Root mean square

Root mean square

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{{other uses|RMS}} In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

 measure of the magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....

 of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids. RMS is used in various fields, including electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

. It can be calculated for a series of discrete values or for a continuously varying function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. The name comes from the fact that it is the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

 of the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

 of the squares of the values. It is a special case of the generalized mean
Generalized mean
In mathematics, a generalized mean, also known as power mean or Hölder mean , is an abstraction of the Pythagorean means including arithmetic, geometric, and harmonic means.-Definition:...

 with the exponent p = 2.

Definition

The RMS value of a set of values (or a continuous-time waveform
Waveform
Waveform means the shape and form of a signal such as a wave moving in a physical medium or an abstract representation.In many cases the medium in which the wave is being propagated does not permit a direct visual image of the form. In these cases, the term 'waveform' refers to the shape of a graph...

) is the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

 of the arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

 (average
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....

) of the squares of the original values (or the square of the function that defines the continuous waveform). In the case of a set of n values \{x_1,x_2,\dots,x_n\}, the RMS value is given by: 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 \le t \le 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 {2T}} {\int_{-T}^{T} {[f(t)]}^2\, dt}}. The RMS over all time of a periodic function
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

 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. 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
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

 is used instead of the mean.

RMS of common waveforms

NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
WaveformEquationRMS
DC
Direct current
Direct current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...

, constant
y=a\,a\,
Sine wave
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

y=a\sin(2\pi ft)\,\frac{a}{\sqrt{2}}
Square wave
Square wave
A square wave is a kind of non-sinusoidal waveform, most typically encountered in electronics and signal processing. An ideal square wave alternates regularly and instantaneously between two levels...

y=\begin{cases}a & \{ft\} < 0.5 \\ -a & \{ft\} > 0.5 \end{cases}a\,
Modified square wave
Inverter (electrical)
An inverter is an electrical device that converts direct current to alternating current ; the converted AC can be at any required voltage and frequency with the use of appropriate transformers, switching, and control circuits....

y=\begin{cases}0 & \{ft\} < 0.25 \\ a & 0.25 < \{ft\} < 0.5 \\ 0 & 0.5 < \{ft\} < 0.75 \\ -a & \{ft\} > 0.75 \end{cases}\frac{a}{\sqrt{2}}
Sawtooth wave
Sawtooth wave
The sawtooth wave is a kind of non-sinusoidal waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a saw....

y=2a\{ft\}-a\, a \over \sqrt 3
Triangle wave
Triangle wave
A triangle wave is a non-sinusoidal waveform named for its triangular shape.Like a square wave, the triangle wave contains only odd harmonics...

y=>2a\{ft\}-a\,| a \over \sqrt 3
Pulse train y=\begin{cases}a & \{ft\} < D \\ 0 & \{ft\} > D \end{cases}a \sqrt{D}
Notes:
t is time
f is frequency
a is amplitude (peak value)
D is the duty cycle
Duty cycle
In engineering, the duty cycle of a machine or system is the time that it spends in an active state as a fraction of the total time under consideration....

 or the percent(%) spent high of the period (1/f)
{r} is the fractional part
Fractional part
All real numbers can be written in the form n + r where n is an integer and the remaining fractional part r is a nonnegative real number less than one...

 of r
NEWLINENEWLINE Complex wave forms made from common known wave forms have a 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_{Total} = \sqrt {{other uses|RMS}} In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

 measure of the magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....

 of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids. RMS is used in various fields, including electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

. It can be calculated for a series of discrete values or for a continuously varying function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. The name comes from the fact that it is the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

 of the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

 of the squares of the values. It is a special case of the generalized mean
Generalized mean
In mathematics, a generalized mean, also known as power mean or Hölder mean , is an abstraction of the Pythagorean means including arithmetic, geometric, and harmonic means.-Definition:...

 with the exponent p = 2.

Definition

The RMS value of a set of values (or a continuous-time waveform
Waveform
Waveform means the shape and form of a signal such as a wave moving in a physical medium or an abstract representation.In many cases the medium in which the wave is being propagated does not permit a direct visual image of the form. In these cases, the term 'waveform' refers to the shape of a graph...

) is the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

 of the arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

 (average
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....

) of the squares of the original values (or the square of the function that defines the continuous waveform). In the case of a set of n values \{x_1,x_2,\dots,x_n\}, the RMS value is given by: 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 \le t \le 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 {2T}} {\int_{-T}^{T} {[f(t)]}^2\, dt}}. The RMS over all time of a periodic function
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

 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. 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
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

 is used instead of the mean.

