Linear Graphs and Electrical Networks
| Date: 09 June 2011, 04:42
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This text has grown out of a graduate course entitled "Foundations of Electric Network Theory," organized at the University of Illinois by the second author in 1949. Such a course has since been taught by the two authors regularly at Illinois, Syracuse, and Michigan State Universities. Over the period of years, the material has naturally evolved into a shape quite different from the original. However, the basic philosophy of mathematical precision, coordinated with the objective of establishing the foundation of network theory, has remained unchanged throughout.
For many years, an intensive search was made (especially by the second author) for a way to determine precisely, rather than dimly suspect, the mathematical properties of the Kirchhoff equations of electrical network theory. In themselves, these equations seemed to be infinitely varied and to fit into no detectable pattern. Darkly at first, but with accelerated clarity as linear graph concepts were brought to bear, it became evident that here was the tool for the Kirchhoff-equations problem. In retrospect, it seems obvious that since the linear graph determines the coefficient matrices of these equations, it is in the linear graph that the properties of the equations are to be found.
Theory of graphs depends on the mathematical discipline of linear algebra, which is not very familiar to electrical engineers. We have kept this fact in mind and at least tried to explain briefly such concepts as field, ring, linear vector space, etc., that are used. However, we assume knowledge of matrix algebra and use it without explanation. Similarly, in the applications presented, Laplace transformation and theory of functions are assumed in the network theory, and Boolean algebra in the switching theory.
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