O objetivo deste trabalho é apresentar uma metodologia de cálculo da distribuição de correntes e dos campos eletromagnéticos e tensões induzidas associadas em estruturas atingidas diretamente por descargas atmosféricas que, agregada à filosofia geral de compatibilidade eletromagnética (CEM), constitua uma abordagem inovadora e abrangente quando comparada às recomendações de proteção contra descargas atmosféricas atuais. Propõe-se, portanto, desenvolver uma metodologia que: · Constitua uma ferramenta de auxílio nas tomadas de decisões, aplicável desde a etapa de definição da arquitetura da estrutura e do leiaute dos equipamentos até a definição final da configuração do sistema de proteção contra descargas atmosféricas (S.P.C.D.A); · Possibilite a implementação de uma filosofia de proteção dos equipamentos e sistemas elétricos e eletrônicos contra descargas atmosféricas na qual se integre as recomendações tradicionais às necessidades atuais relacionadas à área de compatibilidade eletromagnética; · Seja abrangente e com grau de precisão satisfatório, quando comparadas com as recomendações atuais, com possibilidade de exploração dos valores das grandezas elétricas envolvidas no domínio do espaço e do tempo, e · Apresente simplicidade na implementação em computadores pessoais. / The main purpose of this work is to present a suitable method for calculating the current distribution, the electric and magnetic field, and the induced voltage on electric circuits regarding a structure directly struck by lightning. Moreover, when the method is used together with the electromagnetic compatibility (EMC) requirements, it can provide the designer with a new and helpful methodology meeting the recent aspects or the state of the art on lightning protection systems (L.P.S). Thus, a methodology is proposed, which provides the L.P.S. design team with: · The availability to take technical decisions in all steps of the project, from the definition of the architecture of the structure to the definition of the best layout of L.P.S.; · Recommendations to achieve the EMC of the electric and electronic system, in addition to the traditional ones; · Reliable numerical results concerning simulations in the space and time-domain, and · Simple implementation in personal computers.

O objetivo deste trabalho é apresentar uma metodologia de cálculo da distribuição de correntes e dos campos eletromagnéticos e tensões induzidas associadas em estruturas atingidas diretamente por descargas atmosféricas que, agregada à filosofia geral de compatibilidade eletromagnética (CEM), constitua uma abordagem inovadora e abrangente quando comparada às recomendações de proteção contra descargas atmosféricas atuais. Propõe-se, portanto, desenvolver uma metodologia que: · Constitua uma ferramenta de auxílio nas tomadas de decisões, aplicável desde a etapa de definição da arquitetura da estrutura e do leiaute dos equipamentos até a definição final da configuração do sistema de proteção contra descargas atmosféricas (S.P.C.D.A); · Possibilite a implementação de uma filosofia de proteção dos equipamentos e sistemas elétricos e eletrônicos contra descargas atmosféricas na qual se integre as recomendações tradicionais às necessidades atuais relacionadas à área de compatibilidade eletromagnética; · Seja abrangente e com grau de precisão satisfatório, quando comparadas com as recomendações atuais, com possibilidade de exploração dos valores das grandezas elétricas envolvidas no domínio do espaço e do tempo, e · Apresente simplicidade na implementação em computadores pessoais. / The main purpose of this work is to present a suitable method for calculating the current distribution, the electric and magnetic field, and the induced voltage on electric circuits regarding a structure directly struck by lightning. Moreover, when the method is used together with the electromagnetic compatibility (EMC) requirements, it can provide the designer with a new and helpful methodology meeting the recent aspects or the state of the art on lightning protection systems (L.P.S). Thus, a methodology is proposed, which provides the L.P.S. design team with: · The availability to take technical decisions in all steps of the project, from the definition of the architecture of the structure to the definition of the best layout of L.P.S.; · Recommendations to achieve the EMC of the electric and electronic system, in addition to the traditional ones; · Reliable numerical results concerning simulations in the space and time-domain, and · Simple implementation in personal computers.

