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Modal Analysis of Weakly Coupled Electromagnetic Systems Using Tridiagonal Toeplitz Matrices

Jose Brandao Faria(1*)

(1) Department of Electrical Engineering , Instituto de Telecomunicações Instituto Superior Técnico University of Lisbon, Portugal
(*) Corresponding author


DOI: https://doi.org/10.15866/iree.v10i6.8012

Abstract


In the analysis of weakly coupled electromagnetic systems, consisting of several identical equidistant components, the coupling matrix is often approximated by tridiagonal Toeplitz matrices, whose eigenvalues and eigenvectors can be determined in closed-form. The theory of tridiagonal Toeplitz matrices is not new, it can be found in several linear algebra articles and math books, but its presentation is usually very dense and far from appealing to the electrical engineer. This paper has two goals. The first is of tutorial nature —we present a concise review of the modal analysis of tridiagonal matrices. The second goal is to call the reader’s attention to the inherent fragilities of the tridiagonal matrix approximation —we discuss and work out application examples, dealing with multiconductor transmission line systems, where the approximation is shown to clearly fail.
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Keywords


Matrix Methods; Multiconductor Transmission Lines; Multimodal Propagation; Weakly Coupled Systems

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References


P. Ashwin, J. W. Swift, The dynamics of n weakly coupled identical oscillators, J. Nonlinear Sci., vol. 2, 1992, pp. 69-108.
http://dx.doi.org/10.1007/bf02429852

S. Y. Elnaggar, R. J. Trevo, S. M. Mattar, Coupled mode theory applied to resonators in the presence of conductors, IEEE Trans. Microw. Theory Tech., vol. 63, 2015, pp. 2124-2132.
http://dx.doi.org/10.1109/tmtt.2015.2432766

P. B. Catrysse, V. Liu, S. Fan, Complete power concentration into a single waveguide in large-scale waveguide array lenses, Sci. Report, vol. 4, 2014, p. 6635.
http://dx.doi.org/10.1038/srep06635

A. Grayzel, A useful identity for the analysis of a class of coupled transmission-line structures, IEEE Trans. Microw. Theory Tech., vol. 22, 1974, pp. 904-907.
http://dx.doi.org/10.1109/tmtt.1974.1128378

F. Romeo, M. Santomauro, Time-domain simulation of n coupled transmission lines, IEEE Trans. Microw. Theory Tech., vol. 35, 1987, pp. 131-136.
http://dx.doi.org/10.1109/tmtt.1987.1133615

J. A. Faria, Normal mode synthesis of n-coupled coplanar stripline structures, Microw. Opt. Tech. Lett., vol. 4, 1991, pp. 131-134.
http://dx.doi.org/10.1002/mop.4650040312

S. Noschese, L. Pasquini, L. Reichel, Tridiagonal Toeplitz matrices: properties and novel applications, Num. Linear Algebra Appl., vol. 20, 2013, pp. 302-326.
http://dx.doi.org/10.1002/nla.1811

F. R. Gantmacher, The Theory of Matrices - Vol. 1 (Chelsea, 1977).
http://dx.doi.org/10.1126/science.131.3408.1216-a

J. A. Faria, Multiconductor Transmission-Line Structures (Wiley, 1993).
http://dx.doi.org/10.1109/tpwrs.2004.831692

J. A. Faria, Multimodal propagation in multiconductor transmission lines, J. Electromag. Waves Appl., vol. 28, 2014, pp. 1677-1702.
http://dx.doi.org/10.1080/09205071.2014.960536

A. Bottcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices (Springer, 1999).
http://dx.doi.org/10.1007/978-1-4612-1426-7

W. C. Yueh, Eigenvalues of several tridiagonal matrices, Appl. Mathematics E-Notes, vol. 5, 2005, pp. 66-74.

S. Lipschutz, M. Spiegel, J. Liu, Schaum’s Outline of Mathematical Handbook of Formulas and Tables - 4th Edition (McGraw-Hill, 2012).

C. R. Paul, Analysis of Multiconductor Transmission Lines (Wiley, 1994).
http://dx.doi.org/10.1109/9780470547212

J. A. Faria, Electromagnetic Foundations of Electrical Engineering (Wiley, 2008).
http://dx.doi.org/10.1002/9780470697498

R. F. Harrington, Field Computation by Moment Methods (Wiley-IEEE Press, 1993).
http://dx.doi.org/10.1109/9780470544631

M. V. Schneider, Microstrip lines for microwave integrated circuits, Bell Systems Tech. J., vol. 48, 1969, pp.1421-1444.
http://dx.doi.org/10.1002/j.1538-7305.1969.tb04274.x

S. R. Nelatury, M. N. Sadiku, V. K. Devabhaktuni, CAD models for estimating the capacitance of a microstrip interconnect: comparison and improvisation. PIERS Proceedings, 2007, pp. 18-23, Prague, Czech Republic.


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