Calcul des courants induits : des équations locales aux effets macroscopiques.
BARRE Olivier 01/04/2013 Ingénieur A&M Agrégé G.E. Dr és sciences Enseignant en Techno Techno (collège DESCARTES - Mons en Baroeul – Nord/59 )
-1-
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Préambule Pour exploiter les propriétés des champs magnétiques, il faut des matériaux ayant des perméabilités élevées. Les matériaux magnétiques idéaux n’existent pas. Les perméabilités ne sont pas infinies et les comportements ne sont pas linéaires par rapport à l’excitation magnétique (H). De plus, des phénomènes parasites apparaissent apparaissent lors des mises en œuvre et se traduisent par des réalisations dont les rendements sont certes très bons, mais n’arriveront jamais à 100 %. Dans le cas concret des machines électriques, les tôles fines au silicium sont utilisées pour réaliser les parties magnétiques massives telles que la tôlerie active du rotor et du stator. Ces constituants ont un comportement non linéaire. Ils présentent toujours des imperfections comme un phénomène d’hystérésis et ne peuvent empêcher à de faibles courants induits de se développer. Ainsi, les efforts déployés depuis des décennies pour limiter les pertes ont finalement conduit à la réalisation de machines présentant des rendements de plus de 97%. Maintenant, les augmentations de rendement sont surtout issues d’une meilleure maîtrise des pertes à l’extérieur de ces parties massives. Une ventilation plus efficace permet de diminuer la puissance mécanique prise au moteur pour se refroidir. Une recherche précise de la circulation des flux de fuite à l’extérieur des parties actives (Rotor + stator) conduit à identifier les éléments de la machine qui consomment inutilement la puissance active produite par le champ magnétique. Finalement, pour des équipements de très forte puissance comme les turbo-alternateurs, les constructeurs arrivent à proposer des équipements de plus de 900 MW avec des rendements supérieurs à 99 %. Les machines électriques existent depuis plus d’un siècle. Elles sont régulièrement améliorées et cela explique les forts rendements rencontrés aujourd’hui. De plus, les puissances mises en jeux en font des équipements clefs des dispositifs liés à l’énergie électrique. Le rendement n’est pas le seul objectif à atteindre, la fiabilité est toute aussi importante. Que penser d’une centrale électrique ou le générateur d’un millier de Mégawatt devrait être remplacé tous les cinq ans ? De même, peut-on imaginer un réseau de distribution ou l’on considère les transformateurs haute-tension au même titre que les fusibles ? Bien au contraire, ces équipements sont associés à des contraintes fortes, ils sont et resteront couteux, ils ne peuvent donc pas être redondants. L’histoire montre que les utilisateurs leur demandent toujours plus au niveau durée de vie, rendement... Les efforts consentis en R&D pour répondre à ces attentes leur ont ouverts aussi de nouveaux champs d’application. Les moteurs électriques propulsent maintenant des bateaux de commerce, des paquebots ou même des bâtiments militaires autres que les sous-marins. L’arrivé de l’électronique n’a pas changé cet état de fait, il suffit de prendre un exemple tiré du quotidien, vous pouvez toujours changer très facilement de téléphone portable mais il n’en est pas de même pour les moteurs électriques qui équipent votre voiture. Ce n’est pas la simplicité apparente de ces équipements qui doit conditionner leur importance. Ne voir en ces objets qu’un assemblage de cuivre, fer et isolant revient à considérer l’être humain comme un mélange d’oxygène (65%), carbone (18%), hydrogène (10%), azote (3%)… « On fait la science avec des faits, comme on fait une maison avec des pierres : mais une accumulation de faits n'est pas plus une science qu'un tas de pierres n'est une maison. maison. » Henri Poincaré
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Sommaire.
Introduction.
Page 7
Chapitre 1 : Les courants courants induits vus vus comme une application application du vecteur vecteur de Poynting Poynting .
Page 9
Chapitre 2 : Effets des courants induits induits (Cas du blindage électromagnétique) électromagnétique)..
Page 15
Chapitre 3 : Equations locales locales des courants induits. induits.
Page 23
Chapitre 4 : Cas d’étude en éléments finis.
Page 26
Chapitre 5 : Les difficultés difficultés d’interprétati d’interprétation on et utilisation utilisation des sources de tension.
Page 35
Chapitre 6 : Limites d’étude d’étude des éléments éléments finis finis 2D.
