In a sinusoidal wave, average current is always ________ rms current.

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Examoogle · Accepted Answer

Less than

Explanation: For a pure sinusoidal current $I(t) = I_m \sin(\omega t)$, the average value over a full cycle is zero; however, in electrical engineering, 'average current' typically refers to the average value over a half-cycle ($I_{avg} = \frac{2I_m}{\pi} \approx 0.637I_m$). The root-mean-square (RMS) current is defined as $I_{rms} = \frac{I_m}{\sqrt{2}} \approx 0.707I_m$, making the average value strictly less than the RMS value.

Working Principle: The RMS value is derived from the heating effect of the current, calculated by taking the square root of the mean of the squares of the instantaneous values. Because the current spends more time at lower instantaneous values compared to the squaring process, the RMS value is mathematically weighted higher than the arithmetic mean of the half-cycle. This difference is defined by the Form Factor, which is the ratio of $I_{rms}$ to $I_{avg}$, being approximately $1.11$ for a sine wave.

Key Formulas: $I_{avg} = \frac{2I_m}{\pi} \approx 0.637 I_m$ — Average current over a half-cycle; $I_{rms} = \frac{I_m}{\sqrt{2}} \approx 0.707 I_m$ — RMS current for sinusoidal waveform

Key Points: The average value of a sinusoidal wave over a full cycle is always 0.; The term 'average current' in AC circuits context refers to the mean of the absolute value over a half-cycle.; The Form Factor ($K_f = \frac{I_{rms}}{I_{avg}}$) is always greater than 1 for a sine wave.; RMS value is always higher because it accounts for the power/heating effect.

Advantages: RMS values allow direct comparison between AC and DC power dissipation.; Simplifies power calculations ($P = I_{rms}^2 R$).

Disadvantages: Average values are less useful for calculating actual power delivered to a load.; Average and RMS definitions vary significantly for non-sinusoidal waveforms.

Applications: AC motor and generator design.; Power transmission efficiency calculations.; Rectifier circuit analysis.

Additional Info: Form Factor = 1.11 is a standard value used for sinusoidal AC calculations.. If the waveform is not sinusoidal, the relationship between average and RMS changes based on the Crest Factor and Form Factor.

Answer: B is correct — The RMS value is mathematically larger than the average value of a sine wave because the squaring operation in RMS effectively weights the peak portions of the waveform more heavily.

Exam Tip: Remember the mnemonic 'RMS is Root Mean Square', which is always greater than or equal to the average for non-constant waveforms in AC circuits.

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