{{other uses|RMS}}
In

mathematicsMathematics 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

statisticalStatistics 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

magnitudeThe 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 engineeringElectrical 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

functionIn 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 rootIn 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

meanIn 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 meanIn 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

waveformWaveform 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 rootIn 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 meanIn 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...

(

averageIn 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

$\backslash \{x\_1,x\_2,\backslash dots,x\_n\backslash \}$, the RMS value is given by:

$x\_\{\backslash mathrm\{rms\}\}\; =\; \backslash sqrt\; \{\backslash frac\{1\}\{n\}\backslash left(\{\{x\_1\}^2\; +\; \{x\_2\}^2\; +\; \backslash cdots\; +\; \{x\_n\}^2\}\backslash right)\}$
The corresponding formula for a continuous function (or waveform)

$f(t)$ defined over the interval

$T\_1\; \backslash le\; t\; \backslash le\; T\_2$ is

$f\_\{\backslash mathrm\{rms\}\}\; =\; \backslash sqrt\; \{\{1\; \backslash over\; \{T\_2-T\_1\}\}\; \{\backslash int\_\{T\_1\}^\{T\_2\}\; \{[f(t)]\}^2\backslash ,\; dt\}\},$
and the RMS for a function over all time is

$f\_\backslash mathrm\{rms\}\; =\; \backslash lim\_\{T\backslash rightarrow\; \backslash infty\}\; \backslash sqrt\; \{\{1\; \backslash over\; \{2T\}\}\; \{\backslash int\_\{-T\}^\{T\}\; \{[f(t)]\}^2\backslash ,\; dt\}\}.$
The RMS over all time of a

periodic functionIn 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 valueIn 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

Waveform | Equation | RMS |
---|

DCDirect 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 | $y=a\backslash ,$ | $a\backslash ,$ |

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...
| $y=a\backslash sin(2\backslash pi\; ft)\backslash ,$ | $\backslash frac\{a\}\{\backslash sqrt\{2\}\}$ |

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...
| $y=\backslash begin\{cases\}a\; \&\; \backslash \{ft\backslash \}\; <\; 0.5\; \backslash \backslash \; -a\; \&\; \backslash \{ft\backslash \}\; >\; 0.5\; \backslash end\{cases\}$ | $a\backslash ,$ |

Modified square waveAn 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....
| $y=\backslash begin\{cases\}0\; \&\; \backslash \{ft\backslash \}\; <\; 0.25\; \backslash \backslash \; a\; \&\; 0.25\; <\; \backslash \{ft\backslash \}\; <\; 0.5\; \backslash \backslash \; 0\; \&\; 0.5\; <\; \backslash \{ft\backslash \}\; <\; 0.75\; \backslash \backslash \; -a\; \&\; \backslash \{ft\backslash \}\; >\; 0.75\; \backslash end\{cases\}$ | $\backslash frac\{a\}\{\backslash sqrt\{2\}\}$ |

Sawtooth waveThe 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....
| $y=2a\backslash \{ft\backslash \}-a\backslash ,$ | $a\; \backslash over\; \backslash sqrt\; 3$ |

Triangle waveA 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\,| | $a\; \backslash over\; \backslash sqrt\; 3$ |

Pulse train | $y=\backslash begin\{cases\}a\; \&\; \backslash \{ft\backslash \}\; <\; D\; \backslash \backslash \; 0\; \&\; \backslash \{ft\backslash \}\; >\; D\; \backslash end\{cases\}$ | $a\; \backslash sqrt\{D\}$ |

Notes:
*t* is time
*f* is frequency
*a* is amplitude (peak value)
*D* is the duty cycleIn 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 partAll 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).

$RMS\_\{Total\}\; =\; \backslash sqrt\; \{\{other\; uses|RMS\}\}\; Inmathematics$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

statisticalStatistics 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

magnitudeThe 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 engineeringElectrical 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

functionIn 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 rootIn 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

meanIn 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 meanIn 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

waveformWaveform 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 rootIn 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 meanIn 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...

