A cylindrical dipole antenna is numerically analyzed by the moment method. The surface of the antenna is approximated by triangular patches and the electric field integral equation is used for direct calculation of the surface current distribution. Therefore, the cylinder antenna can be treated in open or closed boundary form. The current expansion functions and the testing functions of the electric field boundary condition are of the triangular type. The surface integrals are numerically solved by a 33-point Gaussian quadrature approximation. The current distribution on a flat plate illuminated by a plane wave and the input admittance of a hollow cylindrical dipole as the near field quantities has been investigated. The convergence of the input admittance against the number of the triangular patches is presented, and the admittance solution is compared with the thin-wire approximation and theoretical results. Finally the CPU time and memory storage size for different numbers of patches are presented. Rapid admittance convergence and few required unknowns per square wavelength are the advantages of surface patch modeling </p

Analoui, M.

Nakata, Takayoshi

Tsuboi, H.

M. Analoui

H. Tsuboi

T. Nakata

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Numerical analysis of antenna by a surface patch modeling

Okayama University Scientific Achievement Repository

PhysicsElectricity & Magnetism fieldsOkayama University Year 1990Numerical analysis of antenna by asurface patch modelingM. Analoui H. TsuboiOkayama University Okayama UniversityTakayoshi NakataOkayama UniversityThis paper is posted at eScholarship@OUDIR : Okayama University Digital InformationRepository.http://escholarship.lib.okayama-u.ac.jp/electricity and magnetism/164IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 2, MARCH 1990 NUMERICAL ANALYSIS OF ANTENNA BY A SURFACE PATCH MODELING M. Analoui*, H. Tsuboi** and T. Nakata** *The Graduate School of Natural Science and Technology, **Department of Electrical and Electronic Engineering, Okayama University, Tsushima, Okayama 700, Japan Abstract -  In this paper, the cylindrical dipole antenna is numerically analyzed by the moment method. Surface of the antenna is approximated by triangular patches and the electric field integral equation is  used for direct calculation of the surface current distribution. Therefore we can treat the cylinder antenna in open or closed boundary form. The current expansion functions and the testing functions of the electric field boundary condition are triangular type. The surface integrals are numerically solved by a 33-point Gaussian quadrature approximation. The current distribution on a flat plate illuminated by a plane wave and the input admittance of a hollow cylindrical dipole as the near field quantities have been investigated. The convergence of the input admittance against the number of the triangular patches is presented and also the admittance solution is compared with the thin-wire approximation and theoretical results. Finally the CPU time and memory storage size for different number of patches are presented. Rapid admittance convergence and few required unknowns per square wavelength are the advantages of the surface patch modeling. The surface current distribution is the most important parameter in the antenna analysis. All the near or far field quantities may be derived from the current distribution. The input admittance or impedance reflects the current at feeding region. The cylindrical antenna may be treated by a thin- wire approximation which is  a solution for axial component of the current distribution. The surface patch and wire-grid techniques investigate all components of the current on the antenna surface. The wire-grid solution is more simple than the surface patch technique but its validity on the near field solution is still doubtful [l], [2]. Advantages of the surface patch approximation from view point of the antenna analysis are a) direct and simple relation between vectors of calculated current matrix and the current distribution on the antenna surface, b) no restriction for treating very thin surfaces and c) very few unknowns per wavelength square. Recently several surface patch approximation techniques have been developed for modeling solid surfaces as an accurate and direct solution of the surface current and charge distributions. N.N. Wang et al. used piecewise-sinusoidal reaction technique and rectangular surface patches for treating scattering and radiation problem of arbitrary shaped conducting bodies [3]. J.J. Wang introduced a triangular surface patch technique to calculate the scattering characteristics of a three- dimensional arbitrary-shaped closed conducting body 141 and Rao et al. used the electric field integral equation and triangular patches to develop a simple and