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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
11

Modélisation du captage des polluants lors des opérations de meulage / Numerical simulation of polutant capture in grinding operations

Tafnout, Fatna 01 October 2012 (has links)
Ce travail à la fois expérimental et numérique s'inscrit dans le cadre des études menées sur le captage des polluants sur les machines tournantes et porte plus particulièrement sur la caractérisation de la dispersion des fines particules d'usinage : i.e. ces particules sont assimilées à un traceur passif du fait de leur temps de réponse aérodynamique négligeable. L'objectif recherché à travers cette étude est de développer une méthode numérique de conception des captages des polluants sur machines d'usinage. Le premier travail a donc consisté à réaliser un dispositif expérimental recréant une situation similaire à une opération de meulage, mais dans un cadre parfaitement contrôlé. L'écoulement retenu est celui engendré par un cylindre en rotation dans une veine d'essai ventilée. Un polluant particulaire, représenté par des microbilles de verre sphériques, est généré dans la veine d'essai (en utilisant un système d'ensemencement en particules), recréant ainsi les effets aérauliques des plus grosses particules d'usinage. Un gaz traceur, représentant la fraction fine de particules, est injecté simultanément avec le jet de particules via un capillaire placé à la source. Les propriétés du jet de particules obtenu sont caractérisées par vélocimétrie par suivi de particules (PTV) afin d'obtenir des données d'entrée et de validation pour les simulations numériques. La méthode de suivi de particules, mise au point dans le cadre de cette étude reste robuste même dans les zones fortement chargées en particules (i.e. la région source du jet). Le champ de vitesse de la phase gazeuse a quant à lui été caractérisé par vélocimétrie par images de particules (PIV), au moyen d'un code développé spécifiquement (Belut 2006 [4]). La dispersion d'un gaz traceur (hexafluorure de soufre - SF6) émis simultanément avec le jet de particules est ensuite étudiée expérimentalement : la dispersion d'un tel gaz est en effet jugée représentative de celle des plus fines particules d'usinage dont le temps de réponse aérodynamique est négligeable. Enfin, une modélisation complète du banc d'essai est réalisée afin de permettre une comparaison avec les résultats expérimentaux pour ainsi progresser dans la validation des modèles utilisés pour décrire la dispersion d'un traceur gazeux représentatif des fines particules. Le logiciel commercial Fluent est employé pour les simulations numériques / This study based at the same time in experimental and numerical aspect, is part of the framework to understand deeply the capture of pollutants on rotating machines and focuses especially on the characterization of the dispersion of fine particles generated by machining operations : i.e. these particles are assimilated as a passive tracer due to their negligible aerodynamic response time. The main objective covered by this study is to develop a numerical method of close capture exhausts systems for machining devices. The first task was to achieve an experimental device re-creating a similar grinding operation in a perfectly controlled environment. The flow used is defined as the one generated by a rotating cylinder in a ventilated test rig. A pollutant of particles, represented by spherical glass beads, is produced inside the test rig (by using a seeding system of particles), recreating therefore the aerodynamic effects of largest machining particles. A tracer gas, representing the fraction of fine particles, is injected simultaneously with the jet of particles through a capillary tube placed at the source. The properties of the jet of particles obtained are characterized by particle tracking velocimetry (PTV) allowing then to obtain data input and validation of numerical simulations. This particle tracking technic, developed in this study remains as robust method even in heavily loaded particles (i.e. the source region of the jet). The velocity field of the gas phase has meanwhile been characterized by particle image velocimetry (PIV), using a code developed specifically for this assay (Belut 2006 [4]). The dispersion of a tracer gas (sulfur hexafluoride - SF6) emitted simultaneously with the jet of particles is then studied experimentally : the dispersion of such a gas is indeed considered representative of finest particles which aerodynamic response time is negligible. Finally, a complete modeling of the experimental test rig is performed to allow comparison with experimental results in order to progress in the validation of models used to describe the dispersion of a tracer gas representative of finest particles. The commercial soft ware FLUENT is used for numerical simulations
12

Melhoramento da tensão disruptiva de cadeias de isoladores de vidro para linhas de transmissão