(

averageIn 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

$\backslash \{x\_1,x\_2,\backslash dots,x\_n\backslash \}$, the RMS value is given by:

$x\_\{\backslash mathrm\{rms\}\}\; =\; \backslash sqrt\; \{\backslash frac\{1\}\{n\}\backslash left(\{\{x\_1\}^2\; +\; \{x\_2\}^2\; +\; \backslash cdots\; +\; \{x\_n\}^2\}\backslash right)\}$
The corresponding formula for a continuous function (or waveform)

$f(t)$ defined over the interval

$T\_1\; \backslash le\; t\; \backslash le\; T\_2$ is

$f\_\{\backslash mathrm\{rms\}\}\; =\; \backslash sqrt\; \{\{1\; \backslash over\; \{T\_2-T\_1\}\}\; \{\backslash int\_\{T\_1\}^\{T\_2\}\; \{[f(t)]\}^2\backslash ,\; dt\}\},$
and the RMS for a function over all time is

$f\_\backslash mathrm\{rms\}\; =\; \backslash lim\_\{T\backslash rightarrow\; \backslash infty\}\; \backslash sqrt\; \{\{1\; \backslash over\; \{2T\}\}\; \{\backslash int\_\{-T\}^\{T\}\; \{[f(t)]\}^2\backslash ,\; dt\}\}.$
The RMS over all time of a

periodic functionIn 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 valueIn 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

Waveform | Equation | RMS |
---|

DCDirect 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 | $y=a\backslash ,$ | $a\backslash ,$ |

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...
| $y=a\backslash sin(2\backslash pi\; ft)\backslash ,$ | $\backslash frac\{a\}\{\backslash sqrt\{2\}\}$ |

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...
| $y=\backslash begin\{cases\}a\; \&\; \backslash \{ft\backslash \}\; <\; 0.5\; \backslash \backslash \; -a\; \&\; \backslash \{ft\backslash \}\; >\; 0.5\; \backslash end\{cases\}$ | $a\backslash ,$ |

Modified square waveAn 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....
| $y=\backslash begin\{cases\}0\; \&\; \backslash \{ft\backslash \}\; <\; 0.25\; \backslash \backslash \; a\; \&\; 0.25\; <\; \backslash \{ft\backslash \}\; <\; 0.5\; \backslash \backslash \; 0\; \&\; 0.5\; <\; \backslash \{ft\backslash \}\; <\; 0.75\; \backslash \backslash \; -a\; \&\; \backslash \{ft\backslash \}\; >\; 0.75\; \backslash end\{cases\}$ | $\backslash frac\{a\}\{\backslash sqrt\{2\}\}$ |

Sawtooth waveThe 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....
| $y=2a\backslash \{ft\backslash \}-a\backslash ,$ | $a\; \backslash over\; \backslash sqrt\; 3$ |

Triangle waveA 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\,| | $a\; \backslash over\; \backslash sqrt\; 3$ |

Pulse train | $y=\backslash begin\{cases\}a\; \&\; \backslash \{ft\backslash \}\; <\; D\; \backslash \backslash \; 0\; \&\; \backslash \{ft\backslash \}\; >\; D\; \backslash end\{cases\}$ | $a\; \backslash sqrt\{D\}$ |

Notes:
*t* is time
*f* is frequency
*a* is amplitude (peak value)
*D* is the duty cycleIn 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 partAll 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).

$RMS\_\{Total\}\; =\; \backslash sqrt\; \{\{other\; uses|RMS\}\}\; Inmathematics$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

statisticalStatistics 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

magnitudeThe 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 engineeringElectrical 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

functionIn 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 rootIn 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

meanIn 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 meanIn 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

waveformWaveform 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 rootIn 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 meanIn 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...

