Question

Derive expression for frequency of parallel resonance circuit. Where this circuit is used?

Answer 1

Consider a circuit consisting of an inductance L and a capacitance C, both connected in parallel with an alternating source of e.m.f., E as shown in Fig. 

The instantaneous value of alternating e.m.f. is given by

E = E₀ sin ωt ...........(1)

Let I_L. be current passing through the inductor. Since current in an inductor lags behind the e.m.f. by a phase π/2.

Let I_C be current passing through the capacitor. Since current in a capacitor leads the e.m.f. by a phase π/2,

∴ I_C = \(\frac{E_0}{X_C}\)  sin (ωt - π/2)

= E₀ ωC cos ωt .............(3)

∴ Total current I in the circuit at any instant will be

I = I_L + I_C

= E₀ (\(\frac{1}{\omega L }+ \omega C\)) . cos ωt

Thus we find that in parallel resonant circuit, resonance occurs when the applied frequency becomes equal to natural frequency of oscillation of the circuit.

From eq. (4), we find that at parallel resonance, the current in the circuit becomes zero i.e. I = 0 Fig. and the impedance becomes maximum. Due to this reason the circuit is called rejected circuit and work as perfect-choke for a.c. Fig. shows the variation of current with the frequency.

Uses: These circuits are used as filter circuits in radio to block the currents of undesirable frequencies and is also called oscillatory circuit.