Define the power for $AC$ circuit. Obtain an equation of average power for $L-C-R$ series $AC$ circuit.

(N/A) Electric power is the rate of energy consumption in an electric circuit. Instantaneous power cannot be measured in an $AC$ circuit,hence true power is measured. True power in an $AC$ circuit means the value of average power over a full period.
Let a voltage $V = V_{m} \sin \omega t$ applied to an $AC$ circuit drive a current in the circuit given by:
$I = I_{m} \sin (\omega t + \phi)$
where $I_{m} = \frac{V_{m}}{Z}$ and $\phi = \tan^{-1} \left( \frac{X_{L} - X_{C}}{R} \right)$.
The instantaneous power supplied by the source is:
$P = VI = (V_{m} \sin \omega t) [I_{m} \sin (\omega t + \phi)]$
$P = V_{m} I_{m} \sin \omega t \cdot \sin (\omega t + \phi)$
Using the identity $2 \sin A \sin B = \cos (A - B) - \cos (A + B)$:
$P = \frac{V_{m} I_{m}}{2} [\cos \phi - \cos (2 \omega t + \phi)]$
The average power over a cycle is the average of the two terms on the $R$.$H$.$S$. The average of the time-dependent term $\cos (2 \omega t + \phi)$ over a full cycle is zero.
Therefore,the average power is:
$P_{avg} = \frac{V_{m} I_{m}}{2} \cos \phi = \frac{V_{m}}{\sqrt{2}} \cdot \frac{I_{m}}{\sqrt{2}} \cos \phi$
$P_{avg} = V_{rms} I_{rms} \cos \phi$
Here,$\cos \phi$ is called the power factor,where $\cos \phi = \frac{R}{Z}$.