Root Mean Square

Root mean square (RMS) values are used in electrical engineering to describe AC signal quantities and make them comparable to DC signals.

Continuous Waveforms

Given a continuous waveform y=f(t), its RMS quantity is defined as the following:

\begin{aligned} y_{rms} = \sqrt{\frac{1}{t_2-t_1}\int_{t_1}^{t_2} {|f(t)|^2 dt}} \end{aligned}

The RMS value over a period of an AC signal is identical to the RMS value over the entire signal. Therefore, RMS calculations can be simplified using the periodic nature of AC signals.

Now supposed y=f(t) is a continuous periodic signal with the period T

\begin{aligned} y_{rms} = \sqrt{\frac{1}{T}\int_{a}^{a+T} {|f(t)|^2 dt}} \end{aligned}

where:

a = is an arbitrary point in the continuous signal

Discrete Waveforms

RMS also applies to discrete waveforms. Given a discrete waveform y=f[t], its RMS quantity is defined as the following:

\begin{aligned} y_{rms} = \sqrt{\frac{1}{n}\sum_{k=1}^{n} {|f[t_k]|^2}} \end{aligned}

Now supposed y=f[t] is a discrete periodic signal with the period T

\begin{aligned} y_{rms} &= \sqrt{\frac{1}{T}\sum_{k=a}^{a+T} {|f[t_k]|^2}} \\ \end{aligned}

where:

a = is an arbitrary sequence number in the discrete signal

RMS Values for Common Waveforms

WaveformRMS Value
y=C y_{rms}=C
y=Asinx y_{rms}=\frac{A}{\sqrt{2}}
y=Acosxy_{rms}=\frac{A}{\sqrt{2}}