5.2 Average power

  Alternating Current

The average rate of doing work (power) in one cycle is known as average power.

It is also known as true power.

Mathematically, $${P_{av}} = \frac{{\int\limits_0^T {Pdt} }}{{\int\limits_0^T {dt} }}$$$${P_{av}} = {V_0}{i_0}\frac{{\int\limits_0^T {\left[ {{{\sin }^2}\omega t\cos \phi - \frac{1}{2}{{\sin }^2}\omega t\sin \phi } \right]} dt}}{{\int\limits_0^T {dt} }}$$$${P_{av}} = \frac{{{V_0}{i_0}\cos \phi \int\limits_0^T {{{\sin }^2}\omega tdt} - \frac{{{V_0}{i_0}\sin \phi }}{2}\int\limits_0^T {\sin 2\omega t} dt}}{{\int\limits_0^T {dt} }}$$$${P_{av}} = \frac{{{V_0}{i_0}\cos \phi \int\limits_0^T {\left( {\frac{{1 - \cos 2\omega t}}{2}} \right)dt} - \frac{{{V_0}{i_0}\sin \phi }}{2}\int\limits_0^T {\sin 2\omega t} dt}}{{\int\limits_0^T {dt} }}$$$${P_{av}} = \frac{{\frac{{{V_0}{i_0}\cos \phi }}{2}\left[ {t - \frac{{\sin 2\omega t}}{{2\omega }}} \right]_0^T + \frac{{{V_0}{i_0}\sin \phi }}{2}\left[ {\frac{{\cos 2\omega t}}{{2\omega }}} \right]_0^T}}{{\left[ t \right]_0^T}}$$$${P_{av}} = \frac{{\frac{{{V_0}{i_0}\cos \phi }}{2}\left[ {\left( {T - \frac{{\sin 2\omega T}}{{2\omega }}} \right) - \left( {0 - 0} \right)} \right] + \frac{{{V_0}{i_0}\sin \phi }}{2}\left[ {\frac{{\cos 2\omega T}}{{2\omega }} - \frac{1}{{2\omega }}} \right]}}{{\left[ {T - 0} \right]}}$$As $T = \frac{{2\pi }}{\omega }$,$${P_{av}} = \frac{{\frac{{{V_0}{i_0}\cos \phi }}{2}\left[ {T - 0} \right] + \frac{{{V_0}{i_0}\sin \phi }}{2}\left[ {\frac{1}{{2\omega }} - \frac{1}{{2\omega }}} \right]}}{T}$$$${P_{av}} = \frac{1}{2}{V_0}{i_0}\cos \phi $$Also, $${P_{av}} = \frac{1}{2}{V_0}{i_0}\cos \phi $$$${P_{av}} = \frac{{{V_0}}}{{\sqrt 2 }}\frac{{{i_0}}}{{\sqrt 2 }}\cos \phi $$$${P_{av}} = {V_{rms}}{i_{rms}}\cos \phi $$

The term $\cos \phi $ is known as power factor.