The perfect electromagnetic conductor medium (PEMC) is a recently introduced concept that generalizes the well-known PEC and PMC materials. Inside PEMC material, a linear combination of electric and magnetic field vanishes, their relation being the PEMC parameter. PEMC emerges as the most simple material, fully isotropic, from the four-dimensional formulation of electromagnetics.

A lossless PEMC does not allow electromagnetic waves or energy to penetrate into it. The effect of PEMC becomes visible as a boundary effect. The boundary condition between "ordinary" matter and PEMC is that a linear combination of the tangential components of electric and magnetic fields vanish. Using this relation, not much more complicated than the PEC boundary condition, many problems involving PEMC media can be solved. A very efficient way to solve such problems is a transformation method, which changes the complex materials to known ones. Fields are also transformed, but when the fields of the new problem have been solved, and backward transformation returns us the unknown fields of the original problem.

In this presentation, we review all the various structures and constellations involving PEMC media that we have analyzed. These include reflection from planar layers, waveguides and resonators, image theory, scattering and diffraction from small and large inclusions, material modelling of composites, losses; all these with PEMC boundaries involved. Also generlizations of the basic PEMC medium will be discussed.

Large finite periodic structures such as large antenna arrays or finite crystals can be analyzed by methods developed for infinite periodic structures using windowing techniques. An iterative scheme can be formulated, in which each step involves solving a problem of infinite periodicity, and subsequently filtering the result through a window function representing the finite periodic structure. This method converges unconditionally if the antenna array is large enough and no surface waves are present.

If surface waves are present, the estimates proving convergence break down, but an integral equation can be formulated which solves the entire finite problem in the subspace spanned by the surface waves only. The surface waves can be rigorously characterized by a singular value decomposition as the null space of Maxwell's equations for the infinite periodic case. These functions can be precomputed and reformulated as coefficients in an integral equation, which is (almost) a Fredholm integral equation of the second kind.

In this contribution, we demonstrate the resulting integral equation and the iterative scheme solving the finite periodic problem, and provide physical interpretations of the mathematical results. Ideally, the computational requirements of the method do not grow with the size of the periodic structure, and most computationally intensive parts can be precomputed from the infinite periodic problem. Thus, the method is well suited for evaluating numerous apertures for the finite periodic problem while keeping the individual elements and element spacing fixed.

The Finite-Difference Time-domain (FDTD) method proposed by Yee [2] is a popular approach to solve time-domain Maxwell equations because of its high flexibility and efficiency. However, its time step size is limited by the CFL stability condition. Some unconditionally stable methods like the 'one-step' method proposed by De Raedt et al in 2002 [1] have been developed. The efficiency of this approach was demonstrated by De Raedt when dealing with domains containing dielectrics without losses and for obtaining the solution at a given time t. In this paper, we propose to develop an iterative time scheme based on the one step method for applications related to bioelectromagnetic problems. This method, which is unconditionally stable, can be interesting for such problems where small details like heterogeneous tissues makes CFL condition as a limiting factor for the use of the FDTD method.

In the 'one-step' method, we consider the spatially discretized time-domain Maxwell equations represented in a matrix form, without current source, namely spatial terms are grouped in a matrix and the electric and magnetic fields are represented by a vector.
Then the 'one-step' method consists in solving the obtained ordinary differential equation by the Chebyshev polynomial expansion. The main contributions of this work are two folds.
Firstly, in order to help the introduction of losses, an iterative formula in conjunction with the implementation of hard sources is established. The results are compared with FDTD results for the same test cases. The results of this comparison are in good agreement, e.g. for a Gaussian pulse, the relative difference is less than 3% and for a sinusoidal pulse, it is less than 6%.
Secondly, the losses can be inserted if we stay in the ellipse of convergence of the Chebyshev polynomial extension. We show that with the iterative method we can use losses having a value of sigma smaller than 123 S/m. This is sufficient for the study of electromagnetic wave interactions with the human body. The PML conditions exposed in [2] are then introduced in the 'one-step' iterative method. The reflected signal amplitude is less than 9.10-4 for an incident signal of magnitude 1, a performance which is as good as that when using the FDTD method.

This method is not limited by the CFL condition. So the time-step can be increased in order to improve its efficiency. It is demonstrated that it is possible to increase the time step dto, without affecting the signal with dispersion effects, up to dto =3 dt depending on the presence of dielectrics (dt is the time step for the classical FDTD). Thus the iterative 'one-step' method is faster than the FDTD one.

References:

[1] H. De Raedt et al, 'One-Step finite-difference time-domain algorithm to solve the Maxwell equations', Phys. Rev. E 67, 056706, 2003.
[2] A. Taflove, 'Computational Electrodynamics - The Finite-Difference Time Domain Method', Artech House, Boston, 1995.

Recently, a technique called SM/AIM, based on a modification of the classical method of moments (MoM) approach, has been developed for the analysis of large patch arrays. The key feature of this method is to introduce a neighbourhood distance r_d separating two regions: a near-interaction region and a weak-interaction region. The MoM matrix is then divided into the sum of a strong- and a weak-matrix. The near-interaction region requires that several reaction integrals be evaluated and stored in the strong-matrix, which is a sparse matrix. The computational efficiency is then fully achieved when the Green's function can be approximated, in weak region, by using a convenient factorized form. This allows us to operate a canonical grid expansion of the exact weak-matrix elements into a series of translationally invariant terms times a function of the height differences between the basis and weighting functions. When an iterative solver is used, the matrix-vector product can be performed rapidly due to the sparse nature of the strong-matrix and the use of the fast Fourier transform (FFT) and its inverse for the weak-matrix multiplies.

It is worth noting that, in order to make the algorithm efficient, it is important to maintain as low as possible both the number of terms of the series expansion and rd The former condition allows us to reduce the number of FFTs and inverse FFTs involved in the solving process. If instead we use a small strong-interaction region extension, we keep down the number of non-zero elements of the strong-matrix (i.e., the dynamic memory needed to store it, or the computation time when its elements are recalculated on each conjugate gradient iteration), as well as the related products in the iterative solver algorithm. Frequently, this feature is essential for the technique to be used on a simple workstation.

For structures in free space the required form is simply obtained through a Taylor series expansion [A. Freni, et al., MMET*02, pp. 48-53]. In the case of multilayered structures, to maintain the accuracy without increasing the complexity, we need a closed form expression for the spatial domain Green's functions. In particular, we can use again a Taylor series expansion by approximating the Green's function using the Complex Image Method (DCIM) and applying the Sommerfeld identity [F. De Vita, et al., Jina 2004, pp.142-144].

Even though the above-mentioned procedure seems to work well with thin stratified media, it appears unstable when applied to structures with an increasing number of layers.

Here, we investigate a different technique based on a spectral domain approach, and we use an orthogonal function expansion to obtain an appropriate closed form for the Green's function while maintaining the accuracy and the efficiency of the SM/AIM method both in free space and in stratified media. Comparing to the DCIM approach, the spatial domain Green's function is now obtained through a spectral domain integral.This slightly increases the evaluation time required to perform the Green's function factorization, but the overall number of significant terms of the Green's function expansion, the computational complexity, is considerably reduced.

In recent years, a wide range fast methods have been developed for accelerating the iterative solution of the electromagnetic integral equations discretized by Method of Moments (MoM) [1]. Most of them are based on multilevel subdomain decomposition and require a computational cost of order NlogN or Nlog²N. One of these methods is the Multilevel Matrix Decomposition Algorithm (MLMDA), originally developed by Michielsen and Boag for 2-D TM_z scattering problems [2].

The authors of this presentation were attracted by the MLMDA for its simplicity, ease of implementation and, above all, because it can be programmed on top of any MoM code that uses subdomain basis functions much smaller than the wavelength, as is usually the case, and with most of the commonly used Green's functions. For that reason, the authors applied the MLMDA to 3D problems and found that it was particularly efficient and accurate for piece-wise planar objects [3]. The application of MLMDA to the analysis of printed antennas [4] lead to excellent results: for example, a 16x16 microstrip patch array of size 15x11.5 wavelengths with N=58,884 unknowns is analyzed in less than 12min with and error of 0.1% in the induced current and 0.6% in the input impedance compared to conventional MoM. However, when [3] was published the performance of MLMDA for general 3D problems was worse than MLFMA [1], needing more computation time and storage memory.

The aim of this paper is to present a new and improved formulation of MLMDA with similar computation time and accucary than MLFMA. For example, the solution of an X-band horn of N=68,904 unknowns using symmetries takes 6min 41sec with MLMDA and 7min 2sec with MLFMA, while the error in the computation of J is 6.6% with MLMDA and 7.2% with MLFMA.

The great advantage of MLMDA over the widely used MLFMA is that MLMDA implementation on top of an existing MoM code is independent on the kind of subdomain basis functions and the Green’s function used, making MLMDA the method of choice for implementation into antenna modelling codes.

[1] W. C. Chew, J-M. Jin, C.-C. Lu, E. Michielssen, and J. M. Song, "Fast Solution Methods in Electromagnetics," IEEE Transactions on Antennas and Propagation, Vol. 45, No. 3, pp. 533-543, March 1997.

[2] E. Michielsen and A. Boag, `A Multilevel Matrix Decomposition Algorithm for Analyzing Scattering from Large Structures,´IEEE Trans A&P, Vol 44, No 8 Aug. 1996.

[3] J.M. Rius, J. Parrón, E. Úbeda, J.R. Mosig, "Multilevel Matrix Decomposition Algorithm for Analysis of Electrically Large Electromagnetic Problems in 3-D", Microwave and Optical Technology Letters, Vol. 22, No. 3, pp. 177-182, 5th August 1999.

[4] Josep Parrón, Juan M. Rius, and Juan R. Mosig, "Application of the Multilevel Decomposition Algorithm to the Frequency Analysis of Large Microstrip Antenna Arrays", IEEE Trans. on Magnetics, Vol.38, No.2, pp. 721-724, March 2002.

This work deals with the full-wave analysis of electrically large and/or complex antennas. The starting point is the electric field integral equation formulation, transformed into a finite-dimensional problem by the application of the Method of Moments (MoM). The geometry of the arbitrarily-shaped conductors is modelled with triangular cells over which the unknown electric current density is represented by the Rao-Wilton-Glisson functions.

The application of MoM produces a linear system with a fully populated matrix. The number of necessary unknowns increases with the electrical size of the structure and strongly depends on its geometrical complexity. With standard MoM, matrix filling has a complexity of O(N²) for both the operation count and the storage and the direct LU solution is of the order O(N³). Because of this, the full-wave analysis of the structures of present interest is barred with the conventional MoM.

The Synthetic Function Expansion approach, a technique previously introduced by the authors, allows the analysis of large and complex problem alluded above at a reduced memory and CPU cost, bounded within the resources provided by a standard (32 bit) personal computer. In this approach, the structure is decomposed into smaller portions (blocks) and on each portion one numerically generates basis functions with support on the entire block; these are subsequently used as basis functions for the analysis of the entire structure. Since only a small number of Synthetic Functions (SF) is required to obtain an accurate solution, the overall number of unknowns is drastically reduced with a consequent impact on storage and solution time.

A further improvement to the approach described above is the use of two meshes (with two levels of accuracy) modeling the same structure. The main idea is to avoid the computation of the parts of the matrix of moments for the dense grid which correspond to the interactions between distant blocks of the structure, and to use instead the coarser grid which has a much smaller number of unknowns. In order to achieve this one needs to compute the projection matrix between the coarse and dense grid for each block and use the SF on the dense grid which were already found previously on each block, obtaining thus an opportune set of SF defined on the block’s coarse grid which will be used to compress the interaction matrix of moments between distant blocks of the structure.

Apart from the strong reduction of the MoM matrix filling time, the double-grid scheme allows a further reduction of the time employed for the matrix compression, due to the reduced number of unknowns on the coarse mesh. It is worth to mention that the efficiency of the scheme increases with the electrical size of the structure hence with the number of unknowns.

Introduction. Electric field integral equation (EFIE) is widely used for the numerical analysis of electromagnetic radiation and scattering. Essentially, when broad-band information is desired, it is more efficient to solve the electromagnetic problem in the time domain (TD). Several formulations have been presented, the marching-on in time method (MOT) - the explicit and implicit schemes [1], and the marching-on in order method (MOO) - the scheme with weighted Lagguerre polynomials [2], [3].

Problem. The principle of the analysis in the time domain is that an analyzed structure is excited by a desired pulse and a transient response is computed. The length of transient response depends on an analyzed structure and also on the way of modeling of feeding ports. From the calculation point of view, the length of the transient response is proportional to a number of time steps (MOT method) or a number of Laguerre's polynomials (MOO method) and so it has influence on time needed for the calculation. There is an issue. How should the ports be modeled in the time domain to be the length of the transient response shorter and the modeling less time-consuming?

Methods Used. TD-EFIE is solved by the method of moments. RWG functions are used as spatial basis and testing functions. The scheme with weighted Laguerre polynomials, MOO method, was used because this one does not suffer from late-time instability as the explicit or implicit scheme (MOT method).

Original Results. The discussion and proposals, how the feeding ports (for modeling antennas in the time domain) could be modeled for obtaining shorter transient responses, are the original ideas of the proposed paper. The comparison of the time-consume of MOT methods and MOO method for modeling antennas is another original contribution of this paper.

Verification. The proposed approach was verified on two antennas: the strip dipole and the bow tie antenna. The comparisons were made with data which were obtained by solving these structures in the frequency domain. The agreements of the transient responses in the time domain and of the input impedance in the frequency domain, by both examples, are very good. In addition, by using the proposal feeding port, the length of the transient responses was about a quarter of the original values.

Conclusions. A feeding port, when an antenna is analyzed in the time domain, has significant influence on the length of transient responses. By using a suitable feeding port, such as the proposal one, the length of transient response can be reduced up to a quarter of the original values and make MOT and MOO methods less time-consuming.

References

[1] RAO, S. M. Time Domain Electromagnetics. London: Academic Press, 1999.

[2] CHUNG, Y., S., SARKAR, T., etc. Solution of Time Domain Electric Field Integral Equation Using the Laguerre Polynomials. IEEE Transac. on A&P. 2004, vol. 52, no. 9, p. 319–2328.

[3] LACIK, J., RAIDA, Z. Transient Analysis of Scatterers and Antennas: Laguerre Polynomials Scheme with Improved Efficiency. In Proceedings of ICEAA 2005. Torino: Polytecnico di Torino, 2005, p. 257-260.

