Derive an expression for the average power consumed in a series LCR circuit connected to ac source in which the phase-difference between the voltage and current in the circuit is $\varphi$

A series LCR circuit is connected to an ac source having voltage $v=v_m\sin \omega t$. Derive the expression for the instantaneous current $I$ and the phase relationship to the applied voltage. Obtain the condition for resonance to occur. Define power factor. State the conditions under which it is

(i) Maximum,

A series LCR circuit is connected to an ac source having voltage $v=v_m\sin \omega t$. Derive the expression for the instantaneous current $I$ and the phase relationship to the applied voltage. Obtain the condition for resonance to occur. Define power factor. State the conditions under which it is

(ii) Minimum.

Explaining the nature of variation. A series L.C.R circuit is connected to an ac source using phasor diagram, derive the expression for the impedance of the circuit. Plot a graph to show the variation of current with the frequency of the source.

A voltage $V=V_0\sin \omega t$ applied to a series LCR circuit drives a current $i=i_0\sin \omega t$ in the circuit. Derive the expression for the average power dissipated in the circuit.

For currents used for transporting electric power, a low power factor implies large power loss in transmission. Explain.