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Parallel Resonance Circuit

Parallel Resonance Circuit :- In a parallel resonant circuit, an inductor and a capacitor are connected in parallel with an alternating voltage source, as shown in the figure below :

Assume that the alternating electromotive force (emf) applied by the source is as follows :

$\displaystyle E={{E}_{0}}sin\omega t$ …..(1)

Since in an inductor the current lags behind the alternating voltage by (π/2) radians, therefore the current flowing in the inductor is :

$\displaystyle {{I}_{L}}=\frac{{{E}_{0}}}{{{X}_{L}}}sin\left( \omega t-\frac{\pi }{2} \right)$ …..(2)

Again, in a capacitor, the current leads the alternating voltage by (π/2) radians; therefore, the current flowing in the capacitor is :

$\displaystyle {{I}_{\text{C}}}=\frac{{{E}_{0}}}{{{X}_{C}}}sin\left( \omega t+\frac{\pi }{2} \right)$ …..(3)

The total current (I) flowing in the circuit,

$\displaystyle I={{I}_{L}}+{{I}_{\text{C}}}$

$\displaystyle I=\frac{{{E}_{0}}}{{{X}_{L}}}sin\left( \omega t-\frac{\pi }{2} \right)+\frac{{{E}_{0}}}{{{X}_{C}}}sin\left( \omega t+\frac{\pi }{2} \right)$

$\displaystyle \Rightarrow I=\frac{{{E}_{0}}}{{{X}_{L}}}cos\left( -\omega t \right)+\frac{{{E}_{0}}}{{{X}_{C}}}cos\left( \omega t \right)$

$\displaystyle \Rightarrow I=\left[ \frac{-1}{{{X}_{L}}}+\frac{1}{{{X}_{C}}} \right]{{E}_{0}}cos\omega t$

$\displaystyle \Rightarrow I=\left[ -\frac{1}{\omega L}+\omega C \right]{{E}_{0}}cos\omega t$ …..(4)

In a parallel resonance circuit, at the condition of resonance (ω = ω_r), the current drawn from the external source becomes zero, therefore

$\displaystyle -\frac{1}{{{\omega }_{r}}L}+{{\omega }_{r}}C=0$

$\displaystyle \Rightarrow {{\omega }_{r}}C=\frac{1}{{{\omega }_{r}}L}$

$\displaystyle \Rightarrow \omega _{r}^{2}=\frac{1}{LC}$

$\displaystyle \therefore {{\omega }_{r}}=\frac{1}{\sqrt{LC}}$ …..(5)

and

$\displaystyle {{\nu }_{r}}=\frac{1}{2\pi \sqrt{LC}}$ …..(6)

Thus, at resonance, the frequency of the applied alternating voltage becomes equal to the natural oscillating frequency of the L-C parallel resonant circuit. At this condition, the circuit is called a parallel resonance circuit, and the frequency given in equation (6) is called the parallel resonance frequency or parallel resonant frequency.

Because at the parallel resonant frequency the current in the circuit becomes zero, therefore the impedance of the circuit becomes maximum at the parallel resonant frequency.

Note :-

A series resonant circuit is called an acceptor circuit, whereas a parallel resonant circuit is called a rejector circuit.

Example 1.

An AC source of variable frequency is connected, as shown in the figure, across an L–C parallel combination having inductance 0.01 H and capacitance 1 μF. For the change in frequency from 1 kHz to 3 kHz, draw the approximate graph showing the variation of current.

Parallel Resonance Circuit - Curio Physics

Solution :

Here, L and C are connected in parallel with the AC source. The resonant frequency of the circuit is :

$\displaystyle {{\nu }_{r}}=\frac{1}{2\pi \sqrt{LC}}=\frac{1}{2\pi \sqrt{0.01\times {{10}^{-6}}}}=\frac{{{10}^{4}}}{2\pi }\simeq 1.6\text{ }kHz$

In case of parallel resonance circuit, at resonance, the current in the L–C circuit becomes zero; therefore, the required graph will be as shown in the figure :-

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