The work in this thesis concerns the estimation of the electromagnetic, elastodynamic and piezoelectric properties of homogenized composite materials (HCMs). A composite may be considered homogeneous if wavelengths are suficiently large in comparison to the size of the particles of each component material. This thesis examines HCMs constructed from two component materials and several methods of estimating the HCMs constitutive properties. Firstly, the Maxwell Garnett estimates and Bergman-Milton bounds on the electromagnetic properties of HCMs are examined. While both are widely used, we re-examine them, for isotropic dielectric HCMs, in light of recent advancements in material manufacture. Secondly, we examine the strong-property-fluctuation theory (SPFT). The SPFT estimate is calculated using iterations upon an initial ansatz, these iterations being dependent on statistical cumulants of the spatial distribution of the particles of the component materials. The zeroth-order SPFT estimate is identical to the first-order and both are taken to be identical to a comparison material. For the second-order estimate a two-point correlation function along with its associated correlation length are used to characterize the component materials' particle distribution. The general framework for the elastodynamic SPFT was established in 1999 by Zhuck and Lakhtakia. Here we further develop the elastodynamic SPFT for orthotropic HCMs, in order to undertake numerical studies. We simplify certain integrals in order to make them amenable to numerical computation. Also, we establish the piezoelectric SPFT for orthorhombic mm2 materials. The general theory is developed first in a manner analogous to the elastodynamic SPFT. We then implement a two-point covariance function, perform similar integral simplifications to those done in the elastodynamic SPFT and carry out numerical experiments. From the numerical studies it is clear that, for both the elastodynamic and piezoelectric HCMs, the lowest-order SPFT estimate is similar to that provided by the corresponding Mori-Tanaka formalism. It is also apparent that the second-order SPFT estimate provides a significant correction to the lowest-order estimate, which reflects dissipative losses due to scattering.

The terahertz frequency range is a relatively unstudied region of the electromagnetic spectrum. However with the emergence of numerous applications for terahertz light in diverse areas such as security scanning, biological imaging, gas spectroscopy and astrophysics there has been considerable recent growth in the volume of research activity in this area. The studies presented in this thesis aim to introduce the physics of surface plasmons to the terahertz frequency range, and on the way to use some of the unique capabilities of terahertz spectroscopy to try and find new information about fundamental surface-plasmon based electromagnetic structures. Four distinct experiments are described in this work, all of them underpinned by the technique of terahertz time-domain spectroscopy (Chapter 2). This is a very powerful and adaptable spectroscopic method which allows us to measure the electric field of pulsed terahertz radiation as a function of time. This in turn allows us to directly extract both phase and amplitude of the terahertz light as a function of frequency, over a broad frequency range. Furthermore, this method of terahertz spectroscopy can be combined with photoexcitation pulses of visible/NIR light which can be used to make dynamic changes to the properties of materials in the terahertz beam. The first experiment reported (Chapter 3) measures the propagation of coupled surface plasmons in a resonant slit cavity. We use terahertz time-domain spec- troscopy to determine the characteristics of the cavity resonances in a semiconductor slit near the surface plasma frequency of the material, where we are able to mea- sure very large red-shifts in the frequency of the cavity resonance. By considering the phase information which can be extracted directly from time-resolved terahertz measurements we are able to link the behaviour of the resonances to the propagation characteristics of the surface plasmon modes inside the slits. The second experiment (Chapter 4) is a more direct measurement of surface plasmons, propagated over the surface of a semiconductor wafer. We show that the electric field of the surface plasmon is confined to a subwavelength region around the surface, and that the confined field is useful for spectroscopy of very thin layers above the surface. We are able to measure films with thickness less than 1/600th of the wavelength of the terahertz light. After these two experiments with confined semiconductor surface plasmons we move on to a pair of experiments looking at terahertz surface modes mediating the transmission of light through holes in metal films. In the initial experiment (Chapter 5) we use the time-domain data from terahertz spectroscopy to determine the role that surface mode lifetime plays in modifying the amplitude and width of Extraor- dinary Optical Transmission (EOT) resonances, which arise from the periodicity of a hole-array lattice. By changing the temperature of the lossy dielectric semicon- ductor substrate we are able to modify the surface mode lifetime, and link this to the resonant transmission characteristics. In Chapter 6 we extend the hole array EOT experiment by making dynamic changes to the propagation of the surface mode which mediates the transmission. This is achieved by photo-exciting the semiconductor substrate inside the holes and forming a thin layer of material with high charge carrier density on the surface. Interaction of the surface mode with the photoexcited region quenches the resonant transmission. We show that by changing the hole size so that the surface-mode mediated transmission pathway predominates in the spectrum it is possible to use optical pulses to modulate the transmission of terahertz radiation with very high efficiency. In the conclusions (Chapter 7) we link together some of the insights and infer- ences which can be drawn from the above results, as well as evaluating the efficacy of the experimental and simulation methodology.

