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CHAPTER 10 Magnetic Coupling. Transformers
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It should be noted that the three-phase problems we ve solved here, using the method of symmetrical components, could also have been solved by the ordinary method of loop currents. It should, however, also be noted that other types of three-phase problems exist in which application of the method of symmetrical components provides the only practical way of obtaining exact and rigorous solutions.
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Matrix Algebra. Two-Port Networks
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Here we take up the subject of matrix algebra, which has important applications in the study of electric networks. It should be noted, however, that matrix algebra nds wide use in many elds of endeavor, from economics to computer graphics, for example. Your time will therefore be well spent in mastering this interesting and useful subject.
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Introduction to Matrix Algebra
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Let us begin by de ning that a matrix ( MAY triks ) is a rectangular array of elements, the elements being arranged in a de nite order in horizontal rows and vertical columns. The location of any element in a matrix is always speci ed by giving rst the ROW and then the COLUMN that the element is located in. Thus, using subscripts, a notation such as a23 denotes the element at the intersection of the second row and the third column (a symbol such as a23 can be read as a, two, three ). When it is deemed necessary, the row and column subscripts are separated by a comma (for example, a16;11 ). A matrix is usually identi ed as such by enclosure in square brackets. Figure 268 is an example of a 3 by 4 matrix, meaning it has 3 rows and 4 columns.
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Fig. 268
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Matrix Algebra. Networks
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In the above, note that the subscripts used with each element denote rst the row and then the column in which the element appears. This convention, of giving rst the row number and then the column number, is always used. A matrix having m rows and n columns is said to be an m by n matrix. An m by n matrix therefore consists of mn elements. Thus, the 3 by 4 matrix above consists of a total of 12 elements. The presence of an m by n matrix is often denoted by the symbol m n , where the cross is read as by. Figure 268 represents a 3 4 matrix. A matrix consisting of only a single row of elements is called a row matrix, while a matrix consisting of only a single column of elements is called a column matrix. Thus Fig. 269 is an example of a 1 by 3 row matrix and Fig. 270 is a 3 by 1 column matrix.
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Fig. 269
Fig. 270
A matrix having the same number of rows as columns, that is, an m by m matrix, is called a SQUARE matrix. The general example of a 3 3 square matrix is shown below in Fig. 271.
Fig. 271
You will recall, from Chap. 3, that a determinant is also a square array of elements. Let us emphasize, however, that a square matrix and a determinant are two entirely di erent things. A matrix, including a square matrix, is simply an ordered array of elements; it is a mathematical symbol and, taken as a whole entity, it has no numerical value. A determinant, on the other hand, represents a single number or value, which can be found by expanding the determinant according to the rules laid down in Chap. 3. It is true that, in certain circumstances, a determinant is formed from a square matrix, but this is a result of a special operation, as we ll learn later on. The main diagonal of a square matrix consists of all the elements lying on the diagonal line drawn from the upper left-hand element down to the lower right-hand element. Thus the main diagonal of the 3 by 3 square matrix above consists of the elements a11 , a22 , and a33 . If the elements in the main diagonal of a square matrix are all ones, (all 1 s), and all the other elements are zeros, the square matrix is then called a unit or identity matrix. Figure 272 is a unit matrix of order 5. The unit matrix, which may of course be of any order n (n 5 in the above), is usually denoted by the symbol I, and will be useful in some of our later work.
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