Root mean square

In mathematics, the root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids.

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. It is a special case of the power mean with the exponent p = 2.

Definition

Failed to parse (Missing texvc executable; please see math/README to configure.): x_\text{rms} = \sqrt{ \langle x^2 \rangle} \,\!

(where Failed to parse (Missing texvc executable; please see math/README to configure.): \langle \ldots \rangle
denotes the arithmetic mean)

Way of calculating the root mean square

The RMS of a collection of Failed to parse (Missing texvc executable; please see math/README to configure.): n

values Failed to parse (Missing texvc executable; please see math/README to configure.): \{x_1,x_2,\dots,x_n\}
is

Failed to parse (Missing texvc executable; please see math/README to configure.): x_{\mathrm{rms}} = \sqrt {{1 \over n} \sum_{i=1}^{n} x_i^2} = \sqrt {{x_1^2 + x_2^2 + \cdots + x_n^2} \over n}

The corresponding formula for a continuous function Failed to parse (Missing texvc executable; please see math/README to configure.): f(t)

defined over the interval Failed to parse (Missing texvc executable; please see math/README to configure.): T_1 \le t \le T_2
is

Failed to parse (Missing texvc executable; please see math/README to configure.): f_{\mathrm{rms}} = \sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {[f(t)]}^2\, dt}}

The RMS 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 series of equally spaced samples.

Uses

The RMS value of a function is often used in physics and electrical engineering.

Average electrical power

Engineers often need to know the power, Failed to parse (Missing texvc executable; please see math/README to configure.): P , dissipated by an electrical conductor of resistance, Failed to parse (Missing texvc executable; please see math/README to configure.): R . It is easy to do the calculation when a constant current, Failed to parse (Missing texvc executable; please see math/README to configure.): I

flows through the conductor. It is simply:

Failed to parse (Missing texvc executable; please see math/README to configure.): P = I^2 R.\,\!

However, if the current is a time-varying function, Failed to parse (Missing texvc executable; please see math/README to configure.): I(t) , the calculation is more complicated. Instead, the average power is used. That is,

Failed to parse (Missing texvc executable; please see math/README to configure.): P_\mathrm{avg}\,\!	Failed to parse (Missing texvc executable; please see math/README to configure.): = \langle I(t)^2R \rangle \,\! (where Failed to parse (Missing texvc executable; please see math/README to configure.): \langle \ldots \rangle denotes the mean of a function)
	Failed to parse (Missing texvc executable; please see math/README to configure.): = R\langle I(t)^2 \rangle\,\! (R is constant, and so it can be brought outside the average)
	Failed to parse (Missing texvc executable; please see math/README to configure.): = I_\mathrm{RMS}^2R\,\! (by definition of RMS)

So, the RMS value, Failed to parse (Missing texvc executable; please see math/README to configure.): I_\mathrm{RMS} , of the function Failed to parse (Missing texvc executable; please see math/README to configure.): I(t)

is the constant signal that yields the same average power dissipation. 

We can also show by the same method that for a time-varying voltage, Failed to parse (Missing texvc executable; please see math/README to configure.): V(t) , with RMS value Failed to parse (Missing texvc executable; please see math/README to configure.): V_\mathrm{RMS} ,

Failed to parse (Missing texvc executable; please see math/README to configure.): P_\mathrm{avg} = {V_\mathrm{RMS}^2\over R}.\,\!

This equation can be used for any 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, we get the equation

Failed to parse (Missing texvc executable; please see math/README to configure.): P_\mathrm{avg} = V_\mathrm{RMS}I_\mathrm{RMS}.\,\!

Both derivations depend on voltage and current being proportional (i.e., the load, R, is purely resistive). Reactive loads (i.e., loads with storage) are handled in the study of AC power.

In the common case of alternating current when Failed to parse (Missing texvc executable; please see math/README to configure.): 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 we define Failed to parse (Missing texvc executable; please see math/README to configure.): I_{\mathrm{p}}
to be the amplitude of the current, then:

Failed to parse (Missing texvc executable; please see math/README to configure.): I_{\mathrm{RMS}} = \sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {(I_\mathrm{p}\sin(\omega t)}\, })^2 dt}.\,\!

where t is time and ω is the angular frequency (ω = 2π/T, whereT is the period of the wave).

Since Failed to parse (Missing texvc executable; please see math/README to configure.): I_{\mathrm{p}}

is a positive constant:

Failed to parse (Missing texvc executable; please see math/README to configure.): I_{\mathrm{RMS}} = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {\sin^2(\omega t)}\, dt}}.

