Question

Explain electrical resonance in an LCR series circuit. Deduce the expression for the resonant frequency of the circuit.

Answer 1

Suppose a sinusoidally alternating emf e, of peak value e₀ and frequency f, is applied to a circuit containing an inductor of inductance L, a resistor of resistance R and a capacitor of capacitance C, all in series, from figure (a) The inductive reactance, X_L , and the capacitive reactance, X_C , are

X_L = ωL and X_C = \(\cfrac1{ωC}\)

where ω = 2πf.

The rms values i_rms and e_rms of current and emf are proportional to one another.

i_rms = \(\cfrac{e_{rms}}Z\)

where Z = \(\sqrt{R^2+(X_L-X_C)^2}\) = the impedance of the circuit.

(a) An LCR series circuit (b) The variation of the current near the resonant frequency

The impedance Z drops to a minimum at the frequency f_r for which the inductive and capacitive reactances are equal (and opposite, in a phasor diagram); i.e., when

At this frequency, Z = R and the phase angle Φ = 0, i.e., the combination behaves like a pure resistance, and the current and emf are in phase. If R is small, the loss is small. Then, the current may be very large. At any other frequency, the impedance is greater than R. If a mixture of frequencies is applied to the circuit, the current only builds up to a large value for frequencies near the one to which the circuit is ‘tuned’, as given by Eq. (5). The resonance curve, from figure (b), shows the variation of the rms current with frequency. This is an example of electrical resonance. Equations (3) or (4) give the resonance condition and f_r is called the resonant frequency of the LCR series circuit.

At the resonant frequency, the potential differences across the capacitor and inductor are equal in magnitude but in exact antiphase; the current is in quadrature, i.e., 900 out of phase with them. The energy stored in the electric field of the capacitor changes periodically as the square of the potential difference across it; while the energy stored in the magnetic field of the inductor changes periodically as the square of the current.

At moments when the potential difference across the capacitor is a maximum and the current through the inductor zero, there is then a maximum of energy stored in the electric field of the capacitor. At moments the potential difference across the capacitor is zero and the current through the inductor a maximum, there is then a maximum of energy stored in the magnetic field of the inductor.

At resonance, the total energy stored in the L-C system is constant, and is simply passed back and forth between the electric and magnetic fields. When the resonant current is first building up, this energy is drawn from the ac supply. After that, the supply only needs to make up the energy lost as heat in the resistor.