Question

In series LCR circuit $R=18\Omega$ and $Z=30\Omega$. An rms voltage 210 V is applied. The true power consumed is nearly

• A. 210 W
• B. 400 W
• C. 800 W
• D. 900 W

Hint:

Only resistance consumes "true" power in an AC circuit; L and C consume zero average power.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The true power in an AC circuit is only consumed by the resistor. Reactances do not consume power on average.

Step 2: Key Formula or Approach:
\[ P_{true} = V_{rms} I_{rms} \cos \phi \] where \( I_{rms} = \frac{V_{rms}}{Z} \) and Power Factor \( \cos \phi = \frac{R}{Z} \).
Combining these:
\[ P_{true} = \frac{V_{rms}^2 R}{Z^2} \]

Step 3: Detailed Explanation:
Given:
\( V_{rms} = 210 \text{ V} \)
\( R = 18 \Omega \)
\( Z = 30 \Omega \)
\[ P_{true} = \frac{(210)^2 \times 18}{(30)^2} \] \[ P_{true} = \frac{44100 \times 18}{900} \] \[ P_{true} = 49 \times 18 \] \[ P_{true} = 882 \text{ W} \] Rounding to the nearest multiple provided in options: \( 882 \approx 900 \text{ W} \).

Step 4: Final Answer:
The true power consumed is nearly 900 W.