Este trabalho discute o projeto de compensadores multivariáveis robustos com especificações no domínio do tempo. Primeiramente faz-se a análise dos compensadores por observadores de estados como forma de atingir tais objetivos. Mostra-se que, em certas condições, essa estrutura equivale à dos observadores proporcionais-integrais e apresentam-se as condições de estabilidade nominal. Evidencia-se também que é possível tratar esse problema de controle como um problema de "model matching" ou como um problema de controle com dois graus de liberdade. Mostra-se também que o projeto do compensador é equivalente ao projeto de sistemas de controle por realimentação estática da saída. Essa equivalência implica que, embora os compensadores por observadores sejam cômodos à incorporação de especificações temporais, sua estrutura é limitada para garantir que especificações gerais sejam satisfeitas. Contorna-se então essa limitação estendendo-se o estudo ao caso dos compensadores sem essa restrição estrutural. O problema de "model matching" e o problema de controle 2D são considerados como forma indireta de incorporar-se as especificações temporais e condições de projeto são obtidas reduzindo-se os possíveis conservadorismos dos projetos usuais. Ainda neste sentido, formula-se o problema denominado de "model tracking", podendo ser capaz de reduzir ainda mais esses aspectos de conservadorismo. Nessa estrutura, o erro de rastreamento entre a saída do modelo de referência e a saída medida da planta é realimentado, fazendo com que o modelo de referência faça parte explícita do controlador, permitindo ajuste pós-projeto da mesma forma que faz com a estrutura 2D. Toda as condições de projeto são obtidas no domínio da freqüência, em termos de restrições da malha aberta e da malha fechada, permitindo que técnicas padrões de projeto multivariável como LQG/LTR e H¥ possam ser diretamente utilizadas na obtenção do compensador. Obtiveram-se assim metodologias de projeto capazes de garantir robustamente a rejeição do erro de rastreamento de um modelo de referência, além de permitir que especificações usuais como a rejeição dos erros de medida e rejeição dos distúrbios sejam incorporadas no mesmo procedimento de projeto. O controle de um tanque de mistura é usado como exemplo numérico para ilustrar as metodologias de projeto. / This work is concerned with the design of robust multivariable controllers with time domain specifications. As a first step an analysis of controllers based on state observers has been done. It has been shown that this structure is equivalent to the proportional-integral observers. The conditions for nominal stability have then been presented. It has also been shown that this control problem can be seen as a model matching control design with two degrees of freedom. Furthermore it has been shown that compensator design can be reduced to the traditional static output feedback problem. This fact implies that although compensators based on state observers seem to be suitable to deal with time domain specifications, their structure has limitations to guarantee that more general specifications be satisfied. In order to overcome such limitations both the model matching and the 2-D control structures have been considered to include time domain specifications in the design problem. Design conditions have been derived in order to reduce the eventual conservatism associated to the usual design procedure. A control design problem named model tracking has also been formulated with the aim of reducing even more such conservatism. In this new structure the tracking error between the output of the reference model and the measured plant output is fed back. The reference model is then an explicit part of the controller. As a consequence small adjustments can be done on it during system start-up in the same way as with the 2-D structure. All the design conditions have been written in the frequency domain as constraints on both open-loop and closed-loop transfer matrices. Standard multivariable design techniques like LQG/LTR and H¥ can then be used. The proposed design methodology can robustly guarantee tracking of the reference model output as well as both measurement error and disturbance rejections. The control of a mixture tank has been used as a numeric example to illustrate the design methodologies.

Les travaux effectués durant cette thèse s’inscrivent dans le cadre d’une collaboration entre l’équipe CEM de l’Institut Pascal et l’équipe EDPAN du Laboratoire de Mathématiques de l’Université Blaise Pascal de Clermont-Ferrand. Nous présentons ici une étude qui, partant de l’analyse du processus de Retournement Temporel en électromagnétisme, a débouché sur le développement d’une méthode originale baptisée Linear Combination of Configuration Fields (LCCF) ou, en français, Combinaison Linéaire de Configurations de Champs. Après avoir introduit l’ensemble des outils et méthodes utilisés dans ces travaux, ce mémoire détaille le processus de Retournement Temporel de base ainsi qu’un ajout apporté à celui-ci. Par la suite, la méthode LCCF s’étant révélée applicable à plusieurs problèmes d’identification de sources en électromagnétisme, nous nous consacrons à la présentation détaillée des différentes variantes de celle-ci et nous illustrons son utilisation sur de nombreux exemples numériques. / This Ph.D thesis is the result of a collaboration between the CEM team of Pascal Institute and the EDPAN team of the Laboratory of Mathematics of the Blaise Pascal University in Clermont-Ferrand. We present here a study based on Time Reversal process in Electromagnetics. This work led to the development of a novel method called Linear Combination of Configuration Field (LCCF). This thesis first introduces the tools and the numerical methods used during this work. Then, we describe the Time Reversal process and a possible improvement to the basic technic. Afterwards, several possible applications of the LCCF method to electromagnetic source identification problems are detailed and we illustrate each of it on various numerical examples.