Roberta Pereira Leite, Luiza January 2006 (has links)
Made available in DSpace on 2014-06-12T17:40:15Z (GMT). No. of bitstreams: 2 arquivo7614_1.pdf: 2492587 bytes, checksum: 9fdb3dceda854ab374cc2e324e109e2a (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2006 / As cadeias de isoladores de vidro são largamente utilizadas tanto para isolar os condutores de eletricidade da terra como suportá-los mecanicamente nas torres das linhas de transmissão. A quebra parcial ou total das partes vítreas dos isoladores de vidro, por exemplo, por vandalismo, reduz drasticamente a suportabilidade à tensão elétrica devido à diferença de potencial e é um problema mundial porque é responsável por um número significativo de desligamentos não programados pelas empresas do setor elétrico. Na presente dissertação, visou-se estudar a tensão de disrupção numa cadeia de isoladores de vidro com diferentes unidades com a parte vítrea quebrada simulando uma linha de transmissão de 230 kV. Visou-se também desenvolver técnicas para aumentar a suportabilidade à tensão de disrupção aumentando a rigidez dielétrica da atmosfera em torno da cadeia de isoladores, assim como dos isoladores com aplicação de materiais dielétricos. Os seguintes experimentos foram realizados visando atingir os objetivos: (i) caracterizar os isoladores por radiografia digital para conhecer detalhes de sua estrutura interna; (ii) estudar a influência da distribuição dos isoladores de vidro com as saias vítrea quebradas ao longo da cadeia com dezesseis unidades, (iii) estudar a influência de atmosferas de ar e de gás SF6 sob pressões de até 1,3 atm numa câmara de acrílico utilizando até três isoladores e à freqüência industrial (60 Hz), (iv) estudar a utilização da jaqueta polimérica para aumentar a tensão disruptiva de isoladores. Estudos das tensões disruptivas foram realizadas em cadeias com 12, 10 e 9 isoladores de vidro com parte vítrea quebrada e distribuídos ao longo da cadeia em cinco diferentes configurações. Para efeito de comparação, a temperatura, pressão e umidade relativa foram corrigidas de acordo com a norma NBR 6936. As medidas de tensão disruptiva foram realizadas à freqüência industrial e sob impulso de manobra nas polaridades positiva e negativa. Como resultado, nas cadeias de isoladores de vidro com 6 unidades inteiras e 10 quebradas, observou-se uma diferença significante nos valores das tensões disruptivas em função da mudança na distribuição dos isoladores inteiros e quebrados ao longo da cadeia. Um resultado interessante ocorreu quando utilizamos 6 isoladores inteiros uniformemente distribuídos ao longo da cadeia e intercalados com 10 isoladores quebrados, onde obteve-se a tensão disruptiva de 459 kV, maior que quando utilizamos 7 isoladores inteiros intercalados com 9 quebrados, onde obteve-se o valor de tensão disruptiva de 448 kV. Este resultado pode ser atribuído à distribuição não uniforme de campo elétrico ao longo da cadeia e aumento do caminho. Nos ensaios com a câmara de atmosfera controlada, não se observou influência significativa da atmosfera de ar e SF6 e da pressão no valor da tensão disruptiva nos isoladores de vidro com as saias quebradas, que foi em torno de 15 kV. Por outro lado, o valor da tensão disruptiva para isoladores inteiros sob atmosfera de SF6 e sob a pressão de 1.3 atm alcançou um valor de 120 kV, enquanto que o valor da tensão disruptiva na atmosfera de ar e pressão local foi de 80 kV. Estudo comparativo sobre a tensão disruptiva da cadeia de 16 isoladores de vidro com as partes de vidro totalmente quebradas mas com pinos revestidos com jaqueta de nylon mostrou que a tensão disruptiva foi de 375 kV, que é cerca de 3,5 vezes superior do que uma cadeia com 16 isoladores quebrados convencionais. Consequentemente, os resultados positivos obtidos no presente trabalho poderão, com estudos mais detalhados, assegurar no futuro: um maior segurança na atuação das equipes que trabalham na manutenção de linhas energizadas e minimizar a quantidade de desligamentos das linhas de transmissão
13

Surface charge accumulation and partial discharge activity for small gaps of electrode/epoxy interface in sf6 gas

Okubo, Hitoshi, Mansour, Diaa-Eldin A., Kojima, Hiroki, Hayakawa, Naoki, Endo, Fumihiro 08 1900 (has links)
No description available.
14

