The extraction uses special asymptotes closely connected to the actual Green's functions. These asymptotes can be used both for single elements and periodic.
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Modeling of Infinite Periodic Arrays with Dielectric Volumes and Quasi-3D Oriented Conductors V. Volski and G. A. E. Vandenbosch ESAT-TELEMIC, Katholieke Universiteit Leuven, Belgium
Abstract— A new asymptotic extraction technique is used within the Method of Moments to model periodic quasi-3D arrays consisting of complex elements including vertical/horizontal conductors and dielectric volumes. The extraction uses special asymptotes closely connected to the actual Green’s functions. These asymptotes can be used both for single elements and periodic arrays of elements. This step facilitates the simultaneous development of new features for both topologies.
1. INTRODUCTION
The modeling of periodic arrays composed of complex elements including horizontal/vertical conductors and dielectric volumes is of interest in different applications including antenna design, frequency selective surfaces and so on. The design of such arrays in many cases is facilitated if the modeling of a single element is also available. These two topologies can be very useful to estimate the mutual coupling in arrays. Nowadays there are several software tools available based on different techniques that can be used to treat these two topologies. Some of the software tools are better suited to model single elements, while others are better suited to model antennas enclosed in an environment with periodic or waveguide boundary conditions. The switch between different software tools is not always straightforward because the input/output formats are very often incompatible. Thus, it is of great interest to have a single code that is able to treat efficiently these topologies using the same approach in both cases. The MoM method in combination with Mixed Potential Integral Equations (MPIE) has proven to be very efficient in the modeling of conductors in layered structures [1]. This approach is based on the construction of spatial Green’s functions (GFs). This method can be used for a single element and a periodic array of elements. The GFs construction for a single element and for a periodic array encounters different types of problems despite that in both cases it is based on the Inverse Fourier Transform (IFT) of the same spectral GFs, which are known in a closed form. The IFT is expressed in terms of double integrals for a single element or in terms of double series for a periodic array. The convergence of the spectral GFs is normally insufficient for a direct numerical evaluation. In the case of a single element, moreover, a special precaution should be taken to deal correctly with branch cuts and poles. In the case of a periodic array, these problems are avoided because the needed spectrum is discrete. This last difference contributes enormously to the fact that the construction of the GFs is typically treated separately for a single element and a periodic array resulting in different algorithms and software codes. As a consequence available software tools are normally better suited for a single element or a periodic array. The aim of this paper is to report an efficient algorithm and its implementation within a single code, with only minor differences between a single element and a periodic array. This technique was consecutively applied to model periodic arrays only with horizontal conductors, and then with a combination of vertical and horizontal conductors. The application of this technique was shown at EUCAP 06 for horizontal conductors and at EUCAP07 for horizontal and vertical conductors. In this paper, dielectric volumes are included. The combination of vertical and horizontal conductors with dielectric volumes widens considerably the variety of topologies that can be modeled. 2. THEORY
The MPIE in the MoM formulation requires the calculation of the GFs in the spatial domain for different source types (electric horizontal and vertical conductors and volumes). The algorithm to calculate them is quite cumbersome and its implementation needs a lot of effort. The approach chosen in this paper relies on a strong resemblance between the two algorithms to calculate GFs for a single source and a periodic array of sources. The original algorithm for a single source is slightly adjusted in such a way that the periodicity of the elements can be implemented very easily.
