Question

Draw a curve for showing variation in alternating current with frequency in LCR resonant circuit. Hence obtain an expression of bandwidth.

Answer 1

Here $f_1,f_2$ are half power frequencies(since $I=\frac{I_0}{\sqrt{2}}$)

At maximum power, $I_0=\frac{V}{Z_{m}in}$

$=\frac{V}{R}$( since this is the resonant frquency)

At half power,

$\frac{V^2}{Z^{\prime }}=\frac{1}{2}\times \frac{V^2}{R}$

$⟹Z^{\prime }=\sqrt{2}R$

Hence for $f_1,f_2$,

$(\omega L-\frac{1}{\omega C})=\pm R$

$⟹{\omega }_1=-\frac{R}{2L}+\sqrt{\frac{R^2}{4L^2}+\frac{1}{LC}}$

${\omega }_2=\frac{R}{2L}+\sqrt{\frac{R^2}{4L^2}+\frac{1}{LC}}$

Thus bandwidth=$\Delta \omega ={\omega }_2-{\omega }_1$

$=\frac{R}{L}$