We propose a new method for RCS computation of long internally complex, dielectric objects such as wind turbine blades. RCS computation of these objects is important for the aviation community to determine the wind farms impact on radio navigation systems. These objects are far greater than the wavelength but asymptotic methods cannot be applied to solve this type of complex problem. A popular choice for solving electromagnetic problems with an arbitrary object is the method of moment (MoM) but direct MoM suffers from the storage requirement and computational complexity for large scale problems. These limitations impose on the one hand the use of a cluster to share out the storage and the computational complexity cost and on the other hand the use of an iterative method combined with an accelerating method to reduce the computational complexity cost.

The method proposed here is an improved iterative algorithm whose convergence is proven. Moreover the algorithm can be easily adapted to parallel computation. We use the formulation per block of the classical MoM . Thanks to the geometrical decomposition technique three accelerating methods can be used efficiently in an iterative algorithm. The first accelerating method allows the elimination of the internal degrees of freedom that do not participate in RCS computation. The second accelerating method accelerates all the matrix vector products used in the preconditioning procedure (reduction from O(N^2) to O(N^1.5)) as well as in the iterative resolution (reduction from O(N^2) to O(N^1.5)). This accelerating method also reduces the memory requirement (reduction from O(N^2) to O(N^1.5)). The third one consists in using a 'geometric-neighboring' preconditionner adapted to the physical aspect of the problem.

Figure 1 represents a simple model of a wind turbine blade used to validate our algorithm. The decomposition technique consists in introducing a separation interface (additional number of degrees of freedom) which divides the simple model into several sections. In the article we demonstrate how the number of unknowns is significantly reduced with minimal computation cost. The separation interfaces lead to the definition of concentrated geometrical blocks. This property allows us to efficiently use the low-rank QR factorisation as well as a block Jacobi preconditioner. This 'geometric-neighboring' preconditioner has an interesting property displayed in figure 2: the number of iterations increases slowly when the number of fixed size blocks increases and that much more favourably than expected. Actually the condition number must scale as D^0.5, where D is the block size. As the maximum size of the geometrical block is fixed by the memory available, this property allows to process a large object by increasing the number of blocks.

Thanks to the well-conditioned linear system, GMRES converges in only a few iterations. GMRES is used without restart, thus iterative resolution convergence is proven.

Numerical results are presented for the simple model wind turbine blade cut into 8 sections. RCS computed by our method is compared with the solution of direct method in figure 3.

A full-wave methodology has been used to analyse the GALILEO System Navigation Antenna for the space segment of the global positioning system. It is a multilayer array structure composed by cavity-backed microstrip patches in stacked configuration, and coaxial probe feeding performing in circular polarization. Details of this antenna can be found in [1].

The array antenna follows a hexagonal regular lattice which makes possible to perform a modular analysis. As a first step we have modelled each element of the array considering a finite hexagonal metallic flange. For this purpose the Generalized Scattering Matrix (GSM) of each isolated element has been obtained in terms of coax and spherical modal expansion. In order to obtain these matrices, a hybrid 3-D Finite Elements-Modal Analysis Method is employed [2]. For identical elements this analysis is carried out only once. The radiation pattern for the isolated element, including the hexagonal ground plane, has been compared with measurements to validate the proposed model.

In the following step, the GSM´s of the isolated elements are analytically connected by using properties of rotation and translation of spherical modes to obtain the overall GSM of the array [2]. This matrix provides the mutual coupling parameters between elements and a transmission matrix that allows us to compute the radiation pattern of the finite array on a finite ground plane including the coupling effects. The overall GSM characterizes the array as a circuit so that the radiation characteristics can be directly obtained for any selected excitation in a few seconds in a Pentium 2.66 GHz 512 MB. This property can be used to optimise a radiation pattern including mutual coupling and finite ground plane effects. Radiation patterns of the array for different frequencies have been obtained and compared with measurements. A good agreement has been found.

References:

[1] A. Montesano, F. Monjas, L.E. Cuesta, A. Olea, "Galileo System Navigation Antenna for Global Positioning", 28th ESA Antenna Workshop on Space Antenna Systems and Technologies 2004, p.p. 247-252, ESTEC, Noordwijk, Netherlands, May. 31-Jun. 3, 2005.
[2] J. Rubio, M.A. González and J. Zapata, "Generalized-Scattering-Matrix Analysis of a Class of Finite Arrays of Coupled Antennas by Using 3-D FEM and Spherical Mode Expansion," IEEE Trans. on Antennas and Propagation, vol.53, no. 3, pp. 1133-1144, March 2005.

A full-wave numerical study of a 2-D model of dual-reflector PACO (parabolic subreflector and conical reflector) antenna is performed. Analysis tools are the integral equations and the Method of Discrete Singularities (MDS) for both polarizations. Numerical results are shown.

The 3-D sketch of a PACO antenna is illustrated in Fig.1. Axially symmetric design of the PACO antenna, together with a TEM horn feed, is aimed at providing omnidirectional coverage with a horizontally directed or slightly tilted main beam, suited for the use in the base stations of the point-to-multi-point radio and TV links.

This configuration is based on quasi-optical considerations and usually is modeled with Geometrical Optics (GO) applied to cross-sectional 2-D geometry. As GO is a high-frequency approximation, realistic antenna modeling can be greatly improved by using a full-wave analysis method. Here, due to large electric size of reflectors, application of both conventional moment-method simulations and FDTD hit prohibitively large computer resources. As a remedy, we use MDS to discretize singular electric-field integral equations (SEFIEs) to economic-size matrix equations possessing fast convergence and controlled accuracy.

To simulate PACO antenna in 2-D, the scattering of the E and H-polarized waves by two curved PEC strips is studied as an electromagnetic boundary-value problem. Both reflectors are assumed zero-thickness and perfectly electric conducting (PEC), and the right-angle corner edge is rounded. The feed is a line current placed at the complex-valued source point (CSP) whose projection to the real space coincides with GO focus and is slightly shifted above the corner edge. Thus, the incident field resembles a directive beam.

The problem is reduced to two coupled SEFIEs for the currents on the strips. Then, we transform SEFIEs to Cauchy-singular ones in order to apply discretization provided by the MDS. This is done by using the quadrature formulas of interpolation type in the nodes of the Chebyshev polynomials of the 1st and 2nd type. As a result we get a set of linear equations, solving which yields the desired surface currents given by their interpolation polynomials.

We have applied MDS to examine the performance of large-size 2-D PACO models with the E-polarized CSP feeds selected to provide –18 dB subreflector edge illumination.

Fig.2 shows the main-beam directivities of two 2-D PACO-like antennas having d1=50 lambda (straight line) and d1=30 lambda (dashed line) as a function of the corner base length, d2 in wavelengths. One can see that the maximum directivity is reached at the values of d2 = d1 + 4 lambda. The minimum directivity corresponds to the strongest diffraction by the reflector. Note that corner reflectors smaller in size than this value cannot produce near to horizontal main beams of the PACO-like antennas.

Fig.3 illustrates near fields for d1=50 lambda, kb=5 and d2=18 (left) and d2=54 lambda (right) antennas. Strong diffraction by the corner reflector edge is well visible in the former case. Radiation pattern for latter case (i.e., with the optimal reflector) is shown in the insert in Fig.2.

The azimuthally magnetized circular ferrite waveguide, operating in the normal TE01 mode is a geometry, eligible for nonreciprocal digital phase shifter [1,2]. Formulae for computation of differential phase shift (DPS) it provides have recently been suggested in terms of the A, B, C numbers [1]. As the exact evaluation of the latter is a tedious task [1], an approximate method for specifying them has been proposed by the L numbers [1] - the limits to which the products of purely imaginary zeros of complex Kummer function [2-6] (wave function for propagation) by the modulus of imaginary part k of its first complex parameter tend when k goes to minus infinity [2]. The calculation of exact values of quantities L however is also a grave problem [2].

Here an approximate method for finding the L numbers (the A, B, C ones, resp. the DPS) is proposed. First, a theorem for the identity of purely imaginary zeros of the complex Kummer function and the real ones of the real Kummer function is proved numerically. It holds if the parameters of both functions meet certain conditions and the parameter k and the first one a of real function get large negative. Then, the Tricomi approximate formulae, giving the real zeros of real Kummer function for large negative a by the real ones of a definite Bessel function [3-6] are used to get the purely imaginary zeros of complex Kummer function for large negative k, employing the theorem for identity of zeros. Next, a formula for approximate reckoning of the L numbers is obtained, yielding approximate expressions for the A, B, C ones that involve the zeros of Bessel function and the newly defined K numbers (the products of purely imaginary zeros of complex Kummer function by the modulus of k). To fix the values of the latter needed, an iterative procedure is harnessed. The expressions mentioned give results, corresponding to specific phase states of waveguide, relevant to large negative k. To find A, B, C for any phase state (for any k), the assumption for independence of these quantities of certain structure parameters [1] is employed. Finally, putting the values of the three numbers in one of the aforesaid formulae [1], the DPS is determined approximately. The method is very simple and is applicable in the whole range of phase shifter operation of configuration [1,2], introducing an error less than one percent.

References [1] Georgiev G.N. and Georgieva-Grosse M.N., "Some new simple methods for differential phase shift computation in the circular waveguide with azimuthally magnetized ferrite", in Proc. 28th ESA Antenna Worksh. Space Antenna Syst. Technol., ESA/ESTEC, Noordwijk, The Netherlands, May 31 - June 3, 2005, Pt. 2, pp. 1159-1166.

[2] Georgiev G.N. and Georgieva-Grosse M.N., "A new property of the complex Kummer function and its application to waveguide propagation," IEEE Antennas Wirel. Propagat. Lett., vol. AWPL-2, pp. 306-309, Dec. 2003.

[3] Tricomi F.G., Lezioni sulle Funzioni Ipergeometriche Confluenti. Torino, Italy: Editore Gheroni, 1952.

[4] Tricomi F.G., Funzioni Ipergeometriche Confluenti. Rome, Italy: Edizioni Cremonese, 1954.

[5] F.G. Tricomi, Fonctions Hypergéométriques Confluentes, Gauthier-Villars, Paris, France, 1960.

[6] Riekstins E.Ya.: Asymptotics and Evaluations of Roots of Equations, Riga, USSR: Zinatne, 1991 (in Russian).

The process of installing VHF,UHF, TCAS and Satcom antennae systems on civil aircraft (AIRBUS, ATR,..) and their operability certification phase are mainly carried out on the basis of campaigns of measurements on reduced scale models or even expensive flight tests. In the framework of the European IPAS (Installed Performances of Antennae on Aerostructures of the 6th PCRD) project run in partnership with European industrial companies and laboratories (in particular BaeS, EADS, ATR, NLR, DLR, ...), ONERA has developed multi-domain calculation methodologies for the numerical simulation of antennae and isolation diagrams that take into account the complete structures of aircraft. The objective is, on the one hand, to propose a consistent and collaborative calculation procedure for the aircraft builder and the antenna maker that ensures that industrial secrets are protected and, on the other, to guarantee large gains in accuracy and calculation time in the parametric study phase inherent in the search for an optimal installation of the radiating parts.

The procedure proposed consists of introducing several calculation domains for which the electromagnetic response is obtained either by integral equations (EFIE or CFIE), or by asymptotic methods (OG,PO,PTD), then condensed in the form of Scattering matrices. The different transfer functions characterizing each sub-domain are then assembled by resolving a network equation thus giving the global electromagnetic behavior of the antennae systems installed on the aircraft.

ONERA is experimenting with a class of global basic functions defined on geometric interfaces covering VHF and UHF antennae. These functions take into account the close couplings between the antenna and the structure and are then used as source terms during the electromagnetic calculations of the antennae and aircraft volumes by integral equations.

These basic functions are subsequently used to condense the sub-domains with very few functions and, by masking the complexity of the antennae (wires, thin surfaces, dielectric materials) by a surface, the external domain of the aircraft, a meshing of which commonly generates between 100,000 and 1 million unknowns, can be closed. This point is crucial for the use of a CFIE integral equation, better conditioned than the EFIE equation and guaranteeing a much faster convergence.

Gain diagrams for quarter-wave monopole antennae operating in the VHF band and installed on a 1/15 scale model of a Fokker 100 have been measured by BAE Systems at Great Baddow for a frequency of 1.8 GHz corresponding to a frequency of 120 MHz for the scale 1 aircraft (Figure 1). The gain comparison with the multidomain results shows a good match in the Yaw and Roll plane .

For more than twenty years, especially since the advent of the well known and extensively used Rao-Wilton-Glisson (RWG) subdomain functions, a lot of accurate electromagnetic modelling with the Method of Moments (MoM) has been accomplished, firstly and mainly for Perfect Electric Conductors (PEC), later also for dielectric bodies. The problem of composite structures made of adjacent homogeneous PEC and dielectric bodies has received much fewer attention so far . To date, though quite general approaches have been presented , no full treatment of this problem has been given yet. It is the purpose of this paper to make new steps towards such a full treatment. First of all, the PEC are presented along with their dual counterpart, the PMC. Next, both volumic bodies and plates are combined in every possible ways : for example a plate sliced between two dielectric volumes, a homogeneous body made of a volume having plate extensions, and fully embedded bodies. The specific case of branched bodies is clearly identified and analyzed in details; a general approach is presented to deal systematically with this singularity. It is then shown how this new concept of branched bodies is naturally related to the question of null surface currents inside a partitioned homogeneous body. To provide a complete answer to this important question, the treatment of composite structures is made even more general through the use and comparison of many formulations consisting of one of the three well known testing schemes, point matching, f and nXf, coupled to one of the following redundancy reduction schemes : PMCHWT or Müller for dielectrics and EFIE or MFIE or CFIE for PEC and PMC. Finally, 3D plates are also revisited and it is shown how some of the above formulations allow computation of the currents independently on both sides of plates.

