derivation of formula for the rms value of current
Root mean square - wikipedia Root mean square Jump to : navigation , search This article needs additional citations for verification . Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed . ( March 2010 ) ( Learn how and when to remove this template message ) In statistics and its applications , the root mean square ( abbreviated RMS or rms ) is defined as the square root of mean square ( the arithmetic mean of the squares of a set of numbers ) . The RMS is also known as the quadratic mean and is a particular case of the generalized mean with exponent 2 . RMS can also be defined for a continuously varying function in terms of an integral of the squares of the instantaneous values during a cycle . For a cyclically alternating electric current , RMS is equal to the value of the direct current that would produce the same average power dissipation in a resistive load . In Estimation theory , the root mean square error of an estimator is a measure of the imperfection of the fit of the estimator to the data . Contents ( hide ) 1 Definition 2 In common waveforms 2.1 In waveform combinations 3 Uses 3.1 In electrical engineering 3.1. 1 Voltage 3.1. 2 Average electrical power 3.2 Speed 3.3 Error 4 In frequency domain 5 Relationship to other statistics 6 See also 7 References 8 External links Definition ( edit ) The RMS value of a set of values ( or a continuous - time waveform ) is the square root of the arithmetic mean of the squares of the values , or the square of the function that defines the continuous waveform . In Physics , the RMS current is the `` value of the direct current that dissipates power in a resistor . '' In the case of a set of n values ( x 1 , x 2 , ... , x n ) ( \ displaystyle \ ( x_ ( 1 ) , x_ ( 2 ) , \ dots , x_ ( n ) \ ) ) , the RMS x r m s = 1 n ( x 1 2 + x 2 2 + ⋯ + x n 2 ) . ( \ displaystyle 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 ≤ t ≤ T 2 ( \ displaystyle T_ ( 1 ) \ leq t \ leq T_ ( 2 ) ) is f r m s = 1 T 2 − T 1 ∫ T 1 T 2 ( f ( t ) ) 2 d t , ( \ displaystyle 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 r m s = lim T → ∞ 1 T ∫ 0 T ( f ( t ) ) 2 d t . ( \ displaystyle f_ ( \ mathrm ( rms ) ) = \ lim _ ( T \ rightarrow \ infty ) ( \ sqrt ( ( 1 \ over ( T ) ) ( \ int _ ( 0 ) ^ ( T ) ( ( f ( t ) ) ) ^ ( 2 ) \ , dt ) ) ) . ) The RMS over all time of a periodic function is equal to the RMS of one period of the function . The RMS value of a continuous function or signal can be approximated by taking the RMS of a sequence 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 is used instead of the mean . In common waveforms ( edit ) Further information : RMS amplitude Sine , square , triangle , and sawtooth waveforms . A rectangular pulse wave of duty cycle D , the ratio between the pulse duration ( τ ( \ displaystyle \ tau ) ) and the period ( T ) ; illustrated here with a = 1 . Graph of a sine wave 's voltage vs. time ( in degrees ) , showing RMS , peak ( PK ) , and peak - to - peak ( PP ) voltages . If the waveform is a pure sine wave , the relationships between amplitudes ( peak - to - peak , peak ) and RMS are fixed and