View source for Root mean square

{{Short description|Square root of the mean square}}
In [[mathematics]], the '''root mean square''' (abbrev. '''RMS''', '''{{sc|rms}}''' or '''rms''') of a [[set (mathematics)|set]] of numbers is the [[square root]] of the set's [[mean square]].<ref name=dicphys>{{cite book
|title=A Dictionary of Physics (6 ed.)
|chapter=Root-mean-square value
|publisher=Oxford University Press
|year=2009
|isbn=9780199233991
|url=https://www.oxfordreference.com/view/10.1093/acref/9780199233991.001.0001/acref-9780199233991-e-2676
}}</ref>
Given a set <math>x_i</math>, its RMS is denoted as either <math>x_\mathrm{RMS}</math> or <math>\mathrm{RMS}_x</math>. The RMS is also known as the '''quadratic mean''' (denoted <math>M_2</math>),<ref>{{cite book |last1=Thompson |first1=Sylvanus P. |title=Calculus Made Easy |date=1965 |publisher=Macmillan International Higher Education |isbn=9781349004874 |page=185 |url=https://books.google.com/books?id=6VJdDwAAQBAJ&pg=PA185 |access-date=5 July 2020 }}{{Dead link|date=August 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref>{{cite book |last1=Jones |first1=Alan R. |title=Probability, Statistics and Other Frightening Stuff |date=2018 |publisher=Routledge |isbn=9781351661386 |page=48 |url=https://books.google.com/books?id=OvtsDwAAQBAJ&pg=PA48 |access-date=5 July 2020}}</ref> a special case of the [[generalized mean#Quadratic|generalized mean]]. The RMS of a continuous [[function (mathematics)|function]] is denoted <math>f_\mathrm{RMS}</math> and can be defined in terms of an [[integral]] of the square of the function.
In [[estimation theory]], the [[root-mean-square deviation]] of an estimator measures how far the estimator strays from 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 the case of a set of ''n'' values <math>\{x_1,x_2,\dots,x_n\}</math>, the RMS is
:<math>
x_\text{RMS} = \sqrt{ \frac{1}{n} \left( {x_1}^2 + {x_2}^2 + \cdots + {x_n}^2 \right) }.
</math>

The corresponding formula for a continuous function (or waveform) ''f''(''t'') defined over the interval <math>T_1 \le t \le T_2</math> is
:<math>
f_\text{RMS} = \sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {[f(t)]}^2\, {\rm d}t}},
</math>

and the RMS for a function over all time is
:<math>
f_\text{RMS} = \lim_{T\rightarrow \infty} \sqrt {{1 \over {2T}} {\int_{-T}^{T} {[f(t)]}^2\, {\rm d}t}}.
</math>

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#Integral calculus|calculus]], as shown by Cartwright.<ref>{{cite journal |last=Cartwright|first=Kenneth V|title=Determining the Effective or RMS Voltage of Various Waveforms without Calculus|journal=Technology Interface|volume=8|issue=1|pages=20 pages|date=Fall 2007|url=http://tiij.org/issues/issues/fall2007/30_Cartwright/Cartwright-Waveforms.pdf}}</ref>

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

==In common waveforms==
[[File:Waveforms.svg|thumb|right|400px|[[Sine wave|Sine]], [[square wave|square]], [[triangle wave|triangle]], and [[sawtooth wave|sawtooth]] waveforms.  In each, the centerline is at 0, the positive peak is at <math>y = A_1</math> and the negative peak is at <math>y = -A_1</math> ]]
[[File:Dutycycle.svg|thumb|right|400px|A rectangular pulse wave of duty cycle D, the ratio between the pulse duration (<math>\tau</math>) and the period (T); illustrated here with ''a'' = 1.]]
[[File:Sine wave voltages.svg|thumb|right|400px|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 [[period (physics)|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'' <math> = 2 \sqrt{2} \times \text{RMS} \approx 2.8 \times \text{RMS}.</math>

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'' <math> = 2 \sqrt{3} \times \text{RMS} \approx 3.5 \times \text{RMS}.</math>

