Selected Papers from the 7th International Symposium on Electric and Magnetic Fields, June 2006 9781846637315, 9781846637308

Citation preview

compel cover (i).qxd

03/01/2008

11:10

Page 1

ISSN 0332-1649

Volume 27 Number 1 2008

COMPEL The international journal for computation and mathematics in electrical and electronic engineering Selected papers from the 7th International Symposium on Electric and Magnetic Fields, June 2006 Guest Editors: Patrick Dular, Gérard Meunier and Francis Piriou

www.emeraldinsight.com

COMPEL

ISSN 0332-1649 Volume 27 Number 1 2008

The international journal for computation and mathematics in electrical and electronic engineering Selected papers from the 7th International Symposium on Electric and Magnetic Fields, June 2006 Guest Editors Patrick Dular, Ge´rard Meunier and Francis Piriou

Access this journal online _________________________

5

Editorial advisory board __________________________

6

Preface __________________________________________

7

What do voltmeters measure? Alain Bossavit _________________________________________________

9

Electromagnetic torque calculation using magnetic network methods Andrzej Demenko and Dorota Stachowiak __________________________

17

Comparison between torque calculation methods in a non-conforming movement interface O.J. Antunes, J.P.A. Bastos and N. Sadowski ________________________

27

Coupling of finite formulation with integral techniques Aldo Canova, Fabio Freschi, Maurizio Repetto and Giambattista Gruosso ___________________________________________

Access this journal electronically The current and past volumes of this journal are available at:

www.emeraldinsight.com/0332-1649.htm You can search more than 175 Emerald journals in Emerald Management Xtra (www.emeraldinsight.com) See page following contents for full details of what your access includes.

37

CONTENTS

CONTENTS continued

Numerical solutions in primal and dual meshes of magnetostatic problems solved with the finite integration technique J. Korecki, Y. Le Menach, J-P. Ducreux and F. Piriou _________________

47

Identification of ferromagnetic thin sheets magnetization: use of gradient and potential measurements G. Cauffet, J.L. Coulomb, S. Guerin, O. Chadebec and Y. Vuillermet _____

56

A t0-f surface impedance formulation for multiply connected conductors G. Meunier, Y. Le Floch and C. Gue´rin_____________________________

64

Subdomain perturbation finite element method for skin and proximity effects in inductors Patrick Dular, Ruth V. Sabariego and Laurent Kra¨henbu¨hl ____________

72

Coupling of analytical and numerical methods for the electromagnetic simulation of permanent magnet synchronous machines M. Scho¨ning and K. Hameyer ____________________________________

85

On the use of PML for the computation of leaky modes: an application to microstructured optical fibres Y. Ould Agha, F. Zolla, A. Nicolet and S. Guenneau __________________

95

Investigation of the characteristics of conformal microstrip antennas Ralf T. Jacobs, Arnulf Kost, Hajime Igarashi and Alan J. Sangster ______

110

Adaptive time integration for electromagnetic models with sinusoidal excitation Galina Benderskaya, Herbert De Gersem, Wolfgang Ackermann and Thomas Weiland_______________________________________________

122

Computational methods for modeling of complex sources Markus Johansson, Lovisa E. Nord, Rudolf Kopecky´, Andreas Fhager and Mikael Persson ________________________________________________

133

Fundamental investigation of 3D optimal design of open type magnetic circuit producing uniform field Norio Takahashi, Koji Akiyama, Hirokazu Kato and Kanji Kishi _______

144

Design of a double-sided tubular permanent-magnet linear synchronous generator for wave-energy conversion Danson M. Joseph and Willem A. Cronje ___________________________

CONTENTS continued 154

Magnetic shielding of buried high-voltage (HV) cables by conductive metal plates Peter Sergeant, Luc Dupre´ and Jan Melkebeek _______________________

170

Improved AC-resistance of multiple foil windings by varying foil thickness of successive layers D.C. Pentz and I.W. Hofsajer _____________________________________

181

Influence of the magnetic model accuracy on the optimal design of a car alternator J. Cros, L. Radaorozandry, J. Figueroa and P. Viarouge _______________

196

Modeling of a beam structure with piezoelectric materials: introduction to SSD techniques Romain Corcolle, Erwan Salau¨n, Fre´de´ric Bouillault, Yves Bernard, Claude Richard, Adrien Badel and Daniel Guyomar___________________

205

3D micromagnetism-magnetostatic coupling technique for MR reading heads modeling I. Firastrau, L.D. Buda-Prejbeanu, J.C. Toussaint and J-P. Nozie`res ______

215

Analysis of the stray magnetic field created by faulty electrical machines V.P. Bui, O. Chadebec, L-L. Rouve and J-L. Coulomb __________________

224

Analysis of the structure-dynamic behaviour of an induction machine with balancing kerfs C. Schlensok and K. Hameyer ____________________________________

235

Wound magnetic core consequences on false residual currents B. Colin, A. Kedous-Lebouc, C. Chillet and P. Mas____________________

246

Limits and rules of use of a dynamic flux tube model Marie-Ange Raulet, Fabien Sixdenier, Benjamin Guinand, Laurent Morel and Rene´ Goyet ___________________________________

256

Finite element formalism for micromagnetism H. Szambolics, L.D. Buda-Prejbeanu, J.C. Toussaint and O. Fruchart_____

266

A 3D electric vector potential formulation for dynamic hysteresis and losses O. Maloberti, V. Mazauric, G. Meunier and A. Kedous-Lebouc __________

277

CONTENTS continued

New discretisation scheme based on splines for volume integral method: application to eddy current testing of tubes Christophe Reboud, Denis Pre´mel, Dominique Lesselier and Bernard Bisiaux _______________________________________________

288

Hybridization of volumetric and surface models for the computation of the T/R EC probe response due to a thin opening flaw Le´a Maurice, Denis Pre´mel, Jo´zsef Pa´vo´, Dominique Lesselier and Alain Nicolas __________________________________________________

298

Simple and direct calculation of capacitive sensor sensitivity map Je´roˆme Lucas, Ste´phane Hole´ and Christophe Baˆtis ___________________

307

h- and b-conform finite element perturbation techniques for nondestructive eddy current testing Ruth V. Sabariego and Patrick Dular ______________________________

319

www.emeraldinsight.com/compel.htm As a subscriber to this journal, you can benefit from instant, electronic access to this title via Emerald Management Xtra. Your access includes a variety of features that increase the value of your journal subscription.

Structured abstracts Emerald structured abstracts provide consistent, clear and informative summaries of the content of the articles, allowing faster evaluation of papers.

How to access this journal electronically

Additional complimentary services available

To benefit from electronic access to this journal, please contact [email protected] A set of login details will then be provided to you. Should you wish to access via IP, please provide these details in your e-mail. Once registration is completed, your institution will have instant access to all articles through the journal’s Table of Contents page at www.emeraldinsight.com/0332-1649.htm More information about the journal is also available at www.emeraldinsight.com/ compel.htm

Your access includes a variety of features that add to the functionality and value of your journal subscription:

Our liberal institution-wide licence allows everyone within your institution to access your journal electronically, making your subscription more cost-effective. Our web site has been designed to provide you with a comprehensive, simple system that needs only minimum administration. Access is available via IP authentication or username and password. Emerald online training services Visit www.emeraldinsight.com/training and take an Emerald online tour to help you get the most from your subscription.

Key features of Emerald electronic journals Automatic permission to make up to 25 copies of individual articles This facility can be used for training purposes, course notes, seminars etc. This only applies to articles of which Emerald owns copyright. For further details visit www.emeraldinsight.com/ copyright Online publishing and archiving As well as current volumes of the journal, you can also gain access to past volumes on the internet via Emerald Management Xtra. You can browse or search these databases for relevant articles. Key readings This feature provides abstracts of related articles chosen by the journal editor, selected to provide readers with current awareness of interesting articles from other publications in the field. Non-article content Material in our journals such as product information, industry trends, company news, conferences, etc. is available online and can be accessed by users. Reference linking Direct links from the journal article references to abstracts of the most influential articles cited. Where possible, this link is to the full text of the article. E-mail an article Allows users to e-mail links to relevant and interesting articles to another computer for later use, reference or printing purposes.

Xtra resources and collections When you register your journal subscription online you will gain access to additional resources for Authors and Librarians, offering key information and support to subscribers. In addition, our dedicated Research, Teaching and Learning Zones provide specialist ‘‘How to guides’’, case studies and interviews and you can also access Emerald Collections, including book reviews, management interviews and key readings. E-mail alert services These services allow you to be kept up to date with the latest additions to the journal via e-mail, as soon as new material enters the database. Further information about the services available can be found at www.emeraldinsight.com/alerts Emerald Research Connections An online meeting place for the world-wide research community, offering an opportunity for researchers to present their own work and find others to participate in future projects, or simply share ideas. Register yourself or search our database of researchers at www.emeraldinsight.com/connections

Choice of access Electronic access to this journal is available via a number of channels. Our web site www.emeraldinsight.com is the recommended means of electronic access, as it provides fully searchable and value added access to the complete content of the journal. However, you can also access and search the article content of this journal through the following journal delivery services: EBSCOHost Electronic Journals Service ejournals.ebsco.com Informatics J-Gate www.j-gate.informindia.co.in Ingenta www.ingenta.com Minerva Electronic Online Services www.minerva.at OCLC FirstSearch www.oclc.org/firstsearch SilverLinker www.ovid.com SwetsWise www.swetswise.com

Emerald Customer Support For customer support and technical help contact: E-mail [email protected] Web www.emeraldinsight.com/customercharter Tel +44 (0) 1274 785278 Fax +44 (0) 1274 785201

COMPEL 27,1

EDITORIAL ADVISORY BOARD

Professor O. Biro Graz University of Technology, Graz, Austria Professor J.R. Cardoso University of Sao Paulo, Sao Paulo, Brazil

6

Professor C. Christopoulos University of Nottingham, Nottingham, UK Professor M. Clemens Helmut-Schmidt University, Hamburg, Germany Professor J.-L. Coulomb Laboratoire d’Electrotechnique de Grenoble, Grenoble, France Professor X. Cui North China Electric Power University, Baoding, Hebei, China Dr L. Davenport Advanced Technology Centre, BAE Systems, Bristol, UK Professor A. Demenko Poznan´ University of Technology, Poznan´, Poland Professor E. Freeman Imperial College of Science, London, UK Professor Dr.-Ing K. Hameyer RWTH Aachen University, Aachen, Germany Professor N. Ida University of Akron, Akron, USA

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 p. 6 # Emerald Group Publishing Limited 0332-1649

Professor D. Lowther McGill University, Ville Saint Laurent, Quebec, Canada Professor O. Mohammed Florida International University, Florida, USA Professor G. Molinari University of Genoa, Genoa, Italy Professor A. Razek Laboratorie de Genie Electrique de Paris - CNRS, Gif sur Yvette, France Professor G. Rubinacci Universita` di Napoli, Federico II, Napoli, Italy Professor M. Rudan University of Bologna, Bologna, Italy Professor M. Sever The Hebrew University, Jerusalem, Israel Dr J. Sturgess AREVA T&D Technology Centre, Stafford, UK Professor J. Tegopoulos National Tech University of Athens, Athens, Greece Professor W. Trowbridge Vector Fields Ltd, Oxford, UK Professor T. Tsiboukis Aristotle University of Thessaloniki, Thessaloniki, Greece

Professor A. Kost Technische Universitat Berlin, Berlin, Germany

Professor Dr.-Ing T. Weiland Technische Universitat Darmstadt, Darmstadt, Germany

Professor T.S. Low National University of Singapore, Singapore

Professor K. Zakrzewski Politechnika Lodzka, Lodz, Poland

Preface The 7th International Symposium on Electric and Magnetic Fields, EMF 2006, was held in Aussois, France, from June 19-22, 2006. Previous editions were held in Lie`ge (September 1992), Leuven (May 1994), Lie`ge (May 1996), Marseille (May 1998), Ghent (May 2000) and Aachen (October 2003). The purpose of the EMF Symposium is to throw a bridge between the recent advances of research in numerical modelling of electromagnetic fields and the growing number of industrial problems requiring such techniques. Therefore, beside classical sessions on the progress of computational methods, special sessions were devoted to advanced industrial applications of electromagnetic modelling. The topics included numerical methods and techniques, coupled problems (mechanical, thermal, and electric circuits), material modelling, optimisation and specific application oriented numerical problems. A number of 110 papers were presented by participants coming from 27 countries. A limited number of 30 papers was selected by the EMF 2006 Scientific Committee for publication in this special issue of the COMPEL Journal. Special thanks are due to all the members of the Scientific Committee for their valuable reviewing work as well as to the AIM (Association of Engineers from the Montefiore Electrical Institute, University of Lie`ge) secretariat for numerous organisational aspects. We hope that this special issue will provide various interesting information to the readers.

EMF 2006 international scientific committee J.P.A. Bastos, Universidade Federal de Santa Catarina, Brazil R. Belmans, Katholieke Universiteit Leuven, Belgium O. Biro, Technische Universitaet Graz, Austria W.A. Cronje, University of the Witwatersrand, South Africa A. Demenko, Technical University of Poznan, Poland P. Dular, Universite´ de Lie`ge, Belgium J. Gyselinck, Universite´ Libre de Bruxelles, Belgium K. Hameyer, RWTH Aachen, Belgium L. Kettunen, Tampere University of Technology, Finland A. Kost, Universita¨t Cottbus, Germany V. Mazauric, Schneider Electric, France J. Melkebeek, Universiteit Gent, Belgium G. Meunier, Laboratoire d’Electrotechnique de Grenoble, France G. Molinari, Universita Degli Studi di Genova, Italy S.I. Nabeta, Universidade de Sao Paulo, Brazil A. Nicolas, Centre de Ge´nie Electrique de Lyon, France A. Nicolet, Universite´ d’Aix-Marseille III, France F. Piriou, Universite´ des Sciences et Technologies de Lille, France

Preface

7

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 7-8 q Emerald Group Publishing Limited 0332-1649

COMPEL 27,1

8

A. Razek, Laboratoire de Ge´nie Electrique de Paris, France M. Repetto, Politecnico di Torino, Italy J.K. Sykulski, University of Southampton, United Kingdom N. Takahashi, Okayama University, Japan M. Trlep, University of Maribor, Slovenia K. Van Reusel, Katholieke Universiteit Leuven, Belgium T. Weiland, Universita¨t Darmstadt, Germany Patrick Dular, Ge´rard Meunier and Francis Piriou Guest Editors

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

What do voltmeters measure? Alain Bossavit

What do voltmeters measure?

LGEP, Gif-sur-Yvette, France

9

Abstract Purpose – The paper aims at justifying the operational rule “a voltmeter’s reading is the electromotive force, as it existed before branching it, along the path g traced out by the connectors between the contact points”. Design/methodology/approach – A simple application of Faraday’s law is enough to make the result plausible. Then it is shown that by an asymptotic analysis with the radius of the leads (assumed perfectly conductive) as small parameter that the current derived by the voltmeter is negligible. Findings – The rule is valid in spite of the considerable modification of the electric field that branching a voltmeter entails. Originality/value – “Voltage drop”, between A and B simply does not make sense. The threads of a voltmeter should carefully be placed in order to measure exactly what one has in view. This is explained by a few examples. Keywords Voltage, Measurement Paper type Conceptual paper

Should the question in the title sound preposterous, let us stress its legitimacy by discussing the following idealization. In otherwise empty 3D Euclidean space E, we have a massive conductor C (Figure 1) and a coil C s, that is the support of a known source current j s (AC at some angular frequency v, and complex-valued). With s . 0 inside C and s ¼0 outside, the eddy current equations, rotH ¼ J ¼ sE þ J s ;

ivB þ rotE ¼ 0;

B ¼ m0 H; ð1Þ R 2 uniquely determine H (subject to the finite energy condition E m0 jHj , 1) and determine E up to some curl-free field supported outside C. For definiteness, we assume that electric charge q ¼ div(10 E ) is specified outside C (yet not on the air-conductor of C ), which lifts the ambiguity, again under the reasonable interface ›C, the boundary R 2 energy condition E 10 jEj , 1: Suppose now a voltmeter and its two connecting leads are branched between two points A and B of ›C. Like all real-life voltmeters, that one has an adjustable inner resistance R, which can be made high enough to reduce the current I through the voltmeter to a negligible value. The reading is then V, equal to RI. How does V relate to the physical situation without the measuring device? Is it correct to say that “V is the voltage drop, between A and B”? (The short answer is: No.) Is “voltage drop”, for that matter, a well defined notion? (Not always.) And even if so, does that make the reading dependent on the position of points A and B only ( No) or is the position of the leads relevant? ( Yes – definitely.) These conclusions can be found in the literature (Moorcroft, 1969, 1970; Klein, 1981; Romer, 1982; Reif, 1982; Peters, 1984) but the underlying rationales are not always convincing, and the very fact that the problem is raised again and again ( Nicholson (2005) and the 2004 web debate, www. physicsforums.com/archive/index.php/t-24022.html) betrays some persistent uneasiness.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 9-16 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836582

COMPEL 27,1 V B

10

Js

A C Cs

Figure 1. The setup

Notes: An induction coil Cs generates eddy currents in conductor C. What's the meaning of V, as displayed by a voltmeter connected between points A and B of the surface?

The paper is in two parts. First, we assume a “negligible” current through the voltmeter, which means that the magnetic field in presence of the connected voltmeter and its threads is about the same as it was before their introduction. (Note, immediately, that no such assumption can be made about the electric field, which is widely changed by the fact of connecting the voltmeter, thin as the leads may be.) Under this hypothesis, we establish the rule quoted in the Abstract. A second part addresses the “negligible” proviso: Assuming the leads are perfect conductors, but with a nonzero radius r, problem (1) is embedded in a family of problems, indexed by the small non-dimensional parameter a ¼ r/ , size of the device . . It is then proved that the corresponding magnetic induction ba converges, in a mathematically definite sense, towards a limit b0. We shall see in conclusion how this results validates the rough analysis, thus making the intended point. A word on notation: although we use vector fields E , B , etc. according to tradition, we put emphasis on the physical entities they are meant to represent, such as electromotive forces, fluxes, etc., i.e. to forms e, b, etc. they “stand R R the differential proxy” for. Hence, notations such as “ g e”, “ S b” etc., meaning “the circulation of E along g”, “the flux of B embraced by surface S”, etc., leaving unmentioned the tangent and/or normal vector fields, dot product, etc., that would be necessary for mathematical definiteness (better achieved, anyway, if e, b, etc. are conceived as differential forms). 1. Rough analysis Call g the path, oriented, from A to B, along the leads and through the voltmeter, and z some path, from B to A, entirely contained in the surface ›C of the conductor. Let S be any (oriented) surface the boundary of which be g þ z. Consider the situation before (Figure 2, left) and after (FigureR2, right) connecting the voltmeter, all other things unchanged. The induction flux F ¼ S b is the same in both cases, since currents do not change. By Faraday’s law: Z Z ivF þ e þ e ¼ 0 ð2Þ g

z

always holds. Thanks to Ohm’s law, the electric field e in the conductor, and hence R its tangential component on the conductor’s surface, does not change either. Therefore, g e is

the same before and after, though e did change. Since, as we assumed, the leads are perfect conductors, where e vanishes, this circulation is the one that exists inside the voltmeter, between its two ports, just what the voltmeter is engineered to show on its dial, as generally agreed (Romer, 1982). We conclude that the voltmeter’s reading is the electromotive force, as it existed before branching it, along the circuit g traced out between contact points A and B by the leads and voltmeter. It is fairly remarkable that a property of the electric field can thus be measured in spite of the considerable modification of this very field brought in by the measuring apparatus. So the naive preconception that V would be “the voltage ‘between’ A and B” is refuted: V does depend on the path g, and hence, on how the voltmeter is connected. Anyway, does this “voltage between” notion make any sense? Only in two cases can this be asserted: (a) when the electric field, or at least, its tangential part on ›C, derives from a potential, so we are talking of a potential difference; and (b) when one has a specific path z in mind, so that by “voltage” one really means the electromotive force, i.e. the integral of e (or circulation[1] of its proxy field E ), along z.

