Half-Power Frequencies in LCR Circuit – Rankers Physics
Topic: Alternating Current
Subtopic: LR, RC and LCR Circuits

Half-Power Frequencies in LCR Circuit

A series LCR circuit containing \(5.0\text{ H}\) inductor, \(80 \mu\text{F}\) capacitor and \(40 \Omega\) resistor is connected to \(230\text{ V}\) variable frequency ac source. The angular frequencies of the source at which power transferred to the circuit is half the power at the resonant angular frequency are likely to be
42 rad/s and 58 rad/s
25 rad/s and 75 rad/s
50 rad/s and 25 rad/s
46 rad/s and 54 rad/s

Solution:

The resonant angular frequency is \(\omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{5 \times 80 \times 10^{-6}}} = 50\text{ rad/s}\) and the bandwidth is \(Delta \omega = \frac{R}{L} = \frac{40}{5} = 8\text{ rad/s}\). The half-power frequencies are \(omega_0 \pm \frac{\Delta \omega}{2} = 50 \pm 4\text{ rad/s}\), which gives \(46\text{ rad/s}\) and \(54\text{ rad/s}\).

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