Question

Current in an ac circuit is given by $i=2\sqrt{2}\sin [\pi t+(\pi /4\left)\right]$. Then the average value of current during time $t=0$ to $t=1s$ is given as $\frac{x}{\pi }$. Find the value of $x$.

Answer 1

Avg value of current $=\frac{1}{1-0}{\int }_0^{1}2\sqrt{2}sin(\pi t+\pi /4)dt=-2\sqrt{2}\frac{1}{\pi }{\left|cos\right(\pi t+\pi /4\left)\right|}_0^1=2\sqrt{2}\frac{1}{\pi }\left[cos\right(\pi /4)-cos(\pi +\pi /4\left)\right]=2\sqrt{2}\frac{1}{\pi }[\frac{1}{\sqrt{2}}-(-\frac{1}{\sqrt{2}}\left)\right]=2\sqrt{2}\frac{1}{\pi }\frac{2}{\sqrt{2}}=\frac{4}{\pi }$
Therefore $x=4$