What do voltmeters measure? 11

Let us review both possibilities. Case (a): it may happen, in special circumstances, that no induction flux crosses ›C, or only a negligible amount of it, in which case the curl of the tangential part of E vanishes. Then, an electric potential v can be defined, up to some additive constant, on ›C – locally[2], atR least, and the voltage drop W, relative to the surface, between A and B (that is, 2 z e) is well defined. Since W ¼ V 2 ivF one can get W if information about F is available. This may happen: for instance, when there is some symmetry plane, containing both points A and B, with respect to which (vectorial) values of B are mirrored; then, placing g in this plane will be the right move (Reif, 1982), since F ¼ 0 in this case, and W ¼ 2 V (Romer, 1982). Or else, the flux F can be negligible for the same reasons the flux through ›C is, just because only a weak, negligible magnetic field exists in the region where A and B lie, and where one branches the voltmeter. Note that, if so,RE is approximately curl-free in this region, hence E ¼ 2 grad v there, so that V ¼ g e ¼ vðAÞ 2 vðBÞ is effectively the voltage drop one wished to measure. γ

B V

B ζ

S A A

Figure 2. V is equal to the emf along g as it existed before branching the voltmeter – the same as after branching it. The effect of the measuring device is, so to speak, to lump this emf between the voltmeter’s terminals

COMPEL 27,1

12

Figure 3. Left: How to effectively R measure W ¼ 2 z e. Right: How non-trivial topology can affect the issue, even when no flux crosses the R conductor’s R surface; R z0 e ¼ z e, but R z00 e – z e, owing to the loop (cf. Note 1), because z00 , contrary to z0 , is not homologous to z.

There is another way to make F vanish, thus radically lifting the uncertainty about its value, as suggested by Figure 3. If a part of g, lying in the air but near the surface of the conductor, closely follows z, and if the other part, joining the voltmeter’s terminals, is properly intertwined, F will be zero. Any preassigned path z can thus be followed, which takes care of case (b). This applies, a fortiori, to case (a), too: if the conductor’s surface lets no induction flux leak through[3], V does equal the surface voltage drop between A and B, whatever the path z followed on the surface. Beware of loops in such reasoning, though (Figure 3, right). So what to do when a specific voltage is one of the quantities a numerical simulation is supposed to deliver? Modelling the whole situation, including voltmeter and wires with their precise geometry, would most often be inconvenient. One will rather ignore them, and just compute the electric field where required, that is, along path g (some chain of edges of the mesh, in practice). Yet, this means E must be computed in the air, which some popular eddy-current methods (based on H , or on J ) avoid to do. With such methods, a complementary computation in the air will be needed (a problem of the form rot E ¼ 2 ivB , D ¼ 10e, div D ¼ q, outside C, with B and the tangential part of E on ›C coming from the first computation). One may therefore prefer an “e-oriented” method (over a large enough computational domain – this also incurs some cost, and is part of the trade-off), in terms of either R e directly or of some a– v combination. In the latter case, remember that V ¼ 2 g ðiva þ grad vÞ; the circulation of the total electric field, not only the so-called “Coulomb voltage” v(A) 2 v(B). (The latter, gauge-dependent, is not physically observable anyway, so its determination may fail to serve any useful purpose.) R Let us now turn a critical eye to the main part of our argument: that “F ¼ S b is “about the same” in both cases (with or without wires and voltmeter)”. To turn this into a dignified mathematical assertion, we must parameterize the situation by a real a . 0, meant to tend to 0, and prove that the corresponding induction R ba tends Rto b0, the one that exists without wires and voltmeter, which entails that S ba tend to S b0 ; and if so we are done. As such a parameterization involves lots of details (size of the voltmeter, its inner resistance, shape of the wires, contact resistances at A and B, etc.), not all relevant to the same degree, we shall rather introduce a clearcut auxiliary problem, of interest in its own right, discuss its relevance, and prove the needed convergence result.

B ζ'' V

V

B

ζ'

ζ

ζ

A

A

2. Detailed analysis Our model problem in this part is magnetostatics in all space: rot H ¼ J;

H ¼ n0 B;

divB ¼ 0;

ð3Þ

with J as data and n0 ¼ 1/m0, under the constraint that B ¼ 0 in Ma, owing to the presence of a perfectly diamagnetic material in Ma. Let us call B (boldface) the functional space {B [ L 2(E): div B ¼ 0} and set B a ¼ {B [ B : B ¼ 0 in M a }: The weak form of (3) consists in finding B a in B a such that: Z Z n0 B a · B 0 ¼ H J · B 0 for all B 0 in B a ; ð4Þ

What do voltmeters measure? 13

E

E J

where H is some “source field” so chosen that rot H J ¼ J . (Alternatively, this is (3) outside the perfect diamagnet, with n · B ¼ 0 on its boundary ›Ma.) There is a unique solution B a, and we investigate its convergence, in the sense of the L 2-norm (or rather, R 2 the norm kBk ¼ ð E n0 jBj Þ1=2 ) towards some limit B 0 when a ! 0. In that case, fluxes R R S ba tend to S b0 ; by well known trace theorems, for any well-behaved surface S. The set Ma is a regular domain intended to model one of the leads[4], and B ¼ 0 stems from the assumed perfect conductivity of this wire (zero skin depth, layer of surface currents which screens out the magnetic field). To account for the “small radius” of the wire, we introduce a closed set M0 , Ma with empty interior (for definiteness, imagine M0 as a smooth curve segment – it will be the above path g, or rather, a half of it) and a family ua of smooth mappings from E to itself, depending smoothly on a, with the following properties: (H1) ua(Ma) ¼ M0, (H2) u0 is the identity. (Figure 4 shows the idea.) Moreover: (H3) the restriction of ua to E 2 Ma is a diffeomorphism (so the derivative Dua(x) is an invertible linear map, from 3D vectors to 3D vectors, at each point), (H4) lima¼ 0 Dua(x) is the identity, (H5) As a map on L 2(E 2 M0), (Dua)2 1 is uniformly bounded. Some clarifications are needed: imagine M0 (meant to model a part of the above path g) cut out from space, and the rest of space “retracted” to E 2 Ma, so that the “buttonhole” M0 “opens up” to a three-dimensional elongated domain Ma: this 1 describes the (restriction to E 2 M0 of the) inverse mapping u2 a . On the other hand, the direct map ua “stitches up the cut” and retracts Ma to M0. The actual thread, with its effective dimensions, is Ma, filled by perfectly diamagnetic stuff, for some definite value of a. The asymptotic analysis is meant to justify the approximation consisting in replacing Ma by M0. Hypotheses H3 to H5 are technical requirements. x uα (x)

y

uα (y)

α z uα (z)

Mα

M0

Figure 4. The mapping ua, shrinking the diamagnetic region Ma onto the subset M0 (meant to represent the path g, or part of it, of the previous section)

COMPEL 27,1

14

Let us recall that, if b is a two-form (with proxy vector B in L 2(E )), its “pullback” ua*b is the two-form defined by kðua*bÞðxÞ; v; wl ¼ kbðxÞ; ðDua ðxÞÞv; ðDua ðxÞÞwl; where v and w are two vectors at x. Abusing the notation, we shall denote the proxy vector of ua*b by ua*B. Pullback and exterior derivative d commute, so if div B ¼ 0, i.e. db ¼ 0, the proxy ua*B is also divergence-free. We note that ua*BðxÞ is well-defined for x in Ma, but null in the case we consider: for if v and w are two vectors anchored at x [ Ma, their images under ua are collinear[5]. We conclude that if B 0 is the solution, in B, of: Z Z 0 n0 B · B ¼ H J · B 0 for all B 0 in B ð5Þ E

E

(i.e. the magnetic field without any diamagnetic inclusion present), then ua*B0 belongs to the above-defined space B a. Now compare (4) and (5), setting B 0 ¼ B 2 B a in (5) and B 0 ¼ Ba 2 ua*B in (4), which results in: Z Z 2 n0 jBa 2 Bj ¼ ðH J þ n0 Ba Þ · ðBa 2 ua*BÞ: E

E

Thanks to H4 and H5 – made for that purpose – the right-hand side tends to 0, therefore B a tends to B 0, as announced, in L 2(E). To sum up: * Proposition 1. Under R hypotheses (H1-H5), and assuming that ua ba vanishes in Ma, R fluxes S ba tend to S b0 for regular surfaces S.

It remains to show that H1– H5 are reasonable. Let us begin with the (not yet realistic) case where M0 is a single point in space E, which we can take as origin. Define then Ma ¼ {x [ E: jxj # a}, and ua(x) ¼ x – ax/jxj, i.e. “draw all points outside Ma towards the origin by the same amount” and shrink M0 to the origin. Call this “spherical stitching”: it clearly satisfies our hypotheses. Second case, an infinite straight line l, and Ma is the tube of radius a around this line: use the previous transformation in planes orthogonal to l, call that “cylindrical stitching”. Next, in the more realistic case of a finite segment from point A to point B: If x projects to line AB between points A and B, bring it closer to its projection by cylindrical stitching, otherwise push it towards A or B, whichever is closer, by spherical stitching. (Then, Ma is a cylinder of radius a with rounded ends.) Last, in the case of a smooth curve from A to B, first build an isotopy (a “good” map from E to itself) to rectify the curve, then compose with the previous transformation. And so on: whatever the shape of the wire (non-uniform radius, varying cross-section, etc.), a suitable family ua that shrinks it to a curve g can be found, such that Proposition 1 hold – in other words, when the perfectly conducting wire reduces to a curve, the magnetic field tends to what it would have been, had this curve not been there. Remark. Therefore, “line defects” and a fortiori, “point defects” with perfect diamagnetism, can be ignored in magnetostatics. This conclusion cannot be drawn for surface defects, however, because the property “ua*B0 [ B a ” does not hold. This is intuitively obvious: A perfectly diamagnetic surface, however thin, will make a barrier to magnetic flux, and force flux lines to turn around it. S

Remark. It may be instructive to interpret our technique of proof in this spirit: The homotopy ua is constructed in order to “warp the flux tubes” of the induction field B 0 so as to push them out of the diamagnetic region. When the latter shrinks to a curve, the “no defect” situation is recovered, but not when the residual defect is a surface. S 3. Discussion and conclusion What precedes should be enough to support the main thesis: what a voltmeter measures is the voltage, as it existed before branching the device, along the path traced out by the two wires and the voltmeter itself between the two measurement points. We have proved the main asymptotic theorem on which to base this conclusion. A lot of details should be dealt with, however, using similar techniques. First, treating the problem as one in magnetostatics, instead of eddy currents, as we did, is justified by the fact that placing the voltmeter does not modify the currents in the main conductor C. Of course, this is not rigorously true, and another asymptotic analysis of this effect (which is a second order perturbation) should be done. Also, to assume perfect conductivity, and hence perfect diamagnetism, is abusive, and again, a limit analysis (with a second small parameter, resistivity of the wires, subordinate to the above a) should take care of that. The voltmeter itself (small and highly resistive) should receive similar attention. What happens at the contact points A and B is also matter for investigation. And beyond convergence results, one might very well push further the Taylor expansions of the fields involved, in terms of the small parameter(s), in order to reach estimates of the errors incurred. Let us just mention these directions of research, and express the hope that this subject, modelling asymptotics, will be developed as a more rigorous substitute to the intuition-based procedures we use nowadays in modelling. Notes 1. Which seems to be the right way to understand “voltage” (Page, 1977; Moorcroft, 1970). “Voltage drop” on the other hand (Reif, 1982), evokes a difference in potential values (also measured in volts), but such a potential may fail to exist. 2. If C presents “loops” v may be multi-valued. Then W depends on the so-called “homology class” of z on the surface, according to around which loops (embracing nonzero flux, as a rule) z goes (Figure 3, right). 3. See Bermu´dez et al. (1999) for an interesting application where this assumption is not warranted. 4. So we need two of them, one on each side of the voltmeter. The latter is also in need of a similar asymptotic study, under hypotheses that factor in its “smallness” (in size) and its high internal resistance. We omit this, which would not bring in new ideas, from the present paper. 5. Because M0 is a curve, one-dimensional, here. In different contexts, if M0 was a flat disk, for instance, a weaker conclusion would follow: that ua*b is orthogonal to the surface of Ma. References Bermu´dez, A., Mun˜iz, M.C., Pena, F. and Bullo´n, J. (1999), “Numerical computation of the electromagnetic field in the electrodes of a three-phase arc furnace”, Int. J. Numer. Meth. Engng., Vol. 46 No. 5, pp. 649-58. Klein, W. (1981), “Experimental ‘paradox’ in electrodynamics”, Am. J. Phys., Vol. 49 No. 6, pp. 603-4.

What do voltmeters measure? 15

COMPEL 27,1

16

Moorcroft, D.R. (1969), “Faraday’s law – demonstration of a teaser”, Am. J. Phys, Vol. 37 No. 3, p. 221. Moorcroft, D.R. (1970), “Faraday’s law, potential and voltage – discussion of a teaser”, Am. J. Phys., Vol. 38 No. 3, pp. 376-7. Nicholson, H.W. (2005), “What does the voltmeter read”, Am. J. Phys., Vol. 73 No. 12, pp. 1194-6. Page, C.H. (1977), “Electromotive force, potential difference, and voltage”, Am. J. Phys., Vol. 45 No. 10, pp. 978-80. Peters, P.C. (1984), “The role of induced emf’s in simple circuits”, Am. J. Phys., Vol. 52 No. 3, pp. 208-11. Reif, F. (1982), “Generalized Ohm’s law, potential difference, and voltage measurements”, Am. J. Phys., Vol. 50 No. 11, pp. 1048-9. Romer, R.H. (1982), “What do ‘voltmeters’ measure? Faraday’s law in a multiply connected region”, Am. J. Phys., Vol. 50 No. 12, pp. 1089-93. About the author Alain Bossavit, PhD in Mathematics from University of Paris 6 (1971), after 30 years with E´lectricite´ de France as Head of the Numerical Analysis Division and Scientific Advisor, is now with the Laboratoire de Ge´nie E´lectrique de Paris (CNRS, University Paris-Sud), where he takes part in work on metamaterials, magnetomechanics, field-matter interactions, and addresses electromagnetic theory from the differential-geometric viewpoint. Alain Bossavit can be contacted at: [email protected]

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

Electromagnetic torque calculation using magnetic network methods Andrzej Demenko and Dorota Stachowiak

Electromagnetic torque calculation 17

Poznan´ University of Technology, Poznan, Poland Abstract Purpose – The aim of the paper is to find the effective algorithms of electromagnetic torque calculation. Design/methodology/approach – The proposed algorithms are related to the analysis of electrical machines using the methods of equivalent magnetic networks. The presented permeance and reluctance networks are formulated using FE methods. Attention is paid to the algorithms of electromagnetic torque calculation for 3D models. The virtual work principle is applied. The principle is adapted to the discrete network models. The network representations of Maxwell’s stress formula are given. Findings – The proposed method of electromagnetic torque calculation can be successfully applied in the 3D calculations of rotating electrical machines. It can be used for scalar and vector potential formulations. The obtained results and their comparison with the measurements show that the method is sufficiently accurate. Originality/value – The presented formulas of electromagnetic torque calculation are universal and can be successfully applied in the FE analysis of electrical machines using nodal and edge elements. Keywords Finite element analysis, Magnetic fields, Electromagnetism, Torque Paper type Research paper

1. Introduction One of the oldest techniques for magnetic field analysis and computation relies on magnetic equivalent circuits (networks). Both reluctance and permeance networks (PNs) can be used to model magnetic fields (Ostovic, 1989; Sykulski et al., 1995; Hecquet and Brochet, 1998). Contemporary network equivalents are often based on finite element formulations and are very detailed and accurate. It has been shown before (Demenko and Sykulski, 2002; Davidson and Balchin, 1983; Demenko et al., 1998) that finite element equations are equivalent to loop or nodal descriptions of appropriate magnetic or electric networks. The edge element (EE) equations using vector potential A represent the loop equations of reluctance network (RN) and the scalar potential V equations for nodal elements are equivalent to the nodal equations of PN. Thus, methods streaming from the finite element approach may help the users of network models to prepare more accurate algorithms. In the paper, the algorithms of electromagnetic torque calculation are considered. The formulas that describe electromagnetic torque have been formed using the virtual work principle. The virtual changes of magnetic energy and/or coenergy caused by “switching over” the network branches are analysed. It has been assumed that the model of electrical machine contains the band with regular homogenous network in the air gap. The band separates the stator from the rotor region.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 17-26 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836591

COMPEL 27,1

18

2. Permeance network PN models may be established using classical circuit approach (Ostovic, 1989; Hecquet and Brochet, 1998), e.g. method of tubes and slices, or using the FE method and EE functions (Demenko and Sykulski, 2002; Demenko et al., 1998). In the case of FE method, the branches of PN are associated with the element edges (Figure 1). The branch magnetomotive forces (mmfs) represent the edge value of electric vector potential T (Demenko and Sykulski, 2002). The PN model of FE has mutual permeances, i.e. the flux in the branch associated with element edge depends on the magnetic voltage related to the others branches of element. Whereas, in the classical network models, mutual permeances do not exist. However, it is interesting to notice that for the FE network composed of rectangular prism, the approximation presented in Demenko et al. (1998) gives PN similar to the classical PN, without mutual permeances. The branch and nodal equations of PN are written in Table I. 3. Reluctance network RN can be formed by the application of the EE method using vector magnetic potential A (Demenko et al., 1998). In the reluctance model of the element, the branches connect the centres of the relevant facets with the centre of the element volume. The RN equivalent with a hexahedron has six branches Figure 1(b). P1

P5

P2 P6 P4

P8

mmf permeance

P7

P3

(a) P5

P1

P6 P2

Figure 1. Network model of hexahedron: (a) permeance (b) reluctance network

P8 Loop flux ϕ7,8 mmf P7

P3

reluctance (b)

The branch fluxes of RN represent the face values of flux. The loop flux wi, j represents the edge value of A, for edge PiPj. The branch reluctances are calculated using the interpolation functions of facet element. In contradistinction to the classical RN, the RN formed by the EE method has mutual reluctances. However, in the case of regular prism, the approximation presented in Demenko et al. (1998) gives a RN without mutual reluctances, equivalent to the classical RN. The branch and loop equations for RN are shown in Table I. The branch mmfs are defined by the edge values of T. However, when using the loop method, it is not necessary to know the branch sources, instead the loop sources are needed. The loop mmfs in RN may be established from the facet values of current density.

Electromagnetic torque calculation 19

4. Calculation of torgue using permeance network The electromagnetic torque is calculated on the basis of virtual work principle. In formulations related to the virtual work principle, we have applied the ideas presented in Coulomb and Meunier (1984). In accordance with virtual work principle, for scalar potential method, the torque is equal to the magnetic coenergy derivative versus the virtual moving (Coulomb and Meunier, 1984). In the proposed method, an interpolation function is applied to describe this derivative. For rotor angle a, interpolation polynomial is based on the data for three discrete positions ai ¼ a þ ib (i ¼ 2 1, 0, 1), where b is the angular distance between nodes in band (Figure 2). For 2D model the interpolation gives:
›W c ða þ DaÞ

1 ¼ ðW c ða þ bÞ 2 W c ða 2 bÞÞ; ð1Þ TðaÞ ¼
2 ›ðDaÞ b Da¼0 where Wc(a ^ b) is the magnetic coenergy for position a ^ b. From equation (1), using symbols in Figure 2 we obtain: TðaÞ ¼

n  X  1 Lb ðVsi 2 Viþ1 Þ2 2 ðVsi 2 Vi21 Þ2 4b i¼1

n X 1 ¼ Lb ðVsi 2 Vi ÞðVi21 2 Viþ1 Þ: 2b i¼1

ð2Þ

This formula is considered to be Maxwell’s stress tensor formula for 2D PN model with integration surface placed in the band. It should be noticed that equation (2) differs from the popular formulas for PN, e.g. from formulas given in Hecquet and Brochet (1998).