(

averageIn 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

$\backslash \{x\_1,x\_2,\backslash dots,x\_n\backslash \}$, the RMS value is given by:

$x\_\{\backslash mathrm\{rms\}\}\; =\; \backslash sqrt\; \{\backslash frac\{1\}\{n\}\backslash left(\{\{x\_1\}^2\; +\; \{x\_2\}^2\; +\; \backslash cdots\; +\; \{x\_n\}^2\}\backslash right)\}$
The corresponding formula for a continuous function (or waveform)

$f(t)$ defined over the interval

$T\_1\; \backslash le\; t\; \backslash le\; T\_2$ is

$f\_\{\backslash mathrm\{rms\}\}\; =\; \backslash sqrt\; \{\{1\; \backslash over\; \{T\_2-T\_1\}\}\; \{\backslash int\_\{T\_1\}^\{T\_2\}\; \{[f(t)]\}^2\backslash ,\; dt\}\},$
and the RMS for a function over all time is

$f\_\backslash mathrm\{rms\}\; =\; \backslash lim\_\{T\backslash rightarrow\; \backslash infty\}\; \backslash sqrt\; \{\{1\; \backslash over\; \{2T\}\}\; \{\backslash int\_\{-T\}^\{T\}\; \{[f(t)]\}^2\backslash ,\; dt\}\}.$
The RMS over all time of a

periodic functionIn 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 valueIn 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

Waveform | Equation | RMS |
---|

DCDirect 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 | $y=a\backslash ,$ | $a\backslash ,$ |

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...
| $y=a\backslash sin(2\backslash pi\; ft)\backslash ,$ | $\backslash frac\{a\}\{\backslash sqrt\{2\}\}$ |

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...
| $y=\backslash begin\{cases\}a\; \&\; \backslash \{ft\backslash \}\; <\; 0.5\; \backslash \backslash \; -a\; \&\; \backslash \{ft\backslash \}\; >\; 0.5\; \backslash end\{cases\}$ | $a\backslash ,$ |

Modified square waveAn 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....
| $y=\backslash begin\{cases\}0\; \&\; \backslash \{ft\backslash \}\; <\; 0.25\; \backslash \backslash \; a\; \&\; 0.25\; <\; \backslash \{ft\backslash \}\; <\; 0.5\; \backslash \backslash \; 0\; \&\; 0.5\; <\; \backslash \{ft\backslash \}\; <\; 0.75\; \backslash \backslash \; -a\; \&\; \backslash \{ft\backslash \}\; >\; 0.75\; \backslash end\{cases\}$ | $\backslash frac\{a\}\{\backslash sqrt\{2\}\}$ |

Sawtooth waveThe 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....
| $y=2a\backslash \{ft\backslash \}-a\backslash ,$ | $a\; \backslash over\; \backslash sqrt\; 3$ |

Triangle waveA 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\,| | $a\; \backslash over\; \backslash sqrt\; 3$ |

Pulse train | $y=\backslash begin\{cases\}a\; \&\; \backslash \{ft\backslash \}\; <\; D\; \backslash \backslash \; 0\; \&\; \backslash \{ft\backslash \}\; >\; D\; \backslash end\{cases\}$ | $a\; \backslash sqrt\{D\}$ |

Notes:
*t* is time
*f* is frequency
*a* is amplitude (peak value)
*D* is the duty cycleIn 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 partAll 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).

$RMS\_\{Total\}\; =\; \backslash sqrt\; \{\{\{RMS\_1\}^2\; +\; \{RMS\_2\}^2\; +\; \backslash cdots\; +\; \{RMS\_n\}^2\}\; \}$
### Average electrical power

Electrical engineers often need to know the

powerIn 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

currentElectric 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:

$P\; =\; I^2\; R.\backslash ,\backslash !$
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

