Transactions on Modelling and Simulation vol 18, 1997 WIT Press, ISSN X

Transcription

1 Boundary element analysis of resistively loaded wire antenna immersed in a lossy medium D. Poljak and V. Roje Department ofelectronics, University of Split, Rudera Boskovica bb, Split, Croatia dpoljak@adria.fesb.hr Abstract The boundary element procedure is developed for modelling the loaded dipole antenna embedded in a dissipative medium. Such approach provides a simple and efficient algorithm for the calculation of current distribution and input impedance. Numerical results are presented for current distribution and input impedance spectrum for a resistively loaded dipole antenna. 1 Introduction The input impedance of the electrically short wire antennas is strongly frequency dependent. On the other hand, the traveling wave antenna has input impedance which is essentially frequency independent. Some decades ago it was theoretically and experimentally shown that such a traveling wave antenna can be realized by a continuous resistive loading if the resistence per load of the antenna varies as a function of the position along the antenna*"*. For a certain non-uniform resistive and frequency independent loading the reflections at the wire ends can be eliminated to a great extent, so that the equivalent travel ing-wave current decreases rapidly by moving outward from the feedgap. These loaded antennas, due to the non-reflecting resistive loading, are convenient for broadband applications and BMP simulators design. Most of the papers dealing with a loaded wire antennas in free space propose the simplest variant of the moment method - pulse basis functions and point matching for solving the corresponding electric field integral equation (EFIEX % The differential operators appearing inside the integral equation kernel are usually evaluated by the finite difference algorithm in order to avoid the quasi singularity of the kernel. However, it has been proved that the point matching

2 486 Boundary Elements technique suffers from a rather poor convergence rate, and moreover the presentation of second-order differential operator with finite difference is not always satisfactory in numerical sense*. In this paper, a novel weak boundary element approach based on Galerkin-Bubnov procedure is used for solving the corresponding integral equation for loaded antenna radiating in lossy media. After solving the EFIE and obtaining the current distribution along the antenna, the input impedance spectrum is also calculated by a simple and efficient BEM procedure. Actually, the input impedance of the loaded dipole antenna is obtained by simply multiplying the matrices previously formed in current distribution calcualtion. Numerical results are obtained for current distribution and input impedance spectrum for a resistively loaded dipole antenna radiating in free space and lossy medium, respectively. A similar approach can also be applied to a more demanding case of wire antenna radiating in the presence of a lossy half space^. 2 Integral equation for loaded thin wire antenna The straight wire antenna with length 2L and radius a, immersed in a lossy medium is under consideration. In accordance to the theory of the wire antennas the tangential component of the electric field on the antenna surface is given by^: L )dz> CD where I(z) is the equivalent current distribution in the antenna axis while g(z,z') denotes the lossy medium Green function: where 7 is the complex phase constant of the medium: is the complex dielectric constant: we and R is the distance from the source point, (located in the antenna axis), to the observation point, (located on the antenna surface), of the form:

3 Boundary Elements 487 (5) The total tangential electric field can be written as the sum of the strongly localized electric field, produced by the voltage source in the gap E\ and the scattered field on the antenna surface E :.. (6) Since the antenna is loaded, the total tangential electric field is then expressed in terms of the current intensity I(z) and load impedance per unit length of the antenna ZJz): Considering the relations (l)-(7), the EFIE for loaded antenna in a lossy medium can be written in the form: Solving the integral equation (8) the equivalent current distribution is obtained and all the antenna parameters can be calculated. The input impedance is one of the most important parameters in broadband antenna applications. According to the well-known "the induced electromagnetic force" approach" the expression for input impedance is given as follows: \7 (9) where E(z) is the electric field determined by equation (8), I(z)* is the complex conjugate of the axial current distribution and IQ is the value of the current at the antenna input. Substituting the (8) into (9) the functional for input impedance Z^(I) is obtained in the form:

4 488 Boundary Elements The current distribution and the input impedance can be obtained using the certain boundary element procedure. 3. The boundary element solution of EFIE Integro-differential equation (1) is usually modeled by point matching technique^'*. Although it is the simplest way of handling the corresponding integral equation this approach usually does not ensure satisfactory convergence rate'' and there are also problems with quasisingularity^^. In this paper, a variant of boundary element formulation for integrodifferential operators, more fully documented in previous papers by Poljak^ and Poljak & Roje\ is applied to the problem of loaded dipole antenna. Utilizing the weighted residual approach, with the Galerkin-Bubnov procedure and the weak formulation^ applied to the (8) the following expression is obtained: ^ dz \ dz> L -L -L where n denotes the total number of basis functions, I; are the unknown coefficients of the solution, and f> and fj are the basis and test functions, respectively. The boundary conditions for current vanishing at free ends of wire are incorporated subsequently into the global boundary element matrix. Performing the boundary element discretization of the domain of interest the linear equation system arising from (11) is then given in the form: M Z s *- -71 i j* *»> > where [Z^ is the local matrix presenting the interaction with the i-th source boundary element to the j-th observation boundary element: (13) Matrices {f} and {f} contain shape functions 4(z) and4(z'), while {D} and

5 Boundary Elements 489 {[)} contain their derivatives, where: M is the total number of boundary elements, n, is the total number of local nodes per element and Alj, Alj is the width of i-th and j-th boundary element, respectively. Functions 4(z) are the Lagrange's polynomials defined by: Each Zjj represents, in fact, the mutual impedance of i-th and j-th boundary element. {V}j is the local right-side vector for j-th observation finite element in the gap: (15) and represents the local voltage vector. In this paper, the linear approximation over boundary element is used since it was shown^* that this choice provides accurate results. The boundary element right-side vector differs from zero only in the feed-gap area, and it is obtained in the closed form. The impressed (incident) electric field can be expressed as: where V? is the feed voltage and Alg denotes the feed-gap width. Using the linear approximation, it follows for two non-zero terms of the element over feed-gap area: y. f3.-r Thus, each node at the central feed-gap boundary element obtains half the value of the total impressed voltage. This method offers a better convergence than it is possible to achieve by point-matching technique^. Moreover the second-order differential operator from the kernel is replaced by trivial derivatives over shape and test functions, so, the problem of quasi-singularity is avoided. 4 Input impedance evaluation In contrast to the commonly used methods for input impedance evaluation, often complicated and inconvenient for numerical implementation"'^-" even

