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Electromagnetic Propagation into Parallel-Plate Waveguide in the Presence of a Skew Metallic Surface Constantinos A. Valagiannopoulos
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Department of Radio Science and Engineering, School of Electrical Engineering, Aalto University, Espoo, Finland Available online: 31 Oct 2011
To cite this article: Constantinos A. Valagiannopoulos (2011): Electromagnetic Propagation into Parallel-Plate Waveguide in the Presence of a Skew Metallic Surface, Electromagnetics, 31:8, 593-605 To link to this article: http://dx.doi.org/10.1080/02726343.2011.621111
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Electromagnetics, 31:593–605, 2011 Copyright © Taylor & Francis Group, LLC ISSN: 0272-6343 print/1532-527X online DOI: 10.1080/02726343.2011.621111
Electromagnetic Propagation into Parallel-Plate Waveguide in the Presence of a Skew Metallic Surface CONSTANTINOS A. VALAGIANNOPOULOS 1 1
Department of Radio Science and Engineering, School of Electrical Engineering, Aalto University, Espoo, Finland Abstract A metallic edge with arbitrary length and inclination is embedded into a coplanar waveguide forming a simple two-port network. The corresponding boundary value problem is rigorously solved by segmenting this perfectly conducting strip into a large number of consecutive pins; the procedure yields robust expressions for the scattering parameters of the device. Interesting properties characterizing its operation are observed through the corresponding diagrams, such as the correlation between different configurations. Additional conclusions have also been drawn from the discussion of the related graphs. Keywords coplanar waveguide, metallic edge, scattering parameters, two-port network
1. Introduction Scatterers, inhomogeneities, and discontinuities into waveguides and tubes have been extensively examined in numerous theoretical analyses and experimental investigations due to their substantial practical value and direct applicability. In the standard textbook of Marcuvitz (1985), a very large set of waveguide configurations incorporating additional equipment, such as apertures, posts, abrupt terminations, sidewall discontinuities, gratings, and junctions, was rigorously or empirically treated. In Dib and Katehi (1992), a general method was also presented, which can be implemented easily to characterize theoretically certain shielded coplanar waveguide discontinuities. A metallic circular cylinder in the vicinity of a step discontinuity, at a parallel-plate waveguide, was rigorously analyzed in Valagiannopoulos and Uzunoglu (2007), where the related equivalent circuit model was provided. In addition, certain boundary-value scattering problems of conducting objects into a propagating slab was resolved (Chou & Jeng, 1998), while the Green function into a tapered funnel was effectively evaluated in Valagiannopoulos and Uzunoglu (2009b). Configurations containing sharp edges are not only of theoretical interest due to their receptivity to analytical modeling, they are also useful from a practical point of view because devices utilize wedge components to acquire desirable features. A recent study (Perkalskis & Gluck, 2007) demonstrated two experiments on edge diffraction Received 3 May 2011; accepted 4 August 2011. Address correspondence to Constantinos A. Valagiannopoulos, Department of Radio Science and Engineering, School of Electrical Engineering, Aalto University, Otakaari 5A St., FIN-02150, Espoo, Finland. E-mail:
[email protected]
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using commonly available microwave equipment in the physics department, while the complete theory of the finite-difference time-doman (FDTD) equations for the sharp diagonal metal edges in three dimensions has been also presented (Lu & Chen, 2005). In Valagiannopoulos (2009), the line source scattering by a metallic wedge with a lossy cylindrical cap was considered, while a comparison between several numerical methods treating a knife-edge metallic obstacle was performed in Simons et al. (1999). Finally, sharp edges are utilized in modern applications, such as RFID in Xiaohua and Hanbin (2011), where a prediction radio-wave propagation loss model was additionally established. The present work combines the two aforementioned concepts (waveguide inclusions, metallic edges) to treat a structural configuration that has not yet been examined using the same set of mathematical techniques. In particular, a perfectly conducting (PEC) edge internal to a coplanar waveguide is considered, which affects the wave propagation. To manipulate the problem, it is assumed that the edge is comprised by a large number of tiny metallic cylinders. Accordingly, the scattering theorem has been implemented and the necessary boundary condition imposed from which the impressed current on the edge is designated. Compact forms for the scattering parameters of the formulated two-port network are derived, whose variations with respect to the parameters of the tube and the edge are presented and discussed.
