Average value of current is given by

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Examoogle · Accepted Answer

$0.637I_m$

Explanation: The average value of a sinusoidal alternating current over a half-cycle is defined as the arithmetic mean of all instantaneous values. For a symmetrical sinusoidal waveform, this is calculated as $\frac{2I_m}{\pi}$, which evaluates to approximately $0.637I_m$.

Working Principle: The average value is derived by integrating the instantaneous current $i = I_m \sin(\omega t)$ over the interval from $0$ to $\pi$ and dividing by the interval duration $\pi$. Mathematically, $I_{avg} = \frac{1}{\pi} \int_{0}^{\pi} I_m \sin(	heta) d	heta = \frac{I_m}{\pi} [-\cos(	heta)]_0^{\pi} = \frac{2I_m}{\pi} \approx 0.637I_m$. Over a full cycle, the average value of a pure sinusoidal wave is zero due to the cancellation of the positive and negative half-cycles.

Key Formulas: $I_{avg} = \frac{2I_m}{\pi}$ — Average value for a half-cycle; $I_{rms} = \frac{I_m}{\sqrt{2}} \approx 0.707I_m$ — Root Mean Square value

Key Points: The average value is strictly defined for a half-cycle in AC circuits.; Form factor is defined as the ratio of RMS value to Average value, which is $\approx 1.11$ for a sine wave.; The average value over a complete period for a symmetrical AC signal is zero.

Advantages: Useful in calculating DC equivalent output in half-wave and full-wave rectifiers.; Simplifies analysis of pulsating DC components in electrical machines.

Disadvantages: Not representative of the actual power-delivering capability of the AC signal.; Mathematically zero over a full cycle, requiring half-cycle rectification for meaningful measurement.

Applications: Moving coil (PMMC) instrument calibration.; Design and analysis of rectifier circuits.

Additional Info: Option B (0.5Im) is incorrect and holds no standard physical significance for sine waves.. Option C (0.707Im) refers to the RMS (Root Mean Square) value, not the average value.

Answer: D is correct — The average value of a sinusoidal current over one half-cycle is mathematically derived as $2I_m/\pi$, resulting in $0.637I_m$.

Exam Tip: Always distinguish between Average and RMS values; recall that Form Factor = RMS/Average = 1.11 for sinusoidal signals.

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