Page 37
Chapitre 7 : Application Application aux machines machines électriques. électriques.
Page 39
Conclusion.
Page 41
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Introduction.
Les champs magnétiques sont les supports à privilégier pour lier des circuits électriques entre eux. En effet, l’absence de contact physique découple électriquement et mécaniquement les circuits qui peuvent être à des niveaux de tension différents ou en mouvement. Pour l’absence de contact électrique, elle se nomme « isolation galvanique » et elle se rencontre régulièrement dans les systèmes de mesure. Toutefois, tout conducteur parcouru par un courant, engendre un champ magnétique et ce champ est parfois un élément non désiré dans le montage que l’on réalise. Il existe, bien sûr, des moyens de limiter les effets néfastes. Les blindages, utilisant les courants induits, sont des solutions faciles à mettre en œuvre et particulièrement efficaces. Les principes physiques avaient été compris très tôt et les solutions techniques n’avaient pas attendu les logiciels de calcul de champ pour être opérationnelles. Pour tous les équipements où interviennent les champs magnétiques, que se soit l’effet principal recherché ou un effet parasite indésirable, le cheminement est toujours le même. Un champ magnétique variable circule dans l’air ou un matériau magnétique. Des circuits apparaissent, dans les matériaux conducteurs et ils voient ainsi passer un flux variable. Une force électromotrice se développe et un courant peut apparaitre. Ce principe est exploité dans les machines électriques et les transformateurs. Il est aussi recherché dans les blindages utilisés en électronique et précédemment précédemment cités. Malheureusement, Il est aussi un effet indésirable pour les autres pièces constituant un équipement. En reprenant l’exemple du Turbo-alternateur, les concepteurs concepteurs vont vont surtout dimensionner la partie active de la machine pour fournir la puissance requise avec le moins de perte possible. Ces parties actives sont obligatoirement maintenues en position par d’autres parties métalliques qui apporteront apporteront des pertes indésirables car car elles subissent subissent aussi les champs magnétiques. Pour illustrer cette dissertation sur les courants induits, la première partie de cette présentation reprendra une ancienne note technique de la NASA (Datée de 1965) dans laquelle l’auteur montre que les matériaux présentent, de manière générale, deux types de pertes quand ils sont soumis à un champ variable. Les conclusions de cette note sont toujours d’actualité et reviennent régulièrement lors des pré-dimensionnements des machines. De par l’objectif de la note technique, l’aspect positif des courants induits n’est pas abordé et c’est pour cette raison que la deuxième partie de la présentation se recentrera sur les blindages utilisés en électronique. Ici aussi, il apparaitra que le résultat espéré est
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1)_Les courants induits vus comme une application du vecteur de Poynting.
En 1965, la NASA publie une note technique sur les moyens de calculer les pertes par hystérésis et par par courant induit dans dans les tôles laminées laminées des transformateurs. transformateurs. Cette note technique est issue d’un plus vaste programme axé sur la distribution et l’utilisation de l’énergie dans un véhicule spatial. Déjà, le souci d’économie et de bonne utilisation de l’énergie électrique dans un véhicule autonome et isolé était une préoccupation des ingénieurs de cette agence américaine. Par la suite, l’aventure l ’aventure d’Apollo 13 (13 avril 1970) remettra sur le devant de la scène, les contraintes extrêmes qui étaient associées à ce genre de véhicule. Cette note technique n’utilise que des calculs analytiques relativement simples pour arriver à des conclusions qui sont encore celles que l’on utilise par défaut pour le calcul des pertes dans les tôles magnétiques. A partir des équations de Maxwell et du vecteur de Poynting, l’auteur, John BARRANGER arrive à prouver que les pertes que l’on rencontre dans les tôles sont de deux types : - Les pertes par courants induits - Les pertes par Hystérésis ;
Pour les courants induits , ceux-ci existent dans tous les matériaux conducteurs, à partir du moment où ils sont soumis à des champs variables. De plus, les densités de courant ne sont pas homogènes et un effet pelliculaire apparait. Ainsi, lors de la résolution des équations différentielles, un terme que l’on désignera sous le nom d’épaisseur de peau sera mis en évidence. C’est la valeur associée à ce terme qui délimitera la zone de circulation des courants. Si cette épaisseur est faible, les courants circuleront à la périphérie du volume soumis au champ variable.