Type of network

Branch equations

Substitution

Nodal or loop equations

Permeance (PN)

f b ¼ LðuV þ QÞ

uV ¼ k n V

kTn Lk n V ¼ 2kTn LQ

Reluctance (RN)

uVf ¼ R m f f 2 Q

ff ¼ kew

kTe R m k e w ¼ kTe Q

Notes: V is the vector of nodal potentials; L is the matrix of branch permeances; kn is the transposed nodal incidence matrix for PN; w is the vector of loop fluxes; Rm is the matrix of branch reluctances; ke is the transposed loop (mesh) matrix for RN; Q is the vector of branch mmfs

Table I. Magnetic network equations

COMPEL 27,1

Stator Ωsn–1 Band

Λb Ωn–1

20

β Ωs1

Ωsn φn–1

Ωn

φn

Ω1

φ1

Λb Ω2

Ωs2 φ2

Ωs3 Ω3

φ3

Band

Rotor α

Rotor in positon α

Stator Ωsn–1

Ωsn β Ωs1 Band Λ b φn φ1 φn–1 Ωn–2 Ωn Ωn–1

Λb Ω1

Ωs2 φ2

Ωs3 Ω2

φ3

Band

Rotor α

Rotor in positon α–β

Stator Ωsn–1 Band Λ b Ωn

Figure 2. Permeance network model of a band for three discrete rotor positions

β Ωs1

Ωsn φn–1

Ω1

φn

Ω2

φ1

Λb Ω3

Ωs2 φ2

Ωs3 Ω4

Band φ3

Rotor Rotor in positon α+β

α

In the case of 3D multilayer models, formula (2) gives a part of torque related to the single layer of 3D PN. The total torque for 3D PN model composed of m layers has a following form: TðaÞ ¼

m n X 1 X Lb;q ðVsq;i 2 Vq;i ÞðVq;i21 2 Vq;iþ1 Þ; 2b q¼1 i¼1

ð3Þ

where subscript q denotes the permeances and potentials related to the q-th layer. 5. Calculation of torgue using reluctance network The formula that describes torque for RN is obtained using the approach that is typical for vector potential formulation (Ren, 1994). The magnetic energy derivative versus the virtual moving is considered (Coulomb and Meunier, 1984). The derivative is approximated by the finite differences. Thus, the expressions that describe magnetic energy W for discrete rotor positions are analysed. As a result, for 2D model we obtain:

1 ðW ða þ bÞ 2 W ða 2 bÞÞ 2b n  X  1 Rb ðwsi 2 wiþ1 Þ2 2 ðwsi 2 wi21 Þ2 ¼2 4b i¼1 n X 1 ¼ Rb ðwi 2 wsi Þðwi21 2 wiþ1 Þ: 2b i¼1

TðaÞ ¼ 2

ð4Þ

21

Here, W(a ^ b) is the magnetic energy for position a ^ b other symbols are shown in Figure 3. The RN model in Figure 3 has been formed by means of FE method. The finite element grids for models in Figures 2 and 3 are identical. In the case of 3D calculations, formula (4) must be supplemented by additional components with loop fluxes that flow in the reluctances associated with the band (reluctances Rb and reluctances Rz in the branches perpendicular to the surface shown in Figure 3). Usually, in the 3D calculations, it is advantage to form band of nine-edge prisms or curved rectangular parallelepipeds. The element edges are parallel to the axis of a cylindrical co-ordinate system r, z, c. The circular band of length l is placed inside the air gap; where l is the distance between the boundary surfaces z ¼ 0, z ¼ l (w(z ¼ 0) ¼ 0, w(z ¼ l ) ¼ 0). The band is subdivided into layers of thickness Dlq (Demenko, 1998). The 3D representation of the formula (4) has been expressed as follows: 1 TðaÞ ¼ 2 4b 1 2 4b

( (

m X q¼1

Rbq

n  X

f2cq;iþ1 2 f2cq;i21



)

i¼1

m 21 X

) n   X 2 2 Rzq fzq;iþ1 2 fzq;i21 :

q¼1

ð5Þ

i¼1

Here, fci^ 1,q is the value of flux in branch Pbq for position a ^ b and fzi^ 1,q is the value of flux in branch Pzq for position a ^ b (Figure 4). Subscript q denotes the reluctances and the branch fluxes related to the q-th layer. The branch fluxes have been expressed by loop fluxes – see Figure 4 and equation (5) has been transformed into the following equation: 1 TðaÞ ¼ 2b

(

1 þ 2b

m X

Rb;q

q¼1

(

m 21 X q¼1

n X



fcq;i wq;i21 2 wq;iþ1



)

i¼1

Rz;q

n X



fzq;i wcq;i21 2 wcq;iþ1



Electromagnetic torque calculation

)

ð6Þ :

i¼1

As a result, we obtain equation that represents the stress tensor formula for torque calculation using RN and EE method (Stachowiak, 2004).

COMPEL 27,1

Stator ϕsn–1

Band

ϕsn

ϕs1

ϕn

ϕ1

ϕn–1

22

β

ϕs2

Band

ϕ2

ϕs3 ϕ3

Rotor loop flux α

Rotor in position α

Stator ϕsn–1

Band

ϕsn ϕn–1

ϕn–2

ϕs1

β

ϕn

ϕs2

Band

ϕs3 ϕ2

ϕ1

Rotor α

Rotor in position α–β

Stator ϕsn–1 ϕn

Figure 3. Reluctance network model of a band for three discrete rotor positions

Band

ϕsn

ϕs1

ϕ1

ϕ2

β

ϕs2 ϕ3

Band

ϕs3 ϕ4

Rotor Rotor in position α+β

α

6. Example The above presented formulas have been used in the calculations of permanent magnet motor (PMM). Motors with magnets mounted on the cylindrical rotor are analysed. The 3D model has been applied. The magnetic networks are formed by means of the FE method. The considered region has been subdivided into curved rectangular parallelepipeds and nine-edge prisms (Dular et al., 1994) (total number of elements < 200,000 per pole). The same example has been calculated using the classical FE scalar potential method and OPERA-3d Software – version 7.1. In the calculations using this software, the Maxwell’s tensor formula has been applied. In Figure 5, the results of cogging torque calculation are given. Results of OPERA-3d Software are denoted by abbreviation OPM. It can be seen that the results of PN and RN method are very close

Electromagnetic torque calculation

ϕsq,i ϕrq–1,i branch Pbq Rzq–1 φψq,i

Rzq ϕrq,i

Rbq

23

r ψ

φzq,i φ q,i

z

(a) ϕsψq,i

φψq,i

Rzq

ϕrq,i

Rbq ϕrq,i–1 r branch Pzq

φzq,i

ϕψq,i

Figure 4. Part of the band with branch Pbq (a), branch Pzq (b)

ψ z

(b)

T cog [Nm] 1.5 OPM 1.0 0.5 0.0 0 – 0.5 –1.0

5

10

15

20

25

α [°]

30

PN RN

–1.5

and differs from the results of OPERA-3d. The calculation using OPERA-3d takes one order of magnitude more cpu time than for PN method and RN method. The calculated function T(a) is periodical with the period equal to the angular dimension of slot pitch. Here, this dimension is 158. Therefore, it is easy to determine the position of T ¼ 0. For this position, the results of PN and RN method are exactly equal to zero. The methods have been used for the total torque calculations. The calculations have been performed for given values of stator currents and for different rotor position.

Figure 5. Results of cogging torque calculation

COMPEL 27,1

24

As a result, we obtain torque-angle characteristics. Figure 6 shows calculated torque-angle characteristics for following values of phase current: ia ¼ 2 10 A, ib ¼ 5 A, ic ¼ 5 A. The T(a) characteristics in Figure 6 have been interpolated by trigonometric polynomials. The amplitude of calculated harmonics are shown in Figure 7. It can be seen that the OPERA-3d Software gives the results with “unfounded” harmonics, i.e. harmonics of order 2, 3, 4, 8, 9, 10 that do not exists in the considered machine. The validity of our method is investigated by comparing with the measured cogging torque. The proposed RN method has been applied in the calculations of cogging torque of DC brushless motor. The eight-pole 24-slots motor with the permanent magnets mounted on a cylindrical rotor is considered. The results in Figure 8 show that the applied RN method is accurate. 7. Conclusion The proposed method of electromagnetic torque calculation can be successfully applied in the 3D calculations of rotating electrical machines. The method guarantees T [Nm] 18

PN RN

12

OPM

6 0 0

30

60

90

150 α [°] 180

120

–6

Figure 6. Torque-angle characteristics

–12 –18 14.898 T [Nm]

15.292 14.527

RN

PN

OPM

1.5 PN

1.0 RN

OPM

0.5

Figure 7. Amplitude of harmonics of function T(a)

0.0 1

2

3

4

5

6 7 8 9 harmonic order

10

11

12

13

Electromagnetic torque calculation

Tcog [Nm] 1.5 1.0 calculations

0.5

25 0.0 – 0.5

measurement

–1.0 –1.5 0.0

5.0

10.0

15.0

20.0

25.0

α [°]

30.0

a good accuracy. For the rotor positions of T ¼ 0, which have been determined analytically, the calculated torque is equal to zero with eight-decimal-place accuracy. The obtained results and their comparison with the measurements show that the method is sufficiently accurate. The presented formulas can be adapted to the nodal element method for scalar potential formulation and to the EE method for vector potential formulation.

References Coulomb, J. and Meunier, G. (1984), “Finite element implementation of virtual work principle for magnetic or electric force and torque computation”, IEEE Trans. on Magn., Vol. 20 No. 5, pp. 1894-6. Davidson, J. and Balchin, M. (1983), “Three dimensional eddy currents calculation using a network method”, IEEE Trans. on Magn., Vol. 19 No. 6, pp. 2325-8. Demenko, A. (1998), “3D edge element analysis of permanent magnet motor dynamics”, IEEE Trans. on Magnetics, Vol. 34 No. 5, pp. 3620-3. Demenko, A. and Sykulski, J. (2002), “Network equivalents of nodal and edge elements in electromagnetics”, IEEE Trans. on Magn., Vol. 38 No. 2, pp. 1305-8. Demenko, A., Nowak, L. and Szela˛g, W. (1998), “Reluctance network formed by means of edge element method”, IEEE Trans. on Magn., Vol. 34 No. 5, pp. 2485-8. Dular, P., Hody, J.Y., Nicolet, A., Genon, A. and Legros, W. (1994), “Mixed finite elements associated with a collection of tetrahedra, hexahedra and prisms”, IEEE Trans. on Magn., Vol. 30 No. 5, pp. 2980-3. Hecquet, M. and Brochet, P. (1998), “Time variation of forces in a synchronous machine using electric coupled network model”, IEEE Trans. on Magn., Vol. 34 No. 5, pp. 3214-7. Ostovic, V. (1989), Dynamics of Saturated Machines, Springer-Verlag, Berlin. Ren, Z. (1994), “Comparision of different force calculation methods in 3D finite element modelling”, IEEE Trans. on Magn., Vol. 30 No. 5, pp. 3471-5.

Figure 8. Measured and calculated cogging torque as a function of rotor position ða0 # a # a0 þ 2tÞ

COMPEL 27,1

26

Stachowiak, D. (2004), “Edge element analysis of brushless motors with inhomogeneously magnetized permanent magnets”, COMPEL, Vol. 23 No. 4, pp. 1119-28. Sykulski, J., Stoll, J., Sikora, R., Pawluk, K., Turowski, J. and Zakrzewski, K. (1995), Computational Magnetics, Chapman & Hall, London. About the authors Andrzej Demenko received the MS and PhD degrees in Electrical Engineering from Poznan´ University of Technology in 1970 and 1975, respectively, and the Doctor of Science degree from Warsaw Institute of Electrical Engineering in 1986. Since, 1970, he has been employed in research and education. He is currently a Professor of Electrical Engineering at Poznan´ University of Technology. He has published five books and over 200 conference and journal papers on electromagnetics, electrical machines, and applied physics. His research interests include electromagnetic field calculation, numerical methods, field-circuit approaches in electrical machines design and analysis. E-mail: [email protected] Dorota Stachowiak received the MS and PhD degrees in Electrical Engineering from Poznan´ University of Technology in 1999 and 2004, respectively. Since, 2000, she has been employed in research and education. Her fields of interest are electrical machines and electromagnetics. She has published over 25 conference and journal papers. E-mail: [email protected]

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

Comparison between torque calculation methods in a non-conforming movement interface

Torque calculation methods 27

O.J. Antunes, J.P.A. Bastos and N. Sadowski UFSC, Grucad/EEL/CTC, Florianopolis, Brazil Abstract Purpose – The purpose of this paper is to compare torque calculation methods when a non-conforming movement interface is implemented by means of Lagrange multipliers. Design/methodology/approach – The following methods are here used for computing the torque in a synchronous machine and in a switched reluctance motor: Arkkio’s method (AM), local Jacobian matrix derivative (LJD) method, Maxwell stress tensor method (MST) and co-energy variation method. Findings – This paper shows that, the numerical stability produced by Lagrange multipliers yields a stable torque result, even in thin airgap machines if AM, LJD method or MST method are used. Originality/value – This work presents a comparative study to indicate the performance of the most commonly used torque calculation methods, when a non-conforming technique is used, considering a small displacement of the rotor, which is necessary for dynamic cases or coupling with circuit. Keywords Torque, Measurement Paper type Research paper

Introduction Although many works in the literature deals with torque calculation when considering the movement, the majority of them consider the rotor displacement step equal to the length of the elements in the airgap. There is a complete comparative study of torque calculation methods (Sadowski et al., 1992) considering a small displacement of the rotor, which is necessary for dynamic cases or coupling with circuit, but using a conforming method: the moving band. In Antunes et al. (2006b) the torque is studied with high-order interpolation for small displacements of the rotor in small airgaps, but the torque is calculated only by Maxwell stress tensor method (MST). The obtained results in Antunes et al. (2006b) shows that a conforming technique based on Lagrange multipliers produces more stable torque results than the moving band. This work presents a comparative study to indicate the performance of the most commonly used torque calculation methods when a non-conforming technique is used. Only first order interpolation is employed because, as shown in Antunes et al. (2006b), it produces better results in thin airgaps. The Lagrange multipliers method is used to take into account the movement. Lagrange multipliers This work uses a Lagrange multipliers formulation similarly as shown in Antunes et al. (2006a). The whole domain is decomposed in two sub-domains Va and Vb connected by a sliding interface G.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 27-36 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836609

COMPEL 27,1

28

One way to find the LM formulation is to minimize the energy functional associated with the magnetostatic problem added to a new functional to ensure the continuity of the vector potential at the interface. In this way, the following variational formulation is obtained: 

 Z Z Z › 1 a › 1 a ~ a · 1 7A ~ a dVa ¼ 7v lva dGþ va J a þ B oy 2 B ox dVa ð1Þ ›x ma ›y ma ma G Va Va 

 Z 1 ~ › 1 b › 1 b ~ 7vb · 7Ab dVb ¼ 2 lvb dGþ vb J b þ B 2 B dVb ð2Þ ›x mb oy ›y mb ox mb G Vb Vb Z ð3Þ ðAa 2 Ab Þvc dG ¼ 0: Z

Z

G

In the equations above va, vb and vc are test functions, Ja and Jb the current densities, Aa and Ab the vector potentials in each domain. Baoy , Baox , Bboy and Bbox are remanent induction components and ma and mb the magnetic permeabilities. l is identified either as the Lagrange multiplier or as the tangential component of magnetic field on G. The discretization of the equations above produces a symmetric system. This system is ill-conditioned and non-positive definite. However, it is possible to reformulate the final system in a well conditioned and positive definite form equivalent to the mortar element method (Antunes et al., 2006a). In this way, the matrices to couple the sub-domains are calculated as shown in Antunes et al. (2005). Torque calculation methods Maxwell stress tensor The torque acting on the rotor of an electrical machine may be calculated by integrating the MST along a surface S, placed in the airgap: I  h i ~ · n~ ÞH ~ 2 mo H 2 n~ ds ~r £ mo ðH ð4Þ T¼ 2 s !

!

Where, H is the magnetic field, r is the vector connecting the origin to the midpoint of the integration segment ds and mo is the air magnetic permeability. The method of local Jacobian matrix derivation Based on the virtual work principle, the LJD, proposed by Coulomb (1983), allows the torque evaluation by means of the relationship: T¼L

Nvd I  X i¼1

!T

2B · J 21 ·

Ve !

›J ! ·H þ ›u

Z 0

H!

!

21

B · dH · jJj

 ›jJj dVe ›u

ð5Þ

in which J is the Jacobian matrix, B is the magnetic induction, u the rotor displacement and L is the length of the machine. The sum is performed in the airgap and for the Nvd elements placed between the fixed and mobile parts that can be virtually deformed when the mobile part turns.

The method proposed by Arkkio This method (Arkkio, 1988) is a variant of the MST and consists in integrating the torque given by equation (4) in the whole volume of the airgap comprised between the radii rr and rs. The torque is computed with the following expression: Z L rBr Bt dS a ð6Þ T¼ mo ðr s 2 r r Þ Sa

Torque calculation methods 29

where Sa is the region between the radii rr and rs. Br and Bt are the magnetic radial and tangential inductions. Co-energy variation method The torque can be calculated by deriving the magnetic co-energy W0 by maintaining the current constant: T¼L

dW 0 jI ¼cte du

ð7Þ

In the numerical modeling, this derivation is approximated by the difference between two successive calculations: ðW 0uþdu 2 W 0u Þ T¼L ð8Þ du where du represents the displacement step. Results For the synchronous machine of Figure 1, which has an airgap thickness of 0.3 mm, the discretization shown in Figure 2 is used. The results shown in Figure 3 are obtained with two phases connected in series with 10 A excitation. The rotor displacement step is 0.12 degrees. To overpass one edge on G ten rotor displacement steps are necessary.

iron magnet

Figure 1. Thin airgap synchronous machine with polar pieces

COMPEL 27,1

S stator

30

G

Figure 2. Detail of the mesh for the synchronous machine with polar pieces with 81 edges on G. 1,772 elements and 1,015 nodes

rotor

Torque [Nm]

10 AM=LJD=MST

5 0 –5 –10

0

20

40

60 80 100 120 Rotor displacement [degrees]

140

160

180

140

160

180

140

160

180

10 Torque [Nm]

CV 5 0 –5 –10

0

20

40

60 80 100 120 Rotor displacement [degrees]

Figure 3. State torque for two phases connected in series with 10 A excitation for AM, LJD method, MST and CV method

Torque [Nm]

10 Experimental Result

5 0 –5 –10

0

20

40

60 80 100 120 Rotor displacement [degrees]

With MST, LJD and AM equivalent results are obtained, while with the co-energy variation (CV) method strong oscillations are produced. For a current of 40 A, the saturation is high and the results shown in Figure 4, for a non-linear case, show-again the same results for MST, LJD and AM. Only with a zoom (Figure 5) a little difference may be noted among them. Another thin airgap machine is simulated: the switched reluctance motor shown in Figure 6, which has an airgap of 0.25 mm.

Torque calculation methods 31

40 AM=LJD=MST

30

Torque [Nm]

20 10 0 –10 –20 –30 –40

0

20

40

60 80 100 120 Rotor displacement [degrees]

140

160

180

40 30 Experimental Result

Torque [Nm]

20 10 0 –10 –20 –30 –40

0

20

Note: Non-linear case

40

60 80 100 120 Rotor displacement [degrees]

140

160

180

Figure 4. Static torque for two phases connected in series with 40 A excitation for AM, LJD method and MST method

COMPEL 27,1

Torque [Nm]

32

24.32

AM

24.3

LJD

24.28

MST

24.26 24.24 24.22 24.2 24.18

Figure 5. Detail of Figure 4

24.16 71.2

71.25

71.3

71.45 71.5 71.35 71.4 Rotor displacement [degrees]

4

71.55

71.6

3

5

2 phase 1

+I

+I

6

1

–I

–I

7

10

8

Figure 6. Five-phase 10/8 switched reluctance motor

9

Note: Phase 1 (stator poles 1 and 6) under excitation

qr

For this machine, two discretizations (Figure 7) are used: Mesh 1 (with 2,486 elements, 1,605 nodes, 120 edges on G and four layers of elements in the airgap) and Mesh 2 (with 4,861 elements, 2,730 nodes, 240 edges on G and two layers of elements in the airgap). The airgap thickness is 0.25 mm. The interface G is placed in the middle of the airgap. The rotor displacement step is 0.15 degrees for all results. To overpass one edge on G 10 (for Mesh 1) and five (for Mesh 2) rotor displacement steps are necessary. The simulations with Mesh 1 presented in Figure 8 shows equivalent results with MST, LJD and AM, while with the CV method strong oscillations are produced (Figure 9). With Mesh 2, which is denser than Mesh 1 and with better shaped quadrilaterals in the airgap, similar results are obtained: the simulations for MST, LJD and AM are equivalent (Figure 10). But, we may observe that the simulation is closer to the experimental result, specially near 20 degrees for the rotor displacement, when compared with Mesh 1 (Figure 8). We can also observe in Figure 11 that the oscillations with CV method are reduced with Mesh 2.

Torque calculation methods 33

stator

stator S

S

G

G

Figure 7. Detail of Mesh 1 (a) and Mesh 2 (b). Slipping interface G for LM and MST integration surface S

rotor

rotor (a)

(b)

0.7 0.6 Experimental result Torque [Nm]

0.5 AM=LJD=MST

0.4 0.3 0.2 0.1 0

0

5

10 15 Rotor displacement [degrees]

20

Figure 8. Static torque for a 2 A excitation for AM, LJD method and MST method. Mesh 1

COMPEL 27,1

0.7 0.6

34

Torque [Nm]

0.5 0.4 CV 0.3 0.2 0.1 0

Figure 9. Static torque for a 2 A excitation for Mesh 1 for CV method. Mesh 1

–0.1

0

5

10 15 Rotor displacement [degrees]

20

0.7 0.6 Experimental result

Torque [Nm]

0.5 AM=LJD=MST

0.4 0.3

0.2

Figure 10. Static torque for a 2 A excitation for AM, LJD method and MST method. Mesh 2

0.1 0

0

5

10 15 Rotor displacement [degrees]

20

Conclusion This paper shows that the numerical stability produced by Lagrange multipliers or mortar methods yields a stable torque result, even in thin airgap machines if MST, LJD or AM are used. It is important to point out that AM may produce oscillations when the moving band is used (Sadowski et al., 1992) due to the elements deformation in the airgap.

