Derive an expression for the resonant frequency of a parallel circuit, one branch consisting of a coil of inductance L and resistance R and the other branch of capacitance C.
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
\(I=I_1+I_C\)
\(I=\frac{V}{R+j\omega L}+jV\omega C\)
\(I=V\left[\frac{1}{R+j\omega L}+j\omega C\right]\)
\(I=V\left[\frac{(R-j\omega L)}{R^2+{(\omega L)}^2}+j\omega C\right]\)
\(I=V\left[\frac{R}{R^2+{(\omega L)}^2}+j\left(-\frac{\omega L}{R^2+{(\omega L)}^2}+\omega C\right)\right]\)
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 \(f_0=\frac{1}{2\pi L}\sqrt{\frac{L}{C}-R^2}\)
Find the current in each branch of the network using Kirchhoff’s law
Fig.23