Os tempos de relaxação longitudinal (T1) e transversal (T2) são largamente utilizados em medidas qualitativas e quantitativas em Ressonância Magnética Nuclear no domínio do tempo (RMN-DT). As constantes de tempo T2 e T1 são medidas com as sequências de pulsos Carr-Purcel-Meiboom-Gill (CPMG) e Inversão Recuperação (IR), respectivamente. Além dessas sequencias, também têm sido desenvolvidos outros métodos que permitem a medição de T1 e T2 em um único experimento. Dentre esses, destacam-se os provenientes da condição Continuous Wave Free Precession: CWFP e CP-CWFP (Carr-Purcel-CWFP) com ou sem alternância de fase entre os pulsos que compõe o trem de pulsos de 90º desses métodos. Baseando-se no fato de que os métodos provenientes da condição CWFP têm sido promissores para a determinação das constantes de tempo de uma forma rápida, esse trabalho visou aprofundar o estudo das sequências de pulsos oriundas dessa condição a fim de desenvolver um método de determinação rápida da constante de tempo T1 em um único experimento e também implementar a condição CWFP em sequências de pulsos 2D (2D-CWFP). Utilizando-se de pulso de baixo ângulo de rotação (θ ~10 graus) foi possível determinar a constante de tempo T1 utilizando a condição CWFP, esse método foi denominado de CWFP-T1. Contudo, este apresentou um sinal com baixa razão sinal ruído (S/R) devido aos baixos valores de q. Para minimizar esse efeito foram estudados cinco métodos de redução de ruído que podem ser aplicados em sinais de RMN-DT após a aquisição, sendo que, dentre os métodos estudados foi comprovado que os filtros wavelet (WA) e o Savistiky-Goaly (SG) são os mais eficazes para redução do ruído sem distorcer as informações da amostra. Além disso, o sinal obtido pelo método CWFP-T1 associado ao filtro WA demonstrou ser eficiente para determinação de gordura em carne bovina. Dentre os métodos 2D-CWFP desenvolvidos neste trabalho, o CPMG-CWFP-T1 apresentou vantagem em tempo de experimento sobre o método IR-CPMG. Além disso, o sinal adquirido pelo método CPMG-CWFP-T1 quando associado ao filtro SG propiciou a obtenção de mapas com melhor resolução na dimensão de T1, que as outras sequências de pulsos abordadas neste estudo, e também demonstrou ser um método promissor para a análise da carne e gordura bovina. Esta foi a primeira vez em que a constante de tempo T1 foi utilizada como dimensão de aquisição de uma forma conveniente em experimento 2D de correlação dos tempos de relaxação, visto que aquisições dessa natureza geralmente requerem longos tempos de experimento. / The longitudinal (T1) and transverse (T2) relaxation times are widely used in the qualitative and quantitative measurements in time domain Nuclear Magnetic Resonance (TD-NMR). Those relaxation times are generally measured by the Carr-Purcel-Meiboom-Gill (CPMG) and Inversion Recovery (IR) pulse sequences, respectively. In addition to these sequences, other methods have been development, which makes it possible to measure T1 and T2 in a single experiment. Among these, a special reference is made those ones coming from the Continuous Wave Free Precession condition: CWFP and CP-CWFP (Carr-Purcel-CWFP) with or without phase alternation between the pulses that make up the pulse train of 90º of these methods. Based on the fact that the methods from the CWFP condition have been promising to determine the time constants fast, the purpose of this work was to deepen the study of the sequences from that condition in order to develop a rapid method to determine the T1 relaxation time in a single experiment, and also to implement the CWFP condition in 2D sequences (2D-CWFP). Using a small flip angle (θ ~10 degrees) it was possible to determine the T1 relaxation time using the CWFP condition; this method was named CWFP-T1­. Nevertheless, this sequence has a signal with a low signal-to-noise ratio (SNR) because of the small values of q. In order to minimize this effect was studied five denoise methods to applied in the post-acquisition TD-NMR signals. Among the denoise methods studied, the wavelet (WA) and the Savistiky-Goaly (SG) are the most effective to reduce the noise of the signal without distorting the sample information. In addition, the CWFP-T1 signal associated to the WA has shown to be effective for determining the fat in beef samples. Among the 2D-CWFP methods developed in this work, the CPMG-CWFP-T1 had advantage in the experimental time compared with the IR-CPMG method. Furthermore, the signal acquired by CPMG-CWFP-T1 sequence, when associated with the SG filter, had 2D maps with the best resolution in the T1 dimension than the other sequences addressed in this work and also proved to be effective to analyze fat beef. This was the first time that T1 time constant was encoded in the acquisition dimension at a convenient way in the 2D experiments of relaxation times correlation, once such acquisitions generally require extensive experimental time.

