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?
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.
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