An inductor, a capacitor and a resistor are connected in series with an ac source v = v_m sin ω t. Derive an expression for the average power…

Deriving an expression for the average power dissipated in series LCR circuit Obtaining expression for the resonant frequency \[\mathrm{v}=\mathrm{v}_{\mathrm{m}} \sin \omega \mathrm{t} \] \[\mathrm{i}=\mathrm{i}_{\mathrm{m}} \sin (\omega \mathrm{t}+\varphi) \] \[\text { Power, } \begin{aligned} \mathrm{P}=\mathrm{v} \mathrm{i} & =\left(\mathrm{v}_{\mathrm{m}} \sin \omega \mathrm{t}\right) \times\left[\mathrm{i}_{\mathrm{m}} \sin (\omega \mathrm{t}+\varphi)\right] \\ & =\frac{\mathrm{v}_{\mathrm{m}} \mathrm{i}_{\mathrm{m}}}{2}[\cos \varphi-\cos (2 \omega \mathrm{t}+\varphi)] \end{aligned} \] The average power over a cycle is given by the average of the two terms in RHS of eqn ($\displaystyle 1$). It is only the \(\displaystyle 2^{\text {nd }}\) term which is time dependent. It's average is zero. Therefore, \[\mathrm{P}=\frac{\mathrm{v}_{\mathrm{m}} \mathrm{i}_{\mathrm{m}}}{2} \cos \varphi \]