RMS of common waveforms

NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
WaveformEquationRMS
DC
Direct current
Direct current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...

, constant
y=a\,a\,
Sine wave
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

y=a\sin(2\pi ft)\,\frac{a}{\sqrt{2}}
Square wave
Square wave
A square wave is a kind of non-sinusoidal waveform, most typically encountered in electronics and signal processing. An ideal square wave alternates regularly and instantaneously between two levels...

y=\begin{cases}a & \{ft\} < 0.5 \\ -a & \{ft\} > 0.5 \end{cases}a\,
Modified square wave
Inverter (electrical)
An inverter is an electrical device that converts direct current to alternating current ; the converted AC can be at any required voltage and frequency with the use of appropriate transformers, switching, and control circuits....

y=\begin{cases}0 & \{ft\} < 0.25 \\ a & 0.25 < \{ft\} < 0.5 \\ 0 & 0.5 < \{ft\} < 0.75 \\ -a & \{ft\} > 0.75 \end{cases}\frac{a}{\sqrt{2}}
Sawtooth wave
Sawtooth wave
The sawtooth wave is a kind of non-sinusoidal waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a saw....

y=2a\{ft\}-a\, a \over \sqrt 3
Triangle wave
Triangle wave
A triangle wave is a non-sinusoidal waveform named for its triangular shape.Like a square wave, the triangle wave contains only odd harmonics...

y=>2a\{ft\}-a\,| a \over \sqrt 3
Pulse train y=\begin{cases}a & \{ft\} < D \\ 0 & \{ft\} > D \end{cases}a \sqrt{D}
Notes:
t is time
f is frequency
a is amplitude (peak value)
D is the duty cycle
Duty cycle
In engineering, the duty cycle of a machine or system is the time that it spends in an active state as a fraction of the total time under consideration....

 or the percent(%) spent high of the period (1/f)
{r} is the fractional part
Fractional part
All real numbers can be written in the form n + r where n is an integer and the remaining fractional part r is a nonnegative real number less than one...

 of r
NEWLINENEWLINE Complex wave forms made from common known wave forms have a 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_{Total} = \sqrt {{other uses|RMS}} In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

 measure of the magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....

 of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids. RMS is used in various fields, including electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

. It can be calculated for a series of discrete values or for a continuously varying function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. The name comes from the fact that it is the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

 of the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

 of the squares of the values. It is a special case of the generalized mean
Generalized mean
In mathematics, a generalized mean, also known as power mean or Hölder mean , is an abstraction of the Pythagorean means including arithmetic, geometric, and harmonic means.-Definition:...

 with the exponent p = 2.

Definition

The RMS value of a set of values (or a continuous-time waveform
Waveform
Waveform means the shape and form of a signal such as a wave moving in a physical medium or an abstract representation.In many cases the medium in which the wave is being propagated does not permit a direct visual image of the form. In these cases, the term 'waveform' refers to the shape of a graph...

) is the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

 of the arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

 (average
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....

) of the squares of the original values (or the square of the function that defines the continuous waveform). In the case of a set of n values \{x_1,x_2,\dots,x_n\}, the RMS value is given by: 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 \le t \le 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 {2T}} {\int_{-T}^{T} {[f(t)]}^2\, dt}}. The RMS over all time of a periodic function
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

 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. 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
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

 is used instead of the mean.

RMS of common waveforms

NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
WaveformEquationRMS
DC
Direct current
Direct current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...

, constant
y=a\,a\,
Sine wave
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

y=a\sin(2\pi ft)\,\frac{a}{\sqrt{2}}
Square wave
Square wave
A square wave is a kind of non-sinusoidal waveform, most typically encountered in electronics and signal processing. An ideal square wave alternates regularly and instantaneously between two levels...

y=\begin{cases}a & \{ft\} < 0.5 \\ -a & \{ft\} > 0.5 \end{cases}a\,
Modified square wave
Inverter (electrical)
An inverter is an electrical device that converts direct current to alternating current ; the converted AC can be at any required voltage and frequency with the use of appropriate transformers, switching, and control circuits....

y=\begin{cases}0 & \{ft\} < 0.25 \\ a & 0.25 < \{ft\} < 0.5 \\ 0 & 0.5 < \{ft\} < 0.75 \\ -a & \{ft\} > 0.75 \end{cases}\frac{a}{\sqrt{2}}
Sawtooth wave
Sawtooth wave
The sawtooth wave is a kind of non-sinusoidal waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a saw....

