Radio Science

Efficient full-wave characterization of arrays of antennas embedded in finite dielectric volumes

Authors


Abstract

[1] This paper shows how the Macro Basis Functions (MBFs) approach can be applied to the solution of arrays of complex elements, made of metallic parts embedded in disconnected dielectric volumes. It is shown how the reduced system of equations can be obtained very efficiently with the help of a multipole approach. We also provide a subsequent formulation for the array impedance matrix and a fast method for the computation of embedded element patterns, with the help of a decomposition into a finite series of pattern multiplication problems, for which the fast Fourier transform can be exploited. Examples are provided for arrays of tapered slot antennas in separate dielectric boards. Comparisons are provided with brute-force solutions for a 4 × 4 array and increasing number of MBFs, and with the infinite-array solution for a 16 × 16 array. An appendix provides new insight into the problem of completeness of the MBFs bases by establishing a link between the MBF approach and Krylov subspaces.

1. Introduction

[2] The numerical simulation of the electromagnetic behavior of large finite antenna arrays remains a challenge [Holter et al., 2000; Lee et al., 2005], especially when the elements of the array have complex shapes and involve finite dielectric parts. This is also true for the analysis of periodic bandgap materials [Crocco et al., 2007] or metamaterials [Shelby et al., 2001]. For this type of structures, methods based on Finite Elements, or on a combination of Finite Elements and Integral Equations [see, e.g., Vouvakis et al., 2006] are well suited. For large structures, as long as the materials involved are piecewise homogeneous, the method of moments (MOM), together with its fast extensions, remains an excellent candidate, mainly because unknowns are limited to the interfaces between homogeneous regions and because of its ability to accurately handle dielectric materials with high contrast.

[3] Over the last few years, noniterative solution techniques have been increasingly investigated. Such methods are known under different names, like the Macro Basis Functions approach (MBF) [Suter and Mosig, 2000], or the Characteristic Basis Functions approach (CBF) [Prakash and Mittra, 2003], Aggregate Basis Functions [Matekovits et al., 2001], or Standard distributions [Craeye et al., 2004]. In each case, slight variants appear in the way to determine such functions. In the following, we will use the name “Macro Basis Functions” (MBFs), while the functions used to test the fields will be named “Macro Testing Functions” (MTFs). MBFs are generally obtained through the solution of smaller problems; for instance very small arrays, or a single element under a large number of excitations. In a first class of methods [Crocco et al., 2007; Matekovits et al., 2007], MBFs are obtained from excitation by a number of auxiliary sources distributed over an equivalence surface and then selected based on the level of their linear independence, with the help of a SVD procedure. In another class of methods, a “primary” MBF is obtained by direct excitation of a given antenna, while “secondary” MBFs can be obtained by solving for the field on neighboring elements, excited by the fields radiated by the primary solution [Yeo et al., 2003]. Infinite-array solutions can also be exploited as MBFs, as explained by Craeye and Sarkis [2008], where the Array Scanning Method [Munk and Burrell, 1979] is exploited to select the (real and imaginary) scan angles to be considered for the infinite-array solutions.

[4] The two main contributions of this paper are the efficient computation of interactions between MBFs and MTFs and the fast evaluation of embedded element patterns. The interactions will be obtained with a complexity proportional to N, or even less for elements with complex shapes, where N is the number of elementary basis functions on a single antenna within the array. This method can be regarded as an extension of the approach presented by Craeye [2006] to unit cells containing dielectric material. Further time saving is also obtained by eliminating redundant terms in the formulation proposed by Craeye [2006] for metallic antennas. Time savings obtained by combinations of MBFs and multipoles can also be found in the work of Lu et al. [2007], where scattering by metallic periodic structures is considered in an iterative scheme. Using the MBF approach, the number of unknowns can be reduced by about one to two orders of magnitude. This also enables the fast computation of the array impedance matrix, which is obtained, as will be shown below, by a further reduction of the reduced system of equations.

