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, f₀ = 50 Hz

As Resonant frequency f0 = 1 /2π  √(LC)

Figure

 (a)       At resonant frequency capacitance

      C   = 1/4π (f₀ )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 = ω₀ L/R = (2π f₀ L )/ R         

= (2π × 50 × 0.4) /5

= 25.1327