Root mean square

In mathematics and its applications, the root mean square (RMS or rms) is defined as the square root of the mean square (the arithmetic mean of the squares of a set of numbers). The RMS is also known as the quadratic mean and is a particular case of the generalized mean with exponent 2. RMS can also be defined for a continuously varying function 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 direct current that would produce the same average power dissipation in a resistive load.

In estimation theory, the root mean square error 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 is the "value of the direct current that dissipates power in a resistor."

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

$x_{\mathrm {rms} }={\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)}}.$ The corresponding formula for a continuous function (or waveform) f(t) defined over the interval $T_{1}\leq t\leq T_{2}$ is

$f_{\mathrm {rms} }={\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{[f(t)]}^{2}\,dt}}},$ and the RMS for a function over all time is

$f_{\mathrm {rms} }=\lim _{T\rightarrow \infty }{\sqrt {{1 \over {T}}{\int _{0}^{T}{[f(t)]}^{2}\,dt}}}.$ 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.

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

In common waveforms A rectangular pulse wave of duty cycle D, the ratio between the pulse duration ($\tau$ ) 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 $=2{\sqrt {2}}\times {RMS}\approx 2.8\times {RMS}.$ 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 $=2{\sqrt {3}}\times {RMS}\approx 3.5\times {RMS}.$ Waveform Equation RMS
DC, constant $y=A_{0}\,$ $A_{0}\,$ Sine wave $y=A_{1}\sin(2\pi ft)\,$ ${\frac {A_{1}}{\sqrt {2}}}$ Square wave $y={\begin{cases}A_{1}&\operatorname {frac} (ft)<0.5\\-A_{1}&\operatorname {frac} (ft)>0.5\end{cases}}$ $A_{1}\,$ DC-shifted square wave $y=A_{0}+{\begin{cases}A_{1}&\operatorname {frac} (ft)<0.5\\-A_{1}&\operatorname {frac} (ft)>0.5\end{cases}}$ ${\sqrt {A_{0}^{2}+A_{1}^{2}}}\,$ Modified sine wave $y={\begin{cases}0&\operatorname {frac} (ft)<0.25\\A_{1}&0.25<\operatorname {frac} (ft)<0.5\\0&0.5<\operatorname {frac} (ft)<0.75\\-A_{1}&\operatorname {frac} (ft)>0.75\end{cases}}$ ${\frac {A_{1}}{\sqrt {2}}}$ Triangle wave $y=\left|2A_{1}\operatorname {frac} (ft)-A_{1}\right|$ $A_{1} \over {\sqrt {3}}$ Sawtooth wave $y=2A_{1}\operatorname {frac} (ft)-A_{1}\,$ $A_{1} \over {\sqrt {3}}$ Pulse train $y={\begin{cases}A_{1}&\operatorname {frac} (ft)D\end{cases}}$ $A_{1}{\sqrt {D}}$ Phase-to-phase voltage $y=A_{1}\sin(t)-A_{1}\sin \left(t-{\frac {2\pi }{3}}\right)\,$ $A_{1}{\sqrt {\frac {3}{2}}}$ Notes:
• 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 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).

$RMS_{\text{Total}}={\sqrt {{RMS_{1}}^{2}+{RMS_{2}}^{2}+\cdots +{RMS_{n}}^{2}}}$ (If the waveforms are in phase, then their RMS amplitudes sum directly.)

Uses

In electrical engineering

Voltage

A special case of RMS of waveform combinations is:

$RMS_{\text{AC+DC}}={\sqrt {{RMS_{\text{DC}}}^{2}+{RMS_{\text{AC}}}^{2}}}$ where $RMS_{\text{DC}}$ refers to the direct current, or average, component of the signal and $RMS_{\text{AC}}$ 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:

$P=I^{2}R.$ 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:

{\begin{aligned}P_{\text{avg}}&=\left\langle I(t)^{2}R\right\rangle &&{\text{where }}\langle \cdots \rangle {\text{ denotes the mean of a function}}\\[5pt]&=R\left\langle I(t)^{2}\right\rangle &&{\text{(as }}R{\text{ does not vary over time, it can be factored out)}}\\[5pt]&=I_{\text{RMS}}^{2}R&&{\text{by definition of root-mean-square}}\end{aligned}} So, the RMS value, IRMS, 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 VRMS,