[5] Other important characteristics of finite arrays are the “embedded element patterns”, which correspond to the radiation pattern obtained when one element is excited, while the other elements of the array are passively terminated. As a result of array truncation, these patterns differ between all elements of the array. Even if current distributions can be rapidly obtained for all possible excitations at individual elements, the computation of embedded element patterns in all directions can be very time consuming and it can become dominant in the overall computation time. We will show that, thanks to the MBF approach, the embedded element patterns, as well as the patterns for any excitation law, can be obtained as a finite series of pattern multiplications, which allows the exploitation of the fast Fourier transform in the pattern computation. Some of the results presented in this paper have already been shown in recent conferences [Craeye, 2007; Craeye and Dardenne, 2007; Craeye et al., 2007], generally with focus on some subtopics and without mathematical proof of formulas, because of limited available space. The present paper provides full details of mathematical proofs, along with additional examples for larger arrays. Comparisons with brute-force solutions for a 4 × 4 array will also show the convergence of solutions when the number of MBFs is increased. The MBFs are constructed with the help of primary, secondary and higher-order scattering products; the relationship between these MBF and Krylov subspaces will be briefly highlighted in Appendix A.

[6] This paper is organized as follows. In section 2, the MBF approach is recalled and comments are made on the approach adopted for the determination of the set of MBFs. The method is then particularized to arrays of complex elements involving finite dielectric material. A formulation for the subsequent computation of the array impedance matrix is provided in section 3. Section 4 explains how the computation of the MoM impedance matrix can be accelerated with the help of the Multipole approach in the case of dielectric volumes containing metallic surfaces. Fast pattern evaluation is detailed in section 5, and numerical examples for the case of complex elements are provided in section 6. Finally, comments, concluding remarks and further prospects are provided in section 7.

2. MBFs for Complex Elements

[7] Let us consider a planar array made of M = MxMy identical antennas; each antenna is made of a metallic surface located inside a dielectric volume. The antenna surface, as well as the air-dielectric interface, are discretized with the help of elementary basis functions, of Rao-Wilton-Glisson [Rao et al., 1982] and rooftop types, for instance. The unknowns are, for each antenna m, the coefficients that multiply elementary decomposition functions in the solutions of electric surface currents on the metallic part (index f) and of equivalent electric and magnetic surface currents on the dielectric-air interface Sd (index s). The currents on a unit cell are sketched in Figure 1. The coefficients are represented by the following column vector:

equation image

where jm,f, jm,s and mm,s are vectors of coefficients describing solutions with respect to elementary basis functions. They correspond to electric currents on the metal, equivalent electric currents on Sd and equivalent magnetic currents on Sd, respectively.

Figure 1.

General representation of unit cell. Arrows represent electric current on metallic plate and equivalent electric and magnetic currents on air-dielectric interface.

[8] At the element level, the problem will be solved by imposing zero tangential electric fields on the metallic surface, as well as continuity of tangential electric and magnetic fields across Sd [Guissard, 2003; Djordjevic and Notaros, 2004; Yla-Oijala et al., 2005]. At the array level, the resulting MoM “impedance” matrix contains blocks standing for interactions between pairs of antennas. (Not all blocks have units of impedance.) For interactions between antennas m1 and m2, such a block can be written as:

equation image

where index f or s indicates whether the basis or testing function is on the metallic plate or on the dielectric-air interface Sd. The exponents j and m refer to the type of source (electric or magnetic), while the exponents e and h stand for the type of field under test. It is important to note that, when m1m2 (nondiagonal block), there is no direct interaction between the metallic plate and the outer world, which means that, in this case, only the subblocks with index ss are nonzero.

[9] MBFs are obtained by solving small problems; in the examples given below, they correspond to excitation of an isolated antenna (primary MBF) and excitation of neighboring antennas by the primary MBF (secondary MBFs [Yeo et al., 2003; Wan et al., 2005]). The secondary MBF can be written as Zmm−1Zmnx1, where x1 describes the primary MBF and Zmn is a block of the MoM impedance matrix corresponding to basis functions located on antenna n and testing functions located on antenna m. When mutual coupling is very strong, for instance when there is little spacing between adjacent antennas, MBFs resulting from a higher-order multiple-scattering process may need to be considered. In this case, MBFs of order Q + 1 can be written as products of the type:

equation image

with nq = mq−1 for q > 1 and mqnq. As briefly outlined in Appendix A, the Krylov subspace [Greenbaum, 1997], which has finite dimension and contains the exact solution of the global discretized problem, can be constructed from solutions of type (3). One may wonder, however, about the number of such solutions to involve for obtaining errors below a given threshold. In the following, we limited second-order MBFs to interactions between adjacent elements (8 elements around a given antenna) and third-order MBFs to interactions between pairs of elements separated by less than a tenth of the wavelength. In the examples given in section 6, the improvement of the solution will be illustrated versus the number of MBFs considered. This multiple-scattering approach, which takes benefit from the geometry of the array, is entirely based on interactions between elements and hence avoids the introduction of fictitious sources [Crocco et al., 2007], whose placement around 3-D elements needs to be carried out with caution, especially if the separation between elements is very small (smallest spacing of 0.03 wavelengths in the examples below).