Torque calculation methods

0.7 0.6

Torque [Nm]

0.5

35

0.4

CV

0.3 0.2 0.1 0

0

5

10 15 Rotor displacement [degrees]

20

References Antunes, O.J., Bastos, J.P.A., Sadowski, N., Razek, A., Santandrea, L., Bouillault, F. and Rapetti, F. (2005), “Using high-order interpolation with mortar element method for electrical machines analysis”, IEEE Transaction on Magnetics, Vol. 41 No. 5, pp. 1472-5. Antunes, O.J., Bastos, J.P.A., Sadowski, N., Razek, A., Santandrea, L., Bouillault, F. and Rapetti, F. (2006a), “Comparison between non-conforming movement methods”, IEEE Transaction on Magnetics, Vol. 42 No. 4, pp. 599-602. Antunes, O.J., Bastos, J.P.A., Sadowski, N., Razek, A., Santandrea, L., Bouillault, F. and Rapetti, F. (2006b), “Torque calculation with conforming and non-conforming movement interface”, IEEE Transaction on Magnetics, Vol. 42 No. 4, pp. 983-6. Arkkio, A. (1988), “Time-stepping finite element analysis of inductiuon motors”, paper presented at ICEM-88. Coulomb, J.L. (1983), “A methodology for the determination of global quantities from finite element analysis and its application to the evaluation of magnetic forces, torques and stiffness”, IEEE Transaction on Magnetics, Vol. 19 No. 6, pp. 2514-9. Sadowski, N., Lefe`vre, Y., Lajoe-Mazenc, M. and Cros, J. (1992), “Finite element torque calculation in electrical machines while considering the movement”, IEEE Transaction on Magnetics, Vol. 28 No. 2, pp. 1410-3.

About the authors O.J. Antunes finished his PhD thesis at Universidade Federal de Santa Catarina in 2005. He works in a technical school (Centro Federal de Educac¸a˜o Tecnolo´gica de Santa Catarina – Brazil). His researches relates to the modelling of electrical machines by means of finite elements method take into account the movement. O.J. Antunes is the corresponding author and can be contacted at: [email protected]

Figure 11. Static torque for a 2 A excitation for Mesh 1 for CV method. Mesh 2

COMPEL 27,1

36

J.P.A. Bastos finished his doctoral thesis (Docteur d’Etat) at Universite´ Pierre et Marie Curie, Paris VI in 1984. He is a full time Professor at Universidade Federal de Santa Catarina. He founded Grupo de Concepc¸a˜o e Ana´lise de Dispositivos Eletromagne´ticos (GRUCAD) in 1985, group with an important role in the development of the area of electromagnetic field analysis in Brazil. E-mail: [email protected] N. Sadowski finished his PhD thesis at the Institut National Polytechnique de Toulouse in 1993. He is a full time Professor at Universidade Federal de Santa Catarina and a researcher at GRUCAD. His research specializes in electrical machines and numerical methods. He has intensive consulting activities with electrical industries in Brazil and foreign. E-mail: [email protected]

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

Coupling of finite formulation with integral techniques

Coupling of finite formulation

Aldo Canova, Fabio Freschi and Maurizio Repetto Dipartimento di Ingegneria Elettrica, Politecnico di Torino, Torino, Italy, and

37

Giambattista Gruosso Dipartimento di Elettronica e Informazione, Politecnico di Milano, Milano, Italy Abstract Purpose – The paper aims to describe the coupling of magnetostatic finite formulation of electromagnetic field with two integral methods. Design/methodology/approach – The first hybrid scheme is based on Green’s function applied to magnetization source while the other one is based on a magnetic scalar potential boundary element method. A comparison of the two techniques is performed on a benchmark case with analytical solution, on a 2D multiply-connected problem and on an industrial case where measurements are available. Findings – The proposed hybrid approaches have proved to be effective techniques to solve open boundary non-linear magnetostatic problems. Similar convergence speed with respect to the number of unknowns is found for both schemes Originality/value – The paper shows the effectiveness of hybrid schemes applied to the finite formulation, assessing their performances on various test cases. Keywords Electromagnetism, Numerical analysis, Electromagnetic fields Paper type Research paper

1. Introduction In the last few years, geometrical formulations of electromagnetic field have collected a great interest because they allow to write field equations directly in algebraic form, ready to be solved by numerical techniques. Leaving the description of the basic technique proposed by Tonti to some other publications (Tonti, 2002; Repetto and Trevisan, 2004), in this paper the coupling of finite formulation of electromagnetic fields (FFEF) with integral techniques is proposed. The aim of the coupling is mainly to limit the discretization of the domain to its active parts only, avoiding the discretization of the infinite surrounding homogeneous region. The main rationales of this process can be found in: . easier discretization of complex domains, avoiding meshing of conductors of complex shape; . exact satisfaction of far field regularity conditions; and . overall smaller dimensions of solution matrix. On the other hand, it must be remarked that, as well as in hybrid FEM-based techniques, hybrid schemes handle infinite domains at the price of loosing the sparsity of the solution matrix. The paper is organized as follows: in Section 2 the hybridization of FEFF has been obtained by means of two different coupling schemes, one based on the evaluation of the

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 37-46 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836618

COMPEL 27,1

38

magneto-motive force along boundary edges by means of Green’s function, the other based on the boundary element method (BEM). Some details about multiply connected domains are also provided in Section 2. The performances of the two coupling schemes are assessed on three test cases in Section 3. Finally, Section 4 proposes some conclusions. 2. Hybrid formulations Starting from the standard formulation of magnetostatic problem, in terms of line integral of magnetic vector potential on primal edges p, and considering the Tonti diagram shown in Figure 1, the solution equation is expressed by: ~ M ~ n Cp ¼ i þ CF CM

ð1Þ

~ are primal and dual curl matrices, Mn , is the reluctance matrix (Tonti, where C and C 2002). The symbols used for variables are described in Table I. Equation (1) requires the discretization over the whole space in order to get proper boundary conditions for magnetic vector potential terms p. The hybrid schemes couple equation (1), defined on ferromagnetic materials without electric currents flowing through it, to an integral technique which gives the solution of the field problem in the exterior region. The decomposition of the magnetic field in two parts is crucial for the definition of ~ can be expressed as: an hybrid technique. Magnetic field H configuration variables

Figure 1. Tonti diagram of magneto-static problem

Table I. Symbols used in Tonti’s diagram of Figure 1

source variables

~ ~ ~ Notes: L and S are respectively primal edges and faces, P, L and S are respectively dual nodes, edges and faces

Symbol

Description

Space association

P F i F c

Line integral of magnetic vector potential Magnetic flux Electric current Magnetomotive force Magnetic scalar potential

Primal lines Primal faces Dual faces Dual lines Dual points

~ ¼H ~S þ H ~M H

ð2Þ

~ S is the magnetic field due to impressed current sources, lying outside the where H ~ M can be obtained by resorting discretized domain; the magnetization dependent term H to magnetic scalar potential c. By making use of free space Green function Gð~r 2 ~r 0 Þ, c can be written as: Z Z rM ð~r 0 Þ dVM cð~rÞ ¼ rM ð~r 0 ÞGð~r 2 ~r 0 ÞdVM ¼ ð3Þ r 2 ~r 0 j V V j~ ~ 2 7S · M ~ is the equivalent magnetic charge distribution over where rM ¼ 27 · M integration domain VM. Magnetization sources can be subdivided in: . configuration magnetization, due to the polarization of permeable materials: ~ ¼ xH ~ ¼ ðmr 2 1Þ B: ~ M m0

.

Coupling of finite formulation

39

ð4Þ

~ is computed by means of magnetic vector potential and these In this case, since B terms are included in the solution matrix. ~ is known and imposed magnetization, due to permanent magnets. In this case M independent from the magnetic vector potential solution and thus they belong to the known RHS term.

Magnetic scalar potential is used as a coupling term between the internal FFEF solution of equation (1) and the integral one taking care of the surrounding isotropic non-magnetic space. 2.1 FFEF coupled with Green’s function In this case the magnetic scalar potential, computed by means of equation (3), is used to impose a boundary condition on the Ampere theorem for the edges lying on the border of the mesh. This boundary condition is needed because the dual face associated to these edges is not entirely contained inside the mesh. By making reference to Figure 2, the exterior part of the integration path can be efficiently treated by a magnetic scalar potential difference as: trace of boundary edge a

b

contour of dual face associated to boundary edge

Figure 2. 2D trace of an edge lying on the boundary of ferromagnetic material and contour of its dual face

COMPEL 27,1

40

Z

b

~ ¼ c ðaÞ 2 c ðbÞ ~ M · dl H

ð5Þ

a

where a and b are the points where the dual face contour intersects the mesh ~ S contribute can be computed by means of the usual boundary. Besides this term, the H Biot-Savart’s formula along any path linking a and b which does not cross any current carrying region. This condition can be seen as a constraint on the continuity of tangential magnetic field at the interface expressed in finite form. This integral coupling links all the internal edges to the matrices rows belonging to boundary edges. 2.2 FFEF coupled with BEM formulation An integral equation can be written for the external part of the domain as: Z Z r; ~r 0 Þ ›cð~r 0 Þ 0 ›Gð~ ds þ Gð~r; ~r 0 ÞdS ¼ 0 dcð~rÞ 2 cð~r Þ ›n S S ›n

ð6Þ

where d is a constant which is equal to p if point ~r lies on a plane surface (Rucker and Richter, 1988). In this case, the magnetic scalar potential c plays the same role as in the previous scheme while its normal derivative ›c=›n can be linked to the internal solution by the continuity of magnetic flux density on the triangular faces lying on the surface of the mesh:
 ›c þ H k;n ¼ Fk m0 S k ð7Þ ›n ~ · n~ S is the normal where k is the generic boundary face, Sk is its area, H k;n ¼ H k component of magnetic field due to impressed current sources and Fk is the magnetic flux through Sk. In this way all boundary edges are linked together by integral relations and thus their rows are partially filled up. 2.3 Multiply connected domains Magnetic vector potential solution must be gauged, this can be done by setting to a constant value magnetic vector potential integrals on the edges belonging to a tree of the mesh (Repetto and Trevisan, 2004). The use of hybrid formulations often requires the analysis of multiply connected meshed domains, for instance, if a closed ferromagnetic core is studied. Structures can have also more than one hole, as for instance in a three-phase transformer core. This situation requires to take into account additional conditions on magnetic vector potential besides the usual gauging, in fact, the magnetic flux flowing through the hole, or through each of the holes, is not known and must be imposed by an additional constraint. One technique is to create a closed path of edges around the hole which correspond to a belted tree as proposed in Kettunen et al. (1998). The automatic building of such a sub-graph has been extensively treated by different authors, (Ren and Razek, 1993; Kettunen et al., 1998). A semi-automatic procedure has been set up to treat multiply connected structures. From the user definition of: . a cutting surface in each of the hole of the structure; and . a cutting surface through the core.

a tree on the edges sub-sets identified by these cuts is automatically selected and a belted tree (Kettunen et al., 1998) is obtained for each hole in the mesh. This approach avoids the formation of spurious turns around the core, as highlighted as a possible problem in Ren and Razek (1993). 3. Test cases 3.1 Test case 1: comparison with analytical solution The two hybrid schemes have been compared on a test case with analytical solution made of a cylindrical permanent magnet with known imposed magnetization along its z-axis, Figure 3. For each hybrid technique, the convergence is evaluated by the RMS error between the analytical and computed values of magnetic flux density in tetrahedra centers. By evaluating this quantity as function of the number of unknowns, an estimation of the convergence speed can be obtained (Giuffrida et al., 2006). The graph of Figure 4 shows

Coupling of finite formulation

41

PERMAG Solution: DualINT Facets: 4826 Tethr: 3528

Figure 3. Cylindrical permanent magnet used for comparison of the two hybrid schemes

z x

y

5 FFEF+Green FFEF+BEM

RMS error (× 10–6)

4

3

2

1

0 0.5

1.0

1.5

2.0 2.5 3.0 3.5 4.0 4.5 number of unknowns (kunits)

5.0

5.5

6.0

Figure 4. RMS error between hybrid solutions and analytical one as function of number of unknowns

COMPEL 27,1

42

that in the test case the hybrid scheme FFEF þ Green has the same convergence rate as FFEF þ BEM but with a lower value. This trend is also confirmed by other test cases not referenced in this paper. 3.2 Test case 2: multiply connected domain In this case, the procedure for the management of multiply connected regions is tested on an axisymmetric structure and comparisons are made versus a standard 2D FEM code (Meeker, 2006). The axisymmetric trace of the structure is shown in Figure 5(a), whereas Figure 5(b) shows the non-linear magnetic characteristic. No significant differences between the two techniques are observed in the field computation. Figure 6 1 cm

1 cm

1 cm

1 cm

coil

2.5

1 cm

2.0 1.5

B (T)

nonlinear iron

2 cm

1.0 0.5

1 cm 0.0

Figure 5. Axisymmetric structure used for test

0

1 cm

5,000 10,000 15,000 20,000 25,000 30,000 35,000

H (A/m) (a)

(b)

AXISYMMETRIC Solution: DualINT Facets: 2912 Tethr: 1920

Figure 6. Mesh of the axisymmetric multiply connected test case and vector plot of magnetic flux density

z x

y

shows the mesh used and the vector plot of magnetic flux density on a radial plane. The comparison of results obtained by the hybrid codes and the 2D axisymmetric code reported in Figure 7 shows a satisfactory agreement.

Coupling of finite formulation

3.3 Test case 3: industrial application The proposed technique is finally used to evaluate the magnetic field outside a 5 kVA, three-phase dry transformer (Figure 8). In this case the analysis of the domain requires to handle a multiply-connected domain with two holes. The structure under study is

43

0.12 2D axisymmetric FEM code SALLY3D

magnetic flux density (T)

0.10 0.08 0.06 0.04 0.02 0.00 0.00

0.01

0.02

0.03 0.04 radial line (m)

0.05

0.06

0.07

Figure 7. Magnetic flux density along a radial line 1 cm over the coil: comparison between hybrid procedure (SALLY3D) and a standard 2D FEM code

Figure 8. Experimental set-up for environmental magnetic field measurements

COMPEL 27,1

44

shown in Figure 9, together with the field distribution on an orthogonal plane. Figure 10 compares the field values obtained by measurements and simulations, when only the central internal winding is supplied with the magnetization current. The computed field values are in good agreement versus experimental ones. It must be remarked that the use of an hybrid scheme allows to efficiently model the transformer far field. 4. Conclusions The work performed on the coupling of FFEF and integral techniques allows to draw some conclusions: TRANSFORMER Solution: DualINT Facets: 9136 Tethr: 4224

Figure 9. 3D model of the transformer

z x y

3.2

magnetic flux density (µT)

2.8

Figure 10. Comparison between measurement, FFEF þ Green

measurements SALLY3D

2.4 2.0 1.6 1.2

z = 50 cm z = 75 cm z = 100 cm

0.8 0.4 0.0 –0.5 –0.4 –0.3 –0.2 –0.1 0.0 0.1 X axis (m)

0.2

0.3

0.4

0.5

.

.

. .

The coupling of two different techniques can be performed by imposing the continuity of tangential and normal component of fields. In finite form, these conditions are integrated along lines or surfaces. Both schemes have the same convergence speed but, on the case tested, the FFEF þ Green scheme seems to have better convergence performances than the FFEF þ BEM one. The FFEF þ BEM scheme couples all the surface edges among them. The FFEF þ Green scheme couples all the internal edges to the ones on the mesh surface if polarizable materials are present and thus magnetization depends on solution. If only imposed magnetization sources are considered with m0 permeability, the terms coming from hybrid formulation take place at the RHS term only and the solution matrix keeps its sparse structure. This fact is particularly important in the case of micro-magnetic computations.

References Giuffrida, C., Gruosso, G. and Repetto, M. (2006), “Finite formulation of nonlinear magneto-statics with integral boundary conditions”, IEEE Transactions on Magnetics, Vol. 42 No. 5, pp. 1503-11. Kettunen, L., Forsman, K. and Bossavit, A. (1998), “Formulation of the eddy current problem in multiply connected regions in terms of H”, International Journal for Numerical Methods in Engineering, Vol. 41 No. 5, pp. 935-54. Meeker, D. (2006), “Finite element method magnetics”, Version 4.0.1, available at: femm.fostermiller.net Ren, Z. and Razek, A. (1993), “Boundary edge elements and spanning tree technique in three-dimensional electromagnetic field computation”, International Journal for Numerical Methods in Engineering, Vol. 36, pp. 2877-93. Repetto, M. and Trevisan, F. (2004), “Global formulation for 3D magneto-static using flux and gauged potential approaches”, International Journal for Numerical Methods in Engineering, Vol. 60, pp. 755-72. Rucker, W. and Richter, K. (1988), “Three-dimensional magnetostatic field calculation using boundary element method”, IEEE Transactions on Magnetics, Vol. 24, pp. 23-6. Tonti, E. (2002), “Finite formulation of electromagnetic field”, IEEE Transactions on Magnetics, Vol. 38, pp. 333-6. About the authors Aldo Canova is an Associate Professor in Fundamentals of Electrical Engineering at Politecnico di Torino, Italy. He is involved in research activities related to the numerical computation of electromagnetic fields in the area of power devices and he is author of more than 100 papers. He is a Member of the Comitato Elettrotecnico Italiano (CEI) and of the Conference Intenationale des Grands Reseaux Electriques a Haute Tension (CIGRE) on magnetic field mitigation techniques. Fabio Freschi received the Laurea (summa cum laude) and the PhD degree in Electrical Engineering from the Politecnico di Torino, Italy, where he is currently working as Assistant Professor in Fundamentals of Electrical Engineering. His main research interests are related to optimization, inverse problems, computational electromagnetics and environmental electromagnetic fields. He published more than 30 papers on these topics. Fabio Freschi is the corresponding author and can be contacted at: [email protected]

Coupling of finite formulation

45

COMPEL 27,1

46

Maurizio Repetto is a Full Professor of Fundamentals of Electrical Engineering at Politecnico di Torino. He is author of more than 100 publications in the area of computational electromagnetics. In particular, he is involved in research projects about the analysis of ferromagnetic hysteresis and automatic optimization of power devices. Giambattista Gruosso is an Assistant Professor in the Dipartimento di Elet-tronica ed Informazione, Politecnico di Milano, Italy. He is author of more than 30 publications in the area of computational electromagnetics. He has worked on analysis and optimization of electromechanical devices, environmental electromagnetic fields and shielding, and magnetic equivalent circuit analysis.

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

Numerical solutions in primal and dual meshes of magnetostatic problems solved with the finite integration technique

Numerical solutions

47

J. Korecki and Y. Le Menach Lamel-L2EP – USTL, Villeneuve d’Ascq, France

J-P. Ducreux Lamel, Electricite De France, R&D Division, Clamart, France, and

F. Piriou Lamel-L2EP – USTL, Villeneuve d’Ascq, France Abstract Purpose – To compare the numerical solutions in primal and dual meshes of magnetostatic problems solved with the finite integration technique. Design/methodology/approach – The development of the whole set of magnetostatic discrete formulations is proposed. Four formulations are computed: two in terms of fields and two in terms of potentials. Moreover, each computation is carried out on the primal and dual mesh. Two applications are presented and the results are analysed and discussed. Findings – The whole set of magnetostatic formulations gives only two solutions. The solutions do not depend of the formulation, but they depend of the choice of the field discretisation in primal or dual mesh. Originality/value – The computation is carried out on the dual mesh. Keywords Numerical analysis, Meshes, Electromagnetism Paper type Research paper

1. Introduction To solve the Maxwell equations in the magnetostatic case, two formulations in terms of potentials are classically used: the scalar potential V-formulation and the vector potential A-formulation. Using the finite element method (FEM), each solution is computed on a so-called primal mesh (Bossavit, 1998). These solutions are complementary (Albanese and Rubinacci, 1990), which can be used to estimate the discretization error (Marmin et al., 2000). But according to the Tonti diagram, both scalar and vector potentials should not be defined on the same mesh. If the scalar potential is defined on a primal mesh then the vector potential should rather be defined on the associated dual mesh (Tonti, 2001), and vice versa. As a consequence, the FEM does not compute the fields on the same discretized Tonti diagram, which leads to differences of numerical results. This work has been supported by the French CNRT project (Centre National de Recherche Technologique).