known , as they are for any continuous periodic wave . However , this is not true for an arbitrary waveform , which may or may not be periodic or continuous . For a zero - mean sine wave , the relationship between RMS and peak - to - peak amplitude is : Peak - to - peak = 2 2 × R M S ≈ 2.8 × R M S . ( \ displaystyle = 2 ( \ sqrt ( 2 ) ) \ times ( RMS ) \ approx 2.8 \ times ( RMS ) . ) For other waveforms the relationships are not the same as they are for sine waves . Waveform Equation RMS DC , constant y = A 0 ( \ displaystyle y = A_ ( 0 ) \ , ) A 0 ( \ displaystyle A_ ( 0 ) \ , ) Sine wave y = A 1 sin ⁡ ( 2 π f t ) ( \ displaystyle y = A_ ( 1 ) \ sin ( 2 \ pi ft ) \ , ) A 1 2 ( \ displaystyle ( \ frac ( A_ ( 1 ) ) ( \ sqrt ( 2 ) ) ) ) Square wave y = ( A 1 frac ⁡ ( f t ) < 0.5 − A 1 frac ⁡ ( f t ) > 0.5 ( \ displaystyle y = ( \ begin ( cases ) A_ ( 1 ) & \ operatorname ( frac ) ( ft ) < 0.5 \ \ - A_ ( 1 ) & \ operatorname ( frac ) ( ft ) > 0.5 \ end ( cases ) ) ) A 1 ( \ displaystyle A_ ( 1 ) \ , ) DC - shifted square wave y = A 0 + ( A 1 frac ⁡ ( f t ) < 0.5 − A 1 frac ⁡ ( f t ) > 0.5 ( \ displaystyle y = A_ ( 0 ) + ( \ begin ( cases ) A_ ( 1 ) & \ operatorname ( frac ) ( ft ) < 0.5 \ \ - A_ ( 1 ) & \ operatorname ( frac ) ( ft ) > 0.5 \ end ( cases ) ) ) A 0 2 + A 1 2 ( \ displaystyle ( \ sqrt ( A_ ( 0 ) ^ ( 2 ) + A_ ( 1 ) ^ ( 2 ) ) ) \ , ) Modified sine wave y = ( 0 frac ⁡ ( f t ) < 0.25 A 1 0.25 < frac ⁡ ( f t ) < 0.5 0 0.5 < frac ⁡ ( f t ) < 0.75 − A 1 frac ⁡ ( f t ) > 0.75 ( \ displaystyle y = ( \ begin ( cases ) 0& \ operatorname ( frac ) ( ft ) < 0.25 \ \ A_ ( 1 ) &0. 25 < \ operatorname ( frac ) ( ft ) < 0.5 \ \ 0&0. 5 < \ operatorname ( frac ) ( ft ) < 0.75 \ \ - A_ ( 1 ) & \ operatorname ( frac ) ( ft ) > 0.75 \ end ( cases ) ) ) A 1 2 ( \ displaystyle ( \ frac ( A_ ( 1 ) ) ( \ sqrt ( 2 ) ) ) ) Triangle wave y = 2 A 1 frac ⁡ ( f t ) − A 1 ( \ displaystyle y = \ left 2A_ ( 1 ) \ operatorname ( frac ) ( ft ) - A_ ( 1 ) \ right ) A 1 3 ( \ displaystyle A_ ( 1 ) \ over ( \ sqrt ( 3 ) ) ) Sawtooth wave y = 2 A 1 frac ⁡ ( f t ) − A 1 ( \ displaystyle y = 2A_ ( 1 ) \ operatorname ( frac ) ( ft ) - A_ ( 1 ) \ , ) A 1 3 ( \ displaystyle A_ ( 1 ) \ over ( \ sqrt ( 3 ) ) ) Pulse train y = ( A 1 frac ⁡ ( f t ) < D 0 frac ⁡ ( f t ) > D ( \ displaystyle y = ( \ begin ( cases ) A_ ( 1 ) & \ operatorname ( frac ) ( ft ) < D \ \ 0& \ operatorname ( frac ) ( ft ) > D \ end ( cases ) ) ) A 1 D ( \ displaystyle A_ ( 1 ) ( \ sqrt ( D ) ) ) Phase - to - phase voltage y = A 1 sin ⁡ ( t ) − A 1 sin ⁡ ( t − 2 π 3 ) ( \ displaystyle y = A_ ( 1 ) \ sin ( t ) - A_ ( 1 ) \ sin \ left ( t - ( \ frac ( 2 \ pi ) ( 3 ) ) \ right ) \ , ) A 1 3 2 ( \ displaystyle A_ ( 1 ) ( \ sqrt ( \ frac ( 3 ) ( 2 ) ) ) ) <Td_colspan="3"> Notes : t is time f is frequency A is amplitude ( peak value ) D is the duty cycle or the proportion of the time period ( 1 / f ) spent high . frac ( r ) is the fractional part of r In waveform combinations ( edit ) Waveforms made by summing known simple waveforms have an RMS that is the root of the sum of squares of the component RMS values , if the component waveforms are orthogonal ( that is , if the average of the