Abstract: The most common design aid for designing microwave antenna is the Electromagnetic simulator where the physical antenna structure is simulated and its response calculated. However, to complete the design cycle further understanding of the antenna characteristics is required. Since most modern communication systems are digital, a model compatible with digital system simulation would be a valuable design aid. Furthermore, an equivalent circuit model would help the designer relate the antenna response to the physical structure and to study the effects of varying the antenna dimensions on the response. For these reasons we have integrated CAD procedures in the antenna design cycle in order to identify both system and circuit models from either measurement or from the results of the Electromagnetic simulator. The System Model :This model is used to represent the antenna in the simulation of entire communication system. Since almost all modern communication systems are digital, the model has to be compatible with digital system simulation software such as Simulink. Unlike models based on equivalent electrical networks, the model does not have to obey Kirchoff’s and Ohm’s laws. This makes the model more flexible and low order models can be identified for complex responses. The system identification process assumes that the response of any linear time invariant system can be represented as the ratio of two polynomials. The model can then be incorporated in the simulation of a complete digital communication system as will be presented at the conference. The advantages of the system model are:

Transient finite-element methods based on Whitney elements represent powerful techniques for solution of the Maxwell equations. These methods are normally used on unstructured, body conforming grids, and therefore do not suffer from the staircasing errors present in e.g. the finite-difference time-domain (FDTD) method. In principle, a tetrahedral grid could be used to resolve fine system details. However, in practice, the number of unknowns can be prohibitive and may lead to worse conditioning of the system. Thus, the development of accurate models that characterize the physics of small features without the need for a highly resolved grid is essential. In recent publications accurate and stable subcell models for the finite element method have been developed for thin wires and thin slots. In this paper we focus on modeling of thin material sheets and coatings. Important applications include among others: complex antennas etched on thin dielectric substrates, structures coated with thin layers of radar absorbing material (RAM) and radomes used to enclose antennas.

A drawback with previously published methods based on first-order, Leontovich, impendance boundary conditions (IBCs) is that only the tangential electric field component at the sheet is affected by these boundary conditions and the fact that the normal electric field component is discontinuous across a dielectric sheet is not taken into account. In this paper we follow a different approach where thin structures are modeled through the use of degenerated prism elements, so-called shell elements. The use of shell elements makes it possible to take the discontinuous normal electric field component into account by introducing additional degrees of freedom.

As a test case we consider scattering by a dielectric sphere covered with a thin dielectric coating. The material parameters are taken to be ε_sph = 4ε₀ for the sphere and ε_sph = 10ε₀ for the coating. The thickness of the coating is 1/30:th of the radius of the sphere which is denoted by a. The incident plane wave has vertical polarization and the scattered field was computed for a horizontal azimuthal sweep, starting from the monostatic angle. The figure shows the bistatic radar cross section (RCS) for ka=1. The details of the method and some more results will be included in the full length paper.

In this paper, the problem of electromagnetic scattering from a large dielectric-coated cone is studied by using two methods, namely the three-dimensional Finite Difference Time Domain technique (FDTD), and the Method of Moments (MoM) specialized for bodies of revolutions (BORs). The MoM/BOR is numerically more efficient than the FDTD at low frequencies, not only because it uses the surface integral equation formulation as opposed to a volumetric sampling of the unknown electric and magnetic fields in the FDTD, but also because it takes the advantage of the rotational symmetry of the structure. However, the FDTD becomes faster at high frequencies and wide incident angles, because for these cases the MoM/BOR algorithm requires the inclusion of many azimuth modes, each of which is computationally intensive to process numerically. The 3-D FDTD code also has the advantage that it can generate the results for the entire frequency band of interest in a single run. However, unlike the MoM/BOR, the FDTD method does require that the code be run anew for each incident angle. Also the BOR version of the FDTD code requires the use of a very small time step for higher order modes to obtain stable results and, consequently, the required run time can also be quite large for this method.

In this paper, we simulate a metallic cone with grooves that has a partial dielectric coating, as shown in Fig.1. The height of this cone is 12.2 wavelengths at the 3dB cutoff frequency of the incident Gaussian pulse used for the FDTD simulation, and its diameter is 3.4wavelengths. We need to use a fine resolution in the vicinity of the grooves in the FDTD simulation to capture the details, especially since the depth of the smallest groove is only 0.0094wavelengths. We validate the results, by comparing those generated by the FDTD and the MoM/BOR codes for the axial incidence case (which excites only M=1modes). Next, we used the FDTD to simulate the oblique incidence cases and investigated the different diffraction mechanisms by evaluating the near fields at different points located close to the cone surface, (see Fig.2). We see from Fig.2 that the tip, near base edge and far base edge contribute to the field at point (1). The paper will present the results with explanation for the scattering phenomena and the creeping field around the outer surface of the cone.

This paper presents the design of a dual band microstrip antenna for wireless communication. The antenna is designed to resonate at the two frequencies 5.06 GHz and 5.85 GHz. This antenna has a bandwidth of 25% with center frequency 5.56 GHz. The antenna is designed as a patch with two slots, as shown in Fig.1. The feeding of the patch is a coaxial probe and it is air-filled between the patch and the ground plane. The outer dimensions of the patch and the height are designed so that the antenna resonates at the upper resonant frequency. The slots' position, length and width are designed to control the lower resonant frequency and the bandwidth. The antenna has been simulated using the two techniques to validate the results: the method of moments (MoM) based on the IE3D commercial software and the finite-difference time-domain (FDTD) technique. The S11 results of the designed antenna are shown in Fig.2. The paper will present how to choose the dimensions of the patch and the slots and the positions of the slots to control the resonant frequencies and the bandwidth of the antenna. A design and the simulation of a patch antenna based on this geometry for WiMax technology, the radiation pattern, and the directivity of the antenna will be presented also.

Nowadays it is well known that combline filters can achieve the specifications of the satellite communication systems: low loss performance, sharp frequency responses, and enduring high power levels. These filters consist essentially of metallic cavities whose access ports are in general rectangular or circular coaxial waveguides. Typically, they include metallic and/or dielectric obstacles. Within this study, we consider an empty rectangular cavity with perfectly conducting walls as geometry for the metallic enclosure. For the shake of simplicity, we assume that the excitation can be modeled with electric currents. When solving the problem with a MPIE-MoM strategy we need to compute the Green's functions G_A and G_v for the empty rectangular enclosure. It is well known that, if inside this context we calculate G_A and G_v using the Lorentz Gauge, G_A is a diagonal dyadic, and its components G_A^xx, G_A^yy, G_A^zz as well as G_v have resonances corresponding to the allowed modes of the cavity. A study of these four Green's functions reveals that the relationship between type of Green's function, existence of resonances and eventual modal degeneracy is very complex. This relationship has been understood through a rigorous classification scheme of allowed modes, to be presented in this communication. With this scheme we can determine a priori which the modal resonances to be present in each type of Green's function are, and link these resonances to classical waveguide modes. This allows us a very efficient development of a software tool for combline filters where all the possible singularities associated to resonances are carefully dealt with. As a preliminary result we present results for G_A^xx where the singularities are due only to a fraction of the possible TE modes and to a set of degenerate modes (figure).

This study has been done during the development of a software simulation tool which will be capable of analyzing combline filters. The software is developed in the framework of a European Space Agency (ESA-ESTEC) project (reference number: 16332/02/NL/LvH).

It is common to use a phase center for characterizing an antenna in the Frequency Domain (if it has the one). Often this center is frequency dependent so this concept can’t be directly moved to the Time Domain (TD). In TD antennas are usually characterized with a normalized impulse response that relates radiated field with the input voltage:

There are several R in this formula. The R in the lhs is the observation point coordinate,R’ in the denominator is the distance from the observation point to the so-called amplitude center of the antenna – this is the point from which the fields decay as 1/R. There is another R” in the argument of delta function that describes impulse arrival delay. This distance in fact is measured from another point – let’s call it a time center. This point corresponds to the phase center in FD. The main issue is that such a definition is meaningful if the impulse response itself is assumed to be with almost identical waveform but changing amplitude within main beam. It is always true for some (may be quite narrow) angle range. Thus the time center is defined via curvature of the wave front in the main lobe direction where the wavefront can be considered locally spherical. The actual calculation of this characteristic is convenient to perform considering arrival time distribution over some plane perpendicular to the beam axis. The arrival time is calculated using correlation of the waveform with that on axis.

To illustrate this idea we consider a 3D wavebeam radiated by current distribution over plane. We take linearly polarized currents with radially-Gaussian spatial distribution excited simultaneously with some bipolar pulse (to avoid static fields). The radiated fields are calculated analytically with Mode Basis Method. Then we study the positions of amplitude and time centers as functions of the ratio of the pulse duration to sources diameter.

Further we consider diffraction of the wavebeam at a plane interface of dielectric medium with some conductivity. This problem was also treated analytically with TD diffraction operators and modal decomposition. We studied transformation of the wavebeam centers at diffraction, namely how the centers of the reflected and transmitted beams relates to those of the incident beam.

The introduced concept of time center provides good physical insight into the processes of impulse wavebeams propagation and diffraction considered in the Time Domain.

"New generation" of MoM-based codes have been recently introduced to reduce the standard MoM computational effort; among them there is the MAGMAS code developed at the Katholieke Universiteit Leuven (KUL) for the analysis of printed antennas, circuits and packaging [1] that employes several techniques expedient to achieve this goal. In these types of problems, a very dense mesh is often necessary, and the increase of the associated MoM matrix condition number might endanger the accuracy of the solution, especially when approximate techniques are employed to reduce the computational burden. For this reason it becomes very important to control the matrix condition number.

Here we propose the integration of the MAGMAS code with the MultiResolution (MR) approach described in [2]; the latter can be viewed, inter alia, as a powerful pre-conditioner. The MR scheme essentially consists in the generation and use of vector basis functions, applicable to the analysis of planar and/or 3D antennas, having properties similar to those of scalar wavelet. The spectral localization of the MR functions allows the control of the condition number of the MoM matrix in the MR basis simply by application of a diagonal preconditioner. The MR functions are expressed as a linear combination of the sub-domain functions defined on the same mesh (actually, RWG functions defined on a triangular mesh) and therefore they could be quite easily integrated with other MoM-based codes that used the same sub-domain functions. In this communication we will specifically refer to the integration with the MAGMAS code, but similar consideration can be extended to the integration of the MR scheme with any other MoM-based code.

As preliminary examples of application of the integration between the MR scheme and MAGMAS, we considered two very simple structures, consisting both in a square patch: in the first case, it is located in air and excited by an incident field radiated by an electric dipole, while in the second case it is located on a grounded dielectric slab with permittivity ĺ = 2.2 and thickness 7.5 mm and it is fed by a probe. In the first situation, the unknown current on the patch is modeled only with RWG functions, while in the second one a special attachment mode is used to describe the smooth transition of the electric current between the probe and the patch.

Table. 1. 1-norm Condition Number of the MoM matrix

	[ZPC]	[ZMR_PC]
Patch with incident field excitation	1429	477
Probe fed patch	23302	5901

The effectiveness of the integration of the MR scheme with MAGMAS is shown by the results reported in Tab. 1, in which the 1-norm condition number of the MoM matrix in the original basis [ZPC] and in the MR basis [ZMR_PC], both after the application of a diagonal preconditioner, is reported, for the two configurations: comparing the first with the second row it becomes evident that the use of the MR functions reduces noticeably the matrix condition number.

References:

[1] www.esat.kuleuven.be/telemic/antennas/magmas/ (list of scientific references available there)

[2] F. Vipiana, P. Pirinoli, G. Vecchi, , IEEE Trans. Antennas Propag., vol. 53, No.7, pp.2247-2258.

Scattering in structures which are large in terms of wavelengths is usually solved by a ray optical technique such as GTD (Uniform Geometrical Theory of Diffraction). The principle is that the power radiates along rays and near-field effects are not taken into account.

This is a problem when the radiating antenna is close to parts of the scattering structure. Here MoM (Method of Moments) yields a better solution. Unfortunately, the computing time required for the MoM solution increases drastically with the size of the scatterer.

The paper describes determination of the scattering from the LICEF antenna elements on the SMOS satellite taking into account that the solar panels rotate during the flight of the satellite. This is a task well suited for a ray analysis by GTD since the method is fast and this is essential due to the ever changing scattering structure.

However, the body of the satellite is not large in terms of wavelengths and, furthermore, near-field effects must be included in describing the source radiation because the source antennas are mounted upon deployable arms which shields for the direct illumination of the satellite body and the solar panels. Thus GTD is not suitable for determining the scattering in these structures.

The scattering in these structures may then be solved by MoM. The structures are affordable in size and, which is important, have an orientation which is fixed in time.

The paper will describe how the two methods are combined, especially the problem of delimiting the structures handled by MoM from the structures handled by GTD as the latter must be in the far field of the former while they nevertheless are connected parts of the same satellite structure.

The presented work is carried out for ESA on ESTEC contract 11514/05/NL/FF.

The goal of this paper is to extend the effective approach that is used for the calculation of GFs in planar multilayered structures for a single dipole to the case of a periodic array of dipoles. In both cases the main problem consists of the calculation of the GFs in the spatial domain. In the spectral domain, for layered structures the GF is identical in both cases, a single dipole or a periodic array of dipoles, and it is known in closed form. However, the transform between the spectral and the spatial domain requires considerable efforts and in the case of a periodic array is not so straightforward. The transform is expressed in terms of double infinite integrals for a single dipole or a double infinite series for a periodic array. The main problem is the poor convergence of the integrals or of the terms in the series. A widely used solution for this problem consists of the modification of the GFs in the spectral domain using special asymptotes. The asymptotes have the same leading terms as the spectral GFs but in contrast to the GFs the asymptotes are known in closed form, not only in the spectral domain but also in the spatial domain. By subtracting the asymptotes from the spectral GFs and adding them again but to the spatial GFs, the numerically calculated part of the transform becomes much easier to handle. One of the interesting questions is whether it is possible to use the same asymptotes in both cases (for a single dipole and a periodic array).

Each asymptote in the spectral domain is linked to a corresponding function in the spatial domain. For a single dipole it is sufficient to choose the spectral asymptote so that its spatial counterpart is known in closed form. In contrast to the single dipole, the asymptote in case of a periodic array in the spatial domain is also expressed in terms of double series, including contributions from all cells. In order to avoid numerical problems it is logical to impose an additional condition, namely to require a good convergence of the function in the spatial domain. Then the evaluation of the spectral GFs in the spatial and spectral domain can be performed numerically. There are several asymptotes that are widely used and indeed satisfy this additional condition. Unfortunately, when we deal with vertical currents or volume currents the number of GFs becomes prohibitively large. It is possible to reduce drastically the number of required GFs if we consider the z-dependency of the currents on these components and perform a primary integration over the z-direction in the spectral domain. It is important to understand that in this case, if we want to keep the efficiency of the technique, we should also know in a closed form the counterpart of the integrated asymptotes. In many cases the asymptotes that are used for horizontal currents cannot be effectively used any more. However, the problem can be solved by using the asymptotes introduced in [1]. In the final paper we will present several numerical examples and a comparison with periodic GFs calculated using other methods.

