Radio Science

A study of edge effects in triangular grid tapered-slot arrays using coupling coefficients

Authors


Abstract

[1] Edge effects for tapered-slot elements in triangular grid array antennas are studied by using finite difference time domain codes. The S parameters for finite-by-infinite arrays are computed and evaluated for two different edge models and, to evaluate the edge effects, the results are compared with the S parameters for the infinite array. As expected, the largest difference between the results occurs for the elements closest to the edges, because of the missing coupling from nearby elements and the perturbed element currents due to the edge geometry. By using the proposed method it is possible to distinguish between these two edge effects. A method to combine the finite-by-infinite array results and the infinite array results is presented and used to characterize the perturbation caused by the truncation of the infinite array.

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.

2. Finite Excitation of an Infinite Array

[9] The array considered is a triangular grid array, as shown in Figure 1. The antenna elements position relative a reference element is denoted by the integers p and q, where p denotes the number of elements from the reference element along the x axis, and q the number of elements from the reference element along the oblique direction that is inclined at an angle γ with respect to the x axis. The reference element is denoted as the m:th element and the element described by p and q is the n:th element. Since the lattice is triangular the element spacings and angle fulfill the relation 2b = atanγ. The phase shifts between elements along these two axes are denoted ψx and ψy, and the reflection coefficient for the Floquet excitation is denoted

equation image
Figure 1.

Geometry of the array antenna.

[10] The unprimed Floquet excitation is used for the phase shifts along the x and y axis and the primed Floquet excitation uses the natural axis of the lattice. The phase shifts are denoted

equation image
equation image
equation image

Using this notation it is possible to calculate the scattering parameters for the infinite array by doing a Fourier series expansion [Roscoe and Perrott, 1994]:

equation image

These S parameters can then be used to calculate the active reflection coefficient for an infinite array with its excitation restricted to the elements within a finite region:

equation image

where m is the element for which the active reflection coefficient is computed, xn and yn denote the positions of the element with index n, an is the complex valued excitation weights, and N is the total number of excited elements. In this paper only uniform excitations with linear phase shifts are considered, thus an = 1 for all of the excited elements.

2.1. Arrays That Are Finite in Only One Direction

[11] When using equation (5), the active reflection coefficient has to be known for a large set of scan directions. If we allow the array to be infinite in one direction and fix the phase shift in this direction we can reduce the required number of evaluations of the integrand.

[12] The antenna elements are oriented so that the E plane coincides with the zx plane and the H plane with the yz plane; see Figure 1. The simplest truncation is an edge perpendicular to the H plane, denoted as “H plane edge” since, when scanning in the H plane, the beam is scanned toward or from that edge. The other edge is the “E plane edge,” obtained when the array is truncated perpendicular to the E plane. This edge is more difficult to analyze since the elements are shifted half an element spacing every other element row along the edge.

[13] For the infinite array the coupling coefficients are given by equation (5). If the excitation of the array is truncated along only one of the edges and the excitation is uniform with only a linear phase shift along the direction of the edge, the coupling coefficients can be grouped together to reduce the required data in the integrand. The resulting coupling coefficients will be between a linear array, excited with a uniform amplitude and a linear phase shift, and an element in an identical infinite array; see Figures 2 and 3. The H plane edge is natural for the grid, and if the suggested excitation is used equation (5) is easily simplified by using the Poisson summation formula.

Figure 2.

The H plane edge model where coupling is computed between a uniformly excited linear array and a single element.

Figure 3.

The E plane edge model where coupling is computed between a uniformly excited linear array and a single element. Due to the triangular grid, two types of linear arrays are required for the E plane edge model.

[14] If the infinite linear array is uniformly excited with no phase shift between the elements, the coupling coefficient between an element qb from the linear array becomes

equation image

[15] For the more complicated E plane edge, the easiest way to achieve a similar result is to use two linear arrays, one with a pa separation distance between the linear array, and the element along the x axis and one linear array with a separation distance of (p + equation image)a; see Figure 3. To simplify the description of the linear elements position, p′ is used to describe the linear array.