Although the Maxwell equations govern the propagation of EM waves at all frequencies, the methods required to generate, direct, analyse and detect radiation differ from band to band. This thesis is concerned with the development and demonstration of 'Quasi-optical' techniques for millimetre wavelengths, which involve the propagation of Gaussian profiled beams a few wavelengths across, and the realization of a general circuit approach to mm-wave measurement. Gaussian beam-mode analysis, which is used in later chapters to understand propagation of quasi-optical beams, is reviewed in chapter 1. Chapter 2 outlines the design, manufacture and testing of corrugated feed horns which generate fundamental Gaussian beam-modes. The design and manufacture of lenses which control the spreading of the beams and directional couplers which provide desirable signal processing functions in quasi-optical circuits is discussed in chapter 3. Chapter 4 traces the development of a Faraday isolator which operates in free-space and can suppress unwanted reflections in quasi-optical circuits. Chapter 5 discusses a reusable circuit board upon which systems can be easily and quickly constructed. A null reflectometer built using quasi-optical components is outlined in chapter 6 and reflection measurements from lenses and horns presented. Chapter 7 describes a corrugated feed horn/lens antenna used in a balloon-borne nw-wave cosmic background experiment. Finally, Chapter 8 demonstrates the use of quasi-optical components in a 115 GHz receiver circuit.

The Thesis opens with an introduction to electrical resonators and some instruments which are special to the centimeter region of wavelengths. The excitation of resonators by the principle of the velocity modulation of electrons is then given and it is followed by a brief mention of velocity modulated oscillators and magnetron oscillators. Little specific mention is made in these sections of the author's work. A reference is however made to an early split anode 3 cm. magnetron and also to a tunable velocity modulated oscillator. These were made by him before such tubes had passed the experimental stage. The development of magnetrons required some simple but reliable method of measuring pulsed high tension power. Some satisfactory methods developed by the author are described. To prevent there being any possible doubt, these power measurements were checked by careful measurement of the heat developed in the magnetron. To examine the electric fields within resonators and feeder lines small test probes and loops are necessary. These probes and loops require a consideration of the various types of rectifiers, detector instruments and high frequency matching sections. All are discussed both from their simplicity of design and from their performance. For the design of 3 cm. magnetrons, whose output system was to be matched to the standard fitting used in the Services, an accurate standing wave measuring instrument was built. The errors occurring in these instruments are examined and shown to be negligible only under certain conditions of operation. The general properties of wave guides and coaxial cables are given prior to an account of the author's experiments on the measurements of magnetron output powers. Mention of preliminary work on power measurement is made and is followed by a full description of a rugged glass contained helical absorber suitable for low or for high power routine measurements. Full details of the calibration of the absorbers is given. There are also details of a calorimeter for coaxial output valves and also for wave guide output valves. These instruments have given a trouble free performance for nearly two years and are eminently suited for use in factories for testing the quality of the valves produced. The increase in the output powers tram magnetrons soon necessitated a complete redesign in the output system. The author's experiments on what was, as far as he is aware, the first 3 cm. wave guide output valve are included.