[1] M. Vrancken, G.A.E. Vandenbosch, IEEE MTT, Vol.51, No. 1, pp.216-225, Part 1, Jan 2003

This abstract introduces a method for the reconstruction of electromagnetic currents from known radiated near field data. The algorithm, based on Electric Field Integral Equation (EFIE), solves the Inverse Radiation Problem over arbitrary radiating structures. The purpose is the characterization of any kind of antennas, despite the complexity of the antenna geometry, for diagnostic purposes as well as for NF-FF transformation.

Calculation of sources from radiated field involves the Inverse Radiation Problem resolution. Some of the methods for the reconstruction of the equivalent electric and magnetic currents are based on the Fourier Transform properties for planar domains both for fields and currents. Others are based on Integral Equation algorithms. The method presented in this paper uses matrix formulation derived from discrete resolution of integral equations, after the application of Equivalence Principles. By this way, it is possible to reconstruct equivalent electric and magnetic currents from spherical acquisition of near field.

Field and geometry data conform the equation system, which is solved through a Conjugated-Gradient (CG) method, due to its good convergence in a few iterations. In addition, several regularization parameters related to matrix equation system have been tuned, obtaining a better convergence of the CG method and reducing the error function.

The amount of data involved in the Inverse Radiation Problem resolution over arbitrary surfaces supposes a limitation of the problem size. Thus, to avoid matrix storage, an iterative method for solving very large equation systems has been developed. Consequently, it is not necessary matrix storage, reducing memory consumption, which allows a considerable increase of data required to reconstruct currents over complex geometries.

In order to validate the algorithm, comparisons with other software for retrieving sources over 2D and 3D domains have been done, obtaining satisfactory results. From this point, several kinds of antennas, whose near-field has been measured at anechoic chamber, have been analyzed, retrieving successfully currents distribution from near field measurements.

The described method appears to be very efficient for determination of currents over complex geometry antennas. On one hand, the calculation of both electric and magnetic currents over the 3D surface of the antenna supposes a better characterization of the antenna especially for diagnostic task. On the other hand, although the method could have computational efficiency problems because of using electric and magnetic currents, the proposed matrix formulation and the proposed regularization parameters avoid such problem.

References:

[1] F. Las-Heras, B. Galocha, J.L. Besada. "Equivalent Source Modeling and Reconstruction for Antenna Measurement and Synthesis", 1997 IEEE AP-S Intern. Symposium, Montreal, Canada, Digest, vol. 1, pp. 156-159, 1997.

[2] F. Las-Heras, T.K. Sarkar, "Radial field retrieval in spherical scanning for current reconstruction and NF-FF transformation", IEEE Transactions on Antenna and Propagation, Vol. 50, No. 6, June 2002, pp. 866-874

Intoduction

Antenna performance of mobile phones depends strongly on the electromagnetic properties of the chassis, typically more than on the nominal antenna element itself [1]. Antenna independent analysis of a mobile phone chassis is therefore of high interest. It provides insight for purposeful tuning of the chassis and for optimum placement and design of the antenna element which is not available from the conventional "all-in-one" analysis approach which focuses on the feed point input impedance only [2]. The present paper describes a numerical approach for antenna independent analysis of a mobile phone chassis based on the concept of characteristic modes [3] and its application to practical design issues.

Method and Application Results

The application of the modern planar antenna systems needs light and rigid radomes with low insertion losses, good matching in the operation frequency range and number of other requirements like mechanical and temperature stability, water, sun-shine and dust resistance, etc. Therefore, the successful solution of this problem is not possible without wide use of well-designed composite materials. There exist appropriate analytical models [1-2] for calculation of the insertion and reflection losses in multilayer radomes according to the number of layers, incident angle, wave polarization, etc. in a wide frequency range.

A key circumstance for the accurate RF design of the composite radomes is the right knowledge of the dielectric parameters of the used materials - the dielectric constant and the dielectric loss tangent of each layer. The most of the modern materials used for the antenna radome fabrications are as a rule anisotropic ones. The fact means that their dielectric parameters depend on the directions. We have proposed a measuring method, two-resonator method, for determination of the material anisotropy (for the dielectric constant and for the dissipation factor) (see [3, 4]).

The method is based on determination of the longitudinal parameters (in the surface plane) by cylindrical TE011-mode resonator and the transversal parameters (normal to the surface plane) by cylindrical TM010-mode resonator. The main benefit of this method is the possibility to determine the dielectric parameters of each layer in the multilayer samples (from the composite antenna radome), if the other layers have known parameters. This extraction procedure allows accurate determination of the dielectric anisotropy even of layers, which could not form self-depending samples (thin coating, absorber and hydrophobic films, different types of radome cores (e.g. Parabeam® 3D glass radomes), etc.

In the full text we consider the following problems:

◦ We modify the existed theory [1] to the case of anisotropic layer materials [2] to calculate the insertion and return losses of multilayer radomes and give several examples to check applicability of the present theoretical model for flat radomes;

◦ We discuss in details the dielectric anisotropy of a wide kind of antenna radomes: one- or multilayer; laminated or composited with honeycomb or foam cores, etc. Measured results for dielectric parameters of these materials are given.

◦ Appropriate models of the antenna radomes are presented and experimentally verified: coarse one average layer model, classical three-layer models and improved models with additional layer (glue-fillets layer, resin-filled cores, coating layers, etc. (Fig. 1).

References:

[1] D. T. Paris, IEEE Trans. Antennas Propagat., AP-18, pp. 7-15, Jan. 1970

[2] V. Peshlov, S. Alexandrov and P. Dankov, 15th MIKON-2004, Warsaw, Poland, May 2004, vol. 2, pp. 562-566

[3] Plamen I. Dankov et. all.., 35th EuMC’2005, Paris, France, Oct. 2005, pp. 517-520

[4] Plamen I. Dankov, IEEE Trans. on Microwave Theory and Tech., MTT-54, April 2006 (in print)

The efficient analysis of shielded environments, and more specifically rectangular cavities coupled to radiating elements is treated in this paper. The technique presented uses a MPIE/EFIE formulation for the part located outside of the cavity (antenna for instance), where full 3-D conductors and finite dielectrics embedded in a traditional multilayered structure can be taken into account. Inside the cavity, we use also an MPIE formulation, where the cavity Green's functions are used.

The finite dielectrics are represented by equivalent electric and magnetic surface currents. The Green's function inside the cavity are expressed as an infinite series of modal contributions. This series has a very slow convergence, as it contains singular terms. In order to speed up the computation of this series, we do first extract the singularities by expressing the series into two series, the first one a static series containing the singularities, the second a fast converging dynamic series. Ewald's transform is then applied to accelerate the convergence of the static series, which has to be computed only once for each problem, as it is frequency independent.

An example results is shown below. The structure analysed consists of a dielectric resonator antenna coupled to a rectangular cavity through a slot. As we can see in the figure, the results presented in this paper (solid line) agree very well with measurements shown in dashed line.

The time-domain analysis of resonant structures, among which are resonance antennas, must be grounded on stable and reliable numerical algorithms since the resonance systems are acutely sensitive to the variation of parameters. At the same time, the physical treatment of the numerical results here is based both on analytical representations for the solutions of the relevant initial boundary-value problems and on the results obtained in the frequency domain. We propose a novel methodology for studying open resonant structures based on the finite-difference time-domain algorithms that utilize the 'fully absorbing' boundary conditions for truncating the computational domain [1,2]. The resonant modes are highly sensitive to the influence of the virtual fields caused by reflections from the artificial boundaries. The 'fully absorbing' conditions have been constructed in [1] without any heuristic assumptions about a fine field structure in the vicinity of the artificial boundary. The errors introduced by these conditions are less by an order than the finite-difference approximation error, what is extremely important in the resonance situations.

We consider basic stages in the study of open resonators by the example of the transient E-polarized fields in the near zone of compact inhomogeneities of the R²-space. Similar reasoning holds in the case of H-polarization and in 3-D vector case as well. In the context of our approach, the basic electrodynamic characteristics (the eigen frequencies, Q factor, and the fields of free oscillations in open resonators, the radiation efficiency, the directional patterns, and the spatial field distributions in the near-field, intermediate, and far-field zones of resonance antennas) can be determined over a wide frequency band. The objects under investigation are axially-symmetrical radiators of TE₀- and TM₀-waves (an antenna with a semi-confocal resonator, an antenna with a resonator in the form of a section of a radial waveguide (Fig. 1), a dielectric disc resonator in the aperture of a monopole antenna) and plane resonance antennas with waveguide feeding.

In this paper an accurate method to analyze frequency selective surfaces (FSS) illuminated by a non-uniform field, generated by an horn, is described. Frequency selective surfaces are used to combine and separate electromagnetic beams at different frequencies and they are key devices in the beam waveguide feeding system of large reflector antennas for deep-space mission [1].

FSS are usually designed under uniform plane-wave approximation, at the nominal incidence angle; anyway, this hypothesis often is not realistic because of the impinging field which is far from being a uniform plane-wave. For this reason we developed a novel approach based on the local plane-wave approximation of the illumination on the FSS [2]. In particular the incidence angle is locally obtained by considering the propagation direction of the impinging field radiated by the horn. This technique has been applied to analyze and design FSS for deep-space applications, showing an excellent agreement with measurements [2].

Nevertheless, in some cases, where the FSS is seen by the horn under large angles or when the FSS transmission/reflection coefficients strongly vary with the incidence angle, the method described above may result inaccurate, since in some cases, the transmitted field for a given direction may exceed the incident field. In fact, for interference reasons, the field beyond the dichroic may give a transmitted radiation field with maxima and minima in different angular position with respect to the original incidence pattern.

In this paper we propose a more rigorous approach, based on a plane-wave expansion of the incident field. In particular, the field impinging on the FSS is thought having a periodicity given by a square cell of appropriate dimension T (larger than the dichroic), as shown in Fig. 1, and is approximated by a discrete plane wave spectrum. After that the FSS transmission/reflection coefficients are obtained for each plane-wave by means of the MoM/BI-RME method [3] and then the scattered field is obtaining by summing up the transmitted/reflected plane-waves. As an example, this technique can be applied to analyze a curved FSS such as the one proposed in [4] shown in Fig 2a, which exhibits a very critical behavior (Fig 2b).

Fig. 1 – Incident field and its fictitious replicas, used in the plane-wave expansion method.

Fig. 2 – Feed horn with dichroic mirror: a) geometry of the structure; b) radiation pattern with and without the dichroic.

REFERENCES

[1] P. Besso, M. Bozzi, L. Perregrini, L. Salghetti Drioli, and W. Nickerson, "Deep Space Antenna for Rosetta Mission: Design and Testing of the S/X-Band Dichroic Mirror," IEEE Trans. Antennas Propag., Vol. AP-51, No. 3, pp. 388-394, March 2003.

[2] P. Besso, M. Bozzi, M. Formaggi, S. Germani, M. Pasian and L. Perregrini, "A Novel Approach for the Design of Dichroic Mirrors for Deep Space Antennas", 35th European Microwave Conference 2005, Paris, France, October 3-7, 2005.

[3] M. Bozzi, L. Perregrini, J. Weinzierl and C. Winnewisser, "Efficient Analysis of Quasi-Optical Filters by a Hybrid MoM/BI-RME Method", IEEE Trans. Antennas Propag., Vol. AP-49, No. 7, pp. 1054-1064, July 2001.

[4] B. Philips, E.A. Parker, R.J. Langley, "Ray tracing analysis of the transmission performance of curved FSS", Microwaves, Antennas and Propagation, IEE Proceedings, Vol. 142, Is. 3, pp. 193-200, June 1995.

This paper addresses the analysis and design of axisymmetric horn antennas with arbitrary shape. The horns can be empty or loaded with one or more dielectric materials. These materials can even be uniaxially anisotropic ones. Corrugated horns can also be studied. The analysis method based on a Segmentation technique, the Finite-Element and a Matrix Lanczos-Padč algorithm (SFELP [1]) efficiently finds the radiation patterns, the directivity and the return losses. This efficiency allows the use of this technique in connection with a Simulated Annealing algorithm adapted to electromagnetic problems.

The antenna is surrounded by a spherical surface, which constitutes the radiation port. An expansion of spherical modes describes the electromagnetic field at these points. The optimal diameter of the sphere is decided with convergence criteria, at the same time reducing the computation domain and hence the size of the mesh. To achieve this goal, we assume that the sphere intercepts the conical surface of the horn far enough from the aperture to have negligible fields (and currents). The excitation port can be a circular or a coaxial waveguide.

We will present several examples of use of this method to proof its validity. The first one constitutes the reproduction of a dielectric loaded horn taken from the bibliography. The second example is an empty horn with curved profile designed to satisfy a predefined radiation pattern. In the third example we design a horn with curved profile loaded with a dielectric in its core with its own curved profile too to achieve the maximum aperture efficiency. This result is compared with the one obtained from a dielectric loaded horn with the same length but with straight profiles. Next we will show a more realistic horn design with two dielectrics. The goal of the optimization is the same of the previous case. The last example will be the design of a corrugated horn, that achieves excellent results compared with dielectric loaded ones, that modify the deep and profile of the corrugations following two independent smooth curves.

The axisymmetric devices have the advantage of its simplicity in the manufacture. In this paper we have presented examples of horn antennas but monopoles, dielectric, resonators or reflectors have also been studied and designed with the current method in a efficient way.

[1] J.Rubio, J. Arroyo, J. Zapata, "SFELP-An Efficient Methodology for Microwave Circuit Analysis", IEEE Trans. on Microwave Theory and Techn., vol. 9, pp.509-516, Mar., 2001.

In this paper we review two methods to solve the layered sphere problem. In both cases, the solution is expressed as a multipole expansion. In the first approach, the boundary conditions at each interface is expressed, yielding a linear system of equations that can be solved to compute the coefficients of the multipole expansion. In the second approach, the formalism of transmission and reflection matrices is used. The reflection matrices are computed recursively from the inner to the outer layer and in the reverse direction. The fields are then computed in a second step.