This thesis details the application of a medium frequency (MF) Doppler radar to observations of meteoroids entering the Earth's atmosphere. MF radars make possible a greater height coverage of the meteor region (70 to 160 km) than conventional meteor radars. However this type of radar has generally been under-utilised for meteor observations, primarily due to the less than ideal radio environment associated with MF systems. This situation demanded selection of the most appropriate radar meteor techniques and in this respect a variety of techniques are evaluated for application at this frequency. The 2 MHz radar system used in this study is located at the Buckland Park research facility (35.6 deg. S, 138.5 deg. E), near Adelaide, South Australia and is operated by the Department of Physics of the University of Adelaide. This radar has the largest antenna of any MF radar with 89 crossed dipoles distributed over an area of about 1 km in diameter. Beam forming is achieved by varying the phase to groups of elements of the array. The array was constructed in the 1960's, and while having several upgrades, a preliminary examination of the array and associated systems indicated that a significant amount of maintenance work would be required to enable the system to be used for meteor observations. It was also apparent that the software used with the radar hardware for atmospheric studies was not suitable for processing meteor data. Thus a major refurbishment of the radar hardware, as well as the development of appropriate software, was initiated. The complete radar system was divided into its constituent components of antenna array, transmitter, receiver and computer systems. The transmitter and receiver systems were examined and various improvements made including increasing total output power and enhancing beam steering capability. Time domain reflectometry (TDR) techniques were extensively used on the antenna array, as many feed cables showed the presence of moisture. New hardware in the form of a portable power combining system was designed, constructed and tested to further increase radar experimental capabilities. Techniques were developed that verified system performance was to specification. Extensive night time observations of sporadic and shower meteor events were made over a two and a half year period. A particular study was made of the Orionids shower as well as other meteor activity on the night of 22 October 2000. Using the upgraded beam swinging features of the array, a narrow radar beam was used to track the shower radiant in an orthogonal sense so as to maximise the number of shower meteors detected. From each echo, various intrinsic meteoroid parameters were determined, including meteor reflection point angle-of-arrival using a five-element interferometer, echo duration and height; meteoroid speeds were determined using the Fresnel phase time technique. Meteor echoes belonging to the Orionids radiant were selected using a coordinate transform technique. The speed was then used as an additional discriminant to confirm the Orionid shower members. A second radiant, observed at a slightly higher declination is classified as also part of the Orionid stream. The sporadic meteor component in the data set was examined and found to exhibit speeds much higher than expected for sporadic meteors at the time of the observations. However, these results are consistent with a selection bias based on meteoroid speed, that is inherent in radar observations. The Orionid observations indicate that the refurbishment of the radar system and the introduction of new software for meteor analysis has been successfully achieved and that radar meteor studies can now be carried out routinely with the Buckland Park 2 MHz radar. Moreover it has been shown for the first time that meteoroid speeds can be determined with a MF radar operating on a PRF as low as 60 Hz. / Thesis (Ph.D.)--School of Chemistry and Physics, 2003.

