Write an equation of average power for $L-C-R$ series $AC$ circuit and discuss its various cases.

(A) Average power for a given circuit is given by,
$P = VI \cos \phi$
where $V = \frac{V_{m}}{\sqrt{2}}$ and $I = \frac{I_{m}}{\sqrt{2}}$ are the $RMS$ values of voltage and current.
Cases:
$(1)$ Resistive circuit (Circuit containing only pure resistor):
In such a circuit,current and voltage are in the same phase. So,phase difference $\phi = 0^{\circ}$.
$\therefore$ Average power $P = VI \cos 0^{\circ} = VI$.
There is maximum power dissipation.
$(2)$ Purely inductive or capacitive circuit:
In a purely inductive circuit,current lags behind voltage by $\frac{\pi}{2}$. In a purely capacitive circuit,current leads voltage by $\frac{\pi}{2}$. In both cases,$\phi = \frac{\pi}{2}$.
Average power $P = VI \cos \frac{\pi}{2} = 0$.
No power is dissipated; this current is known as wattless current.
$(3)$ $L-C-R$ series circuit:
In an $L-C-R$ series circuit,power dissipated is $P = VI \cos \phi$,where $\phi = \tan^{-1}\left(\frac{X_{C} - X_{L}}{R}\right)$. Power is dissipated only in the resistor.
$(4)$ Power dissipated at resonance in $L-C-R$ circuit:
At resonance,$X_{C} - X_{L} = 0$,so $\phi = 0^{\circ}$ and $\cos \phi = 1$.
Therefore,$P = VI = I^{2}R$ (since $Z = R$).
Maximum power is dissipated in the circuit at resonance.