Les cintes superconductores d'alta temperatura han sorgit com un material molt prometedor per aplicacions de potència en alts camps magnètics ja que s'ha demostrat la possible utilització del material YBaCuO-123 (YBCO) per produir cintes superconductores amb prestacions de densitat de corrent crítica de l'ordre de 1MA/cm2 a 77K sota camps magnètics elevats (1-5T). Les cintes superconductores de YBCO estan formades per una estructura bi-dimensional de fronteres de gra de baix angle (FGBA) (~4º-8º) induïdes per la textura del substrat metàl·lic flexible sobre el qual estan dipositades amb previ condicionament mitjançant el dipòsit de capes tampó. La granularitat present en aquests materials representa la limitació més important al pas de corrent elèctrica i estableix la presencia de dos mecanismes de dissipació interactuants associats per una banda al pas de corrent percolativa a través de les FGBA (limitada per la densitat de corrent crítica associada a les fronteres de gra JcFG) i per altra banda a la corrent que flueix al interior dels grans superconductors (limitada per la densitat de corrent crítica associada als grans, JcG). Si be JcG esta clarament determinada per l'ancoratge dels vòrtex Abrikosov en els grans superconductors, la dissipació associada a JcFG es encara un interrogant. Estudis teòrics han predit l'existència d'una nova classe de vòrtex superconductors associats a les fronteres de gra que han estat anomenats vòrtex Abrikosov-Josephson.En aquesta tesi doctoral hem analitzat la resposta electromagnètica de les cintes superconductores de YBCO mitjançant mesures magnètiques de cicles de histèresi menors. A partir de la forma característica dels cicles de histèresi s'ha desenvolupat un formalisme que ens ha permès determinar de manera independent i simultània les densitats de corrent crítica JcG i JcFG a baix camp magnètic, en el marc del model d'estat crític. La metodologia desenvolupada ens ha permès estudiar la relació existent entre els efectes de granularitat, l'ancoratge de vòrtex Abrikosov als grans i la dissipació associada al moviment de vòrtex Abrikosov-Josephson a les FGBA i per tant ens ha permès inferir en els mecanismes físics que regeixen el transport elèctric d'aquests materials. Hem realitzat un estudi complert de la relació entre aquests mecanismes variant diferents paràmetres característics de les cintes superconductores tal com l'angle mig entre fronteres de gra, el tamany de gra, el gruix de la capa superconductora o el procés de creixement. Addicionalment, mitjançant mesures de susceptibilitat ac, hem identificat i caracteritzat diferents dominis de percolació de corrent en diverses cintes superconductores de YBCO associats a diferents valors de JcFG. Aquest estudi ens ha permès complementar l'anàlisi magnètic de la granularitat inherent a aquests materials i caracteritzar la dissipació associada. A partir de l'estudi de les corbes de susceptibilitat ac mesurades a diferent freqüència hem determinat l'energia d'activació tèrmica a diferents camps magnètics i diferents densitats de corrent. Comparant els valors obtinguts amb els valors de Jc mesurats mitjançant magnetització dc hem pogut determinar un diagrama de fases magnètic (camp magnètic/temperatura) per a les cintes superconductores de YBCO on es poden distingir diferents règims de moviment de vòrtex. / High temperature coated conductors has emerged as a very promising material for power applications at high magnetic fields since it has been demonstrated the possible utilization of YBaCuO-123 (YBCO) for performing coated conductors able to support current densities higher than 1MA/cm2 at 77K and high magnetic fields (1-5T). YBCO coated conductors consist on a bi-dimensional structure of low angle grain boundaries (LAGB) (~4º-8º) induced by the texture of the metallic tape used as a substrate, which is usually covered with several buffer layers.The granularity present in this materials represent the major limitation for the electrical current to flow and establish the presence of two different interactive dissipation mechanisms associated on one hand to the percolative current flowing through the LAGB (limited by the grain boundary critical current density, JcGB) and on the other hand to the current flowing inside the superconducting grains (limited by the grain critical current density, JcG). While JcG is clearly limited by the Abrikosov vortex pinning inside the superconducting grains, the dissipation associated to JcGB is still an open issue. Theoretical studies has predicted the existence of a new kind of vortex located at the grain boundaries called Abrikosov-Josephson vortices.In this Thesis we have analyzed the electromagnetic response of YBCO coated conductors by means of magnetic measurements of minor hysteresis loops. From the characteristic feature of the hysteresis loops we have developed a new formalism which has allowed us to determine independently and simultaneously both critical current densities, JcG i JcGB, at low applied magnetic field, in the framework of the Critical state Model. The methodology devolved has allowed us to study the relationship between granularity effects, Abrikosov vortex pinning inside the grains and the dissipation associated to the motion of Abrikosov-Josephson vortices at the LAGB and thus, it has allowed us to infer in the physical mechanisms which governing the electric transport in these materials. We have performed a complete study of the relationship between these mechanisms by changing different characteristic parameters of the YBCO coated conductors such as the average misorientation angle in the grain boundaries, the grain size, the thickness of the YBCO layer or the growing process.Additionally, by means of ac susceptibility measurements, we have identified and characterized different current percolative domains in several YBCO coated conductors associated to different values of JcGB. This study has allowed us to complement the analysis of magnetic granularity inherent at these materials and characterize the associated dissipation. By means of the study of ac susceptibility curves measured at different frequency we have determined the thermal activation energy of YBCO coated conductors at several magnetic fields and current densities. By comparing the resultant values with the values of Jc measured by means of dc magnetization we have determined a general magnetic phase diagram (magnetic field/temperature) for YBCO coated conductors where we have distinguished different vortex motion regimes.