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In practice, this means that major steps like the calculation of GFs in the spectral domain can remain almost the same and only a few additional procedures are required. The main advantage of this approach is that it is a modular analysis technique, which allows to develop and to test new features in the global modeling scheme simultaneously for a single element and a periodic array of elements. It is important to mention that the MPIE formulation is widely used to model periodic arrays consisting of elements composed of horizontal conductors only. The reason for this is that the number of required GFs is small in this case and there are several efficient approaches for their calculation available [2]. In the case of 3D conductors in the layered medium, the number of GFs starts to depend on the chosen approach. In our approach, the number of GFs remains relatively small due to a partial implementation of the current dependency in the GFs. All current components on the vertical conductors are expressed in terms of rooftop basis functions and these vertical rooftop basis functions are considered as elementary sources for the GFs components. However, these more complex sources do not alter our general procedure. The GFs in the spatial domain for periodic arrays can be expressed in the following form [2] q XX ˜ ij (βmn ) e−j(kmx x+kmy y) , β = k 2 + k 2 Gij (x, y) = (1) G mx my kmx kny
˜ ij (βmn ) is the spectral GF. In general, this series has a poor convergence. Using a sowhere G called Kummer’s transformation, the poor convergence of the GF in (1) is improved by subtracting specially selected asymptotes. Gij (x, y) = Gspectral (x, y) + Gspatial (x, y)) ij ij i h X X ˜ ij (βmn ) − G ˜ as (βmn ) e−j(kmx x+kmy y) Gspectral G (x, y) = ij ij m
Gspatial (x, y) = ij
n
XX m
as gij (rmn ) =
n
XX m
(2)
˜ as (βmn ) e−j(kmx x+kmy y) G ij
n
In order to construct an efficient algorithm, this double series in the spatial domain should also have a good convergence. These conditions can be satisfied by selecting the following asymptotes: ¡ ¢m 1 − e−βt e−β∆ as ˜ Gij (β, t) = (3) βm The asymptote in (3) has the necessary leading term for large values of β, that annihilates the ˜ ij (β), and it has no singularity at β = 0. The Fourier transform of (3) is known leading term of G in a closed form. 1
g as (r) p
ρ+∆2
1
−q ρ2
2
,
for m = 1 and
+ (∆+t) µq ¶2 2 ρ2 + (∆+t) + ∆ + t as µ ¶ g (x, y, z) = ln ³p , ´ q 2 2 2 2 ρ +∆ + ∆ ρ + (∆+2t) + ∆ + 2t
(4a)
for m = 2.
(4b)
Asymptotes for different sources (horizontal and vertical conductors and volume) can be constructed using these basic asymptotes. Each asymptote can be considered as a static combined source of 2 anti-phased sources separated in the z-direction by a small distance t. These asymptotes are relatively simple and it is possible to perform the integration over the source and observation domains if it is necessary. Although the convergence of the series is improved with the Kummer’s transformation, the efficiency can be increased further using special acceleration routines. Acceleration algorithms can reduce the required number of terms in series. Series with oscillating terms in the spectral domain are accelerated using Shank’s transform [3]. Series in the spatial domain are accelerated using the rho-algorithm [4], a technique more suited for monotone behavior. The general idea works
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very well. However there are some practical problems that maybe require particular attention. The convergence of series in the spectral and in the spatial domains depends on many factors like the periodicity of the cells, the chosen parameter t. Only for GFs associated with horizontal conductors, this problem is well investigated [2]. Although the main guidelines remain valid for vertical conductors, the situation is more complex because these GFs are expressed in terms of more complex functions making the definition of the spectral threshold above which the leading asymptotic term becomes dominant more complex. The convergence of the series and the choice of the parameters are of interest in ongoing research. For instance, the minimal number of terms that has to be actually calculated in a series is about 10 by 10 in the spectral and spatial domains for accuracy normally required in practical applications. For structures containing only horizontal conductors this gives normally a good approximation. If the period of an array is decreasing, then the convergence in the spectral domain is improving and the convergence in the spatial domain is decreasing. This can be partially corrected by adjusting the parameter t in (3) and (4). 3. NUMERICAL RESULTS