$P\_\backslash mathrm\{avg\}\backslash ,\backslash !$ | $=\; \backslash langle\; I(t)^2R\; \backslash rangle\; \backslash ,\backslash !$ (where $\backslash langle\; \backslash ldots\; \backslash rangle$ denotes the mean of a function) |

| $=\; R\backslash langle\; I(t)^2\; \backslash rangle\backslash ,\backslash !$ (as *R* does not vary over time, it can be factored out) |

| $=\; (I\_\backslash mathrm\{RMS\})^2R\backslash ,\backslash !$ (by definition of RMS) |

NEWLINENEWLINE
So, the RMS value,

$I\_\backslash mathrm\{RMS\}$, 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

voltageVoltage, 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

$V\_\backslash mathrm\{RMS\}$,

$P\_\backslash mathrm\{avg\}\; =\; \{(V\_\backslash mathrm\{RMS\})^2\backslash over\; R\}.\backslash ,\backslash !$
This equation can be used for any periodic

waveformWaveform 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

sinusoidalThe 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 waveThe 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

$P\_\backslash mathrm\{avg\}\; =\; V\_\backslash mathrm\{RMS\}I\_\backslash mathrm\{RMS\}.\backslash ,\backslash !$
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 powerPower 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 currentIn 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

sinusoidalThe 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

$I\_\{\backslash mathrm\{p\}\}$ to be the peak current, then:

$I\_\{\backslash mathrm\{RMS\}\}\; =\; \backslash sqrt\; \{\{1\; \backslash over\; \{T\_2-T\_1\}\}\; \{\backslash int\_\{T\_1\}^\{T\_2\}\; \{(I\_\backslash mathrm\{p\}\backslash sin(\backslash omega\; t)\}\backslash ,\; \})^2\; dt\}.\backslash ,\backslash !$
where

*t* is time and

*ω* is the

angular frequencyIn 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*, where

*T* is the period of the wave).
Since

$I\_\{\backslash mathrm\{p\}\}$ is a positive constant:

$I\_\{\backslash mathrm\{RMS\}\}\; =\; I\_\backslash mathrm\{p\}\backslash sqrt\; \{\{1\; \backslash over\; \{T\_2-T\_1\}\}\; \{\backslash int\_\{T\_1\}^\{T\_2\}\; \{\backslash sin^2(\backslash omega\; t)\}\backslash ,\; dt\}\}.$
Using a

trigonometric identity to eliminate squaring of trig function:

$I\_\{\backslash mathrm\{RMS\}\}\; =\; I\_\backslash mathrm\{p\}\backslash sqrt\; \{\{1\; \backslash over\; \{T\_2-T\_1\}\}\; \{\backslash int\_\{T\_1\}^\{T\_2\}\; \{\{1\; -\; \backslash cos(2\backslash omega\; t)\; \backslash over\; 2\}\}\backslash ,\; dt\}\}$
$I\_\{\backslash mathrm\{RMS\}\}\; =\; I\_\backslash mathrm\{p\}\backslash sqrt\; \{\{1\; \backslash over\; \{T\_2-T\_1\}\}\; \backslash left\; [\; \{\{t\; \backslash over\; 2\}\; -\{\; \backslash sin(2\backslash omega\; t)\; \backslash over\; 4\backslash omega\}\}\; \backslash right\; ]\_\{T\_1\}^\{T\_2\}\; \}$
but since the interval is a whole number of complete cycles (per definition of RMS), the

$\backslash sin$ terms will cancel out, leaving:

$I\_\{\backslash mathrm\{RMS\}\}\; =\; I\_\backslash mathrm\{p\}\backslash sqrt\; \{\{1\; \backslash over\; \{T\_2-T\_1\}\}\; \backslash left\; [\; \{\{t\; \backslash over\; 2\}\}\; \backslash right\; ]\_\{T\_1\}^\{T\_2\}\; \}\; =\; I\_\backslash mathrm\{p\}\backslash sqrt\; \{\{1\; \backslash over\; \{T\_2-T\_1\}\}\; \{\{\{T\_2-T\_1\}\; \backslash over\; 2\}\}\; \}\; =\; \{I\_\backslash mathrm\{p\}\; \backslash over\; \{\backslash sqrt\; 2\}\}.$
A similar analysis leads to the analogous equation for sinusoidal voltage:

