Determine the value of capacitance to give resonance:
A circuit shown in Figure having a resistance of 5 Ω, an inductance of 0.4 H and a variable capacitance in series is connected across a 110 V, 50 Hz supply. Determine:
1. The value of capacitance to give resonance
2. Current
3. Voltage across the inductance
4. Voltage across the capacitance
5. Q-factor of the circuit.
Solution
Applied Voltage, V = 110 V Resistance,
R = 5 Ω Inductance, L = 0.4 Henry
Resonant frequency, f_{0} = 50 Hz
As Resonant frequency f0 = 1 /2π √(LC)
Figure
(a) At resonant frequency capacitance
C = 1/4π (f_{0 })^{2} L = 1/ (4π × (50)^{2} × 0.4)
= 2.53303 × 10^{- 5}
= 25.3303 μF
(b) At resonant condition I = V/ R = 110/5.
= 22 A
(c) Voltage across inductance = I X_{L} = 22 × 2π × 50 × 0.4 = 2764.60 V.
(d) Voltage across capacitance = Voltage across inductance
= 2764.60 V
(e) Q-factor = ω_{0} L/R = (2π f_{0 }L )/ R
= (2π × 50 × 0.4) /5
= 25.1327