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Recently, Radio Frequency Identification (RFID) technology has become commonplace in many applications. It is based on storing and remotely retrieving the data embedded on the tags. The tag structure can be chipped or chipless. In chipped tags, an integrated IC attached to the antenna is biased by an onboard battery or interrogating signal. Compared to barcodes, the chipped tags are expensive because of the existence of the chip. That was why chipless RFID tags are demanded as a cheap candidate for chipped RFID tags and barcodes. As its name expresses, the geometry of the tag acts as both modulator and scatterer. As a modulator, it incorporates data into the received electric field launched from the reader antenna and reflects it back to the receiving antenna. The scattered signal from the tag is captured by the antenna and transferred to the reader for the detection process. By employing the singularity expansion method (SEM) and the characteristic mode theory (CMT), a systematic design process is introduced by which the resonant and radiation characteristics of the tag are monitored in the pole diagram versus structural parameters. The antenna is another component of the system. Taking advantage of ultra-wideband (UWB) technology, it is possible to study the time and frequency domain characteristics of the antenna used in chipless RFID system. A new omni-directional antenna element useful in wideband and UWB systems is presented. Then, a new time-frequency technique, called short-time matrix pencil method (STMPM), is introduced as an efficient approach for analyzing various scattering mechanisms in chipless RFID tags. By studying the performance of STMPM in early-time and late-time responses of the scatterers, the detection process is improved in cases of multiple tags located close to each other. A space-time-frequency algorithm is introduced based on STMPM to detect, identify, and localize multiple multi-bit chipless RFID tags in the reader area. The proposed technique has applications in electromagnetic and acoustic-based detection of targets. / Ph. D.

Les freins aéronautiques sont soumis à des instabilités vibratoires induites par le frottement. Il en résulte des vibrations qui présentent un risque pour la structure du frein et de l’atterrisseur et posent des problèmes d’intégration. Safran Landing Systems doit donc répondre à des spécifications avionneurs strictes sur les niveaux des vibrations générées par son équipement. Le respect de ces spécifications est actuellement contrôlé par la réalisation d’essais de freinage longs et coûteux. L’objectif de ces travaux de recherche est de reproduire numériquement ces phénomènes vibratoires via des outils intégrables au processus de conception d’un frein. Le crissement de frein, bien qu’il soit l’objet de recherches depuis le début du XXe siècle, demeure un phénomène assez mal compris, notamment dans l’aéronautique. Des vibrations instables apparaissent régulièrement sur l’ensemble de la plage fréquentielle 0-2 kHz. Au cours de la dernière décennie, une instabilité vibratoire vers 200 Hz dénommée whirl 2 s’est manifestée de manière récurrente et souvent critique sur la plupart des nouveaux freins développés. On cherche donc à mettre en place une méthode permettant de simuler l’apparition et les amplitudes des instabilités vibratoires, notamment du mode de whirl 2. Dans une première partie, on présente des analyses d’essais vibratoires réalisés en conditions opérationnelles et expérimentales. On décrit ensuite la modélisation par la méthode des éléments finis du frein instable au sens de Lyapunov. La stabilité du système linéarisé est étudiée et on montre une corrélation en fréquence et déformée entre le modèle et les essais. Ce modèle éléments finis est trop volumineux en l’état pour permettre la simulation d’amplitudes de vibrations non linéaires. On propose donc dans une seconde partie deux méthodes de réduction adaptées à l’architecture complexe d’un système de freinage aéronautique et permettant la prise en compte du frottement. La première est une méthode semi-analytique qui se révèle très performante jusqu’à 500 Hz. La seconde méthode de réduction mise en oeuvre est la double synthèse modale. Elle est implémentée dans sa version classique, puis une amélioration est proposée avec succès : la double synthèse modale complexe. La troisième partie est consacrée à l’étude de la dynamique non linéaire du whirl 2 par la réalisation d’intégrations temporelles. La simulation des amplitudes de vibration nécessite la prise en compte réaliste du comportement non linéaire du frein. Or, on fait d’abord le constat que, contrairement à une hypothèse communément admise, les non-linéarités de contact situées aux interfaces frottantes