Question

What is resonant electrical circuit? What are its types? Find out an expression for resonant frequency for series L-C-R circuit.

Answer 1

Resonance Circuit: In an alternating current, if the phase of the applied potential difference and the current flowing in the circuit are same, then the circuit is called resonance circuit. The phenomenon shown by these types of circuit is called resonance.
Resonance L.C.R. circuit are of two types:
(i) Series resonance circuit.
(ii) Parallel resonance circuit.
Resonant Frequency for series L-C-R circuit.
We know that the impedance of the circuit is $Z=\sqrt{R^2+{(\omega L-\frac{1}{\omega L})}^2}$ .......(i)
If $V$ is the potential difference and $I$ is the current and $\varphi$ is the phase difference between then,
$\tan \theta =\frac{\omega L-\frac{1}{\omega C}}{R}$
Or $\varphi =\tan \left\{\frac{\omega L-\frac{1}{\omega C}}{R}\right\}$
In the condition of impedance the current flows through this equation
$I=I_0\sin (\omega t-\varphi )$
where $I_0=\frac{V_0}{Z}=\frac{V_0}{\sqrt{R^2+{(X_L-X_C)}^2}}$
$I_0=\frac{V_0}{\sqrt{R^2+{(\omega L-\frac{1}{\omega C})}^2}}$ ......(ii)
If $\omega L=\frac{1}{\omega C}$ then from equation (ii)
$Z=R$ (minimum) and $I_0=\frac{V_0}{R}$ (maximum)
Thus, in this condition the impedance is minimum so that the current flowing in the circuit is maximum.
This condition is known as resonance.
In the condition of resonance, $\omega L=\frac{1}{\omega C}$
${\omega }^2=\frac{1}{LC}$
Or $\omega =\frac{1}{\sqrt{LC}}$
$2\pi f=\frac{1}{\sqrt{LC}}$
$f=\frac{1}{2\pi \sqrt{LC}}$
This is expression for resonant frequency for series L-C-R circuit.