what is positive sequence negative sequence and zero sequence current
Symmetrical components - wikipedia Symmetrical components In electrical engineering , the method of symmetrical components simplifies analysis of unbalanced three - phase power systems under both normal and abnormal conditions . The basic idea is that an asymmetrical set of N phasors can be expressed as a linear combination of N symmetrical sets of phasors by means of a complex linear transformation . In the most common case of three - phase systems , the resulting `` symmetrical '' components are referred to as direct ( or positive ) , inverse ( or negative ) and zero ( or homopolar ) . The analysis of power system is much simpler in the domain of symmetrical components , because the resulting equations are mutually linearly independent if the circuit itself is balanced . Contents ( hide ) 1 Description 2 The three - phase case 2.1 A matrix 2.2 Decomposition 2.3 Intuition 3 Poly - phase case 4 Contribution of harmonics to symmetrical components in 3 - phase power systems 5 Consequence of the zero sequence component in power systems 6 See also 7 References Description ( edit ) Set of three unbalanced phasors , and the necessary symmetrical components that sum up to the resulting plot at the bottom . In 1918 Charles Legeyt Fortescue presented a paper which demonstrated that any set of N unbalanced phasors ( that is , any such polyphase signal ) could be expressed as the sum of N symmetrical sets of balanced phasors , for values of N that are prime . Only a single frequency component is represented by the phasors . In 1943 Edith Clarke published a textbook giving a method of use of symmetrical components for three - phase systems that greatly simplified calculations over the original Fortescue paper . In a three - phase system , one set of phasors has the same phase sequence as the system under study ( positive sequence ; say ABC ) , the second set has the reverse phase sequence ( negative sequence ; ACB ) , and in the third set the phasors A , B and C are in phase with each other ( zero sequence , the common - mode signal ) . Essentially , this method converts three unbalanced phases into three independent sources , which makes asymmetric fault analysis more tractable . By expanding a one - line diagram to show the positive sequence , negative sequence and zero sequence impedances of generators , transformers and other devices including overhead lines and cables , analysis of such unbalanced conditions as a single line to ground short - circuit fault is greatly simplified . The technique can also be extended to higher order phase systems . Physically , in a three phase winding a positive sequence set of currents produces a normal rotating field , a negative sequence set produces a field with the opposite rotation , and the zero sequence set produces a field that oscillates but does not rotate between phase windings . Since these effects can be detected physically with sequence filters , the mathematical tool became the basis for the design of protective relays , which used negative - sequence voltages and currents as a reliable indicator of fault conditions . Such relays may be used to trip circuit breakers or take other steps to protect electrical systems . The analytical technique was adopted and advanced by engineers at General Electric and Westinghouse and after World War II it became an accepted method for asymmetric fault analysis . As shown in the figure to the above right , the three sets of symmetrical components ( positive , negative , and zero sequence ) add up to create the system of three unbalanced phases as pictured in the bottom of the diagram . The imbalance between phases arises because of the difference in magnitude and phase shift between the sets of vectors . Notice that the colors ( red , blue , and yellow ) of the separate sequence vectors correspond to three