Question:

Published on: 10 August, 2022

**What is resonance? Deduce the expression of frequency in a series RLC circuit and parallel RLC circuit at resonance.**

Answer:

In general , the resonance means a particular situation in a physical system where a sinusoidal force having a constant amplitude gives rise to an enhanced response of maximum amplitude varying with time. When resonance occur an exchange of energy takes place between the forcing system and the source.

**Expression of frequency in a series RLC circuit at resonance:**

Fig. 3 Series RLC circuit

The series RLC circuit is given in Fig. 3. The designed circuit consists of resistance R, inductance L and capacitance C in series configuration. Let us consider the current I flowing through the series RLC circuit by the a.c. voltage source with constant value of \(V_S\). The complex impedance in the circuit is given by

\(Z=R+j\omega L+\frac{1}{j\omega c}=R+j(\omega L-\frac{1}{\omega C})\)

The current in the circuit

\(I=\frac{V_S}{Z}=\frac{V_S}{R+j(\omega L-\frac{1}{\omega C})}\)

The magnitude of impedance of inductive and capacitive are \(X_L=\omega L\) and \(X_C=\frac{1}{\omega C}\)

The magnitude of impedance

\(\left|Z\right|=\sqrt{R^2+\left(X_L+X_C\right)^2}\)

It can be easily seen from above the expressions that the inductive impedance value is increase linearly with the frequency and capacitive impedance \(X_C\) decreases with the increase of frequency. The resistance R does not depend on the frequency. At a certain frequency current flowing through the RLC series circuit is maximum and in – phase with the applied voltage and the circuit behaves like a resistive circuit and series resonance is obtained. At the series resonance, inductive and capacitive impedance numerical values are same. Therefore we can write

\(X_L=X_C\)

At \(\omega=\omega_0\), we have,

\(\omega_0L=\frac{1}{\omega_0C}\)

\(2\pi f_0L=\frac{1}{2\pi f_0C}\)

Hence the resonance frequency of series RLC circuit is given by:––

\(f_0=\frac{1}{2\pi\sqrt{LC}}\)

**Expression of frequency in a parallel RLC circuit at resonance:**

Fig.26

According to the Fig. (a), we have

\(I=I_R+I_L+I_C\)

Or \(I=\frac{V}{R}-j\frac{V}{\omega L}+jV\omega C\)

Or \(I=V\left[\frac{1}{R}+j(\omega C-\frac{1}{\omega L})\right]\)

\(I=V\left[\frac{1}{R}+j\left(\frac{1}{X_C}-\frac{1}{X_L}\ \right)\right]=\frac{V}{Z}=VY\)

Therefore the complex admittance \(Y=\frac{1}{Z}=\frac{1}{R}+j(\omega C-\frac{1}{\omega L})\).

The complex impedance \(Z=\frac{1}{Y}=\frac{1}{\frac{1}{R}+j(\omega C-\frac{1}{\omega L})}\)

If the conductance and susceptance be G and B respectively

Thus \(Y=G+jB=\frac{1}{R}+j(\omega C-\frac{1}{\omega L})\)

\(G=\frac{1}{R},\ B=\omega C-\frac{1}{\omega L}=B_C-B_L\)

At the resonance \(B_C=B_L\)

\(\omega C=\frac{1}{\omega L}\)

\(\omega^2=\frac{1}{LC}\)

The resonance frequency of parallel RLC circuit is \(f_0=\frac{1}{2\pi\sqrt{LC}}\)

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