Root mean square

The root mean square or rms is a statistical measure of the magnitude of a varying quantity. It can be calculated for a series of discrete values or for a continuously varying function. The name comes from the fact that it is the square root of the mean of the squares of the values.

The rms for a collection of N values {x₁, x₂, ... , x_N} is:

<math>

x_{rms} = \sqrt {{1 \over N} \sum_{i=1}^{N} x_i^2} = \sqrt {{x_1^2 + x_2^2 + ... + x_N^2} \over N} </math>

and the corresponding formula for a continuous function f(t) defined over the interval T₁ ≤ t ≤ T₂ is:

<math>

x_{rms} = \sqrt {{1 \over {T_2 - T_1}} {\int_{T_1}^{T_2} {[f(t)]}^2\, dt}} </math>

Uses

The rms value of a function is often used in physics. For example, we may wish to calculate the power P dissipated by an electrical conductor of resistance R. It is easy to do the calculation when a constant current I flows through the conductor. It is simply,

(1) <math>P = I^2 R</math>

But what if the current is a varying function I(t)? This is where the rms value comes in. It may be shown that the rms value of I(t) can be substituted for the constant current I in the above equation to give the mean power dissipation, thus:

(2) <math>P = I_{rms}^2 R</math>

In the common case when I(t) is a sinusoidal current, as is approximately true for mains power, the rms value is easy to calculate from equation (2) above. The result is:

<math>I_{rms} = {I_p \over {\sqrt 2}}</math>

where I_p is the amplitude.

The rms value can be calculated using equation (2) for any waveform, for example an audio or radio signal. This allows us to calculate the mean power delivered into a specified load.

It is important to note that rms is a mean value and not an instantaneous measurement. Therefore expressions such as "peak rms power", sometimes used in advertisements for audio amplifiers, are misleading. See also PMPO.