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CHAPTER 10 Magnetic Coupling. Transformers
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constant, independent of time. If, however, you have the patience to carefully apply the following trigonometrical identities sin x sin y 1 cos x y cos x y 2 and cos x y cos x cos y sin x sin y note 6 in Appendix note 25 in Appendix
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to the foregoing equation for p, you ll nd that the expression actually reduces to p 1:5V 0 I 0 cos  445
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which, since V 0 , I 0 , and  are all constants in any given case, shows that p is also constant in any given case in any balanced three-phase system. Problem 206 Show that p, in eq. (445), is EQUAL to PT in eqs. (440) and (444).
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The Unbalanced Case; Symmetrical Components
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Let us begin by observing that a SINGLE plane vector is de ned in terms of two independent variables, its MAGNITUDE and its ANGULAR POSITION relative to an agreed-upon reference axis. The independent variables are also referred to as degrees of freedom ; thus, a single " plane vector is said to have two degrees of freedom. Such a vector, A A=h, is illustrated in Fig. 259, where A and h are the two degrees of freedom. (As always, positive angles are measured in the ccw direction from the reference axis.)
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Fig. 259
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Next consider a balanced set of three plane vectors. As we know, this is any set of three vectors having EQUAL MAGNITUDES and EQUAL PHASE DISPLACEMENTS. " " " Now let A1 , B1 , and C1 be such a balanced set, in which the equal phase displacement is "1 as the reference vector, displaced an angle h from the reference 1208, and let us take A " " " axis, as illustrated in Fig. 260, where jA1 j jB1 j jC1 j A1 B1 C1 : We are already familiar with the fact that the vector sum of such a balanced set of vectors is equal to zero. In this regard, note that a balanced set of plane vectors has just TWO degrees of freedom, these being the common magnitudes of the vectors and the angular displacement h of the reference vector from the reference axis. Thus the common magnitude, A1 B1 C1 , and the reference angle h are the two degrees of freedom in Fig. 260.
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CHAPTER 10 Magnetic Coupling. Transformers
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Fig. 260
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Next, in Fig. 260, assuming the diagram to be drawn on the complex plane, note that* " " B1 A1  j120 and " " C1 A1  j240
where 120 and 240 are understood to be angles in degrees{ (1208 and 2408), and thus that general form of the algebraic equation for the balanced case of Fig. 260 can be written as " " " A1 A1  j120 A1  j240 0 446
the right-hand side re ecting the fact that the vector sum of such a balanced set of vectors is zero. " " " Now consider an unbalanced set of three plane vectors A, B, and C, such as is illustrated in Fig. 261, in which let " " " " S A B C 447
Fig. 261
" " * Let V V ja be a vector quantity on the complex plane. Now multiply V by  jb , thus " V  jb V ja  jb V j a b " " " showing that multiplying a vector V by  rotates V through the angle b but does not change the magnitude of V .
{ See footnote in connection with eq. (159) in Chap. 6.
CHAPTER 10 Magnetic Coupling. Transformers
" " where S is the vector sum of the three vectors, in which S MAY or MAY NOT be equal to zero, depending upon the particular circumstances. It follows that such an unbalanced set of three vectors will, in general, have SIX degrees of freedom (two for each of the individual vectors). As you would expect, unbalanced conditions sometimes do occur in practical threephase work. Fortunately, the solution of such problems can be expedited by means of what is called symmetrical components. This is an algebraic procedure based upon the fact an UNBALANCED set of three vectors can be expressed as the sum of THREE BALANCED SETS of three vectors each. The procedure is important because it allows the solution of a more di cult unbalanced problem in terms of the superposition of three easier balanced problems. In this regard, let us remark that the procedure is not only of great practical value but is also an interesting example of the application of the algebra of the complex plane to electric circuit problems. Let us begin our explanations as follows. First, we ve seen that a balanced set of plane vectors possesses just two degrees of freedom, while an unbalanced set of three plane vectors possesses, in general, six degrees of freedom. Now, in regard to physical systems, it is a fundamental fact that the number of degrees of freedom must remain the same in any valid equivalent description of a system. It thus follows that it will, in general, require the sum of three balanced sets to replace one unbalanced set. OUR PROBLEM, therefore, is to nd three balanced sets that are vectorially equivalent to A GIVEN UNBALANCED SET of three vectors. In the method of symmetrical components the problem is solved by resolving the given unbalanced set into three balanced sets called the positive sequence set, the negative sequence set, and the zero sequence set. Let us rst consider the positive sequence and negative sequence sets, which we ll de ne in connection with Figs. 262 and 263.
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