Pour les pertes par hystérésis , celles-ci sont liées à la matière et plus précisément à son comportement sous champ variable. La variation de l’induction B n’est pas linéaire par rapport à l’excitation magnétique H. Cet état de fait se retrouve dans la forme de la courbe B=f(H) d’un matériau ferromagnétique. Les pertes par hystérésis sont proportionnelles à la surface de la courbe B=f(H) et aussi proportionnelles à la fréquence.
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Ce vecteur (1) est le produit vectoriel entre E et B. A tout instant, il représente le flux d’énergie associé à l’onde électromagnétique. Toutefois, ce qui est le plus intéressant, c’est la divergence de ce vecteur pour un volume donné. En effet, si cette divergence est nulle, cela veut dire que le volume n’absorbe aucune énergie électromagnétique et que le transfert s’effectue sans perte. Par contre, si cette divergence est non nulle, cela nous informe que le volume considéré absorbe de l’énergie. Il y a donc des pertes lors du transfert. -
La divergence du vecteur de Poynting. Cette opération ne présente un intérêt que si elle est associée aux équations de Maxwell (2 et 3).
(2)
(3) L’équation (2) n’est que la génération de la force électromotrice à partir d’un champ variable. L’équation (3) est la relation entre le champ et les courants qui l’engendrent. Toutefois, dans cette équation, les courants de déplacement ont été omis car ils sont généralement négligeables devant les courants de conduction.
Ces deux équations sont importantes car elles permettent d’écrire la divergence du vecteur de Poynting sous une forme plus exploitable pour le problème concerné (4).
(4) En faisant intervenir le temps, comme par exemple deux instants t1 et t2, on peut poursuivre en écrivant l’énergie perdue dans le volume concerné (5) entre ces deux
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(6) (terme 1)
(terme 2)
Pertes par courants induits. Les courants induits se comportent d’une manière très particulière, ils ne sont pas uniformément répartis dans le volume. Bien au contraire, ils se retrouvent, dans certains cas à la surface des volumes ou ils apparaissent. C’est donc cette particularité qui va être mise en évidence. Le point de départ, pour cette démonstration est l’équation (7) qui n’est .E ). que l’association des équations (2) et (3) et de la loi de comportement ( J = σ E ).
(7) De plus, en utilisant l’opérateur rotationnel (8), on arrive à une équation similaire à celle associée à la conduction ou la diffusion (9) :
(8)
(9)
L’auteur poursuit son analyse dans un cas simple bidimensionnel. Le problème est réduit à une simple tôle de largeur a et de longueur l (infinie) dans laquelle passe un champ B selon l’axe Y, des courant induits J0 vont apparaitre sur les surfaces extérieures
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Pour un régime sinusoïdal, la densité de courant J (Ici J z) peut se mettre sous la forme d’une équation différentielle (10) :
(10)
avec
(11)
Et le terme T est caractéristique du volume contenant les courants induits. Ce n’est rien d’autre que l’inverse de l’épaisseur de peau (12).
(12) Finalement, l’auteur, John BARRANGER, arrive à proposer une formule générale pour calculer les pertes liées aux courants induits dans les tôles fines des transformateurs. Dans le cas où l’épaisseur de peau est grande par rapport à l’épaisseur de la tôle, les pertes s’écrivent :
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Pertes par hystérésis.
L’entité à évaluer est :
(14) En plus, si on considère un régime sinusoïdal établi, il suffit de faire l’intégration sur une période complète.
(15) Ce qui donne un résultat bien connu, c’est l’aire du cycle d’ hystérésis.
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réalisation de la machine et ne sont sont donc pas disponibles pour pour les phases de conception et de dimensionnement dimensionnement (M.V. ( M.V. DESHPANDE).
Pour aller plus loin :
NASA TECNICAL NOTE / NASA TN D-3114 -HYSTERESIS AND EDDY-CURRENT LOSSES OF A TRANSFORMER LAMINATION VIEWED AS AN APPLICATION OF THE POYNTING THEOREM By John John Barranger Barranger / Lewis Research Center Cleveland, Ohio November 1965
EDDY CURRENTS IN SOLID RECTANGULAR CORES S. K. Mukerji, M. George, M. B. Ramamurthy and K. Asaduzzaman Asaduzzaman Faculty of Engineering & Technology, Multimedia University Melaka 75450, Malaysia DESIGN and TESTING TESTING of ELECTRICAL MACHINES. MACHINES. (Seconde Edition) M.V. DESHPANDE
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2)_Effets des courants induits (dans l es blindages).
