The rms value is _________ times he maximum value

Question

Examoogle · Accepted Answer

0.707

Explanation: The Root Mean Square (RMS) value of a sinusoidal alternating current or voltage is defined as $0.707$ times the peak (maximum) value. This factor is derived from the square root of the mean of the squares of the instantaneous values over one complete cycle.

Working Principle: For a sinusoidal waveform $v(t) = V_m \sin(\omega t)$, the RMS value represents the equivalent DC power dissipation in a resistive load. It is calculated by taking the square root of the average of the squared function over a period $T$: $V_{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} (V_m \sin(\omega t))^2 dt}$. Solving this integral yields $V_{rms} = \frac{V_m}{\sqrt{2}}$, where $\frac{1}{\sqrt{2}} \approx 0.707$.

Key Formulas: $V_{rms} = \frac{V_m}{\sqrt{2}} \approx 0.707 V_m$ — Relationship between RMS and Peak voltage; $I_{rms} = \frac{I_m}{\sqrt{2}} \approx 0.707 I_m$ — Relationship between RMS and Peak current

Key Points: RMS value is also known as the effective value of AC.; It produces the same amount of heat in a resistor as a DC value of the same magnitude.; The crest factor is the ratio of peak value to RMS value, which is $\sqrt{2} \approx 1.414$ for a pure sine wave.

Advantages: Provides a standardized way to compare AC and DC power.; Easily measurable with standard multimeters.

Disadvantages: Only strictly accurate for sinusoidal waveforms; non-sinusoidal waveforms require different coefficients.; Does not provide information about the instantaneous peak or average value directly.

Applications: Power system ratings (e.g., 230V AC supply).; Analysis of power dissipation in electrical heating elements.; General AC circuit analysis and design.

Additional Info: The value 1.414 is the reciprocal, representing the Peak to RMS ratio.. Option B (0.5) corresponds to the Average value of a half-wave rectified sine wave in relation to its peak.

Answer: D is correct — The RMS value of a sinusoidal wave is mathematically expressed as $\frac{1}{\sqrt{2}}$ of its peak value, which equals approximately 0.707.

Exam Tip: Remember that for a sine wave, the Form Factor is $1.11$ (RMS/Average) and the Crest Factor is $1.414$ (Peak/RMS).

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