Which statement is true about RMS voltage relative to peak voltage in an AC cycle?

Which statement is true about RMS voltage relative to peak voltage in an AC cycle?

• A. RMS is less than peak
• B. RMS is greater than peak
• C. RMS equals peak
• D. RMS is twice peak

Answer: (A)

For a sinusoidal AC voltage, the RMS value is the effective DC value that would produce the same heating effect. Mathematically, v(t) = Vp sin(ωt). The RMS over a full cycle is V_RMS = sqrt(1/T ∫ v(t)^2 dt) = sqrt(1/T ∫ Vp^2 sin^2(ωt) dt). Since the average of sin^2 over a cycle is 1/2, this becomes V_RMS = Vp / √2 ≈ 0.707 Vp. So the RMS voltage is smaller than the peak voltage. For example, a peak of 170 V corresponds to about 120 V RMS. The peak is √2 times the RMS, not equal to or lesser than it beyond that ratio.

For a sinusoidal AC voltage, the RMS value is the effective DC value that would produce the same heating effect. Mathematically, v(t) = Vp sin(ωt). The RMS over a full cycle is V_RMS = sqrt(1/T ∫ v(t)^2 dt) = sqrt(1/T ∫ Vp^2 sin^2(ωt) dt). Since the average of sin^2 over a cycle is 1/2, this becomes V_RMS = Vp / √2 ≈ 0.707 Vp. So the RMS voltage is smaller than the peak voltage. For example, a peak of 170 V corresponds to about 120 V RMS. The peak is √2 times the RMS, not equal to or lesser than it beyond that ratio.