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 47-55 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836627

COMPEL 27,1

48

In this paper, we propose to solve the Maxwell equations using the finite integration technique (FIT), in particular with hexahedral elements (Clemens and Weiland, 2001; Albertier et al., 2005). The interest of the FIT is the possibility to carry out the computation on both primal and dual meshes. Consequently, both formulations in terms of potentials will be solved on both meshes. In addition, the formulations in terms of fields (B and H) are presented and computed also on both primal and dual meshes. Finally, an academic problem for which the analytical solution is known will be studied with the whole set of possible formulations. The different solutions will be analysed and discussed. An iron core coil will be also studied with both complementary formulations in terms of potentials. 2. Magnetostatic formulations In the magnetostatic case, the Maxwell equations take the following form: curlH ¼ J with H £ njGh ¼ 0

ð1Þ

divB ¼ 0 with B · njGb ¼ 0

ð2Þ

where H is the magnetic field, J the current flux density and B the magnetic flux density. To take the material properties into account, the magnetic constitutive law has to be introduced: B ¼ mðjHjÞH

ð3Þ

To obtain the formulations in terms of fields B and H, the constitutive law (3) is introduced in equations (1) or (2), respectively. The following equations are then obtained: curl

B ¼ J and divB ¼ 0 m

curlH ¼ J and divmH ¼ 0

ð4Þ ð5Þ

Both formulations are not often solved because after the discretization using FIT or FEM, the matrix system is not square. We can explain this in a next paragraph about the discretization. Another solution consists in using the formulations in terms of potentials. From equation (1), the magnetic scalar potential V-formulation can be defined as: divmðH s 2 gradVÞ ¼ 0

ð6Þ

where Hs is a source field defined such that curlH s ¼ J with help of a tree technique and J is imposed. On the other hand, from equation (2), it is possible to introduce the magnetic vector potential A-formulation, given by: curl

1 curlA ¼ J m

ð7Þ

3. Discrete formulations According to the classical works concerning the discretization of magnetic quantities, the scalar potential V is defined on the nodes, the magnetic field H and the magnetic vector potential A on the edges (circulation) and the magnetic flux density B on the facets (flux). But to define the discrete form of the constitutive law we must introduce some conditions between two sequences of spaces. Indeed, it is necessary to introduce a primal mesh M and a dual mesh. Figure 1 shows a 2D example of primal and dual meshes with square elements. The primal mesh is orthogonal then the dual mesh is also orthogonal. The primal nodes, edges, facets and volumes are associated to dual elements, facets, edges and volumes, respectively. For the next the primal nodes, edges, facets and volumes are noted N, E, F and V. On the ~ E; ~ F~ and V~ correspond to dual nodes, edges, facets and volumes. other hand, the sets N; In these conditions, the magnetic field and the scalar potential can be discretized on primal mesh (noted, respectively, Vd and hd) then the magnetic flux density and the vector potential (noted b~ d and a~ d ) will be expressed on facets and edges of the dual ~ respectively. A priori the choice of the discretisation of magnetic quantities on mesh M, the primal or dual meshes is arbitrary. Using the properties of the incidence matrix the discrete operators of gradient, curl and divergence can be defined. These operators are denoted G, C and D on the primal mesh and ~ C~ and D~ on the dual mesh. It can be noted that the following properties occur: G; G t ¼ 2D~

ð8Þ

C ¼ C~ t

ð9Þ

D ¼ 2G~ t

ð10Þ

Numerical solutions

49

where X t is the matrix transposition of X. The discrete constitutive law for magnetic material will be defined with the mass matrix M such as: ð11Þ b~ d ¼ Mhd If the mesh is constituted of only hexahedra, the relation between hd on edge “e” and ~ is given by: the flux b~ d through the dual facet “f” b~ d ¼ mav

S~ f~ Le

ð12Þ

hd Primal edge Dual edge Primal node Dual node

Figure 1. 2D example of a primal mesh M associated to a ~ dual mesh M

COMPEL 27,1

50

where Le represents the length of primal edge, S~ f~ the surface of the dual facet. mav is the average value of the magnetic permeability of the flux tube of length Le and cross section S~ f (Clemens and Weiland, 2001). In fact the problem can be established in two ways according to the Tonti diagram: ~ ðVd ; hd ; jd Þ [ M and ðb~ d ; a~ d Þ [ M

ð13Þ

~ d ; h~ d ; ~jd Þ [ M ~ and ðbd ; ad Þ [ M ðV

ð14Þ

or:

From these assumptions, the computation with the fields (B, H) or the potentials (A, V) are only a transformation to be able to solve the problem. Let us take again the Maxwell equations and apply the discretisation, we can obtain two possibilities according to the assumptions (13) and (14): ( Chd ¼ jd ð15Þ D~ b~ d ¼ 0 or:

8 < C~ h~ d ¼ ~jd : Dbd ¼ 0

ð16Þ

From discrete Maxwell equations, two solutions are possible for each formulation (B, H, V and A). Indeed, each formulation can be expressed on primal or dual meshes.. Using equations (11), (15) and (16), the whole set are obtained of discrete formulations, which is summarized in Table I. All these formulations can be solved with the FIT. Two equations are necessary to obtain B and H formulations. Taking the case of H formulation on the primal mesh, the equation set C and 2 G t M give a matrix which the numbers of line is equal to node number plus facet numbers, but the column number is equal to edge number. Consequently, as written before, the matrix system for B and H formulations are not square. We can represent all the involved quantities and operators in two Tonti diagrams (Figure 2). The first one (Figure 2(a)) corresponds to the definition (13): the scalar potential, the magnetic field and the flux current density are defined on the Form H

B Table I. Whole set of magnetostatic discrete formulations

Primal mesh ( Chd ¼ jd

Dual mesh 8 < C t h~ d ¼ ~jd

2G t Mhd ¼ 0 8 < C t M 21 bd ¼ ~jd

: DM h~ d ¼ 0 8 < CM 21 b~ d ¼ jd : 2G tss b~ d ¼ 0

:

Dbd ¼ 0

V

2G t M ðhs 2 GVd Þ ¼ 0

~ dÞ ¼ 0 DM ðh~ s þ D t V

A

C t M 21 Cad ¼ ~jd

CM 21 C t a~ d ¼ jd

N

Ωd

–G E

~ bd = Mhd

hd

R F

Numerical solutions

~ V ~ bd

~ ad

jd

D D

– Gt ~ F

51

Rt ~ E Dt ~ N

V V

~ V

N –G E

~ jd

ad

R F

~ bd = Mhd

bd

D

~ hd

– Gt ~ F Rt ~ E

Figure 2. Two Tonti diagrams of the magnetostatic case

Dt ~ Ωd ~ N

V V

primal mesh. On the second diagram (Figure 2(b)) the magnetic vector potential and the magnetic flux density belong to the dual mesh, which fulfils the definition (14). Let us take the Maxwell equations (1) and (2) associated with the boundary conditions. If we consider Figure 1, the boundary GðG ¼ Gb < Gh Þ of studied domain are discretize on primal mesh. In this case the boundary G are only constituted by a set of N, E and F. If the magnetic field formulation is computed on the primal mesh, the boundary condition H £ njGh ¼ 0 is obvious to impose. Moreover, it is imposed strongly because hd ¼ 0 on Gh. On other hand, as there are not dual facets on Gb ; B · njGb ¼ 0 is free and is ensured weakly. Now if hd is discretised on dual mesh, H £ njGh ¼ 0 is verified weakly where as B · njGb ¼ 0 can now be imposed strongly. The summary of this analysis is presented in Table II for each formulation. Remark: Classically with the FEM both formulations in term of potentials are solved only on the primal mesh. In these cases the first Tonti diagram is used for V-formulation (Figure 2(a)) and the second for A-formulation (Figure 2(b)). Form H B V A

Primal mesh hd jGh hd jGh hd jGh hd jGh

¼ 0 bd jGb < 0 bd jGb ¼ 0 bd jGb < 0 bd jGb

Dual mesh

¼ :

96

where curl b Eðx; yÞ ¼ curl ðEðx; yÞeibz Þe2ibz and where curlb H is defined in the same way. For a given b, these equations can be seen as an eigenproblem: Lb F ¼ k0 F;

ð2Þ

with: 0

0

B B Lb :¼ B B @ iZ 0 1r 21 curlb ¼

21 curlb 2iZ 21 0 mr ¼

0

1 C C C: C A

ð3Þ

Of course, from this latest system of equations two equivalent eigenproblems can be derived: 8 21 mr curlb ð1r 21 curlb H Þ ¼ k20 H > > ¼ >

1r 21 curlb ðmr 21 curlb EÞ ¼ k20 E: > > ¼ :¼

ð4Þ

2.2 Spectral problem In the literature concerning the fibres, at least three kinds of modes are studied; the guided modes, the leaky modes and the radiative modes. All these modes are governed, of course, by the same formal spectral problem (2) but the functional spaces in which they live are different. For this purpose, we have to define accurately these functional spaces. Definition 1. We say that (E, H) is a guided mode if the following three conditions are fulfilled: . (b, v) [ (Rþ )2 . (E, H) – (0, 0) . (E, H) [ [L 2(R2)]3. Definition 2. We say that (E, H) is a leaky mode if we can find a strictly positive real number G in order to fulfil the following three conditions: . (b, v) [ Cþ £ Rþ , where Cþ ¼ {z [ C; Im{z} . 0; Re{z} . 0} . (E, H) – (0, 0) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ðe2GR E; e2GR H Þ [ ½L 2 ðR2 Þ
3 , where R ¼ x 2 þ y 2 .

Definition 3. We say that (E, H) is a radiative mode if, for any strictly positive real number G the following four conditions are fulfilled: . (b, v) [ (Rþ )2 . (E, H) – (0, 0) . (E, H)
[L 2(R2)]3. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ðe2GR E; e2GR H Þ [ ½L 2 ðR2 Þ
3 , where R ¼ x 2 þ y 2 . For the sake of simplicity, in this paragraph, we assume that the fibre is made of isotropic and non-magnetic materials. Moreover, the cladding is supposed to be homogeneous. In other words, for a sufficiently large R0, the permittivity is constant, 1r(x, y) ¼ 11. In this case, provided that: ffi 1 ðx; yÞ; 1Max ¼ maxðx;yÞ[R2 =pffiffiffiffiffiffiffiffiffi x 2 þy 2 ,R r 0

is greater than 11 and also that finite energy eigenvectors are considered (Definition 1), the spectrum of the operator Lb consists of a discrete set of eigenvalues belonging to [b 2/1Max; b 2/11 [and of continuous spectrum [b 2/11; þ 1] (Zolla et al., 2005). On the other hand, when 1Max , 11, only leaky modes may exist and complex valued propagation constants have to be considered. If we focus our attention only on leaky modes propagating along the increasing z, it turns out that b0 ¼ Re{b} and b00 ¼ Im{b} are both positive. Note that, if the first condition is fulfilled, the field (H,E) cannot be of finite energy. Actually, within the cladding every component of the electromagnetic field is solution of Helmholtz equation, namely: 2

DU þ k~ 1 U ¼ 0;

ð5Þ

2 where U is one component of either E or H and k~ 1 ¼ k21 11 2 b 2 . The function U can be written in cylindrical co-ordinates as a Fourier Bessel expansion:

U ðx; yÞ ¼ U c ðr; wÞ ¼

X

inw ~ cn H ð1Þ n ðk1 rÞe ;

ð6Þ

n[Z

where H ð1Þ n refers to the Hankel function of the first kind of order n. This latest function has a well known asymptotic behaviour, namely: sffiffiffiffiffiffiffiffiffiffiffi 2 2iðnðp=2Þþðp=4ÞÞ ik~ 1 r ~ e H ð1Þ e þ Oðr 23=2 Þ; n ðk1 rÞ ¼ ~ pk1 r

ð7Þ

and leads to: ~0

eik 1 r ~ 00 Uc ðr; wÞ ¼ s ðwÞ pffiffiffi e2k 1 r ; r

ð8Þ

Use of PML for the computation of leaky modes 97

COMPEL 27,1

with: sffiffiffiffiffiffiffiffiffi X 2 s ð wÞ ¼ e2ip=4 cn einðw2p=2Þ ; pk~ 1

ð9Þ

n[Z

98

where k~ 01 ¼ Re{k~ 1 } and k~ 001 ¼ Jm{k~ 1 }. It is therefore of prime importance to know the sign of k~ 001 . First of all the outgoing wave condition leads to the positiveness of the real part of k~ 1 . And if we let b ¼ b 0 þ ib 00 , we deduce: 2 k~ 1 ¼ k20 11 2 b02 þ b002 2 2ib0 b00

ð10Þ

As a result we have:

b 0 b 00 ¼ 2k~ 01 k~ 001 : Bearing in mind that b0 and b00 are both positive together with k~ 01 , we have to conclude that k~ 001 is a real strictly negative number and consequently the electromagnetic field does diverge exponentially at infinity and the coefficient of this divergence (the smallest G satisfying the aforementioned condition (c) in Definition 2) is linked precisely with the imaginary part of the propagation constant b. From both theoretical and practical points of view, the non-finiteness of energy of the electromagnetic field leads to dramatic consequences. And especially when dealing with the weak form of Maxwell equations: the field does not vanish at infinity. Last, the real part of b is of the same magnitude as k0 ¼ v/c whereas its imaginary part can be extraordinary smaller, say 102 8 k0 in the following numerical experiments and sometimes for very low leakage as small as 102 15 k0! And for experimental reasons this latest information is of prime importance. 2.3 Finite element method and PML 2.3.1 Circular PMLs. In the finite element analysis of wave problems in open space, one of the main difficulties is to truncate the unbounded domain. A common approach is to surround a finite region of interest with absorbing boundary conditions at finite distance. An alternative to conditions defined on the boundary is to introduce a special layer of finite thickness surrounding the region of interest such that it is non-reflecting and completely absorbing for the waves entering this layer under any incidence. Such regions have been introduced by Berenger (1994) and are called perfectly matched layers (PML). Nowadays, the most natural way to introduce PML is to consider them as maps on a complex space (Lassas et al., 2001; Lassas and Somersalo, 2001) so that the corresponding change of (complex) co-ordinates leads to equivalent 1 and m (that are complex, anisotropic, and inhomogeneous even if the original ones where real, isotropic, and homogeneous). This leads automatically to an equivalent medium with the same impedance than the one of the initial ambient medium since 1 and m are transformed in the same way and this ensures that the interface with the layer is non-reflecting. Moreover, a correct choice of the complex map leads to an absorbing medium able to dissipate the outgoing waves. The problem can therefore be properly truncated under the condition that the artificial boundary is situated in a region where the field is damped to a negligible value. To sum up, we have a problem in an unbounded region with outgoing propagating waves or with exponentially diverging waves. A change of co-ordinates is performed such that it corresponds to the identity map in a region of

interest (bounded, convex and, for all practical purposes, with a simple shape) and to complex co-ordinates for the surrounding region. These complex co-ordinates are chosen to turn propagating waves to evanescent waves (i.e. exponentially decreasing at infinity) so that this outer domain can be truncated. The geometry of most MOF leads to rather use cylindrical PML. In this case, the PML corresponds to a complex stretch of the radial co-ordinate r, the region of interest is a disk defined by r , R * and the PML region is a circular annulus around the region of interest defined by R * , r , R trunc. R * and R trunc are real constants. As the expressions of the material tensors in Cartesian co-ordinates are needed, the whole setting requires a transformation between Cartesian and cylindrical co-ordinates. The recipe involves a sequence of co-ordinate systems. We start here with the physical coordinates and we finish with the modelling co-ordinates. The mapping will therefore be from the last system of the list to the first one while the pull back maps will be from the first system to the last one. (1) (x, y, z) are real valued classical Cartesian co-ordinates. (2) (~x; y~ ; z~) are a complex stretch of the previous Cartesian co-ordinates. They are complex valued and it is fundamental to understand that this change is an active transformation rather than a mere change of co-ordinates in the sense that the ambient space is changed. (x, y, z) are a parametrization of R3 and the complex stretch corresponds to an extension of the problem to C3 and more precisely to a three dimensional subspace G of C3 (in terms of real dimensions C3 is six dimensional and R3 and G are three dimensional) (Lassas et al., 2001; Lassas and Somersalo, 2001). The map from G to R3 is chosen in such a way that the restriction of this map to the region of interest is the identity map. The solution of the original problem on R3 can be extended analytically to C3 and then restricted to G. If the complex stretch is correctly chosen, this “complexified” solution on G is evanescent where the physical solution involves outgoing or even exponentially diverging waves. (3) (r~; u~; z~) is a cylindrical representation of (~x; y~ ; z~). (4) (rc, uc, zc) are real-valued cylindrical co-ordinates on G. They are related to (r~; u~; z~) via u~ ¼ uc , z~ ¼ zc , and a radial complex stretch:

r~ ¼

Z

rc

sr ðr0 Þdr0 ;

ð11Þ

0

where sr is a complex-valued function of a real variable, i.e. sr ¼ 1 in the central region of interest defined by rc , R * (the complex stretch corresponds to an identity map in this region) and sr has a complex value in the PML defined by R * , r , R trunc. (5) (xc, yc, zc) are the Cartesian representation of ( rc, uc, zc) and are also real-valued co-ordinates that will be called modelling co-ordinates. This is the modelling space where the numerical approximations are written, where the finite element mesh is defined, and where all the outgoing waves are turned to evanescent ones so that the computation domain can be truncated. In the end, only the real-valued co-ordinates x, y, z and xc, yc, zc are involved but the complex map corresponds to a complex valued Jacobian. In the case of cylindrical

Use of PML for the computation of leaky modes 99

COMPEL 27,1

coordinates, r~ and rc are just introduced to compute the radial stretch. Note also that uc ¼ u~ and therefore will be simply denoted u. The Cartesian to cylindrical co-ordinates transformation is just used to obtain the Cartesian expression of the corresponding metric tensor. The Jacobian associated to these changes of co-ordinates are: Jx~ r~ ¼ Jxr ðr~; uÞ; Jrr~ c ¼ diagðð›r~=›rc Þ; 1; 1Þ ¼ diag ðsr ðrc Þ; 1; 1Þ; Jrc xc

100

¼ Jrx ðrc ðxc ; yc Þ; uðxc ; yc ÞÞ: The global Jacobian Js is the product of the individual Jacobians: Js ¼ Jx~ r ~Jrr~ c Jrc xc

  r~ ¼ RðuÞ diag sr ; ; 1 Rð2uÞ; rc

ð12Þ

where R(u) denotes the following matrix of rotation: 0

cos u

2sin u

B RðuÞ ¼ B @ sin u

cos u

0

0

0

1

C 0C A:

1

Note that we solve in fact numerically the extended problem obtained by the complex stretch equation (11) and defined on G that has the very remarkable property to coincide with our original problem in the region of interest. In order to comply with traditional notation in the PML context and to avoid cumbersome notations, we drop the c subscript associated with the modelling co-ordinates that will subsequently be denoted as r and (x, y, z) without any ambiguity. For isotropic uniform media outside the region of interest, the cylindrical PML characteristics are obtained by multiplying 1 and m by the following complex matrix: 

21 2T T21 s ¼ Js Js detðJs Þ ¼ RðuÞdiag

0

 r~ sr r sr r~ ; ; Rð2uÞ ¼ sr r r~ r

ððrsr sinðuÞ2 Þ=r~Þ þ ððr~cosðuÞ2 Þ=rsr Þ

B B sinðuÞcosðuÞððr~=rsr Þ 2 ðrsr =r~ÞÞ @ 0

sinðuÞcosðuÞððr~=rsr Þ 2 ðrsr =r~ÞÞ ððrsr cosðuÞ2 Þ=r~Þ þ ððr~sinðuÞ2 Þ=rsr Þ 0

0

1

C 0 C: A r~sr =r

This latest expression is the metric tensor in Cartesian co-ordinates (x, y, z) for the cylindrical PML and u, r, r~, and sr( r) are explicit functions of the variables x and y. Another remarkable property of the PML is that they provide the correct extension to non-Hermitian operators (since Ts is complex and symmetric) that allow the computation of the leaky modes and this may be obtained via a correct choice of the PML parameters, namely R *, R trunc, and sr( r). The method used to compute these parameters is accurately described hereunder (Figure 1).