ne suffisent pas à expliquer à elles seules la saturation des amplitudes vibratoires constatée expérimentalement. La recherche des phénomènes physiques non linéaires influents nous amène a considérer l’interaction de la structure vibrante avec le circuit hydraulique de commande du frein. La modélisation du couplage hydrodynamique fournit alors des éléments de compréhension inédits et permet de formuler des règles de conception. Enfin on étudie l’impact du frottement sec dans les contacts périphériques des disques de freinage avec la structure. Ce phénomène, jusque là négligé, apparaît largement prépondérant. Des études d’influences, présentant une bonne corrélation avec les essais, permettent de mettre en évidence de manière robuste l’influence du design et des scénarios de freinage sur les amplitudes vibratoires. / These vibrations are a threat for the brake and landing-gear structural integrity and represent an issue in terms of integration. Thus Safran Landing Systems has to comply with aircraft manufacturers’ strict requirements on the vibration amplitude its product is likely to generate. Compliance to these requirements is assessed by long and costly braking test campaigns. The objective of the research presented here is to reproduce by simulation the brake dynamic instabilities with numerical tools that could be integrated in the design process. Brake squeal has been a research topic since the early XXth century. However it remains a rather ill-understood phenomenon, especially in aeronautics. Unstable vibrations regularly appear on the whole 0-2kHz frequency spectrum. In the last decade, an instability located around 200 Hz called whirl 2 persistently appeared on the newly developed wheel and brake assemblies, sometimes exhibiting critical vibration amplitudes. Consequently, Safran Landing Systems wishes to develop numerical tools able to simulate both the occurrence and the amplitudes associated with friction-induced instabilities, especially with the whirl 2 mode. In the first part of this report, an experimental analysis of the brake is conducted, on both laboratory and in operational set-ups. The modelling of the wheel and brake assembly using the finite element method is then described. The system stability in a Lyapunov’s sense is studied and shows good correlation in both frequencies and mode shapes with the experiments. This finite element model is too big to be used to perform the transient simulation of the nonlinear amplitudes. In the second part, two reduction methods, tailored to the complex aircraft brakes architectures, are thus presented. The first method is a semi-analytical. It shows excellent performances up to 500 Hz. The second reduction method is the double modal synthesis, implemented under its classical version. It is then successfully improved and called "complex double modal synthesis". The third part is dedicated to the study of the nonlinear dynamics of the whirl 2 through transient analyses. The nonlinear amplitudes simulation requires taking into account the relevant nonlinear brake behavior. However, it is first observed that, contrary to a commonly accepted hypothesis, the contact nonlinearities located at the friction interfaces cannot single-handedly account for the vibration amplitudes saturation observed in the tests. The need to identify the relevant physical phenomena leads then to consider the interaction between the squealing brake structure and its hydraulic command circuit. The modelling of the hydro-mechanical coupling provides an unprecedented insight and allows to prescribe design rules. Finally, we study the impact of dry friction in the peripheral contacts between the braking discs and the structure. This phenomenon, neglected until now, appears to have a major influence. Sensitivity studies exhibit a good correlation with tests, allowing to highlight, in a robust manner, the impact of brake design and braking scenarii on the nonlinear vibration amplitudes.

Flextensional actuators assembled in association with piezoceramics feature the amplification of nanometric displacements generated by the ceramics energy conversion. For applications that require high precision positioning or vibration response attenuation, such as hard disc reading or atomic force microscopy, a response tracking control needs to be implemented. Shell and plate piezoactuators with vibration control have been extensively studied in literature, however the design of controlled piezoelectric systems by means of the Topology Optimization Method (TOM) has not been fully explored in literature yet, and is generally focused on the frequency domain transient analysis, which employs a model reduction method for the sake of computational implementation. Dealing with transient analysis of flextensional