Partial Discharges and Associated Mechanisms for Micro Gap Delamination at Epoxy Spacer in GIS

Okubo, Hitoshi, Endo, Fumihiro, Hayakawa, Naoki, Kojima, Hiroki, Mansour, Diaa-Eldin A 06 1900 (has links)
No description available.
15

Estimation Of Steady-state Temperature Distribution In Power Transformer By Using Finite Difference Method

Gozcu, Ferhat Can 01 February 2010 (has links) (PDF)
Estimating the temperature distribution in transformer components in the design stage and during the operation is crucial since temperatures above the thermal limits of these components might seriously damage them. Thermal models are used to predict this vital information prior to actual operations. In this study, a two-dimensional, steady-state model based on the Finite Difference Method (FDM) is proposed to estimate the temperature distribution in the three-phase, SF6 gas insulatedcooled power transformer. The model can predict the temperature distribution at the specific discredited locations in the transformer successfully. This study also compares predicted temperatures of the model proposed in this study with the results of the previous study which is based on Finite Element Method (FEM) and the results of the research performed by the designers of the transformer. The results show that FDM model proposed in this study can be used to estimate the temperature distribution in the transformer with an acceptable accuracy and can be an alternative of the previous study which is based on FEM.
16

Breakdown Voltage of Compressed Sulfur Hexafluoride (SF6) at Very Low Frequency / Low Frequency (30 kHz)

January 2010 (has links)
abstract: The U.S. Navy is interested in evaluating the dielectric performance of SF6 at 30 kHz in order to develop optimal bushing designs and to ensure reliable operation for the Very Low Frequency/ Low Frequency (VLF/LF) transmitting stations. The breakdown experiments of compressed SF6 at 30 kHz in the pressure range of 1-5 atm were conducted in both the uniform field (plane-plane gap) and the non-uniform field (rod-plane gap). To understand the impact of pressure on the breakdown voltage of SF6 at VLF/LF, empirical models of the dielectric strength of SF6 were derived based on the experimental data and regression analysis. The pressure correction factors that present the correlation between the breakdown voltage of SF6 at VLF/LF and that of air at 50/60 Hz were calculated. These empirical models provide an effective way to use the extensively documented breakdown voltage data of air at 60 Hz to evaluate the dielectric performance of SF6 for the design of VLF/LF high voltage equipment. In addition, several breakdown experiments and similar regression analysis of air at 30 kHz were conducted as well. A ratio of the breakdown voltage of SF6 to that of air at VLF/LF was calculated, from which a significant difference between the uniform gap and the non-uniform gap was observed. All the models and values provide useful information to evaluate and predict the performance of the bushings in practice. / Dissertation/Thesis / M.S. Electrical Engineering 2010
17

Elimination of SF6 from transmission system equipment

Cai, Xiaolei January 2013 (has links)
Sulphur hexafluoride gas is the dominant insulation and interruption material in high voltage gas insulated substation. Its usage remains a concern of transmission system operators owing to the global warming potential of the gas. The work carried out in this thesis aims to find the environment-friendly materials that can replace SF6. These candidates are required to have a strong dielectric strength for high voltage busbar insulation and well arc extinguishing capability necessary for high voltage circuit breaker.A range of alternative insulation types including CF3I gas and its mixture, high pressure air and solid insulating foam are considered as substitute of SF6. Theoretical studies on the dimensions of busbars used in substations are carried out for these options. The dimension of the dielectric system and its ampacity of respect system are calculated using heat transfer models considering their boiling point and proper working pressure which is related with the dielectric strength of some gas.On the other hand, SF6 gas circuit breaker is extremely popular on the medium and high voltage power networks owning to its effective arc extinguishing performance. It would be ideal if a substitute material could be found for SF6 as an interruption material. Biodegradable oil PTFE ablation, other gas candidates including N2, CF3I are investigated as possible replacement of SF6 through literature study.The usage of vacuum circuit breaker is eventually capable to operate in high voltage transmission system. Simulations have been carried out with software ATP/EMTP to investigate the influence of different characteristics of vacuum circuit breaker including chopping current level, the dielectric strength of vacuum gap and the opening time. And then the probability of overvoltages when vacuum circuit breakers installed is studied by statistical study in MATLAB.
18

