Question

Prove for an a.c. circuit:
$P_{av}=V_{rms}\times I_{rms}\times \cos \varphi$

Answer 1

Let the alternating potential is $V$ then at the same instant of time $I$ is the current. Where $\varphi$ is the phase between e.m.f. and current.
Let
Voltage $V=V_0\sin \omega t$ .....(i)
and Current $I=I_0\sin (\omega t-\varphi )$ ......(ii)
Then instantaneous power is $P=V\times I$
Then from equation (i) and (ii) $P=V_0\sin \omega t\times I_0\sin (\omega t-\varphi )$
$=V_0{I}_0\sin \omega t\cdot \sin (\omega t-\varphi )$
$=V_0{I}_0\sin \omega t(\sin \omega t\cos \varphi -\cos \omega t\sin \varphi )$
$[\because \sin (A-B)=\sin A\cos B-\cos A\sin B]$
$=V_0{I}_0({\sin }^2\omega t\cos \varphi -\sin \omega t\cos \omega t\sin \varphi )$
$=V_0{I}_0({\sin }^2\omega t\cos \varphi -\frac{1}{2}\sin 2\omega t\sin \varphi )$
In one complete cycle ${\sin }^2\omega t=\frac{1}{2}$ and $\sin 2\omega t=0$
$\therefore$ Average Power is $P_{av}=\frac{1}{2}{V}_0{I}_0\cos \theta$
Or $P_{av}=\frac{V_0}{\sqrt{2}}\times \frac{I_0}{\sqrt{2}}\times \cos \varphi$
Or $P_{av}=V_{rms}\times I_{rms}\times \cos \varphi$