Skip to main content
SpringerLink
Log in

Method of solving the generalized problem on eigenvalues of multidimensional circuits, I

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

This work is devoted to the development of a new method for solving the complete generalized problem of large-scale electrical circuit eigenvalues. We prove that the determinant of the conductance node matrix of an arbitrary k-loop LC-circuit is the Weinstein function for the loop impendance matrix of this circuit, and vice versa. A recurrent method of imposing constraints on the basic problem was defined, which permitted us to separate roots of characteristic polynomials in the recurrent process. Also a condition for conservativity of multiple eigenvalues was defined. A solution algorithm for the problem of defining a full range of eigenvalues of an oscillatory system with a finite number of degrees of freedom was suggested.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. Aronshein, “The Rayleigh–Ritz and A. Weinstein methods for approximation of eigenvalues,” Proc. Natl. Acad. Sci. USA, 474–594 (1943).

  2. S. H. Gould. Variational Methods for Eigenvalue Problems. An Introduction to the Weinstein Method of Intermediate Problems, Oxford Univ. Press, London (1966).

    MATH  Google Scholar 

  3. G. Kron, Tensor Analysis of Networks, John Wiley and Sons, New York, (1959).

    Google Scholar 

  4. A. Mylnikov, “Tensor-geometric methods for problems of circuit theory,” J. Math. Sci., 148, No. 2, 192–258 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. A. Mylnikov and A. I. Prangishvili, “Homological and cohomological invariants of electrical circuits,” Autom. Remote Control, No. 4 67–76 (2002).

    MathSciNet  Google Scholar 

  6. A. Weinstein, “Bounds for eigenvalues and the method of intermediate problems,” in: Partial Differential Equations in Continuum Mechanics, University of Wisconsin Press, Madison (1961), pp. 39–53.

    Google Scholar 

  7. G. V. Zeveke, P. A. Ionkin, A. B. Netushil, and S. V. Strakhov, Foundaions of Circuits Theory [in Russian], Energia, Moscow (1965).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Georgian Technical University, Tbilisi, Georgia

    A. A. Mylnikov

Authors
  1. A. A. Mylnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Mylnikov.

Additional information

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 74, Proceedings of the International Conference “Modern Algebra and Its Applications” (Batumi, 2010), Part 1, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mylnikov, A.A. Method of solving the generalized problem on eigenvalues of multidimensional circuits, I. J Math Sci 186, 785–794 (2012). https://doi.org/10.1007/s10958-012-1034-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-012-1034-6

Keywords

Search

Navigation