The rms value is ______ times the maximum value.

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Examoogle · Accepted Answer

0.707

Explanation: The Root Mean Square (RMS) value, also known as the effective value, is defined as 0.707 times the peak (maximum) value for a pure sinusoidal waveform. It represents the equivalent DC current that would produce the same heating effect in a resistor as the AC current.

Working Principle: For a sinusoidal signal $i(t) = I_m \sin(\omega t)$, the RMS value is calculated as the square root of the mean of the squared values over one complete cycle. Mathematically, $I_{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} (I_m \sin(\omega t))^2 dt}$. Solving this integral yields $I_{rms} = \frac{I_m}{\sqrt{2}}$, which evaluates to approximately 0.707 $I_m$.

Key Formulas: $V_{rms} = \frac{V_m}{\sqrt{2}} \approx 0.707 V_m$ — Relation between RMS and Peak value for sine waves; $V_m = \sqrt{2} V_{rms} \approx 1.414 V_{rms}$ — Relation between Peak and RMS value for sine waves

Key Points: RMS value is the standard measure for AC voltage and current in power systems.; The 0.707 factor applies specifically to sinusoidal waveforms.; Form factor for a sine wave is defined as the ratio of RMS value to Average value ($\approx 1.11$).

Advantages: Allows direct comparison of AC heating effect to DC power.; Simplifies power calculations in resistive circuits.

Disadvantages: The factor 0.707 is invalid for non-sinusoidal waveforms like square or triangular waves.; Requires integration for complex non-periodic signals.

Applications: Domestic AC supply rating (e.g., 230V AC is RMS value).; Measuring instruments like AC Voltmeters and Ammeters.

Additional Info: Option A (1.414) represents the peak factor for a sine wave (Peak/RMS).. Option C (2) is irrelevant to standard sinusoidal RMS relationships.

Answer: D is correct — The rms value of a sinusoidal waveform is exactly $\frac{1}{\sqrt{2}}$ times its maximum value, which is approximately 0.707.

Exam Tip: Remember that 0.707 is derived from $1/\sqrt{2}$, while 1.414 is $\sqrt{2}$. Always distinguish between Peak-to-RMS (0.707) and Peak-to-Peak ratios.

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