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In mathematics and its applications, the root mean square of a set of numbers ${\displaystyle x_{i}}$ (abbreviated as RMS, RMS or rms and denoted in formulas as either ${\displaystyle x_{\mathrm {RMS} }}$ or ${\displaystyle \mathrm {RMS} _{x}}$) is defined as the square root of the mean square (the arithmetic mean of the squares) of the set.^[1] The RMS is also known as the quadratic mean (denoted ${\displaystyle M_{2}}$)^[2][3] and is a particular case of the generalized mean. The RMS of a continuously varying function (denoted ${\displaystyle f_{\mathrm {RMS} }}$) can be defined in terms of an integral of the squares of the instantaneous values during a cycle.

For alternating electric current, RMS is equal to the value of the constant direct current that would produce the same power dissipation in a resistive load.^[1] In estimation theory, the root-mean-square deviation of an estimator is a measure of the imperfection of the fit of the estimator to the data.

Definition

The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform. In physics, the RMS current value can also be defined as the "value of the direct current that dissipates the same power in a resistor."

In the case of a set of n values ${\displaystyle \{x_{1},x_{2},\dots ,x_{n}\}}$, the RMS is

${\displaystyle x_{\text{RMS}}={\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)}}.}$

The corresponding formula for a continuous function (or waveform) f(t) defined over the interval ${\displaystyle T_{1}\leq t\leq T_{2}}$ is

${\displaystyle f_{\text{RMS}}={\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{}^{2}\,{\rm {d}}t}}},}$

and the RMS for a function over all time is

${\displaystyle f_{\text{RMS}}=\lim _{T\rightarrow \infty }{\sqrt {{1 \over {2T}}{\int _{-T}^{T}{}^{2}\,{\rm {d}}t}}}.}$

The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a sample consisting of equally spaced observations. Additionally, the RMS value of various waveforms can also be determined without calculus, as shown by Cartwright.^[4]

In the case of the RMS statistic of a random process, the expected value is used instead of the mean.

In common waveforms

Sine, square, triangle, and sawtooth waveforms. In each, the centerline is at 0, the positive peak is at ${\displaystyle y=A_{1}}$ and the negative peak is at ${\displaystyle y=-A_{1}}$

A rectangular pulse wave of duty cycle D, the ratio between the pulse duration (${\displaystyle \tau }$) and the period (T); illustrated here with a = 1.

Graph of a sine wave's voltage vs. time (in degrees), showing RMS, peak (PK), and peak-to-peak (PP) voltages.

If the waveform is a pure sine wave, the relationships between amplitudes (peak-to-peak, peak) and RMS are fixed and known, as they are for any continuous periodic wave. However, this is not true for an arbitrary waveform, which may not be periodic or continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak amplitude is:

Peak-to-peak ${\displaystyle =2{\sqrt {2}}\times {\text{RMS}}\approx 2.8\times {\text{RMS}}.}$

For other waveforms, the relationships are not the same as they are for sine waves. For example, for either a triangular or sawtooth wave

Peak-to-peak ${\displaystyle =2{\sqrt {3}}\times {\text{RMS}}\approx 3.5\times {\text{RMS}}.}$

Waveform	Variables and operators	RMS
DC	${\displaystyle y=A_{0}\,}$	${\displaystyle A_{0}\,}$
Sine wave	${\displaystyle y=A_{1}\sin(2\pi ft)\,}$	${\displaystyle {\frac {A_{1}}{\sqrt {2}}}}$
Square wave	${\displaystyle y={\begin{cases}A_{1}&\operatorname {frac} (ft)<0.5\\-A_{1}&\operatorname {frac} (ft)>0.5\end{cases}}}$	${\displaystyle A_{1}\,}$
DC-shifted square wave	${\displaystyle y=A_{0}+{\begin{cases}A_{1}&\operatorname {frac} (ft)<0.5\\-A_{1}&\operatorname {frac} (ft)>0.5\end{cases}}}$	${\displaystyle {\sqrt {A_{0}^{2}+A_{1}^{2}}}\,}$
Modified sine wave	${\displaystyle y={\begin{cases}0&\operatorname {frac} (ft)<0.25\\A_{1}&0.25<\operatorname {frac} (ft)<0.5\\0&0.5<\operatorname {frac} (ft)<0.75\\-A_{1}&\operatorname {frac} (ft)>0.75\end{cases}}}$	${\displaystyle {\frac {A_{1}}{\sqrt {2}}}}$
Triangle wave	${\displaystyle y=\left|2A_{1}\operatorname {frac} (ft)-A_{1}\right|}$	${\displaystyle A_{1} \over {\sqrt {3}}}$
Sawtooth wave	${\displaystyle y=2A_{1}\operatorname {frac} (ft)-A_{1}\,}$	${\displaystyle A_{1} \over {\sqrt {3}}}$
Pulse wave	${\displaystyle y={\begin{cases}A_{1}&\operatorname {frac} (ft)<D\\0&\operatorname {frac} (ft)>D\end{cases}}}$	${\displaystyle A_{1}{\sqrt {D}}}$
Phase-to-phase voltage	${\displaystyle y=A_{1}\sin(t)-A_{1}\sin \left(t-{\frac {2\pi }{3}}\right)\,}$	${\displaystyle A_{1}{\sqrt {\frac {3}{2}}}}$
where: y is displacement, t is time, f is frequency, Ai is amplitude (peak value), D is the duty cycle or the proportion of the time period (1/f) spent high, frac(r) is the fractional part of r.

In waveform combinations

Waveforms made by summing known simple waveforms have an RMS value that is the root of the sum of squares of the component RMS values, if the component waveforms are orthogonal (that is, if the average of the product of one simple waveform with another is zero for all pairs other than a waveform times itself).^[5]

${\displaystyle {\text{RMS}}_{\text{Total}}={\sqrt {{\text{RMS}}_{1}^{2}+{\text{RMS}}_{2}^{2}+\cdots +{\text{RMS}}_{n}^{2}}}}$

Alternatively, for waveforms that are perfectly positively correlated, or "in phase" with each other, their RMS values sum directly.

Uses

In electrical engineering

Voltage

A special case of RMS of waveform combinations is:^[6]

${\displaystyle {\text{RMS}}_{\text{AC+DC}}={\sqrt {{\text{V}}_{\text{DC}}^{2}+{\text{RMS}}_{\text{AC}}^{2}}}}$

where ${\displaystyle {\text{V}}_{\text{DC}}}$ refers to the direct current (or average) component of the signal, and ${\displaystyle {\text{RMS}}_{\text{AC}}}$ is the alternating current component of the signal.

Average electrical power

Electrical engineers often need to know the power, P, dissipated by an electrical resistance, R. It is easy to do the calculation when there is a constant current, I, through the resistance. For a load of R ohms, power is defined simply as:

${\displaystyle P={}^{}}$Zdroj:https://en.wikipedia.org?pojem=Root-mean-square
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