No description available.

Durante las últimas décadas, la construcción de anillos de almacenamiento de electrones exclusivamente dedicados a la producción de radiación sincrotrón ha sido la clave para justificar el gran desarrollo de los componentes ópticos para rayos X. Se requieren nuevos elementos ópticos para una explotación óptima de las propiedades de esta luz, que puede usarse para descubrir los secretos de la materia y para revelar el mundo microscópico. El uso de radiación sincrotrón como sonda ha hecho posible una gran cantidad de experimentos para expandir el conocimiento de muchas áreas científicas. Paulatinamente, la radiación sincrotrón se ha convertido en un instrumento indispensable para muchos científicos, que trabajan en disciplinas muy diferentes como la biología, la química, la ciencia de materiales o incluso la arqueología. La microscopía de rayos X ha emergido como técnica para observar estructuras que no son accesibles con microscopia óptica convencional, y que tiene ventajas respeto a la microscopía electrónica debido a la mayor longitud de penetración y a la sensibilidad química de la radiación X. La óptica de los microscopios de rayos X incluye componentes como las lentes zonales de Fresnel que se producen con técnicas de microfabricación. En este trabajo, se han fabricado lentes zonales de Fresnel utilizando distintas técnicas y se han testado en diversas Fuentes de Luz Sincrotrón. Describiremos en detalle las técnicas de micro- y nanofabricación que son necesarias para la producción de estos elementes, des de la litografía por haz de electrones a la transferencia del patrón a distintos materiales. En particular, presentamos lentes para rayos X blandos hechas de silicio. Mostraremos que éstas funcionan bien en las fuentes de luz existentes y que debido a su robustez serán también apropiadas para las fuentes de rayos X de 4a generación. También preparamos un elemento óptico difractivo que produce una mancha iluminación cuadrada y llana, y que puede usarse como lente condensadora en microscopía de rayos X de transmisión. Finalmente, también demostramos un nuevo método de fabricación que puede mejorar la resolución espacial última de las lentes difractivas para rayos X. Se fabricaron lentes zonales de Fresnel con una última zona de 20 nm y líneas de 15 nm han sido claramente resueltas en microscopía de rayos X de rastreo. Este trabajo se ha realizado en el Laboratorio de Luz Sincrotrón en Barcelona, con la participación del Centro Nacional de Microelectrónica de Barcelona (CSIC-CNM) y del Grupo de Óptica del Departamento de Física de la Universidad Autónoma de Barcelona. Al mismo tiempo, partes esenciales de este trabajo se han realizado con la colaboración del Dr. C. David y el Dr. K. Jefimovs del Labor fur Mikro- und Nanotechnologie al Paul Scherrer Institut en Villigen (Suiza). / During the last decades, the construction of electron storage rings exclusively dedicated to the production of synchrotron radiation has been a key reason to explain the large development of x-ray optics. New optical elements are required for an optimal exploitation of the properties of this light, which can be used to find out the secrets of matter and to reveal the microscopic world. The use of synchrotron light as a probe has made possible a large quantity of experiments to expand the knowledge in many scientific areas. Little by little, synchrotron radiation sources have become an indispensable tool for the research of lots of scientists, who work in very different disciplines such as biology, chemistry, physics, material science or even archaeology. X-ray microscopy has emerged as a technique to observe structures which are not accessible with conventional optical microscopy, and that has advantages in respect to electron microscopy due to the longer penetration depth and chemical sensitivity of the x-ray radiation. The optics of the x-ray microscopes includes components such as the Fresnel zone plate lenses which are made by means of microfabrication techniques. Within this work, Fresnel zone plate lenses were produced using different approaches and they have been tested in several Synchrotron Light Sources. We will describe in detail the micro- and nanofabrication techniques that are necessary for the production of such elements, from the electron beam lithography to the pattern transfer into different materials. In particular, we will present lenses for soft x-rays made of silicon. We show that they perform well at the current light sources and we think that due to their robustness they will also be suitable for the 4th generation x-ray sources. We also prepared a diffractive optical element which produces a square flat top illumination spot, and that can be used as a condenser lens in full-field transmission x-ray microscopy. Finally, we will also demonstrate a novel fabrication method which can push the ultimate spatial resolution of x-ray diffractive lenses. Fresnel zone plates with an outermost zone width of 20 nm have been fabricated and 15 nm lines have been clearly resolved in scanning transmission x-ray microscopy. This work has been carried out in the Laboratori de Llum Sincrotró in Barcelona, with the participation of the Centro Nacional de Microelectrònica de Barcelona (CSIC - CNM) and the Grup d'Òptica del Departament de Física de la Universitat Autònoma de Barcelona. At the same time, essential parts of this work have been done in close collaboration with Dr. C. David and Dr. K. Jefimovs from the Labor für Mikro- und Nanotechnologie at the Paul Scherrer Institut in Villigen (Switzerland).

