Symmetrical Components and Fortescue's Theorem
An unbalanced set of N related phasors can be resolved into N systems of phasors known as symmetrical components. For a three-phase system, these components include Positive Sequence, Negative Sequence, and Zero Sequence. These components help simplify complex scenarios in electrical systems.
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5. Symmetrical Components. Fortescue's Theorem: An unbalanced set of N related phasors can be resolved into N systems of phasors called the symmetrical components of the original phasors. For a three-phase system (i.e. N = 3), the three sets are: Positive Sequence (indicated by + or 1 ) - three phasors, equal in magnitude, 120o apart, with the same sequence (a-b-c) as the original phasors. Negative Sequence ( or 2 ) - three phasors, equal in magnitude, 120o apart, with the opposite sequence (a-c-b) of the original phasors. Zero Sequence ( 0 ) - three identical phasors (i.e. equal in magnitude, with no relative phase displacement). Symmetrical Components and their relationship to one another The original set of phasors ~ V ~ V ~ V ~ V ~ V ~ V ~ V ~ V ~ V ~ V ~ V ~ V ~ V ~ V ~ V = + + = = 0 1 2 a a a a 0 0 0 a b c ~ V ~ V ~ V ~ V ~ V ~ V ~ V ~ V o o = + = + + = 1 120 1 120 1 1 1 1 c a 0 1 2 b a b b b b o o = + + = + = 1 120 1 120 2 2 2 2 0 1 2 b a c a c c c c Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 5, March 31, 2015 1
5. Symmetrical Components, cont. The symmetrical components of all a-b-c voltages are usually written in terms of the symmetrical components of phase a by defining complex number a as 120 1 + = a o o 2 120 1 0 4 2 1 = + = a o o 3 0 1 0 6 3 1 = + = a , o ~ ~ ~ , , V V V Substituting into the previous equations for yields ~ ~ ~ ~ a a a a V V V V + + = 2 0 a a a b V a V a V V + + = ~ ~ ~ ~ a a a c V a V a V V + + = a b ~ V c ~ V 1 1 a 1 a 0 1 2 0 a a ~ V ~ V ~ ~ ~ ~ = 2 1 1 2 1 b a ~ V ~ V 2 1 a a 2 2 c a 0 1 2 1 1 1 which in simplified form is 1 1 a 1 a ~ V ~ V ~ V ~ V 1 1 2 = = 1 , , T T = 1 = T a a 2 1 , T 012 012 abc abc 3 2 2 1 a a 1 a a Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 5, March 31, 2015 2
5. Symmetrical Components, cont. Why do all this? Answer: to simplify most situations. ~ V 0 ~ V = 1 , V ~ T 012 abc a ~ ~ 1 1 1 V ~ a a 1 = 2 1 V ~ a V ~ 1 a b 3 2 2 1 V a a V 2 a c For balanced set, have only positive-sequence ~ V ~ V + + + + 2 2 1 1 a 1 a 1 1 a 1 a 1 a 1 1 0 0 a a a a ~ V ~ V ~ V ~ V 0 a a 1 ~ V ~ V ~ V = = = + + = + + = = 2 2 2 2 3 3 3 3 1 1 1 1 3 1 a a a a a a a a a 1 a a a 3 3 3 3 3 ~ V ~ V + + + + 2 2 4 2 2 1 1 1 1 0 0 a a a a a a a a a a 2 a a Swap phases b and c (negative rotation), have only negative-sequence ~ V ~ V + + + + 2 2 1 1 a 1 a 1 1 a 1 a 1 a 1 1 0 0 a a a a ~ V ~ V ~ V ~ V 0 a a 1 ~ V ~ V ~ V = = = + + = + 1 + = = 2 2 2 4 2 1 1 1 1 0 0 a a a a a a a a a 1 a a a 3 3 3 3 3 ~ V ~ V + + + + 2 2 2 2 3 3 1 1 1 1 1 3 1 a a a a a a a a 2 a a Common mode (a, b, c identical), have only zero-sequence ~ V ~ V 1 1 a 1 a 1 1 a 1 a 1 3 3 1 ~ V ~ V ~ V 0 a a 1 ~ V ~ V ~ V = = = + + = = 2 2 2 1 1 1 1 0 0 a a a a a 1 a a a 3 3 3 3 ~ V ~ V + + 2 2 2 1 1 1 1 0 0 a a a a a a 2 a a Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 5, March 31, 2015 3
5. Symmetrical Components, cont. Zero-Sequence (i.e., Common Mode) Transformer Models, again Zero-sequence current can flow ONLY when there is a ground path R + jX Grounded Wye - Grounded Wye Grounded Wye - Delta R + jX Grounded Wye - Ungrounded Wye R + jX Ungrounded Wye - Delta R + jX Delta - Delta R + jX Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 5, March 31, 2015 4
5. Symmetrical Components, cont. Three-phase circuits can be solved in full a-b-c, or in using 0-1-2 sequence components. For a balanced circuit, solve it in positive-sequence because negative- and zero- sequences are zero, so that the positive-sequence solution IS the phase A solution. Hence, the one-line diagram. Consider the following a-b-c equations for a symmetric network. Symmetric means that self-impedances S for each phase are equal, and mutual-impedances M between phases are equal. ~ ~ V ~ S M M I ~ a a ~ ~ = abc V Z I = V ~ M S M I ~ abc abc b b V M M S I c c Converting to 0-1-2 ~ V ~ = abc T Z I T 012 012 ~ V ~ = 1 1 T ~ V T T ~ I Z I T 012 012 abc = = 1 , Z Z T Z T where 012 012 012 012 abc Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 5, March 31, 2015 5
5. Symmetrical Components, cont. = 1 Z T Z T 012 abc 1 1 a 1 a 1 1 a 1 a S M M 1 = 2 2 1 1 M S M 3 2 2 1 1 a a M M S a a + + S + S 2 2 2 1 1 a 1 a S M S M S M 1 = 2 2 ( ) ( S ) 1 S M a M a M 3 2 2 ( ) ( ) 1 S M a S M a M a a + 0 ( 3 2 ) 0 0 S M 1 = 0 ( 3 ) 0 S M 3 0 ( 3 ) S M This is huge! If network impedances are balanced, then a coupled a-b-c network can be solved using three simple uncoupled 0-1-2 networks. + ( 2 ) 0 0 S M = 0 0 ( ) 0 S M 0 ( ) S M Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 5, March 31, 2015 6
5. Symmetrical Components. Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 5, March 31, 2015 7
5. Symmetrical Components. Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 5, March 31, 2015 8
5. Symmetrical Components. Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 5, March 31, 2015 9
