NationMaster - Encyclopedia: Root mean square

In mathematics, 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's especially useful when variates are positive and negative, e.g. waves. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Template:Otherusescccc A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... In science, a magnitude is the numerical size of something: see orders of magnitude. ...

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 power mean with the power $t = 2$ . Partial plot of a function f. ... In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... In algebra, the square of x is written x2 and is defined as the product of x with itself: x × x. ... A generalized mean, also known as power mean or HÃ¶lder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means. ...

1 Calculating the root mean square
2 Uses
3 Relationship to the arithmetic mean and the standard deviation
4 See also
5 External links

Calculating the root mean square

The rms for a collection of $N$ values ${x_1,x_2,dots,x_N}$ is:

$x_{mathrm{rms}} = sqrt {{1 over N} sum_{i=1}^{N} x_i^2} = sqrt {{x_1^2 + x_2^2 + cdots + x_N^2} over N}$

and the corresponding formula for a continuous function $f (t)$ defined over the interval $T_1 le t le T_2$ (for a periodic function the interval should be a whole number of complete cycles) is:

$f_{mathrm{rms}} = sqrt {{1 over {T_2-T_1}} {int_{T_1}^{T_2} {[f(t)]}^2, dt}}$

Uses

The RMS value of a function is often used in physics and electronics. For example, we may wish to calculate the power $P$ dissipated by an electrical conductor of resistance $R$ . It is easy to do the calculation when a constant current $I$ flows through the conductor. It is simply: Physics (from the Greek, (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time. ... The field of electronics comprises the study and use of systems that operate by controlling the flow of electrons (or other charge carriers) in devices such as thermionic valves (vacuum tubes) and semiconductors. ... In physics, power (symbol: P) is the rate at which work is performed. ... In electricity, current refers to electric current, which is the flow of electric charge. ...

$P = I^2 R,!$

But what if the current is a varying function $I (t)$ ? This is where the rms value comes in. It may be trivially shown that the rms value of $I (t)$ can be substituted for the constant current $I$ in the above equation to give the average power dissipation:

$P_mathrm{avg},!$	$= langle I^2R rangle ,!$ (where $langle cdot rangle$ denotes the arithmetic mean)
	$= Rlangle I^2 rangle,!$ (R is constant so we can take it outside the average)
	$= I_mathrm{rms}^2R,!$ (by definition of RMS)

We can also show by the same method In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...

$P_mathrm{avg} = {V_mathrm{rms}^2over R},!$

By taking the square root of both these equations and multiplying them together, we get the equation

$P_mathrm{avg} = V_mathrm{rms}I_mathrm{rms},!$

However, it is important to stress that this is based on the assumption that voltage and current are proportional (that is the load is resistive) and is not true in the general case (see AC power for more information). Usually hidden to the unaided eye, the 60Hz blinking of (non-incandescent) lighting powered by AC mains is revealed in this motion-blurred long exposure of city lights. ...

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 we define $I p$ to be the peak amplitude: City lights viewed in a motion blurred exposure. ... In trigonometry, an ideal sine wave is a waveform whose graph is identical to the generalized sine function y = Asin[ω(x − α)] + C, where A is the amplitude, ω is the angular frequency (2π/P where P is the wavelength), α is the phase shift, and C...

$I_{mathrm{rms}} = sqrt {{1 over {T_2-T_1}} {int_{T_1}^{T_2} {(I_mathrm{p}sin(omega t)}, })^2 dt},!$

Since $I p$ is a positive real number:

$I_{mathrm{rms}} = I_mathrm{p}sqrt {{1 over {T_2-T_1}} {int_{T_1}^{T_2} {sin^2(omega t)}, dt}}$

Using a trigonomentric identity to eliminate squaring of trig function:

$I_{mathrm{rms}} = I_mathrm{p}sqrt {{1 over {T_2-T_1}} {int_{T_1}^{T_2} {{1 - cos(2omega t) over 2}}, dt}}$