piezoelectric actuators, an active closed loop control design is more suited for the positioning and vibration problem, which consists on measuring the outputs of the system by the closed loop sensor layer, whose signal is modified by a control gain and eventually inputted into the actuator layer so the system response signal is modulated. Aiming to enhance the active feedback control in piezoelectric actuators (PEAs), this work targets the design of the flextensional microstructure considering an active velocity feedback control (AVFC), where the active piezoelectric sensing and actuating cycles imply in an extra damping to the system. Therefore, the flextensional mechanism compliance shall be distributed within the design domain by the allocation of void regions where there should be the flexible hinges. Such a design can be accomplished by means of the TOM, which employs a systematic analysis of the dynamic model through the finite element method (FEM). In this work, the finite element (FE) system model takes into account the piezoelectric ceramics intermediate nodes, what is denominated as non-collapsed piezoelectric nodes model, and whose induced voltage during the time domain dynamic response contributes to the active control of the system. The topology optimization (TO) problem is formulated for the system vibration suppression at the restoring position and at the actuated position (positioner) subject to material volume and design variables constraints. The TOM implemented is based on the solid isotropic material with penalization (SIMP), the dynamic adjoint sensitivity, and on the optimization solver known as sequential linear programming (SLP). To illustrate the method, bidimensional examples of optimized topologies are numerically obtained by employing different velocity feedback control gains, and the topologies efficiency are compared and contrasted. / Atuadores piezoelétricos flextensionais funcionam como amplificadores nanométricos dos deslocamentos gerados pela piezocerâmica. Em sistemas que necessitam de alta precisão de posicionamento final ou baixa energia de vibração após sofrer um impacto, como na leitura de um disco rígido ou na microscopia de força atômica, requer-se que o atuador conte com um fator de correção de posicionamento, o qual pode ser obtido através de uma lei de controle. A utilização de material piezoelétrico para o controle de vibração em dispositivos de casca e placa foi amplamente abordado na literatura, porém o projeto de sistemas piezoelétricos controlados utilizando-se do Método da Otimização Topológica (MOT) foi pouco explorado e em geral é focado na análise transiente no domínio da frequência, o qual necessita que o problema tenha que ser reduzido para que a implementação computacional torne-se viável. Tratando-se de análise transiente em atuadores piezoelétricos flextensionais pode-se considerar o emprego de um controle ativo, o qual captura informações do dispositivo através de piezo-sensores e as realimenta na forma de um sinal de entrada em piezo-atuadores para modulação do sinal de resposta. Visando aprimorar o efeito do controle ativo em atuadores piezelétricos, este trabalho é voltado para o projeto de sua estrutura flextensional considerando o controle ativo de realimentação de velocidade, em que o ciclo envolvendo sensoreamento e atuação piezoelétricos fornece um amortecimento extra ao sistema. Deseja-se portanto que a flexibilidade do mecanismo flextensional seja distribuída no domínio de projeto pré-definido alocando vazios em regiões ao redor de onde devem estar presentes articulações da estrutura flexível, o que é obtido pelo MOT. Para encontrar a distribuição otimizada de material no domínio de projeto, o MOT emprega a análise sistemática do modelo dinâmico através do Método dos Elementos Finitos (MEF). Neste trabalho a modelagem do sistema para o MEF leva em consideração a presença dos nós intermediários das cerâmicas piezoelétricas, denominada modelagem para nós não colapsados, cuja tensão gerada ao longo da resposta dinâmica temporal influencia no controle ativo do sistema. O problema de Otimização Topológica (OT) é formulado para a atenuação da vibração do sistema em posição neutra e em posição atuada (posicionador) sujeito a restrições de volume e a valores máximo e mínimo que as variaveis de projeto assumem. A implementação do MOT é baseada no modelo de material denominado Material Isotrópico Sólido com Penalização (MISP), no cálculo da sensibilidade dinâmica adjunta, e na rotina de otimização conhecida como Programação Linear Sequencial (PLS). Para ilustrar o método, são projetados dispositivos bidimensionais e diferentes ganhos de controle de realimentação de velocidade são utilizados para obtenção da topologia otimizada, analisando-se a eficiência em cada caso.

