Vrms2

Vrms2 is the square of the root-mean-square (RMS) value of an electrical voltage signal. It is often used in power analysis because it directly relates to average power dissipated in a resistor through the relation P = Vrms^2 / R.

Mathematical definition for periodic signals: if v(t) is periodic with period T, Vrms2 equals the time average of the squared voltage, Vrms2 = (1/T) ∫_0^T [v(t)]^2 dt. For a stochastic or stationary process, Vrms2 can be written as the expected value E[v(t)^2].

Relation to Vrms: Vrms is the square root of Vrms2, i.e., Vrms = sqrt(Vrms2). For a sine wave v(t) = Vp sin(ωt), Vrms = Vp/√2 and Vrms2 = Vp^2/2.

DC offset and decomposition: If a signal has a DC component Vdc and a fluctuating part v'(t) with mean square Vrms'2, then Vrms2 = Vdc^2 + Vrms'2. This reflects that the total mean square includes both the constant offset and the varying content.

Applications and interpretation: Vrms2 is a measure of the energy content of a voltage signal, with direct relevance to power calculations in linear impedances. For a resistor R, average power is P = Vrms2 / R. In Fourier analysis, Vrms2 is related to the sum of the squares of the harmonic amplitudes (Parseval's theorem), linking Vrms2 to the distribution of power among frequency components.

See also: root-mean-square, mean square, power, Fourier analysis.