{| class="wikitable"
|-
! Waveform !! Variables and operators !! RMS
|-
| [[Direct current|DC]] || <math>y = A_0\,</math> || <math>A_0\,</math>
|-
| [[Sine wave]]
| <math>y = A_1 \sin(2\pi ft)\,</math>
| <math>\frac{A_1}{\sqrt{2}}</math>
|-
| [[Square wave]]
| <math>y = \begin{cases}
 A_1 & \operatorname{frac}(ft) < 0.5 \\
-A_1 & \operatorname{frac}(ft) > 0.5
\end{cases}</math>
| <math>A_1\,</math>
|-
| DC-shifted square wave
| <math>y = A_0 + \begin{cases}
 A_1 & \operatorname{frac}(ft) < 0.5 \\
-A_1 & \operatorname{frac}(ft) > 0.5
\end{cases}</math>
| <math>{\sqrt{A_0^2 + A_1^2}}\,</math>
|-
| [[Inverter (electrical)#Modified sine wave|Modified sine wave]]
| <math>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}</math>
| <math>\frac{A_1}{\sqrt{2}}</math>
|-
| [[Triangle wave]]
| <math>y = \left|2 A_1 \operatorname{frac}(ft) - A_1\right|</math>
| <math>A_1 \over \sqrt 3</math>
|-
| [[Sawtooth wave]]
| <math>y = 2 A_1 \operatorname{frac}(ft) - A_1\,</math>
| <math>A_1 \over \sqrt 3</math>
|-
| [[Pulse wave]]
| <math>y = \begin{cases}
 A_1 & \operatorname{frac}(ft) < D \\
 0   & \operatorname{frac}(ft) > D
\end{cases}</math>
| <math>A_1 \sqrt D</math>
|-
| [[Three-phase electric power|Phase-to-phase sine wave]]
| <math>y = A_1 \sin(t) - A_1 \sin\left(t - \frac{2\pi}{3}\right)\,</math>
| <math>A_1 \sqrt{\frac{3}{2}}</math>
|-
| colspan=3 | where: {{unbulleted list
|''y'' is displacement,
|''t'' is time,
|''f'' is frequency,
|''A{{sub|i}}'' 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''.
}}
|}
<!--fixme: add more waveforms-->

===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 functions|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).<ref>
{{cite web
| last1=Nastase |first1=Adrian S.
| title=How to Derive the RMS Value of Pulse and Square Waveforms
| url=https://masteringelectronicsdesign.com/how-to-derive-the-rms-value-of-pulse-and-square-waveforms/
| website=MasteringElectronicsDesign.com
| access-date=21 January 2015
}}</ref>
:<math>\text{RMS}_\text{Total} =\sqrt{\text{RMS}_1^2 + \text{RMS}_2^2 + \cdots + \text{RMS}_n^2}</math>

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

==Uses==

===In electrical engineering===

====Current{{Anchor|Electrical current}}====
The RMS of an [[alternating current|alternating electric current]] equals the value of constant [[direct current]] that would dissipate the same power in a [[Resistor|resistive load]].<ref name=dicphys/>

====Voltage{{Anchor|Voltage}}====
{{Further|Root mean square AC voltage}}
A special case of RMS of waveform combinations is:<ref>{{cite web |title=Make Better AC RMS Measurements with your Digital Multimeter |url=https://literature.cdn.keysight.com/litweb/pdf/5988-6916EN.pdf |website=Keysight |access-date=15 January 2019 |archive-date=15 January 2019 |archive-url=https://web.archive.org/web/20190115182511/https://literature.cdn.keysight.com/litweb/pdf/5988-6916EN.pdf |url-status=dead }}</ref>
:<math>\text{RMS}_\text{AC+DC} = \sqrt{\text{V}_\text{DC}^2 + \text{RMS}_\text{AC}^2}</math>

where <math>\text{V}_\text{DC}</math> refers to the [[direct current]] (or average) component of the signal, and <math>\text{RMS}_\text{AC}</math> is the [[alternating current]] component of the signal.

====Average electrical power{{Anchor|Average power}}====
{{Further|AC power}}
Electrical engineers often need to know the [[power (physics)|power]], ''P'', dissipated by an [[electrical resistance and conductance|electrical resistance]], ''R''. It is easy to do the calculation when there is a constant [[electric current|current]], ''I'', through the resistance. For a load of ''R'' ohms, power is given by:
:<math>P = I^2 R.</math>

However, if the current is a time-varying function, ''I''(''t''), this formula must be extended to reflect the fact that the current (and thus the instantaneous power) is varying over time. If the function is periodic (such as household AC power), it is still meaningful to discuss the ''average'' power dissipated over time, which is calculated by taking the average power dissipation:

:<math>\begin{align}
P_\text{Avg}
&= \left( I(t)^2R \right)_\text{Avg}  &&\text{where } (\cdots)_\text{Avg} \text{ denotes the temporal mean of a function} \\[3pt]
&= \left( I(t)^2 \right)_\text{Avg} R &&\text{(as } R \text{ does not vary over time, it can be factored out)} \\[3pt]
&= I_\text{RMS}^2R                     &&\text{by definition of root-mean-square}
\end{align}</math>

So, the RMS value, ''I''<sub>RMS</sub>, of the function ''I''(''t'') is the constant current that yields the same power dissipation as the time-averaged power dissipation of the current ''I''(''t'').

