Symmetrical components


Symmetrical components

In electrical engineering, the method of Symmetrical components is used to simplify analysis of unbalanced three phase power systems.

Description

Charles Legeyt Fortescue in a paper presented in 1918 ("Method of Symmetrical Co-Ordinates Applied to the Solution of Polyphase Networks") demonstrated that any set of N unbalanced polyphase quantities could be expressed as the sum of N symmetrical sets of balanced phasors. Only a single frequency component is represented by the phasors.

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 - BAC), 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 asymmetric fault analysis more tractable.

By expanding a one-line diagram to show the positive sequence, negative sequence and zero sequence impedances of generators and transformers and other devices, 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 phase order 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. Since these effects can be detected physically, the mathematical tool became the basis for the design of protection 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 was an accepted method for asymmetric fault analysis.

The three-phase case

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

V_{abc} = egin{bmatrix} V_a \ V_b \ V_c end{bmatrix}

and the three symmetrical components phasors arranged into a vector as

V_{s} = egin{bmatrix} V_0 \ V_1 \ V_2 end{bmatrix}

where the subscripts 0, 1, and 2 refer respectively to the zero, positive, and negative sequence components.

A phase rotation operator alpha is defined to rotate a phasor vector forward by 120 degrees or frac{2 pi}{3} radians. A matrix A can be defined using this operator to transform the phase vector into symmetrical components.

A = egin{bmatrix}1 & 1 & 1 \ 1 & alpha^2 & alpha \ 1 & alpha & alpha^2 end{bmatrix}

V_{abc} = A cdot V_s

ee also

* Symmetry

References

* 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.
* [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.
* Westinghouse Corporation, "Applied Protective Relaying", 1976, Westinghouse Corporation, no ISBN, Library of Congress card no. 76-8060 - a standard reference on electromechanical protection relays


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