[16] The coupling coefficient between the element and the linear array at the distance pa then becomes

equation image

The coupling coefficient between the element and the linear array at the distance a(p + equation image) becomes

equation image

For both of these results the symmetry of the scattering parameters, SFL(ψx, −π) = SFL(ψx, π), is used. Due to the triangular grid the active reflection coefficient can also be written as SFL(ψx, π) = SFL(ψx + 2π, 0).

2.2. Numerical Method

[17] The FDTD code by Holter and Steyskal [1999] employing periodic boundaries with time shift between the boundaries was used to calculate the active reflection coefficient for the E plane edge model (−2πψx ≤ 2π, ψy = 0), and for the H plane edge model (ψx = 0, −πψyπ). Since this code is restricted to phase shifts between the boundaries corresponding to scan directions in visual space, the computational domain was extended to include multiple elements, in order to be able to calculate the required phase shifts. If the coupling coefficients are computed for the elements within the computational domain it is possible to compute the required phase shifts between the elements if the computational domain is larger than λ/2 in the scan direction [Ellgardt, 2009]. For low frequencies the domain becomes large and some phase shifts are not spanned using this method. These phase shifts were instead computed using an additional FDTD code employing phase shift boundaries [Turner and Christodoulou, 1999; Pettersson et al., 2004].

[18] The finite array results were computed by using open boundaries in one direction and periodic boundaries with zero phase shift in the other direction. The elements were excited one at a time, to compute the finite-by-infinite S parameters, while the other elements were terminated with their characteristic impedances. All computations where made with the time shift code. However, this is of minor importance since as long as there is no phase shift in the excitation between the elements in the infinite direction there is no numerical difference between the phase shift and time shift codes.

3. Phased Array Element

[19] The antenna element considered in this study resembles a design made for an experimental antenna [Ellgardt and Wikström, 2009]. The tapered-slot element consists of three metallic layers separated by a dielectric material. The outer metal layers are shaped as a bilateral tapered slot and the intermediate layer is a stripline. Unlike the experimental element, a stripline impedance transformer is not included in this element. Therefore, instead of 50 Ω the characteristic impedance of the stripline is 66.4 Ω at the port. The dielectric constant is chosen to ɛr = 2.5 and the thickness of the elements is chosen to 1.6 mm. The antenna elements protrude through a ground plane and the dielectric substrates are fastened on the backside of the ground plane where electric contact is made with the ground plane.

[20] This type of antenna is intended for military applications, where it is important that the antenna has a small radar cross section. Hence, in order to reduce the backscattered field at cross-polarized incidence, an absorber, with the thickness of 19 mm, is positioned on top of the ground plane. As a consequence, the antenna element must be extended from the ground plane by the same length as the absorber is thick; see Figure 4. Note, an absorber may reduce the antenna efficiency but for this design it is a minor problem for the infinite array [see Ellgardt and Wikström, 2009]. The substrates create a parallel waveguide structure that is below cutoff at the operational frequencies. The unwanted losses are therefore expected at high frequencies or for scan blindness phenomena, when the field extends down to the absorber. It is possible that the finite effects cause further losses, but this problem is left for future studies. To add extra stability to the array, the space between the substrates is filled with a foam material, Rohacell 71 HF, that has the dielectric constant ɛr = 1.09. To protect the whole antenna, the foam is covered by a 0.3 mm thin glass fiber layer with ɛr ≈ 3.5.

Figure 4.

Geometry of the phased array element. (a) The antenna element without the absorber, the foam, and the glass fiber layer. (b) The antenna element as it is in the phased array. The dimensions for the experimental antenna are l = 36.4 mm, h = 63 mm, a = 14.1 mm, b = 12.4 mm, and ht = h + 2.2 mm. Tapered-slot opening rate R = 0.06 mm−1; see Shin and Schaubert [1999] for definition.