$P_{\text{Avg}}={V_{\text{RMS}}^{2} \over R}.$ This equation can be used for any periodic waveform, such as a sinusoidal or sawtooth waveform, 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:

$P_{\text{Avg}}=V_{\text{RMS}}I_{\text{RMS}}.$ Both derivations depend on voltage and current being proportional (i.e., the load, R, is purely resistive). Reactive loads (i.e., 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 sinusoidal current, as is approximately true for mains power, the RMS value is easy to calculate from the continuous case equation above. If Ip is defined to be the peak current, then:

$I_{\text{RMS}}={\sqrt {{1 \over {T_{2}-T_{1}}}\int _{T_{1}}^{T_{2}}\left[I_{\text{p}}\sin(\omega t)\right]^{2}dt}}.\,\!$ where t is time and ω is the angular frequency (ω = 2π/T, where T is the period of the wave).

Since Ip is a positive constant:

$I_{\text{RMS}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{\sin ^{2}(\omega t)}\,dt}}}.$ Using a trigonometric identity to eliminate squaring of trig function:

{\begin{aligned}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}}}\\I_{\text{RMS}}&=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{aligned}} but since the interval is a whole number of complete cycles (per definition of RMS), the sin terms will cancel out, leaving:

$I_{\text{RMS}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}\left[{t \over 2}\right]_{T_{1}}^{T_{2}}}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{{T_{2}-T_{1}} \over 2}}}={I_{\text{p}} \over {\sqrt {2}}}.$ A similar analysis leads to the analogous equation for sinusoidal voltage:

$V_{\text{RMS}}={V_{\text{p}} \over {\sqrt {2}}}.$ Where IP represents the peak current and VP represents the peak voltage.

Because of their usefulness in carrying out power calculations, listed voltages for power outlets (e.g., 120 V in the USA, or 230 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 VP = VRMS × 2, assuming the source is a pure sine wave. Thus the peak value of the mains voltage in the USA is about 120 × 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 day represents an RMS current of 5 amps in the long term.

The term "RMS power" is sometimes erroneously used 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

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 calculated using the following equation:

${v_{\text{RMS}}}={\sqrt {3RT \over {M}}}$ where R represents the ideal gas constant, 8.314 J/(mol·K), T is the temperature of the gas in kelvins, and M is the molar mass of the gas in kilograms per mole. The generally accepted terminology for speed as compared to velocity is that the former is the scalar magnitude of the latter. Therefore, although the average speed is between zero and the RMS speed, the average velocity for a stationary gas is zero.

Error

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 how far on average the error is from 0. The mean of the pairwise differences does not measure the variability of the difference, and the variability as indicated by the standard deviation is around the mean instead of 0. Therefore, the RMS of the differences is a meaningful measure of the error.

In frequency domain

The RMS can be computed in the frequency domain, using Parseval's theorem. For a sampled signal $x[n]=x(t=nT)$ , where $T$ is the sampling period,

$\sum _{n=1}^{N}{x^{2}[n]}={\frac {1}{N}}\sum _{m=1}^{N}{\bigl |}X[m]{\bigr |}^{2},$ where $X[m]=\operatorname {FFT} \{x[n]\}$ and N is the sample size, i.e. the number of observations in the sample and FFT coefficients.

In this case, the RMS computed in the time domain is the same as in the frequency domain:

$\mathrm {RMS} \{x[n]\}={\sqrt {{\frac {1}{N}}\sum _{n}{x^{2}[n]}}}={\sqrt {{\frac {1}{N^{2}}}\sum _{m}{{\bigl |}X[m]{\bigr |}}^{2}}}={\sqrt {\sum _{m}{\left|{\frac {X[m]}{N}}\right|^{2}}}}.$ Relationship to other statistics

If ${\bar {x}}$ is the arithmetic mean and $\sigma _{x}$ is the standard deviation of a population or a waveform then:

$x_{\mathrm {rms} }^{2}={\overline {x}}^{2}+\sigma _{x}^{2}={\overline {x^{2}}}.$ From this it is clear that the RMS value is always greater than or equal to the average, in that the RMS includes the "error" / square deviation 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, i.e., referring to the square root of the mean squared deviation of a signal from a given baseline or fit. This is useful for electrical engineers in calculating the "AC only" RMS of a signal. Standard deviation being the root mean square of a signal's variation about the mean, rather than about 0, the DC component is removed (i.e. RMS(signal) = Stdev(signal) if the mean signal is 0).