6 490 Boundary Elements for the case of insulated antenna with assumed sinusoidal current, the boundary element method (BEM), discussed so far, provides an efficient procedure for evaluation of input impedance. As a matter of fact, the functional of input impedance through the BEM procedure can be written as follows: (18) A(AZ, or in matrix notation: Z.^-L(/F[Z](7)' (19) rol where [Z] represents the global BEM matrix. It is obvious from relation (18) and (19) that BEM approach provides the input impedance calculation simply by multiplying the previously formed matrices obtained within the current calculation procedure. 4.1 Non-reflecting resistive loading It is well-known that the unloaded antenna, due to the reflections from the free ends, has a frequency sensitive input impedance. In order to retain the input impedance fairly constant versus frequency these reflections have to be reduced. This can be accomplished by using the non-reflecting resistive loading [1], [4]. Such a loading proposed by Wu & King^ and used elsewhere*"* is originally derived as a solution for impedance per unit length of the antenna as a function of distance from the feed point and obtained if the current is represented by an outward traveling wave with no reflected wave. The expressions for such a load are then given by^: L- z (20) where: ^ - _ ** (21) =2f(l --) ^ and Z is the wave impedance of the lossy medium.

7 Boundary Elements Numerical results Calculated numerical results are obtained for the unit voltage excitation in the frequency domain and by usage of linear approximation. According to the previous convergence investigations^'^, the optimal number of 31 elements over the entire antenna length is chosen for satisfactory convergence and computational time. Figure 1 shows a comparison of current magnitude obtained by BEM with Shen expperimental results' and Popovic numerical results^. It is obvious that the current distribution, due to the resistive loading, rapidly decreases as it moves from the feed-point. The influence of resistive loading is obvious, especially from figure 2. Namely, the input impedance magnitude of unloaded antenna varies significantly versus frequency while in the case of loaded antenna it remains fairly constant in a broad frequency range c ^o 1 -t V-E: z(m) Figure 1: Magnitudes of current distribution along dipole antenna in free space with =6, a= m, L=0.5m and f=450mhz.

8 492 Boundary Elements 2500.c 2000 c t frequency (Hz) (Millions) Figure 2: Magnitudes of input impedance of loaded dipole antenna with I*! =5.1083, a=0.05m, L=0.5m and e,= 10, 0-=0.001mhos/m. 6 Concluding notes A new method for the analysis of loaded wire antennas is proposed. The corresponding EFIE is solved by a variant of the boundary elment method, convenient for treating integro-differential operators. Moreover, the usage of BEM in antenna current calculation provides a simple and efficient algorithm for the input impedance evaluation. The proposed method shows advantages over the commonly used pointmatching techniques avoiding the quasi-singularity of the kernel and ensuring the satisfactory convergence rate. Finally, the procedure presented in this work can be easily extended to more demanding antenna geometries. 7 References 1. Shen, L.C. An experimental study of the antenna with non-reflecting resisitive loading, IEEE Trans. AP, 1967, 5, Popovic, B.D. Theory of cylindrical antennas with arbitrary impedance loading, Proc. IEE, 1971, 10,

9 Boundary Elements Wu, T.T. & King, R.W.P. The cylindrical antenna with non-reflecting resisitive loading, IEEE Trans. AP, 1965, 5, Rahmat-Samii, Y., Par ham i, P. & Mittra R. Loaded horizontal antenna over an imperfect ground, IEEE Trans. AP, 1978, 6, pp Parhami, P. & Mittra, R.: Wire antennas over a lossy half-space, IEEE Trans. AP, 1980, 5, Jeng, G. & Wexler, A.: Isoparametric, finite element, variational solution of integral equations for three dimensional fields, Int. Journ. Num. Meth. Eng., 1977, 9, Poljak, D. New numerical approach in the analysis of a thin wire radiating over a lossy half-space, Int. Journ. Num. Meth. in Eng., 1995, 22, Poljak, D. & Roje, V. Boundary-element approach to calculation of wire antenna parameters in the presence of dissipative half-space, IEE Proc. Microw. Antennas andpropag., 1995, 6, Roje, V. & Poljak, D. Electromagnetic wave analysis software, Proc. SofCom'94, pp , Split, Croatia, Werner, D.H., Werner, P.L. & Breakall, J.K. Some computational aspects of Pocklington's electric field integral equation for thin wires, IEEE Trans. AP, 1994, 4, Rudge, A.W. Input impedance of a dipole antena above a conducting half-space, IEEE Trans. AP, 1972, 1, Tiberio, R., Manara, G. & Pelosi, G. A hybrid technique for analyising wire antennas in the presence of a plane interface, IEEE Trans. AP, 1985, 8, Lindell, I.V., Alanen, E. and Mannersalo K. Exact image method for impedance computation of antennas above the ground, IEEE Trans. AP, 1985, 9, Roje, V. & Poljak, D. Various approaches for solving integral equations in electromagnetics, Proc. ICEAA 95, Turin, pp , Richmond, J.H. Radiation and scattering of thin wire structures in the complex frequency domain, Tech. Rep , Ohio State University, Electr.Sci. Lab., 1974.