<p>Electromagnetic transients impinging on electromagnetic power devices - such as electric machines, transformers and reactors - can stress the design severely. Thus the magnitudes of the transients are often decisive for the design of the devices. Further, the operation of a device can be transient in itself. This is the case for the explosive magnetic flux compression generator (EMG) and a ferromagnetic actuator. </p><p>Models are presented that are mainly intended for transients in the millisecond range and faster. Hence, eddy currents and the related skin and proximity effect become significant in windings, magnetic cores and in the armatures of the devices. These effects are important for, e.g., the damping of the transients. Further, the displacement current in the insulation of the winding is significant. It changes the response of the windings dramatically, as it manifests the finite velocity of propagation of the electromagnetic fields. Under such circumstances, reflections and excited resonances can make the transient voltage and current distribution highly irregular. </p><p>Induced voltages are modelled with self and mutual inductances or reluctances combined with winding templates. The displacement currents are modelled with capacitances or coefficients of potential. Cauer circuits and their dual form are used to model eddy currents in laminated cores and in conductors. The Cauer circuit enables one to consider hysteresis and the non-linear response of a magnetic core. It is also used to model the eddy currents in the moving armature of an EMG. </p><p>A set-up is presented that can be used to study the transient voltage and the current distribution along a coil. </p><p>The transient response of coaxially insulated windings is analysed and modelled in detail. A lumped circuit model is developed for a coil, Dryformer^TM - the new high-voltage transformer - and Powerformer^TM, the new high-voltage generator. An alternative model, a combined lumped circuit and FEM model, is presented for a coaxially insulated winding in two slot cores.</p>

Electromagnetic transients impinging on electromagnetic power devices - such as electric machines, transformers and reactors - can stress the design severely. Thus the magnitudes of the transients are often decisive for the design of the devices. Further, the operation of a device can be transient in itself. This is the case for the explosive magnetic flux compression generator (EMG) and a ferromagnetic actuator. Models are presented that are mainly intended for transients in the millisecond range and faster. Hence, eddy currents and the related skin and proximity effect become significant in windings, magnetic cores and in the armatures of the devices. These effects are important for, e.g., the damping of the transients. Further, the displacement current in the insulation of the winding is significant. It changes the response of the windings dramatically, as it manifests the finite velocity of propagation of the electromagnetic fields. Under such circumstances, reflections and excited resonances can make the transient voltage and current distribution highly irregular. Induced voltages are modelled with self and mutual inductances or reluctances combined with winding templates. The displacement currents are modelled with capacitances or coefficients of potential. Cauer circuits and their dual form are used to model eddy currents in laminated cores and in conductors. The Cauer circuit enables one to consider hysteresis and the non-linear response of a magnetic core. It is also used to model the eddy currents in the moving armature of an EMG. A set-up is presented that can be used to study the transient voltage and the current distribution along a coil. The transient response of coaxially insulated windings is analysed and modelled in detail. A lumped circuit model is developed for a coil, DryformerTM - the new high-voltage transformer - and PowerformerTM, the new high-voltage generator. An alternative model, a combined lumped circuit and FEM model, is presented for a coaxially insulated winding in two slot cores.

Current trends in electric power consumption indicate an increasing use of dc in end-user equipment, such as computers and other electronic appliances used in households and offices. With a dc power system, ac/dc conversion within these loads can be avoided, and losses reduced. AC/DC conversion is instead centralized, and by using efficient, fully controllable power-electronic interfaces, high power quality for both ac and dc systems during steady state and ac grid disturbances can be obtained. Connection of back-up energy storage and small-size generation is also easier to realize in a dc power system. To facilitate practical application, it is important that the shift from ac to dc can be implemented with minimal changes. Results from measurements carried out on common household appliances show that most loads are able to operate with dc supply without any modifications. Furthermore, simple, and yet sufficiently accurate, load models have been derived using the measurement results. The models have been used for further analysis of the dc system, both in steady state and during transients. AC microgrids have gained research interest during the last years. A microgrid is a part of power systems which can operate both connected to the ac grid, and autonomously in island mode when the loads are supplied from locally distributed resources. A low-voltage dc microgrid can be used to supply sensitive electronic loads, since it combines the advantages of using a dc supply for electronic loads, and using local generation to supply sensitive loads. An example of a