Editing Root mean square (section)

==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.

=== Audio Engineering ===
RMS is used in [[Audio engineer|audio engineering]] to measure signal volume, particularly in the case of audio processing. The alternative volume measurement is peak volume, in analog as the signal V<sub>pp</sub>, or in digital as the -dB peak below clipping given the encoding format. Signal RMS in this context is often used as the comparator signal for [[Compressor (audio signal processor)|compression]], which produces a "smoothing" effect in compression by responding more slowly to sharp transients like those on drums.<ref>{{Cite web |last=Lessons |first=Alex-Mixing |date=2019-09-23 |title=Peak vs RMS compression: The difference between Peak and RMS |url=https://www.mixinglessons.com/peak-vs-rms-compression/ |access-date=2025-04-07 |website=mixinglessons.com |language=en-GB}}</ref> RMS is also used as a [[Mastering (audio)|mastering]] metric to compare against other time average units such as [[LUFS|LUFs]].<ref>{{Cite web |last=Luthar |first=Margaret |date=June 12, 2024 |title=RMS levels defined: how to get consistent loudness levels in mastering |url=https://www.izotope.com/en/learn/rms-levels.html?srsltid=AfmBOooGSAJqXnQOsjQbXSHrtq3_gZaQ5ssxT-bKCbhxuT0HKR43nw9H |access-date=April 7, 2025 |website=izotope.com}}</ref>