4. Comparison of S Parameters for the Finite Array and the Infinite Array Edge Models

[21] The S parameters for the infinite array will be compared with the S parameters for an array that is finite in one direction and infinite in the other. To compute the S parameters for the finite array, the E plane and H plane edge models in Figure 5 were used. Both edge models use periodic boundaries with no phase shift in the infinite direction. The E plane edge is more problematic to model, since two element rows are required due to the triangular grid. The increased size of the computational domain limits the number of elements in a row to 13. This is a fairly small array, whereby the edges will affect the active reflection coefficient for any element in the array. The H plane edge is easier to model than the E plane edge, since only a single element along the E plane is required in the computational domain. The H plane edge model consists of 24 elements, which is almost the same number of elements as in the E plane edge array. The center elements in this array will be farther away from the edges and will behave almost as the infinite array elements. The edges are designed so that the absorbing layer and foam layer extend approximately one element spacing from both edges.

Figure 5.

The computational domains for the E plane edge and H plane edge models.

[22] In Figure 6 the H plane edge S parameters are shown at 8 GHz for the infinite array and the finite array, for three different elements in the array as a function of q (see Figure 2 for the definition). The frequency was chosen so that it becomes necessary to use data from the invisible space to calculate the S parameters. Furthermore, at this frequency the antenna is fairly well matched. In the finite array, the deviation from the infinite array result is largest for the element at the edge; see Figure 6a. The edge element S parameters are close to the infinite array result in terms of phase but the magnitude is considerable lower than in the infinite array solution. The difference between the two solutions decreases as the distance between the two elements increases. Obviously, the edge element will have a different self contribution S0 than an interior element. It is interesting that for the edge elements the S parameters are lower than the ones for the infinite array. In this case this will lead to that the active reflection coefficient for this port will be smaller compared to the infinite array for most phase shifts. From Figure 6a we can also see that the S parameters for the finite array almost need 8–12 elements to converge to the infinite S parameters. For an element farther from the edge the convergence will be much faster.

Figure 6.

The S parameters for the H plane edge model.

[23] In Figure 6b the S parameters for element 3 from the edge are shown. Just as in the previous case the phase is similar for the finite array and the infinite array solution but the magnitude is different for the elements close to the edge. However, now the difference between the self contributions is small and the difference between the mutual S parameters decreases as the distance from the closest edge increases.

[24] Farther from the edge the differences between the solutions are small. For element 7, counted from the edge the only discrepancy between the solutions is the coupling to the edge element and a slight difference in the self contribution; see Figure 6c.

[25] For the E plane edge we need to calculate the S parameters between a linear array of elements and an element that is separated by pa or (p + equation image)a; see Figure 3. For the first case, which includes the self contribution, one of the elements in the linear array of excited elements is positioned in the same substrate as the element that the linear array couples to. For the second case, no element in the linear array is positioned in the same substrate as the element it couples to. The finite array is designed so that element 1 is closest to its nearest edge and element 13 is one half element spacing from the opposite edge. In Figure 7 the S parameters for the finite and the infinite array are compared. The difference between the two solutions is largest for the elements close to an edge, just as in the H plan edge case. Element 1 has a larger self contribution than in the infinite array, but the difference between the solutions decreases faster than for the H plane edge when the distance between the elements increases. The phase on the other hand does not agree so well between the infinite and finite array solutions for the E plane edge as it did for the H plane edge.

Figure 7.

The S parameters for the E plane edge model.

[26] The results for the elements closer to the edge agree fairly well with the infinite array results. For elements 3 and 7 the biggest difference between the infinite and the finite array results is, as expected, for the coupling to the elements close to the edge. For element 3, the self contribution is almost the same as in the infinite array and the difference between the solutions is largest for the elements closest to the closest edge. The agreement between the infinite array solution and element 7 in the finite array is very good; the largest difference is in the phase for the elements close to the edge.

5. Active Reflection Coefficient

[27] To evaluate the impact of the difference in the S parameters the active reflection coefficient was calculated for the finite array and the infinite array. The excitation was chosen to be uniform in amplitude and with a linear phase shift ϕ from one edge to the other. The excitation of the infinite array was truncated to the corresponding elements in the finite array. In Figures 8 and 9 a positive phase shift corresponds to that the beam is steered in the direction of increasing element number.

Figure 8.

Active reflection coefficient for the H plane model compared with the finite excitation and finite-by-infinite array results.

Figure 9.