From a numerical point of view, the two approaches are not equivalent. We compare the accuracy and complexity of both algorithms on a test case representative for electromagnetic induction sensors.

We also show that if we define an impedance for the forward and backward propagating modes, the reflection matrices can be expressed as function of those impedances. The expressions obtained are a generalization of the classical expressions that must be used when the forward and backward impedances are not equal.

A Reverberation Chamber (RC) is a test facility which for many years has been used for EMC-testing and recently also for high accuracy antenna measurements. Basically, a RC is a three dimensional resonant metallic cavity in which the electromagnetic fields are stirred to provide an environment that is statistically isotropic and homogenous. The chamber has to be large in terms of the wavelength so that several cavity modes are present at the frequency of operation. By changing the interior of the cavity, i.e. mode stirring, we can change the field distribution in the cavity. In this way we can create the wanted statistically isotropic and homogenous environment that has similarities with a uniform multipath environment.

In order to create a good environment in the chamber and thereby make it possible to perform high accuracy measurements the key is efficient mode stirring. Several different stirring techniques can be used, such as mechanical plate stirring, platform stirring, frequency stirring and polarization stirring. A convenient method to study the performance of a stirring technique (as well as several other properties of the RC) is to perform computer simulations.

In this paper a 3D model of the RC is presented. A model which will help us to model mechanical wire stirrers and antennas inside the RC by using a Method of Moments solution combined with the thin wire approximation. Instead of resolving the entire cavity numerically with all its details, we include the walls of the cavity directly in the Green's function. Thereby, the unknowns in the MoM formulation become few, so that the time needed to invert the MoM matrix is short. Instead, more time is needed to compute the coefficients of the matrix, but this is rather independent of cavity size. The cavity Green's function is basically created using a spectral domain technique combined with imaging and periodic boundary conditions. To adjust the Q-value of the RC a lossy material is specified inside the RC. Finally, asymptotic extraction is used in order to speed up the calculations.

We will present several validation cases and comparisons with measurements as well as other computer codes. Results where the statistics of the modelled RC are verified against the theory of RC's will also be presented.

Most of the MoM methods use the free space Green's function for treating the EM behaviour of arbitrary bodies under the receiving or emitting mode. This kind of solver is now popular and used in space, aeronautic and defense companies for the treatment of EMC & antenna engineering problems on large systems in the mid-frequency range.

In the high frequency limit, the switching to the asymptotic methods (PTD or GTD) or to the extension of the MoM method, i.e the FMM allows to extend the frequency range of the traditional MoM method. On the other hand, difficulties remains at very high frequency:

- to treat the behaviour of illuminated objects in confined environment such as the reverberating test chamber or - to evaluate the transfert of energy between some rectangular cavities coupled by means of an aperture or going through wires.

We present in this paper an alternative to solve such problems at high frequency using the exact MoM method with its traditional 3 basis functions (wire, surface, wire to surface attachment) of electric and magnetic kind. The behind idea is based on a revision of the free space Green's function operations inside the MoM numerical kernel and to extend the numerical kernel to accept various Green's function regions in the same problem. In the present case, we focus on the rectangular cavity Green’s function. This paper is bdivided in tho parts.

The first part presents the backgrounds of the method, i.e the Green's dyadic formalism applied to the metallic cavity equiped with arbitrary metallic scatterers. The second part focus on the formalism to treat multiple cavities connected via arbitrary shape apertures.

The derivation of the dyadics of the mixed potential integral form of Maxwell's equations is done for electric and magnetic currents. In the present case, no extra additional auxilliary-potential is required to make unique the scalar potentials.

Convergence problems due to the damping nature of the obtained Floquet’s terms are investigated. Their treatment by means of the Ewald method is explicited.

The inherent difficulties of the MoM method due to the singularities of the self term are treated. It is shown how the Ewald method allows to extract the singular term, quite similar to the free space singular kernel.

Results are presented for a metallic cavity excited with a metallic wire. If available at the date of the congress, coupling results throug multiple cavities will be shown.

Complex Source Point (CSP) method is based on the analytical continuation of the point source field where the source position is allowed to be complex. It allows the extension of the Gaussian Beams (GBs) solution of Maxwell equations also outside the paraxial region, thus providing a useful tool to represent arbitrary antenna radiation patterns. Such a GBs/CSPs representation of actual radiators can be used within a ray code to predict the interaction between antennas and the operating environment, if standard diffraction formulation results are extended to the CSP case, i.e., when the illuminating source comes from a complex point. The CSP extension of "discrete" ray techniques like Geometrical Theory of Diffraction (GTD) and its uniform version (UTD) appears to be substantially more problematic than the extension to incremental theories, such as Physical Theory of Diffraction (PTD) and Incremental Theory of Diffraction (ITD). Unlike ray techniques, PTD/ITD incremental waves construct the diffracted field by superposing contributions outcoming from every point on the edge. At each incremental point the contribution is calculated by invoking the high-frequency localization principle, and resorting to a local canonical problem. The superposition is performed via a numerical integration, which substitutes the ray tracing operation and, although generally more time consuming, can overcome some GTD/UTD impairments usually associated with caustic regions.

When performing the analytical continuation of the UTD for CSP, the diffracted field description requires a "complex ray tracing"; i.e., the diffracted ray arises from a complex diffraction point whose determination and geometrical interpretation is definitely no more straightforward. This aspect can be a strong drawback when applying UTD formulas to implement ray-tracing algorithms, where the estimate of diffraction points must be performed at first. Conversely, this drawback does not appear when one performs the analytic continuation of an incremental theory. PTD/ITD contributions are readily calculated allowing geometrical source coordinates to be complex and by integrating them on the structure edges, as usual.

In this paper we analyze this CSP extension of ITD technique. First, we examine the canonical problem of the half-plane for various complex positions of the source and observation scans. Subsequently, in order to apply the ITD solution to a finite object, the field scattered from a perfectly conducting circular disc is presented and compared with those obtained by more established alternative techniques.

In the past, multipole (or modal) techniques have often been employed in the context of antenna fields. For example, they have been used to derive general properties of linear antennas as the relations between the field modal amplitudes and the elements of the antenna’s equivalent circuit [1], and to derive expressions for the antenna Q and the bandwidth [2], [3]. Moreover, multipole expansions in spherical, in cylindrical, and in Cartesian coordinates (the latter is often referred to as a plane-wave expansion) are the basis to calculate the probe-corrected far-field from measured or estimated values of the antenna near field [4].

The present approach deals with a systematic technique to obtain the spherical-multipole amplitudes of the field from numerically computed near-field data. The method works in the frequency-domain as well as in the time-domain and is exemplarily being demonstrated for the Finite-Difference Time-Domain (FDTD) method. To this end, first the linear antenna near-field is simulated for the case of a wide-band impulse, using FDTD with a Perfectly Matched Layer (PML). The tangential near-field on an arbitrarily chosen (Huygens-) surface, which completely encloses the antenna structure, is then replaced by a number of equivalent electric and magnetic elementary dipoles located on the Huygens surface and with amplitudes which are linearly interpolated by a method described and applied in [5] for scattering problems. In parallel to the progress of the FDTD time-stepping scheme (as ‘on the fly’) , the corresponding time-domain spherical-multipole amplitudes are then deduced by means of the (in the far-field exact) inverse Fourier transform of the bilinear free-space dyadic Green’s function in spherical coordinates. The obtained multipole amplitudes allow a convenient post-processing of the time-domain antenna far-field for any direction. Finally, a numerically performed (discrete of fast) Fourier transform of the time-domain multipole amplitudes directly yields the frequency-domain multipole amplitudes of the antenna field for the entire frequency spectrum of the above mentioned wide-band impulse. These multipole amplitudes can be used to analyze the antenna field at any point outside a minimum sphere enclosing the antenna structure, and to conveniently deduce all quantities of interest (e.g., the exact antenna Q) during the post-processing stage. First numerical results for a simple wire antenna prove the usefulness and efficiency of the method.

References:

[1] Chu, L,J.: Physical Limitations of Omni-Directional Antennas. J. App. Physics, 19, pp. 1163-1175, Dec 1948.
[2] Fante, R.L.: Quality Factor of General Ideal Antennas. IEEE Trans. Antennas Propagat., 17, 151-155, March 1969.
[3] Yaghjian, A.D., and S. Best: Impedance, Bandwidth, and Q of Antennas. IEEE Trans. Antennas Propagat., 53, 1298-1324, April 2005.
[4] Hansen, J.E. (Ed.): Spherical Near-Field Antenna Measurements. London: Peter Peregrinus Ltd., 1988.
[5] Oetting, C.-C., and L. Klinkenbusch: Near-to-Far-Field Transformation by a Time-Domain Spherical-Multipole Analysis. IEEE Trans. Antennas Propagat., 53, pp.2054-2063, June 2005.

A new Fast Physical Optics method is introduced for computing backscattered fields at high frequencies. An analytic manipulation is presented for computing highly oscillatory integrals. A deep mathematical theory called Morse Theory is applied to accelerate the solution time. The computational cost of the scattering integral behaves as O(1) with frequency. The accuracy is excellent.

The solution of very large electromagnetic scattering problems is complicated in general. There are different asymptotic methods that allow to obtain approximate solutions at high frequencies. In this article a new 2D Fast Physical Optics (2DFPO) method is used to solve electrically large scattering problems. Physical Optics consists of approximating induced currents to the scaterer by PO currents. Therefore Scattered field is obtained by means of the PO integral over the scatterer. For some extremely large problems the PO integral is highly oscillatory, so the brute force numerical integration may become very time consuming. The numerical integration of highly oscillatory integrals is a problem with a large number of applications within different fields of science such as: quantum cosmology, nuclear physics, acoustic scattering or, in the matter at hand, electric scattering.

This paper shows a new method of integration by means Morse Theory. In virtue of the Morse lemma, PO integral, which is highly oscillatory, can be approximated with an excellent accuracy by another highly oscillatory integral easily solved by means of the analytic calculation of an antiderivative of the integrand. This method can be applied to any one dimensional highly oscillatory integral.

The above mentioned method is applied to calculate the backscattered field produced by one cylindrical surface. The following figure (fig 1) shows the PO integral for scattered field at R=16 Lambdas. The following table shows the results at different frequencies with the brute force numerical integration and the 2DFPO method. We can see the agreement between the numerical brute force integration and the method above mentioned. The total computation time is 0.203 seconds for 2DFPO and 23.589 seconds for numerical integration. We observe that the computation time does not increase with frequency with 2DFPO.

Table1
Radius (lambdas)·········2DFPO (V/m)·········Numerical integration (V/m)······Error (%)
····160···············0.022288 + 0.051427i······0.022286 + 0.051428i·············0.0025295
···1600··············0.017486 - 0.002897i·······0.017486 - 0.002897i··············0.0002064
··16000·············0.003670 - 0.004236i·······0.003670 - 0.004236i··············0.0000204
·160000············0.000124 - 0.001768i·······0.000124 - 0.001768i··············0.0000023

Fig 1

In present time a serious attention is paid to the design of artificial materials the properties of which are not encountered in any natural substance (e.g., left-handed materials). Our goal within this work is the development of the solution methods for this class of problems. Nowadays usual numerical methods of electromagnetics such as FEM, FDTD are utilized in metamaterials science. The common feature of these approaches is the use of local approximations of electromagnetic (EM) fields, i.e. the unknown in the solution method presents a field quantity at a given point in space. Thus, one unfavorable property holds: the enlargement of structural complexity of the material (e.g., the enlargement of the number of "atoms") leads to essential rise of algebraic complexity of the problem. We next inquire "what should investigator do with the solution method if he or she wants to consider large (but finite) ensembles of atoms of complicated geometry and at the same time to consume sufficiently small computer resources?" The answer is seen in utilizing the methods based on analytic approximations of wave fields. The benefit offered by these techniques is evident: each harmonic of the expansion does not relate to any concrete element of the structure, it describes the behavior of EM system in whole. The basic problem of known semi-analytic approaches is the requirement to analytic approximation to remain convergent right up to the boundary of the domain in which the solution is looked for by means of this approximation. Unfortunately, this instance is automatically fulfilled for relatively narrow class of geometrical objects. Complicated inhomogeneous structures investigated in metamaterials science do not easily satisfy this requirement.

Recently, on the ground of the ideas of pattern equation method and null-field method we have derived certain integrodifferential equations for the solution of scattering problems with 3d perfectly conducting (PEC) bodies. Further the method has been generalized for dielectric bodies and scatterers with boundary conditions of miscellaneous kind: PEC-dielectric. For the sake of validation analytical solutions have been derived for PEC and dielectric spheres that coincide with Mie solutions. Further the generalization of the method has been proposed for the cases in which difficulties arise with the validity of Rayleigh hypothesis (e.g., infinitesimally thin plates and shallow shells). In this work the method is applied to the analysis of scattering properties and spectral response of electrically dense gratings of finite extent composed of split ring resonators (SRR). It is worth noting that it turned out possible to use only two multipole expansions for a whole ensemble of these inclusions. The quantitative criterion of the validity of Rayleigh hypothesis has been derived for circular ring that remains also valid for multilayered array of SRRs. It has been also shown that scattering pattern of a single SRR as well as the array of SRRs has the properties of Huygens source. The scattering patterns and spectral responses have been found for arrays consisting of 100-1000 SRRs. Qualitative agreement of spectral responses with literature data have been demonstrated.

In known paper (1948), Chu ascertained that the bounds of antennas gain were depending from possibility of antenna matching to the spherical harmonics of field. Chu discovered the sequence of 2-Port circuits that were been equivalent to the spherical harmonics in this sense. Chu and authors of the further publications were usually limiting analysis to approximation of the 2-Ports by the resonant circuits. In this message, I offer the new algorithm for computation the limiting gain of the antenna that is enclosed in an imagined sphere and acts in the preassigned frequency band. The algorithm is based on the exact analysis of the constraints conditioned by the properties of the Chu 2-Ports. There are absent explicit restrictions associated with number of the radiated spherical waves. The new version of the broadband matching method is used for computing of the power transfer upper bound of a ladder 2-Port. The computed upper bound of gain depending on an antenna size are given. The results are correct for any antennas, in particular, for supergain ones.

