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| In [[electrical engineering]], the method of '''symmetrical components''' is used to simplify analysis of unbalanced [[three-phase]] power systems under both normal and abnormal conditions. The basic idea is that an asymmetrical set of ''N'' [[phasor]]s can be expressed as a [[linear combination]] of ''N'' symmetrical sets of phasors by means of a [[complex number|complex]] [[linear transformation]].<ref>{{cite book|url=http://books.google.com/books?id=bpEeycYeWJIC&pg=PT244 |title=Power Systems and Restructuring |last1=Hadjsaïd |first1=Nouredine |first2=Jean-Claude |last2=Sabonnadière |publisher=John Wiley & Sons |year=2013 |ISBN=9781118599921 |page=244}}</ref> In the most common case of three-phase system, 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 circuit|balanced]].
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| The principle of symmetrical components resembles the [[RGB color model|RGB transformation of colors]], where a broad spectrum of colors can be transformed into three independent components, which can be then manipulated independently and transformed back into the visible spectrum using a display such as [[cathode ray tube]].
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| ==Description==
| | Elderly video games ought in order to not be discarded. They may be worth some money at several video retailers. Means positivity . buy and sell in several game titles, you may possibly get your upcoming distinction at no cost!<br><br>Game titles are fun to practice with your kids. Aids you learn much another recommendation of your kid's interests. Sharing interests with your kids like this can on top of that create great conversations. It also gives you an opportunity to monitor growth and development of their skills.<br><br>Pay attention to a mission's evaluation when purchasing something special. This evaluation will allow you to discover what age level clash of [http://Imageshack.us/photos/clans+hack clans hack] tool is perfect for and will let you know when the sport is violent. It figure out whether you should buy the sport.<br><br>Guilds and clans have already been popular ever since the very beginning of first-person gift idea shooter and MMORPG avid gamers. World of WarCraft develops fot it concept with their own World associated Warcraft guilds. A real guild can easily always stay understood as a in respect of players that band away for companionship. People in the guild travel together again for fun and pleasure while improving in ordeal and gold.<br><br>Regardless of the reason, computer game secret sauce are widespread and dotted fairly rapidly over the online market place. The gaming community is intending to find means cease cheaters from overrunning any sort of game; having lots involved with cheaters playing a one particular game can really produce honest players to quit playing, or play just with friends they trust. This poses a extensive problem particularly for price games for example EverQuest, wherein a loss connected with players ultimately result within a loss of income.<br><br>Casino is infiltrating houses around the world. Some play these games for work, rather others play them by enjoyment. This company is booming and won't go away completely anytime soon. Refer to for some fantastic recommendations on gaming.<br><br>Pc games or computer games elevated in popularity nowadays, not only with the younger generation, but also with grownups as well. If you cherished this article and you simply would like to receive more info regarding [http://circuspartypanama.com clash of clans hack cydia] please visit our web-page. Quite a bit games available, ranging at a intellectual to the each day - your options get limitless. Online character playing games are amongst the most popular games anywhere on the earth. With this popularity, plenty of folks that are exploring and finding ways to go along with whole game as very fast as they can; reasons for using computer How to hack in clash of clans range from simply trying to own your own good friends stare at you in awe, or getting whole lot of game money anyone really can sell later, or simply just - rid the game of the fun factor for the other players. |
| [[File:Unbalanced symmetrical components.pdf|thumb|400px|Set of three unbalanced phasors, and the necessary symmetrical components that sum up to the resulting plot at the bottom.]]
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| In 1918 [[Charles Legeyt Fortescue]] presented a paper<ref>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 July 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.</ref> which demonstrated that any set of N unbalanced [[phasors]] (that is, any such ''[[Polyphase system|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.
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| 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). Essentially, this method converts three unbalanced phases into three independent sources, which makes [[Fault (power engineering)|asymmetric fault]] analysis more tractable.
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| By expanding a [[one-line diagram]] to show the positive sequence, negative sequence and zero sequence impedances of [[Electrical generator|generator]]s, [[transformer]]s 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.
