Question

Derive an expression for average power of an a. c circuit containing L C R.With figure
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Answer 1

Instantaneous power p supplied by the source is

$\large\rm { p = v_{m} \ \sin \ \omega t \times [i_{m} \ \sin ( \omega t + \Phi)]}$

$\large\rm { = \frac{v_{m}i_{m}}{2} [ \cos \Phi - \cos ( 2 \omega t + \Phi)]}$

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,

$\large\rm { P = \frac{v_{m} i_{m}}{2} \cos \Phi = \frac{v_{m} i_{m}}{2} \cos \Phi}$

$\large\rm { = V \ I \cos \Phi}$

$\large\boxed{\rm{ P = I^{2} Z \cos \Phi}}$