Symmetrical components
In electrical engineering, the method of symmetrical components simplifies analysis of unbalanced three-phase power systems under both normal and abnormal conditions.
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.
[3] In 1918 Charles Legeyt Fortescue presented a paper[4] 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.
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.
Essentially, this method converts three unbalanced phases into three independent sources, which makes asymmetric fault analysis more tractable.
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.
Notice that the colors (red, blue, and yellow) of the separate sequence vectors correspond to three different phases (A, B, and C, for example).
Symmetrical components are most commonly used for analysis of three-phase electrical power systems.
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.
If the original unbalanced set of voltage phasors have positive or abc phase sequence, then: meaning that Thus, where If instead the original unbalanced set of voltage phasors have negative or acb phase sequence, the following matrix can be similarly derived: The sequence components are derived from the analysis equation where The above two equations tell how to derive symmetrical components corresponding to an asymmetrical set of three phasors: 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.
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.
The synchronous component is in the same manner 3 times the deviation from the "inverse equilateral triangle".
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 zero sequence represents the component of the unbalanced phasors that is equal in magnitude and phase.
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.
