# A series LCR circuit with R = 20 Ω,

Question:

A series $L C R$ circuit with $R=20 \Omega, L=1.5 \mathrm{H}$ and $C=35 \mu \mathrm{F}$ is connected to a variable-frequency $200 \mathrm{~V}$ ac supply. When the frequency of the supply equals the natural frequency of the circuit, what is the average power transferred to the circuit in one complete cycle?

Solution:

At resonance, the frequency of the supply power equals the natural frequency of the given LCR circuit.

Resistance, R = 20 Ω

Inductance, L = 1.5 H

Capacitance, C = 35 μF = 30 × 10−6 F

AC supply voltage to the LCR circuit, V = 200 V

Impedance of the circuit is given by the relation,

$Z=\sqrt{R^{2}+\left(\omega L-\frac{1}{\omega C}\right)^{2}}$

At resonance, $\omega L=\frac{1}{\omega C}$

$\therefore Z=R=20 \Omega$

Current in the circuit can be calculated as:

$I=\frac{V}{Z}$

$=\frac{200}{20}=10 \mathrm{~A}$

Hence, the average power transferred to the circuit in one complete cycle= VI

= 200 × 10 = 2000 W.