[10] A limited number P of MBFs is constructed in this way. The coefficients of the P MBFs with respect to elementary decomposition functions, given as column vectors xp°, are grouped in a matrix:

equation image

For a given excitation, the solution on any given cell is then written as a superposition of these solutions, with coefficients Cm,p to be determined:

equation image

[11] As for the testing of fields, aggregate functions will be used as well. These “Macro Testing Functions” (MTFs) will correspond here to the set of MBFs (Galerkin testing procedure). Since the MTFs are made of both electric and magnetic current distributions, several choices are possible for the testing procedure. A possible extension of the testing procedure used for MBFs applied to metallic structures consists of computing the following interactions:

equation image

where equation imaget is the equivalent magnetic current associated with the MTF, while equation images is the magnetic field scattered by the MBF. The surface S covers both the metallic surface and the air-dielectric interface. Similarly, equation imaget is the equivalent electric current associated with the MTF, while equation images is the electric field scattered by the MBF. In its discretized form, this interaction can be computed with the help of the MoM impedance blocks as follows:

equation image

where j and m are the coefficients of electric and magnetic equivalent currents on the surfaces, in terms of RWG decompositions functions. Indices t and b refer to testing and basis functions, respectively. In section 4, the four contributions to the interaction between a macro basis function and a macro testing function will be denoted by:

equation image

[12] The interactions between all MBFs associated with antennas m1 and m2 can be written in compact form as [Yeo et al., 2003]:

equation image

where exponent H stands for transposed conjugate. As will be explained in section 4, if m1 and m2 are sufficiently far away from each other (typically separated by two cells), a much faster formulation can be used to compute the reduced block equation image of the array impedance matrix.

[13] The MoM excitation vector v is reduced in a similar way. The segment corresponding to a given antenna j is reduced as vjr = QHvj. The reduced system of equations then reads:

equation image

where c is a column vector in which the vectors of unknowns with respect to all MBFs on successive antennas, cm, are concatenated.

3. Array Impedance Matrix

[14] The impedance matrix of the antenna array, seen as an M port, can be obtained from the reduced MoM impedance matrix Zr. Let us denote the reduced MoM admittance matrix by Yr. The latter can also be subdivided into M × M blocks (M is the number of antennas) of size P × P (P is the number of MBFs). The block Yi,jr, stands for interaction between antennas j and i in the array. From there, the array admittance matrix Y can be obtained through the following reduction:

equation image

where xs is the sth line of matrix Q, and s is the index of the basis function supporting the delta-gap voltage source, while W is the width of that basis function, taken normal to current flow.

[15] Result (11) can be proven as follows. Y(i,j) corresponds to the current obtained at the feeding point of antenna i when antenna j is excited with a unit voltage source, while the other antennas of the array are short-circuited. In that configuration, the currents on antenna i, in terms of coefficients of elementary basis functions, can be written as:

equation image

Y(i,j) is then the port current, obtained by keeping only the sth line of xi, i.e., by replacing Q by xs in (12), while multiplying by the width W. Besides this, since antenna j is excited only through a delta-gap source with unit voltage, overlapped by testing function s only, the corresponding segment of the excitation vector vj is a column vector with only one nonzero entry, with value −W; hence, we have vjr = QHvj = −WxsH. These two comments explain how (12) leads to (11).

[16] For the determination of the embedded element patterns (see section 5), it may be convenient to include in the computation the series impedance ZL attached to each antenna (generator impedance on transmit, input impedance of amplifier on receive). In that case, a term W2ZL is subtracted from entry Zffej(s,s) of (2) (see Craeye et al. [2004] for details). In that case also, the admittance matrix Y is not exactly the array admittance matrix, but it corresponds to (Z + ZLU)−1, where U is a M × M unit matrix and Z is the array impedance matrix.

4. Fast Matrix Reduction

[17] When based on formula (7), the reduction of the original MoM impedance matrix is particularly slow, mainly because all different blocks (for different spacings between cells) of the original matrix need to be computed. If Ms is the number of unknowns on the air-dielectric interface of a given unit cell, the computational complexity is proportional to Ms2, with a quite large proportionality constant. A multipole approach allows us to perform these computations much faster for MBFs located on cells located not too close to each other (typically, for distances larger than two array spacings). This will be explained below in a few steps.