y=2a\{ft\}-a\, a \over \sqrt 3
Triangle wave
Triangle wave
A triangle wave is a non-sinusoidal waveform named for its triangular shape.Like a square wave, the triangle wave contains only odd harmonics...

y=>2a\{ft\}-a\,| a \over \sqrt 3
Pulse train y=\begin{cases}a & \{ft\} < D \\ 0 & \{ft\} > D \end{cases}a \sqrt{D}
Notes:
t is time
f is frequency
a is amplitude (peak value)
D is the duty cycle
Duty cycle
In engineering, the duty cycle of a machine or system is the time that it spends in an active state as a fraction of the total time under consideration....

 or the percent(%) spent high of the period (1/f)
{r} is the fractional part
Fractional part
All real numbers can be written in the form n + r where n is an integer and the remaining fractional part r is a nonnegative real number less than one...

 of r
NEWLINENEWLINE Complex wave forms made from common known wave forms have a 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_{Total} = \sqrt {{{RMS_1}^2 + {RMS_2}^2 + \cdots + {RMS_n}^2} }

Average electrical power

Electrical engineers often need to know the power
Power (physics)
In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

, P, dissipated by an electrical resistance, R. It is easy to do the calculation when there is a constant current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

, 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 nonetheless still meaningful to talk about the average power dissipated over time, which we calculate by taking the simple average of the power at each instant in the waveform or, equivalently, the squared current. That is,NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
P_\mathrm{avg}\,\!= \langle I(t)^2R \rangle \,\! (where \langle \ldots \rangle denotes the mean of a function)
= R\langle I(t)^2 \rangle\,\! (as R does not vary over time, it can be factored out)
= (I_\mathrm{RMS})^2R\,\! (by definition of RMS)
NEWLINENEWLINE So, the RMS value, I_\mathrm{RMS}, of the function I(t) is the constant signal that yields the same power dissipation as the time-averaged power dissipation of the current I(t). We can also show by the same method that for a time-varying voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

, V(t), with RMS value V_\mathrm{RMS}, P_\mathrm{avg} = {(V_\mathrm{RMS})^2\over R}.\,\! This equation can be used for any periodic waveform
Waveform
Waveform means the shape and form of a signal such as a wave moving in a physical medium or an abstract representation.In many cases the medium in which the wave is being propagated does not permit a direct visual image of the form. In these cases, the term 'waveform' refers to the shape of a graph...

, such as a sinusoidal
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

 or sawtooth wave
Sawtooth wave
The sawtooth wave is a kind of non-sinusoidal waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a saw....

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, we get the equation 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 capable of not just dissipating energy but also storing it) are discussed under the topic of AC power
AC power
Power in an electric circuit is the rate of flow of energy past a given point of the circuit. In alternating current circuits, energy storage elements such as inductance and capacitance may result in periodic reversals of the direction of energy flow...

. In the common case of alternating current
Alternating current
In alternating current the movement of electric charge periodically reverses direction. In direct current , the flow of electric charge is only in one direction....

 when I(t) is a sinusoidal
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

 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_{\mathrm{p}} to be the peak current, then: 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
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

 (ω = 2π/T, whereT is the period of the wave). Since I_{\mathrm{p}} is a positive constant: 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 trigonometric 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(2\omega t) \over 2}}\, dt}} 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 \sin terms will cancel out, leaving: 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 sinusoidal voltage: V_{\mathrm{RMS}} = {V_\mathrm{p} \over {\sqrt 2}}. Where I_{\mathrm{P}} represents the peak current and V_{\mathrm{P}} represents the peak voltage. It bears repeating that these two solutions are for a sinusoidal wave only. Because of their usefulness in carrying out power calculations, listed voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

s 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 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 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
Audio power
Audio power is the electrical power transferred from an audio amplifier to a loudspeaker, measured in watts. The electrical power delivered to the loudspeaker, together with its sensitivity, determines the sound power level generated .Amplifiers are limited in the electrical energy they can...

.

