ELECTROMAGNETICTRANSMISSIONTHROUGHAPERTURES通过孔电磁波传输.docx

Hybrid Method Analysis of Electromagnetic Transmission through Apertures of Arbitrary Shape in a Thick Conducting ScreenABUNGU N ODERO1, DOMINIC B O KONDITI1ALFRED V OTIENO21Jomo Kenyatta University of Agriculture and Technology 2University of NairobiAbstract In this paper a hybrid numerical technique is presented, suitable for analyzing transmission properties of an arbitrarily shaped slot in a thick conducting plane. The slot is excited by an electromagnetic source of arbitrary orientation. The analysis of the problem is based on the generalized network formulation for aperture problems. The problem is solved using the method of moments(MOM) and the finite element method(FEM) in a hybrid format. The finite element method is applicable to inhomogeneously filled slots of arbitrary shape while the method of moments is used for solving the electromagnetic fields in unbounded regions of the slot. The cavity region has been subdivided into tetrahedral elements resulting in triangular elements on the surfaces of the apertures. Validation results for rectangular slots are presented. Close agreement between our data and published results is observed. Thereafter, new data has been generated for cross-shaped, H-shaped and circular apertures. Index Terms FiniteElement Method, Method of Moments, Perfect Electric Conductor (PEC) 11 IntroductionTHE problem of electromagnetic transmission through apertures has been extensively investigated 1-6. Many specific applications, such as apertures in a conducting screen, waveguide-fed apertures, cavity-fed apertures, waveguide-to-waveguide coupling, waveguide-to-cavity coupling, and cavity-to-cavity coupling have been investigated in the literature.In order to analyze electromagnetic scattering or radiation from apertures, it is desirable to develop efficient evaluation schemes. The past seventeen years have witnessed an increasing use of computational methods for the solution of electromagnetic problems. Traditional integral-equation methods continue to be used for many applications. However, recently, the greatest progress in computational electromagnetics has been in the development and application of hybrid techniques, such as FEM/MOM and (finite-difference time-domain)FDTM/MOM . The FDTM requires a fine subdivision of the computational domain for good resolution and is quite computationally intensive. The method of moments is an integral equation method which handles unbounded problems very effectively but is computationally intensive when complex inhomogeneities are involved. In contrast, inhomogeneities are easily handled by the finite element method, which requires less computer time and storage capacity because of its sparse and banded matrix. The matrix filling time is also negligible when simple basis functions are used 3. However, it lacks the ability to satisfy Sommerfeld radiation condition and requires an absorbing boundary. Since the investigation to be carried out encompasses unbounded problems as well as bounded problems with complex inhomogeneities, a procedure combining the finite element method and the method of moments would be effective.In this paper, we discuss the hybrid formulation of the aperture problem followed by a presentation and discussion of results for the various apertures. Rectangular, circular, cross-shaped and H-shaped apertures are some of the apertures studied.2 Problem Formulation Consider the 3dimensional (aperturecavityaperture) structure illustrated in Fig.1. Fig. 1. Problem geometry: (a) Top view (b) Cross-sectional viewupper aperturelower aperturecavity wall(a)yxoRegion A(b)Region CRegion BconductorconductorJi, Mizx oThe problem consists of a cavity in a thick conducting plane having an apertures at the top surface and bottom surfaces. The free space region above the plane ( z 0 ) is denoted as region A and that below the plane ( z -d ) as region B. Also, the volume occupied by the cavity ( -d z 0 )Region B ( z -d )Region CS1S2dxFig. 2: Equivalent currents introduced in the aperture planes.zz = 0M1(x,y)S1xJi, Mi-M1(x,y)M2(x,y)S1S2Fig. 3: Problem geometry decomposed into three regions; A, B, and C.-M2(x,y)S2- zx = -dThe total magnetic field in regions A and B can be expressed as a superposition of the short-circuited magnetic field (i.e., the incident magnetic field) plus the specular magnetic field , produced by the equivalent surface currents. The tangential components of the total magnetic field in the aperture planes are then expressed as (1)In the interior region C, the closed cavity is perturbed by the equivalent currents and , as illustrated by Fig. 3. The total tangential magnetic field that is produced on is expressed as the superposition of the fields produced by each current (2)with a similar expression on . To ensure the uniqueness of the solution, the tangential magnetic field must be continuous across the aperture planes. To this end, continuity of the total tangential magnetic fields across the aperture planes is enforced. Therefore, on (3)At this point, the equivalent currents are still unknown. Their approximate solution is derived using the method of moments since the surfaces S1 and S2 are assumed to be infinitesimally thin. Hence, the equivalent magnetic currents are expanded into a series of known basis functions weighted by unknown coefficients. (4) (5)The approximate equivalent currents are substituted into equation (3). Two sets of vector testing functions are also introduced: and in the and planes, respectively. Taking the inner product of equation (3) with and taking advantage of the linearity of the operators results in (6)where, a similar expression is derived on (7)where the subscript denotes the tangential fields. In terms of the aperture admittance matrix, the coupled equations (6) and (7) are expressed as (8)The advantage of using this function is that the admittance matrices of the interior and the exterior regions can be constructed independently. The aperture admittance matrices and are computed by solving the problem of the equivalent magnetic currents radiating into a half-space. The aperture admittance matrix is computed by solving the interior cavity problem. 