[18] The first step consists of recalling the plane wave decomposition of the free-space scalar Green's function [Coifman et al., 1993]. Let us denote by equation image an observation point on antenna i, and by equation image a source point on antenna j, and we assume that equation image is close to equation image, the vector starting at the reference point on antenna j and ending on the corresponding reference point on antenna i. Then,

equation image

where equation imagei and equation imagej are positions of observation and source points with respect to reference points on corresponding antennas. With these definitions, we can approximate the scalar Green's function as:

equation image

where k is the free-space wave number (k2 = ω2equation imageμ, with equation image and μ the permittivity and permeability of the surrounding medium) and the integration is carried out over the domain of directions equation image pointing on the unit sphere U. Function T corresponds to the multipole translation function:

equation image

where hl(2) is the spherical Hankel function of second kind, Pl is the Legendre polynomial of order l, and equation image = requation image. In the examples shown in section 6, L could be kept to a relatively small value, L = 14.

[19] The second step consists of writing the interactions between macro basis and macro testing functions located on different antennas, in the same way as in the case of elementary basis and testing functions [Yla-Oijala et al., 2005]:

equation image
equation image

where equation imageb and equation imaget denote the macro basis and macro testing functions limited to the air-dielectric interface Sd. To write (16), we assumed that these distributions do not involve currents perpendicular to the edge of the domain on which they are defined, which is true if they are made of aggregations of RWG or rooftop functions. Besides this, we also have:

equation image

[20] The third step consists of replacing Q and ∇′Q in (16) and (17) by their expressions found from the multipole decomposition (14). The integral over the unit sphere in (14) is then moved outside the integrals over surfaces S and S′. The latter integrals can then be written as Fourier transforms of the macro basis and testing functions and of their divergences. After a few transformations, the result can be written as:

equation image

where Pe and Ph are specific products of patterns of macro basis (index b) and macro testing (index t) functions and of their divergences:

equation image
equation image

where equation image is the pattern of the conjugate of the part of the macro (basis or testing) function corresponding to equivalent electric surface currents, limited to Sd:

equation image

and equation imagen is a scalar quantity obtained from the same transform applied to the divergence of the macro function. These patterns are obtained in a straightforward way from the patterns of the elementary basis functions and from the vectors of coefficients js. equation image and equation image are obtained in a similar way from the equivalent magnetic part of the macro function, which is described by the vector of coefficients ms.

[21] Finally, the equation above can be simplified with the help of the following relationship:

equation image

where we again exploited the fact that MBFs have zero currents normal to the boundaries of the domain on which they are defined. This leads to:

equation image

and to a similar expression for interactions that involve patterns of magnetic parts of the macro functions. The exponent u refers to the component along direction equation image. The identity (24) leads to a simplification of (20) and (21), where patterns of divergences and components along equation image have disappeared. Those components can also be omitted for the terms in which a cross product with equation image is taken. Finally, the products of patterns appearing in (19), can be computed as:

equation image
equation image

where the ‘o’ exponents remind us that only components of fields orthogonal to the directions of observation need to be taken into account. In other words, the interactions can be computed with the help of only the far field patterns of MBFs, where contributions of electric and magnetic fields are calculated separately.

5. Embedded Element Patterns

[22] Besides the array impedance matrix, the reduction of the MoM impedance matrix with the method presented above allows a fast computation of the array fields (equivalent electric and magnetic currents on the metallic plate and on interface Sd) for any excitation law. It is well known that, in case of mutual coupling, the pattern multiplication procedure cannot be applied. Indeed, the current distributions on different antennas cannot be considered proportional to each other. However, under the MBF approximation, the current distributions are described as a superposition of a few predetermined distributions. This means that, provided that the contributions of different MBFs are treated separately, the pattern multiplication principle can be used again.

[23] This procedure may be applied for any excitation law for the array. A way to obtain rapidly the radiation pattern for all possible excitations consists of computing and storing all embedded element patterns, obtained for excitation of one element at a time, while the other elements are passively terminated. The excited element is fed by a unit voltage source, in series with the generator impedance. The loads terminating other elements also correspond to the generator impedances.

