Question

Assuming expression for impedance in a parallel resonant circuit, state the condition for parallel resonance. Define resonant frequency and obtain an expression for it.

Answer 1

Hint: Resonance is defined as the tendency of the system to oscillate at greater amplitude at some frequencies than at others. It is common among the systems that have a tendency to oscillate at a particular frequency and that frequency is known as natural frequency.

Complete step by step solution:
Impedance of the circuit is given by
\[Z = \dfrac{e}{i}\]
\[\therefore Z = \dfrac{1}{{\omega l - \omega c}}\]
At a particular frequency of applied emf if \[{X_l} = {\text{ }}{X_c}\]; \[{I_l} = {\text{ }}{I_c}\]and net r.m.s. current i of the circuit is zero and the impedance (Z) of the circuit is infinite.
In practice, the impedance (Z) of the circuit is maximum and not infinite because of resistance of the coil and hence the r.m.s. current given to the circuit is minimum (tends to zero),this is the condition of parallel resonance.
An ac current is given by,
\[i = {i_0}sin\omega t\]
 \[i = {i_c} - {i_l}\]
\[\dfrac{e}{z} = \dfrac{e}{{{X_c}}} - \dfrac{e}{{{X_l}}}\]
\[\dfrac{1}{z} = \dfrac{1}{{{X_c}}} - \dfrac{1}{{{X_l}}}\]
for parallel resonance
\[{X_l} = {\text{ }}{X_c}\]
\[
  \dfrac{1}{Z} = 0 \\
  Z = \infty \\
 \]
Resonant frequency (fr) :
The frequency at which imprudence (L) becomes infinity and current becomes zero is called resonant frequency :
\[\omega l = \omega c\]
\[\omega = \sqrt {\dfrac{1}{{lc}}} \]
 \[
  2\pi {f_r} = \sqrt {\dfrac{1}{{lc}}} \\
  {f_r} = \dfrac{1}{{2\pi \sqrt {lc} }} \\
 \]
The frequency of A.C. for which resonance takes place and minimum current flows through the circuit is called resonant frequency. At resonant frequency, current should be minimum and impedance

Note: An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor L, and a capacitor C, connected together. An LC circuit is an idealized model since there is no dissipation of energy due to resistance. The purpose of an LC circuit is to oscillate with minimal damping, so the resistance is made as low as possible.