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Publication# Electromagnetic modelling of planar circuits in bounded layered media

Abstract

Printed circuits in bounded media encompass a wide range of practical structures such as discontinuities in waveguides, planar circuits embedded in shielded multilayered media or even two-dimensional printed periodic structures. The Electromagnetic (EM) modeling of printed circuits in layered bounded media is performed via an Integral Equation (IE) technique. Green's functions (GFs) are specially defined to satisfy both the Boundary Conditions (BCs) imposed by the layered media and by the transverse boundary enclosing the entire structure. Finally, a system of IEs on the equivalent sources can be solved numerically by means of the Method of Moments (MoM). Each of the problems enumerated above has already been solved by other authors using IE-MoM techniques. Nevertheless, our formulation introduces a unified approach applicable to all the aforementioned problems. Due to the symmetry presented by a bounded layered media, the GF problem can be reduced into a two-dimensional transverse boundary problem and a one-dimensional transmission line problem in the normal direction. Both problems can be treated independently. This thesis proposes and fully develops an efficient technique that encompasses different laterally bounded multilayered problems with a seamless transition between them. The method is based on a modal representation of the transverse boundary problem and on the expansion of the equivalent surface currents by zero-curl & constant-charge Basis Functions (BFs). It offers a unified and versatile approach that, on one hand eliminates redundancy in the formulation and on the other hand simplifies each particular problem to the evaluation of constant coefficients or basic line integrals. Analytical solutions can be found for the combination of linear subsectional basis functions in rectangular and circular Perfect Electric Conductor (PEC) boundaries as well as for periodic lattices. This thesis then solves the problem of transmission line model in the longitudinal direction by proposing an efficient algorithm that guarantees numerical stability under a variety of known critical conditions where other already known formulations fail. In addition, it introduces alternate equivalent expressions of this formulation that allow new interpretations of the problem. Due to its practical interest, the method is applied for the EM modeling of multilayered boxed printed circuits. This motivated the implementation of a dedicated software tool for the efficient analysis of these topologies including losses. Extensive numerical experiments have been carried out to assess the validity of the aforementioned theory and some properties of test-structures (losses, mesh, etc).

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To meet strict requirements of the information society technologies, antennas and circuit elements are becoming increasingly complex. Frequently, their electromagnetic (EM) properties cannot be anymore expressed in closed-form analytical expressions mainly because of the multitude of irregular geometries found in actual devices. Therefore, accurate and efficient (in terms of computational time and memory) electromagnetic models coupled with the robust optimization techniques, are needed in order to be able to predict and optimize the behavior of the innovative antennas in complex environments. The contribution of this thesis consists in the development and improvement of accurate electromagnetic modeling and optimization algorithms for an ubiquitous class of antennas, the planar printed antennas. The approach most commonly used to model and analyze this type of structures is the Integral Equation (IE) technique numerically solved using the Method of Moments (MoM). From the computational point of view, the main challenge is to develop techniques for efficient numerical evaluation of spatial-domain Green's functions, which are commonly expressed in terms of the well-known Sommerfeld integrals (SIs), i.e., semi-infinite range integrals with Bessel function kernels. Generally, the analytical solution of the SIs is not available, and their numerical evaluation is notoriously difficult and time-consuming because the integrands are both oscillatory and slowly decaying, and might possess singularities on and/or near the integration path. Due to the key role that SIs play in many EM problems, the development of fast and accurate techniques for their evaluation is of paramount relevance. This problem is studied in detail and several efficient methods are developed. Finally, the applicability of one of these methods, namely the Weighted Averages (WA) technique, is extended to the challenging case appearing in many practical EM problems: the evaluation of semi-infinite integrals involving products of Bessel functions. However, the development of effective analysis codes is only one aspect. At least equally important is the availability of reliable optimization techniques for an adequate design of antennas. For that purpose, the Particle Swarm Optimization (PSO) algorithm is introduced in the context of our analysis codes. Moreover, the innovative hybrid version of the PSO algorithm, called the Tournament Selection PSO, has been proposed with the aim of even further improving convergence performances of the classical PSO algorithm. Detailed theoretical description of this socially inspired evolutionary algorithm is given in the thesis. Finally, the characteristics of both algorithms are compared throughout several EM optimization problems.

