Fig. 19

The series RL circuit with parallel C is given in Fig. 19. The designed circuit consists of resistance R, inductance L are in series with parallel configuration of capacitance C. Let us consider the current I flowing through the RLC circuit by the a.c. voltage source with constant value of V.

The current in the circuit

$I=I_1+I_C$

$I=\frac{V}{R+j\omega L}+jV\omega C$

$I=V[\frac{1}{R+j\omega L}+j\omega C]$

$I=V[\frac{(R-j\omega L)}{R^2+{(\omega L)}^2}+j\omega C]$

$I=V[\frac{R}{R^2+{(\omega L)}^2}+j(-\frac{\omega L}{R^2+{(\omega L)}^2}+\omega C)]$

Here at the resonance

$\frac{\omega L}{R^2+{(\omega L)}^2}=\omega C$

Or $R^2+{(\omega L)}^2=\frac{L}{C}$

Or ${(\omega L)}^2=\frac{L}{C}-R^2$

$\omega L=\sqrt{\frac{L}{C}-R^2}$

The resonance frequency of the RLC circuit is $f_0=\frac{1}{2\pi L}\sqrt{\frac{L}{C}-R^2}$