Editing Alternating current (section)

=== Root mean square voltage{{anchor|Root mean square}} ===
{{further|RMS amplitude}}
{{broader|Root mean square voltage}}
[[File:Sine wave 2.svg|thumb|A sine wave, over one cycle (360°). The dashed line represents the [[root mean square]] (RMS) value at <math>{\sqrt {0.5}}</math> (about 0.707).|alt=A graph of sin(x) with a dashed line at y=sin(45)]]

Below an AC waveform (with no [[DC component]]) is assumed.

The RMS voltage is the square root of the [[mean of a function|mean]] over one cycle of the square of the instantaneous voltage.

{{unordered list
| For an arbitrary periodic waveform <math>v(t)</math> of period <math>T</math>:
: <math>V_\text{rms} = \sqrt{\frac{1}{T} \int_0^{T}{[v(t)]^2 dt}}.</math>
| For a sinusoidal voltage:
: <math>\begin{align}
  V_\text{rms} &= \sqrt{\frac{1}{T} \int_0^{T}[{V_\text{peak}\sin(\omega t + \phi)]^2 dt}}\\
               &= V_\text{peak}\sqrt{\frac{1}{2T} \int_0^{T}[{1 - \cos(2\omega t + 2\phi)] dt}}\\
               &= V_\text{peak}\sqrt{\frac{1}{2T} \int_0^{T}{dt}}\\
               &= \frac{V_\text{peak}}{\sqrt {2}}
\end{align}</math>

where the [[trigonometric identity]] <math>\sin^2(x) = \frac {1 - \cos(2x)}{2}</math> has been used and the factor <math>\sqrt{2}</math> is called the [[crest factor]], which varies for different waveforms.
| For a [[triangle wave]]form centered about zero
: <math>V_\text{rms} = \frac{V_\text{peak}}{\sqrt{3}}.</math>
| For a [[Square wave (waveform)|square wave]]form centered about zero
: <math>V_\text{rms} = V_\text{peak}.</math>
}}