Which formula defines the root-mean-square (Vrms) value in terms of Vmax?

Which formula defines the root-mean-square (Vrms) value in terms of Vmax?

• A. Vrms = Vmax × 0.5
• B. Vrms = Vmax × 0.7071
• C. Vrms = Vmax
• D. Vrms = Vmax × .707

Answer: (D)

For a sinewave, the root-mean-square value is the peak value divided by the square root of 2. If v(t) = Vmax sin(ωt), then Vrms^2 = (1/T) ∫ v(t)^2 dt over a full cycle, which evaluates to Vmax^2/2. Taking the square root gives Vrms = Vmax/√2, which is about 0.7071 times Vmax. Expressing this as Vrms = Vmax × 0.707 captures that relationship. The form with 0.5 would be incorrect for a sinewave, since it would imply Vrms = Vmax/2, not Vmax/√2. Vrms = Vmax would only apply to a constant (DC) value, not a varying sine.

For a sinewave, the root-mean-square value is the peak value divided by the square root of 2. If v(t) = Vmax sin(ωt), then Vrms^2 = (1/T) ∫ v(t)^2 dt over a full cycle, which evaluates to Vmax^2/2. Taking the square root gives Vrms = Vmax/√2, which is about 0.7071 times Vmax. Expressing this as Vrms = Vmax × 0.707 captures that relationship. The form with 0.5 would be incorrect for a sinewave, since it would imply Vrms = Vmax/2, not Vmax/√2. Vrms = Vmax would only apply to a constant (DC) value, not a varying sine.