Question

For an ac source rated at 220 V, 50 Hz, which of the following statements is correct?

• A. The peak value over a period of (1/50) s is 220 V.
• B. The average value over a period of (1/50) s is 220 V.
• C. The average value over a period of (1/50) s is 0 V.
• D. The average value over a period of (1/50) s is 220\(\sqrt{2}\) V.

Hint:

Always remember that the standard voltage rating for AC circuits is the RMS value. For a sine wave, the average value over a full cycle is 0, and the average value over a half cycle is \(\frac{2V_0}{\pi}\). Don't confuse RMS, peak, and average values.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The rated voltage of an AC source (like household mains) is its Root Mean Square (RMS) value. We need to analyze the properties of this sinusoidal AC voltage, such as its peak and average values over a full cycle.

Step 2: Key Formula or Approach:
For a sinusoidal AC voltage: - The RMS voltage is related to the peak voltage (\(V_0\)) by \(V_{rms} = \frac{V_0}{\sqrt{2}}\). - The frequency \(f\) is the reciprocal of the time period \(T\), i.e., \(T = 1/f\). - The average value of a sinusoidal function over a complete cycle is zero.

Step 3: Detailed Explanation:
Given data:
RMS voltage, \(V_{rms} = 220 \, \text{V}\).
Frequency, \(f = 50 \, \text{Hz}\).
The time period of the AC source is \(T = 1/f = 1/50 \, \text{s}\). The question asks for properties over this period, which is one full cycle.
Analysis of the options:
1. The peak value over a period of (1/50) s is 220 V.
The peak value is \(V_0 = V_{rms} \times \sqrt{2} = 220\sqrt{2}\) V. The given statement is incorrect.
2. The average value over a period of (1/50) s is 220 V.
The period of (1/50) s is one full cycle. The average value of a sine wave over a full cycle is zero. 220 V is the RMS value, not the average value. The statement is incorrect.
3. The average value over a period of (1/50) s is 0 V.
For any sinusoidal AC quantity (voltage or current), the waveform is symmetric about the time axis. The positive half-cycle has an area equal to the negative half-cycle. Therefore, the average value over one complete cycle is always zero. This statement is correct.
4. The average value over a period of (1/50) s is 220\(\sqrt{2}\) V.
This is the peak value, not the average value. The statement is incorrect.

Step 4: Final Answer:
The only correct statement is that the average value over a full period is 0 V.