For a sine wave, the peak value is equal to the RMS value multiplied by which factor?

For a sine wave, the peak value is equal to the RMS value multiplied by which factor?

• A. 1.0
• B. 0.707
• C. 2.0
• D. √2 (approximately 1.414)

Answer: (D)

The peak value relates to the RMS value by a constant factor because RMS measures the heating effect, while the peak is the maximum instantaneous value of the waveform. For a sine wave described by v(t) = Vm sin(ωt), the RMS value is found from the average of v(t)^2 over one period: Vrms^2 = (1/T) ∫0^T Vm^2 sin^2(ωt) dt. The average value of sin^2 over a full cycle is 1/2, so Vrms^2 = Vm^2/2, giving Vrms = Vm/√2. Rearranging, Vm = Vrms · √2. Since √2 is approximately 1.414, the peak value is the RMS value multiplied by √2.

The peak value relates to the RMS value by a constant factor because RMS measures the heating effect, while the peak is the maximum instantaneous value of the waveform. For a sine wave described by v(t) = Vm sin(ωt), the RMS value is found from the average of v(t)^2 over one period: Vrms^2 = (1/T) ∫0^T Vm^2 sin^2(ωt) dt. The average value of sin^2 over a full cycle is 1/2, so Vrms^2 = Vm^2/2, giving Vrms = Vm/√2. Rearranging, Vm = Vrms · √2. Since √2 is approximately 1.414, the peak value is the RMS value multiplied by √2.