Active reflection coefficient for the E plane model compared with the finite excitation and finite-by-infinite array results.

[28] From the S parameters for element 1 in the H plane edge geometry one might suspect that the match is better for the finite array edge element than for the truncated infinite array. It is indeed the case, as can be seen in Figure 8a. The elements far from the edge share this trait; the active reflection coefficient is in general lower for the finite array than for the truncated excitation of the infinite array. In Figures 8b and 8c the active reflection coefficients for elements 3 and 7 are shown. The active reflection coefficient varies similarly for the finite array and the truncated infinite array, the local maxima and minima are basically located at the same phase shift. For element 7 the difference between the truncated and the finite array is so small that it probably has very little practical importance. However, the difference between the infinite array and the truncated infinite array is noticeable. The active reflection coefficient for the infinite array has local minima at ϕ = ±1.78, these minima are difficult to resolve for an array with 24 elements. The finite excitation and the finite array results are closer to the infinite array solution when the element is steered from the closest edge than when it is steered toward the edge.

[29] As compared with the H plane edge, the E plane edge behaves differently in that the matching of element 1 is bad at most scan angles and worse than for the infinite array with truncated excitation. Therefore, it is likely that the edge elements for the E plane edge would benefit from using dummy elements to a larger extent than the elements at the edge in the H plane edge model. From the S parameters one can observe that the solutions for the finite excitation and the finite array are fairly similar for elements farther from the edge. The phase shifts that are larger than π in Figure 9 correspond to the invisible space for the infinite array. For a lossless infinite array these phase shifts will cause total reflection, but in this case energy is dissipated in the absorbing layer covering the ground plane. The agreement between the two solutions is in general worse for these phase shifts than for the visible space.

[30] The observations regarding the two different edges are similar for higher as well as lower frequencies; see Figures 10 and 11. The difference is small between the active reflection coefficient for the finite array and the truncated excitation for the middle element in the array. For the edge elements the truncated excitation model is not a good approximation of the finite array. The active reflection coefficient may have local maxima and minima at the same frequencies but the magnitude of the active reflection coefficient is in general far from the results for finite arrays.

Figure 10.

Active reflection coefficient for element 1 for the H plane model.

Figure 11.

Active reflection coefficient for element 1 for the E plane model.

5.1. Combined Results

[31] Since the small difference between the finite array and the truncated array is caused by the contributions from the edges, which are largest when the element is close to an edge, the elements within the array can be described accurately by the truncated array. These effects are included in the finite array results but they are time consuming to compute and the size of the problem is restricted by the hardware. To compute active reflection coefficients for arbitrary sized arrays it would be useful if an edge perturbation could be added to the infinite array result.

[32] An idea to include the edge perturbation is to use the infinite array solution for elements within the array and finite array results for excitations of elements close to an edge. This can be done by computing the outgoing voltage in a port using the infinite array coupling coefficient for the incident waves at the ports of the internal elements and the finite coupling coefficients for the incident waves at the ports for the r elements closest to the edge. To avoid discontinuities in the results both methods are used for s elements in between the edge and interior region and then weighted together. The weights are chosen to vary linearly from one region to the other, e.g., if s = 2 the weights for the two intermediate elements are (2/3, 1/3) for the finite array coefficients and (1/3, 2/3) for the infinite array coefficients. This method requires that the finite array is sufficiently large, so that multiple reflections between the edges are negligible. When the frequency exceeds 12 GHz the array can guide surface waves, giving rise to multiple reflections between the opposite edges. Below 12 GHz there are also guided waves in the array, but these waves are attenuated by the absorber and radiation and will not cause multiple reflections between the edges as long as the antenna is sufficiently large. To evaluate this procedure the active reflection coefficient was computed for the previous examples with r = 2 and s = 2; see Figures 8 and 9. The solution obtained using the combined results shows a substantial improvement in the agreement with the finite array result. The largest improvement is for the edge elements and the elements closest to the edge. For the elements close to the center of a large finite array the agreement with the infinite array is already very good and the improvement is small.