Average power can also be found using the same method that in the case of a time-varying [[voltage]], ''V''(''t''), with RMS value ''V''<sub>RMS</sub>,
:<math>P_\text{Avg} = {V_\text{RMS}^2 \over R}.</math>

This equation can be used for any periodic [[waveform]], such as a [[sine wave|sinusoidal]] or [[sawtooth wave]]form, allowing us to calculate the mean power delivered into a specified load.

By taking the square root of both these equations and multiplying them together, the power is found to be:
:<math>P_\text{Avg} = V_\text{RMS} I_\text{RMS}.</math>

Both derivations depend on voltage and current being proportional (that is, the load, ''R'', is purely resistive). [[Electrical reactance|Reactive]] loads (that is, loads capable of not just dissipating energy but also storing it) are discussed under the topic of [[AC power]].

In the common case of [[alternating current]] when ''I''(''t'') is a [[sine wave|sinusoidal]] current, as is approximately true for mains power, the RMS value is easy to calculate from the continuous case equation above. If ''I''<sub>p</sub> is defined to be the peak current, then:
:<math>I_\text{RMS} = \sqrt{{1 \over {T_2 - T_1}} \int_{T_1}^{T_2} \left[I_\text{p} \sin(\omega t)\right]^2 dt},</math>

where ''t'' is time and ''ω'' is the [[angular frequency]] (''ω''&nbsp;=&nbsp;2{{pi}}/''T'', where ''T'' is the period of the wave).

Since ''I''<sub>p</sub> is a positive constant and was to be squared within the integral:
:<math>I_\text{RMS} = I_\text{p} \sqrt{{1 \over {T_2 - T_1}} {\int_{T_1}^{T_2} {\sin^2(\omega t)}\, dt}}.</math>

Using a [[list of trigonometric identities|trigonometric identity]] to eliminate squaring of trig function:
:<math>\begin{align}
I_\text{RMS} &= I_\text{p} \sqrt{{1 \over {T_2 - T_1}} {\int_{T_1}^{T_2} {{1 - \cos(2\omega t) \over 2}}\, dt}} \\[3pt]
		   &= I_\text{p} \sqrt{{1 \over {T_2 - T_1}} \left[ {t \over 2} - {\sin(2\omega t) \over 4\omega} \right]_{T_1}^{T_2} }
\end{align}</math>

but since the interval is a whole number of complete cycles (per definition of RMS), the sine terms will cancel out, leaving:
:<math>I_\text{RMS} = I_\text{p} \sqrt{{1 \over {T_2 - T_1}} \left[ {{t \over 2}} \right]_{T_1}^{T_2} } = I_\text{p} \sqrt{{1 \over {T_2 - T_1}} {{{T_2 - T_1} \over 2}} } = {I_\text{p} \over \sqrt{2}}.</math>

A similar analysis leads to the analogous equation for sinusoidal voltage:
:<math>V_\text{RMS} = {V_\text{p} \over \sqrt{2}},</math>

where ''I''<sub>P</sub> represents the peak current and ''V''<sub>P</sub> represents the peak voltage.

Because of their usefulness in carrying out power calculations, listed [[voltage]]s for power outlets (for example, 120{{nbsp}}V in the US, or 230{{nbsp}}V in Europe) are almost always quoted in RMS values, and not peak values. Peak values can be calculated from RMS values from the above formula, which implies ''V''{{sub|P}}&nbsp;=&nbsp;''V''<sub>RMS</sub>&nbsp;×&nbsp;{{radic|2}}, assuming the source is a pure sine wave. Thus the peak value of the mains voltage in the USA is about 120&nbsp;×&nbsp;{{radic|2}}, or about 170 volts. The peak-to-peak voltage, being double this, is about 340 volts. A similar calculation indicates that the peak mains voltage in Europe is about 325 volts, and the peak-to-peak mains voltage, about 650 volts.

RMS quantities such as electric current are usually calculated over one cycle. However, for some purposes the RMS current over a longer period is required when calculating transmission power losses. The same principle applies, and (for example) a current of 10 amps used for 12 hours each 24-hour day represents an average current of 5 amps, but an RMS current of 7.07 amps, in the long term.

The term ''RMS power'' is sometimes erroneously used (e.g., in the audio industry) as a synonym for ''mean power'' or ''average power'' (it is proportional to the square of the RMS voltage or RMS current in a resistive load). For a discussion of audio power measurements and their shortcomings, see [[Audio power]].