$I_{mathrm{rms}} = I_mathrm{p}sqrt {{1 over {T_2-T_1}} left [ {{t over 2} -{ sin(2omega t) over 4omega}} right ]_{T_1}^{T_2} }$

but since the interval is a whole number of complete cycles (per definition of rms for a periodic function) the sin terms will cancel

$I_{mathrm{rms}} = I_mathrm{p}sqrt {{1 over {T_2-T_1}} left [ {{t over 2}} right ]_{T_1}^{T_2} } = I_mathrm{p}sqrt {{1 over {T_2-T_1}} {{{T_2-T_1} over 2}} } = {I_mathrm{p} over {sqrt 2}}$

The peak amplitude is half of the peak-to-peak amplitude. When the peak-to-peak amplitude is known, the same formula is applied by using half of the p-p value.

The RMS value can be calculated using equation (2) for any waveform, for example an audio or radio signal. This allows us to calculate the mean power delivered into a specified load. For this reason, listed voltages for power outlets (e.g. 110 V or 240 V) are almost always quoted in RMS values, and not peak values. International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ...

From the formula given above, we can calculate also the peak-to-peak value of the mains voltage which is about 310 volts (USA) and 677 volts (Europe) respectively.

It is 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 and no practical use. However, the term "RMS power" is sometimes used in the consumer audio industry as an incorrect synonym for "mean power" or "average power". For a discussion of audio power measurements and their shortcomings, see Audio power. The term sine power is used in the specification or measurement of audio amplifiers or loudspeakers. ...

In chemistry, 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: Chemistry (from Greek Ï‡Î·Î¼ÎµÎ¯Î± khemeia[1] meaning alchemy) is the science of matter at the atomic to molecular scale, dealing primarily with collections of atoms, such as gases, molecules, crystals, and metals. ... A gas is one of the four major phases of matter (after solid and liquid, and followed by plasma, that subsequently appear as a solid material is subjected to increasingly higher temperatures. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...

${u_mathrm{rms}} = {sqrt{3RT over {M}}}$

where $R$ represents the ideal gas constant (in this case, 8.314 J/(mol⋅K)), $T$ is the temperature of the gas in kelvins, and $M$ is the molar mass of the compound in kilograms per mole. Molar gas constant (also known as universal gas constant, usually denoted by symbol R) is the constant occurring in the universal gas equation, i. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zeroâ€”the lowest possible temperature where nothing could be colder and no heat energy remains in a substanceâ€”is defined as zero kelvin (0 K). ... Molar mass is the mass of one mole of a chemical element or chemical compound. ...

Relationship to the arithmetic mean and the standard deviation

If $bar{x}$ is the arithmetic mean and $σ x$ is the standard deviation of a population then In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is defined as the square root of the variance. ...

$x_{mathrm{rms}}^2 = bar{x}^2 + sigma_{x}^2$

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

External links

RMS calculator
RMS values and measurement
An explanation of why RMS is a misnomer when applied to power
RMS Power Explained
RMS, Peak and Average for some waveforms

Category: Statistical deviation and dispersion

Results from FactBites:

NationMaster - Encyclopedia: Root mean square (1527 words)
	In mathematics, root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity.
	In chemistry, the root mean square velocity is defined as the square root of the average velocity-squared of the molecules in a gas.
	Root mean square speed is a measure of the velocity of particles in a gas.

Smart Computing Encyclopedia Entry - root mean square (rms) (230 words)
	Root mean square, usually referred to simply as rms, is a mathematical term prevalent in many different fields, including electrical engineering and statistics.
	The term refers to the process of deriving the square root of the average of a group of numbers and is one of the primary methods of measuring electrical output in volts and watts of an AC wave.
	However, obtaining a root mean square measurement is considered a far better indicator of the quality of the speakers, not to mention a better way to rate the amplifier power.

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Contents

Calculating the root mean square GA_googleFillSlot("encyclopedia_square");

Uses

Relationship to the arithmetic mean and the standard deviation

See also

External links

Calculating the root mean square