Derive an expression for the average power consumed in a series LCR circuit connected to a.c., source in which phase difference between the voltage and the current in the circuit is ϕ. 

Average power in LCR circuit: 

Let, the alternating e.m.f. applied to an LCR circuit is
E = E₀ sin ωt                  ...(i) 

If alternating current developed lags behind the applied e.m.f. by a phase angle ϕ then, 

I = I₀ sin (ωt – ϕ)

Total work done over a complete cycle is

           $\mathrm{W} = \underset{0}{\overset{\mathrm{T}}{\int }}\mathrm{EI} \mathrm{dt}$ 

              $= \underset{0}{\overset{\mathrm{T}}{\int }}{\mathrm{E}}_0 \sin \mathrm{\omega t}. {\mathrm{I}}_0 \sin (\mathrm{\omega t} - \mathrm{\varphi }) \mathrm{dt}$
              $= {\mathrm{E}}_0{\mathrm{I}}_0\underset{0}{\overset{\mathrm{T}}{\int }}\sin \mathrm{\omega t} \sin (\mathrm{\omega t}-\mathrm{\varphi })\mathrm{dt}$ 
              $= \frac{{\mathrm{E}}_0 {\mathrm{I}}_0}{2} \underset{0}{\overset{\mathrm{T}}{\int }} 2 \sin \mathrm{\omega t} \sin (\mathrm{\omega t} - \mathrm{\varphi }) \mathrm{dt}$
              $= \frac{{\mathrm{E}}_0{\mathrm{I}}_0}{2}\underset{0}{\overset{\mathrm{T}}{\int }}\left[\cos (\mathrm{\omega t} \pm \mathrm{\omega t} + \mathrm{\varphi }) - \cos (\mathrm{\omega t}+\mathrm{\omega t}-\mathrm{\varphi }\right]\mathrm{dt}$
                                                         $\left[\because 2 \sin \mathrm{A} \sin \mathrm{B} = \cos (\mathrm{A}-\mathrm{B}) - \cos (\mathrm{A}+\mathrm{B})\right]$

Thus,    $\mathrm{W} = \frac{{\mathrm{E}}_0{\mathrm{I}}_0}{2}\underset{0}{\overset{\mathrm{T}}{\int }}\left[\cos \mathrm{\varphi } - \cos (2\mathrm{\omega t} - \mathrm{\varphi })\right] \mathrm{dt}$
            $\mathrm{W} = \frac{{\mathrm{E}}_0{\mathrm{I}}_0}{2}{\left[\mathrm{t} \cos \mathrm{\varphi } - \frac{\sin (2 \mathrm{\omega t} - \mathrm{\varphi })}{2\mathrm{\omega }}\right]}_0^{\mathrm{T}}$ 

              $= \frac{{\mathrm{E}}_0 {\mathrm{I}}_0}{2}[\mathrm{T} \cos \mathrm{\varphi }]$ 

           $\mathrm{W} = \frac{{\mathrm{E}}_0{\mathrm{I}}_0}{2}. \cos \mathrm{\varphi }. \mathrm{T}$ 
∴ Average power in LCR circuit over a complete cycle is
                                 $\mathrm{P} = \frac{\mathrm{W}}{\mathrm{T}} = \frac{{\mathrm{E}}_0 {\mathrm{I}}_0}{2} \cos \mathrm{\varphi } = \frac{{\mathrm{E}}_0}{\sqrt{2}}.\frac{{\mathrm{I}}_0}{\sqrt{2}}\cos \mathrm{\varphi }$ 

$\therefore$     $\mathrm{P} = {\mathrm{E}}_{\mathrm{v}}{\mathrm{I}}_{\mathrm{v}} \cos \mathrm{\varphi }.$