product of one simple waveform with another is zero for all pairs other than a waveform times itself ) . R M S Total = R M S 1 2 + R M S 2 2 + ⋯ + R M S n 2 ( \ displaystyle RMS_ ( \ text ( Total ) ) = ( \ sqrt ( ( RMS_ ( 1 ) ) ^ ( 2 ) + ( RMS_ ( 2 ) ) ^ ( 2 ) + \ cdots + ( RMS_ ( n ) ) ^ ( 2 ) ) ) ) A special case of this , particularly helpful in electrical engineering , is given # In electrical engineering below . Another special case , useful in statistics , is given in # Relationship to other statistics . ( If the waveforms are in phase , then their RMS amplitudes sum directly . ) Uses ( edit ) In electrical engineering ( edit ) Voltage ( edit ) Further information : Root mean square AC voltage In electrical engineering , a special case of # RMS of waveform combinations ( see also # Relationship to other statistics ) is : R M S Total = R M S DC 2 + R M S AC 2 ( \ displaystyle RMS_ ( \ text ( Total ) ) = ( \ sqrt ( ( RMS_ ( \ text ( DC ) ) ) ^ ( 2 ) + ( RMS_ ( \ text ( AC ) ) ) ^ ( 2 ) ) ) ) where R M S DC ( \ displaystyle RMS_ ( \ text ( DC ) ) ) refers to the direct current component of the signal and R M S AC ( \ displaystyle RMS_ ( \ text ( AC ) ) ) is the alternating current component of the signal . Average electrical power ( edit ) Further information : AC power Electrical engineers often need to know the power , P , dissipated by an electrical resistance , R. It is easy to do the calculation when there is a constant current , I , through the resistance . For a load of R ohms , power is defined simply as : P = I 2 R . ( \ displaystyle P = I ^ ( 2 ) R . ) However , if the current is a time - varying function , I ( t ) , this formula must be extended to reflect the fact that the current ( and thus the instantaneous power ) is varying over time . If the function is periodic ( such as household AC power ) , it is still meaningful to discuss the average power dissipated over time , which is calculated by taking the average power dissipation : P avg ( \ displaystyle P_ ( \ text ( avg ) ) \ , \ ! ) = ⟨ I ( t ) 2 R ⟩ ( \ displaystyle ( ) = \ left \ langle I ( t ) ^ ( 2 ) R \ right \ rangle \ , \ ! ) ( where ⟨ ... ⟩ ( \ displaystyle \ langle \ ldots \ rangle ) denotes the mean of a function ) = R ⟨ I ( t ) 2 ⟩ ( \ displaystyle ( ) = R \ left \ langle I ( t ) ^ ( 2 ) \ right \ rangle \ , \ ! ) ( as R does not vary over time , it can be factored out ) = I RMS 2 R ( \ displaystyle ( ) = I_ ( \ text ( RMS ) ) ^ ( 2 ) R \ , \ ! ) ( by definition of RMS ) So , the RMS value , I , of the function I ( t ) is the constant current that yields the same power dissipation as the time - averaged power dissipation of the current I ( t ) . Average power can also be found using the same method that in the case of a time - varying voltage , V ( t ) , with RMS value V , P Avg = V RMS 2 R . ( \ displaystyle P_ ( \ text ( Avg ) ) = ( V_ ( \ text ( RMS ) ) ^ ( 2 ) \ over R ) . ) This equation can be used for any periodic waveform , such as a sinusoidal or sawtooth waveform , allowing us to calculate the mean power delivered into a specified load . By taking the square root of both these equations and multiplying them together , the power