$V\_\{\backslash mathrm\{RMS\}\}\; =\; \{V\_\backslash mathrm\{p\}\; \backslash over\; \{\backslash sqrt\; 2\}\}.$
Where

$I\_\{\backslash mathrm\{P\}\}$ represents the peak current and

$V\_\{\backslash mathrm\{P\}\}$ 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

voltageVoltage, 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

*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 powerAudio 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 amplifierAn 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 frequencyRadio 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

voltageVoltage, 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

$P\_\backslash mathrm\{input\}(t)\; =\; I\_Q\; V\_\{CC\}\; +\; I\_\backslash mathrm\{out\}(t)\; V\_\{CC\}\backslash ,$
where

$I\_Q$ is the amplifier's operating current. Clearly, because

$V\_\{CC\}$ is constant, the time average of

$P\_\backslash mathrm\{input\}$ depends on the time

*average* value of

$I\_\backslash mathrm\{out\}$ and not its RMS value. That is,

$\backslash langle\; P\_\backslash mathrm\{input\}(t)\; \backslash rangle\; =\; I\_Q\; V\_\{CC\}\; +\; \backslash langle\; I\_\backslash mathrm\{out\}(t)\; \backslash rangle\; V\_\{CC\}.\backslash ,$
### Root-mean-square speed

{{main|Root-mean-square speed}}
In the

physicsPhysics 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

gasGas 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:

$\{v\_\backslash mathrm\{RMS\}\}\; =\; \{\backslash sqrt\{3RT\; \backslash over\; \{M\}\}\}$
where

$R$ represents the ideal gas constant, 8.314 J/(mol·K),

$T$ is the temperature of the gas in

kelvinThe 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 massMolar 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

meanIn 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 deviationStandard 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 theoremIn 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:

$\backslash sum\backslash limits\_\{n\}\{\{\{x\}^\{2\}\}(t)\}=\backslash frac\{\backslash sum\backslash limits\_\{n\}\{\{\{\backslash left|\; X(f)\; \backslash right|\}^\{2\}\}\}\}\{n\}$, where

$X(f)=FFT\backslash \{x(t)\backslash \}$,

$n$ is number of

$x(t)$ samples.
In this case, the RMS computed in time domain is the same as in frequency domain:

$RMS\; =\backslash sqrt\{\backslash frac\{1\}\{n\}\backslash sum\backslash limits\_\{n\}\{\{\{x\}^\{2\}\}(t)\}\}\; =\; \backslash frac\{1\}\{n\}\backslash sqrt\{\backslash sum\backslash limits\_\{n\}\{\{\{\backslash left|\; X(f)\; \backslash right|\}^\{2\}\}\}\}\; =\; \backslash sqrt\{\backslash sum\backslash limits\_\{n\}\{\{\{\; \backslash left|\backslash frac\{X(f)\}\{n\}\backslash right|\; ^\; 2\; \}\}\}\}$
## Relationship to the arithmetic mean and the standard deviation

If

$\backslash bar\{x\}$ is the

arithmetic meanIn 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

$\backslash sigma\_\{x\}$ is the

standard deviationStandard 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

populationA 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

waveformWaveform 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:

$x\_\{\backslash mathrm\{rms\}\}^2\; =\; \backslash bar\{x\}^2\; +\; \backslash sigma\_\{x\}^2.$
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 deviationStandard 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

NEWLINE

NEWLINE- L2 norm
NEWLINE- 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
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
The root-mean-square deviation is the measure of the average distance between the atoms of superimposed proteins...

NEWLINE- 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
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
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...

NEWLINE

## External links

NEWLINE

NEWLINE
{{DEFAULTSORT:Root Mean Square}}