[24] Writing the embedded element pattern equation image as a sum of contributions of different MBFs, for each of them, the pattern multiplication is used:

equation image

where equation imagep(equation image) is the element pattern of the equivalent current distribution described by the pth macro basis function, computed with the help of the equivalent currents js and ms on Sd only. equation imagep can be obtained very fast from side products of the MBF + Multipole computations. Indeed, it can be written as:

equation image

where equation imagex is the polarization vector and the “primes” (′) sign indicates that the patterns have been computed from the current distributions themselves, not from their conjugate value, as was done in (22). Nevertheless, the patterns of elementary basis functions, required to compute (22), can be used in parallel to evaluate the pattern (28).

[25] The factor Ap in (27) corresponds to the array factor resulting from the array excited with the coefficients associated with the pth MBF:

equation image
equation image
equation image

where ux and uy are the horizontal projections of the unit vector equation image which indicates the direction of observation:

equation image
equation image

with MxMy the dimension of the two-dimensional inverse discrete Fourier transform, carried out with low complexity with the help of the FFT algorithm, and r ∈ [0, 1, …, Mx], s ∈ [0, 1, …, My], while a and b are the array spacings along x and y.

[26] A finer sampling in the domain of directions ux and uy is easily obtained by artificially augmenting the size N of the array, through a zero-padding procedure. From (30), it can be seen that the solution will be periodic in the (ux, uy) plane with periods (2π/ka, 2π/kb) = (λ/a, λ/b), respectively. The values of ux and uy, as given by (32) and (33), are limited to ux ∈ [−1/2 λ/a, 1/2 λ/a] and uy ∈ [−1/2 λ/b, 1/2 λ/b]. This means that, when the array spacing, a or b, is larger than half a wavelength, the values of ux and uy do not cover the whole visible space, defined by ux2 + uy2 < 1. The pattern in the remaining part of the visible space is then obtained with the help of the periodicity referred to above, leading to the onset of grating lobes.

6. Numerical Examples

[27] In this section, examples will be shown for an array of tapered-slot antennas [Shin and Schaubert, 1999], made of a metallic fin embedded in a dielectric slab of relative permittivity equal to 4. In the examples below, the slabs enclosing the antennas will be disconnected. However, the results obtained by Craeye and Sarkis [2008] for connected arrays with the help of infinite-array solutions for the case of purely metallic materials allow us to expect the MBF approach to be extensible to connected dielectric structures; such approaches may benefit from infinite-array solution methods as the one described by Dardenne and Craeye [2008]. The width of the metallic part of the antenna is 12.7 cm, its height is 20.8 cm (that metallic part is similar to the one considered for simulations by Schaubert et al. [2003], with a 12% smaller height). The slab has a thickness of 0.5 cm and exceeds the antenna dimensions by 0.2 cm in the two other dimensions. The mesh, made of RWG [Rao et al., 1982], rooftop and hybrid basis functions, is shown in Figure 2. The array structure is shown in Figure 3. The array spacing is 14 cm in both directions, and the antennas are fed by a delta-gap source located just above the cavity. The free-space wavelength of operation is 30 cm. A first validation test consists of verifying that the power Pin fed to the impedance of an isolated antenna corresponds to the total power Prad radiated in the fields. The antenna impedance is 40.9 − j 11.7 Ω. The relative error of Prad with respect to Pin is 1.15%, which may be considered as sufficiently low, in view of the relatively coarse meshing of the structure (about 8 rooftop basis functions per wavelength in the dielectric medium).

Figure 2.

Mesh considered for the metallic part of the antenna (thick lines), and mesh of the slab in which it is inserted (thin lines).

Figure 3.

Geometry of the array of tapered slot antennas. Array spacings: a = b = 14 cm.

[28] The first array computations have been carried out with the help of one primary and eight secondary MBFs only. Another validation point consisted of comparing the active input impedances with a brute-force solution for a 4 × 4 array (Figure 3), considering constant excitation with unit voltage sources with 100 Ω series impedances. The result is shown in Figure 4. (This comparison was already shown by Craeye et al. [2007], where an uncertainty remained, however, because of incomplete convergence of the brute-force solution.) It can be seen that the correspondence is very good, since the impedance variations due to array truncation, of the order of 10 Ohm, are represented within a maximum error of the order of 2 Ohm, by the MBF-based solution, with 9 MBFs only. The largest part of the solution time consists of estimating the MBFs, requiring each of the order of 1 min on a 1.6 GHz laptop computer. The multipole formulation has been used for elements separated by at least two cells. In view of the wavelength, and according to Hastriter et al. [2003], in (15), the maximum index of the sum could be kept to L = 14, which provides at least 3 digits of accuracy. This level of accuracy is sufficient because the interactions looked for are related to pairs of antennas that are not very close to each other.