The Finite Element-Boundary Integral (FE-BI) method is used in the context of a stationary iteration scheme to solve for the radiation patterns of a finite array of identical elements. The radiating element is a cavity-backed slot/patch (CBS/P) antenna which is decomposed into the interior region and the aperture region. The array elements are coupled to each other through the aperture unknowns and the free-space Green’s function. This field interaction between array elements is taken into account using a BI formulation. The resulting matrix system consists of the main block diagonal, which involves the finite-element self-matrix, as well as the lower and upper block triangular matrices which consist of the coupling matrices. For identical elements, the block entries of the main diagonal will be the same, thus saving memory space by storing just a single matrix, which is sparse. The coupling matrices, on the other hand, are full matrices but smaller in size compared to the self-matrix since they correspond to the aperture unknowns. In case that the array elements are equally spaced, then only the first block row of the system matrix must be stored in memory because of its Toeplits structure.

Solution of this block matrix system can be obtained using any iterative technique including Conjugate Gradient (CG) methods. Instead, it was decided to use stationary iterative techniques in conjunction with a sparse Incomplete LU (ILU) decomposition. The ILU decomposition is used to factorize the self-matrix before entering the iteration process. The iteration process begins by solving the block matrix system as if the upper and lower block triangular matrices were zero. This is equivalent to a finite array whose elements are totally isolated; i.e., as if there were no coupling among elements. This is considered as the initial solution which is continuously updated through the iteration process by taking into account coupling among nearby elements. The algorithm allows the user to define an area in the vicinity of an element where field interaction takes place. No field interaction is allowed with array elements that reside outside this area in order to further improve convergence and reduce memory allocation.

The first iterative technique to be implemented is the Jacobi method, which uses the solution from the previous iteration and a back-substitution of the factorized self-matrix to update the fields of the radiating elements. The Gauss-Seidel method uses current updates of the fields as well as previous updates, thus providing theoretically an improvement in the convergence of the algorithm. The third iterative method is the Successive Over Relaxation (SOR) method which is a generalization of the Gauss-Seidel method. Based on preliminary results, a 100-element linear array of CBS antennas was run on a 3 GHz, 500Mb RAM PC, and it was found that the Jacobi method converges a bit faster than the Gauss-Seidel method. Specifically, the Jacobi method took 15 iterations and a total time of 11 minutes to converge to a residual norm of less than 0.01, whereas the Gauss-Seidel method took 25 iterations and a total time of 13 minutes. Coupling was allowed with the neighboring 20 elements on each side.

The finite-difference time-domain (FDTD) method has been extensively used to simulate open problems. However, the computation can be long when problems such as antennas with their environment are involved. Indeed, the antenna geometry requires a fine description to have good accuracy on the near-field parameters such as the impedance. As a consequence, a simple FDTD simulation leads to a spatialy oversampled volume that increases the simulation time.

One way to overcome this problem is to use subgriding FDTD. It consists in dividing the FDTD volume in various areas that are meshed with different cell size. However, subgriding schemes imply interpolation and/or extrapolation of the fields on the boundaries of the areas at each time step of the simulation. Those mathematical operations can introduce undesired instability along the time.

In this paper we propose a new multiresolution approach to simulate this kind of structures. The FDTD simulation of the structure is now divided in two parts (figure 1):
- First of all, the antenna and its environment are simulated with a fine FDTD till the switch time T_sw. During this part of the simulation, a fine mesh is used to take into account the near-field variations. As a consequence, the impedance of the radiating element is evaluated with a good accuracy.
- Then, the near-fields in the volume are interpolated and injected in a coarse FDTD. The structure simulation is ended using this coarse grid, accurate enough to traduce the contribution of the environment.

This approach enables the simulation to be faster than the fine FDTD, and more accurate than the coarse FDTD only. It is clear that this method is slower when compared with a subgriding scheme, but the interpolation is made only once. Therefore, the generated instability can be controlled by correcting the near-field values during the switching time.

Simulations have been performed for an ultra-wide band (UWB) antenna surrounded by another UWB antenna (figure 2). A significant gain in simulation time has been obtained with good accuracy on the various results such as the impedance, the radiated fields and the transmission coefficient between each antenna. For example, figure 2 presents the resulting S parameters for the fine FDTD, the coarse FDTD and our multiresolution approach with a switch time T_sw equals to 3/10.T_obs. The simulation is 12 min 34 sec long for the fine FDTD, 18 sec for the coarse one, and 4 min 21 sec for our multiresolution scheme.

We present in this paper a fictitious domain approach based on a Discontinuous Galerkin Time-Domain (DGTD) method, on a cartesian grid of parallelepipedic elements.

The fictitious domain method was designed to solve partial differential equations in exterior domains by embedding a perfect conducting metallic surface in a regular grid. The grid is artificially extended inside the obstacle and surface currents on the triangularised surface of the conductor have to be taken into account). The method therefore ends up as a local perturbation of a numerical scheme based on a cartesian grid, and it turned out to be really efficient for time dependent problems, in particular for the electromagnetic scattering by a perfect conductor using Yee's FDTD method ([1], [2]). In the general case, the convergence of the method depends on a uniform inf-sup condition, leading to a compatibility condition between the boundary mesh and the volumic mesh ([4]).

This method allows to take into account the geometry of metallic structures without staircase. Moreover, A DGTD method is perfectly adapted to calculate the main characteristics of antennas, such as impedances, electric and magnetic currents at the same time and localisation, radiation patterns.... ([3]).

This method is applied to some radiating devices analysis such as line patch antennas horns and wave guides. The results are compared to those obtained with measurements and with an harmonic integral equation method (SR3D).

References
[1] Sylvain Garcés. Une méthode de domaines fictifs pour la modélisation des structures rayonnantes tridimensionnelles.Thesis. Sup'aero. november 1997.
[2] Francis Collino et al. Fictitious Domain Method for Unsteady Problems: Application to electro-magnetic Scattering. INRIA RR-2963. 1996.
[3] Nicolas Canouet. Méthodes de Galerkin Discontinu pour la résolution du systčme de Maxwell sur des maillages localement raffinés non-conformes. Thesis. Ecole nationale des Ponts et Chauss\'ees. december 2003.
[4] V.Girault and R.Glowski. Error analysis of a fictitious domain method applied to a Dirichlet problem. 1994.

A general formulation entailing efficient treatment of frequenct dependent material dispersion in the transmission line modeling (TLM) technique is developed. The method uses more complicated shunt and series loadings than conventional TLM in order to handle such dispersion. The formulation is different from currently proposed methods. There are a number of advantages of our approach: ability to model directly frequency dependent materials based on measured data, satisfaction of Kramers-Kronig relations. In addition, susceptiblity and convolution are not used in the formulation which makes its numerical implementation easy and fast. The numerical dispersion using this method is evaluated. The computational model developed has potential applications for modeling frequency dependent materials in a number of areas such as power systems, absorbing structures used to line anechoic chambers, industrial dielectric substrates, and plasma.

Radiating cables were originally developed in order to provide a signal coverage in such areas as tunnels, mines, underground etc. Nowadays radiating cables start to be of use in many different applications. One of the most important applications is indoor signal coverage. The radiating cables almost uniformly distribute the signal energy along its length and therefore do not produce so called hot spots in the vicinity of the source as point antennas do. Also the signal strength is decaying slower with the radial distance comparing to the point antennas. Thus in the case of radiating cables the coverage is more uniform and the electromagnetic exposure can be smaller.

A radiating cable is a coaxial cable with specially designed slots in its outer conductor. The electromagnetic field traveling inside the cable is through these slots coupled outside and by reciprocity the field outside the cable influences the inner field of the radiating cable. The amount of the radiated energy is determined by the dimensions, shape and number of slots. The frequency band is dependent on the positions and orientation of the slots.

The main parameters of the radiating cables are insertion and coupling loss. Insertion loss is attenuation of the signal traveling inside the radiating cable and is measured in dB per unit length. Coupling loss represents the ration of the power inside the cable to the power received by a dipole antenna at a specified distance from the cable. Both quantities are in general frequency dependent. Frequency bandwidth of the particular radiating cable is determined as a frequency interval where both above mentioned quantities have the required values.

Using a full-wave electromagnetic simulator CST Microwave Studio the frequency bandwidth, insertion and coupling loss of coaxial radiating cables were investigated based on dimensions, positions and shape of the slots. An extensive parametric study was conducted to examine all the dependencies. As an illustration, Fig. 1 a) depicts some of the parameters describing the slots in a coaxial cable. Example of the simulation results is shown in Fig. 1 b). Results of the parametric study as well as a discussion of optimal shapes and dimensions of the slots will be presented in the full paper. The presented work is a part of activities of CTU Prague in the frame of ACE2.

Fig. 1: a) Some of the parameters describing the slots in a coaxial cable. b) Computed S-parameters of one particular coaxial radiating cable.

Unknown current distribution (UCD) and input impedance of the vertical dipole antenna (VDA) placed above linear, isotropic and homogenous lossy half-space, are determined in this paper. For the purpose of solving of this problem, the system of integral equations of Hallen's type (SIE-H) is used, which was numerically solved using the point-matching method and the polynomial current approximation. The influence of electrical parameters of the lossy half-space (permittivity and finite ground conductivity), expressed by the Sommerfeld's integral kernel (SIK), is modelled in a new and simple manner, giving the results of satisfying accuracy. The modelling of the SIK corresponds to introduction of two fictitious images.

This kind of approach to SIK modelling can be found in the available literature (e.g. [Yang, C., Zhou, B., IEEE Trans. on EMC, Vol.46, pp. 133-141, 2004.]). However, the SIK approximation proposed there, although simple, generates significant error whilst evaluating the SIK in the surroundings of the VDA, making its application in the majority of real earth parameter cases practically unjustifiable.

In this paper, an approximate expression for the SIK evaluation was assumed in a form that corresponds to the two-image approximation, with unknown constants that are determined in the process of approximation in a new and simple way. The approximation procedure is similar to the one used by the authors in [Rančić, P., Kitanović, M., AEÜ, Vol. 51, pp. 155-162, 1997.] and [Rančić M., Rančić P., AEÜ, 2004. (accepted for publishing)].

In the case of a vertical Hertz’s dipole (VHD), the spectral reflection coefficient (SRC) in the integrand of the SIK can be written in a two-image approximation form, which consists of two addends. The first one, denoted as B, is an unknown constant, and the second one is in a form of an exponential function with two more unknown constants. All three of the mentioned unknown constants are obtained by matching the SRC at certain points.

Once unknown constants are determined, the approximate form of the SRC is substituted into the SIK, which now gains an approximate form that can be easily calculated. The model of the SIK consists of two terms: one that corresponds to the image in the flat mirror, and the other one that relates to the fictitious image placed at real distance.

Besides simplicity, the proposed approximation is satisfyingly accurate in the surroundings of the antenna for the wide range of electrical parameters of the lossy half-space (i.e., permitivitty range - [1,81], conductivity range - (0,∞]). This conclusion is confirmed by comparison to accurate SIK calculations from [Petrović V., ETF Belgrade, 2005.] and [Đorđević A. et all, AWAS for Windows, Artech House, 2002.]. A part of the obtained numerical results is presented in Figs. 1a and 1b.

In radio-astronomical applications one aims to study very weak galactic sources using sensitive antenna (array) receiver systems. The signal-to-noise ratio is a measure for the receiver system sensitivity and needs to be maximized. For designing such receiver systems, a tool is required to model both the scatter and noise characteristics of the antenna array and its noisy environment, as well as the RF beamforming network. Currently, none of the commercial tools provides us with the capability to model, e.g., a non-uniform sky noise temperature distribution and simulate the entire antenna system (efficiently). We developed an antenna system simulator comprising an ElectroMagnetic (EM) simulator and a MicroWave simulator (MW) for modeling the signal and noise characteristics of the entire antenna array system. In addition, the EM simulator makes use of the Characteristic Basis Function Method (CBFM) to compute large array antennas e±ciently. All receiver components are characterized by scattering matrices and noise wave correlation matrices. The simulator has been verified against commercially available codes and measurements. Results will be presented for a 4-monopole reference array system that we measured and modeled using the previously described simulator. The figure below illustrates the pertaining antenna system where the inner 2 elements are beamformed and the outer 2 elements are terminated by noisy 50­ resistors. For this measurement we performed a "hot-cold" test and obtained excellent agreement between the measured and predicted absolute output noise power, both for the "hot" and "cold" situation. Figure 1: Absolute output noise level simulated and measured; a comparison between measurements and simulations.

When employing commercially available software in antenna design the computational complexity often prohibits the inclusion of the antenna surroundings such as a large finite ground plane. In many cases antenna designs are thus based on assumptions of infinite ground planes and hence the impact of the finite ground plane on the antenna radiation is not always known prior to actual fabrication and testing of the antenna. However, from such analyses certain quantities, e.g., impedance characteristics, currents on wires, or fields in apertures, can typically be extracted. From this information the antenna can be approximately represented by an impressed incident field whose interaction with the ground plane forms a scattering problem. In principle this requires that the above-mentioned antenna characteristics can be assumed to be unaffected by the introduction of the finite ground plane which, however, is a reasonable assumption for large ground planes.

In the present work the Method of Auxiliary Sources (MAS) is employed to solve the thus established scattering problem and determine the influence of finite ground planes on the antenna radiation. The standard impedance boundary condition (SIBC) is assumed to hold on the ground plane surface and thus general surface impedances can be modelled. Pairs of electric or magnetic crossed Hertzian dipoles, with independent excitations, are used as auxiliary sources, and the SIBC is tested in test points on the ground plane surface. Examples of the approach are given in virtue of two practical antenna cases. The first is a cavity-backed annular slot antenna and the second a small planar phased array antenna with printed dipole elements. The antennas are investigated with ground planes of different shapes and sizes and the obtained far-field results are compared with reference simulations and measurements. Comparison of the calculated and measured results generally shows good agreement and two examples are given in the figures.

Dielectric rod protruding from a waveguide is one of the attractive phased-array element due to its design simplicity. Such elements allow providing good array match in a wide sector of scanning. They can also shape flat-topped radiation patterns corresponding to minimization of controlled elements in the limited field-of-view arrays. In practice, planar waveguide arrays with conical rods are usually used. However, there are no numerical results for such arrays in the literature. At present, they can be simulated by commercial software of general type like HFSS. However, the calculations in this case usually require great expenses of computer resources, and, for this reason, the indicated software is not always convenient for multiple calculations usually required in the process of numerical optimization of the array geometry.