The study of the lightning interaction with tall strike objects has attracted considerable attention of lightning researchers lately. Many lightning measurements including current and associated electromagnetic fields were recently made all over the globe namely in Russia, South Africa, Germany, Brazil, Japan, and Austria. It is a novel area of studies, and the resolution of associated questions will have an impact upon many lightning-related applications such as lightning protection and the determination of lightning parameters from remote field measurements. The main objective of the thesis is to carry out further theoretical investigations and experimental measurements to understand and elucidate recently raised questions on the characteristics of lightning return-strokes to tall structures and their associated electromagnetic radiation. Chapter 2 presents a review on recent progress in the modeling of lightning strikes to tall towers and associated experimental data obtained during the last decade or so. Two types of return stroke models namely the Engineering Models, and the Electromagnetic or Antenna-Theory (AT) models, extended to take into account the presence of a tall strike object are discussed. The Chapter contains also a description of the computational methods for the evaluation of electromagnetic fields generated by a lightning strike to a tall structure, as well as an overview of available data on lightning current and associated electromagnetic fields. The chapter finally highlights some important questions raised by different research groups in the past few years which call for further investigations. These questions are as follows: No systematic theoretical analysis nor experimental data are available for electromagnetic fields in the immediate vicinity of a tall structure struck by lightning. The characterization of nearby electromagnetic fields is particularly important in the analysis of the interaction to nearby electrical and electronics systems. Why do lightning return stroke models not reproduce the far-field zero crossing associated with lightning to tall structures? How should these models be revised to be able to reproduce such an effect? How should the engineering models be revised in order to remove the associated current discontinuity at the return stroke wavefront? It is well-known that the measurements of electromagnetic fields from lightning are affected by the presence of nearby buildings and metallic structures. However, no systematic and quantitative analysis of such an effect is presently available in the literature. The work presented in this thesis addresses all of the above questions. The main original contributions of this thesis, consisting of both theoretical and experimental work, are presented in Chapters 3 through 6. Chapter 3 is devoted to a theoretical description of the signature of electric and magnetic fields at very close distance associated with lightning strikes to a tower. It is shown that the electric field generated by a lightning return stroke to a tall structure can change polarity at very close distance range. This change in the polarity seems to be a specific signature of the very close vertical electric field. A simple equation is derived which provides an estimate of the critical distance below which such an inversion of polarity might occur. It is also shown that the inversion of polarity depends on the value of the reflection coefficient at the base of the tower and disappears for reflection coefficients close to 1. On the other hand, other parameters such as the return stroke speed, the reflection coefficient at the top of the strike object, and the adopted return stroke model seem not to have an impact on the inversion of polarity. Simulation results also showed that the electric field peak at distances beyond the height of the tower or so exhibits the typical 1/r dependence. At closer distances, however, the E-field peak features a saturation, due to the so-called tower shadowing effect. This shadowing effect results in a substantial decrease of the nearby electric field. On the other hand, the magnetic field peak varies inversely proportional to the horizontal distance and does not depend significantly on the presence of an elevated strike object. Chapter 4 introduces an improved version of the engineering models for return-strokes to tall structures which accounts for (1) the presence of possible reflections at the return stroke wavefront, and, (2) a return stroke initiation above the structure due to an upward connecting leader. We also propose an elegant iterative solution that can be easily implemented into computer simulation programs to take into account in a straightforward way multiple reflections occurring at the discontinuities at the tower ends and at the return stroke wavefront. Simulation results for the magnetic fields are compared with experimental waveforms associated with lightning strikes to the CN Tower (553 m). It is shown that taking into account the reflections at the return-stroke wavefront results in better reproducing the fine structure of the magnetic field waveforms. Chapter 5 presents and discusses obtained measurements of electric (vertical and radial) and magnetic fields from leaders and return strokes associated with lightning strikes to the