Question

The average value of current $i=I_m\sin \omega t$ from $t=\frac{\pi }{2\omega }$ to $t=\frac{3\pi }{2\omega }$ is how many times of $I_m$?

Answer 1

$i_{avg}=\frac{{\int }_{\pi /2\omega }^{3\pi /2\omega }idt}{{\int }_{\pi /2\omega }^{3\pi /2\omega }dt}$
$=\frac{{\int }_{\pi /2\omega }^{3\pi /2\omega }{I}_m\sin \left(\omega t\right)dt}{\frac{3\pi }{2\omega }-\frac{\pi }{2\omega }}$
$=\left(\frac{\omega }{\pi }\right)\frac{I_m}{\omega }[-\cos \left(\frac{3\pi }{2}\right)+\cos \left(\frac{\pi }{2}\right)]=0$