Cartesian co-ordinates y (x,y)

R*

Use of PML for the computation of leaky modes

Polar co-ordinates y

x

( r,j)

x

R*

PML

101

PML

Complex cartesian co-ordinates

Complex polar co-ordinates

yc

yc ( rc ,jc)

(xc ,yc)

R*

xc

R*

PML

xc

PML

Notes : The physical space in Cartesian (x,y) and polar (r,j) co-ordinates are linked by x = r cos j and y = r sin j. The stretched complex polar co-ordinates (rc,jc) r and the real polar co-ordinates are linked by r = Ú 0 c sr(r′ ) dr′ (where sr is a complex valued function) and j = jc. Finally, the complex Cartesian co-ordinates which characterizes the modelling space and the complex polar co-ordinates are linked by rc = ÷x2c + y2c yc and jc = 2 arctan x + ÷x2 + y2

(

c

c

Figure 1. Four different systems of co-ordinates in order to obtain circular PML

)

c

2.3.2 How to choose the complex stretch coefficient?. For this purpose, we first introduce new tensor fields 1s ðx; yÞ and ms ðx; yÞ defined as follows: ¼

¼

2T zs :¼ J21 detð Js Þ for z ¼ {1; m}; s zr Js ¼

¼

ð13Þ

We are now in a position to introduce a new electromagnetic field (called substituted field in the sequel) Fs ¼ (Hs, Es)T which is solution of equation (2) except that we have replaced 1r and mr by 1s and ms . The main feature of this latest field is the remarkable ¼ ¼ ¼ ¼ correspondence with the first field F; whichever the function sr provided that it equals 1 for r , R *, the two fields F and Fs are identical in the region r , R * (Berenger, 1994). In other words, the PML is completely reflectionless. In addition, for complex valued functions sr (Im{sr} strictly positive in PML), the field Fs may converge exponentially towards zero although its counterpart F diverges exponentially: Fs is of

COMPEL 27,1

102

finite energy and for this substituted field a weak formulation can be easily derived which is essential when dealing with finite element method. Moreover, when dealing with simple stretched functions it is possible to give a simple criterion which ensures the exponential decreasing of the field Fs. For instance, take the following example: ( 1 for r0 # R * 0 sr ðr Þ ¼ j in PML where j is a complex number. In that case, the complex function r(rc) is given by: 8 < rc for rc # R * r¼ : R * þ jð r c 2 R * Þ in PML The function U can be reexpressed in the stretched polar co-ordinates in the PML as per: X   inw * * c ~ U cs ðrc ; wc Þ ¼ cn H ð1Þ : n k1 ðR þ jðrc 2 R ÞÞ e n[Z

  For large values of rc, the behaviour of Ucs is governed by exp ik~ 1 ðR * þ jðrc 2 R * ÞÞ which exponentially converges towards zero if k~ 1 j has a strictly negative imaginary part. Consequently, the field Fs converges exponentially towards zero as well (and therefore is of finite energy) if the simple following criterion is fulfilled:

g :¼ k~ 001 j 0 þ k~ 01 j 00 . 0:

ð14Þ

Keeping in mind that k~ 001 is negative whereas k~ 01 is positive, a straightforward solution could be j 0 , 0 and j 00 ¼ 0. In other words, j would be a real negative number. But, in that case, the function r(rc) is no longer monotonic and therefore we are “doomed” to choose the number j within the upper complex plane. As a conclusion, for a given couple (k0, b) in Rþ £ Cþ , we compute k~ 1 , from equation (10) and j is chosen in such a way that g is sufficiently large in order to ensure that the skin layer thickness of the substituted electromagnetic field is of the same order than the thickness of the PML (Figure 2). In order to illustrate the decreasing of the substituted field within the PML, let us consider a six hole MOF shown in Figure 3. The Figure 2 shows the exponential decreasing of the z component of the electric field for a parameter j ¼ 1 2 2i and for a wavelength l ¼ 1.55 mm: we find b ¼ 5.8323 £ 105 þ 0.173092 i m2 1. The decreasing of the field which is characterized by g is therefore computed by two different ways: The first one (denoted gth) through the equation (14) and the second one (denoted gnum), by numerical computing of the field itself. Finally, we find gth ¼ 4.98105 m2 1 and gnum ¼ 4.97 £ 105 m2 1. 2.3.3 Weak formulation for the substituted field Fs. If j verifies the inequality (14) the field converges towards zero at infinity and, as a result, a weak formulation can be easily derived. For instance, with the electric field Es, we obtain: Z Z 2 21 * ðms curlb £ E s Þ · ðcurlb
£ E s Þd xdy 2 k0 ð1s E s Þ · E
*s d xdy ¼ 0 ð15Þ R2 ¼

R2

¼

log | Es,z(x,0) | –22

Numerical decreasing for the substituted field

Use of PML for the computation of leaky modes

Theoretical decreasing for the substituted field

–23

103

–24

–25

–26

–27

–28

x( mm) 20

22

24

26

28

30

Notes : The logarithm of the z component of the substituted electric field, log | Es,z(x,0) |, is plotted versus x in the PML (for R* < x < Rtrunc). The structure is depicted in Fig. 3

Figure 2. Behaviour of the substituted field within the PML

y V e Si

2rs

e2

O

x

Notes : A bulk of silica is drilled by six air holes distant each other from = 6.75 mm. Each hole is circular with a radius equal to rs = 2.5 m m. Note that for such as structure no propagating mode can propagate V

Figure 3. Six hole MOF structure

COMPEL 27,1

for any test function E*s . Then, we split the electric field Es into a transverse part Et and a longitudinal part Ez: E s ¼ E t þ E z z^ from which, we are led to: curlb £ E s ¼ ibE t £ z^ þ 7t E z £ z^ þ ð7t £ E t Þz^:

104 Finally, by letting:

(

L0 ðE t Þ ¼ E t £ z^ L1 ðE s Þ ¼ 7t E z £ z^ þ ð7t £ E t Þz^

we derive the following weak formulation for the substituted electric field: Z R2

b 2 F 0 þ ibðF 1;1 þ F 1;2 Þ þ F 2 d xdy ¼

Z R2

k20 G d xdy

ð16Þ

with: 8   > F 0 ¼ ðms 21 L0 ÞðE t Þ · ðL0 Þ E*t > > ¼ > > > > >   > > > 21 * > F ¼ ð m L ÞðE Þ · ðL Þ s 0 t 1 Es > 1;1 > ¼ > > > > >   > < F 1;2 ¼ ðms 21 L1 ÞðE s Þ · ðL0 Þ E*t ¼ > > >   > > > 21 > F ¼ ð m L ÞðE Þ · ðL Þ E*s s s 2 1 1 > > ¼ > > > > > Q > > > G ¼ ð1s E s Þ · E* > > s ¼ > : In most applications in optics, we have to find the dispersion curves, i.e. to look for b’s for a given k0. In such a case, the eigenproblem described in (16) leads to a generalized eigenproblem owing to the presence of both b and b 2 (Tisseur and Meerbergen, 2001, for instance). 2.3.4 Generality on finite elements. The discretisation of the equations is obtained via finite elements (Zolla et al., 2005; Nicolet et al., 2004, 2006; Guenneau et al., 2001, 2002; Lasquellec et al., 2002; Nicolet and Zolla, 2007). The cross-section of the guide is meshed with triangles and Whitney finite elements are used, i.e. edge elements for the transverse field and nodal elements for the longitudinal field: Et ¼

#edges X j¼1

etj w je ðx; yÞ and E z ¼

#nodes X j¼1

e zj w jn ðx; yÞ;

where etj denotes the line integral of the transverse component Et on the edges, e zj denotes the line integral of the longitudinal component Ez along one unit of length of the z-axis (which is equivalent to the nodal value), and w je and w jn are, respectively, the basis functions of Whitney 1-forms and 0-forms on triangles. 3. Numerical results 3.1 Comparison with the multipole method One of our former challenge was to compare our results with those obtained with other methods such as the multipole method. The philosophy of this latest method is completely different from those described in this paper and the reader can refer to Zolla et al. (2005) for a comprehensive review of this method. This method is indeed well-suited for the step index MOF. Be that as it may, we consider the hexagonal structure shown in Figure 3 for a given wavelength l0 ¼ 1.55 mm for which the index of silica is about 1Si ¼ 1.4440. For this structure, we give a map representing one mode in Figure 3 and the corresponding complex effective index, namely neff ¼ b/k0 for the two different methods: 1.438774 þ 4.32 £ 102 8i for the multipole method and 1.438773 þ 4.28 £ 102 8i for the finite element method. Besides, it is worth knowing that the practical implementation of the model has been performed thanks to the COMSOL software with about 40,000 elements and the computation takes a few seconds on an ordinary laptop. Note that regarding its smallness with respect to the real part, the imaginary part is computed with an amazing accuracy. 3.2 Leaky modes for gradient index MOF In this paragraph, we present some numerical results for more sophisticated fibres. Besides of the six air holes, the permittivity of the bulk varies continuously as per:

1r ðx; yÞ ¼ 11 þ

1i1

exp

x2 þ y2 2 r 20

! ð17Þ

The case corresponding to the precedent paragraph is therefore 11 ¼ 1Si and 1i1 ¼ 0. By way of example, we take r0 ¼ 10 mm (r0 is therefore of the same magnitude as L) and we give a curve representing neff versus 1i1 with 11 ¼ 1Si for the same wavelength as before i.e. l0 ¼ 1.55 mm. Note that the central symmetry of 1r does not break the C6v symmetry of the fibre (Figures 4 and 5). 3.3 Leaky modes for elliptical hole MOF In this paragraph, a six elliptical hole structure is considered as shown in Figure 6 for a given wavelength l0 ¼ 1.55 mm for which the index of silica is about 1Si ¼ 1.4440. The elliptical holes are distant from each other from L ¼ 6.75 mm and are characterized by their semi-major axis a and their semi-minor axis b. Moreover, the orientation and the choice of a and b are chosen in such a way that both the C6v structure and the area of these ellipses are preserved (ab ¼ r 20 ). Eventually, for r0 ¼ 2.5 mm, and for different values of b, effective indices (real part in Figure 7 and imaginary part in Figure 8) are given.

Use of PML for the computation of leaky modes 105

COMPEL 27,1

×10–5 4

×104

8 2

106

6 0 4

–2

Figure 4. Modulus of the Poynting vector for the C6v six hole MOF shown in Figure 3

–4

2

–4

–2

4

2

0

×10–5 Note : The structure is surrounded by cylindrical PML

4.5 108

4 3.5 3 2.5 2

Figure 5. The complex effective index neff as a function of 1i1

1.5 1 0.0001

0.001 e1i

0.01

4. Conclusion The search for leaky modes is far from being a simple task. The first way of obtaining these modes consists in computing the scattering matrix and looking for poles of this latest matrix in the complex plane. This numerical stage is a delicate operation and has two major drawbacks: the “pole hunting” in the complex plane is generally performed in a point-by-point fashion and this method is merely devoted to step index fibres. On the other hand, the method used in this paper as shown before is a versatile and efficient method and may be useful for obtaining leaky modes in delicate situations. Finally, we hope to obtain leaky modes for the challenging nonlinear MOFs.

Use of PML for the computation of leaky modes

y V eSi e0 b

107 O

a

x

Figure 6. Six elliptical hole MOF structure

Note : The C6u structure is preserved 1.4405 1.44 1.4395 1.439 1.4385 1.438 1.4375 1.437 0.4

0.6

0.8

1 a

1.2

1.4

1.6

1 a

1.2

1.4

1.6

Figure 7. Real part of effective index versus the parameter a ¼ b/a

1e-5 1e-6 1e-7 1e-8 1e-9 1e-10 1e-11 1e-12 1e-13 1e-14 0.4

0.6

0.8

Note : For the value a = 0.44 (Ellipses are close to touching) leakage is extremely weak = 5.10–13!!

Figure 8. Imaginary part of effective index versus the parameter a ¼ b/a

COMPEL 27,1

108

References Berenger, J-P. (1994), “A perfectly matched layer for the absorption of electromagnetic waves”, Journal of Computational Physics, Vol. 114, pp. 185-200. Cregan, R.F., Mangan, B.J., Knight, J.C., Birks, T.A., Russell, P.S., Roberts, P.J. and Allan, D.C. (1999), “Single-mode photonic band gap guidance of light in air”, Science, Vol. 285, pp. 1537-9. Guenneau, S., Nicolet, A., Zolla, F. and Lasquellec, S. (2002), “Modeling of photonic crystal optical fibers with finite elements”, IEEE Transactions on Magnetics, Vol. 38 No. 2, pp. 1261-4. Guenneau, S., Nicolet, A., Zolla, F., Geuzaine, C. and Meys, B. (2001), “A finite element formulation for spectral problems in optical fibers”, COMPEL, Vol. 20 No. 1, pp. 120-31. Guenneau, S., Lasquellec, S., Nicolet, A. and Zolla, F. (2002), “Design of photonic band gap optical fibers using finite elements”, COMPEL, Vol. 21 No. 4, pp. 534-9. Lassas, M. and Somersalo, E. (2001), “Analysis of the PML equations in general convex geometry”, Proceedings of the Royal Society of Edinburgh, Vol. 131 No. 5, pp. 1183-207. Lassas, M., Liukkonen, J. and Somersalo, E. (2001), “Complex Riemannian metric and absorbing boundary condition”, Journal de Mathematiques Pures et Appliquees, Vol. 80 No. 7, pp. 739-68. Nicolet, A. and Zolla, F. (2007), “Finite element analysis of helicoidal waveguides”, IET Science, Measurement & Technology, Vol. 1 No. 1, pp. 67-70. Nicolet, A., Zolla, F. and Guenneau, S. (2004), “Modelling of twisted optical waveguides with edge elements”, The European Physical Journal-Applied Physics, Vol. 28 No. 5, pp. 153-7. Nicolet, A., Movchan, A.B., Guenneau, S. and Zolla, F. (2006), “Asymptotic modelling of weakly twisted electrostatic problems”, Comptes Rendus Me´canique, Vol. 334 No. 2, pp. 91-7. Ranka, J.K., Windeler, R.S. and Stentz, A.J. (2000), “Visible continuum generation in air-silica microstructured optical fibers with anomalous dispersion at 800 nm”, Opt. Lett., Vol. 25, pp. 25-7. Tisseur, F. and Meerbergen, K. (2001), “The quadratic eigenvalue problem”, Siam Review, Vol. 43 No. 2, pp. 235-86. Zolla, F., Renversez, G., Nicolet, A., Kuhlmey, B., Guenneau, S. and Felbacq, D. (2005), Foundations in Photonic Crystal Fibres, Imperial College Press, London.

About the authors Y. Ould Agha was born in Nouakchott, Mauritania, on the 31st December 1977. He obtained the Master in Electronics, Power Systems and Automatics from Nouakchott University in 2002. In July 2004, he received the Master of Research in Components and Systems for the data processing, at the University of Blaise Pascal (Clermont-Ferrand-France). He is currently a PhD student in Physics at the Fresnel Institute (University of Provence). E-mail: yacoub.ould-agha@ fresnel.fr F. Zolla received a PhD thesis on Electromagnetic Diffraction at the University of Marseille in 1993. He is currently a Professor at the University of Provence and is the author of many papers on theoretical and numerical study of gratings, photonic crystals and homogenization of Maxwell equations. An emerging research topic is that of spectral analysis of dielectric waveguides and numerical schemes for the modelling of exotic materials such as Photonic Crystal Fibers, Twisted Fibers, and Invisibility Cloaks. F. Zolla is the corresponding author and can be contacted at: [email protected] A. Nicolet has received his Engineering Degree (Inge´nieur Civil Electricien (Electronique)) in 1984 and his PhD in Applied Sciences in 1991 both from the University of Lie`ge, Belgium.

From 1984 to 1994, he was a Research Engineer at the Department of Applied Electricity of the University of Lie`ge. From 1994 to 1996, he was the Head of the Department of Electrical Engineering and Power Electronics of the Ecole Supe´rieure des Inge´nieurs de Marseille(ESIM) France. In 1996, he received the “Habilitation a` Diriger des Recherches” from the University Paul Ce´zanne of Aix-Marseille III. Since, 1996, he is a Professor at the University Paul Ce´zanne of Aix-Marseille III (63rd section: electronics, optronics and systems) and he has joined the Fresnel Institute (UMR CNRS 6133) at his creation in 2000. His main research interests are the numerical computations of electromagnetic fields and he presently focuses on the modelling of micro-structured optical fibres. He is author or co-author of more than 80 papers and of a book entitled Foundations of Photonic Crystal Fibres (Zolla, Renversez, Nicolet & al., Imperial College Press, 2005). E-mail: [email protected] S. Guenneau graduated with a PhD thesis on Models of Photonics from University of Aix-Marseille in 2001. He subsequently worked as a Research Assistant in 2001-2004 with Professor Alexander Movchan at Liverpool University and Sir John Pendry at Imperial College London. He was then appointed as an Applied Mathematics Lecturer in 2004 at Liverpool University. In January 2006, he took over a Research Scientist position at Fresnel Institute (Marseille) within the Centre National de la Recherche Scientifique (CNRS) E-mail: sebastien. [email protected]

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

Use of PML for the computation of leaky modes 109

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

COMPEL 27,1

Investigation of the characteristics of conformal microstrip antennas

110

Ralf T. Jacobs and Arnulf Kost Brandenburg University of Technology, Cottbus, Germany

Hajime Igarashi Hokkaido University, Sapporo, Japan, and

Alan J. Sangster Heriot-Watt University, Edinburgh, UK Abstract Purpose – The purpose of this paper is the analysis of the radiation and impedance characteristics of cavity backed patch antennas embedded in a curved surface. Single patch elements and small scale array antennas are considered. The impact of curvature on the performance of the patch antenna is investigated, and the effect of mutual coupling between the elements in an array is examined. Design/methodology/approach – A finite element-boundary integral procedure has been implemented to accurately determine the performance characteristics of the patch radiators on planar and cylindrical surfaces. Simulated results will be shown to be in good agreement with measurements. Findings – Mutual coupling effects between array elements are examined and it can be observed that an active element primarily interacts with the nearest neighbour elements. A comparison of an array element with a single patch radiator shows that the mutual coupling effects cause no significant mismatch between a patch and a feed network in practical applications. Originality/value – The characteristics of conformal microstrip antennas are investigated for single patch radiators and patch elements in array environments. Simulations are supported by measurements. Keywords Boundary-element method, Finite element analysis, Antenna impedance Paper type Research paper

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 110-121 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836681

1. Introduction Patch antenna arrays in stripline technology are used as radiating and receiving elements in wide range of microwave systems such as radar, remote sensing, communication, radio frequency identification and biomedical systems. In many potential developments, the surface on which the radiating elements are mounted may be subject to flexing as a result of mechanical forces acting on the structure, and the radiating elements need to be tolerant to this. A finite element-boundary integral procedure has been developed to accurately determine the performance characteristics of the patch radiators on planar and cylindrical surfaces. The discretization is performed with edge elements which match the coordinate system of the underlying problem domain in order to circumvent geometrical approximation errors. Good agreement between theory and measurement is demonstrated.

2. Method of analysis The geometry of a single element is displayed in general (j, h, z)-coordinates shown in Figure 1. The patch of length ðl pj ; l ph Þ is backed by a cavity of length ðl cj ; l ch Þ and is excited using a coaxial probe feed positioned at ðl Fj ; l Fh Þ. The height of the cavity domain D is determined by the thickness l cj of the dielectric substrate, which is characterised by the relative permittivity 1r and the loss tangent tand. The relative permeability mr of the substrate equals unity. In case of a planar antenna, the general coordinates coincide with (x, y, z), and in case of a curved antenna with (f, z, r). The computation of the radiation characteristics is performed employing the finite element-boundary integral method (Kost, 1994; Volakis et al., 1997; Kempel et al., 1995). The electric field E~ in the cavity domain D satisfies the wave equation: 7£

1 7 £ E~ 2 k20 1r E~ þ jk0 Z 0 sE~ ¼ 2jk0 Z 0 ~J mr

Characteristics of microstrip antennas 111

ð1Þ

subject to the boundary condition n~ £ E~ ¼ 0 on the conducting cavity walls. Z0 represents the characteristic impedance and k0 the wavenumber of free space, s the conductivity of the medium and ~J denotes the electric current density. The vector wave equation is converted into weak form applying the weighted residual method, which leads to:

lxc p

lx

lhp

lhc

lhF lxF

h z

x s→• D m, e

z h

S

Æ

J

s→• x

lzc

Figure 1. Geometry of a single patch antenna in general coordinates

COMPEL 27,1

ððð


 1 ~ · ~vi þ jk0 Z 0 sE~ · ~vi dV ~ · ð7 £ ~vi Þ 2 k2 1r E ð7 £ EÞ 0 D mr ðð ððð ~J · ~vi ·dV ; i [ ½1; N
~ rad
· ~vi dS ¼ 2jk0 Z 0 2 jk0 Z 0 O ½n~ £ H ›D

112

ð2Þ

D

where ~vi represents the weighting function. The contour integral reduces to a surface integral over the aperture S, since the conducting cavity walls provide a ~ The radiated magnetic field H ~ rad can be Dirichlet condition for the electric field E. ~~ ~r; ~r0 Þ for the outer equated with the aid of the appropriate dyadic Green’s function Gð half-space: ðð ~~ 0 ~ ~ rad ¼ 2jk0 Y 0 ~r; ~r0 Þ · ðn~ £ EÞdS ð3Þ H Gð S

~0

~r denotes the field vector, r the source vector, n~ the outward pointing unit normal, and Y0 ¼ 1/Z0 the characteristic admittance of free space. Explicit expressions for the dyadic Green’s functions are given in the appendix. The coaxial probe feed is modelled as current filament since the size of the probe is only a small fraction of the operating wavelength. The current density in the probe is modelled as: ~Jz ¼ I 0 dðx 2 xF Þdð y 2 yF Þ~ez   ~Jr ¼ I 0 dðw 2 wF Þ dðz 2 zF Þ~er r

for planar

ð4Þ

for cylindrical

ð5Þ

antenna structures. The angular feed position wF is determined through the arc length l Fw and the radius r of the cylinder. I0 represents the current in the probe and ~ei the unit vector in i-direction. Expansion of the electric field in equation (2) with the same set of functions as the weighting functions: E~ ¼

