Fig. 262. POSITIVE SEQUENCE set.

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Fig. 263. NEGATIVE SEQUENCE set.

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" " " In the gures, note that both of the sets are balanced, meaning that A1 , B1 , C1 have " " " equal magnitudes, also A2 , B2 , C2 have equal magnitudes, with the phase displacements between vectors being 1208 in both sets. As usual, the vectors can represent either rms values of sinusoidal voltages and currents or impedances. The vectors themselves, in any given case, always remain xed in position, with positive angles measured in the ccw sense, as shown in the gures. Note that both sets are speci ed relative to the same common origin and reference line.

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CHAPTER 10 Magnetic Coupling. Transformers

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In the gures, note that the positive sequence set is identi ed by the subscript 1, while the negative sequence set is identi ed by the subscript 2. We ll use this method of identi cation IN ALL OF THE WORK that follows. Since both sets are balanced, it follows that the vector sum in each case is zero; that is ) " " " " S1 A1 B1 C1 0 448 " " " " S2 A2 B2 C2 0 Next, in regard to Figs. 262 and 263, the term sequence refers to the order in which the letters appear in the diagram in the ccw sense, as follows. If (in going around the diagram in the ccw sense) the order of the letters is ABC we are said to have a positive sequence of vectors, but if the order is ACB we have a negative sequence. Thus, in accordance with this de nition, Fig. 262 is a positive sequence of vectors and Fig. 263 is a negative sequence set. The concept of sequence is important for the following reason. First note that, by eq (448), the SUM of the two sequences can be written in the form " " " " " " " " S1 S2 A1 A2 B1 B2 C1 C2 449

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in which the left-hand side is the sum of two BALANCED sets of three vectors each, and (as we ll show in problem 208) the right-hand side represents an UNBALANCED set of three vectors; thus eq. (449) shows that it s possible to represent an unbalanced set of three vectors (the right-hand side) as the sum of two balanced sets of three vectors each. This is the basic principle behind the method of symmetrical components. The following two problems will clarify this point. Problem 207 " 0 "0 " 0 " " " Let A1 , B1 , C1 and A1 , B1 , C1 be two sets of positive sequence vectors (note the 1 subscripts). Using eqs. (446) and (449), show that the vector sum of the two sets is equivalent to a single balanced set of three vectors. Problem 208 " " " " " " Let A1 , B1 , C1 be a positive sequence set of vectors, and A2 , C2 , B2 be a negative sequence set. Show that the vector sum of the two sets is a single unbalanced set of three vectors. The above two problems show that an unbalanced set of three vectors can be represented as the sum of two balanced sets of three vectors each ONLY if one of the sets is a positive sequence set and the other a negative sequence set, where positive sequence and negative sequence are de ned in connection with Figs. 262 and 263 where, algebraically, " " B1 A1 j120 " " C1 A1 j240 " " and B2 A2 j240 " " and C2 A2 j120

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To bring out another important point let us begin by writing eq. (449) in the form 9 " " " A1 A2 A 0 > = " " " 450 B1 B2 B 0 > "1 C2 C 0 ; " " C " " " in which A 0 , B 0 , and C 0 are the three components of the unbalanced set of vectors.

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