root-mean-square voltage

root-mean-square voltage, equivalent direct current (DC) voltage of an alternating current (AC) source.

Certain electric circuits include sources of alternating electromotive forces of the sinusoidal form V = V₀ cos (ωt) or V = V₀ sin (ωt). The sine and cosine functions have values that vary between +1 and −1; either of the equations for the voltage represents a potential that varies with respect to time and has values from +V₀ to −V₀. The voltage varies with time at a rate given by the numerical value of ω; ω, which is called the angular frequency, is expressed in radians per second. The figure shows an example with V₀ = 170 volts and ω = 377 radians per second, so that V = 170 cos (377t). The time interval required for the pattern to be repeated is called the period T, given by T = 2π/ω. In the figure, the pattern is repeated every 16.7 milliseconds, which is the period. The frequency of the voltage is symbolized by f and given by f = 1/T. In terms of ω, f = ω/2π, in hertz.

The root-mean-square (rms) voltage of a sinusoidal source of electromotive force (V_rms) is used to characterize the source. It is the square root of the time average of the voltage squared. The value of V_rms is V₀/√2, or, equivalently, 0.707V₀. Thus, the 60-hertz, 120-volt alternating current, which is available from most electric outlets in American homes and which is illustrated in the figure, has V₀ = 120/0.707 = 170 volts; that is, 120 volts is the rms voltage. The potential difference at the outlet varies from +170 volts to −170 volts and back to +170 volts 60 times each second. The rms values of voltage and current are especially useful in calculating average power in AC circuits.