===Speed===
{{Main article|Root-mean-square speed}}

In the [[physics]] of [[gas]] molecules, the '''root-mean-square speed''' is defined as the square root of the average squared-speed. The RMS speed of an ideal gas is [[Maxwell–Boltzmann distribution#Distribution for the velocity vector|calculated]] using the following equation:
:<math>v_\text{RMS} = \sqrt{3RT \over M}</math>

where ''R'' represents the [[gas constant]], 8.314&nbsp;J/(mol·K), ''T'' is the temperature of the gas in [[kelvin]]s, and ''M'' is the [[molar mass]] of the gas in kilograms per mole. In physics, speed is defined as the scalar magnitude of velocity. For a stationary gas, the average speed of its molecules can be in the order of thousands of km/h, even though the average velocity of its molecules is zero.

===Error===
{{Main article|Root-mean-square deviation}}
When two data sets — one set from theoretical prediction and the other from actual measurement of some physical variable, for instance — are compared, the RMS of the pairwise differences of the two data sets can serve as a measure of how far on average the error is from 0. The mean of the absolute values of the pairwise differences could be a useful measure of the variability of the differences. However, the RMS of the differences is usually the preferred measure, probably due to mathematical convention and compatibility with other formulae.

==In frequency domain==
The RMS can be computed in the frequency domain, using [[Parseval's theorem]]. For a sampled signal <math>x[n] = x(t=nT)</math>, where <math>T</math> is the sampling period,
:<math>\sum_{n=1}^N{x^2[n]} = \frac{1}{N}\sum_{m=1}^N \left| X[m] \right|^2,</math>

where <math>X[m] = \operatorname{DFT}\{x[n]\}</math> and ''N'' is the sample size, that is, the number of observations in the sample and DFT coefficients.

In this case, the RMS computed in the time domain is the same as in the frequency domain:
:<math>
\text{RMS}\{x[n]\}
= \sqrt{\frac{1}{N}\sum_n{x^2[n]}}
= \sqrt{\frac{1}{N^2}\sum_m{\bigl| X[m] \bigr|}^2}
= \sqrt{\sum_m{\left| \frac{X[m]}{N} \right|^2}}.
</math>

==Relationship to other statistics==
{{see also|Accuracy}}
{{QM_AM_GM_HM_inequality_visual_proof.svg}}
The [[standard deviation]] <math>\sigma_x=(x-\overline{x})_\text{rms}</math> of a [[statistical population|population]] or a [[waveform]] <math>x</math> is the RMS deviation of <math>x</math> from its [[arithmetic mean]] <math>\bar{x}</math>.  They are related to the RMS value of <math>x</math> by <ref>
{{cite book
| title=Digital signal transmission
| edition=2nd
| author1=Chris C. Bissell
| author2=David A. Chapman
| publisher=Cambridge University Press
| year=1992
| isbn=978-0-521-42557-5
| page=64
}}</ref>
:<math>\sigma_x^2 = \overline{(x-\overline{x})^2} = x_\text{rms}^2 - \overline{x}^2 </math>.  

From this it is clear that the RMS value is always greater than or equal to the average, in that the RMS includes the squared deviation (error) as well.

Physical scientists often use the term ''root mean square'' as a synonym for [[standard deviation]] when it can be assumed the input signal has zero mean, that is, referring to the square root of the mean squared deviation of a signal from a given baseline or fit.<ref>
{{cite encyclopedia
| title=Root-Mean-Square
| url=https://mathworld.wolfram.com/Root-Mean-Square.html
| last=Weisstein|first=Eric W.
| encyclopedia=[[MathWorld]]
}}</ref><ref>{{cite web
| title=ROOT, TH1:GetRMS
| url=http://root.cern.ch/root/html/TH1.html#TH1:GetRMS
| access-date=2013-07-18
| archive-date=2017-06-30
| archive-url=https://web.archive.org/web/20170630033713/http://root.cern.ch/root/html/TH1.html#TH1:GetRMS
| url-status=dead
}}</ref> This is useful for electrical engineers in calculating the "AC only" RMS of a signal. Standard deviation being the RMS of a signal's variation about the mean, rather than about 0, the [[DC component]] is removed (that is, RMS(signal) = stdev(signal) if the mean signal is 0).

==See also==
* [[Average rectified value]] (ARV)
* [[Central moment]]
* [[Geometric mean]]
* [[Glossary of mathematical symbols]]
* [[L2 norm]]
* [[Least squares]]
* [[Mean squared displacement]]
* [[Pythagorean addition]]
* [[True RMS converter]]

== Notes ==
{{reflist|group=note}} 
==References==
{{Reflist}}

==External links==
*[http://www.hifi-writer.com/he/misc/rmspower.htm A case for why RMS is a misnomer when applied to audio power]
*[http://phy.hk/wiki/englishhtm/Rms.htm A Java applet on learning RMS]
{{Machine learning evaluation metrics}}
[[Category:Means]]
[[Category:Statistical deviation and dispersion]]

[[it:Valore efficace]]