In an L-R-C series circuit, the current is given by $\mathrm{i}=\mathrm{I} \cos \mathrm{\omega t}$. If the voltage amplitudes for the resistor, inductor and capacitor are V_R, V_L and V_C respectively, then:

Instantaneous power into resistor

${\mathrm{P}}_{\mathrm{R}}={\mathrm{V}}_{\mathrm{R}}{\mathrm{I}}_{\mathrm{R}}=\left({\mathrm{V}}_{\mathrm{R}}\cos \mathrm{\omega t}\right)(\mathrm{Icos}\mathrm{\omega t})={\mathrm{V}}_{\mathrm{R}}{\mathrm{Icos}}^2\mathrm{\omega t}$

V_R is in phase with current.

${\mathrm{P}}_{\mathrm{L}}=-{\mathrm{V}}_{\mathrm{L}}\mathrm{Isin}\mathrm{\omega tcos}\mathrm{\omega t},$

V_L leads current by $\frac{\mathrm{\pi }}{2}$

Voltage across inductor

$$\begin{gathered} ={\mathrm{V}}_{\mathrm{L}}\cos \left(\mathrm{\omega t}+\frac{\mathrm{\pi }}{2}\right)=-{\mathrm{V}}_{\mathrm{L}}\sin \mathrm{\omega t} \\ {\mathrm{P}}_{\mathrm{C}}={\mathrm{V}}_{\mathrm{C}}\mathrm{Isin}\mathrm{\omega tcos}\mathrm{\omega t}, \end{gathered}$$

V_C lags current by $\frac{\mathrm{\pi }}{2}$

Voltage across capacitor$={\mathrm{V}}_{\mathrm{C}}\cos \left(\mathrm{\omega t}-\frac{\mathrm{\pi }}{2}\right)={\mathrm{V}}_{\mathrm{C}}\sin \mathrm{\omega t}$

$\therefore$ Total power$={\mathrm{P}}_{\mathrm{R}}+{\mathrm{P}}_{\mathrm{L}}+{\mathrm{P}}_{\mathrm{C}}$