Figure 4.

Active input impedances from brute-force approach and from MBF approach for a 4 × 4 array for uniform excitation. Wavelength: 30 cm. Nine MBFs used.

[29] We analyzed the improvement of the accuracy of the solution when the number of MBFs is increased. This has been done at the level of port currents and of embedded element patterns. Five sets of MBFs were considered: (1) one MBF, consisting of the primary MBF only (e.g., element 6 in Figure 3); (2) three MBFs, with two secondaries, closest to the primary (elements 5 and 7 in Figure 3); (3) five MBFs, with four secondaries (elements 2, 5, 7 and 10 in Figure 3); (4) nine MBFs, with eight secondaries (1 to 3, 5, 7 and 9 to 11 in Figure 3); (5) eleven MBFs, those from case 4 and two third-order (or tertiary) MBFs resulting from the current distributions excited on a given element by fields radiated by secondaries. Here, the third-order MBFs have been defined such that secondaries and third-order MBFs correspond to closest interactions (e.g., in Figure 3, secondary on element 5 with tertiary on element 6, and secondary on 7 with tertiary on 6).

[30] Figure 5 shows the magnitudes of port currents obtained in the 4 × 4 array with element 6 excited and the other elements passively terminated. The improvement from the case of 5 MBFs (case 3 above) to the case of 11 MBFs (case 5 above) is very clear. Results for the same choices of MBFs are shown in Figures 6 and 7for the E plane embedded element patterns for excitation at elements 1, 2, 5 and 6 in the 4 × 4 array of Figure 3. It can be seen that embedded element patterns exhibit very wide variations. These variations are already well represented when 5 MBFs are used, except in the sidelobes (|θ| > 50 deg.); the latter are much better represented when 11 MBFs are used (Figure 7). The behavior of the error versus number of unknowns is provided in Table 1. For the five cases referred to above, the r.m.s. error between patterns, as well as the r.m.s. error between port currents is provided. The first is normalized w.r.t. the maximum of the pattern and the latter is normalized w.r.t. to the value at the excited element. (The relatively high value of the port current at the excited element is the reason for the apparently lower errors for port currents.) It is expected that errors keep decreasing when the number of MBFs is increased, for instance by considering second-order MBFs extending over larger values of interelement spacing, or by considering more third-order MBFs. In such case, in order to maintain a well-conditioned system of equations, it may be useful to exclude some MBFs based on their too strong linear dependence with other MBFs, with the help of a SVD analysis [Matekovits et al., 2007; Crocco et al., 2007].

Figure 5.

Port currents in 4 × 4 array of Figure 3 with element 4 excited at 1 GHz. Brute-force solution, and solution obtained with 5 and 11 MBFs, and corresponding errors (magnitude of difference between complex quantities).

Figure 6.

E plane embedded element patterns in 4 × 4 array of Figure 3 with elements 1, 2, 5 and 6 excited successively. Exact solutions (dashed) and solutions obtained with 5 MBFs (solid). Wavelength: 30 cm.

Figure 7.

E plane embedded element patterns in 4 × 4 array of Figure 3 with elements 1, 2, 5 and 6 excited successively. Exact solutions (dashed) and solutions obtained with 11 MBFs (solid). Wavelength: 30 cm.

Table 1. Relative Errors in Terms of Patterns and Port Currents for Five Different Sets of MBFs
 Pattern ErrorPort Current Error
1 MBF0.1940.0495
3 MBFs0.1100.0324
5 MBFs0.0780.0251
9 MBFs0.0470.0175
11 MBFs0.0290.0055

[31] Simulations have been carried out for a 16 × 16 array under the same configuration as above, and with uniform excitation at 1 GHz. The set of 11 MBFs described above has been considered. Impedance results are compared in Figure 8 with the infinite-array active impedance, computed with results obtained with the method described by Dardenne and Craeye [2008]. Wide variations, due to array truncation, are visible. Near the middle of the array (middle of row 8), the finite-array result resembles the infinite-array solution. A slight offset, of the order of 2 Ohm, seems to appear between reactances in the middle and the infinite-array reactance. This comparison provides a validation point in the large-array limit for the method described here. As for the computation of the reduced MoM impedance matrix, the computation time needed for the 16 × 16 array exceeds by 1 min only the time needed for a 4 × 4 array; this additional time would be more than 1 hour when based on explicit computations of the blocks of the MoM impedance matrix. The computation of all embedded element patterns requires about 1 min.