is found to be : P Avg = V RMS I RMS . ( \ displaystyle P_ ( \ text ( Avg ) ) = V_ ( \ text ( RMS ) ) I_ ( \ text ( RMS ) ) . ) Both derivations depend on voltage and current being proportional ( i.e. , the load , R , is purely resistive ) . Reactive loads ( i.e. , loads capable of not just dissipating energy but also storing it ) are discussed under the topic of AC power . In the common case of alternating current when I ( t ) is a sinusoidal current , as is approximately true for mains power , the RMS value is easy to calculate from the continuous case equation above . If I is defined to be the peak current , then : I RMS = 1 T 2 − T 1 ∫ T 1 T 2 ( I p sin ⁡ ( ω t ) ) 2 d t . ( \ displaystyle I_ ( \ text ( RMS ) ) = ( \ sqrt ( ( 1 \ over ( T_ ( 2 ) - T_ ( 1 ) ) ) \ int _ ( T_ ( 1 ) ) ^ ( T_ ( 2 ) ) \ left ( I_ ( \ text ( p ) ) \ sin ( \ omega t ) \ right ) ^ ( 2 ) dt ) ). \ , \ ! ) where t is time and ω is the angular frequency ( ω = 2π / T , where T is the period of the wave ) . Since I is a positive constant : I RMS = I p 1 T 2 − T 1 ∫ T 1 T 2 sin 2 ⁡ ( ω t ) d t . ( \ displaystyle I_ ( \ text ( RMS ) ) = I_ ( \ text ( p ) ) ( \ sqrt ( ( 1 \ over ( T_ ( 2 ) - T_ ( 1 ) ) ) ( \ int _ ( T_ ( 1 ) ) ^ ( T_ ( 2 ) ) ( \ sin ^ ( 2 ) ( \ omega t ) ) \ , dt ) ) ) . ) Using a trigonometric identity to eliminate squaring of trig function : I RMS = I p 1 T 2 − T 1 ∫ T 1 T 2 1 − cos ⁡ ( 2 ω t ) 2 d t I RMS = I p 1 T 2 − T 1 ( t 2 − sin ⁡ ( 2 ω t ) 4 ω ) T 1 T 2 ( \ displaystyle ( \ begin ( aligned ) I_ ( \ text ( RMS ) ) & = I_ ( \ text ( p ) ) ( \ sqrt ( ( 1 \ over ( T_ ( 2 ) - T_ ( 1 ) ) ) ( \ int _ ( T_ ( 1 ) ) ^ ( T_ ( 2 ) ) ( 1 - \ cos ( 2 \ omega t ) \ over 2 ) \ , dt ) ) ) \ \ I_ ( \ text ( RMS ) ) & = I_ ( \ text ( p ) ) ( \ sqrt ( ( 1 \ over ( T_ ( 2 ) - T_ ( 1 ) ) ) \ left ( ( t \ over 2 ) - ( \ sin ( 2 \ omega t ) \ over 4 \ omega ) \ right ) _ ( T_ ( 1 ) ) ^ ( T_ ( 2 ) ) ) ) \ end ( aligned ) ) ) but since the interval is a whole number of complete cycles ( per definition of RMS ) , the sin terms will cancel out , leaving : I RMS = I p 1 T 2 − T 1 ( t 2 ) T 1 T 2 = I p 1 T 2 − T 1 T 2 − T 1 2 = I p 2 . ( \ displaystyle I_ ( \ text ( RMS ) ) = I_ ( \ text ( p ) ) ( \ sqrt ( ( 1 \ over ( T_ ( 2 ) - T_ ( 1 ) ) ) \ left ( ( t \ over 2 ) \ right ) _ ( T_ ( 1 ) ) ^ ( T_ ( 2 ) ) ) ) = I_ ( \ text ( p ) ) ( \ sqrt ( ( 1 \ over ( T_ ( 2 ) - T_ ( 1 ) ) ) ( ( T_ ( 2 ) - T_ ( 1 ) ) \ over 2 ) ) ) = ( I_ ( \ text ( p ) ) \ over ( \ sqrt ( 2 ) ) ) . ) A similar analysis leads to the analogous equation for sinusoidal voltage : V RMS = V p 2 . ( \ displaystyle V_ ( \ text ( RMS ) ) = ( V_ ( \ text ( p ) ) \ over ( \ sqrt ( 2 ) ) ) . ) Where I represents the peak current and V represents the peak voltage . Because of their usefulness in carrying out power calculations , listed voltages for power outlets ( e.g. , 120 V in the USA , or 230 V in Europe ) are almost always quoted in RMS values , and not peak values . Peak values can be calculated from RMS values from the above formula , which implies V = V × √ 2 , assuming the source is a pure sine wave . Thus the peak value of the mains voltage in the USA is about 120 × √ 2 , or