Symmetrical components
Encyclopedia
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.
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 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; CBA), 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 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.
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.
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.
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
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
radians or 120°. Define the operator 
phasor vector forward by that angle.
Note that α3 = 1 so that α−1 = α2.
The zero sequence components are in phase; denote them as:
and the other phase sequences as:
Thus,
where
Conversely, the sequence components are generated from the analysis equations
where
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.
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
, the method of symmetrical components is used to simplify analysis of unbalanced three phase power systems under both normal and abnormal conditions.
Description
In 1918 Charles Legeyt FortescueCharles Legeyt Fortescue
Charles LeGeyt Fortescue was an electrical engineer. He was born in York Factory, in what is now Manitoba where the Hayes River enters Hudson Bay...
presented a paper which demonstrated that any set of N unbalanced phasors (that is, any such polyphase
Polyphase
Polyphase may refer to:* Polyphase matrix, in signal processing* Polyphase system, in electrical engineering* Polyphasic sleep...
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 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; CBA), 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
Fault (power engineering)
In an electric power system, a fault is any abnormal flow of electric current. For example, a short circuit is a fault in which current flow bypasses the normal load. An open-circuit fault occurs if a circuit is interrupted by some failure. In three-phase systems, a fault may involve one or more...
analysis more tractable.
By expanding a one-line diagram
One-line diagram
In power engineering, a one-line diagram or single-line diagram is a simplified notation for representing a three-phase power system. The one-line diagram has its largest application in power flow studies. Electrical elements such as circuit breakers, transformers, capacitors, bus bars, and...
to show the positive sequence, negative sequence and zero sequence impedances of generator
Electrical generator
In electricity generation, an electric generator is a device that converts mechanical energy to electrical energy. A generator forces electric charge to flow through an external electrical circuit. It is analogous to a water pump, which causes water to flow...
s, transformer
Transformer
A transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying current in the first or primary winding creates a varying magnetic flux in the transformer's core and thus a varying magnetic field...
s and other devices including overhead lines
Overhead lines
Overhead lines or overhead wires are used to transmit electrical energy to trams, trolleybuses or trains at a distance from the energy supply point...
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 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
Protective relay
In electrical engineering, a protective relay is a complex electromechanical apparatus, often with more than one coil, designed to calculate operating conditions on an electrical circuit and trip circuit breakers when a fault is detected...
s, which used negative-sequence voltages and currents as a reliable indicator of fault conditions. Such relays may be used to trip circuit breaker
Circuit breaker
A circuit breaker is an automatically operated electrical switch designed to protect an electrical circuit from damage caused by overload or short circuit. Its basic function is to detect a fault condition and, by interrupting continuity, to immediately discontinue electrical flow...
s or take other steps to protect electrical systems.
The analytical technique was adopted and advanced by engineers at General Electric
General Electric
General Electric Company , or GE, is an American multinational conglomerate corporation incorporated in Schenectady, New York and headquartered in Fairfield, Connecticut, United States...
and Westinghouse and after World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
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 phasorPhasor
Phasor is a phase vector representing a sine wave.Phasor may also be:* Phasor , a stereo music, sound and speech synthesizer for the Apple II computer* Phasor measurement unit, a device that measures phasors on an electricity grid...
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
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
radians or 120°. Define the operator phasor vector forward by that angle.Note that α3 = 1 so that α−1 = α2.
The zero sequence components are in phase; denote them as:
and the other phase sequences as:
Thus,
where
Conversely, the sequence components are generated from the analysis equations
where
An intuitive feeling
The phasors 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.