A series LCR circuit with L = 0.12 H, C = 480 nF, R = 23 Ω is connected to a 230 V variable frequency supply. (a) What is the source frequency for which current amplitude is maximum? Obtain this maximum value. (b) What is the source frequency for which average power absorbed by the circuit is maximum? Obtain die value of this maximum power. (c) For which frequencies of the source is the power transferred to the circuit half the power at resonant frequency? What is the current amplitude at these frequencies? (d) What is the Q-factor of the given circuit?

Inductance, L = 0.12 H

Capacitance, C = 480 nF = 480 x 10⁻⁹ F

Resistance, R = 23 Ω

Supply voltage, V = 230 V

Peak voltage is given as: Vo =√2 x 230 = 325.22 V

Ans (a).

Current flowing in the circuit is given by the relation,

I₀=  V₀/√[(R² + (ωL- 1/ωC)²]

Where, Io = maximum at resonance

At resonance, we have

ω_R– 1/ω_RC =0

Where, ω_R  = Resonance angular frequency

Where , ω_R = Resonance angular frequency

Therefore ω_{R = 1/√(LC) = 1/ √ (0.12 x 480 x 10⁻⁹ ) = 4166.67 rad/s}

Resonant frequency , ν_R = ω_R/2π = 4166.67 /(2 x 3.14) = 663.48 Hz

And ,maximum current,   (I₀)_Max  =   V₀/R = 325.22/23 = 14.14 A

Ans (b).

Maximum average power absorbed by the circuit is given as:

 (P_av)_Max= 1/2 x  (I₀)²_MaxR

= 1/2 x (14.14)² x 23 = 2299.3 W

Hence, resonant frequency is 663.48 Hz.

Ans (c).

The power transferred to the circuit is half the power at resonant frequency.

Frequencies at which power transferred is half,

=  ω_R ± Δω

= 2π (ν_R ± Δν)

Where,

Δω=R/2L= 23/(2 x 0.12)= 95.83 rad/s

Hence, change in frequency,

Δν = 1/2π    x    Δω  =95.83 /2π = 15.26 Hz

Therefore

ν_R +Δν  = 663.48 + 15.26 = 678.74 Hz

and ν_R – Δν  = 663.48 – 15.26 =648.22 Hz

Hence, at 648.22 Hz and 678.74 Hz frequencies, the power transferred is half. At these frequencies, current amplitude can be given as:

I’ = 1/√ 2 x (I₀)_Max

= 14.14/√ 2 = 10A

Ans (d).

Q-factor of the given circuit can be obtained using the relation,

Q = ω_R L/R  = 4166.67 x 0.12 /23 = 21.74

Hence, the Q-factor of the given circuit is 21.74.