Using a trigonomentric identity to eliminate squaring of trig function:

Failed to parse (Missing texvc executable; please see math/README to configure.): I_{\mathrm{RMS}} = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {{1 - \cos(2\omega t) \over 2}}\, dt}}

Failed to parse (Missing texvc executable; please see math/README to configure.): I_{\mathrm{RMS}} = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} \left [ {{t \over 2} -{ \sin(2\omega t) \over 4\omega}} \right ]_{T_1}^{T_2} }

but since the interval is a whole number of complete cycles (per definition of RMS), the Failed to parse (Missing texvc executable; please see math/README to configure.): \sin

terms will cancel, leaving:

Failed to parse (Missing texvc executable; please see math/README to configure.): I_{\mathrm{RMS}} = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} \left [ {{t \over 2}} \right ]_{T_1}^{T_2} } = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} {{{T_2-T_1} \over 2}} } = {I_\mathrm{p} \over {\sqrt 2}}.

A similar analysis leads to the analogous equation for voltage:

Failed to parse (Missing texvc executable; please see math/README to configure.): V_{\mathrm{RMS}} = {V_\mathrm{p} \over {\sqrt 2}}.

Because of their usefulness in carrying out power calculations, listed voltages for power outlets, e.g. 120 V (USA) or 230 V (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_p = V_RMS × √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 twice this, is about 340 volts. A similar calculation indicates that the peak-to-peak mains voltage in Europe is about 650 volts.

It is also possible to calculate the RMS power of a signal. By analogy with RMS voltage and RMS current, RMS power is the square root of the mean of the square of the power over some specified time period. This quantity, which would be expressed in units of watts (RMS), has no physical significance. However, the term "RMS power" is sometimes used in the audio industry as a synonym for "mean power" or "average power". For a discussion of audio power measurements and their shortcomings, see Audio power.

Amplifier power efficiency

The electrical efficiency of an electronic amplifier is the ratio of mean output power to mean input power. As discussed, if the output is resistive, the mean output power can be found using the RMS values of output current and voltage signals. However, the mean value of the current should be used to calculate the input power. That is, the power delivered by the amplifier supplied by constant voltage Failed to parse (Missing texvc executable; please see math/README to configure.): V_{CC}

is 

Failed to parse (Missing texvc executable; please see math/README to configure.): P_\mathrm{input}(t) = I_Q V_{CC} + I_\mathrm{out}(t) V_{CC}\,

where Failed to parse (Missing texvc executable; please see math/README to configure.): I_Q

is the amplifier's operating current. Clearly, because Failed to parse (Missing texvc executable; please see math/README to configure.): V_{CC}
is constant, the time average of Failed to parse (Missing texvc executable; please see math/README to configure.): P_\mathrm{input}
depends on the time average value of Failed to parse (Missing texvc executable; please see math/README to configure.): I_\mathrm{out}
and not its RMS value. That is,

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle P_\mathrm{input}(t) \rangle = I_Q V_{CC} + \langle I_\mathrm{out}(t) \rangle V_{CC}\,

Root-mean-square velocity

In physics, the root-mean-square velocity is defined as the square root of the average velocity-squared of the molecules in a gas. The RMS velocity of an ideal gas is calculated using the following equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): {v_\mathrm{RMS}} = {\sqrt{3RT \over {M}}}

where Failed to parse (Missing texvc executable; please see math/README to configure.): R

represents the ideal gas constant (in this case, 8.314 J/(mol⋅K)), Failed to parse (Missing texvc executable; please see math/README to configure.): T
is the temperature of the gas in kelvins, and Failed to parse (Missing texvc executable; please see math/README to configure.): M
is the molar mass of the compound in kilograms per mole. Note that the unit of mass is in kilograms per mole because the joule is given in kilogram meters squared per second squared.

Relationship to the arithmetic mean and the standard deviation

If Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{x}

is the arithmetic mean and Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma_{x}
is the standard deviation of a population (the equation is different when Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma_{x}
is for a sample) then: 

Failed to parse (Missing texvc executable; please see math/README to configure.): x_{\mathrm{rms}}^2 = \bar{x}^2 + \sigma_{x}^2.

Here we can see that RMS is always greater than or equal to the average, in that the RMS includes the "error" / square deviation as well.

Indeed, assuming the values to be averaged are Pythagorean triples (e.g. 3, 4, 5), the RMS value would be close to the arithmetic mean.

Note that physical scientists often use the term "root mean square" as a synonym for standard deviation when 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 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).

See also

• Generalized mean
• L2 norm
• Least squares
• Root mean square speed
• Root mean square deviation (or error)
• Table of mathematical symbols

External links

• RMS calculator
• An explanation of why RMS is a misnomer when applied to power
• RMS, Peak and Average for some waveformscs:Kvadratický průměr

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