Question

An AC voltage \( V = 20 \sin 200 \pi t \) is applied to a series LCR circuit which drives a current \( I = 10 \sin \left( 200 \pi t + \frac{\pi}{3} \right) \). The average power dissipated is:

• A. \( 21.6 \, \text{W} \)
• B. \( 200 \, \text{W} \)
• C. \( 173.2 \, \text{W} \)
• D. \( 50 \, \text{W} \)

Answer 1

The Correct Option is D

To calculate the average power dissipated, we first identify the given AC voltage and current expressions:

• Voltage: \(V = 20 \sin 200 \pi t\) 
• Current: \(I = 10 \sin \left(200 \pi t + \frac{\pi}{3}\right)\)

The formula for average power in an AC circuit is:

\(P_{\text{avg}} = V_{\text{rms}} \cdot I_{\text{rms}} \cdot \cos \phi\)

Where:

• \(V_{\text{rms}}\) is the RMS voltage.
• \(I_{\text{rms}}\) is the RMS current.
• \(\phi\) is the phase angle between voltage and current.

Step 1: Calculate RMS Values

For sinusoidal waveforms, the RMS value is the peak value divided by \(\sqrt{2}\):

• \(V_{\text{rms}} = \frac{20}{\sqrt{2}} = 10\sqrt{2} \, \text{V}\)
• \(I_{\text{rms}} = \frac{10}{\sqrt{2}} = 5\sqrt{2} \, \text{A}\)

Step 2: Determine the Phase Angle (\(\phi\))

The phase difference is explicitly given as \(\frac{\pi}{3}\), so \(\phi = \frac{\pi}{3}\).

Step 3: Calculate Cosine of Phase Angle

\(\cos \phi = \cos \left( \frac{\pi}{3} \right) = \frac{1}{2}\)

Step 4: Calculate Average Power

Substitute the calculated values into the average power formula:

\(P_{\text{avg}} = V_{\text{rms}} \cdot I_{\text{rms}} \cdot \cos \phi = (10\sqrt{2}) \cdot (5\sqrt{2}) \cdot \frac{1}{2}\)

Simplifying the expression:

\(P_{\text{avg}} = 10 \cdot 5 \cdot 2 \cdot \frac{1}{2} = 50 \, \text{W}\)

Conclusion:

The average power dissipated in the circuit is \(50 \, \text{W}\).

• \(50 \, \text{W}\)