3-D EM Simulation of Infinite Periodic Arrays and Finite Frequency Selective Horns

Frequency Selective Surfaces (FSS) are predominantly passive electromagnetic filters formed by thin conducting elements on a dielectric substrate or periodic aperture elements in a conducting sheet. Depending on the type of elements (conducting or apertures), they exhibit bandstop or bandpass properties when excited with an incident electromagnetic wave. These filtering properties of FSS have been successfully utilised in curved FSS antenna systems as FSS sub–reflectors, radomes, Frequency Selective Guides (FSG) and Frequency Selective Horns (FSH) [1]. Recent advances in various aspects of FSS and applications along with modelling and analysis of FSS are documented in [2, 3]. The most common analysis is the unit cell based approach using the Floquet modes, well suited for infinite planar arrays. However, arbitrarily curved or finite surfaces present greater problems because of the numerical complexities and the intricacies in modeling 3-D structures. In a curved array, it becomes necessary to perform the computations for groups of elements or individual elements conformed to the profile of the full 3-D structure. A quasi-static approach using a finite current model has previously been successfully employed in [4] and also a Pocklington equation technique in [5]. In this paper a finite conical FSH antenna consisting of conducting elements on a dielectric substrate modelled using the CST MICROWAVE STUDIO® is presented. The full 3-D EM simulation tool is based on the Finite Integration Technique (FIT). The versatility of FIT allows for problems to be formulated on a Cartesian (Hexahedral) or general non-orthogonal (Tetrahedral) grids both in the time domain as well as the frequency domain. Initially, the frequency domain solver using the full Floquet modal expansion and periodic boundaries was used to calculate the resonant frequency of the unit cell. The time domain solver using the hexahedral mesh with a finite conductor thickness was used to model the FSH. 1.0 Infinite Arrays and Analysis The Frequency Selective Surfaces (FSS) are periodic structures in a one or two dimensional array and as the name suggests, perform frequency filtering. Thus, depending on their physical construction, material and the element geometry, they can be divided into band-pass and band-stop filters. In 1919 Marconi first patented periodic structures [6], but work on FSS started with earnest in the early 60s [7] and concentrated on the use of FSS in Cassegainian subreflectors in parabolic dish antennas. Today FSS are employed in radomes (terrestrial and airborne), missiles and electromagnetic shielding applications. The analysis of FSS started with mode matching techniques which were first applied in waveguide problems. The mode matching method led to the approximate method of equivalent circuit analysis which gave a lot of insight into the behaviour of FSS since it was partly based on the transmission line principles. The modelling capability however was limited by the inability of the Mode Matching Method to model any FSS geometry and the inaccuracy the equivalent circuit method. With the advent of computers more accurate numerical techniques were developed for the analysis of FSS such as the method of moments (with entire or subdomain basis functions) the finite difference method and the finite element method. In this paper, we investigate modeling infinite periodic FSS arrays and finite FSS structures using CST MICROWAVE STUDIO®. The full 3-D EM simulation tool is based on the Finite Integration Technique (FIT). 2.0 Finite Integration Technique (FIT) The Finite Integration Technique (FIT) was first proposed by Weiland in 1976/1977 [8,9]. This numerical method provides a universal spatial discretisation scheme applicable to various electromagnetic problems ranging from static field calculations to high frequency applications in time or frequency domain. Unlike most numerical methods, FIT discretises Maxwell’s equations in the integral form of rather than the differential one. FIT generates exact algebraic analogues to Maxwell’s equations, that guarantee physical properties of computed fields and lead to a unique solution. Maxwell's equations and the related material equations are transformed from continuous domain into a discrete space by allocating electric voltages on the edges of a grid G and magnetic voltages on the edges of a dual grid G . The allocation of the voltage, flux components on the grid can be seen in Figure 1. The discrete equivalent of Maxwell's equations, the socalled Maxwell’s Grid Equations are shown in Eqs.(1)-(4). This description is still an exact representation and does not contain any approximation errors.