Higher Dimensional Electrical Circuits

Abstract

In this paper, we describe a physical problem, based on electromagnetic fields, whose topological constraints are two-dimensional versions of Kirchhoff’s laws, involving 2-simplicial complexes embedded in $${\mathfrak {R}}^3$$ rather than graphs. The topological constraints are on flux and mmf vectors, which we show satisfy a ‘generalized Tellegen’s Theorem,’ i.e., that they constitute complementary orthogonal vector spaces analogous to voltage and current spaces for ordinary graph-based electrical circuits. We show that the problem of solving this two-dimensional electrical circuit reduces to that of solving an ordinary resistive electrical circuit that can be regarded as dual to it. We present a linear time algorithm for constructing, the above mentioned, dual electrical network. This algorithm is based on the construction of a triangle adjacency graph of the original embedded 2-simplicial complex. This graph contains the information of the order in which we encounter the triangles incident at an edge, when we rotate say clockwise with respect to the orientation of the edge. The triangle adjacency graph is processed through a ‘sliding’ algorithm which simulates sliding on the surfaces of the triangles, moving from one triangle to another which shares an edge with it but which also is adjacent with respect to the embedding of the complex in $${\mathfrak {R}}^3$$. The connectedness information provided by this algorithm is used to construct, in linear time on the size of the 2-simplicial complex, its dual graph.

This is a preview of subscription content, access via your institution.

References

1. 1.

V. Belevitch, Classical Network Theory (Holden-Day, San Francisco, 1968)

2. 2.

F.H. Branin Jr., The network concept as a unifying principle in engineering and the physical sciences, in Problem Analysis in Science and Engineering, ed. by F.H. Branin, K. Huseyin (Academic Press, New York, 1977), pp. 41–111

3. 3.

F.H. Branin Jr., N.Y. Kingston, The Telation Between Network Theory, Vector Calculus, and Theoretical Physics, in Proceedings of the International Symposium on Operator Theory of Networks and Systems, vol 2 (1977) 97–102

4. 4.

V. Del Toro, Electrical Engineering Fundamentals (Prentice Hall, Upper Saddle River, 1972)

5. 5.

A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002)

6. 6.

G. Kron, Tensor Analysis of Networks (J. Wiley, New York, 1939)

7. 7.

J.R. Munkres, Elements of Algebraic Topology (Benjamin Cummings Publishing Company, Inc., Menlo Park, 1984)

8. 8.

H. Narayanan, Submodular Functions and Electrical Networks, Annals of Discrete Mathematics, vol 54 (North Holland, Amsterdam, 1997). Open 2nd edition http://www.ee.iitb.ac.in/~hn/

9. 9.

J. Oxley, Matroid Theory, vol. 3, Oxford Graduate Texts in Mathematics (Oxford University Press, Oxford, 2006)

10. 10.

P. Penfield Jr., R. Spence, S. Duinker, Tellegen’s Theorem and Electrical Networks (Cambridge Press, Cambridge, 1970)

11. 11.

S. Seshu, M.B. Reed, Linear Graphs and Electrical Networks (Addison-Wesley Publishing Company, Reading, 1961)

12. 12.

D. A. Spielman, Algorithms, graph theory, and linear equations in Laplacian matrices, in Proceedings of the International Congress of Mathematicians (Hyderabad 2010)

13. 13.

D.A. Spielman, N. Srivastava, Graph sparsification by effective resistances. SIAM J. Comput. 40, 1913–1926 (2011)

14. 14.

S. Theja, H. Narayanan, On the notion of generalized minor in topological network theory and matroids. Linear Algebra Appl 458, 1–46 (2014)

15. 15.

B.D.H. Tellegen, A general network theorem, with applications. Philips Res. Rep. 7, 259–269 (1952)

16. 16.

W.T. Tutte, Lectures on matroids. J. Res. Natl. Bur. Stand. B 69, 1–48 (1965)

17. 17.

A. J. van der Schaft, B.M. Maschke, Conservation laws and open systems on higher-dimensional networks, in Proc. 47th IEEE Conf. on Decision and Control, Cancun, Mexico (2008) pp. 799–804

18. 18.

A.J. van der Schaft, B.M. Maschke, Conservation laws and lumped system dynamics, in Model-Based Control: Bridging Rigorous Theory and Advanced Technology, ed. by P.M.J. Van den Hof, C. Scherer, P.S.C. Heuberger (Springer, Berlin, 2009), pp. 31–48

19. 19.

L. Weinberg, Matroids, generalized networks and electric network synthesis. J. Comb. Theory 23, 106–126 (1977)

20. 20.

D.J.A. Welsh, Matroid Theory (Academic Press, Cambridge, 1976)

21. 21.

H. Whitney, Geometric Integration Theory (Princeton University Press, Princeton, 1957)

Acknowledgements

The authors would like to acknowledge helpful discussions with Arvind Nair. They would also like to acknowledge the valuable suggestions of the anonymous reviewer for improving the readability of the paper and for pointing out important references. Hariharan Narayanan was partially supported by a Ramanujan fellowship.

Author information

Authors

Corresponding author

Correspondence to H. Narayanan.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

Narayanan, H., Narayanan, H. Higher Dimensional Electrical Circuits. Circuits Syst Signal Process 39, 1770–1796 (2020). https://doi.org/10.1007/s00034-019-01236-5

• Revised:

• Accepted:

• Published:

• Issue Date:

Keywords

• Kirchhoff’s laws
• Nonplanar graph
• Simplicial complex
• Matroid dual