Signature infrarouge et modélisation pour la télédétection de deux gaz : SF6 et RuO4 / Infrared signature and modeling for remote sensing of two gases : SF6 and RuO4

Faye, Mbaye 07 July 2017 (has links)
Ce travail est une contribution à l'étude spectroscopique de deux gaz d'origine anthropique pour lesquels la modélisation de la signature infrarouge pourra permettre une quantification dans rejets dans l'atmosphère. L'hexafluorure de soufre (SF₆) est principalement rejeté par l'industrie électrique, qui l'utilise comme isolant gazeux et sa concentration augmente rapidement dans l'atmosphère (de l'ordre de 7 % par an). C'est un gaz à effet de serre puissant listé dans le protocole de Kyoto car ses absorptions intenses vers 10 μm placées dans une fenêtre de transparence atmosphérique et sa durée de vie élevée dans la haute atmosphère(environ 3200 ans) lui confèrent un coefficient de transfert radiatif exceptionnellement élevé (Global Warming Potential, ou GWP égal à 23900 fois celui du dioxyde de carbone, CO₂). La détection et la quantification de ce gaz par sa signature infrarouge via des mesures par satellite nécessitent une très bonne modélisation des spectres d'absorption, incomplètement connus. SF6 est un molécule lourde qui présente un grand nombre de niveaux vibrationnels de basse énergie, donnant lieu à des bandes chaudes dans le domaine des absorptions atmosphériques autour de sa vibration fondamentale la plus intense et celles-ci impliquent des états excités peu ou pas caractérisés. Ainsi, à température ambiante, seules 32 %des molécules se trouvent dans l'état vibrationnel de base. Les bases de données actuelles ne contiennent que les données des bandes de vibration fondamentales. Pour pallier ce manque, il est donc nécessaire d'observer un certain nombre de transitions depuis l'état fondamental vers les niveaux excités de basse énergie servant d'état initial aux bandes chaudes et d'acquérir des données permettant de modéliser aussi les niveaux à deux quanta de vibration constituant les niveaux d'arrivée de ces bandes chaudes.Ce travail est une contribution à l'étude spectroscopique de deux gaz d'origine anthropique pour lesquels la modélisation de la signature infrarouge pourra permettre une quantification dans rejets dans l'atmosphère. L'hexafluorure de soufre (SF6) est principalement rejeté par l'industrie électrique, qui l'utilise comme isolant gazeux et sa concentration augmente rapidement dans l'atmosphère (de l'ordre de 7 % par an). C'est un gaz à effet de serre puissant listé dans le protocole de Kyoto car ses absorptions intenses vers 10 μm placées dans une fenêtre de transparence atmosphérique et sa durée de vie élevée dans la haute atmosphère(environ 3200 ans) lui confèrent un coefficient de transfert radiatif exceptionnellement élevé (Global Warming Potential, ou GWP égal à 23900 fois celui du dioxyde de carbone, CO₂). La détection et la quantification de ce gaz par sa signature infrarouge via des mesures par satellite nécessitent une très bonne modélisation des spectres d'absorption, incomplètement connus. SF₆ est un molécule lourde qui présente un grand nombre de niveaux vibrationnels de basse énergie, donnant lieu à des bandes chaudes dans le domaine des absorptions atmosphériques autour de sa vibration fondamentale la plus intense et celles-ci impliquent des états excités peu ou pas caractérisés. Ainsi, à température ambiante, seules 32 %des molécules se trouvent dans l'état vibrationnel de base. Les bases de données actuelles ne contiennent que les données des bandes de vibration fondamentales. Pour pallier ce manque, il est donc nécessaire d'observer un certain nombre de transitions depuis l'état fondamental vers les niveaux excités