}{\pi}\left[\cos(0)-\cos(\pi)\right]I_P =\frac{2}{\pi}I_P\;=0.637 I_P$$