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| 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 relay]]s, which used negative-sequence voltages and currents as a reliable indicator of fault conditions. Such relays may be used to trip [[circuit breaker]]s or take other steps to protect electrical systems.
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| The analytical technique was adopted and advanced by engineers at [[General Electric]] and [[Westinghouse Electric Corporation|Westinghouse]] and after [[World War II]] it was an accepted method for asymmetric fault analysis.
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| As shown in the figure to the 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.
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| ==The Three-Phase Case==
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| Symmetrical components are most commonly used for analysis of three-phase electrical power systems. If the phase quantities are expressed in [[phasor]] notation using complex numbers, a vector can be formed for the three phase quantities. For example, a vector for three phase voltages could be written as
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| :<math>V_{abc} = \begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix} =
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| \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}</math>
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| 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 <math>\scriptstyle\frac{2}{3}\pi</math> radians or 120°. Define the operator <math>\scriptstyle\alpha</math> phasor vector forward by that angle.
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| :<math>\alpha \equiv e^{\frac{2}{3}\pi i}</math>
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| Note that α<sup>3</sup> = 1 so that α<sup>−1</sup> = α<sup>2</sup>.
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| The zero sequence components are in phase; denote them as:
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| :<math>V_0 \equiv V_{a,0} = V_{b,0} = V_{c,0}</math>
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| and the other phase sequences as:
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| :<math>\begin{align}
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| 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}
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| \end{align}</math>
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| Thus,
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| :<math>\begin{align}
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| V_{abc}
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| &= \begin{bmatrix} V_0 \\ V_0 \\ V_0 \end{bmatrix} +
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| \begin{bmatrix} V_1 \\ \alpha^2 V_1 \\ \alpha V_1 \end{bmatrix} +
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| \begin{bmatrix} V_2 \\ \alpha V_2 \\ \alpha^2 V_2 \end{bmatrix} \\
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| &= \begin{bmatrix}1 & 1 & 1 \\ 1 & \alpha^2 & \alpha \\ 1 & \alpha & \alpha^2 \end{bmatrix}
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| \begin{bmatrix} V_0 \\ V_1 \\ V_2 \end{bmatrix} \\
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| &= \textbf{A} V_{012}
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| \end{align}</math>
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| where
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| :<math>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}</math>
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| Conversely, the sequence components are generated from the analysis equations
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| :<math>V_{012} = \textbf{A}^{-1} V_{abc} </math> | |
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| where
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| :<math>\textbf{A}^{-1} = \frac{1}{3} \begin{bmatrix}1 & 1 & 1 \\ 1 & \alpha & \alpha^2 \\ 1 & \alpha^2 & \alpha \end{bmatrix}</math>
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| ==An intuitive feeling==
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| The phasors <math>\scriptstyle V_{(ab)}= V_{(a)}-V_{(b)}; \;V_{(bc)}= V_{(b)}-V_{(c)}; \; V_{(ca)}= V_{(c)}-V_{(a)}</math> 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 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.
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| For an illustration see [[Napoleon's Theorem]].
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| ==Poly-phase Case==
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| 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.
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| ==See also==
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| * [[Symmetry]]
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| ==References==
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| ;Notes
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| {{reflist}}
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| ;Bibliography
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| * J. Lewis Blackburn ''Symmetrical Components for Power Systems Engineering'', Marcel Dekker, New York (1993). ISBN 0-8247-8767-6
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| * William D. Stevenson, Jr. ''Elements of Power System Analysis Third Edition'', [[McGraw-Hill]], New York (1975). ISBN 0-07-061285-4.
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| * [http://www.ieee.org/organizations/pes/public/2004/nov/peshistory.html History article] from [[IEEE]] on early development of symmetrical components, retrieved May 12, 2005.
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| * Westinghouse Corporation, ''Applied Protective Relaying'', 1976, Westinghouse Corporation, no ISBN, Library of Congress card no. 76-8060 - a standard reference on electromechanical protective relays
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| [[Category:Electrical engineering]]
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| [[Category:Components]]
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