Les blindages sont apparus apparus en même même temps que les premiers émetteurs/récepteurs. Ces simples plaques de tôles conductrices ont un rôle important à jouer dans la stabilité des montages. Le principe lié à leur fonctionnement est celui du courant induit : tout conducteur plongé dans un champ magnétique variable voit se développer des courants induits. L’origine de ces courants se retrouve dans la force électromotrice développée dans un circuit par un champ magnétique variable. Utilisation du blindage électromagnétique .
Généralement, les équipements électroniques travaillant dans des domaines de fréquences dépassant le Khz sont pourvus de blindage électromagnétique. L’objectif est d’interdire l’arrivée de signaux parasites sur les circuits à l’intérieur de la zone protégée. Pour réaliser cet objectif, le circuit à protéger est enveloppé par des matériaux conducteurs qui se comporteront comme des écrans (Photo-1). Ce procédé est utilisé depuis les débuts de l’industrie électronique. Il est aussi utilisé dans l’autre sens, c'est-à-dire qu’il peut interdire aux perturbations électromagnétiques associées à un circuit de se propager au voisinage immédiat. Les perturbations sont alors arrêtées par le blindage. Les normes liées à la CEM (Compatibilité Electromagnétique) ont remis sur le devant de la scène ces simples tôles qui ne sont pas si simples à mettre en œuvre.
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Un champ magnétique (vert) arrive sur une surface conductrice qui est assimilée à une spire conductrice (noire). Un courant va apparaitre dans cette spire fictive pour s’opposer à la variation du champ magnétique incident (rouge). Le résultat est, qu’idéalement, le courant induit va annuler le champ magnétique au niveau de la surface conductrice. La partie ainsi protégée ne ressentira absolument pas les effets du champ extérieur.
Il devient évident qu’un champ magnétique non variable (Champ permanent) n’est pas altéré par ce type de blindage. Par contre, cette interprétation est une image i déalisée de ce que l’on espère avoir au niveau du blindage.
La réalité est tout autre. En effet, les parasites extérieurs arrivent toujours à passer l’écran et se retrouvent au niveau du montage à protéger. Les électroniciens connaissent bien ce phénomène et caractérisent les blindages par rapport à leur niveau d’atténuation. La représentation précédente n’est qu’une image bien imparfaite des phénomènes qui se déroulent dans cette simple plaque de tôle conductrice. Il faut donc rechercher dans l’origine des courants induits, les explications à ce phénomène d’atténuation. Si le comportement décrit précédemment était le comportement réel, une couche mince de matériau conducteur nous protégerait de toutes les perturbations électromagnétiques, pour toutes fréquences fréquences et pour toutes amplitudes. amplitudes. Le comportement comportement réel est un peu plus complexe ; pour mieux l’appréhender, il faut suivre le cheminement chronologique des effets et des causes.
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toutefois plus compréhensible lors de l’examen, d’une bobine à noyau de fer. Tout bobinage (avec ou sans noyau) possède une caractéristique inductive par le simple fait que le courant qui le parcourt produit un champ magnétique.
Ainsi, pour calculer le courant induit, il faut faire apparaitre la spire fictive comme étant un récepteur composé d’une résistance et d’une inductance en série.
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Celle-ci est interprétée comme étant une spire fermée. Un flux magnétique incident va apparaitre et sera proportionnel à la surface de la spire. Il est possible de représenter ce flux incident comme étant une fonction du temps.
Evolution du flux dans la spire fictive (l’évolution est aussi sinusoïdale). Ce flux va engendrer une force électromotrice dans la spire. Celle-ci sera déphasée de 90° par rapport au flux car elle est associée à -dB/dt.