2. Mathematical Formulations The physical configuration of the analyzed problem is shown in Figure 1, where the utilized cylindrical coordinate system (; '; y) is also defined. An empty parallel-plate waveguide with PEC walls of height h is investigated in the presence of an inclined metallic strip with length r , positioned along the radius ' D 0 and nailed at the lower planar boundary. The structure is excited by the dominant mode of the tube, while a time dependence of the form e Cj!t is suppressed throughout the analysis. The incident electric wave is y-polarized and possesses the form sin ' Ei nc .; '/ D sin e g1 cos ' ; (1) h
Figure 1. Physical configuration of the analyzed device. A skew metallic strip affects the propagation through a parallel-plate waveguide. (color figure available online)
Metallic Strip into Parallel-Plate Waveguide
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q where gn D .n= h/2 k02 is evaluated with ReŒgn , ImŒgn 0. The same polarization property is owned by the scattering components of the electric field. The notation k0 D p ! "0 0 is used for the free space wavenumber. Green’s function of the considered structure has been computed rigorously in Valagiannopoulos and Uzunoglu (2009a) and is comprised of two terms. The first, which is singular and corresponds to the radiation of an infinite dipole source at .; '/ D .P; ˆ/ in the presence of the lower metallic plane, is given by Gsingular.; '; P; ˆ/ D
j h .2/ p 2 H0 k0 C P 2 2P cos.' ˆ/ 4 p i .2/ H0 k0 2 C P 2 2P cos.' C ˆ/ ;
(2)
.2/
where the symbol H0 denotes the Hankel function of zeroth order and second kind. The other term expresses the effect of the upper metallic boundary and concerns the infinite number of images developed along the vertical axis. The most suitable expression for such a term, when the observation point .; '/ is relatively close to the source point .P; ˆ/, is found out equal to the integral Gimage.; '; P; ˆ/ D
4
Z
C1
1
0
2
sinhŒk0 P sin ˆ. C j /
e
2k0 h. 2 Cj /
e
2k0 h. 2 Cj /
cosŒk0 . cos '
sinhŒk0 sin '. 2 C j /
P cos ˆ/ p 2 C 2j
p
2 C 2j
d :
(3)
That is because the exponential term makes the integrand vanish rapidly for increasing , even when .; '/ D .P; ˆ/. It is well known that a rudimentary analytic tool for solving wave diffraction problems is the scattering integral (Valagiannopoulos, 2011). According to this important formula, the scattering field is given by Escat D j!0
Z
.S/
K.C /ŒGsingular C GimagedC;
(4)
where the current induced on the metallic edge (S ) is denoted by K.C /, and C is the integration variable along the surface. If one assumes that the strip is separated in a large r number U D Π2a of thin wires (with small radius a, as demonstrated in Figure 1) posed one next to the other along the line f' D ; 0 < < r g, an alternative expression for the scattering field is obtained: Escat .; '/ D
j!0
U X uD1
u
Z
.Su /
ŒGsingular.; '; PC ; ˆC / C Gimage.; '; PC ; ˆC /dC: (5)
The surface of the uth cylinder is a circle, denoted by Su , centralized at .; '/ D .u ; / D .2au; / with small radius a, flowed by a constant y polarized current u . By imposing the boundary condition for vanishing electric field on the surfaces of the thin cylinders,
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which almost coincide with their centers, one obtains the following U U linear system: U X uD1
u
Z
.Su /
ŒGsingular.v ; ; PC ; ˆC / C Gimage.v ; ; PC ; ˆC /dC D
1 Ei nc .v ; / j!0
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(6) for v D 1; : : : ; U . Due to the limited extent of each rod’s cross-section, the image component of Green’s function exhibits negligible variation, namely Z
.Su /
Gimage.v ; ; PC ; ˆC /dC D 2 aGimage.v ; ; u; /:
(7)
As far as the integral of the singular component is concerned, it is explicitly evaluated with help from standard techniques (Valagiannopoulos, 2008): Z
.Su /
Gsingular.v ; ; PC ; ˆC /dC D
8 .2/ ˆ ˆ ˆ