efficient numerical procedure for treating problems of scattering from conducting bodies of arbitrary shape [5]. In this work we have embarked upon to use triangular patches [5] for the antenna analysis. We have treated a plane wave illuminated flat square plate as a comparison with the result of [5]. Then the input admittance of a hollow cylinder dipole has been investigated for different number of the patches. The 905 computed results have been compared with analytical and thin-wire approximation data. The required CPU time and memory storage size also have been presented. In the next section the surface patch technique for the moment solution of the electric field integral equation is briefly presented. The numerical results and conclusion are presented in the next sections consecutively. THEORY Consider an arbitrary shaped perfectly conducting body (closed or open) illuminated by an incident electric field Ei. A frequency domain expression for scattered field is given by Es = -joA -V @ (1)  where the vector potential A and the scalar potential @ are defined as in which J is the surface current distribution, k is the wave number given by k=21rA and R= I r-r' I is the distance between a field point r and a source point r'. o is the frequency in radians per second and E and are the permittivity and permeability of the medium respectively. The electric field boundary condition results in an integrodifferential equation with unknown J as - Ei, = ( -joA - V@ ) (4) Referring to a triangulated conducting scatterer, every two triangles with a common side are supposed as a source-pair T n f  , or field-pair Tm+ as shown in Fig. 1. The electric current flows along radial direction p n f in triangles T,f .  In this literature the subscripts m and n mean a field and a source triangle-pair respectively and their superscript plus or minus sign mean a positive assumed current direction from plus triangle to minus triangle. Any point in the triangles can be defined either with respect to global origin 0 or with respect to the triangle vertices Om* and On&. Here a surface current expansion function associated with nth pair is given by 0018-9464/90/0300-0905$01.00 1990 IEEE 906 and the divergence of the expansion function in the polar co-ordinates system may be given by la E a +  ; r on Tn+ V-F,(r) = -1, /Sa- ; r on Ta- 1 .  ; otherwise where 1, is the length of the common side of n t  h triangle-pair and Snf are the areas of triangles. Let N represents the total number of pairs. Then n=l (7) where I, is constant and unknown. In fact each In can be interpreted as the normal components of the current density to pass the common side of the nth pair. t- lm Fig. 1 Geometry representation of nth source-pair and mth field-pair. In order to find the current coefficients, the electric field integral equation is tested with respect to testing functions F m . Testing functions Fm and expansion functions Fa are the same. Equation (4) is tested based on the following symmetric product Then, we obtain <F, Fm> = j o <  A, Fm> + < V@, Fm> In Eq. (9) the products may be simplified by evaluating the vectors at the centroids of respect triangles (rc,* ) a s  follows A -  Consequently Eq.(9) becomes which Eq. (13) is the equation enforced at each triangle-pair, m=1, 2. ..., N. The expanded surface current Eq. (7 )  is now substituted into Eq.(13) to make an NxN system of linear equations which is written in matrix form a s  where elements of [Z] and [VI are given by -1 e-jkRmnf 0 m . P  = ~ V. Fa@') ds' (18) 4xjoe Rmn* r' on Taf in which E& = Ei(rc,f) (19) Rmn* = kcm- r',l (20) As Eqs. (17) and (18) show the vector potential A and the scalar potential @ should be calculated at the centroid of triangles where for self impedances there are the ( 9 )  907 singularities. It seems for numerical calculation of the integrals, at first, the singular points should be removed. Since only the centroid points are singular, high degree Gaussian quadrature formulas which have no sampling point at the centroid are capable to accomplish the singular integrals. In this work we used a 33-sampling point Gaussian quadrature formula [a] and no anomaly was observed. Every triangle may be one part of at most three source-pairs and its field should be calculated at all triangles. One may compute the vector and scalar potentials of every individual triangle patch on the centroid of all the patches and then according to the membership of the patches in the source and field pairs the elements of Zmn are computed. Fig. 2 shows the surface current distribution on the square flat plate illuminated by a normally incident plane wave. For comparison the result of [SI is also presented. To demonstrate the antenna numerical analysis applicability of the procedure numerical results are presented for the input admittance of a hollow