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B max (incident) B = 0 (induit)
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Hélas, la caractéristique inductive du circuit n’est absolument pas maîtrisable. Il n’est pas possible d’augmenter le nombre de spire de la spire fictive. Il n’est pas possible d’augmenter la perméabilité magnétique du milieu sans modifier la caractéristique résistive. Utiliser de l’acier ferait revenir les problèmes de saturation et de rémanence. Même les alliages spéciaux comme le Mumetal à forte perméabilité sont limités en fréquence. La grandeur qui caractérise le mieux, les capacités du matériau à être utilisé comme blindage, est le produit σr.µr car il prend en compte l’effet inductif et l’effet résistif. Finalement, en haute fréquence, la seule solution est de diminuer au maximum la résistivité, pour rendre la spire fictive équivalente la plus inductive possible. Le principe du blindage par couche mince conductrice montre maintenant ses limites, il ne peut fonctionner efficacement qu’en haute fréquence, seule domaine où la partie inductive de la spire équivalente augmente naturellement par rapport à l’effet résistif. Mais cela n’explique pas encore pourquoi il faut aussi utiliser l’épaisseur de la tôle de blindage comme paramètre d’affaiblissement du champ incident.
La pénétration pénétration du champ. champ.
Pour une fréquence donnée, la spire équivalente se comporte comme une résistance en
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plus le volume du conducteur qui permet la circulation du courant diminue. La zone de conduction est faible et se concentre près de la surface du conducteur (Effet de peau). Ainsi, en comparant comparant un conducteur conducteur en cuivre avec son homologue en aluminium, les valeurs d’épaisseur de peau augmentent pour l’aluminium. Il a la même perméabilité magnétique que le cuivre mais il est de résistivité légèrement supérieure.
Exemples de valeurs Pour un conducteur en cuivre, on a les valeurs ci-dessous.
fréquence
δ
50 Hz
9,38 mm
60 Hz
8.57 mm
10 kHz
0.66 mm
100 kHz
0.21 mm
1 MHz
66
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solutions pour éviter aux aux montages de subir subir les perturbations extérieures. extérieures. Il existe, bien sûr, d’autres effets qui n’ont pas été abordés. En particulier, les phénomènes mécaniques qui pourraient se produire sur les tôles. Le courant induit et le champ magnétique incident vont engendrer des forces de Laplace sur le blindage. Effet réel mais qui reste suffisamment marginal pour être négligé.
Conclusion sur l’effet de peau dans les conducteurs.
Ce phénomène est bien connu des électroniciens qui travaillent t ravaillent en haute fréquence. Ils
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3)_Equations locales des courants induits. La première partie a permis de montrer que les pertes dans les matériaux sont de deux types : courants induits et hystérésis. Pour mettre en évidence les phénomènes associés aux courants induits et aux principes du blindage en Electronique, nous avons utilisé un exemple simple de surface conductrice. Le cheminement employé peut être exploité et peut servir de référence à un calcul local par élément fini. Cette méthode de calcul repose sur l’écriture des équations locales associées aux champs électromagnétiques. Les équations équations de Maxwell
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Ainsi, en 2D, sous condition de linéarité et de composantes variables sinusoïdalement, le problème de magnétodynamique se met sous une for me réduite avec un Laplacien:
Le premier terme n’est que rot( rot(A)).
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4)_Cas d’étude en éléments finis.
Les méthodes analytiques ne peuvent résoudre les problèmes posés que si les outils mathématiques sont disponibles. Les propriétés géométriques ainsi que des hypothèses simplificatrices peuvent contribuer à l’obtention du résultat. Bien souvent, ces méthodes ont montré leurs aptitudes à dimensionner tout à fait correctement des équipements électriques comme les machines tournantes ou les transformateurs. Des résultats précis sont finalement atteints quand, l’expérience aidant, les concepteurs sont arrivés à estimer les effets qu’ils ne pouvaient pas calculer en toute rigueur.
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Cette structure se comporte comme une inductance. Pour rester dans un problème simple à quantifier, le matériau est considéré comme linéaire avec une perméabilité relative de 1000, la bobine est constituée de 1000 spires et le courant sera de 1 A (valeur crête). Pour toutes les valeurs de tension ou de courant, les valeurs qui seront données par la suite, le seront en valeur crête.
Les simulations numériques en régime permanent nous donnent une inductance dont la valeur est de 1,85 H. Pour obtenir cette valeur, la méthode utilisée consiste à calculer le rapport entre le flux et le courant car : Ф= L × I
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L’axe horizontal est l’axe du courant et l’axe vertical est l’axe de la tension. Le
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Watts. La puissance apportée par la source est de 10,5 Watts (2,61 + 7,92). L’écran
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Pour un courant de 1 A, la tension aux bornes de la bobine est de 82,36 volts et le
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Ajout d’un anneau interne.
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En régime variable 50 Hz et présence de l’anneau dans l’entrefer.
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