Cette thèse constitue un ensemble de travaux et de réflexions sur la question de la modélisation des applications électromagnétiques en nano-optique en utilisant la méthode des différences finies dans le domaine temporel (FDTD). Dans un premier temps, des codes FDTD bidimentionnels pour le calcul de bandes interdites photoniques ont été mis en oeuvre. Ces algorithmes tiennent comptes de la dispersion des métaux nobles dans la gamme optique décrite par le modèle de Drude ou de Drude-Lorentz. Ces programmes FDTD permettent de tenir compte de la propagation soit dans le plan perpendiculaire au plan d'invariance (appelé "cas dans le plan" ou "in-plane" en anglais) pour les deux polarisations TE et TM ainsi que le cas d'une propagation quelconque hors du plan (ou off-plane). Plusieurs diagrammes de bandes sont calculés et présentés pour les structures carrées et triangulaires dans les cas diélectriques et métalliques. Ensuite, nous avons implémenté un code BOR-FDTD, basé sur la discrétisation des équations de Maxwell exprimées en coordonnées cylindriques, pour la modélisation des guides d'ondes (ou d'autres objets) à symétrie de révolution. Les conditions absorbantes PML pour décrire l'espace libre sont intégrées à la BOR-FDTD ainsi que les deux modèles de Drude et de Drude-Lorentz. Des simulations ont été effectuées pour le calcul de modes propres de guides d'ondes coaxiaux et cylindriques sub-longueurs d'ondes faits en métal parfait et en métal réel (argent par exemple). Les résultats montrent la possibilité de guider des signaux optiques sans beaucoup de pertes dans un guide coaxial fait en argent de dimensions sublongueur d'onde. Ce dernier résultat est original et constitue une très importante avancée dans le domaine de la "nanoconnectique" en optique, plus particulièrement pour l'optique intégrée. Puis, un autre code numérique FDTD-3D a été élaboré pour la modélisation des structures périodiques (type cristaux photoniques tridimensionnels) éclairées en incidence oblique. Ce code intègre aussi les couches absorbantes PML ainsi que les modèles de dispersion de Drude et de Drude-lorentz. Les résultats obtenus sont comparés à ceux issus d'autres modèles théoriques. Les applications de ce code à l'étude de radôme, à l'excitation du mode TEM de la structure métallique à ouvertures annulaires et aux calculs des spectres d'extinction Raman montrent l'efficacité de la FDTD pour la modélisation de telles structures.

A distributed sensor using an optical fiber for detecting and locating intruders over long perimeters (>10 km) is described. Phase changes resulting from either the pressure of the intruder on the ground immediately above the buried fiber or from seismic disturbances in the vicinity are sensed by a phase-sensitive optical time-domain reflectometer (φ−OTDR). Light pulses from a cw laser operating in a single longitudinal mode and with low (MHz/min range) frequency drift are injected into one end of the single mode fiber, and the backscattered light is monitored with a photodetector. In laboratory tests with 12 km of fiber on reels, the effects of localized phase perturbations induced by a piezoelectric fiber stretcher on φ−OTDR traces were characterized. In field tests in which the sensing element is a single mode fiber in a 3-mm diameter cable buried in an 8 to 18 inch deep, 4 inch wide trench in clay soil, detection of intruders on foot up to 15 ft from the cable line was achieved. In desert terrain field tests in which the sensing fiber is in a 4.5-mm diameter cable buried in a 1 ft deep, 2.5 ft wide trench filled with loose sand, high sensitivity and consistent detection of intruders on foot and of vehicles traveling down a road near the cable line was realized over a cable length of 8.5 km and a total fiber path of 19 km in real time. In a final series of field tests in clay soil, phase changes produced by the steps of a person walking up to 15 ft away from the buried cable were observed, and vehicles traveling at 10 mph were consistently detected up to 300 ft away. Based on these results, this technology may be regarded as a candidate for providing low-cost perimeter security for nuclear power plants, electrical power distribution centers, storage facilities for fuel and volatile chemicals, communication hubs, airports, government offices, military bases, embassies, and national borders.