Figure 8.

Active input (top) resistances and (bottom) reactances for the uniformly excited 16 × 16 array, with 100 Ohm termination, as well as infinite-array solutions (straight lines). Wavelength: 30 cm. Abscissa gives index along columns (linear arrays in the plane of antennas), numbers on plot give indices along rows.

7. Conclusion

[32] In this paper, a fast reduction of the MoM impedance matrix, based on the Macro Basis Functions approach, has been presented for the analysis of finite arrays of antennas made of metallic parts embedded in finite dielectric volumes. Therefore, the testing procedure has been extended by considering that the electric field radiated by the MBF is tested by the electric part of the MTF, while the magnetic field radiated by the MBF is tested by the magnetic part of the MTF. For interactions between MBFs and MTFs located on different antennas, only the equivalent currents on the dielectric-air interface need to be considered. However, even for quite complex antennas, the related reduction of complexity is marginal. Much faster interactions are obtained through a multipole approach, without aggregation nor disaggregation steps. For the computation of interactions between two MBFs, an approach based on the MoM impedance matrix would have a complexity proportional to Ms2, where Ms is the number of unknowns on the dielectric-air interface. The complexity of the multipole approach presented here is proportional to the number Ω of directions to be considered. For a given accuracy, this number is proportional to the square of the size of the antenna element in terms of wavelength, as can be delineated from Bucci and Franceschetti [1987, equation (33)]. If the element is very small but complex, Ω may be significantly smaller than Ms; hence the time saving is at least equal to Ms.

[33] The main characteristics of a given phased array are its impedance matrix and its embedded element patterns. We showed that the array impedance matrix can be rapidly delineated from the reduced MoM impedance matrix, while the embedded element patterns can be evaluated very fast. It is well known that, due to mutual coupling, the pattern multiplication approach cannot generally be applied. However, we showed that, within the MBF approximation, the embedded element patterns, as well as any array pattern, can be computed as a finite series of pattern multiplication problems, which then enables the exploitation of the FFT, as well as the recuperation of MBF patterns computed in the course of the multipole approach. Examples have been provided for the case of arrays of tapered slot antennas embedded in dielectric slabs; comparisons with brute-force solutions for a 4 × 4 array with an increasing number of MBFs showed us that, when MBFs are obtained in a multiple-scattering procedure, their number may be kept quite low.

[34] Although the completeness of the MBF approach is often questioned, several promising routes have been opened for a systematic choice of the MBFs. First, an approach for disconnected elements, based on the equivalence principle, has been proposed by Crocco et al. [2007]. Second, regarding multiple-scattering approaches, which are themselves an extension of the use of primary and secondary MBFs, a link has been established (Appendix A) with Krylov subspaces, which have proven a powerful tool in the framework of iterative methods. Finally, a systematic way of determining the MBFs, even for connected elements (so far, through conducting material) has been presented by Craeye and Sarkis [2008], where infinite-array solutions are exploited. Hence, we believe that, at least for the case of periodic structures, important prospects are still ahead for the improvement of MBFs-based approaches. Extensions in progress involve the fast analysis of irregular arrays, as well as structures with metallization located just on the air-dielectric interface and arrays of antennas connected through their supporting dielectric structure [Dardenne and Craeye, 2008].

Appendix A:: Krylov Subspaces and MBFs

[35] Assume two antennas, denoted by indices 1 and 2. A multiple-scattering (and generally nonconvergent) refinement of (equivalent) currents, represented here by vectors x1 and x2 can be written as a block Gauss-Seidel iterative process (see Greenbaum [1997, section 2.1] for a reminder on the Gauss-Seidel iteration). If the excitation vectors are v1 and v2, the first approximation can be written as:

equation image

where the Zij matrices are blocks of the MoM impedance matrix standing for basis functions on antenna j and testing functions on antenna i. Then, successive corrections of this simple iteration procedure are [Greenbaum, 1997, equation (2.1)]:

