4.7 Resonance series $L-C-R$ circuit

  Alternating Current

When the frequency of an AC supply is such that the inductive reactance $\left( {{X_L}} \right)$ and capacitive reactance $\left( {{X_C}} \right)$ becomes equal $\left( {{X_L} = {X_C}} \right)$ then the impedance of the series $L-C-R$ circuit is equal to the ohmic resistance in the circuit.

In other words, the $L-C-R$ series circuit will behave as a purely resistive circuit. Due to minimum value of impedance, the current in $L-C-R$ series circuit will be maximum. This condition is known as resonance, and the frequency of the AC supply is known as resonant frequency.

Mathematically, $${X_L} = {X_C}$$$$\omega L = \frac{1}{{\omega C}}$$$$\omega = \frac{1}{{\sqrt {LC} }}$$

So, $$f = \frac{\omega }{{2\pi }}$$ or $$f = \frac{1}{{2\pi \sqrt {LC} }}$$

[Diagram of circuit and phasors, not mentioned in copy but u draw]

If $R$, $L$ and $C$ are constant and the frequency $f$ of the applied voltage is raised continuously from zero, the peak current varies as shown in the figure.

At first, the current is very small. It increases to maximum when the frequency increases to the resonance value $\left( {{f_0}} \right)$ and then fall again.

It is interesting to note that,

1. Before resonance, the current leads voltage

2. At resonance, current is in phase with the voltage

3. After resonance, current lags behind the voltage

Series resonant circuit is also known as acceptor circuit.

• It is used in radio and TV receiver sets for tuning a particular radio station / TV channel.
  

• Resonance phenomenon is exhibited by a circuit only if both $L$ and $C$ are present in the circuit. Then only voltages across $L$ and $C$ cancel each other. We cannot have resonance in a $RL$ or $RC$ circuit.