In electrical engineering, the method of symmetrical components simplifies the analysis of a three-phase power system exhibiting an electrical fault or other unbalanced condition. The symmetrical components corresponding to an asymmetrical set of three phasors are:
Sequence 0 (also known as zero sequence or homopolar) is one-third the sum of the original three phasors. Sequence 1 (positive sequence) is one-third the sum of the original three phasors rotated counterclockwise by 0°, 120°, and 240°. Sequence 2 (negative sequence) is one-third the sum of the original three phasors rotated counterclockwise 0°, 240°, and 120°. The analysis of power systems is much simpler in the domain of symmetrical components, because the resulting equations are mutually linearly independent if the power system itself is balanced. In this case, each symmetrical component can be analyzed separately, similar to the per-phase analysis. Protective relays utilize symmetric components for fault detection. For example, during normal operation, the zero-sequence current is very small, so a high current value is a convenient and reliable indicator of a ground fault.
History The basic idea dates back to 1895, when Ferraris et al. produced an analysis of a single-phase motor by splitting a field set inside it into two components revolving in the opposite directions. The concept now known as the positive and negative sequences was published by Ernst Alexanderson in 1913 in his work on phase balancers, and by L. G. Stokvis in 1912-1915 while investigating the voltage regulation. These works lacked the clear definition of a zero sequence. 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.
Description 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.
The three-phase case 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
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