Question

An alternating current $i$ is given by $=i_0\sin 2\pi (t/T+1/4)$. Then the average current in the first one quarter time period is

• A. $\frac{2i_0}{\pi }$
• B. $\frac{i_0}{\pi }$
• C. $\frac{i_0}{2\pi }$
• D. $\frac{3i_0}{\pi }$

Answer 1

Answer: (A)

$\frac{2i_0}{\pi }$

$i=i_{0}sin2\pi (t/T+1/4)$

Average current $<i>=\frac{{\int }_{t_1}^{t_2}idt}{t_2-t_1}$

$=\frac{{\int }_0^{T/4}{i}_{0}sin2\pi (t/T+1/4)dt}{T/4-0}$

$=\frac{i_0[\frac{-cos2\pi (t/T+1/4)}{2\pi /T}{]}_0^{T/4}}{T/4}$
$=\frac{\frac{i_0}{2\pi /T}}{T/4}[-(-1)-0]$
$=\frac{i_{0}T}{2\pi }\times \frac{4}{T}$
$=\frac{2i_0}{\pi }$