In the present work, a new algorithm is proposed for analysis of infinite planar arrays of stepped circular waveguides with protruding dielectric rods as, in particular, it is shown in the Figure below. The algorithm is based on representing the transverse electric and magnetic field components in the layer comprising the rods in the form of expansions over transverse vector functions of the Floquet channel with unknown coefficients depending on the longitudinal coordinate. The longitudinal field components are determined via the transverse ones from Maxwell equations. Substitution of the fields into the Maxwell equations and projection of the latter on the same transverse functions (the Galerkin method) give a system of ordinary differential equations different from that corresponding to the incomplete Galerkin method described in the literature. The differential equations are solved by the one-dimensional finite-element method with using the triangular basis and testing functions. The fields on the bottom and top boundaries of the rod layer are matched to the ordinary waveguide and Floquet harmonics, respectively, with resulting in a block-banded system of linear algebraic equations. The algebraic system is solved by a fast specialized algorithm based on the Gauss elimination method, and the solution is used for calculation of the array characteristics of practical interest.

High efficiency of the algorithm operation is confirmed by the results of testing including the power balance check and comparison with calculated and measured data available in the literature for some particular cases. Some new results of practical interest for developers of the arrays in question are also presented and discussed.

Faraday's vectors are by definition , where . In view of this replacement and denotations of Maxwell equations become . In a homogenious space the Maxwell equations will be converted to a system of two independent equations about the Beltrami vortexs. Under these circumstances, Faraday's each vector describes a total electromagnetic field of an ideal circular polarization. The mutual transformation of Faraday's different vectors emerges in nonuniform mediums and on boundaries of areas of a homogenious space. The similar replacements of vectors of a field in the Maxwell equations over and over again were occurring in the literary sources during 20 century, however they have not found of wide application in the antenna theory and engineering. A purpose of my message is to show efficiency of application of Faraday's vectors to problems of investigation of polarizing, diffraction and other of the antenna performances.

The application of Faraday's vectors makes simpler the problem of a finding of antenna parts which are source for the cross-polarization defects. The formulating of the criterion of absence of such defects at diffraction on an impedance surfase is becoming also accessible with Faraday's vectors. It is becoming also simpler the asymptotic deciding of the problem of an electromagnetic fields diffraction from a curvilinear reflector edge. Electromagnetic field in the Faraday's vectors form may be represented by atypical spherical wave expansion. Such expansion allows better to understand problems of creating of a supergan antenna, and also to calculate limiting gain and broadbandness of antenna.

PLEASE VIEW ATTACHMENT, AS IMAGE # 10 DOES NOT APPEAR SUCCESSFULLY. (ECB/CK)

In this paper a scheme to obtain an adaptive method in space for the resolution of Maxwell's equations is presented. Using interpolating wavelets it is possible to obtain an adaptive grid allowing a reduction of the computation time and an economy of the computational resources.

Currently, the numerical methods for the resolution of electromagnetic problems have a great acceptance and the obtained results are very good when compared with the measured results. There are a variety of methods, but the Yee's FDTD (Finite Difference Time Domain) scheme is one of the most popular methods in computational electromagnetics.

FDTD models a region of space by dividing it into cells. In each of these cells the value of the three components of the electric and magnetic fields are calculated in different points of the cell (staggered grid) and stored, for a given time. From these values, we can obtain a new set of values in a later instant by solving the equation in a recursive way, thus advancing in time until the steady state is reached. Despite its simplicity and modelling versatility, the technique suffers however from serious limitations due to the use of uniformly dense grids leading to long computation times to simulate such grid. The method is also limited for the Courant stability factor. To improve the performance of FDTD method, we can use the wavelet's theory for the resolution of Maxwell's equations. In this context the first method to appear was the MRTD (Multiresolution Time Domain Technique).

There are two versions: S-MRTD and W-MRTD. The first uses scaling functions only, the second uses scaling functions and wavelets.

In this paper we present another type of adaptive strategy named SPR (Sparse Point Representation) that uses biorthogonal-interpolating wavelets. With this strategy is possible to obtain an adaptive mesh as a function of time that allows to an economy of resources and a relatively short or acceptable time of simulation. This grid is refined only in certain regions of the space, and less refined in other regions where the variation of the fields is smoother. The principle of the method is to represent the solution only through those points values indicated by the significant wavelet coefficients, which are defined as interpolating errors. Using this method it is possible to use a number NS of points less than the N points of the original representation. It is acceptable to assume that this sparse representation could lead to efficient algorithms in terms of computer memory and the number of arithmetic operations. When we apply this technique, there are two options, for the solution of Maxwell's equations: in the first, each component of the field has one independent grid, as in the FDTD case (staggered grids). In the second, the grid is common for both the electric and magnetic field (non-staggered grids). In the present paper we demonstrate some of the potential of interpolating wavelets for the resolution of Maxwell equations, in a non-staggered grids model, in terms of the dispersion and stability proprieties, through the results of a numerical simulation in one dimension.

Electromagnetic solvers based on Finite-Difference Time-Domain (FDTD) methods remain one of the most popular due to their high accuracy, stability, simplicity and attractive computational cost. Traditional orthogonal Yee grid guarantees the second-order accuracy, however it may become a limitation when one is trying to conform it to complex geometries. Unstructured polyhedral tessellation can easily be applied in these cases. Significant modifications to FDTD solvers are needed in order to maintain the accuracy of EM solvers. Discrete divergence-preserving methods applied to the integral form of Maxwell’s equations have been proposed in some works in mid-nineties (e.g. [1,2] and references therein). The numerical examples provided in these papers show only planar examples though.

A second-order accurate scheme with a correction step involving an intermediate evaluation of the magnetic field at the grid nodes is proposed in this paper. In this case the electric field projections are evaluated along the edges with second-order accuracy and the divergence-free condition of the field is enforced. Perfectly-matching layer, full field – scattered field separation, and inhomogeneous formulations are implemented to illustrate the solution for several 3-D benchmark problems. Distribution of the magnetic field in a cylindrical space tessellated with conformal hexaedral elements with resolution 20 samples per wavelength is shown in Figure1. The corresponding normalized maximum error within several cycles is shown in Figure 2.

Figure 1. Field distribution at the cut plane: f = c/(20dx)

Figure 2. Maximum error for the magnetic field in 5 cycles at frequency f = c/20dx

1. S.D. Gedney, F.S.Lansing, and D.L.Rascoe, "Full wave analysis of microwave monolithic circuit devices using a generalized Yee-algorithm based on unstructured grid", IEEE Trans. MTT, 44, no.8, pp.1393-1400.
2. A. Taflove and S.C. Hagness, Computational electrodynamics, Artech House, Boston, London, 2000.

In this work, a periodic finite difference time domain (FDTD) analysis [1] is applied for the first time in the study of a two-dimensional (2-D) leaky-wave planar antenna based on dipole frequency selective surfaces (FSSs). While leaky-wave antennas (LWAs) have been extensively studied, works presenting a periodic FDTD analysis of such structures have only very recently appeared in the literature cite [2].

A 2-D periodic LWA has been proposed in the past comprised of a metallic array, such as a frequency selective surface (FSS), which acts as a Partially Reflective Surface (PRS) for the waves excited inside the cavity. The operation of such an antenna configuration was studied in [3], based on resonant optical cavity theory. In this paper, the FDTD method is used for the dispersion analysis of this antenna. First, the use of the FDTD method for modelling of periodic structures is reviewed and some of its aspects are studied in detail for the first time, in relation to their effect when leaky modes are present. Analyzing the structure's unit cell with the FDTD model, the complex propagation constants of the radiating leaky-wave modes are calculated, and radiation patterns based on the calculated values are derived.

In particular, the sine-cosine method [1] is employed for the periodic analysis of the structure with the FDTD. Our analysis focuses on the study of the leaky-wave modes that are responsible for radiation in the frequency of operation of this antenna. To this end, we examine the effect of the terminating absorbing boundary condition (ABC) and the use of an efficient tool for improvement of the frequency resolution in the prediction of the leaky mode resonant frequency.

Referring to the role of the ABC, a comparison between a second-order Mur ABC and a carefully optimized UPML medium shows that the implemented ABC's efficiency is very important for the accurate prediction of the leaky mode's resonant frequencies. Furthermore, the decaying nature of the temporal waveform corresponding to a leaky mode also poses a problem regarding the time steps available for extraction of the mode's resonant frequency via an FFT of the time-domain responses. As illustrated with a specific example of FDTD-computed waveforms that include both leaky and propagating modes, the former disappear long before they provide sufficient samples for good frequency resolution. To overcome this problem, we apply a Pade model to the early portion of the signal and resolve the leaky mode frequency with the desired accuracy.Dispersion diagrams for propagation along the two axes of the antenna, corresponding to the E- and H- planes of radiation are then derived. We also discuss how the FDTD results can give additional information helpful in understanding this antenna's operation.

References

[1] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Boston, 2000, ch. 13.
[2] T. Kokkinos, C. D. Sarris and G. V. Eleftheriades, "Periodic Finite-Difference Time-Domain modeling of leaky-wave structures applied to the analysis of negative-refractive-index metamaterial-based leaky-wave antennas," IEEE Trans. MicrowaveTheory Tech., in press.
[3] A. P. Feresidis and J. C. Vardaxoglou, "High gain planar antenna using optimized partially reflective surfaces," IEE Proc-Microw. Antennas Propag., vol. 148, No. 6, pp. 345-350, Dec. 2001.

A simplification of the analysis and design of field patterns of multidimensional structures is discussed.

Application of the method of moments (MoM) to the analysis problem yields a matrix description of the structure the admittance matrix. The excitation is represented by a voltage matrix. The product of the admittance matrix with the voltage matrix gives the current on the structure. Use of the MoM has been well established for several decades with known advantages: mutual coupling between array elements is taken into account and no unreasonable assumptions need to be made regarding the current distributions on the wires.

Next, the array factor and the source distribution are related through discrete Fourier transform reducing much of the work done in the analysis and synthesis of point source arrays. With this procedure, a single matrix relates the structure excitation and the far field pattern. With proper manipulation on this matrix, according to the number of field points and feed points, it's possible to obtain the feed voltages for a desired pattern.

This communication describes theory in three dimension structures. The effectiveness of the proposed method is illustrated with known examples in one and two dimensions.

VLF/LF telecommunications (Very Low Frequency band : 3 to 300 kHz) allow informations to be transmitted until thousands of kilometres with a good quality of service. The ability of those transmissions to penetrate seawater permits a fully submerged submarine to receive signals in VLF band. Typically, VLF antennas are electrically large structures with a multitude of arbitrarily oriented thin conductors. Various numerical techniques such as MoM can be used to model these antennas. As their performances are subjected to the influence of the environment (soil characteristics, ground plane, surrounding infrastructures) and components (tuning coil, isolators), finite difference techniques seem to be more suitable to obtain a global analysis. In addition, time domain simulations provide transient analysis which allow to study corona effects due to high voltages involved in these structures.

However, because of the multi-scale modelling problem, finite difference time domain method encounter difficulties to model VLF antennas. Thus, it is necessary to develop an accurate model that describes in detail the physics of the features of wires without it results in very large computations. Several thin wires models based upon physical approximation coupled with empirical parameters have been developed in Transmission Line Matrix method. Nevertheless, all those previous developments are restricted to centred thin wires without offset relative to the Cartesian grid.

An arbitrarily oriented thin wires model using Transmission line Matrix method has been developed [1]. Its accuracy has been evaluated for canonical case as a dipole in various disposals. The results has been successfully compared to theory and to those provided by the FDTD method [2].

In this paper, we provide experimented and simulated data (Fig.1) for typical T-antenna in VLF broadcast. The influence on the input impedance of the oriented arms will be discussed. Crossed loop antennas, typically used in receiving mode, are also simulated. In particular, the influence of the angles will be investigated. Simulation of more complex existing VLF antennas such as large umbrella antennas are also proposed.

Fig. 1 : Comparison between experimented and simulated data for a cage T antenna.
Base resistance (Rb), reactance (-jXb) and radiation resistance (Rr)

[1] B. Larbi, JL. Dubard, C. Pichot, "Implementation arbitrarily oriented thin wire in TLM Method", Proceedings PIERS 2006, 26-29 March 2006, Cambridge

[2] F. Edelvik. "A new technique for accurate and stable modeling of arbitrarily oriented thin wires in the FDTD method". IEEE Transactions on Electromagnetic Compatibility, 45(2):416-423, May 2003.

In the last few years, the Transmission Line Modeling (TLM) method has become a powerful and versatile algorithm for dealing with different types of propagation problems. To do so, the TLM uses unitary circuits, termed nodes, which contain the information described in Maxwell equations. Since the work by Peter B. Johns in 1987 introducing the Symmetrical Condensed Node (SCN), the TLM method has significantly evolved to nowadays become an efficient and powerful numerical tool for the numerical solution of complex electromagnetic field problems.

Although many bounded problems are relatively easy to model, free space simulation by the TLM method is not straightforward because of the limitations of computer resources, such as memory or computation time, which impedes to entirely model a medium with infinite size. Thus, in the simulation of wave propagation in an open space, it is necessary to artificially limit the modeling area by using appropriated boundary conditions.

In this field of modeling infinite size media, the response of an infinite multilayered dielectric slab to an electromagnetic plane wave is a case of particular interest, with applications in the field of new materials analysis and design. Only the case of normal incidence on plane slabs is usually considered in low frequency numerical methods, such as TLM. This is so because infinite plane wave propagation in an infinite size ideal mesh may be simply modeled by using symmetry conditions to allow a mesh reduction to a limited and operative size. Quantities of great interest, such as the reflection or transmission coefficients, for single or multilayered dielectric slabs can easily be numerically obtained with these symmetry conditions. Results may also be applied to model new materials with special electromagnetic properties, such as materials with negative electric permittivity and/or magnetic permeability. Nevertheless, in all these cases, the symmetry of the boundary conditions only allows the study of normal incidence.

In this work, an efficient TLM algorithm for investigating the oblique incidence of an electromagnetic wave on a multilayered dielectric slab is proposed. A technique based on the use of the properties of the constant phase and amplitude planes is implemented using a TLM mesh of three-dimensional SCN. Time-filtering algorithms are also considered to avoid time instabilities produced when the signal is of low amplitude. The validity of the proposed technique is illustrated by calculating reflection and transmission coefficients in particular situations involving multiple internal reflections.