As examples, we consider 2 structures consisting of horizontal, vertical conductors and volumes. The addition of volumes widens the range of structures that can be analyzed enormously. The resonant frequency of an antenna depends not only on the antenna shape but also on the antenna environment. The presence of dielectric layers close to the antenna can change very noticeably its properties. The thickness and permittivity of layers used in the structure have to be chosen from a list of available materials. This rather limited choice can be widened using partial dielectric filling, when part of the dielectric is removed or added. The properties of the obtained structures can be described using some effective permittivities whose values depend on the permittivity and the volume ratio between different inclusions. The effective permittivity can be estimated using effective medium theory [5]. However this theory provides a very good approximation when the size of the inclusions is small in terms of wavelengths. In the other case a full wave modeling is preferable. The examples considered further in the paper demonstrate the efficiency of tuning using partial dielectric volume filling. The first example is an approximation of a magnetic conductor. This structure is inspired by the mushroom cell considered in [6]. A periodic set of patches is mounted on a ground plane. Each patch is fixed on a square stem that is connected also to the ground plane. The period of the array is 2.4 mm, the patch size is 2.25 mm and the thickness is 1.6 mm. Then a square dielectric gasket (dielectric permittivity 2.2 and thickness 1.6 mm) is placed under the patch. Changing the gasket size varies the dielectric filling from 0 mm (no dielectric) to 2.4 mm (full filling). The topology is shown in Fig. 1. The structure is excited by a plane wave. The calculated reflection coefficient is plotted in Fig. 1 for different volume fillings. In contrast to a typical ground plane where the reflection coefficient phase is about 180 degrees, the reflection coefficient phase of the mushroom structure is about zero in a certain frequency band. That corresponds to the behavior of a magnetic conductor. As expected the filling variation allows to tune the frequency very efficiently. The next example is an array of crosses. The period of the array is 10 mm. The cross length L 180
top view
2.4 mm 2.25 mm 1.5 mm 0.75 mm 0 mm
Phase, degrees
90
side view
0
-90
-180 10
15
20 Frequency, GHz
25
Figure 1: Modeling of a magnetic conductor.
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1 0.9
s
0.8
Transmission
0.7 0.6 0.5 0.4 0.3 0.2
L
0.1 10 mm
d
0
10
~3.4 mm ~6.9 mm
15
0 mm
20
25
GHz
Figure 2: Modeling of the array of crosses.
is 6.875 mm and its width is 0.625 mm. This topology is inspired by [7]. As in the previous case, a square dielectric gasket is placed under the cross. The gasket dielectric permittivity is 4 and its thickness is 0.3 mm. The structure is excited again by a plane wave from the top. The calculated transmission is plotted for different dielectric fillings in Fig. 2. This structure behaves like a very good reflector in some frequency band. This band can be tuned using partial dielectric filling. 4. CONCLUSION
In this paper, it is demonstrated that an asymptote extraction technique using the same asymptotes can be used to model a single element and a periodic array of elements in planar layered media. Each element can be composed from horizontal/vertical conductors and dielectric volumes. Our progress in the implementation of dielectric volumes is reported. The combination of different components (horizontal and vertical conductors with dielectric volumes) allows to consider very complex antenna topologies. Several examples illustrate the possibility of our approach. REFERENCES
1. Peterson, A. F., S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics, IEEE Press, 1998. 2. Valerio, G., P. Baccarelli, P. Burghignoli, and A. Galli, “Comparative analysis of acceleration techniques for 2D and 3D Green’s functions in periodic structures along one and two directions,” IEEE Trans. Antennas Propagation, Vol. 55, No. 6, 1630–1643, 2007. 3. Shanks, D., “Non-linear transformations of divergent and slowly convergent sequences,” J. Math. Phys., Vol. 34, 1–42, 1955. 4. Singh, S. and R. Singh, “On the use of rho-algorithm in series acceleration,” IEEE Antennas Propagation, Vol. 39, No. 10, 1514–1516, 1991. 5. Collin, R., Field Theory of Guided Waves, IEEE Press, 1990. 6. Sievenpiper, D., L. J. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “Highimpedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Techniques, Vol. 47, No. 11, 2059–2074, 1999. 7. Cwik, T. and R. Mittra, “The cascade connection of planar periodic surfaces and lossy dielectric layers to form an arbitrary periodic screen,” IEEE Trans. Antennas Propagation, Vol. 35, No. 12, 1397–1405, 1987.
