Question

A series LCR circuit with R = 20 W, L = 1.5 H and C = 35 μF is connected to a variable-frequency 200 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?

Answer 1

At resonance, the frequency of the supply power equalsthe 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,

\(Z=√(R^2+(X_L-X_C))^2 At resonance, X_L=X_C\)

Z=R=20Ω

Current in the circuit can be calculated as:

\(I=\frac{V}{Z}=\frac{200}{20}=10 A\)

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

VI = 200 × 10 = 2000 W.

Answer 2

At resonance, the frequency of supply power equals to naturan frequency of LCR circuit.
Impedence,
\(Z = \sqrt {R^2 + (ω_L - \frac {1}{ω_C})}\)
\(ω_L=\frac {1}{ ωc}\)
So,  \(Z = R\)
Current \(I = \frac VR\)
\(I= \frac {200}{20}\)
\(I= 10\ A\)
Power,
\(P =VI\)
\(P= 200 \times 10\)
\(P= 2000\ W\)

So, the answer is 2000 W.