2.2 Mean or average current and voltage

  Alternating Current

The average value of an alternating current is defined as the average of all instantaneous current during one complete cycle.

Mathematically average current is given by, $${i_{avg}} = \frac{{\int\limits_0^T {idt} }}{{\int\limits_0^T {dt} }}$$$${i_{avg}} = \frac{{\int\limits_0^T {{i_0}\sin \omega t} dt}}{{\int\limits_0^T {dt} }}$$$${i_{avg}} = {i_0}\frac{{\left[ { - \frac{{\cos \omega t}}{\omega }} \right]_0^T}}{{\left[ t \right]_0^T}}$$As $T = \frac{{2\pi }}{\omega }$, $${i_{avg}} = \frac{{{i_0}}}{\omega }\frac{{\left[ { - \cos \left( {\omega \times \frac{{2\pi }}{\omega }} \right) + 1} \right]}}{{\left[ {T - 0} \right]}}$$$${\left\langle {{i_{avg}}} \right\rangle _{full\,cycle}} = 0$$

So, the average value of the alternating current during complete cycle is zero.

Therefore in A.C. the average value of current is defined as its average taken over half the cycle.

So, $${i_{avg}} = {i_0}\frac{{\left[ { - \frac{{\cos \omega t}}{\omega }} \right]_0^{\frac{T}{2}}}}{{\left[ t \right]_0^{\frac{T}{2}}}}$$$${i_{avg}} = \frac{{{i_0}}}{\omega }\frac{{\left[ { - \cos \left( {\omega \times \frac{{2\pi }}{{2\omega }}} \right) + 1} \right]}}{{\left[ {\frac{T}{2} - 0} \right]}}$$$${i_{avg}} = \frac{{T{i_0}}}{{2\pi }}\left( {\frac{4}{T}} \right)$$$${i_{avg}} = \frac{{2{i_0}}}{\pi }$$$${\left\langle {{i_{avg}}} \right\rangle _{half\,cycle}} = 0.637{i_0}$$

Similarly, $${\left\langle {{V_{avg}}} \right\rangle _{half\,cycle}} = \frac{{2{V_0}}}{\pi }$$