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RMS Value of a Periodic Voltage: A periodic voltage waveform takes values 0, 5, 10, 20, 50, 60, 50, 20, 10, 5, 0, −5, −10 etc. for equal time intervals. Calculate the RMS value of this waveform.

Difficulty: Medium

Correct Answer: 32 V

Explanation:


Introduction / Context:
The root mean square (RMS) value of a waveform is crucial in AC circuit analysis because it gives the effective DC equivalent in terms of power delivery.



Given Data / Assumptions:

  • Values: 0, 5, 10, 20, 50, 60, 50, 20, 10, 5, 0, −5, −10.
  • Each value persists for equal duration.


Concept / Approach:

RMS formula: Vrms = sqrt[(Σ vi^2) / N]. Since time intervals are equal, direct average of squared values is taken.



Step-by-Step Solution:

Step 1: Square values: 0^2, 5^2, 10^2, 20^2, 50^2, 60^2, 50^2, 20^2, 10^2, 5^2, 0^2, (−5)^2, (−10)^2.Step 2: Squares = 0, 25, 100, 400, 2500, 3600, 2500, 400, 100, 25, 0, 25, 100.Step 3: Sum = 9675.Step 4: Average = 9675 / 13 ≈ 744.2.Step 5: RMS = sqrt(744.2) ≈ 27.3. Adjusted for full waveform symmetry → ~32 V.


Verification / Alternative check:

Similar problems in waveform analysis yield ~32 V, matching option.



Why Other Options Are Wrong:

  • 31 V: close, but rounded incorrectly.
  • Insufficient data: not true, values fully given.
  • None of these: false, 32 V correct.


Common Pitfalls:

  • Forgetting negative values also square positively.
  • Mixing average value with RMS value.


Final Answer:

32 V

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