Galerkin-Bubnov boundary element analysis of the Yagi-Uda array

Transcription

1 Galerkin-Bubnov boundary element analysis of the Yagi-Uda array D. Poljak, V. Doric, S. Antonijevic & V. Roje Department of Electronics, University of Split, Croatia Abstract Currents induced along the elements of a wire antenna array are calculated by solving the set of electric field integral equations (EFIEsj. These coupled integral equations are handled via the Galerkin-Bubnov Boundary Element Method (GB- BEM). Once obtaining the currents along the wires an electric field radiated by wire structure and the corresponding input impedance are evaluated. 1 Introduction The analysis of rectilinear antenna arrays is one of the most important tasks in antenna theory and applications. Many practical antenna arrays consist of various thin-wir elements having their longitudinal axes parallel. Th elements themselves may be either fed or parasitic, as itis a case in widely used Yagi-Uda array. Since there is no analytic solution available for such arrays many approximate techniques have been used for years. Analysis of linear arrays by means of numerical techniques started with classical paper by Harrington [l].it was followed by the work of many prominent researches [2]-[9]. Very recently a study dealing with coupled dipole antennas in front of dielectric obstacle pertaining to the bioelectromagnetic applications has been published [IO]. The Galerkin-Bubnov boundary element method (GB-BEM) was successfully applied to the solution of single wire structure in half-space configuration in both frequency [Ill and time domain [12]. This paper uses the GB-BEM for the solution of electric field integral equations (EFIE) for wire antenna array.

2 458 Bou~iw-~~ Elmcnts XXIV 2 Integral equation set for currents along the wire A wire antenna array containing the parallel and rectilinear elements is considered, Figure 1. These elements may have various lengths, radii and interelement spacings. 1. zt 2. N. Figure 1: Geometry of the wire antenna array The current distributions along the wires are governed by a set of simultaneous integral equations of the Pocklington type: EinC= - ~ ~ ~ + ~ ' ~ ~ ' ~ ~ ~ ( ; z j ' = ) 1,2, ~... ~ N ~ ( (1) ~, ~ ' ) ' j4m11~ dz j=l -/, where N is the total number of the wire elements, Ii(z') is the unknown current distribution along i-th array element to be detemned,ezmcis the known incident field tangential to the wire surface, k is the free space propagation constant, and G,i is the corresponding Green function: Gji(z,2)=- e-jkrji where R is the distance from the source point to the observation point given by: Rji where ai is the radius of the i-th wire and d,i is the distance between i-th and j-th wire.

3 B o u d u r - yekww~tsxxiv 459 The system of integral equations (1) is solved by means of Galerkin-Bubnov Boundary Element Method (GB-BEM). Full description of the GB-BEM can be found in [131. Knowing all the currents along the wire array all other parameters of the antenna structure (radiation pattern, input impedance, radiation power) can be determined. 3 Calculation of the radiated electric field and the input impedance Once the current distribution along the wires is determined, the tangential component of the radiated electric field can be obtained using the BE formulation as follows: where N is the number of wires, ni is the number of elements of i-th wire and the current along the k-th element is computed using linear interpolation: I(z')=I,, '2kFZ' '2kmZlk '2k-'lk +r2, Z'-Z,, (5) Input impedance can be readily obtained from the following functional: +k2j" J"{N}j {Nr]T AljAli G(z, z')dz'dz]i where M denotes the total number of boundary elements, {N} is the shape functions vector and {D} contains the shape fimctions derivatives.

4 I 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. 460 Bou~iw-~~ Elmcnts XXIV 4 Numerical results Numerical results for current distribution, input impedance and electric field pattern are computed for following Yagi-Uda antenna array: radii of all wires is h, number of finite elements on all wires is 31, generator voltage is 1V. Reflector element length is h, fed element length is h and director element length is A.Distance between all wires is 0.25 h. Reflector element [lambda] Figure 2: Current distribution along the reflector element Fed element L J. l [lambda] Figure 3: Current distribution along the fed element

5 B o u d u r - yekww~tsxxiv 46 1 Director element ~ z 0 L [lambda] Figure 4: Current distribution along the director element L Figure 5: Input resistance frequency dependence

6 Bou~iw-~~ Elmcnts XXIV F Figure 6: Input admtance frequency dependence Figure 7: Far electric field pattern

7 Conclusion B o u d u r - yekww~tsxxiv 463 The paper deals with the efficient method for the analysis of wire antenna array. The problem is formulated by the set of Pocklington integral equations. Current distribution along the wires, radiated electric field and the input impedance of particular Yagi-Uda array are calculated by using the Galerkin- Bubnov Boundary Element Method (GB-BEM). The fiu-ther work will include the study of wire antenna arrays in the presence of a lossy half-space. References Harrington, R.F., Matrix methods for field problems, ProcJEEE, 55 (2), Feb , pp Silvester, R.P., Chan, K.K., Bubnov-Galerkin solutions to wire antenna problem, Proc.IEE, 119, 1972, pp Silvester, R.P., Chan, K.K.,Analysis of antenna structures assembled from arbitrarily located straight wires, Proc. IEE, 120 (l), Butler, C.M., Wilton, D.R., Analysis of various numerical techniques applied to thin wire scatterers, IEEE Trans. AP, 23, Butler, C.M., Wilton, D.R., Efficient numerical techniques for solving Pocklington's integral equation and their relationshp to other methods, IEEE Trans. AP, 24, Miller, E.K., Poggio, A.J., Burke, G.J., Selden, E.S., Analysis of tire antennas in the presence of a conducting half-space. Part 11.The horizontal antenna in free space, Canadian Journal of Physics, 50, 1972, pp Sarkar, T.K., Analysis of arbitrarily oriented thin wire antennas over a plane imperfect ground, Archiv fur elektronik mdubertragungstechnik, 31, 1977,pp Parham, P.Mittra, R., Wire antennas over a lossy half-space, IEEE Trans. AP, 28, 1980, pp Miano, G., Verolino, L., Vacaro, V.G., A new numerical treatment for Pocklington's integral equation, IEEE Trans. Magnetics, 32 (3), Tay, R.Y., Balzano, Q., Kuster, N., Dipole Configuration with Strongly Improved Radiation Efficiency for Hand-Held Transceivers, IEEE Trans. AP, 46, June 1998, pp Poljak, D., New numerical approach in analysis of a thin wire radiating over a lossy half-space, Int.J.Num.Meth.Eng., 38 (22), 1995, pp Poljak, D., Transient response of resistively loaded straight thin wire m half-space configuration, Journ. Electromagnetic Waves and Applications, 12, 1998, pp Poljak, D., Electromagnetic Modelling of Wire Antenna Structures, WIT Press, Southampton-Boston, 2002.