commercial power system which can benefit from using a dc microgrid is data center. The lower losses due to fewer power conversion steps results in less heat which need to be cooled, and therefore the operation costs are lowered. To ensure reliable operation of a low-voltage dc microgrid, well-designed control and protection systems are needed. An adaptive controller is required to coordinate the different resources based on the load-generation balance in the microgrid, and status of the ac grid. The performance of the developed controller has been studied and evaluated through simulations. The results show that it is possible to extend use of the data center dc microgrid to also support a limited amount of ac loads close to the data center, for example an office building. A protection-system design for low-voltage dc microgrids has been proposed, and different protection devices and grounding methods have been presented. Moreover, different fault types and their impact on the system have been analyzed. The type of protection that can be used depends on the sensitivity of the components in the microgrid. Detection methods for different components have been suggested in order to achieve a fast and accurate fault clearing. An experimental small-scale dc power system has been used to supply different loads, both during normal and fault conditions. A three-phase two-level voltage source converter in series with a Buck converter was used to interconnect the ac and the dc power systems. Together the converters have large controllability, high power quality performance, and allow bi-directional power flow. This topology can preferably be used together with energy storage. The tests confirm the feasibility of using a dc power system to supply sensitive electronic loads. / QC 20100908

With the ever advancing state of computer systems, it is imperative to maintain the most up-to-date and reliable safety evaluation data for nuclear power systems. Commonplace now is the practice of updating old accident simulation results with more advanced models and codes using today's faster computer systems. Though it may be quite an undertaking, the benefits of using a more advanced model and code can be significant especially if the result of the new analysis provides increased safety margin for any plant component or system. A series of parametric and sensitivity studies for the Loss of Normal Feedwater Anticipated Transient without Scram (LONF ATWS) for Southern Company's Vogtle Electric Generating Plant (VEGP) Units 1&2 located near Waynesboro, GA was performed using the best-estimate thermal-hydraulics transient analysis code RETRAN-02w. This thesis includes comparison to the results of a generic plant study published by Westinghouse Electric Corporation in 1974 using an earlier code, LOFTRAN, as well as Vogtle-specific analysis. The comparative analysis exposes and seeks to explain differences between the two codes whereas the Vogtle analysis utilizes data from the Vogtle FSAR to generate plant-specific data. The purpose of this study is to validate and update the previous analysis and gather more information about the plant actions taken in response to a LONF ATWS. As a result, now there is a new and updated evaluation of the LONF ATWS for both a generic 4-loop Westinghouse plant and VEGP using a more advanced code. Beyond the reference case analysis, a series of sensitivity and parametric studies have been performed to show how well each type of plant is designed for handling an ATWS situation. These studies cover a wide range of operating conditions to demonstrate the dependability of the model. It was found that both the generic 4-loop Westinghouse PWR system and VEGP can successfully mitigate a LONF ATWS throughout the core's operating cycle.

<p>The Discontinuous Galerkin time domain (DGTD) method is promising in modeling of realistic multiscale electromagnetic systems. This method defines the basic concept for implementing the communication between multiple domains with different scales.</p><p>Constructing a DGTD system consists of several careful choices: (a) governing equations; (b) element shape and corresponding basis functions for the spatial discretization of each subdomain; (c) numerical fluxes onto interfaces to bond all subdomains together; and (d) time stepping scheme based on properties of a discretized</p><p>system. This work present the advances in each one of these steps.</p><p> </p><p>First, a unified framework based on the theory of differential forms and the finite element method is used to analyze the discretization of the Maxwell's equations. Based on this study, field intensities (<bold>E</bold> and <bold>H</bold>) are associated to 1-forms and curl-conforming basis functions; flux densities (<bold>D</bold> and <bold>B</bold>) are associated to 2-forms and divergence-conforming basis functions; and the constitutive relations are defined by Hodge operators.</p><p>A different approach is the study of numerical dispersion. Semidiscrete analysis is the traditional method, but for high order elements modal analysis is prefered. From these analyses, we conclude that a correct discretization of fields belonging to different p-form (e.g., <bold>E</bold> and <bold>B</bold>) uses basis functions with same order of interpolation; however, different order of interpolation must be used if two fields belong to the same p-form (e.g., <bold>E</bold> and <bold>H</bold>). An alternative method to evaluate numerical dispersion based on evaluation of dispersive Hodge operators is also presented. Both dispersion analyses are equivalent and reveal same fundamental results. Eigenvalues, eigenvector and transient results are studied to verify accuracy and computational costs of different schemes. </p><p>Two different approaches are used for implementing the DG Method. The first is based on <bold>E</bold> and <bold>H</bold> fields, which use curl-conforming basis functions with different order of interpolation. In this case, the Riemman solver shows the best performance to treat interfaces between subdomains. A new spectral prismatic element, useful for modeling of layer structures, is also implemented for this approach. Furthermore, a new efficient and very accurate time integration method for sequential subdomains is implemented.