Question

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

Answer 1

Hint: When the current and voltage are in phase then the condition is said to be in resonance. The amplitude of current is maximum at resonant frequency. In the case of LCR circuits which are in series, the inductance and capacitance have phase angles and have equal magnitudes.

Complete step by step answer:
By definition, impedance (Z) of the circuit is given by,
$Z=\dfrac{e}{i}$
$\therefore Z=\dfrac{1}{\left( \omega C-\dfrac{1}{\omega L} \right)}$
At a particular frequency of applied e.m.f if ${{X}_{L}}={{X}_{C}}$,${{i}_{L}}={{i}_{C}}$ and the net rms value current of the circuit is zero and the impedance of the circuit is infinite. In practice, the impedance of the circuit is maximum and not infinite because of resistance of the coil and hence r.m.s current is minimum. This is the condition of parallel resonance.
Resonant frequency
The frequency of A.C for which resonance takes place and minimum current flows through the circuit is called resonant frequency. At resonant frequency the current should be minimum and impedance is maximum.
When the current and voltage are in phase then the condition is said to be in resonance. The frequency at this time is given by,
$f=\dfrac{1}{2\pi \sqrt{LC}}$
Where, L is the inductance and C is the capacitance.

Note: Since in LCR circuits which are in series, the inductance and capacitance have phase angle and have equal magnitudes. So they can cancel each other. In electronics, a parallel resonant circuit is most commonly used. When the resonance condition is achieved we can attain its natural frequency by changing its mass.