Question

A series LCR circuit with $R=40~\Omega$, $L=3H$ and $C=20~\mu F$ is connected to a 400 V AC supply with variable frequency. When the frequency of supply equals the natural frequency of the circuit, the average power transferred to the circuit in one complete cycle is:

• A. 200 W
• B. 4000 W
• C. 6000 W
• D. 800 W

Hint:

At resonance, power is maximized as $Z=R$.

Answer 1

The Correct Option is B

Step 1: The special condition here.
The supply frequency is set equal to the natural (resonant) frequency of the LCR circuit. At resonance the inductive and capacitive reactances cancel: $X_L = X_C$.
Step 2: What this does to impedance.
With the reactances cancelling, the total impedance is just the resistance: \[ Z = R = 40\ \Omega. \]
Step 3: Find the current.
The current is then simply voltage over resistance, \[ I = \frac{V}{R} = \frac{400}{40} = 10\ \text{A}. \]
Step 4: Average power formula.
At resonance the phase angle is zero, so the power factor is $1$ and the average power is $P = I^2 R$ (equivalently $VI$).
Step 5: Plug in the numbers.
\[ P = I^2 R = (10)^2 \times 40 = 100 \times 40 = 4000\ \text{W}. \]
Step 6: Conclusion.
The average power transferred per cycle is $4000\,\text{W}$. \[ \boxed{4000\ \text{W}} \]