Gaisberg tower in Austria obtained in 2007 and 2008. The data include simultaneous records of vertical and radial electric fields, which were obtained for the first time at such close distances. It is found that the vertical and radial electric field waveforms appear as asymmetrical V-shaped pulses. For the vertical electric field, the initial, relatively slow, negative electric field change is due to the downward leader and the following fast positive field change is due to the upward return stroke phase of the lightning discharge. For the horizontal electric fields, however, the bottom of the V is not associated with the transition from the leader to the return stroke. The horizontal field change due to the return stroke is characterized by a short negative pulse of the order of one microsecond or so, starting with a fast negative excursion followed by a positive one. In addition, an analytical expression for the radial electric field, assuming a uniform charge distribution along the leader with constant speed is derived. It is also shown that the return-stroke vertical electric field changes appear to be significantly smaller than similar measurements obtained using triggered lightning. This finding confirms the shadowing effect of the tower predicted by the theoretical analysis of Chapter 3, which results in a significant decrease of the electric field at distances of about the height of the tower or less. Finally, the ability of two different models for the return stroke in reproducing measured vertical and horizontal electric fields is tested using the obtained measured data. The considered models are (1) the engineering MTLE (Modified Transmission Line with Exponential Decay) model, and (2) the electromagnetic model implemented using the Numerical Electromagnetics Code NEC-4. It is shown that both models predict electric field waveforms which are in reasonable agreement with measured waveforms. In general, the predicted fields by the electromagnetic model appear to be in better agreement with measured data, because of the direct use of the measured current waveform as an input and the more accurate representation of the tower. Chapter 6 reports on the effect of nearby buildings on electromagnetic fields from lightning. Indeed, sensors used for the measurement of lightning electric and magnetic fields are often placed close to or on top of buildings or other structures. Metallic beams and other conducting parts in those structures may cause enhancement or attenuation effects on the measured fields. Experimental waveforms radiated from distant natural lightning recorded during the summers of 2006 and 2007 are presented. Electric and magnetic field waveforms were measured simultaneously on the roof of a building and on the ground at different distances away from it. The results suggest that the measured electric field on the roof of the building could be enhanced by a factor of 1.7 to 1.9, whereas the electric fields on the ground experienced a significant reduction by a factor ranging from 5 to 20. Also, it is shown that for a sensor located on the ground close to a building, the magnetic field component perpendicular to the building can experience significant attenuation, presumably due to the effect of the induced currents in the building. The magnetic field on the roof of the building seems not to be significantly affected by the building. Simulations using the Numerical Electromagnetic Code (NEC-4) were also carried out in which the building was represented using a simple wire-grid model. The simulation results support in essence the findings of the experimental analysis, despite quantitative differences which are ascribed, at least in part, to the oversimplified model of the building.

The aim of this thesis is the study of the simulation of electrically large structures and the application of the results to automotive Electromagnetic Compatibility (EMC). The theoretical and experimental work carried out has led to the development of computational tools and to the further understanding of the mechanisms involved in the representation of solid surfaces by means of wire-grid simplifications. The work was done in the context of a European project GUIDELINES FOR ELECTROMAGNETIC COMPATIBILITY MODELLING FOR AUTOMOTIVE REQUIREMENTS (GEMCAR). The first two chapters of the thesis contain a description of the GEMCAR project, a brief overview of some of the existing numerical methods for electromagnetic simulations (particularly, the ones used in GEMCAR), and the explanation of efficient, general simulation strategies that can be applied to different methods. The concept of adaptive sampling and its application are also introduced there. The main original contributions of this thesis are presented in Chapters 3 through 6. They consist of theoretical and experimental work as follows. We present, in Chapter 3, a modified version of the Numerical Electromagnetics Code (NEC). This version, which we have called Parallel NEC, has been adapted to run on parallel supercomputers, taking advantage of the combined processing power and memory of several processors working as a team. Parallel NEC has been implemented in two different supercomputing architectures to test the portability of the code. The original NEC routines in charge of the calculation and filling