Understanding RMS Voltage Calculation | Generated by AI
Explanation
The root mean square (RMS) voltage, \( V_{RMS} \), represents the effective value of an alternating voltage that produces the same power dissipation in a resistive load as a direct current (DC) voltage of the same magnitude. For a sinusoidal waveform \( v(t) = V_{max} \sin(\omega t) \), where \( V_{max} \) is the peak (maximum) voltage, the RMS value is derived as follows:
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Mean square value: The instantaneous voltage squared over one period \( T \) is \( v^2(t) = V_{max}^2 \sin^2(\omega t) \). The average (mean) of \( \sin^2(\omega t) \) over a period is \( \frac{1}{2} \), so the mean square voltage is \( \frac{V_{max}^2}{2} \).
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RMS calculation: Take the square root of the mean square value:
\[ V_{RMS} = \sqrt{\frac{V_{max}^2}{2}} = \frac{V_{max}}{\sqrt{2}} \] -
Numerical approximation: Since \( \sqrt{2} \approx 1.414 \), then \( \frac{1}{\sqrt{2}} \approx 0.707 \). Thus, \( V_{RMS} \approx 0.707 \times V_{max} \).
This confirms option B. The other options are incorrect:
- A overestimates (ignores averaging).
- C inverts the factor (that’s \( V_{max} = \sqrt{2} \times V_{RMS} \)).
- D underestimates (that’s the average value for a full-wave rectified sine, not RMS).