Denna avhandling är fokuserad på modellering av elkraftsystemkomponenter och deras representation vid simuleringar av elkraftsystem. Avhandlingen jämför olika linjäriseringstekniker. Dessa tekniker är såväl numeriska som analytiska och används vid linjärisering av ett dynamiskt system. Efter en linjärisering är det möjligt att beräkna egenvärdena av det linjäriserade systemet samt använda andra verktyg ämnade för studier av linjära system. I avhandlingen visas hur olika linjäriseringtekniker influerar egenvärdesberäkningen av det linjära systemet. I avhandlingen tas fram bandviddsreducerade linjära modeller av en kraftsystemkomponent med hjälp av två tekniker. Senare görs simuleringar med de linjära modellerna tillsammans med ett introducerat gränssnitt. Den studerade kraftsystemkomponenten är en tyristorstyrd seriekondensator (TCSC). En fördel med att använda en linjär representation av en kraftsystemkomponent är att det förenklar simuleringarna. Storleken på komplexiteten av en simulering vid lösandet av ekvationerna minskar och den konsumerade fysiska tiden att simulera minskar. En nackdel med en linjär modell är att dess giltighet kan vara begränsad. Behovet av att bygga linjära modeller av kraftsystemkomponenter torde även finnas i framtiden. Med dagens horisont (år 2006) finns behov av att bygga linjära modeller utgående från detaljerade modeller av bl a högspända likströmslänkar (HVDC-länkar), reaktiva effektkompensatorer (SVC) samt tyristorstyrda seriekondensatorer (TCSC). Hur skall dessa representeras när vi vill studera dynamiken av ett helt kraftsystem och det då är nödvändigt att reducera deras komplexitet? Denna frågeställning uppkommer när vi vill genomföra tidsdomänsimuleringar på en inte alltför detaljerad nivå av de individuella kraftsystemkomponenterna eller när vi vill linjärisera kraftsystemet för att studera dess stabilitet med hjälp av småsignalanalys. / This thesis is focused in modelling of power system components and their representation in simulations of power systems. The thesis compares different linearization techniques. These techniques are both numerical as well as analytical and are utilized when linearization of a dynamic system is desired. After a linearization it is possible to calculate the eigenvalues of the linearized system as well as to perform other applicable activities on a linear system. In the thesis it is shown how the linearization techniques influence the calculation of eigenvalues of the linear system. In the thesis bandwidth-reduced linear models of a power system component are developed using two techniques. The simulations with the linear models are done with an introduced interface system. The studied power system component is a Thyristor-Controlled Series Capacitor (TCSC). One advantage with using a linear representation of a power system component is that it simplifies the simulations. The size of the complexity of a simulation when solving the equations decreases and the consumed physical time to simulate becomes shorter. A disadvantage of a linear model is that its validity might be limited. The need of building linear models of power systems will continue to attract interest in the future. With the horizon of today (year 2006) there is a need of among other models to build linear models of detailed models of High Voltage Direct Current-links (HVDC-links), Static Var Compensators (SVCs), as well as Thyristor-Controlled Series Capacitors (TCSCs). How should these be represented when we want to study the dynamics of a whole power system and it is necessary to reduce their complexity? This question rises when we want to perform time-domain simulations with a not too detailed level of complexity of each individual power system component or if we want to linearize the power system and study it within small-signal stability analysis. / QC 20100915

Simulating wave propagation in microstrip lines with Gyrotropic magnetic substrate is considered in this thesis. Since the static internal field distribution has an important effect on the device behavior, accurate determination of the internal fields are considered as well. To avoid the losses at microwave frequencies it is assumed that the magnetic substrate is saturated in the direction of local internal field. An iterative method to obtain the magnetization distribution has been developed. It is applied to a variety of nonlinear nonuniform magnetic material configurations that one may encounter in the design stage, subject to a nonuniform applied field. One of the main characteristics of the proposed iterative method to obtain the static internal field is that the results are supported by a uniqueness theorem in magnetostatics. The series of solutions Mn,Hn, where n is the iteration number, minimize the free Gibbs energy G(M) in sequence. They also satisfy the constitutive equation M = χH at the end of each iteration better than the previous one. Therefore based on the given uniqueness theorem, the unique stable equilibrium state M is determined. To simulate wave propagation in the Gyrotropic magnetic media a new FDTD formulation is proposed. The proposed formulation considers the static part of the electromagnetic field, obtained by using the iterative approach, as parameters and updates the dynamic parts in time. It solves the Landau-Lifshitz-Gilbert equation in consistency with Maxwell’s equations in time domain. The stability of the initial static field distribution ensures that the superposition of the time varying parts due to the propagating wave will not destabilize the code. Resonances in a cavity filled with YIG are obtained. Wave propagation through a microstrip line with YIG substrate is simulated. Magnetization oscillations around local internal field are visualized. It is proved that the excitation of magnetization precession which is accompanied by the excitation of magnetostatic waves is responsible for the gap in the scattering parameter S12. Key characteristics of the wide microstrip lines are verified in a full wave FDTD simulation. These characteristics are utilized in a variety of nonreciprocal devices like edgemode isolators and phase shifters.