5.2. Dominant Effects

[33] Comparing the results from the finite array computations and finite excitation, they both share the general behavior that, as functions of the frequency, they oscillate around the infinite array solution. The difference between them is the amplitude of these oscillations. In both cases, the period of the oscillations are the same, and it is clear that the infinite array solution is modulated by two oscillating perturbations. Hence, it is logical to assume that it is the same effect that occurs in both of the cases. The periodicity depends on frequency, phase shift, and the distance from the edges. These oscillations as functions of element spacing have previously been modeled as diffracted waves from the edge of the array [Neto et al., 2000; Craeye et al., 2004]. In terms of S parameters the dominant effect is caused by the missing contributions from the nonexcited elements [Holter, 2000].

[34] In this section we are going to present a heuristic model for these oscillations using S parameters, where we are primarily interested in the phase of the perturbation. It is reasonable to assume that the dominant contribution to the active reflection coefficient is caused by the closest elements. The main part of the perturbation of the infinite array solution is therefore caused by the missing scattering from the closest absent element. For a finite antenna scanned using a linear phase shift exp(jϕ0) between two adjacent elements, the infinite array active reflection coefficient would be perturbed by the missing element contribution exp(−jnϕ0) for element n from the edge, with element 1 at the edge. Similarly it will be perturbed by the other edge, with exp(j(Nn) ϕ0) where N is the number of elements along the finite direction of the array. This model is simple and it works remarkably well.

[35] The next part to understand is the period of the oscillations of the active reflection coefficient caused by the change of frequency. The missing contribution in the truncated result is a wave that in the infinite array would travel from the source element to the receiving element. The shortest distance the wave could move would be directly from the port of the source to the port of the receiver. This distance is the number of elements between the ports times the element separation distance d in that direction. The resulting oscillations would be caused by the term exp(jβnf), where β = equation image. However, this assumes that the wave travels the shortest possible distance in vacuum which is reasonable for an array of dipoles in free space. For the array studied here the elements are embedded in a dielectric material with ɛr = 1.09 and it is possible to excite surface waves and leaky waves with different phase velocities. In addition, for the H plane edge the wave needs to travel from the source port through the elements to the top of the elements and then travel toward the receiving element and then through the element to the port. The extra distance that the wave needs to travel would be very similar between all elements. To compensate for the additional path length an extra phase term exp(jκf) is introduced, where κ is proportional to the additional travel time through the element. Using the three phase terms (i.e., exp(−jnϕ0) exp(jβnf) exp(jκf)) one can very efficiently predict the periodicity for the perturbations of the infinite array result for a given element n, provided the coefficients β and κ have been determined.

[36] To determine β and κ for the H plane edge model, the active reflection coefficient for several values of n were computed for an array with 100 excited elements. The local maxima of the perturbation were extracted from the figures. Good agreement was found when the value for β is the time a wave can travel between two elements in free space equation image = 0.252 · 10−9 s and κ = 1.60 · 10−9 s. The value for κ corresponds roughly to the time it takes for a wave to travel two times through the element slot. The accuracy of the extraction method is poor, a 10% change of β is difficult to distinguish. However this value can be improved when we study the perturbation as a function of element number.

[37] In Figures 12a and 13a the difference between the absolute value of the active reflection coefficient for the infinite array and the absolute value of the active reflection coefficient for the combined solution for element 7 are shown. Since the absolute value for the active reflection coefficient can be very small, a difference measure is used instead of a relative measure, making it straight forward to compare the period of the oscillations caused by the truncation. In Figure 12a, 24 elements are used in the array and the perturbation from both edges can be seen clearly. In Figure 13a, 100 elements are used and the perturbation caused by the edge farthest from the element is no longer observable. In Figures 12b and 13b an expected perturbation is shown for element 7 in arrays with 24 and 100 elements, respectively. The perturbation is computed by

equation image

where A and B where set to values close to the corresponding amplitude of the perturbation oscillations at 8 GHz and ∣R∣ > ∣A∣ + ∣B∣. In the visual space domain, the agreement of the perturbed patterns and the difference solution is fairly good. The variable κ is element specific and is not necessarily the same for the H plane edge model and the E plane model. However, if the measured value for κ for the H plane edge is used for the E plane edge model the agreement between the perturbations and the active reflection coefficient as function of frequency and phase shift is as good as for the H plane edge model.

