For the same peak value, which of the following wave has the least mean value?

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Examoogle · Accepted Answer

half wave rectified sine wave

Explanation: The mean (average) value of a periodic waveform is defined as the integral of the instantaneous value over one complete period divided by the period. For a waveform with peak value $V_m$, the average value is significantly reduced when large portions of the waveform are zero or negative, leading to the half-wave rectified sine wave having the lowest mean value among the given options.

Working Principle: The mean value $V_{avg} = \frac{1}{T} \int_{0}^{T} v(t) dt$. For a full sine wave, the positive and negative cycles cancel out (mean of 0). For a half-wave rectified sine wave, only the positive half-cycle exists, resulting in $V_{avg} = \frac{V_m}{\pi} \approx 0.318 V_m$. In contrast, a triangular wave has $0.5 V_m$, and a square wave has $V_m$.

Key Formulas: $V_{avg} = \frac{V_m}{\pi}$ — Half-wave rectified sine wave; $V_{avg} = 0.5 V_m$ — Triangular wave; $V_{avg} = V_m$ — Square wave

Key Points: Average value depends on the area under the curve within one period.; The half-wave rectified sine wave effectively removes the negative cycle, drastically reducing the average compared to a full-cycle wave.; For the same peak, the square wave occupies the most area, followed by the triangular wave, then the rectified sine wave.

Applications: Power electronics rectification analysis; DC motor speed control systems; Instrumentation calibration

Additional Info: Full-wave rectified sine wave mean value is $2V_m/\pi \approx 0.637 V_m$.. Option C (Sine wave) has a mean value of 0 over a complete period.

Answer: A is correct — The half-wave rectified sine wave yields the lowest mean value ($V_m/\pi$) among the choices for a given peak value.

Exam Tip: Always verify if the question asks for the mean value over a full period or half-period; here, the standard periodic definition is assumed.

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