tion of perfectly conducting and impedance parallel plate waveguides in the case where the surface impedances of the upper and lower semi infinite plates are ...
INFLUENCE OF THE JUNCTION OF PERFECTLY CONDUCTING AND IMPEDANCE PARALLEL PLATE SEMI-INFINITE WAVEGUIDES TO THE DOMINANT MODE PROPAGATION ˙ Alinur Büyükaksoy and Ismail Hakkı Tayyar Gebze Institute of Technology, PO. Box:141, 41400, Gebze Kocaeli, TURKIYE Gökhan Uzgören Istanbul Kültür University Ataköy, Istanbul, TURKIYE
ABSTRACT The Wiener-Hopf technique is used to compute the reflection and transmission coefficients related to the junction of perfectly conducting and impedance parallel plate waveguides in the case where the surface impedances of the upper and lower semi infinite plates are different from each other. Key words: Wiener-Hopf; waveguide; reflection. 1. INTRODUCTION In the present work a matrix Wiener-Hopf equation connected with a new canonical scattering problem is solved explicitly. We consider the scattering at the junction of perfectly conducting and impedance parallel plate waveguides in the case where the surface impedances of the upper and lower semi infinite plates are different from each other. This configuration may represents a junction of a smooth (perfectly conducting) and corrugated waveguides with a large number of corrugations (1). The special case in which the impedances of the guiding surfaces are equal has been mentioned by Noble (2) and solved first by Morse and Feshbach (3) and later on by Johansen (4). The representation of the solution to the boundary-value problem in terms of Fourier integrals leads to two simultaneous Wiener-Hopf equations which are uncoupled by using the analytical properties of the functions that occur (see for example,(5), (6) . The solution involves two sets of infinitely many expansion coefficients satisfying two infinite systems of linear algebraic equations. The effects of the spacing between the waveguide plates and the surface impedances on the scattering phenomenon are shown graphically.
_____________________________________________________ Proc. ‘EuCAP 2006’, Nice, France 6–10 November 2006 (ESA SP-626, October 2006)
A time factor e−iωt with ω being the angular frequency is assumed and suppressed throughout the paper. 2. ANALYSIS The geometry of the problem is depicted in Fig.-1. An incident TEM mode travelling in the positive x direction is confined between parallel planes located at y = 0 and y = b. The plates are perfectly conducting for x < 0 and are characterized by constant surface impedances for x > 0. For the sake of generality we assume that the surface impedances of the lower and upper plates are different from each other and denoted by Z1 = η1 Z0 and Z2 = η2 Z0 , respectively, with Z0 being the characteristic impedance of the free space. The aim of this work is to determine the effect of the relative surface impedances η1 and η2 on the reflection and transmission coefficients.
Figure 1. Geometry of the problem
Let the incident TEM mode propagating in the positive x direction be given by ui = exp(ikx)
(1)
where k is the propagation constant which is assumed to have a small imaginary part corresponding of slightly lossy medium. The lossless case can be obtained by letting Im k → 0 at the end of the analysis.
2 The total field uT (x, y) can be written at y ∈ (0, b) and x ∈ (−∞, ∞) as
functions of α in the half-planes Im(α) > Im(−k) and Im(α) < Im(k), respectively.
uT (x, y) = ui (x, y) + u(x, y)
By considering the Fourier transform of the boundary condition (5a) with (6) and (7) one can obtain
(2)
In (2) u(x, y) is the unknown function which will be determined through the boundary value problem consist of the Helmholtz equation ∆u(x, y) + k 2 u(x, y) = 0
(3)
·
F + (α, 0) B(α) = . K(α)
where the dot (·) specifies the derivative with respect to y. Substitution of (10) into (6) and taking the Fourier transform of the (5b) with (6) and (7) one obtains
It is appropriate to use the following Fourier integral representation Z u(x, y) = {A(α) cos [K(α)y] + B(α) sin [K(α)y]} e−iαx dα L
Here, K(α) denotes the square-root function p K(α) = k2 − α2
µ
1 ∂ η1 + ik ∂y
¶
(4b)
(5b)
0
