Question

In a series LCR circuit, obtain an expression for the resonant frequency.

Answer 1

inductive reactance$=X_L=2\pi fL=\omega L$

Capacitive reactance$X_C=\frac{1}{2\pi fC}=\frac{1}{\omega C}$

total circuit reactance$=X_T=X_L-X_C$ or $X_C-X_L$

total circuit impendance $=Z=\sqrt{R^2+{X_T}^2}$

for minimum impendance $X_T$ should be equals to $0$

$\Rightarrow X_T=X_C-X_L=\frac{1}{\omega C}-\omega L=0$

$\Rightarrow \omega =\sqrt{LC}=2\pi f$

$\Rightarrow f=\frac{\sqrt{LC}}{2\pi }Hertz$