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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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We must not conclude from the foregoing, however, that ALL unbalanced sets of three vectors can be expressed as the sum of just two balanced sets of vectors (one positive sequence set and one negative sequence set). This is because inspection of eqs. (448) and (449) shows that eq. (450) is valid only if " " " A0 B0 C0 0 that is, only if the sum of the vectors in the unbalanced set is equal to zero, a condition which MAY or MAY NOT be true in practical work. This indicates that a more general form of eq. (450) is needed, to cover cases in which the sum of the unbalanced vectors is not equal to zero. In this regard, a more general form of eq. (450) can be arrived at by thinking in terms of, degrees of freedom, as follows. We recall that, in general, an unbalanced set of three vectors, such as in Fig. 261, has six degrees of freedom. So far, however, in Figs. 262 and 263 we have only four degrees of freedom. Hence, in addition to Figs. 262 and 263 we must, in order to include the most general unbalanced condition, add one more set of three balanced vectors to bring the degrees of freedom up to six. This is done by de ning what is called a zero sequence set of vectors which consists of three IDENTICAL vectors, meaning that the three vectors all have the same magnitude of amplitude at the same angle k with respect to the reference axis, as shown in Fig. 264.

Thus a zero sequence set of three vectors has two degrees of freedom and, using the subscript 0, is de ned by writing that " " " A0 B0 C0 451

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