different phases ( A , B , and C , for example ) . To arrive at the final plot , the sum of vectors of each phase is calculated . This resulting vector is the effective phasor representation of that particular phase . This process , repeated , produces the phasor for each of the three phases . The three - phase case ( edit ) Three phases where the phase angle of the blue phase is continually increasing . The symmetrical components corresponding to the above voltage phases . Symmetrical components are most commonly used for analysis of three - phase electrical power systems . The voltage or current of a three - phase system at some point can be indicated by three phasors , called the three components of the voltage or the current . This article discusses voltage , however , the same considerations also apply to current . In a perfectly balanced three - phase power system , the voltage phasor components have equal magnitudes but are 120 degrees apart . In an unbalanced system , the magnitudes and phases of the voltage phasor components are different . Decomposing the voltage phasor components into a set of symmetrical components helps analyze the system as well as visualize any imbalances . If the three voltage components are expressed as phasors ( which are complex numbers ) , a complex vector can be formed in which the three phase components are the components of the vector . A vector for three phase voltage components can be written as v a b c = ( V a V b V c ) ( \ displaystyle \ mathbf ( v ) _ ( abc ) = ( \ begin ( bmatrix ) V_ ( a ) \ \ V_ ( b ) \ \ V_ ( c ) \ end ( bmatrix ) ) ) and decomposing the vector into three symmetrical components gives ( V a V b V c ) = ( V a , 0 V b , 0 V c , 0 ) + ( V a , 1 V b , 1 V c , 1 ) + ( V a , 2 V b , 2 V c , 2 ) ( \ displaystyle ( \ begin ( bmatrix ) V_ ( a ) \ \ V_ ( b ) \ \ V_ ( c ) \ end ( bmatrix ) ) = ( \ begin ( bmatrix ) V_ ( a , 0 ) \ \ V_ ( b , 0 ) \ \ V_ ( c , 0 ) \ end ( bmatrix ) ) + ( \ begin ( bmatrix ) V_ ( a , 1 ) \ \ V_ ( b , 1 ) \ \ V_ ( c , 1 ) \ end ( bmatrix ) ) + ( \ begin ( bmatrix ) V_ ( a , 2 ) \ \ V_ ( b , 2 ) \ \ V_ ( c , 2 ) \ end ( bmatrix ) ) ) where the subscripts 0 , 1 , and 2 refer respectively to the zero , positive , and negative sequence components . The sequence components differ only by their phase angles , which are symmetrical and so are 2 3 π ( \ displaystyle \ scriptstyle ( \ frac ( 2 ) ( 3 ) ) \ pi ) radians or 120 ° . A matrix ( edit ) Define a phasor rotation operator alpha , which rotates a phasor vector counterclockwise by 120 degrees : α ≡ e 2 3 π i ( \ displaystyle \ alpha \ equiv e ^ ( ( \ frac ( 2 ) ( 3 ) ) \ pi i ) ) . Note that α = 1 so that α = α . The zero sequence components have equal magnitude and in phase with each other , therefore : V 0 ≡ V a , 0 = V b , 0 = V c , 0 ( \ displaystyle V_ ( 0 ) \ equiv V_ ( a , 0 ) = V_ ( b , 0 ) = V_ ( c , 0 ) ) , and the other phase sequences have the same magnitude , but their phases differ by 120 ° : V 1 ≡ V a , 1 = α V b , 1 = α 2 V c , 1 V 2 ≡ V a , 2 = α 2 V b , 2 = α V c , 2 ( \ displaystyle ( \ begin ( aligned ) V_ ( 1 ) & \ equiv V_ ( a , 1 ) = \ alpha V_ ( b , 1 ) = \ alpha ^ ( 2 ) V_ ( c , 1 ) \ \ V_ ( 2 ) & \ equiv V_ ( a , 2 ) = \ alpha ^ ( 2 ) V_ ( b , 2 ) = \ alpha V_ ( c , 2 ) \ \ \ end ( aligned ) ) ) Thus , v a b c = ( V 0 V 0 V 0 ) + ( V 1 α 2 V 1 α V 1 ) + ( V 2 α V 2 α 2 V 2 ) = ( 1 1 1 1 α 2 α 1 α α 2 ) ( V 0 V 1 V 2 ) = A v 012 ( \ displaystyle ( \ begin ( aligned ) \ mathbf ( v ) _ ( abc ) & = ( \ begin ( bmatrix ) V_ ( 0 ) \ \ V_ ( 0 ) \ \ V_ ( 0 ) \ end ( bmatrix ) ) + ( \ begin ( bmatrix ) V_ ( 1 ) \ \ \ alpha ^ ( 2 ) V_ ( 1 ) \ \ \ alpha V_ ( 1 ) \ end ( bmatrix ) ) + ( \ begin ( bmatrix ) V_ ( 2 ) \ \ \ alpha V_ ( 2 ) \ \ \ alpha ^ ( 2 ) V_ ( 2 ) \ end ( bmatrix ) ) \ \ & = ( \ begin ( bmatrix ) 1&1&1 \ \ 1& \ alpha ^ ( 2 ) & \ alpha \ \ 1& \ alpha & \ alpha ^ ( 2 ) \ end ( bmatrix ) ) ( \ begin ( bmatrix ) V_ ( 0 ) \ \ V_ ( 1 ) \ \ V_ ( 2 ) \ end ( bmatrix ) ) \ \ & = ( \ textbf ( A ) ) \ mathbf ( v ) _ ( 012 ) \ end ( aligned ) ) ) where v 012 = ( V 0 V 1 V 2 ) , A = ( 1 1 1 1 α 2 α 1 α α 2 ) ( \ displaystyle \ mathbf ( v ) _ ( 012 ) = ( \ begin ( bmatrix ) V_ ( 0 ) \ \ V_ ( 1 ) \ \ V_ ( 2 ) \ end ( bmatrix ) ) , ( \ textbf ( A ) ) = ( \ begin ( bmatrix ) 1&1&1 \ \ 1& \ alpha ^ ( 2 ) & \ alpha \ \ 1& \ alpha & \ alpha ^ ( 2 ) \ end ( bmatrix ) ) ) Decomposition ( edit ) The sequence components are derrived from the analysis equation v 012 = A − 1 v a b c ( \ displaystyle \ mathbf ( v ) _ ( 012 ) = ( \ textbf ( A ) ) ^ ( - 1 ) \ mathbf ( v ) _ ( abc ) ) where A − 1 = 1 3 ( 1 1 1 1 α α 2 1 α 2 α ) ( \ displaystyle ( \ textbf ( A ) ) ^ ( - 1 ) = ( \ frac ( 1 ) ( 3 ) ) ( \ begin ( bmatrix ) 1&1&1 \ \ 1& \ alpha & \ alpha ^ ( 2 ) \ \ 1& \ alpha ^ ( 2 ) & \ alpha \ end ( bmatrix ) ) ) The above two equations tell how to derive symmetrical components corresponding to an asymmetrical set of three phasors : Sequence 0 is one - third the sum of the original three phasors . Sequence 1 is one - third the sum of the original three phasors rotated counterclockwise 0 ° , 120 ° , and 240 ° . Sequence 2 is one - third the sum of the original three phasors rotated counterclockwise 0 ° , 240 ° , and 120 ° . Visually , if the original components are symmetrical , sequences 0 and 2 will each form a triangle , summing to zero , and sequence 1 components will sum to a straight line . Intuition ( edit ) Napoleon 's theorem : If the triangles centered on L , M , and N are equilateral , then so is the green triangle . The phasors V ( a b ) = V ( a ) − V ( b ) ; V ( b c ) = V ( b ) − V ( c ) ; V ( c a ) = V ( c ) − V ( a ) ( \ displaystyle \ scriptstyle V_ ( ( ab ) ) = V_ ( ( a ) ) - V_ ( ( b ) ) ; \ ; V_ ( ( bc ) ) = V_ ( ( b ) ) - V_ ( ( c ) ) ; \ ; V_ ( ( ca ) ) = V_ ( ( c ) ) - V_ ( ( a ) ) ) form a closed triangle ( e.g. , outer voltages or line to line voltages ) . To find the synchronous and inverse components of the phases , take any side of the outer triangle and draw the two possible equilateral triangles sharing the selected side as base . These two equilateral triangles represent a synchronous and an inverse system . If the phasors V were a perfectly synchronous system , the vertex of the outer triangle not on the base line would be at the same position as the corresponding vertex of the equilateral triangle representing the synchronous system . Any amount of inverse component would mean a deviation from this position . The deviation is exactly 3 times the inverse phase component . The synchronous component is in the same manner 3 times the deviation from the `` inverse equilateral triangle '' . The directions of these components are correct for the relevant phase . It seems counter intuitive that this works for all three phases regardless of the side chosen but that is the beauty of this illustration . The illustration is from Napoleon 's Theorem . Poly - phase case ( edit ) It can be seen that the transformation matrix above is a discrete Fourier transform , and as such , symmetrical components can be calculated for any poly - phase system . However , by Pontryagin duality , only certain groups have a unique inverse , which is necessary for use in fault analysis . Contribution of harmonics to symmetrical components in 3 - phase power systems ( edit ) Harmonics often occur in power systems as a consequence of non-linear loads . Each order of harmonics contributes to different sequence components . Harmonics of order 2 n ( \ displaystyle \ scriptstyle 2n ) make no contribution . Harmonics of order 3 + 6 n ( \ displaystyle \ scriptstyle 3 + 6n ) contribute to the zero sequence . Harmonics of order 5 + 6 n ( \ displaystyle \ scriptstyle 5 + 6n ) contribute to the negative sequence . Harmonics of order 7 + 6 n ( \ displaystyle \ scriptstyle 7 + 6n ) contribute to the positive sequence . Note that the rules above are only applicable if the phase values ( or distortion ) are exactly the same . Consequence of the zero sequence component in power systems ( edit ) The zero sequence represents the component of the unbalanced phasors that is equal in magnitude and phase . Because they are in phase , zero sequence currents flowing through an n - phase network will sum to n times the magnitude of the individual zero sequence currents components . Under normal operating conditions this sum is small enough to be negligible . However , during large zero sequence events such as lightning strikes , this nonzero sum of currents can lead to a larger current flowing through the neutral conductor than the individual phase conductors . Because neutral conductors are typically not larger than individual phase conductors , and are often smaller than these conductors , a large zero sequence component can lead to overheating of neutral conductors and to fires . One way to prevent large zero sequence currents is to use a delta connection , which appears as an open circuit to zero sequence currents . For this reason , most transmission , and much sub-transmission is implemented using delta . Much distribution is also implemented using delta , although `` old work '' distribution systems have occasionally been `` wyed - up '' ( converted from delta to wye ) so as to increase the line 's capacity at a low converted cost , but at the expense of a higher central station protective relay cost . See also ( edit ) Symmetry Dqo transformation Alpha -- beta transformation References ( edit ) Notes Jump up ^ Hadjsaïd , Nouredine ; Sabonnadière , Jean - Claude ( 2013 ) . Power Systems and Restructuring . John Wiley & Sons . p. 244 . ISBN 9781118599921 . Jump up ^ Charles L. Fortescue , `` Method of Symmetrical Co-Ordinates Applied to the Solution of Polyphase Networks '' . Presented at the 34th annual convention of the AIEE ( American Institute of Electrical Engineers ) in Atlantic City , N.J. on 28 June 1918 . Published in : AIEE Transactions , vol. 37 , part II , pages 1027 -- 1140 ( 1918 ) . For a brief history of the early years of symmetrical component theory , see : J. Lewis Blackburn , Symmetrical Components for Power Engineering ( Boca Raton , Florida : CRC Press , 1993 ) , pages 3 -- 4 . Jump up ^ Gabriele Kass - Simon , Patricia Farnes , Deborah Nash ( ed ) , Women of Science : Righting the Record , Indiana University Press , 1993 , ISBN 0253208130 . pages 164 - 168 Bibliography J. Lewis Blackburn Symmetrical Components for Power Systems Engineering , Marcel Dekker , New York ( 1993 ) . ISBN 0 - 8247 - 8767 - 6 William D. Stevenson , Jr . Elements of Power System Analysis Third Edition , McGraw - Hill , New York ( 1975 ) . ISBN 0 - 07 - 061285 - 4 . History article from IEEE on early development of symmetrical components , retrieved May 12 , 2005 . Westinghouse Corporation , Applied Protective Relaying , 1976 , Westinghouse Corporation , no ISBN , Library of Congress card no . 76 - 8060 - a standard reference on electromechanical protective relays Retrieved from `` https://en.wikipedia.org/w/index.php?title=Symmetrical_components&oldid=845163714 '' Categories : Electrical engineering Three - phase AC power Hidden categories : All articles with unsourced statements Articles with unsourced statements from November 2016 Talk Contents About Wikipedia Deutsch Español فارسی Français हिन्दी Nederlands Polski Português Русский Suomi 3 more Edit links This page was last edited on 9 June 2018 , at 21 : 24 . 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. Symmetrical components in electrical engineering
2. Analysis of unbalanced three-phase power systems
3. Positive, negative, and zero sequence components
4. Simplified calculations for asymmetric fault analysis
5. Contribution of harmonics to symmetrical components