of the interaction matrix have been modified to work in a parallel environment. The matrix is now distributed among the available processors and the elements of the matrix are locally and individually calculated by their "owners". Thus the number of integrals carried out to build the complete matrix equation gets shared, diminishing the necessary runtime for this time-consuming operation. The system of equations is also solved using a parallel version of the Gauss-Doolitle algorithm. However, the most important feature of Parallel NEC is the possibility to use the distributed memory of the processors. This allows the calculation of problems of a size never achieved before using this numerical method without the need of using disk-space as swap memory. The code has been tested with models containing over 20.000 segments, exhibiting execution times comparable to those obtained with a single-processor PC calculating models of one tenth of that size in terms of the number of segments. Parallel NEC is also able to adapt itself automatically to its environment. It will detect the number of available processors and will take advantage of all available memory and calculating resources. The validation of Parallel NEC has been carried out in two steps. First, it was validated using simulation results obtained with other numerical methods. Then, it was validated by using experimental data from the GEMCAR project. The experimental setup as well as the validation are presented in Chapter 4. With the purpose of validating the numerical models developed in GEMCAR, we participated in a number of experimental campaigns carried out at Spiez, Switzerland in 2000 and 2001 using the VERIFY (Vertical EMP Radiating Indoor Facility), an EMP simulator belonging to the Swiss Defense Procurement Agency. Measurements of electric and magnetic fields inside a real vehicle (a Volvo S80) featuring different levels of complexity were carried out. These measurements were performed at 8 different points inside the car and at two points on the surface of the body-shell. The above-mentioned levels of complexity consisted of (1) a "simple test case", comprising the vehicle body-shell (without all doors or glazing), (2) a "medium complexity case" which, this time, included the doors, and (3) a "complex case", consisting of the complete car with all mechanical, electrical and electronic equipment installed. The data used in Chaper 4 refer to the "simple test case", although the "medium test case" measurements are also available . Other partners of the GEMCAR project carried out experimental testing on the three models using other sources of illumination (see Chapters 1 and 4). It is interesting to mention for completeness that, as part of the GEMCAR project, a cable harness was installed following the approximate path of the original cabling of the car, but composed of single wires with 50 R terminations. Current measurements were made at 4 observation points located at the ends of the branches of the harness. These current measurements are not given here as the subject of this thesis was limited to field measurements and simulations only1. The developed code was applied to analyze the penetration of electromagnetic fields inside the vehicle's body shell (i.e., the simple case). The computed results agree well with those obtained with the other methods and with the experimental data obtained from measurements. The application of the code to such a large problem permitted the observation of some issues raised by the application of the so-called Equal Area Rule (EAR) for the calculation of the segments' lengths and radii. In Chapter 5, we discuss the wire-grid representation of metallic surfaces in numerical electromagnetic modeling. We present the origins and the evolution of surface wire-grid modeling and, considering two types of geometries, namely (I) a simple cube, and (2) a complex structure represented by the metallic car shell used in Chapter 4, we show that the Equal Area Rule is accurate as long as the wire-grid consists of a simple square mesh. For more complex body-fitted meshes, such as rectangular and triangular grids, the Equal Area Rule appears to be less accurate in reproducing the electromagnetic field scattered by metallic bodies. In Chapter 6 we present a theoretical development that leads, for the case of a square grid representation of a surface, to the same formula proposed by the Equal Area Rule. This development is, to the best of our knowledge, the first physical and mathematical interpretation of the EAR as of today. Our development shows, however, a different value for the radius of the segments if the representation of the surface uses other polygons, such as in the case of a rectangular or a triangular mesh. To compare the two methods (the traditional versus the new EAR), we carried out a simple numerical test and found that the Equal Area Rule does not always predict the optimum wire radius for the mesh-representation of a surface. ------------------------------ 1The interested reader is referred to: A. Rubinstein, F. Rachidi, D. Pavanello, and B. Reusser. Electromagnetic field interaction with vehicle cable harness: An experimental analysis. In International Conference on Electromagnetic Compatibility, EMC Europe. Sorrento, volume 1, Sep 2002. Proceedings.