## A Simple Method to Find the Number of Branch Points of Propagation Constants of a Loss less Closed Guide without Constructing the Dispersion Curve

Progress in Electromagnetics Research Symposium (PIERS 2009 Moscow), Moscow, Russia, 12 - 21 August 2009, pp.1155-1156 • Publication Type: Conference Paper / Full Text
• Volume:
• City: Moscow
• Country: Russia
• Page Numbers: pp.1155-1156

#### Abstract

Making use of the fact that the characteristic equation that results from application of the moment method, is an algebraic equation for closed and lossless guides filled with inhomogeneous and/or gyrotropic media, we attempt to determine the degree of the discriminant of this equation, assuming we have no explicit expression for the coefficients of this equation in terms of frequency. This is achieved by successive numeric differentiation of the discriminant which we know is a polynomial, and the number of such differentiations to yield a constant value will be the degree of this polynomial. This information gives the number of branch points in the dispersion curve of a lossless closed waveguide, without having to draw the dispersion curve over the full frequency axis. Maxwell's partial differential equations can be transformed into a linear algebraic equation system whose coefficient matrix has squares of the propagation constant functions as eigenvalues. It is also discussed that the discriminant of the algebraic equation is a polynomial in the complex frequency p = sigma + j omega and that the zeroes of this polynomial correspond to singular points (branch points) of the square of the propagation constant function and this function attains multiple values at these zeroes. In our problem, we assume that the coefficients of the characteristic equation are not known analytically. Considering for instance a 20TE + 20TM truncation of the coefficient matrix which has entries that are functions of p, it is pointed out in the literature also that, it is cumbersome to compute these coefficients analytically as functions of p. In our approach, we have chosen to use the numeric evaluation of these coefficients. Hence the aim of this work is to determine the degree of the discriminant polynomial of the characteristic equation when the coefficients of this equation are not known analytically. This degree will give us the total number of multiple roots of the characteristic equation without having to draw the dispersion curve along the full frequency axis. As an example, for a structure consisting in a lossless closed circular guide loaded with a coaxial dielectric rod, we compute derivatives of the discriminant with respect to p in the complex frequency plane numerically with a view to reach a constant value after successive differentiations. The number of such differentiations will give the sought after degree. This procedure is effected through obtaining contour-plots of the various differentiations over a region of the complex frequency plane in the neighborhood of the origin. Then the same procedure is repeated yet for another region of the complex frequency plane. The set of results in both regions confirm each other by yielding the same degree for the discriminant.