RMS (Current and Voltage)

Consider the power versus time graph of an AC signal:

The average power, Pavg, for one cycle is the area under the graph divided by 2π. So if 0≤θ≤2π, p is the instantaneous power, i is the instantaneous current, and I0 is the peak current, then

Pavg = 1/2π×∫0pidθ
    = 1/2π×∫0i2R dθ
    = R/2π×∫0i2dθ
    = R/2π×∫0I02sin2θ dθ
    = I02R/2π×∫0sin2θ dθ
    = I02R/2π×π
    = I02R/2

In the case of sinusoidal current, the Root Mean Square (RMS) of an AC source is the DC source that gives the same average power.

IRMS = I0/√2

and if the peak voltage of the AC signal is V0, then

VRMS = V0/√2

which gives us

Pavg = IRMS2R


Fischer-Cripps. A.C., The Electronics Companion. Institute of Physics, 2005.