Acknowledgement. This work has been supported in part by the 'Ministerio de Educacion y Ciencia' of Spain under project reference FIS2004-03273 cofinanced with FEDER funds of the European Union.

The numerical solution of fields in finite arrays of antennas or scatterers has received a lot of attention over the last few decades. One approach consists of starting from infinite-array solutions and to correct for the effects of array truncation. Other approaches exploit iterative solvers including fast matrix-vector multiplications, based on multipole expansions or on Fast Fourier Transforms. Finally, other methods are based on sub-entire domain basis functions [1]. They rely on the hypothesis that the fields on a given unit cell in the large finite structure can be accurately described as a linear combination of solutions obtained for very small problems. In the present paper, we build on the latter approach. A bottleneck of this method is the computation of the Method-of-Moments impedance matrices for interactions between all macro basis and testing functions located on different cells. We will show how the interactions between macro-basis and macro-testing functions can be computed very efficiently with the help of multipole expansions.

First, as usually done, the macro basis and testing functions are obtained from the solution of a small problem. Second, their patterns are computed. If macro basis and testing functions are located on nearby cells, the interactions are computed as usual. If they are located further away, a multipole formulation is used. This formulation holds for metallic elements as well as for dielectric elements, with the help of the surface equivalence principle. For arrays made of complex elements, the number Ω of directions to be considered for numerical integration of the multipole expansion can be smaller than the number N of decomposition functions in each cell. The numerical complexity of the standard formulation is proportional to N², while that of our formulation is proportional to Ω. Hence, the very large computational gain is obvious. Some more details on the formulation can be found in the attached file "eucap_craeye.pdf"'.

We extensively studied the speed and accuracy of this method for arrays of metallic scatterers, and the formulation for extension to dielectric structures has been numerically verified. In the full paper, simulation results, as well as details about the computational complexity of the proposed method, will be provided for scattering by finite arrays of dielectric spheres.

[1] J. Yeo, V. Prakash, and R. Mittra, ``Efficient analysis of a class of microstrip antennas using the characteristic basis function method (CBFM),'' Microwave Optical Technol. Lett., vol. 38, pp. 444--448, Sep. 2003.

[2] R. Coifman, V. Rokhlin, and S. Wandzuraz, ``The fast multipole method for the wave equation: a pedestrial prescription,'' IEEE Antennas Propagat. Mag., vol. 35, pp. 7--12, June 1993.

This paper presents a combined analytical and a numerical method to simulate the wave propagation in semi-anechoic chambers in the frequency range of 30-200 MHz. The analytical method is based on the homogenization technique, whereas the numerical technique is based on the transmission line modeling known by its acronym TLM. The purely numerical simulation is not an attractive option due to the complex (complicated) geometry of the absorbers used inside the shielded enclosure; So, the technique is to reduce the numerical aspect of the simulation through the use of the homogenization technique until it could fit into a programmable phase of the solution. The homogenisation method yields the effective permittivity and permeability which "average" the absorber/air media and yield an equivalent one-dimensional homogenized plane-layered media instead of modeling a geometrically tapered absorber. The TLM is a cavity oriented technique and is well suited to evaluate the performance of shielded enclosures. In this paper, we suggest a formulation of modeling frequency dependent material that is accurate, reliable, and easy to apply for a general class of commercial absorbing materials. While developing this work, the following features have been observed: 1. The mathematical formulation of the problem should be of general nature; that is, it should not be restricted to particular conditions or cases. 2. The numerical method of the formulation developed is easy to implement 3. The numerical implementation is fast and accurate. The additional time, cost, and effort involved should be justified. In other words, the numerical implementation is considerably faster than the case where frequency by frequency analysis is carried out.

70 m radio telescope in Suffa, Uzbekistan (RT-70) is going to be the world largest MM wave radio telescope [1]. Its construction started in the beginning of 80-ies and it was 65% ready in 1990 before the decay of USSR At present serious efforts are made in Russia and Uzbekistan to complete RT-70 project. Geometry and antenna design of RT-70 is given.

For multibeam simulation in MM band we need to know antenna aberrations and characteristics with feed removal from the radio telescope focus in tens and hundreds wavelengths but the main and nearest sidelobes are often needed in the principal and cross- polarization. PO is the most widely used technique for large reflector analysis which correctly predicts the main antenna beam and nearest sidelobes in the principal and cross-polarization for smooth surface deformations.

Yet, even a middle sized radio telescope in MM band is extremely large in terms of the wavelength and use of PO for simulation even a single beam pattern becomes impossible or very slow. For simulation of the main lobe and nearest sidelobes of RT-70 with a focused and defocused feed we used Scalar Kirchhoff diffraction integral (SKDI)/Projected aperture method and methods of direct Integration of Aperture Field (IAF) in principal polarization [2] which give us acceptable accuracy. The processing time for simulation of 3-D 4x4 beam pattern at 3 mm is 24 hours with Pentium-IV/3.1GHz PC. Gregory optics, the sub-reflector, struts are involved in simulation with GO which is also used to calculate actual focus position. Expected aperture field and its distortions caused by large scale thermal surface deformations are taken into account in the simulation with GO based on results of full-scale 3D FEM analysis [3]. Results of 10x10 beam pattern simulation for RT-70 at 3 mm wavelength are presented.

References

1..B.F.Zabolotny, N.S.Kardashev, Yu.N.Artemenko, A.A.Parshikov, G.I.Shanin. All Russian Astronomical Conference (VAK-2001), Sankt-Petersburg, 2001. 2.V. Khaikin. Simulation of antenna characteristics of MM-wave radio telescope with multibeam focal plane array. In Proceed. of XVII ESA Antenna Workshop, Noordwijk, The Netherlands, May, 2005.
3. A. I. Borovkov, D. V. Shevchenko , V. G. Gimmelman , Yu. I. Machuev and A. V. Gaev . FINITE-ELEMENT MODELING AND THERMAL ANA.LYSIS OF THE RT-70 RADIO TELESCOPE MAIN REFLECTOR International Conference on Antenna Theory and Techniques, 9-12 September, 2003, Sevastopol, Ukraine, pp. 651-654.

The diffraction due to a finite thin slot antenna in a perfectly conducting plane loaded by a dielectric stratified structure is here addressed. This problem is particularly important dealing with both EMC and antenna design applications. In fact, for what concerns EMC applications, the computation of the shielding effectiveness of a structure containing perforated shield is in general not simple also from a numerical point of view, and it is very important to get closed form expressions of the shielding effectiveness of this kind of perforated shields whenever it is possible.

On the other hand, slot antennas have been widely introduced for different applications from microwaves to mobile communications since, compared with the printed ones, they can provide a wider bandwidth, feature that is nowadays particularly appealing for the design of mobile applications of the last generation.

In this paper the diffraction due to the presence of a thin finite slot will be addressed through an analytical approach based on the Wiener-Hopf approach in the spectral domain via a suitable rephrasing of the considered problem in terms of equivalent circuits.

The considered geometry is depicted in Fig.1. A perfectly conducting plane shield is introduced between two volumes, one is filled with air, the other one presents a dielectric stratified structure. An external, incoming from the air filled volume, field impinges( TE wave ) on the shield where a thin slot aperture h long and l high, with h>>l, has been introduced.

In this paper the main idea is to show how it is possible to get a closed form solution to the considered problem through the development. In order to do that the double spatial Fourier transform is introduced, leading to a suitable formalism in the spectral domain rephrasing the electromagnetic problem in an equivalent circuit network model problem. The circuit network approach in the spectral domain yields a sort of Wiener-Hopf equation that can be analytically solved, through a suitable extended version of the Wiener-Hopf approach, or numerically faced and iteratively solved with the requested accuracy through the Method of Moments in the spectral domain

Finally , a Wiener-Hopf approach for solving the dielectrically loaded thin finite slot problem as an extension of the unloaded case has been adopted.

Most methods for numerical calculation of electromagnetic fields are limited in their application to objects that are not very large compared to the wavelength, because of both the storage and execution time these methods require. The method of moments, which is based on integral equations, has the advantage of requiring unknowns only on the body of the scatterer. Even so, the application of MoM to large electromagnetic problems with linear dimensions of many wavelengths poses considerable difficulties, because we need to deal with the large and densely populated MoM matrix equations, which require huge amounts of memory and lead to prohibitively long computer run times.

Recently, researchers have investigated a number of ways to circumvent the problems associated with high memory requirements and computing times for solving the linear systems for large electromagnetic problems. A partial list of these techniques include the Impedance Matrix Localization (IML), the Pre-corrected Fast Fourier Transform (FFT), the Fast Multipole Method (FMM), and the Generalized Sparse Matrix Reduction (GSMR). But, most of these still use iterative methods that can have convergence problems, especially for multi-scale structures. In this paper, the characteristic basis functions method (CBFM) is introduced as a technique for reducing a large MoM matrix to a relatively small and well-conditioned one. This new approach is based on the use of characteristic basis functions (CBFs) that are high level basis functions defined over subdomains, called blocks, that are much larger than those used in the conventional Method of Moments that utilize the subdomain bases. The use of the CBFs leads to a reduction in the number of unknowns in three main steps. In the CBFM, the original geometry is first divided into a number of blocks or macro-domains.

Next, the CBFs are constructed via the SVD approach. Finally, the reduced matrix, the size of which is relatively small, is generated for the weight coefficients of the CBFs by using the Galerkin testing procedure, and is solved directly. The reduced matrix generation can be considerably simplified by taking advantage of the symmetry of the block geometry.

The CBFM is a general approach, and it can be applied to arbitrary geometries including combinations of surfaces and wires. It includes the mutual coupling effects rigorously and thus generates accurate solutions in a systematic manner.

A very important attribute of the CBFM is that it handles multiple right hand sides (incident fields) as well as multiple frequency solutions very efficiently-a feature not found in the other iterative-type approaches for solving large problems that we have mentioned above. Finally, by its very nature, the CBFM algorithm is highly parallelizable, which also distinguishes it from other techniques that have to be adapted for distributed processing. Time-performance for the parallel code is shown in Fig. 1. Figure 2 shows the monostatic RCS of a NASA almond at a frequency of 7GHz.

Introduction:Ray-tracing methods have demonstrated their usefulness in many application fields, from optics to seismology and acoustics in complex environment. Most of this methods rely on a far field assumption, and the source is represented as a radiating point source, from which rays are traced to each observation point. For reflector antennas or radar cross section computations, another method not relying on the same far field assumption has become very popular in the last twenty years, the Shooting and Bouncing Rays , and the Generalized Ray Expansion.

In this paper, we present the formulation of Spectral Ray Tracking method (SRT), validate its concepts in cases where exact solution are known, and demonstrate analytically its potentialities in the multiple reflections for the dielectric waveguide and cavities analysis. SRT is a numerical method, which allows to start from arbitrary source distribution, with a computational effort limited to one 2D, it is based on the discretization of radiating fields not in the spatial domain, but in the spectral domain for an arbitrary source distribution without any approximation, neither far field or asymptotic.

Principal of the Method:To find the field at an observation point P, we sweep the directions of arrival to that point with ray tubes launched backwards from P. In a multi-reflecting and/or refracting environment, it is possible to track a ray launched from the observation point through successive local refractions and reflections. For a given `tube'' of directions of arrival, the ray path is found and saved after this backward launching step. When the four rays tube reaches the source plane, the directions of the rays, projected on the transverse plane of the wave vectors space, define a transverse differential surface in the spectral domain. With the knowledge of both this spectral surface and the aperture distribution plane wave spectrum, we calculate the field associated to the ray tube. This field is then transformed along the ray path previously saved, following the usual Geometrical Optics rules: in multi-reflecting and/or refracting environments, propagation of the field along a ray tube not only changes the phase of the field, but also its amplitude and direction, through reflection and transmission operators, and through phase front transformations at curved interfaces.

Validation

We have already applied this method in the context of dielectric lens antenna analysis, in free space propagation, the rectangular and circular waveguides, and parabolic antenna.

Comparisons between the exact modal solution and numerical results obtained with the SRT method will be presented in the case of a dielectric waveguide excited by a given mode. In this case, a thorough calibration of the method will be presented, versus the spectral domain sampling rate and the number of reflexions to be taken into account, as a function of the distance where fields are calculated. Finally, comparisons between SRT and modal results for a waveguide cavity will be presented, in order to illustrate and validate the SRT method.

The problem of the plain wave diffraction on a slot in an ideally conducting screen is one from a few classical diffraction problems which have not explicit analytical solution up to the date. The only known rigorous solution of this problem in a form of the Mathieu functions series is very difficult for practical applications. This situation gave rise to the numerous works devoted to the approximate and asymptotic solutions of the problem. (The survey of results and appropriate references see for example in [1]). The aim of this work is to present the explicit analytical solution of the more general problem - diffraction of the point-source field on a slot, for which the plane wave case is, as is known, a limiting one. This passage to the limit and general investigation of the solution obtained will be the subject of the subsequent publications. Solution became possible due to the application of a new method of solving the diffraction problems - Partial Scattering Operators Method, - which has been presented in [2]. It has many other applications and seems very promising when applied to the scattering on several bodies.

References.
1. H. Horn, A. W. Maue, K. Westpfahl. Theorie der Beugung. Handbuch der Phycik, Band XXV/1, S. 218. Springer-Verlag, Berlin-Gottingen-Heidelberg; 1961.
2. I.L. Verbitskii. Partial Scattering Operators Method and Some of its Applications, report on Millenium Conference on Antennas and Propogation, April 2000, Davos. Abstracts, vol.1, p.591.

In the present work we consider the propagation of a plane electromagnetic wave by a waveguide formed by two parallel two-part planes consisting of the junction of perfectly conducting and impedance half-planes (See Fig.1). For the sake of generality, the relative surface impedances of the half-planes defined by {x<0,y=0} and {x<0,y=b}are assumed to be different. The representation of the solution to the boundary-value problem in terms of Fourier integrals leads to two simultaneous Wiener-Hopf equations which are uncoupled by using the analytical properties of the functions that occur. The solution involves two infinite sets of unknown coefficients satisfying two infinite systems of linear algebraic equations. These systems are solved numerically and the influence of the parameters such as the waveguide spacing and the surface impedances of the parallel two-part plane on the radiation phenomenon is shown graphically.