Camouflaging electrical -networks as graphs

Author:
Gabriel Kron

Journal:
Quart. Appl. Math. **20** (1962), 161-174

MSC:
Primary 94.30; Secondary 53.45

DOI:
https://doi.org/10.1090/qam/148347

MathSciNet review:
148347

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Abstract: Lately it has become fashionable to develop the theory of electrical networks by starting first with a detailed, scholarly exposition of the theory of graphs (containing both nodes and branches), as given in textbooks on algebraic topology. However, when the graph analysis finally gets down to the study of actual electrically excited networks, it will invariably be found that all traces of the concepts of nodes and their associated ``incidence-matrices'' have disappeared from the final and other equations of state that are to be solved. In their place only the branches and the associated ``connection-matrices'' have been put to use, all still wrapped up however in an ill-fitting graph terminology. Of course such legerdemain had to be resorted to, since graph theorists happened to pick the wrong topological model for an electrical network. A graph possesses enough structure to propagate an electromagnetic wave and not an electrical current.

**[1]**G. Kron,*Tensor analysis of networks*, John Wiley and Sons, New York, (1931)**[2]**G. Kron,*Diakoptics--The piecewise solution of large-scale systems*, a serial of 20 chapters in the June 7, 1957 to Feb. 13, 1959 issues of the Electrical Journal (London)**[3]**G. Kron,*Non-Riemannian dynamics of rotating electrical machinery*, J. of Math, and Physics,**13**, 103-194 (1934)**[4]**G. Kron,*Tensor analysis of multi-electrode tube circuits*Trans, AIEE,**55**, 1220-42 (1936)**[5]**Gabriel Kron,*Equivalent circuit of the field equations of Maxwell. I*, Proc. I. R. E.**32**(1944), 289–299. MR**0010695****[6]**O. Veblen,*Analysis situs*, American Math. Soc. New York, 1931**[7]**W. V. D. Hodge,*The theory and applications of harmonic integrals*, Cambridge, at the University Press, 1952. 2d ed. MR**0051571****[8]**G. Kron,*A generalization of the calculus of finite differences to nonuniformly spaced variables*, Trans. AIEE**1**, Communications and Electronics**77**, 539-44 (1958)**[9]**G. Kron,*Basic concepts of multidimensional space filters*, Trans. AIEE**1**, Communications and Electronics**78**, 554-61 (1959)**[10]**G. Kron,*Self-organizing dynamo-type automata*, Matrix and Tensor Quarterly**11**, 2 (1960)**[11]**G. Kron,*Power-system type self-organizing automata*, to appear in R.A.A.G. Memoirs*III*of the Basic Problems in Engineering and Physical Sciences by Means of Geometry (Japan)**[12]**Gabriel Kron,*Tensors for circuits*, 2nd ed. With an introduction by Banesh Hoffmann, Dover Publications, Inc., New York, 1959. MR**0111460****[13]**Gabriel Kron,*Multidimensional curve-fitting with self-organizing automata*, J. Math. Anal. Appl.**5**(1962), 46–69. MR**0150978**, https://doi.org/10.1016/0022-247X(62)90005-7**[14]**G. Kron,*The misapplication of graph theory to electrical networks*, Trans. AIEE 1, Communications and Electronics,**81**(1962)

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DOI:
https://doi.org/10.1090/qam/148347

Article copyright:
© Copyright 1962
American Mathematical Society