Amplifier power efficiency

The electrical efficiency of an electronic amplifier
Electronic amplifier
An electronic amplifier is a device for increasing the power of a signal.It does this by taking energy from a power supply and controlling the output to match the input signal shape but with a larger amplitude...

 is the ratio of mean output power to mean input power. The efficiency of amplifiers is of interest when the energy used is significant, as in high-power amplifiers, or when the power-supply is taken from a battery, as in a transistor-radio. Efficiency is normally measured under steady-state conditions with a sinusoidal current delivered to a resistive load. The power output is the product of the measured voltage and current (both RMS) delivered to the load. The input power is the power delivered by the DC supply, i.e. the supply voltage multiplied by the supply current. The efficiency is then the output power divided by the input power, and it is always a number less than 1, or, in percentages, less than 100. A good radio frequency
Radio frequency
Radio frequency is a rate of oscillation in the range of about 3 kHz to 300 GHz, which corresponds to the frequency of radio waves, and the alternating currents which carry radio signals...

 power amplifier can achieve an efficiency of 60–80%. Other definitions of efficiency are possible for time-varying signals. 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
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

 V_{CC} is P_\mathrm{input}(t) = I_Q V_{CC} + I_\mathrm{out}(t) V_{CC}\, where I_Q is the amplifier's operating current. Clearly, because V_{CC} is constant, the time average of P_\mathrm{input} depends on the time average value of I_\mathrm{out} and not its RMS value. That is,\langle P_\mathrm{input}(t) \rangle = I_Q V_{CC} + \langle I_\mathrm{out}(t) \rangle V_{CC}.\,

Root-mean-square speed

{{main|Root-mean-square speed}} In the physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

 of gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...

 molecules, the root-mean-square speed is defined as the square root of the average speed-squared. The RMS speed of an ideal gas is calculated using the following equation: {v_\mathrm{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 kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

s, and M is the molar mass
Molar mass
Molar mass, symbol M, is a physical property of a given substance , namely its mass per amount of substance. The base SI unit for mass is the kilogram and that for amount of substance is the mole. Thus, the derived unit for molar mass is kg/mol...

 of the gas in kilograms. 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.

Root-mean-square error

{{Main|Root-mean-square 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
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

 of the pairwise differences does not measure the variability of the difference, and the variability as indicated by the standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

 is around the mean instead of 0. Therefore, the RMS of the differences is a meaningful measure of the error.

RMS in frequency domain

The RMS can be computed also in frequency domain. The Parseval's theorem
Parseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later...

 is used. For sampled signal: \sum\limits_{n}{{{x}^{2}}(t)}=\frac{\sum\limits_{n}{{{\left| X(f) \right|}^{2}}}}{n}, where X(f)=FFT\{x(t)\}, n is number of x(t) samples. In this case, the RMS computed in time domain is the same as in frequency domain: RMS =\sqrt{\frac{1}{n}\sum\limits_{n}{{{x}^{2}}(t)}} = \frac{1}{n}\sqrt{\sum\limits_{n}{{{\left| X(f) \right|}^{2}}}} = \sqrt{\sum\limits_{n}{{{ \left|\frac{X(f)}{n}\right| ^ 2 }}}}

Relationship to the arithmetic mean and the standard deviation

If \bar{x} is the arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

 and \sigma_{x} is the standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

 of a population
Statistical population
A statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest...

 or a waveform
Waveform
Waveform means the shape and form of a signal such as a wave moving in a physical medium or an abstract representation.In many cases the medium in which the wave is being propagated does not permit a direct visual image of the form. In these cases, the term 'waveform' refers to the shape of a graph...

 then: x_{\mathrm{rms}}^2 = \bar{x}^2 + \sigma_{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
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

 when referring to the square root of the mean squared deviation of a signal from a given baseline or fit.{{Citation needed|date=January 2011}} 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).

See also

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    NEWLINE
  • L2 norm
  • NEWLINE
  • Least squares
    Least squares
    The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...

  • NEWLINE
  • Mean squared error
    Mean squared error
    In statistics, the mean squared error of an estimator is one of many ways to quantify the difference between values implied by a kernel density estimator and the true values of the quantity being estimated. MSE is a risk function, corresponding to the expected value of the squared error loss or...

  • NEWLINE
  • Root mean square deviation
    Root mean square deviation
    The root-mean-square deviation is the measure of the average distance between the atoms of superimposed proteins...

  • NEWLINE
  • Table of mathematical symbols
    Table of mathematical symbols
    This is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in , as an image.-Symbols:-Variations:...

  • NEWLINE
  • True RMS converter
    True RMS converter
    When measuring the value of an alternating current signal it is often necessary to convert the signal into a direct current signal of equivalent value . This process can be quite complex...

  • NEWLINE
  • Geometric mean
    Geometric mean
    The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...

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External links

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