Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Camouflaging electrical $ 1$-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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $ e = zi$ 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.

References [Enhancements On Off] (What's this?)

  • [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)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 94.30, 53.45

Retrieve articles in all journals with MSC: 94.30, 53.45

Additional Information

DOI: https://doi.org/10.1090/qam/148347
Article copyright: © Copyright 1962 American Mathematical Society

American Mathematical Society