IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 8, AUGUST 2013 2783 TEM Wave Scattering by a Step Discontinuity on the Outer Wall of a Coaxial Waveguide Sinan Aksimsek, Student Member, IEEE, Gökhan Çinar, Member, IEEE, Börje Nilsson, and Sven Nordebo, Senior Member, IEEE Abstract—In this paper, the propagation of TEM waves along a coaxial waveguide with a step discontinuity on its outer wall is investigated rigorously by applying the direct Fourier trans- form and reducing the problem into the solution of a modified Wiener–Hopf equation. The solution for the field terms are de- termined in terms of an infinite number of unknown coefficients, which satisfy an infinite set of linear algebraic equations. These equations are solved numerically and the effect of area ratio is presented graphically at the end of the analysis. The same problem is also analyzed by applying the mode-matching technique and the results of the two approaches are compared. It is observed numerically that the Wiener–Hopf technique provides a better convergence than the mode-matching technique. Index Terms—Coaxial, discontinuities, electromagnetic (EM) wave propagation and scattering, mode matching, Wiener–Hopf technique. I. INTRODUCTION E LECTROMAGNETIC wave propagation in waveguides has been an interesting topic and subject to various en- gineering problems, such as microwave and transmission line measurement techniques, filters, connectors, and matching de- vices. A typical example is with the low-frequency electromag- netic modeling of a power cable measurement setup [1], where there are many scattering mechanisms, such as different inner and outer radii of two connected coaxial cables (step disconti- nuity on the outer and inner walls), different dielectric media, etc. Among these, scattering by step discontinuities in coaxial waveguides has been drawing interest since many decades. It was first studied by Whinnery et al. in 1944 where they obtained an equivalent circuit by placing an admittance at the plane of discontinuity in the case of TM waves [2]. In 1998, Mongiardo et al. analyzed the same problem with generalized network for- mulation by the use of Green’s function [3]. Yu et al. applied a nonuniform finite-difference time-domain (FDTD) technique to Manuscript received December 06, 2012; revised May 02, 2013 and June 13, 2013; accepted June 17, 2013. Date of publication July 17, 2013; date of cur- rent version August 02, 2013. This work was supported in part by the Swedish Research Council. S. Aksimsek and G. Çinar are with the Electronics Engineering De- partment, Gebze Institute of Technology, Kocaeli 41400, Turkey (e-mail: h.sinanaksimsek@gmail.com; gcinar@gmail.com). B. Nilsson and S. Nordebo are with the Faculty of Technology, Lin- næus University, SE-351 95 Växjö, Sweden (e-mail: borje.nilsson@lnu.se; sven.nordebo@lnu.se). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2013.2271755 study cascaded circularly symmetric discontinuities on waveg- uides in 2001 [4]. Finally, in 2006, Fallahi and Rashed–Mo- hassel considered the dyadic Green’s function approach using the principle of scattering superposition for the problem where there is a step discontinuity on the inner wall [5]. Waveguide discontinuities with axial symmetry were successfully studied in time-domain methods as well as in [6]–[8]. In this paper, TEM wave propagation along a coaxial wave- guide with a step discontinuity on the outer wall is analyzed rigorously by applying the Wiener–Hopf technique in order to understand the effect of the area expansion on the scattering phenomenon as part of the measurement setup in [1]. The Wiener–Hopf technique is applied to scattering problems by considering direct Fourier transform of the Helmholtz equation, boundary conditions, and continuity relations, and for each scattering problem, a unique type of Wiener–Hopf equation is determined (see, e.g., [9]–[13]). In particular, applying this technique to the problem studied in this paper yields a modified Wiener–Hopf equation of the second type involving a certain kernel function, which characterizes the nature of the step discontinuity on the outer wall of a coaxial waveguide. The solution of this Wiener–Hopf equation has an importance in some engineering applications such as microwave filters and power-line measurements mentioned in [1]. It is solved in terms of an infinite number of unknown coefficients, which satisfy an infinite set of linear algebraic equations. These linear algebraic equations are solved numerically and the effect of area ratio on the reflection and transmission coefficients is presented graphically at the end of the analysis. The same problem is then analyzed by applying the mode-matching technique, as described in [14]. This tech- nique has been widely used in previous studies involving step discontinuities at waveguides in general [15]–[17] and at coaxial waveguides [18], [19] when the discontinuities exist both on inner and outer walls. Following a similar procedure as described in [17], the scattering coefficients are determined, and this, together with the Wiener–Hopf analysis, allowed the authors to compare the results on both accuracy and the speed of convergence, which has not been done before in the literature to the best of the authors’ knowledge. It is found that the Wiener–Hopf technique has a faster convergence than the mode-matching technique, while the accuracy of the latter is as good, especially for low frequencies. As the analysis done in this paper is strongly motivated by engineering applications, in particular, power-line measurements, the comparison of these two techniques provides an understanding on the use of the Wiener–Hopf and mode-matching techniques in combination 0018-9480/$31.00 © 2013 IEEE