3 Computing Exterior Admittance Matrices- MOMThe discretization of the MOM computational domain, i.e., region A and region B, is based on triangular patch modeling scheme as proposed by Rao et al 9. The formulation results in the following matrices which are solved following the procedure of Konditi and Sinha 6 (9) (10) (11)The following section describes the solution of the interior problem using the FEM.4 Interior Admittance Matrix - FEMWithin the cavity region, the magnetic fields must satisfy the vector Helmholtz equation (12)where is the total magnetic field. The cavity region is defined as a closed space confined by PEC walls, which will be referred to as the domain . A functional described by the inner product of the vector wave equation and the vectors testing function, , is expressed as (13)is defined. The solution of the boundary-value problem is then found variationally, by solving for the magnetic field at a stationary point of the functional via the first variation (14)The functional has a stationary point ( which occurs at an extremum ) and equations (9) and (10) can be used to derive a unique solution for the magnetic field. For an arbitrary the solution of the above variational expression cannot be found analytically and is derived via the FEM. To this end, the domain is discretized into a finite number of subdomains, or element domains, . It is assumed that each element domain has a finite volume and that the material profile within this volume is constant. Within each element domain the vector magnetic field is expressed as . If there are element domains within , the functional can be expressed in terms of the approximate magnetic fields as (15)Applying the vector identity (16)and the divergence theorem, (12 ) can be written as (17)Also (18)implying contribution from only regions of implied magnetic current sources, i.e, at the apertures only. = surface magnetic current source.So that (19)The approximate vector field in each element domain is represented as (20)with the spacial vector being the Whitney basis function defined by (21and is the barycentric function of node expressed as (22)where is the inwardly directed vectorial area of the tetrahedron face opposite to node , the element volume, and the position vector. Also, is the position vector of the barycenter of the tetrahedron defined as (23)in which is the position vector of node .Using equations (17) in (16) leads to: (24)Letting (25)And (26)equation (21 ) becomes (27)By enforcing the continuity of the magnetic field, a global number scheme can be introduced. The following symmetric and highly sparse matrix results (28)where : are coefficients weighting the edge elements lying in the aperture plane and . are coefficients weighting the remaining edge elements in .5 Results and DiscussionBased on the preceding formulation, a general computer program has been written, incorporating a code developed by Konditi 14, that can be used to analyze apertures of arbitrary shape in an infinite conducting screen of finite thickness. Although the formulation is valid for dissimilar regions on opposite sides of a thick conducting screen coupled via an aperture of arbitrary shape, for the sake of computational simplicity, here only a problem of two similar half spaces has been considered. The formulation has first been validated by considering a problem of a rectangular aperture, Fig. 4, for which results by Auckland and Harrington 8 are available in the literature. The results are seen to be in good agreement. Thereafter, a number of aperture shapes such as circular, cross and H apertures embedded in a screen of varying thickness have been studied. In each case equivalent magnetic current distributions and transmission cross-sections have been calculated. Our results are in good agreement with those of Konditi and Sinha 6 for thin conducting screen (t = 0.001 approximately t = 0.0 as the case in 6).It is further observed that for very thin screens (0.001), the magnitudes and phases of the x-directed magnetic currents on the upper and apertures, M1 and M2, respectively, are equal. However, as the thickness of the screen is increased, the magnitudes of the magnetic current on M2 becomes progressively smaller than that on M1. On the other hand, the current phase on M2 becomes progressively larger than that on M1. As the screen thickness increases the transmission cross-section and the magnitudes of M1 and M2 decrease, Fig 7. The same trend is observed for circular, cross and H-shaped slots, Figs. 8-9. However, the profiles of the magnitudes and phases of the equivalent magnetic currents are different , probably due to resonance phenomenon, as is evident in the above figures. 6 ConclusionIn this paper, a numerical study based on a hybrid FEM/MOM formulation for aperture problems has been presented for apertures of arbitrary shape in a thick conducting screen of infinite extent separating two half spaces. The cavity domain has been discretized into tetrahedral elements which on the apertures resulted in triangular patch modeling with appropriately defined basis functions.Validation of computer code has been achieved by using data available in the literature for rectangular apertures and for circular, cross-shaped, and H-shaped apertures in thin conducting screens Thereafter new data has been generated for circular, cross-shaped, and H-shaped apertures in thick conducting screens which to our knowledge do not exist in the open literature. Some of the parameters computed include transmission cross-section and equivalent magnetic currents. In all the problems investigated, transmission cross-section and magnetic currents magnitudes decrease with increase in thickness of the conducting screen. References:1 R.F. Harrington, J.R. Mautz, and D.T. Auckland: Electromagnetic coupling through apertures, Dept. Elec. Comput. Eng., Syracuse Univ., Syracuse NY, Rep. TR-81-4, Aug. 1981.2 C. M. Butler. Y. Rahmat-Samii. and R. Mittra,. Electromagnetic penetration through apertures in conducting surfaces, IEEE Trans. Electromagn. . vol. EMC-20. pp. 82-93, Feb. 1978.3 Xingchao Yuan, Three-dimensional electromagnetic scattering from inhomogeneous objects by the hybrid moment and the finite element method, IEEE Trans Microwave Theory Tech., Vol. MTT-38, pp. 1053-1058, 1990.4 Harrington, R.F. and Mautz, J.R.: A generalized network formulation for aperture problems, IEEE Trans. Antennas Propagat., Vol. AP-24, pp. 870-873, 19765 S. K. Jeng, Scattering from a cavity-backed slit in a ground plane TE case, IEEE Tram. Antennas Propagat., vol. 38, pp. 1523-1529, 9, Oct. 1990.6 Konditi, D.B.O. and Sinha, S. N.: Electromagnetic Transmission through Apertures of Arbitrary Shape in a Conducting Screen, IETE Technical Review, vol. 18, Nos. 2&3, March-June 2001, pp. 177-190.7 Mautz, J.R. and Harington, R.F.: Electromagnetic transmission through a rectangular aperture in a perfectly conducting plane, Tech. Rept. TR-76-1, Electrical and Computer Engg. Deptt., Syracuse Univ., New York, 13210, 1976.8 David T. Auckland and Harington, R.