What is the correct expression for the rms value of current?

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Examoogle · Accepted Answer

I_rms = I_m/√2 Explanation: The Root Mean Square (RMS) value of a sinusoidal current is defined as the equivalent direct current (DC) that would produce the same amount of heat dissipation in a given resistor as the alternating current over one complete cycle. For a sinusoidal waveform represented by $i(t) = I_m \sin(\omega t)$, the RMS value is mathematically derived by taking the square root of the mean of the squared function over a period. Working Principle: The RMS value is calculated using the integral formula: $I_{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} I_m^2 \sin^2(\omega t) dt}$. Solving this integral yields $I_{rms} = \sqrt{\frac{I_m^2}{T} \int_{0}^{T} \frac{1 - \cos(2\omega t)}{2} dt}$, which simplifies to $I_{rms} = \frac{I_m}{\sqrt{2}}$. This indicates that the RMS value is approximately 0.707 times the peak value of the waveform. Key Formulas: $I_{rms} = \frac{I_m}{\sqrt{2}}$ — Relation between RMS and Peak current for a pure sine wave; $P_{avg} = I_{rms}^2 R$ — Power dissipation using RMS current Key Points: RMS value is also known as the 'effective' value of alternating current.; The factor $1/\sqrt{2}$ applies only to pure sinusoidal waveforms.; For non-sinusoidal waveforms, the RMS value calculation requires integrating the specific waveform function.; Voltmeters and ammeters in AC circuits are generally calibrated to read RMS values. Advantages: Allows direct comparison of AC and DC in terms of energy/power.; Simplifies power calculation in AC circuits. Disadvantages: Cannot be applied directly to non-sinusoidal waveforms without integral calculation.; Misleading if peak-dependent behavior (like insulation breakdown) is being studied. Applications: Power system rating (e.g., 230V AC supply).; Design of heating elements and power dissipation components. Additional Info: The term 'Root Mean Square' is derived from the mathematical process: Squaring the waveform, taking the Mean (average) of the squares, and finally taking the Square root.. Option A ($I_m/2$) is the Average value for a half-wave rectified sine wave, and Option D is the Peak value itself. Answer: B is correct — The RMS value of a sinusoidal current is exactly $I_{rms} = \frac{I_m}{\sqrt{2}}$ or approximately $0.707 I_m$. Exam Tip: Remember that for a sine wave, Average Value = $0.637 I_m$ and RMS Value = $0.707 I_m$. Always distinguish between the 'Form Factor' ($1.11$) and 'Peak Factor' ($1.414$) which relate these values.

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