Question III.B.4: (a) Explain the phenomenon of resonance in an L–R–C series c...
(a) Explain the phenomenon of resonance in an L–R–C series circuit.
(b) A circuit of R =4 Ω , L =0.5 H and a variable capacitor C in series is connected across a 100 V, 50 Hz supply. Calculate
(i) the value of the capacitor for which resonance will occur;
(ii) voltage across the capacitor at resonance
(a) Resonance in L–R–C series circuit: In the series circuit shown in Fig. 47, R is independent of frequency,X_{L}=\omega L =2πfL is proportional to frequency, and X_{C} =\frac{1}{\omega C}=\frac{1}{2πfC} is inversely proportional to frequency. For a variable input frequency, it is interesting to note that at certain frequency, X_{L} will become equal and opposite to X_{C} so that the impedance, Z, of the circuit will become equal to R. This is the minimum value of circuit impedance. Hence, maximum current will flow through the circuit as shown in Fig. 48
When X_{L}= X_{C}
2πfL= \frac{1}{2πfC}or f=\frac{1}{2π \sqrt{LC}}=f_{0}
f_{0} is called the resonance frequency. As shown in Fig. 48, at f_{0}, current I=I_{m} since impedance Z is at the minimum. Quality factor, Q, is the ratio of reactive power to the active power at resonance frequency, f_{0}.
Q-factor=\frac{I^{2}X_{L}}{I^{2}R} or \frac{I^{2}X_{C}}{I^{2}R}thus, Q-factor=\frac{IX_{L}}{IR} = \frac{IX_{C}}{IR} , i.e. Q factor=\frac{Voltage across L or C}{Voltage across R}
Q-factor is the ratio of voltage across L or C to the voltage across R.
A higher quality factor results in a sharper resonance curve and improved ability of the circuit to accept particular power signals at the resonant frequency. By changing the frequency one can tune the circuit so as to receive the particular frequency signal.
The inductor and the capacitor are energy storing devices. The power that is dissipated in the resistor is called active power. The energy which is stored in the inductor and the capacitor are due to reactive power. At resonant frequency the energy stored in the capacitor and inductor oscillates between them and the circuit as a whole appears to be resistive only. High Q-factor resonant circuits called tuned circuits are often used in the field of electronics for receiving signals of particular frequency. Bandwidth is the range of frequencies, f_{2} – f_{1} at resonance in which the current does not drop below 0.707 of the maximum value of current.
(b) At resonance, X_{L}=X_{C}, Resonance frequency, f_{0} = 50 Hz
so, 2πf_{0}L=\frac{1}{2πf_{0}C}
Substituting values,
C =\frac{1}{2π × 50 × 2π × 50×0.5}=\frac{1}{314× 157} F
or, C =\frac{10^{6}}{314×157} μF =20.3 μF
Resonant current, I_{0}=I_{m}=\frac{V}{R}= \frac{100}{4} = 25 A
Voltage across the capacitor, V_{C}=I_{0}X_{C}=\frac{25}{2πf_{0}C}
or, V_{C}=\frac{25 × 10^{6}}{2π×50×20.3}=3.925 V
Q-factor=\frac{V_{C}}{V}=\frac{3925}{100}=39.25.
