## 1. Introduction

[2] When computing the return loss in array antennas, one needs the coupling coefficients between the antenna elements, but for wideband large array antennas it is often difficult and time-consuming to compute these parameters. In large arrays, the internal elements can be approximated to be located in an infinite array, whereby the Floquet theorem is used to reduce the computational domain to a unit cell. In these calculations a phase shift is set between the unit cell boundaries, equivalent to the excitation of an infinite array with uniform amplitude and a linear phase shift along the scanning direction. However, the infinite array approximation holds only for arrays that are large in terms of wavelengths; elements close to an edge will have different radiation patterns and active reflection coefficients. These effects are caused by mainly three factors: (1) the excitation is truncated, (2) the geometry close to an edge will affect the electrical properties of the elements, and (3) unattenuated surface waves will be reflected by the edges and create a standing wave pattern over the aperture [*Janning and Munk*, 2002]. These three effects can be hard to distinguish from each other (especially the second and the third are connected) but all of them must be included in an accurate calculation of the coupling coefficients of the array.

[3] The problem with computing coupling coefficients for a finite array is that the computational domain is very large. Coupling coefficients can be computed from the infinite array data using Fourier series expansion [*Roscoe and Perrott*, 1994; *Bhattacharyya*, 2006]. In some rare cases the coupling coefficients for an infinite array can describe a finite array exactly, provided the coupling coefficient matrix for the finite array is of Toeplitz form, a necessary but not sufficient condition that holds approximately in some cases. For example, an impedance matrix can fairly well describe a finite dipole array, and an admittance matrix can describe a patch element array. However, in most cases the coupling coefficient matrixes cannot be approximated to be of Toeplitz form, for any type of coupling representation between the elements, since on the edge elements the currents will differ from the currents on the central elements, regardless of the port termination. Once the coupling coefficients are computed the active reflection coefficient can be computed for arbitrary excitations of the infinite array. A closely related technique to the Fourier series expansion of the coupling coefficients is the windowing technique [*Ishimaru et al.*, 1985; *Skrivervik and Mosig*, 1993]. However, for the windowing technique the active reflection coefficient is solved for one excitation at a time.

[4] To achieve a better approximation of the finite array one needs to add a perturbation to the truncated array, and several different computer codes have been developed for that purpose [*Neto et al.*, 2000; *Craeye et al.*, 2004; *Lu et al.*, 2004; *Maaskant et al.*, 2007, 2008; *Craeye and Sarkis*, 2008]. In these methods, characteristic currents for the elements are used to reduce the number of unknowns, e.g., the current distribution on any element can be written as a superimposition of current distributions for an edge element in a small array and current distributions on an element in an infinite array. The corresponding coefficients for the characteristic currents are then obtained by solving the integral equations. There are different ways to implement this technique and the complexity of the codes varies. Inspired of these method of moments techniques one might suspect that arbitrarily sized arrays can be analyzed approximately by mixing the scattering parameters (S parameters) for the finite array and the infinite array.

[5] In this paper the edge effects are studied for tapered-slot elements in a triangular grid by comparing a finite array with an infinite array with a finite excitation. To analyze the antenna correctly, dielectric and lossy materials must be included in the numerical model. It is difficult to analyze these elements using the method of moments and therefore the methods based on characteristic currents are not directly applicable. To compute the S parameters for the infinite array and the finite array, finite difference time domain (FDTD) codes furnished with periodic boundaries are used. The Fourier expansion method has previously been shown to be accurate for computing coupling coefficients of linear tapered-slot arrays [*Wang et al.*, 2008] using a finite element method solver.

[6] To simplify the problem the array is finite in only one direction, a model that is useful in three respects. The edge effect is isolated, since the studied elements are only affected by two edges, the computational domain for the finite array is decreased, and the required number of phase shifts to compute the coupling coefficients for the infinite array is reduced.

[7] The results for the finite edge models are used to describe the dominant edge effect in the finite array. The difference between the infinite array and a finite array active reflection coefficient is shown to be caused by the truncation of the excitation. This model of the edge effect is different from previously published results [*Neto et al.*, 2000; *Craeye et al.*, 2004], since it is based on a different approach and studies the frequency dependence of the dominant edge effect.

[8] The paper has been organized as follows. In section 2, the method used to calculate the S parameters for the infinite array is presented. In section 3, the element studied in the paper is presented. In section 4, the S parameters computed using infinite array data and finite array data are compared. In section 5, the S parameters are used to compute active reflection coefficients for the infinite and the finite array, and a method combining the two results is also evaluated. The combined results are used to analyze the perturbations caused by an edge in the array. A summary is given in section 6.