de basse énergie servant d'état initial aux bandes chaudes et d'acquérir des données permettant de modéliser aussi les niveaux à deux quanta de vibration constituant les niveaux d'arrivée de ces bandes chaudes. / This work is a contribution to the spectroscopic study of two gases of anthropogenic origin for which the modeling of the infrared signature may allow a quantification in releases in the atmosphere. Sulfur hexafluoride (SF₆) is mainly rejected by the electrical industry, which uses it as a gaseous insulator and its concentration increases rapidly in the atmosphere (of the order of 7 per cent per year). It is a potent greenhouse gas listed in the Kyoto Protocol because its intense absorption around 10 μm issituated in a window of atmospheric transparency and its lifetime in the upper atmosphere (approximately 3200 years) confers an exceptionally high radiative transfer coefficient (Global WarmingPotential, or GWP equal to 23900 times that of carbon dioxide,CO2). The detection and quantification of this gas by its infrared signature via satellite measurements requires a very accurate modeling of the absorption spectra, incompletely known. For SF₆,in particular because of the fact that this heavy molecule presents a large number of vibrational levels of low energy, there exist hotbands in the region of atmospheric absorption around its most intense fundamental absorption (v₃); these involve excited states little or not characterized. Thus, at ambient temperature, only 32% of the molecules are located in the vibrational ground state. Current databases contain only data for the fundamental vibrationband. To compensate for this lack, it is therefore necessary toobserve a number of transitions from the ground state to the excited levels of low energy used corresponding to initial states of the hot bands and to acquire data for modeling also levels with two quanta of vibration constituting the upper levels of arrival of thes ehot bands.This work is a contribution to the spectroscopic study of two gases of anthropogenic origin for which the modeling of the infrared signature may allow a quantification in releases in the atmosphere.Sulfur hexafluoride (SF₆) is mainly rejected by the electricalindustry, which uses it as a gaseous insulator and its concentration increases rapidly in the atmosphere (of the order of 7 per cent peryear). It is a potent greenhouse gas listed in the Kyoto Protocolbecause its intense absorption around 10 μm is situated in awindow of atmospheric transparency and its lifetime in the upper atmosphere (approximately 3200 years) confers an exceptionally high radiative transfer coefficient (Global Warming Potential, orGWP equal to 23900 times that of carbon dioxide, CO2). The detection and quantification of this gas by its infrared signature via satellite measurements requires a very accurate modeling of theabsorption spectra, incompletely known. For SF6, in particular because of the fact that this heavy molecule presents a largenumber of vibrational levels of low energy, there exist hot bands in the region of atmospheric absorption around its most intensefundamental absorption (v3); these involve excited states little or not characterized. Thus, at ambient temperature, only 32 % of themolecules are located in the vibrational ground state. Current databases contain only data for the fundamental vibration band. Tocompensate for this lack, it is therefore necessary to observe a number of transitions from the ground state to the excited levels oflow energy used corresponding to initial states of the hot bands andto acquire data for modeling also levels with two quanta ofvibration constituting the upper levels of arrival of these hot bands.
19