N X

aj ~vj

ð6Þ

j¼1

yields a set of linear equations with unknown coefficients aj. The cavity is discretized with curl-conforming edge elements which match the coordinate system of the underlying problem domain. Rectangular parallelepipeds are used for planar geometries and cylindrical shell elements for curved structures. This technique circumvents geometrical approximation errors. The elements and the appropriate linear interpolation functions are given in Jin (1993) and Volakis et al. (1998). The contribution of the boundary-integral formulation is determined by the quadruple integration which results from the substitution of equation (3) into equation (2). An analytical solution of these integrals is not attainable due to the Green’s function in the integral kernel, but the application of a coordinate transformation allows a reduction of the quadruple integration to a sum of four double integrations which enables a numerically efficient evaluation. A detailed description of this procedure is outlined in Sangster and Jacobs (2003). After solving the matrix equation for the coefficients aj, the input impedance Zin of the patch element is computed using the reaction concept (Pozar, 1982):

Z in ¼ 2

ððð

N 1X

I 20

~vj · ~JdV :

aj

ð7Þ

D

j¼1

Characteristics of microstrip antennas

The radiation characteristics are evaluated using equation (3): ~ ¼ 2jk0 Y 0 H

NS X

ðð aj

~~ ~r; ~r0 Þ · ðn~ £ ~vj ÞdS 0 ; Gð

ð8Þ

113

S

j¼1

where N S symbolises the number of degrees of freedom on the aperture surface S. The far-field relations Eu ¼ Z0 Hw and Ew ¼ 2Z0 Hu finally determine the radiation pattern, where u and w denote the spherical angles. 3. Simulated results and discussion In order to validate the numerical procedure, the input impedance of a single planar patch element with the dimensions l px ¼ 90 mm and l py ¼ 60 mm is computed for two different feed positions. The impedance loci in Figure 2 shows good agreement with measurement and reflect the tendency of the radiation resistance to increase as the feed element approaches the edge of the patch. Figure 3 shows the E-plane radiation pattern for a circumferentially polarized patch antenna for different cavity sizes. The patch element is mounted on a cylinder of radius r ¼ 149.5 mm. 1 FE-BI 2

0.5

Measured

a)

0.2

+j X Zl 0 R –j X Zl Zl

f

0.2

0.5

5

b)

1

2

5

0.2

a) l Fx = 42mm, l Fy = 6mm b)

l Fx =

42mm,

l Fy =

±∞

5

2

0.5

18mm 1

Notes: f = 1.60 GHz... 1.68 GHz, ∆f = 0.01 GHz, l px = 90 mm, l py = 60mm, l cx = 135 mm, l cy = 90 mm, l cz = 0.762 mm, er = 2.20, tan d = 0.0008, Zl = 50Ω

Figure 2. Input impedance of a planar patch antenna for different feed positions

COMPEL 27,1

0° 30° EjdB(q = 90°, j)

0

60°

114

j –30°

–10 –60°

–20 –30

–90°

90°

120°

Figure 3. E-plane radiation pattern of a circumferentially polarized patch antenna for different cavity sizes

–120°

150°

l cj = 52.5mm l cj = 105mm l cj = 315mm Cavity model Measured

–150° ±180°

Notes: l pj =35 mm, l pz = 35mm, l cz = 70 mm, l cp = 3.175 mm, r = 149.5 mm, l Fj = 30 mm, l Fz = 17.5 mm, er = 2.32, tan d = 0.00015, f = 2.615 GHz, Cavity model and measured results from Sohtell (1989)

The finite element-boundary integral computations show good agreement with measurement and cavity model results from Sohtell (1989). Only marginal deviations are observable between the radiation patterns for the different cavity sizes, which implies that the chosen cavity sizes have very little effect on the antenna performance. This behaviour can be ascribed to the fact that radiation occurs from the edge fields only, which are in the close proximity to the perimeter of the patch. If the cavity walls are sufficiently distant from the patch, such that the fringing fields on the edge are not perturbed, the radiation characteristics of the antenna remain unaltered. The simulations for single patch antennas in the previous figures have established the accuracy of the finite element-boundary integral procedure which provides confidence for the array analysis. For this, a five element single-row array antenna is considered, as shown in Figure 4. In order to eliminate coupling effects of a feed network, only the centre patch is excited whereas the other elements remain passive and unloaded. The excitation of the non-active elements results solely from the electromagnetic fields generated by the active patch. The impact of the passive elements on the radiation pattern therefore reflects the effect of mutual coupling between the elements. Figure 5 shows the H-plane radiation pattern of a planar array with a patch width l px ¼ 24 mm. Simulations are performed for element spacings l dx of 24 and 12 mm. The radiation pattern clearly reflects the effect of the enhanced mutual coupling between the elements as the spacing is reduced. The increased coupling yields a stronger excitation of the passive elements which consequently induce side lobes in the pattern. Only two side lobes are introduced by the five element array, which

l cx l dx

l dx

2 1

2 l px

h

z

l ph

3

4

l Fx

5

l Fh

lch

s→∞

115

lcz

Figure 4. Geometry of a five element array antenna

x

z er h

D s→∞

x

0° –30°

q 30°

–60°

60°

–90°

90° EydB (q, j = 0°)

–50

–120°

–150°

–40 –30

l dx = 24 mm, (a) l dx = 24 mm, (b) l dx = 12 mm, (a) l dx = 12 mm, (b)

120°

–20 –10 0

150°

±180° l px =

Characteristics of microstrip antennas

l py =

24.0 mm, 28.0 mm, l cx = 56.0 mm, l cz = 0.762 mm, l Fx = 12.0 mm, l Fy = 6.0 mm, Notes: er = 2.20, tan d = 0.0008, f = 3.48 GHz, (a) no internal cavity walls, (b) with internal cavity walls

indicates that the active patch interacts primarily with the nearest neighbour elements. Furthermore, computations have been performed for an array where the individual elements have been separated by conducting boundary walls in the substrate, placed in the middle between the patches as shown in Figure 6. In this case, every patch is backed by a separate cavity. The resulting patterns display only marginal differences which shows that mutual coupling predominantly occurs in the outer half-space since no significant alteration is

Figure 5. H-plane radiation pattern of a planar 5 element array antenna excited at centre patch only, for different patch distances

COMPEL 27,1

l cx l dx 2

l dx 1

2

2

3

4

5 l ch

116 h sƕ z

x

z

Figure 6. Array antenna with internal boundary walls

l cz

er h

x

sƕ

provoked by the internal boundary walls. The effect of the mutual coupling in Figure 5 is shown since the passive elements are not connected to a source or load impedance. In order to simulate the radiation characteristics of a practical arrangement, the non-active patches are terminated with a 50 V load at the feed location. The resulting radiation pattern exhibits only minor deviations from the pattern of a single patch antenna as shown in Figure 7, which indicates that mutual coupling effects cause no significant alteration of the radiation characteristics in practical applications. The examination of the input impedance properties in Sangster and Jacobs (2003) confirms this observation. The effect of curvature on the radiation characteristics of an array antenna is examined using a uniformly excited five element array with patch elements of size l pw ¼ 35 mm and l pz ¼ 35 mm. Figure 8 shows the E-plane pattern of a circumferentially polarized array. The radiation pattern broadens if the cylinder radius r decreases, since the ratio of patch length l pw to the circumference of the cylinder increases. Figure 9 shows the H-plane pattern of an axially polarized array. The radiation pattern tends, as expected, towards the pattern of the equivalent planar array as the radius of the cylinder increases. Only the co-polar radiation patterns are depicted for the radiation characteristics since the cross-polar field intensities lie more than 30 dB below the co-polar values. The excitation of the patch antenna is accomplished using a coaxial probe feed, modelled as a current filament under the assumption that the probe size is only a small fraction of the operating wavelength of the antenna. The finite probe diameter, the self-inductance of the feed element, and the coaxial aperture in the ground plane of the antenna are not taken into account. These approximations are susceptible to provoke inaccuracies but the validity of the feed model is supported by the fact that measured impedances of antennas with different probe

0°

Characteristics of microstrip antennas

q

–30°

30°

–60°

60°

–90°

117 90°

–50

Array, l dx = 24 mm Array, l dx = 12 mm Single element

–120°

–150°

EydB (q, j = 0°)

–40 –30

120°

–20 –10 0

150°

±180° Notes : l px = 24.0 mm, l py = 28.0 mm , l cy = 56.0 mm, l cz = 0.762 mm, l Fx =12.0 mm, l Fy = 6.0 mm, er = 2.20, tan d = 0.0008, f = 3.48 GHz 0°

Figure 7. H-plane radiation pattern of a planar five element array antenna excited at centre patch, passive elements terminated with a 50 V load impedance, and compared to a single element

j –30°

30°

–60°

60°

90°

–90°

r = 122.5 mm r = 175 mm

–50 EjdB (q = 90°, j)

r = 350 mm

120°

150°

–40

r = 875 mm

–30

–120°

Planar

–20 –10 0

–150°

±180° Notes : ljp = 35 mm, lzp = 35 mm, lzc = 52.5 mm, lrc = 0.762 mm, ljd =17.5 mm, ljF = 27.5 mm, lzF = 17.5 mm, er = 2.20, tan d = 0.0008, f = 2.79 GHz

Figure 8. E-plane radiation pattern of a circumferentially polarized, uniformly excited five element array antenna for different cylinder radii

COMPEL 27,1

0°

j –30°

30°

–60°

60°

118

90°

–90°

r = 122.5 mm r = 175 mm

–50 EzdB (q = 90° , j)

r = 350 mm

120°

Figure 9. H-plane radiation pattern of an axially polarized, uniformly excited five element array antenna for different cylinder radii

150°

–40

r = 875 mm

–30

–120°

Planar

–20 –10 0

–150°

±180° Notes : ljp = 35 mm, lzp = 35 mm, lzc = 52.5 mm, lrc = 0.762 mm, ljd =17.5 mm, ljF = 17.5 mm, lzF = 7.5 mm, er = 2.20, tan d = 0.0008, f = 2.79 GHz

diameters show no significant variation. The Smith chart in Figure 10 shows computed and measured impedance characteristics of a planar patch antenna. Measurements are performed on patch antennas fed by coaxial probes of diameters 0.75, 1.0, and 1.27 mm. The marginal differences between the measured impedances cannot necessarily be attributed to the different probe diameters, since these could have also been induced through measurement inaccuracies or fabrication tolerances.

4. Conclusion The radiation and impedance characteristics of planar and curved patch antennas have been analysed employing a finite element-boundary integral method. The accuracy of the procedure has initially been established using single elements and the analysis has then been extended to array antennas. Mutual coupling effects between array elements have been examined and it has been observed that an active element primarily interacts with the nearest neighbour elements. A comparison with a single patch radiator has shown that the mutual coupling effects cause no significant mismatch between a patch and a feed network in practical applications. Numerical computations have been validated with measurements and good agreement has generally been observed.

Characteristics of microstrip antennas

1 FE-BI 2

0.5

+j X Zl 0 R –j X Zl Zl

119

f

0.2

0.2

0.5

1

5

2

±•

5

5

0.2 Measured Probe diameter 0.75 mm Probe diameter 1.0 mm

2

0.5

Probe diameter 1.27 mm 1 Notes : f = 2.10 GHz . . . 2.30 GHz, ∆ f = 0.02 GHz, l px = 45 mm, l py = 30 mm, l cx = 60 mm, l cy = 40 mm, lzc= 0.762 mm, l Fx = 11.25 mm, l Fy = 7.5 mm, er = 2.20, tan d = 0.0008, Zl = 50 W

References Bird, T.S. (1984), “Comparison of asymptotic solutions for the surface field excited by a magnetic diplole on a cylinder”, IEEE Transactions on Antennas and Propagation, Vol. 32 No. 11, pp. 1237-44. Jin, J. (1993), The Finite Element Method in Electromagnetics, Wiley, New York, NY. Kempel, L.C., Volakis, J.L. and Silva, R.J. (1995), “Radiation by cavity-backed antennas on a circular cylinder”, IEE Proceedings on Microwaves, Antennas and Propagation, Vol. 142 No. 3, pp. 233-9. Kost, A. (1994), Numerische Methoden in der Berechnung Elektromagnetischer Felder, Springer-Verlag, Berlin. Pathak, P.H. and Wang, N. (1981), “Ray analysis of mutual coupling between antennas on a convex surface”, IEEE Transactions on Antennas and Propagation, Vol. 29 No. 6, pp. 911-22. Pozar, D.M. (1982), “Input impedance and mutual coupling of rectangular microstrip antennas”, IEEE Transactions on Antennas and Propagation, Vol. 30 No. 6, pp. 1191-6. Sangster, A.J. and Jacobs, R.T. (2003), “Mutual coupling in conformal microstrip patch antenna arrays”, IEE Proceedings on Microwaves, Antennas and Propagation, Vol. 150 No. 4, pp. 191-6.

Figure 10. Input impedance of a planar patch antenna with different probe diameters

COMPEL 27,1

120

Sohtell, E.V. (1989), “Microstrip antennas on a cylindrical surface”, in James, J.R. and Hall, P.S. (Eds), Handbook of Microstrip Antennas, IEE Electromagnetic Waves Series,Vol. 2, Peter Peregrinus Ltd., London, pp. 1227-56. Tai, C-T. (1994), Dyadic Green Functions in Electromagnetic Theory, IEEE Press Series on Electromagnetic Waves, IEEE Press, New York, NY. Volakis, J.L., Chatterjee, A. and Kempel, L.C. (1998), Finite Element Method for Electromagnetics, IEEE Press Series on Electromagnetic Wave Theory, IEEE Press, New York, NY. Volakis, J.L., Ozdemir, T. and Gong, J. (1997), “Hybrid finite-element methodologies for antennas and scattering”, IEEE Transactions on Antennas and Propagation, Vol. 45 No. 3, pp. 493-507. Appendix Green’s function for the planar structure is given by: ! 0 e2jk0 j~r2~r j ~~ ~ 1 Gð~ r; ~r0 Þ ¼ 2 ~I þ 2 77 2pj~r 2 ~r0 j k0

ðA1Þ

~ where ~I denotes the unit dyad. The required components of the dyadic Green’s function on the surface of a cylinder with radius r are given by: " #  1 Z 1 H 0 ð2Þ 1 X 1 nkz 2 H ð2Þ 0 0 ww 0 n ðrkr Þ n ðrkr Þ 2 ð2Þ e 2jkz ðz2z Þ e 2jnðw2w Þ dkz ; ðA2Þ G ð~r;~r Þ ¼ rkr k0 H 0 ð2Þ ð2pÞ2 n¼21 21 rkr H n ðrkr Þ n ðrkr Þ G wz ð~r; ~r0 Þ ¼

1 Z 1 21 X nkz H ð2Þ n ðrkr Þ 2jkz ðz2z0 Þ 2jnðw2w0 Þ e e dkz ; 2 ð2pÞ n¼21 21 r 2 kr k20 H 0 ð2Þ n ðrkr Þ

ðA3Þ

1 Z 1 kr H ð2Þ 1 X n ðrkr Þ 2jkz ðz2z0 Þ 2jnðw2w0 Þ e dkz : e 2 ð2pÞ n¼21 21 rk20 H 0 ð2Þ n ðrkr Þ

ðA4Þ

G ð~r; ~r0 Þ ¼ zz

0 ð2Þ H ð2Þ n denotes the Hankel function of second kind and order n, and H n its derivative with respect to the argument. kr and kz represent the radial and axial wave numbers, respectively. G zw ðr; ~r 0 Þ is equal to G wz ðr; ~r 0 Þ. A detailed analysis of Green’s functions is given in Tai (1994) and asymptotic solutions for the cylindrical case are outlined in Bird (1984) and Pathak and Wang (1981).

About the authors Ralf T. Jacobs received a Dipl.-Ing.(FH) from Fachhochschule Dusseldorf, Germany in 1994, a MSc from Heriot-Watt University, Edinburgh, UK in 1996, a Dipl.-Ing. from the University of Siegen, Germany in 1998, and the PhD from Heriot-Watt University in 2001, all in electrical engineering. After a Postdoctoral Research Fellowship in the Department of Mathematics at the University of Strathclyde, Glasgow, he joined the Brandenburg University of Technology at Cottbus, Germany, where he is a Research Associate. Ralf T. Jacobs is the corresponding author and can be contacted at: [email protected]. Arnulf Kost obtained the Dipl.-Ing. and Dr-Ing. degrees at the Technical University of Berlin in 1966 and 1973, respectively. He became Professor at the TU Berlin in 1978 and at the TU Cottbus in 1995. His research field is computational electromagnetics as well as electromagnetic compatibility. He published 160 scientific papers and a monograph Numerische Methoden in der Berechnung elektromagnetischer Felder with the Springer Verlag on this field. He was the

Chairman of COMPUMAG 95 in Berlin. Since, 2001, he is the President of the International COMPUMAG Society. Hajime Igarashi joined Hokkaido University, Japan, as a Research Associate in 1999. He was a Visiting Researcher in Technical University Berlin. Currently, he is with Graduate School of Information Science and Technology in Hokkaido University, where he is a Professor. Alan J. Sangster received a BSc in Electrical and Electronic Engineering in 1963, an MSc in 1964, and a PhD in 1967, all from the University of Aberdeen in Scotland. He spent four years with Ferranti Plc, Edinburgh, and three years with Plessey Radar Ltd, Cowes, UK. Since, 1972, he has been with Heriot-Watt University, Edinburgh, where he became Professor of Electromagnetic Engineering 12 in 1989. He is a Fellow of the Institution of Electrical Engineers (UK), and a Member of the Electromagnetics Academy (USA).

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

Characteristics of microstrip antennas 121

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

COMPEL 27,1

Adaptive time integration for electromagnetic models with sinusoidal excitation

122

Galina Benderskaya CST GmbH, Darmstadt, Germany

Herbert De Gersem Katholieke Universiteit Leuven, Kortrijk, Belgium, and

Wolfgang Ackermann and Thomas Weiland Technische Universita¨t Darmstadt, Institut fu¨r Theorie Elektromagnetischer Felder, Darmstadt, Germany Abstract Purpose – To provide a reliable numerical technique for the time integration of the electromagnetic models with sinusoidal excitation. Design/methodology/approach – The numerical integration of an electrotechnical problem is commonly carried out using adaptive time stepping. For one particular selected time step, Runge-Kutta (RK) adaptive integration methods deliver two approximations to the solution with different order of approximation. The difference between both is used to estimate the local error. Findings – Standard error-controlled RK time integration fails for electromagnetic problems with sinusoidal excitation when the adaptive time step selection relies upon the comparison of a main solution and an embedded solution where the difference of orders is one. This problem is overcome when the embedded solution differs by two orders of approximations. Such embedded solution is efficiently constructed by putting appropriate order conditions on the coefficients of the Butcher table. Originality/value – Using the technique proposed in the paper, electromagnetic problems with sinusoidal dynamics can also be effectively tackled. Keywords Electromagnetism, Magnetic devices, Time study Paper type Research paper

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 122-132 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836690

1. Introduction The numerical integration of an electrotechnical problem is commonly carried out using adaptive time stepping. This technique allows to reduce the computational time considerably guaranteeing at the same time the achievement of a user-prescribed tolerance for the local error (Gustafsson, 1994; Hairer et al., 1993; Hundsdorfer and Vewer, 2003; So¨derlind, 2002). Adaptive time integration based on Runge-Kutta (RK) methods is a well developed and widely highlighted research subject (Gustafsson, 1994, Hairer et al., 1993, Hundsdorfer and Vewer, 2003; So¨derlind, 2002). These adaptive methods have been shown to be reliable and efficient also for very large practical problems. For one particular selected time step, RK adaptive integration methods deliver two approximations to the solution with different order of approximation. The difference between both is used to estimate the local error (Hairer et al., 1993).

For some types of problems, the adaptive schemes commonly proposed in the literature deliver rather surprising results: namely, at time spans, where obviously the step size should be significantly reduced, it is increased. When one would expect an increase of the step size, it appears to be reduced. One of the test examples given below is a plate capacitor, one plate of which is supplied with the voltage V1(t) ¼ 100 V sin(2pft), f ¼ 50 Hz, whereas the other plate is grounded (Figure 1). A suitable mathematical model for this example is the electroquasistatic formulation
 › ð17wÞ þ s7w ¼ 0 7· ð1Þ ›t

Adaptive time integration

123

given in terms of the electric scalar potential w, the conductivity s and the permittivity 1 which is further discretized by the finite integration technique (Clemens et al., 2004). A transient simulation is carried out by a Singly Diagonally Implicit Runge-Kutta (SDIRK) method delivering the main solution of order three and the embedded solution of order two (Cameron, 1999). The solution for one degree of freedom as well as the corresponding dynamics of the step size selection are shown in Figure 2. At time instants

 1 1 þ n; n e N t¼ 4f 2f the excitation signal reaches an extremal value and exhibits a rather fast change. Though, the time step prediction increases the time step length. A similar behavior was observed for other RK adaptive methods.