cylindrical dipole as a thin and open surface. Fig. 3 shows the input - Reference [SI 0.0 0.2 0.4 0.6 0.8 1 .o X / h , Y l h  (b) Fig. 2 (a) A triangulated l k  square flat plate illuminated by a plane wave, (b) Dominant component of current distribution on the plate. n a v 0 10 - 5 -  0 -  0.00702k .I x %patches A 4Opatches 0 ,56 patches 0 72patches 15 0   x 00 0 0 1 2 3 4 5 6 7  kh -5 I I I I I 1 I I 0 1 2 3 4 5 6 7  kh (C) Fig. 3 Input admittance of a center feed rectangular prism for different number of patches: (a) triangulated rectangular prism, (b) input conductance, (c) input susceptance. admittance of a rectangular dipole for different number of the patches. For modeling the surface of dipole 24 patches at least is required. The length of the dipob is divided into three sections. The middle one is supposed as the feeding area and only two other sections carqy the current distribution. The result, as shown in Fig. 3, for kh<3 is very close to the higher patch densities and this is a very sensitive test for the divergence of the surface patch technique. Fig. 4 shows the input admittances of a hexagonal prism dipole derived by the method, the thin-wire approximation (point matching based on the Pocklingtin's integral equation) [7] and the King-Middleton second order solution [8]. Table 1 presents the CPU time and memory storage size required for different number of the patches on a NEC ACOS-2010 mainframe. Note that no advantage of inherent symmetry was made. 908 - 8 1 0 .  5 0 0.007022. 20 #- I - King-Middleton solution [8] 0 0 Thin-wireapproximation 171 I\ 0 Surface-patchappmximation - 0 1 2 3 4 5 6 7  -10 1 I I I I I I I 0 1 2 3 4 5 6 7  kh (c) Fig. 4 Input admittance comparison between patch, thin wire and analytical solutions of center feed dipole: (a) triangulated hexagonal prism (60 patches), (b) input condwme, (c) input susceptance. Table 1 Computation time and storage size for the rectangular prism dipole on a Nu3 ACOS-2010 mainframe computer. Identical triangular type current expansion functions and electric field boundary condition testing functions were used for developing a set of simultaneous equations for calculation of the surface current distribution. The main attention was spent to evaluation of divergence and cost compared with the thin-wire approximation and the wire-grid modeling. The input admittance of a cylindrical dipole was computed for different number of the patches for the antenna surface. The input admittance convergence of a rectangular dipole antenna also was investigated. The numerical results show that the surface patch modeling for calculation of the near field quantities, contrary to the wire-grid approximation, is explicit, the input admittance convergence based on the number of patches per square wavelength is very rapid (this can be observed clearly in the computed input admittance of the cylinder dipole with only 24 patches), and required memory storage size is  lower than for the wire-grid technique .  Beside the above advantages there are complicated mathematics and higher computation time cost. Number of patches 24 40 56 72 References J [l] A.C. Ludwig, "Wire grid modeling of surfaces". Transaction on-nnas and P r o m ,  vol. AP-35, No.9, pp 1045-1048. Sept.1987 [2] K.S.H. Lee, L. Marin and J.P. Castillo, " Limitations of wire-grid modeling of a closed, surface ", 8 tlc CO-, vol.EMC- . . .  18, NO.3, pp 123-129, August 1976 [3] N.N. Wang. J.H. Richmond, and M.C. Gilbreath, "Sinusoidal reaction formulation for radiation and scattering from conducting surfaces", JEEE T r m i o n  May 1975 [4] J.J.H. Wang, "Surface-patch techniques for modeling three-dimensional radiating or scattering objects", Georg ia  Ins t i t u t e  of Technology,  Repor t  [5] S.M. Rao, D.R. Wilton. A.W. Glisson, "Electromagnetic -I T n  * i , vol. AP-30, [6] D.A. Dunavant, "High degree efficient symmetrical Gaussian Quadrature for numerical methods in triangle", o-nnas and ProD &. vol.AP-23. pp.376-382 , No.RADC-TR-81-41, April 1981 scattering surfaces of arbitrary shape", 8 _in r' g, v01.21, pp 1129-1148, 1985 [7] H. Tsuboi, M. Analoui and T. Misaki: "Electromagnetic field Analysis of wire Antenna by moment method," 10th symposium on Computational Electrical and Electronic Engineering, The Japan Society for Simulation Technology, No.11-19. March 1988. [8] R.W.P. King and C.W. Harrison, h n n a s  and Waves: A Modern-, The M.I.T. Press.1969 Total CPU time for each point (ms) The numerical analysis of antenna was investigated by the moment method based on the surface-patch modeling and the electric field integral equation. Surface of the antenna was modeled by triangular patches. - I 

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Numerical analysis of antenna by a surface patch modeling

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