Question

For the circuit shown in the figure. Find the expressions for the impedance of the circuit and phase of current:

Answer 1

Let ac source $V=V_0\sin \omega t$
ie., $I_R=\frac{V_0}{R}\sin \omega t,I_L=\frac{V_0}{X_L}\sin (\omega t-\frac{\pi }{2}),I_C=\frac{V_0}{X_C}\sin (\omega t+\frac{\pi }{2})$
And as all these are in parallel
$I=I_R+I_L+I_C=[\frac{V_0}{R}\sin \omega t+V_0(\omega C-\frac{1}{\omega L})\cos \omega t]$
$\frac{V_0}{R}=I_0\sin \varphi V_0(\omega C-\frac{1}{\omega L})=I_0\cos \varphi I=I_0\sin (\omega t-\varphi )$
$\tan \varphi =(\omega C-\frac{1}{\omega L})/\frac{1}{R}=R(\omega C-\frac{1}{\omega L})$
$I_0=\frac{V_0}{Z}=V_0{[\frac{1}{R^2}+{(\omega C-\frac{1}{\omega L})}^2]}^{1/2}$
$Z=V_0{[\frac{1}{R^2}+{(\frac{1}{X_C}-\frac{1}{X_L})}^2]}^{1/2}$