En esta tesis se diseña una técnica de estimación de parámetros constantes para los modelos de jaula sencilla y doble jaula sin pérdidas en el entrehierro de la máquina de inducción trifásica, que parte de los datos del fabricante para obtener trayectorias de comportamiento teórico, llamada Torque-Speed-Tracking. También se diseña una técnica para la estimación de parámetros variables para el modelo de jaula sencilla cuyo comportamiento resultante se compara con el obtenido con el modelo de doble jaula de parámetros constantes; esta técnica parte de medidas experimentales y modela cada parámetro con una función polinómica de coeficientes constantes. Finalmente se utilizan los parámetros constantes obtenidos para analizar los efectos de los huecos de tensión al utilizar los modelos mencionados y se clasifican dichos huecos en función de su severidad en los picos de intensidad, par y deslizamiento en una máquina de inducción trifásica. / In this doctoral thesis a technique has been designed for the constant parameters estimation for the single- and double-cage models, without air gap losses, of the three-phase induction machine; this technique is called Torque- Speed-Tracking, and obtains a path of theoretical behaviour from the manufacturer's data. Also, a technique for the estimation of four variable parameters for the single-cage model has been designed, whose resulting behaviour is compared with the obtained one with the double-cage model with constant parameters; in this technique each parameter is modelled with a polynomial function with constant coefficients, from experimental measurements. Finally, the obtained constant parameters are used to analysing the effects of voltage dips/sags using the listed models. The voltage dips/sags are classified in terms of their severity on peaks of: intensity, torque and slip, on a three-phase induction machine.

L' ATP es una molecula clau en el metabolisme cel.lular, actuant com a donador d'energia lliure i acoplant reaccions endergoniques i exergoniques. L'ATP es sintetitzat a la mitocondria en un proces anomenat fosforil.lacio oxidativa despres d'una serie de reaccions a la cadena de citocroms que es troba en la membrana interna de l'organel.la. La font d'energia necessaria per aquesta sintesi s'obte en les cel.lules animals dels nutrients de l'ingestio, i de la llum solar en les plantes. Existeix una via alternativa de sintesi d'ATP extramitocondrial, la glicolisi, que s'inicia amb la fosforilacio de la glucosa per l'enzim hexokinasa.Interaccio llum-materiaL'energia electromagnetica l'ona de la qual oscil.la en una longitud d'ona de nanometres es anomenada llum. En aquestes frequencies, l'energia promou excitacio electronica de certs atoms i molecules. Existeix una interaccio diferent de la llum amb molecules, produida per el camp electromagnetic que per definicio la llum provoca en qualsevol medi. El camp electric resultant desplaca els electrons dels enllacos quimics produint una polaritzacio del medi sense que existeixi absorcio de l'energia. Aquest es un mecanisme interactiu que existeix sempre, i es l'unic que es dona en molecules que son transparents (no absorbeixen) per a una determinada frequencia de la llum, com es el cas de l'ATP per l'energia visible i infravermella propera. Experiments, resultats i discussioQuan l'ATP es excitat amb fotons ultraviolats, es produiex una fluorescencia en longituts d'ona visibles. L'io magnesi s'ha utilitzat per estudiar com la llum visible i infravermella propera produiex un desplacament de carregues electriques a la molecula d'ATP. La construccio d'un interferometre de Michelson ha servit per observar l'interaccio no absortiva de la llum i l'ATP. La mesura directa de l'index de refraccio d'una solucio d'ATP dona informacio sobre les caracteristiques electriques del medi. L'observacio de que aquest index canvia despres d'irradiar la solucio amb longituds d'ona visibles i infravermelles properes, confirmen que la llum provoca canvis electrics significatius en l'ATP. En aquest treball tambe s'ha estudiat el comportament bioquimic de l'ATP irradiat quant forma part de dues reaccions quimiques diferents: la de la luciferina-luciferasa i la de la hexoquinasa. En tots dos casos, l'us d'aquest ATP irradiat ha produit una alteracio dels parametres cinetics estudiats, V0 i k en la reaccio de la luciferina-luciferasa, i km i vmax en la reaccio de l'hexoquinasa.ConclusionsAquesta interaccio no absortiva de la llum amb l'ATP es la primera descrita per a una biomolecula. El mecanisme aporta noves dades per explicar