Figure 12.

Difference between the absolute values of the active reflection coefficient for the infinite array solution and the combined method solution for element 7 for an H plane edge model with 24 elements.

Figure 13.

Difference between the absolute values of the active reflection coefficient for the infinite array solution and the combined method solution for element 7 for an H plane edge model with 100 elements.

[38] If we want to illustrate the variation of the perturbation as a function of the element number it is useful to choose a phase shift so that one of the edges perturbation period become easily observed. The amplitude of the oscillations of the perturbation model is chosen so that it corresponds to the closest missing S parameter, i.e., A(n) = ∣Sn∣ exp(jα1), where α1 is real valued constant. This results in a value that is slightly too high for the 10 closest elements to the edge, but seems to be more accurate further away from the edge. The phase factors determined by measurements are compared with the perturbation of the active reflection coefficient as function of element number, and the two curves agree well as seen in Figure 14a. The agreement with the perturbation model for β = equation image shows that the phase velocity in this case is the same as for a plane wave in vacuum. Otherwise the time delay could have been longer since the foam material has ɛr = 1.09 or the path would be longer in terms of wavelengths if the wave bounces between the substrates. If a permittivity correction for the foam is added to β this will require an extra term exp(j) for the model to agree with the data in Figure 14a. Since there is no good explanation why such a term should exist, ζ is assumed to be zero for the correct value of β. The perturbation term for the element closest to element 1 then becomes

equation image

and for the other edge the perturbation is described by

equation image
Figure 14.

The active reflection coefficient compared with ∣SFL + e10, f, n)∣ and ∣SFL + e20, f, n)∣. The amplitudes of the perturbation, A(n) = ∣Sn∣exp(jα1) and B(n) = ∣SNn+1∣exp(jα2), where α1 and α2 are set so that the maxima and minima coincide for the perturbation and the finite result. The linear phase shift toward the edge is set to ϕ0 = 1.2472 and f = 8 GHz.

[39] If this perturbation model is used for the E plane edge model the agreement will be good for the perturbations as function of element number; see Figure 14b. The phase dependance of the perturbation model as function of the element number is the same as suggested by Neto et al. [2000] and Hansen and Gammon [1996].

6. Conclusion

[40] The S parameters have been calculated for two different edges in a triangular grid tapered slot array. To reduce the complexity of the problem the coupling was calculated between an infinite linear array parallel to the edge and one element in the array. These coupling parameters have been computed for a finite-by-infinite array and for an infinite array. The infinite array S parameters were computed from the infinite array results using Fourier series expansion.

[41] The results for the infinite array and the finite-by-infinite array were compared and it was shown that they agree well for elements that are not at an edge or next to it. The largest differences between the two models are found for elements were at least one element is close to an edge. To compensate for this perturbation the S parameters from finite-by-infinite array results were combined with the infinite array results. The combined results are used to give an approximation of the S parameters for larger arrays by using the finite-by-infinite array results for the interactions with the elements closest to the edge and the infinite array results for other elements.

[42] An interesting observation regarding this tapered-slot element is that the match is improved compared to the infinite array for the edge element in the H plane edge. However, the match for the E plane edge is worse for the edge elements. Therefore, the performance of an E plane edge would benefit from introducing dummy elements to a greater degree than for an H plane edge.

[43] The active reflection coefficient were computed as a function of several parameters such as, phase shift, frequency, and element position for the finite-by-infinite array, the finite excited infinite array, and by using a method that combined the infinite and finite results. The active reflection coefficient computed using these methods share the same behavior, and the edge effects can be modeled as a perturbation of the infinite array result. This perturbation was shown to mainly depend on the missing contributions from the absent or nonexcited elements. A simple model that predicts the perturbation period as function of frequency, element position and phase shift was presented. The second-order discrepancy is caused by the geometry around the edge element, which is different and hence modifies the S parameters.

Acknowledgments

[44] The authors would like to thank Henrik Holter, Lars Pettersson, and Torleif Martin for providing the FDTD codes used in the paper.

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