about 170 volts . The peak - to - peak voltage , being double this , is about 340 volts . A similar calculation indicates that the peak mains voltage in Europe is about 325 volts , and the peak - to - peak mains voltage , about 650 volts . RMS quantities such as electric current are usually calculated over one cycle . However , for some purposes the RMS current over a longer period is required when calculating transmission power losses . The same principle applies , and ( for example ) a current of 10 amps used for 12 hours each day represents an RMS current of 5 amps in the long term . The term `` RMS power '' is sometimes erroneously used in the audio industry as a synonym for `` mean power '' or `` average power '' ( it is proportional to the square of the RMS voltage or RMS current in a resistive load ) . For a discussion of audio power measurements and their shortcomings , see Audio power . Speed ( edit ) Main article : Root - mean - square speed In the physics of gas molecules , the root - mean - square speed is defined as the square root of the average squared - speed . The RMS speed of an ideal gas is calculated using the following equation : v RMS = 3 R T M ( \ displaystyle ( v_ ( \ text ( RMS ) ) ) = ( \ sqrt ( 3RT \ over ( M ) ) ) ) where R represents the ideal gas constant , 8.314 J / ( mol K ) , T is the temperature of the gas in kelvins , and M is the molar mass of the gas in kilograms . The generally accepted terminology for speed as compared to velocity is that the former is the scalar magnitude of the latter . Therefore , although the average speed is between zero and the RMS speed , the average velocity for a stationary gas is zero . Error ( edit ) Main article : 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 of the pairwise differences does not measure the variability of the difference , and the variability as indicated by the standard deviation is around the mean instead of 0 . Therefore , the RMS of the differences is a meaningful measure of the error . In frequency domain ( edit ) The RMS can be computed in the frequency domain , using Parseval 's theorem . For a sampled signal x ( n ) = x ( t = n T ) ( \ displaystyle x ( n ) = x ( t = nT ) ) , where T ( \ displaystyle T ) is the sampling period , ∑ n = 1 N x 2 ( n ) = 1 N ∑ m = 1 N X ( m ) 2 ( \ displaystyle \ sum _ ( n = 1 ) ^ ( N ) ( x ^ ( 2 ) ( n ) ) = ( \ frac ( 1 ) ( N ) ) \ sum _ ( m = 1 ) ^ ( N ) ( \ bigl ) X ( m ) ( \ bigr ) ^ ( 2 ) ) , where X ( m ) = FFT ⁡ ( x ( n ) ) ( \ displaystyle X ( m ) = \ operatorname ( FFT ) \ ( x ( n ) \ ) ) and N is number of samples and FFT coefficients . In this case , the RMS computed in the time domain is the same as in the frequency domain : R M S ( x ( n ) ) = 1 N ∑ n x 2 ( n ) = 1 N 2 ∑ m X ( m ) 2 = ∑ m X ( m ) N 2 . ( \ displaystyle \ mathrm ( RMS ) \ ( x ( n ) \ ) = ( \ sqrt ( ( \ frac ( 1 ) ( N ) ) \ sum _ ( n ) ( x ^ ( 2 ) ( n ) ) ) ) = ( \ sqrt ( ( \ frac ( 1 ) ( N ^ ( 2 ) ) ) \ sum _ ( m ) ( ( \ bigl ) X ( m ) ( \ bigr ) ) ^ ( 2 ) ) ) = ( \ sqrt ( \ sum _ ( m ) ( \ left ( \ frac ( X ( m ) ) ( N ) ) \ right ^ ( 2 ) ) ) ) . ) Relationship to other statistics ( edit ) If x _̄ ( \ displaystyle ( \ bar ( x ) ) ) is the arithmetic mean and σ x ( \ displaystyle \ sigma _ ( x ) ) is the standard deviation of a population or a waveform then : x r m s 2 = x _̄ 2 + σ x 2 = x 2 _̄ . ( \ displaystyle x_ ( \ mathrm ( rms ) ) ^ ( 2 ) = ( \ overline ( x ) ) ^ ( 2 ) + \ sigma _ ( x ) ^ ( 2 ) = ( \ overline ( x ^ ( 2 ) ) ) . ) From this it is clear that the RMS value is always greater than or equal to the average , in that the RMS includes the `` error '' / square deviation as well . Physical scientists often use the term `` root mean square '' as a synonym for standard deviation when it can be assumed the input signal has zero mean , i.e. , referring to the square root of the mean squared deviation of a signal from a given baseline or fit . This is useful for electrical engineers in calculating the `` AC only '' RMS of a signal . Standard deviation being the root mean square of a signal 's variation about the mean , rather than about 0 , the DC component is removed ( i.e. RMS ( signal ) = Stdev ( signal ) if the mean signal is 0 ) . A special case of this , particularly helpful in electrical engineering , is given above . See also ( edit ) Central moment Geometric mean L2 norm Least squares Mean squared displacement Table of mathematical symbols True RMS converter Average rectified value ( ARV ) References ( edit ) ^ Jump up to : A Dictionary of Physics ( 6 ed . ) . Oxford University Press . 2009 . ISBN 9780199233991 . Jump up ^ Cartwright , Kenneth V ( Fall 2007 ) . `` Determining the Effective or RMS Voltage of Various Waveforms without Calculus '' ( PDF ) . Technology Interface. 8 ( 1 ) : 20 pages . Jump up ^ Nastase , Adrian S. `` How to Derive the RMS Value of Pulse and Square Waveforms '' . MasteringElectronicsDesign.com . Retrieved 21 January 2015 . Jump up ^ Chris C. Bissell and David A. Chapman ( 1992 ) . Digital signal transmission ( 2nd ed . ) . Cambridge University Press . p. 64 . ISBN 978 - 0 - 521 - 42557 - 5 . Jump up ^ `` Root - Mean - Square '' . Jump up ^ `` ROOT , TH1 : GetRMS '' . External links ( edit ) A case for why RMS is a misnomer when applied to audio power A Java applet on learning RMS Retrieved from `` https://en.wikipedia.org/w/index.php?title=Root_mean_square&oldid=810550387 '' Categories : Statistical deviation and dispersion Means Hidden categories : Articles needing additional references from March 2010 All articles needing additional references Talk Contents About Wikipedia Беларуская ( тарашкевіца ) ‎ Български Català Čeština Dansk Deutsch Eesti Español Esperanto Euskara فارسی Français 한국어 हिन्दी עברית Қазақша Magyar Nederlands 日本 語 Norsk Piemontèis Polski Português Română Русский Slovenčina Српски / srpski Suomi Svenska ไทย Türkçe Українська Tiếng Việt 中文 Edit links This page was last edited on 15 November 2017 , at 23 : 27 . Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply . By using this site , you agree to the Terms of Use and Privacy Policy . 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1. Root mean square (RMS)
2. Definition and calculation of RMS
3. RMS in common waveforms
4. Uses of RMS in electrical engineering
5. Relationship of RMS to other statistics