</p><p>The second approach for solving multidomain cases is based on <bold>E</bold> and <bold>B</bold> fields, which use curl- and divergence-conforming basis functions, respectively, with same order of interpolation. In this way, higher accuracy and lower memory consumption are obtained with respect to the first approach based on <bold>E</bold> and <bold>H</bold> fields. The centered flux is used to treat interfaces with non-conforming meshes, and both explicit Runge-Kutta method and implicit Crank-Nicholson method are implemented for time integration. </p><p>Numerical examples and realistic cases are presented to verify that the proposed methods are non-spurious and efficient DGTD schemes.</p> / Dissertation

Submitted by Jaqueline Silva (jtas29@gmail.com) on 2016-08-29T17:39:21Z No. of bitstreams: 3 Dissertação - Bárbara Pereira Silva - 2016 - parte 1.pdf: 12258499 bytes, checksum: 8af480355ae4ffedb4ed79cbf7484361 (MD5) Dissertação - Bárbara Pereira Silva - 2016 - parte 2.pdf: 12730012 bytes, checksum: d173d49860226f3a2179be40d51ef666 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-08-29T17:39:44Z (GMT) No. of bitstreams: 3 Dissertação - Bárbara Pereira Silva - 2016 - parte 1.pdf: 12258499 bytes, checksum: 8af480355ae4ffedb4ed79cbf7484361 (MD5) Dissertação - Bárbara Pereira Silva - 2016 - parte 2.pdf: 12730012 bytes, checksum: d173d49860226f3a2179be40d51ef666 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-08-29T17:39:44Z (GMT). No. of bitstreams: 3 Dissertação - Bárbara Pereira Silva - 2016 - parte 1.pdf: 12258499 bytes, checksum: 8af480355ae4ffedb4ed79cbf7484361 (MD5) Dissertação - Bárbara Pereira Silva - 2016 - parte 2.pdf: 12730012 bytes, checksum: d173d49860226f3a2179be40d51ef666 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-03-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In the grounding systems analysis, the investigation of the effects caused by lightning discharge is a crucial task, since it is the main electromagnetic phenomena responsible for electricity outages which can be the cause of severe mechanical and thermal stresses on electric installations. This dissertation presents a study of transient behavior for various grounding configurations subject to impulsive currents originated from incident discharges. The model is named hybrid electromagnetic model, because it is based on electromagnetic equations and it is used concepts of electrical circuit analysis. The modeling is performed in frequency domain considering the effects of electromagnetic propagation on the metal electrode and into surrounding soil. In the numerical solution of model is used the method of moments for the discretization of the electrodes and calculus of the impedances. This methodology also considers the variation of electric soil parameters when frequency varies. The modeling comprised the effects of propagation and electromagnetic couplings between the grounding electrodes. Results were obtained for simple electrodes (wire horizontally buried in the soil or a vertical rod) and grounding grid in square format from the computational implementation of the mathematical model. This tool and the results obtained lend itself to the determination of the grounding systems response subjected to an injection of a typical current of lightning discharge. The validation of the tool which resulted from this work is made by comparing the obtained results with results of similar cases reported in the technical literature. / Na análise de sistemas de aterramento elétrico é primordial a investigação dos efeitos causados pela incidência de descarga atmosférica, uma vez que esse é o principal fenômeno eletromagnético responsável pelos desligamentos não programados de instalações elétricas causando severas solicitações mecânicas e térmicas. Nesta dissertação é apresentado um estudo do comportamento transitório para diferentes configurações de eletrodos de aterramentos frente a descargas atmosféricas. O modelo adotado é nomeado de modelo eletromagnético híbrido, pois baseia-se nas equações eletromagnéticas e utiliza-se conceitos da análise de circuitos elétricos. A modelagem é realizada no domínio da frequência, considerando-se os efeitos da propagação eletromagnética no eletrodo metálico e no solo ao redor. Na solução numérica do modelo é empregado o método dos momentos para a discretizacão dos eletrodos e obtenção das impedâncias. É também levada em conta a variação dos parâmetros elétricos do solo com a frequência. São contemplados nesta modelagem os efeitos de propagação e os acoplamentos eletromagnéticos entre os eletrodos do aterramento. A partir da implementação computacional do modelo, foram obtidos resultados para eletrodos simples (em posição horizontal ou vertical) e para malhas quadradas. Esta ferramenta e os resultados obtidos permitem avaliar a resposta do aterramento submetido a uma injeção de uma corrente impulsiva típica de descargas atmosféricas. A validação da ferramenta desenvolvida é feita por meio de comparações dos resultados obtidos com resultados de casos semelhantes publicados na literatura técnica.