In a particular series LR circuit, a driving sinusoidal voltage ${\mathrm{V}}_0\mathrm{sin\omega t}$ is applied across the series combination of resistance R and inductor L. What will be the amplitude of wattless current in this circuit?

$\mathrm{I}={\mathrm{I}}_{\mathrm{o}}\sin \left(\mathrm{\varphi }\right)$

where I₀ is the maximum current value.

Now the impedance of an LR circuit :

$\mathrm{Z}=\sqrt{{\mathrm{R}}^2+{\mathrm{X}}_{\mathrm{L}}^2}$

Hence the current would be :

${\mathrm{I}}_0=\frac{{\mathrm{V}}_0}{\mathrm{Z}}=\frac{{\mathrm{V}}_0}{\sqrt{{\mathrm{R}}^2+{\mathrm{X}}_{\mathrm{L}}^2}}$

Also,

$\Rightarrow \mathrm{sin\varphi }=\frac{{\mathrm{X}}_{\mathrm{L}}}{\mathrm{Z}}$

$\Rightarrow \mathrm{sin\varphi }==\frac{{\mathrm{X}}_{\mathrm{L}}}{\sqrt{{\mathrm{R}}^2+{\mathrm{X}}_{\mathrm{L}}^2}}$

Substituting the value in the above equation for wattless current, we get :

$\Rightarrow I=I_0\sin \varphi$

$\Rightarrow \mathbf{I}=\frac{{\mathrm{V}}_0}{\sqrt{{\mathrm{R}}^2+{\mathrm{X}}_{\mathrm{L}}^2}}·\frac{{\mathrm{X}}_{\mathrm{L}}}{\sqrt{{\mathrm{R}}^2+{\mathrm{X}}_{\mathrm{L}}^2}}$

$\Rightarrow \mathrm{I}=\frac{{\mathrm{V}}_0\mathrm{\omega L}}{{\mathrm{R}}^2+{\mathrm{\omega }}^2{\mathrm{L}}^2} \left(\because {\mathrm{X}}_{\mathrm{L}}=\mathrm{\omega L}\right)$