els efectes observats en el metabolisme cel.lular despres de l'irradiacio d'organismes, teixits i cultius cel.lulars amb llum visible i infravermella propera. / ATP is a key molecule in cellular metabolism. In this thesis, I examined the effects of visible (635 and 655 nm) and near-infrared (810 and 830 nm) light on ATP in solution. I also examined were the biochemical behavior of light-exposed ATP in the luciferine-luciferase reaction and hexokinase reaction, the initial step in glycolysis that begins extra mithocondrial ATP synthesis. Irradiated groups in the luciferine-luciferase reaction showed an improvement in the kinetic parameters V0 and k, and more ATP molecules reacted with the enzyme when they were excited by light. When irradiated ATP was added to the hexokinase reaction, the experimental groups showed significant differences in the Michaelis-Menten kinetic parameters (km for ATP and vmax) and the rate of product synthesis was greater. Changes in both reactions were wavelength and dose dependant. When ATP was excited with UV photons, it fluoresced. This fluorescence decreased when Mg2+ was added, probably because the ion binds the phosphates, which are the part of the molecule responsible for light emission. Irradiating the ATP-Mg2+ solution with 655 nm and 830 nm light increased the fluorescence resulting from a displacement of charges in the phosphor-oxygen bond that repels Mg2+. The refraction of light in an ATP solution was observed by the Michelson interferometer and by directly measuring the refractive index. The refractive index changed after red and near-infrared light interaction due to a change in the electrical permittivity of the medium. Since ATP in water is transparent to visible and near-infrared light, and is therefore not a chromophore for those wavelengths, I conclude that the observed light interaction with ATP is not due to photon absorption but to the electromagnetic disturbance produced by the light, which leads to a polarization of the dielectric molecule that is ATP. This interaction of visible and near-infrared electromagnetic energy with ATP offers new perspectives for explaining light interaction at subcellular level.

Este trabajo se enmarca dentro de la calidad del suministro de energía eléctrica y, en concreto, en el efecto que producen los huecos de tensión en las máquinas de inducción y en los transformadores trifásicos.En primer lugar se ha realizado una introducción al estudio de sistemas trifásicos mediante el método de las componentes simétricas que permiten obtener las ecuaciones de los huecos de tensión analizando las faltas que los originan (trifásica, trifásica-tierra, fase-tierra, fase-fase, fase-fase-tierra), así como su transmisión a otros niveles de tensión y la influencia de la conexión de la carga. Los huecos se han caracterizado mediante su duración, su profundidad, el punto de onda inicial y siete tipologías diferentes de huecos.En un motor de inducción trifásico, un hueco de tensión produce los siguientes efectos: picos de intensidad, picos de par y pérdida de velocidad mecánica. En este trabajo se han determinado los puntos de onda inicial y los valores de duración más desfavorables.También se han obtenido las superficies tridimensionales de los mencionados efectos en función de la profundidad y de la duración para todos los tipos de huecos, midiendo las distancias Euclídeas entre ellas, así como las curvas de nivel de estas superficies que representan la sensibilidad de la máquina a los huecos de tensión. Se ha observado que es posible reducir el número de tipos de huecos a estudiar de siete originales a tres si se utiliza la componente directa de la tensión (en lugar de la profundidad) en las curvas de sensibilidad.En un transformador trifásico, un hueco de tensión puede llegar a saturarlo dando lugar al consumo de una intensidad elevada. Se han determinado los puntos de onda inicial y las duraciones del hueco más desfavorables, así como la relación entre los picos de intensidad y la profundidad. Se ha presentado un método analítico para la determinación de la componente continua del flujo y de los picos de intensidad, que han permitido la obtención de superficies tridimensionales de los picos de intensidad en función del punto de onda inicial y de la duración, así como en función de la profundidad y de la duración para todos los tipos de huecos. Por último el estudio de estas superficies tridimensionales ha permitido agrupar diferentes tipos de huecos con severidad similar, reduciendo a tres las siete tipologías originales.