equation image

It is well known that this stationary process is not always converging. However, defining Z′ = M−1Z, the different corrections can be used to build a subspace, written as equation imagek(Z′, x1) = Span {x1, Zx1, Z2x1, …, Zk−1x1}, which is the Krylov subspace of dimension k for Z′ and x1. It is interesting to note that, if Z′ is not singular, the exact solution to the problem must lie in the Krylov subspace of dimension n, where n is the number of unknowns. This is true, even if the Jacobi or Gauss-Seidel iteration, whose successive corrections can be used to build the Krylov subspace, does not converge [see Greenbaum, 1997, sections 2.1 and 2.4]. In iterative methods like GMRES (see van der Vorst [2000] for a historical perspective), the residue at iteration k is orthogonal to subspace Zequation imagek(Z′, x1), and if Z′ is not singular, the iterative process in principle (i.e., assuming exact arithmetic) converges in at most n steps. In view of the preconditioning effect of the block-Gauss-Seidel operator M−1, such methods may converge below a given accuracy in very few steps, i.e., with a solution in a Krylov subspace of relatively small dimension. In the following, we prove that the Krylov subspace for two identical antennas, with element 1 excited, can be obtained from the MBFs as defined in section 2. The extension to larger arrays is then briefly mentioned.

[36] First, it is easy to verify that, with v2 = 0, x11 and x21 correspond to the primary and secondary MBFs. Then, based on the expression for M−1Z in terms of blocks of the MoM impedance matrix, a straightforward development leads to:

equation image

where X12 = Z11−1Z12 and X21 = Z11−1Z21. From there, by recurrence, we conclude that successive corrections up to order k, and hence the vectors subtending the Krylov subspace of dimension k, can be written in terms of products of the type:

equation image
equation image

with 0 < k′ < k and x21 = −X21x11. It is interesting to notice that t1k′ and t2k′ are nothing else than the MBFs of orders Q = 2k′ + 1 and Q = 2k′ + 2, respectively, as defined in section 2 (see equation (3)), when two elements only are considered. Consequently, the Krylov subspace of dimension k can be constructed with the exclusive help of the MBFs up to order 2k, computed with the multiple-scattering approach referred to in section 2. Since, as recalled above, the Krylov subspace of order n (total number of unknowns) must contain the exact solution, we know that it is possible to write the solution for both antennas as linear superpositions of MBFs, computed up to a finite order. This formally proves the completeness of MBFs obtained in a multiple-scattering way, although the strict upper bound for the number of required MBFs lies very high. In practice, in view of the preconditoning effect of the M−1 operator, very low-order Krylov subspaces, and hence very low-order MBFs, will suffice to achieve a given accuracy. Simulations have been carried out for the case of two antennas very close to each other (configured like antennas 1 and 2 in Figure 3), under the same conditions as for the examples in section 6. In this configuration, the antennas are separated by 0.03 wavelengths only. Results have been obtained for 2, 4, 6 and 8 MBFs, which can be used to describe Krylov subspaces of dimensions 1, 2, 3 and 4, respectively. The relative errors for port currents, computed in the same way as in section 6, are successively 1.3 10−2, 4.2 10−4, 6.1 10−5 and 1.8 10−6. We believe that an important reason for this fast convergence is that, through the multiple-scattering process exploited to determine MBFs, information about the configuration of the array (i.e., on how an element may be illuminated by its neighbors) is taken into account.

[37] For larger arrays, the Krylov subspaces can also be constructed with the help of MBFs obtained in a multiple-scattering approach. With increasing number of antennas, many different MBFs need to be defined for every order k of the Krylov subspace, because of the variety of antenna n-uples that can be involved in the multiple-scattering process. It is expected that higher-order MBFs are only necessary to achieve higher accuracies and should be considered in priority for pairs of antennas very close to each other. Besides this, in large arrays, for any given order, interactions between very distant antennas may be involved. In those cases, however, the resulting MBFs resemble those due to plane wave excitations. Only a few of the MBFs obtained from far-field interactions will be linearly independent within a given threshold. This set of MBFs could be determined with a SVD approach, as done in the work of Crocco et al. [2007] for the case of MBFs obtained from near-field dipole excitations. The number of such MBFs needed to achieve a given accuracy will mainly depend on the size of the elementary antenna. A more systematic analysis of the number of multiple-scattering MBFs needed is currently under investigation.

Acknowledgments

[38] The authors are grateful to the anonymous reviewers for their helpful comments and to Paul Van Dooren (UCL) for discussions about Krylov iteration methods. In memoriam: Albert Guissard (1934–2007), an inspiring researcher, a teacher of rigor and patience.

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