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 complexlinear transformation.[1] Fortescue’s theorem (symmetrical components) is based on superposition principle,[2] so it is applicable to linear power systems only, or to linear approximations of non-linear power systems.

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.^{[citation needed]}

## . . . Symmetrical components . . .

In 1918 Charles Legeyt Fortescue presented a paper[3] 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.[4] 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 system, 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.

## . . . Symmetrical components . . .

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