Marketing Introduction Plan for the New Generation of Sustainable Circuit Breakers LTA 420 kV : A real-life case for implementation at Hitachi ABB Power Grids / Introduktionsplan för marknadsföring av den nya generationen av hållbara strömbrytare LTA 420 kV : Ett verkligt fall för implementering på Hitachi ABB Power Grids

Fradinho Bastos, Ivan January 2021 (has links)
En strömbrytare är en säkerhetsanordning som är utformad för att bryta strömmen om ett problem uppstår. Det finns ett flertal olika typer av brytare beroende vilket spänningsområde som avses. Brytare för lågspänning används för hushållsapparater, medan högspänningstyper används för överföring av spänning i elnätet. Högspänningsbrytare använder idag svavelhexafluorid (SF6)-gas, som ett isolerande medium för att släcka den elektriska ljusbåge som bildas när strömmen bryts. SF6 är dock problematiskt för miljön, då dess globala uppvärmningspotential (GWP) är 23 500 gånger högre jämfört med koldioxid (CO2). Företaget Hitachi ABB Power Grids har utvecklat AirPlus™-tekniken som ersätter SF6-gasen med en koldioxidbaserad gasblandning. Examensarbetet fokuserade på att utvärdera möjligheten att minska användningen av SF6 genom AirPlus™-tekniken och hur tekniken skulle kunna en introduceras på marknaden. Slutligen så utvecklades en strategi för hur detta skulle kunna ske. Studien innefattar den bakomliggande informationsinhämtningen och analysen av data, marknadsutvärdering och utvecklingen av marknadsstrategin. Detta utfördes genom att utvärdera AirPlus™-tekniken och dess konkurrenter, samt genom en kvalitativ och kvantitativ analys av implementeringen av LTA 420 kV-brytaren på marknaden. Sammanfattningsvis så visar studien att marknadsintroduktionen av strömbrytaren LTA 420 kV är genomförbar. Även om koldioxid tekniskt inte har samma prestanda som SF6, är tekniken fortfarande bra och ger starka kundfördelar: GWP minskar med över 99,9%, användningen överensstämmer med miljöregler, den ger lägre ägandekostnader, färre kontroller, lägre kostnad för hantering av koldioxidgasen, och fungerar väl vid extremt låga temperaturer. De viktigaste utmaningarna för Hitachi ABB Power Grids relaterar till konkurrensen på marknaden. Det är därför tillrådligt att företaget arbetar med en effektiv marknadsintroduktion för att säkerställa en stor marknadsandel. / A circuit breaker is a safety device designed to interrupt power if a problem is detected. There are several kinds of circuit breakers for different applications. Low-voltage circuit breakers are used for household appliances, while high-voltage types are used for transmission networks. High-voltage circuit breakers use sulfur hexafluoride (SF6) gas as an insulating medium, which extinguishes the electric arc that is formed when power is cut. However, it is a huge hazard for the environment, as its global warming potential (GWP) is 23,500 times higher than that of CO2 gas. The company Hitachi ABB Power Grids developed the AirPlus™ technology, which replaces the SF6 gas with a carbon dioxide (CO2) based gas mixture. The presented degree project has evaluated the feasibility of reducing the use of SF6 through the AirPlus™ technology and then developed a strategy for the company Hitachi ABB Power Grids for the market introduction of the eco-efficient LTA 420 kV circuit breaker. This study covers the background research, market evaluation, and market strategy.  It was done through research about the AirPlus™ technology and its competitors, so as qualitative and quantitative analysis of the LTA 420 kV circuit breaker implementation in the market. In conclusion, the study shows that the market introduction of the LTA 420 kV circuit breaker is feasible. Although CO2 is not as good an insulation medium as SF6, it is still good and presents strong customer benefits: GWP reduced by over 99.9%, compliance with new regulations, lower cost of ownership, fewer regulatory controls, reduced cost of handling the gas, and well-functioning at extremely low temperatures. The main concerns for Hitachi ABB Power Grids are related to market competition. Thus, it is advisable that the company works on an effective market introduction to assure a large market share.
20

Etude diélectrique d'une isolation hybride gaz-solide pour<br />appareillage haute tension

Hairour, M. 30 October 2007 (has links) (PDF)
Ce travail s'inscrit dans le cadre de la recherche d'une alternative à l'utilisation du gaz hexafluorure de soufre (SF6) dans l'appareillage électrique Haute Tension et plus particulièrement les postes à isolation gazeuse. Ce gaz possède de bonnes propriétés diélectriques mais son potentiel de réchauffement global (GWP) est estimé à 23900 fois celui du dioxyde de carbone (CO2). Malheureusement, les gaz dits simples tels que l'azote (N2) présentent une tenue diélectrique deux à trois fois plus faible que celle du SF6. Dans le but d'améliorer la tenue diélectrique du système, on se propose d'appliquer sur les conducteurs un revêtement isolant à base de silicone constitué d'une sous-couche semi-conductrice d'épaisseur 0.5 mm et d'une couche isolante d'épaisseur 1.5 mm, afin de limiter ou supprimer le rôle des électrons du métal. Les essais diélectriques réalisés sous une tension de choc de foudre dans des conditions expérimentales proches de la pratique industrielle montrent qu'en présence de N2 la tenue diélectrique du système avec revêtement est régie par le gaz. Par conséquent, l'application du revêtement sur les conducteurs diminue la tenue diélectrique du système (effet capacitif). Ces essais montrent, cependant, qu'il est possible d'améliorer la tenue diélectrique du système avec le revêtement, en présence de SF6 en polarité négative. Les paramètres importants semblent être la propreté du gaz, le degré d'ionisation du gaz et surtout les charges qui se déposent à l'interface gaz-revêtement qui, dans certaines conditions, ont un effet bénéfique : ce dernier résultat donne une piste aux travaux futurs.

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