Question

Derive an expression for average power of an AC (alternating current) circuit.

Answer 1

Instantaneous power p supplied by the source is
p = vi = $v_{m}sin\omega t$* $\left[i_{m}sin\right(\omega t+\varphi \left)\right]$
= $\frac{v_m{i}_m}{2}[cos\varphi -cos(2\omega t+\varphi \left)\right]$
The average power over a cycle is given by the average of the two terms in the given equation.It is only the second term which is time independent.
Its average is 0.
Therefore,
P = $\frac{v_m{i}_m}{2}cos\varphi$ = $\frac{v_m{i}_m}{2}cos\varphi$
= $VIcos\varphi$
This can also be written as
P =$I^{2}Zcos\varphi$