V1 (t)

Figure 1. Electroquasistatic model: plate capacitor

COMPEL 27,1

124

A dedicated study was carried out to get more insight into the behavior of adaptive time integration schemes. The results show that the described phenomena take place when sinusoidal/cosinusoidal functions or their modifications are integrated. Our only finding in the literature that concerns this phenomenon was given in Wang et al. (2001): “However, in the case of sine and cosine drives . . . the fixed-time approach will be more efficient . . . ” Sinusoidal excitations are of primordial importance for transient simulations of electromagnetic devices. This paper is dedicated to the analysis of the unexpected behavior described above and to the development of a class of adaptive time integration scheme combined with a reliable time step prediction approach. 2. Adaptive RK methods 2.1 Index of a differential-algebraic equation The most general form of a differential-algebraic equation (DAE) is given by Fðt; x; x0 Þ ¼ 0

ð2Þ

where the Jacobian matrix ›F/›x0 may be singular. For system (2), the index along a solution x(t) is the minimum number of differentiations of the system which would be required to solve for x0 uniquely in terms of x and t (i.e. to define an ordinary differential equation (ODE) for x). Thus, the index is defined with a help of the overdetermined system Fðt; x; x0 Þ ¼ 0 dF 0 00 dt ðt; x; x ; x Þ

¼0 ð3Þ

.. . dm F 0 dt m ðt; x; x ;

. . . ; x mþ1 Þ ¼ 0

as the smallest integer m so that x0 in equation (3) can be solved for in terms of x and t. In practical applications, differentiations of the system (2) as in equation (3) are never computed. However, the notion of an index of a DAE is fundamental since it plays a crucial role in selecting an appropriate time integrator. 2.2×10

100

–3

1.8 j/V

∆t /s 0 1

Figure 2. Simulation results for SDIRK 3(2): (a) voltage distribution for one of the degrees of freedom; (b) selected time steps

–100

0.2 0

0.01

0.02 t /s (a)

0.03

0.04

0

0.01

0.02 t /s (b)

0.03

0.04

2.2 DAEs of index 1 Electroquasistatic and magnetoquasistatic formulations applied to a field model which may be accomplished by a circuit lead to constant coefficient DAEs of the form M

d xðtÞ þ KxðtÞ ¼ rðtÞ dt

ð4Þ

where M and K are coefficient matrices, r is a right hand side vector, x is a vector of unknowns (Nicolet and Delince´ 1996; Benderskaya et al., 2004). Time integrators for such systems cover a substantial part of the joint research on numerical time integration. 2.3 Classical time integration methods The u-type discretization scheme, which is traditionally applied to solve equation (4), reads: 1 Mðx nþ1 2 x n Þ þ Kðux nþ1 þ ð1 2 uÞx n Þ Dt

¼ ur nþ1 þ ð1 2 uÞr n :

ð5Þ

The lower indices n and n þ 1 correspond to the time instants tn and tnþ 1 ¼ tn þ Dt, respectively. Different choices of the parameter u lead to the following classical methods of numerical integration: u ¼ 1 gives the implicit Euler method, u ¼ 1/2 gives the Crank-Nicolson method, u ¼ 2/3 gives the Galerkin method. Although system (4) looks like a regular ODE system, in magnetoquasistatic formulations the M matrix may be singular turning equation (4) into a DAE formulation. In Benderskaya et al. (2004), it is shown that some of the classical u-type methods suffer from stability problems when they are applied for the numerical treatment of the DAE formulations of form (4). It was also proposed to use implicit adaptive RK time-integration methods for simulating such electromagnetic problems. 2.4 Implicit RK methods Using the initial condition x0 at t0, the ith stage of a time step of an implicit RK method is defined by X i ¼ x 0 þ Dt

s X

aij X0j :

ð6Þ

j¼1

The RK method follows from substituting Xi for x and X0i for x0 into system (4) for all stages i ¼ 1, . . . ,s. The solution of the corresponding system of equations leads to the stage derivatives X0i which are then inserted in x 1 ¼ x 0 þ Dt

s X

bi X0i

Adaptive time integration

ð7Þ

i¼1

to obtain the solution at the next time instant. For each RK method, the coefficient matrix A as well as the weight vector b are given by the Butcher table (Hairer et al., 1993). The order of the RK method is the real number p for which the Taylor series for the exact solution x(t0 þ Dt) and the approximate solution x1 coincide up to

125

COMPEL 27,1

126

(and including) the term Dt p, i.e.: kxðt 0 þ DtÞ 2 x 1 k # KDt pþ1

ð8Þ

where Dt is a time step and K is some constant (Hairer et al., 1993). For an RK method to be of one particular order, its coefficients have to satisfy so-called order conditions. The number of the order conditions to be put on the coefficients of the Butcher table to attain an RK-type integration scheme increases significantly with the required order (Table I) (Hairer et al., 1993). 2.5 Adaptive RK methods Together, with the solution x ( p) of order p delivered by a weight vector b, an embedded ^ solution x ðpÞ of a lower order pˆ is obtained by the application of an additional weight ˆ vector b. This allows to construct an error vector ^

y ¼ x ð pÞ 2 x ðpÞ

ð9Þ

which can be used for the adaptive determination of the time step length. A suitable norm for measuring the error vector reads: jyi j  ; kykerr ¼ max   ð pÞ  ð pÞ  i xi þ1x;i 

ð10Þ

where 1ðx;ipÞ is an absolute error tolerance for the component xði pÞ (Benderskaya et al., 2005). The DAE system (4) may describe a coupled formulation incorporating variables having different physical nature which have, consequently, different units. Therefore, it makes sense to choose different absolute tolerances for each solution component:   1ðx;ipÞ :¼ h max xði pÞ ðtÞ ð11Þ te½t 0 ;t i


where h e [102 2, 102 1]. The introduction of the absolute tolerances takes the history of the integration process into account. This leads to the more effective realization of the adaptive time stepping scheme. The adaptive time stepping scheme comprises criteria allowing to decide whether the last calculated integration step has to be repeated or a new simulation step can be performed. According to the standard rule aiming at keeping the error estimate kykerr close to the user-specified tolerance 1tol , the solution is rejected if kykerr . m1tol holds true and a new attempt is made with a smaller step size; otherwise the time step is accepted. In this scheme, m is an accelerating factor usually taken as 1.2 (Gustafsson, 1994).

Order p Order conditions Table I.

1 1

2 2

3 4

4 8

Number of order conditions for RK methods

5 17

6 37

7 85

8 200

9 486

10 1,205

Adaptive time integration

Finally, the length of a next time step is calculated using a formula
Dt nþ1 ¼ r

1tol kykerr

1=ðpþ1Þ ^ Dtn

ð12Þ

where r denotes a safety factor that is usually set to 0.9 (Lang, 1995).

127 3. Dedicated RK methods To our knowledge, all adaptive RK methods developed and implemented up to the present have in common that the difference between the order of the main solution and the order of the embedded solution is equal to one, e.g. SDIRK 3(2) method (Cameron, 1999), Merson 5(4), Zonneveld 4(3) (Hairer et al., 1993). All of them are reported to perform in an effective and reliable way without any exceptions. For problems with sinusoidal dynamics this is, however, not the case as explained below. The Taylor expansion of a sinusoidal function indicates the reason for the failure mentioned in the introduction: sinðvtÞ ¼ sinðvt0 Þ þ cosðvt 0 Þ½vðt 2 t 0 Þ
þ · · · þ

sinðnÞ ðvt 0 Þ ½vðt 2 t 0 Þ
n þ Rnþ1 ðvtÞ n! ð13Þ

where R(nþ 1) (vt) is the reminder. From equation (13) it follows that, e.g. at the point vt0 ¼ p/2 the expansion of the function of third order coincides with the expansion of second order. For common adaptive time integrators implemented, e.g. with SDIRK 3(2), this means that at this very point or in its vicinity no or a negligible difference between the lower and the higher order solutions is detected. As a consequence, the integration will proceed with an enormously large time step. This problem can be successfully overcome by constructing the RK methods where the difference between the main and the embedded orders is more than one. This technique is implemented for the SDIRK 3(2) method (Cameron, 1999) with the A matrix: pffiffiffi 3 2 1 2 2=2 0 0 0 pffiffiffi pffiffiffi 7 6 7 6 2=2 1 2 2=2 0 0 7 6 pffiffiffi pffiffiffi pffiffiffi ð14Þ 7: 6 7 6 523 2 4 226 1 2 2=2 0 4 pffiffiffi pffiffiffi pffiffiffi 5 2=3 þ 1=6 2=6 2 1=3 1=6 1 2 2=2 The fourth and the second rows of the matrix are used as weight vectors to construct a main and an embedded solution of order three and two, respectively. The abscissa vector c is defined as: ci ¼

i X

aij ;

i; j ¼ 1; . . .4:

j¼1

To obtain a weight vector delivering a first order embedded solution, only one order condition on the coefficients of the weight vector

COMPEL 27,1

b~ 5 ½b~ j
;

j ¼ 1; . . . ; 4

has to be satisfied (Table I) (Hairer et al., 1993): s X

128

b~ j ¼ 1:

ð15Þ

j¼1

From Table I one can observe that two order conditions should be applied to obtain a second order RK method. The first one is condition (15) guaranteeing a first order RK method. Another order condition is s X j¼1

1 b~ j cj ¼ : 2

ð16Þ

Conditions (15) and (16) together allow to describe all the variety of possible RK methods of order 2. Now, we turn back again to our technique. Condition (15) allows us to obtain an RK method of the first order. However, to be sure that a weight vector delivering a second order solution will not be obtained by accident, the following inequality has hold as well: s X j¼1

1 b~ j cj – : 2

ð17Þ

Solving together an algebraic system consisting of the equation (15) and inequality (17), the following family of the weight vectors delivering a first order solution is obtained: 8 > b~ 1 ¼ a > > pffiffi > > > pffiffi a > b~ 3 ¼ b – 22221 < 2 > b~ 4 ¼ g > > pffiffi > > > 221 > : b~ 2 ¼ 1 2 a 2 b 2 g – 1 2 a 2 22pffiffi2 a 2 g where a, b, g are parameters. 4. Simulation results The already introduced test example shown in Figure 1 is simulated again with SDIRK method having matrix A given by equation (14) and using an embedded solution of order one instead of order two, i.e. SDIRK 3(1). The corresponding weight vector b˜ is calculated using the technique proposed above and reads  T ~b ¼ 1 1 1 1 : 2 8 4 8

The solution of the higher order is delivered with the vector b which is equal to the fourth row of matrix A given by equation (14). From the results in Figure 3 one can conclude that the time marching process is now adequate: at the time instants t¼

1 1 þ n; 4f 2f

neN

Adaptive time integration

129

and in their vicinity very small time steps are performed. The asymmetry of the time step selection dynamics (Figure 3(b)) is explained by the fact that the adaptive time stepping scheme needs a certain time to recognize the behavior of the transient process and also due to the limits put on the length of the time step. Figure 4 shows the magnitude of the local error, i.e. the difference between the higher and lower order solutions for one selected degree of freedom in case when the simulation is carried out with SDIRK 3(2) method (Figure 4(a)) and with SDIRK 3(1) method (Figure 4(b)). As one can conclude from the results, the SDIRK 3(1) method detects the highest value for the error at the time points t¼

1 1 þ n; 4f 2f

neN 10 ×10

100

–4

8 ∆ t /s 6

j/V 0

4 2 –100

0

0.01

0.02 t /s

0.03

0.04

0

0.01

(a)

0.02 t /s

0.03

0.04

Figure 3. Simulation results for SDIRK 3(1): (a) voltage distribution for one of the degrees of freedom; (b) selected time steps

(b)

0.18 0.06 0.12 yi

0.04 yi 0.06

0.02

0

0

0.01

0.02 t /s (a)

0.03

0.04

0

0

0.01

0.02 t /s (b)

0.03

0.04

Figure 4. Magnitude of the local error: (a) simulation with SDIRK 3(2) method; (b) simulation with SDIRK 3(1) method

COMPEL 27,1

130

and in their vicinity. The SDIRK 3(2) method, however, does not exhibit an adequate behavior for this case. With the user prescribed tolerance 1tol put to 102 3, the total number of time steps for SDIRK 3(2) method is 64, of which 17 steps are rejected. For SDIRK 3(1) 91 steps are computed, but only six are rejected in case when 1tol is equal to 102 2. This allows to conclude, that in spite of the increase in the total number of time steps, the SDIRK 3(1) method is more suitable for the simulations of models with sinusoidal excitation. The usage of the bigger tolerance 1tol ¼ 102 2 in the simulation with SDIRK 3(1) follows from the necessity to avoid extremely small time steps since the magnitude of the error (Figure 4), and since the value of kykerr, increase significantly leading to the decrease of Dtnþ 1 (as in formula (12)). As a second test example consider an electrical circuit shown in Figure 5 where the resonance frequency is determined by the system itself and not by the excitation. The following parameters are used in the simulation: u(t) ¼ U(1 2 e2 t/t), where U ¼ 1 V and t ¼ 0.1 ms. The switch is initially closed and is opened at tswitch ¼ 5 s. The parameters of the circuit elements are as follows: L ¼ 1 mH, C ¼ 10 m F and R ¼ 1 mV. The time span corresponding to the closed switch is simulated with a SDIRK method having matrix A given by equation (14) and using an embedded method of order two specified by the last row of matrix A. With the user specified tolerance 1tol put to 102 3, the number of the accepted and rejected steps is 30 and 2, respectively. The simulation results are shown in Figure 6(a). After 5 s, the switch is opened, and the simulation is continued with the SDIRK 3(1) method, vector b˜ being the same as the one we used in the first test example. The obtained simulated current in the inductor is shown in Figure 6(b). For this part of the simulation, 1tol is equal to 102 2 and the number of the accepted/rejected steps is 40 and 8, respectively.

Figure 5. Second test problem: electrical network

u (t)

+

R

s (t)

1,000

1,000

I/A

I/A 500

Figure 6. Simulated current through an inductor: (a) integration is started with SDIRK 3(2); (b) integration is continued with SDIRK 3(1)

0

L

C

–

0

–1,000 0

2.5

5

5

5.0005

t /s

t /s

(a)

(b)

5.001

5. Conclusion Error-controlled adaptive RK time integrators offer efficient and reliable time step selection for electromagnetic problems with sinusoidal excitation when the difference between the approximation orders of the main and the embedded solution is two. This is not the case for standard time integrators with a difference of one as shown by the examples. Standard error-controlled RK time integration fails for electromagnetic problems with sinusoidal excitation when the adaptive time step selection relies upon the comparison of a main solution and on embedded solution where the difference of orders is one. This problem is overcome when the embedded solution differs by two orders of approximations. Such embedded solution is efficiently constructed by putting appropriate order conditions on the coefficients of the Butcher table. The problem highlighted in the paper is not specific only for the purely sinusoidal functions. It is obviously that the described phenomenon also occurs for the excitation functions that are approximately sinusoidal. Using the technique proposed in the paper, such problems can also be effectively tackled. References Ascher, U.M. and Petzold, L. (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, PA. Benderskaya, G., Clemens, M., De Gersem, H. and Weiland, T. (2005), “Embedded Runge-Kutta methods for field-circuit coupled problems with switching elements”, IEEE Transactions on Magnetics, Vol. 41 No. 5, pp. 1612-5. Benderskaya, G., De Gersem, H., Clemens, M. and Weiland, T. (2004), “Transient field-circuit coupled formulation based on the FI-technique and a mixed circuit formulation”, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 23 No. 4, pp. 968-76. Cameron, F. (1999), Low-order Runge-Kutta Methods for Differential-Algebraic Equations, Tampere University of Technology, Tampere. Clemens, M., Wilke, M., Benderskaya, G., De Gersem, H., Koch, W. and Weiland, T. (2004), “Transient electro-quasistatic adaptive simulation schemes”, IEEE Transactions on Magnetics, Vol. 40 No. 2, pp. 1294-7. Gustafsson, K. (1994), “Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods”, ACM Transactions on Mathematical Software, Vol. 20 No. 4, pp. 496-517. Hairer, E., Nørsett, S.P. and Wanner, G. (1993), Solving Ordinary Differential Equations I. Nonstiff Problems, Springer, Berlin. Hundsdorfer, W. and Vewer, J. (2003), Numerical Solution of Time-dependent Advectiondiffusion-reaction Equations, Springer, Berlin. Lang, J. (1995), “Two-dimensional fully-adaptive solutions of reaction-diffusion equations”, Applied Numerical Mathematics, Vol. 18, pp. 223-40. Nicolet, A. and Delince´, F. (1996), “Implicit Runge-Kutta methods for transient magnetic field computation”, IEEE Transactions on Magnetics, Vol. 32 No. 3, pp. 1405-8. So¨derlind, G.S. (2002), “Automatic control and adaptive time-stepping”, Numerical Algorithms, Vol. 31, pp. 281-310. Wang, H., Taylor, S., Simkin, J., Biddlecombe, C. and Trowbridge, B. (2001), “An adaptive-step time integration method applied to transient magnetic field problems”, IEEE Transactions on Magnetics, Vol. 37 No. 5, pp. 3478-81.

Adaptive time integration

131

COMPEL 27,1

132

About the authors Galina Benderskaya received the diploma degrees in Applied Mathematics and English linguistics from the Ulyanovsk State University, Russia, in 1998. Since August 2002, she was with the Institut fu¨r Theorie Elektromagnetischer Felder at the Technische Universita¨t Darmstadt, Germany, where in 2007 she received the PhD degree in electrical engineering. Since April 2007, she is with the CST (Computer Simulation Technology) GmbH, Germany, working as a Software Development Engineer. Galina Benderskaya is the corresponding author and can be contacted at: [email protected] Herbert De Gersem received the MSc and PhD degrees in electrical engineering from the Katholieke Universiteit Leuven, Belgium, in 1994 and 2001. From 2001 till 2006, he was with the Institut fu¨r Theorie Elektromagnetischer Felder at the Technische Universita¨t Darmstadt, Germany, working as Assistant Professor. Since October 2006, he is an Associate Professor at the Katholieke Universiteit Leuven, Belgium. E-mail: [email protected] Wolfgang Ackermann received the diploma and PhD degrees in electrical engineering from the Universita¨t Siegen, Germany, in 1995 and 2002. From 1995 till 1997 he worked for the interdisciplinary research project “Deep Sea Mining” forming a cooperation between the electrical and mechanical engineering faculties at the Universita¨t Siegen. Since October 2002, he is with the Institut fu¨r Theorie Elektromagnetischer Felder at the Technische Universita¨t Darmstadt, Germany, working as an Assistant Professor. E-mail: [email protected] Thomas Weiland received the MSc and PhD degrees in electrical engineering from the TH-Darmstadt, Germany, in 1975 and 1977. He worked at CERN in Geneva, Switzerland and DESY, Hamburg, Germany where he became head of the Electromagnetic Field Group in 1982. Since 1989, he is Full Professor and Director of the Institut fu¨r Theorie Elektromagnetischer Felder at the Technische Universita¨t Darmstadt. He is Founding President of the CST GmbH, Germany. Weiland holds the Leibniz prize of the German Research Society, the Max Planck research prize and the Philip Morris research prize. E-mail: [email protected]

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

Computational methods for modeling of complex sources Markus Johansson, Lovisa E. Nord, Rudolf Kopecky´, Andreas Fhager and Mikael Persson

Methods for modeling of complex sources 133

Department of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden Abstract Purpose – The purpose of this study is to develop and compare two methods of determining the total field, including phase information, when only field amplitudes have been measured on a set of planes in the near field of a complex electromagnetic source. Design/methodology/approach – The first method is a gradient-based optimization algorithm, based on the adjoint fields. The second method employs an optimization algorithm based on the phase angle gradients of a functional. Findings – The first method, the adjoint field method, is functioning well for a 2D test case. The second method, the phase angle gradient method, gives very good results for 3D test cases. Research limitations/implications – The next step is to test the methods with results from real measurement data. Practical implications – The developed methods are intended for use in dosimetry studies and other applications, where the field distribution from electromagnetic sources are needed. Originality/value – The methods extend previously made constant phase approximations. The present methods are useful in situations where the electromagnetic source is hard to model. Keywords Optimization techniques, Electric fields, Electromagnetism Paper type Research paper

Introduction Electromagnetic near-field modeling of complex sources is useful in various applications, including antenna design and dosimetry studies. In order to determine whether the exposure safety guidelines, such as the EU directive 2004/40/EC, are complied with, it is important to be able to model the field distribution from electromagnetic sources. For this type of problems, source modeling is used to couple the measurements with the numerical calculations. In this study, two different methods to achieve these models are presented. Measured fields on a Huygens surface enclosing a source can be numerically propagated to arbitrary locations on the outside of the surface. The Huygens surface can be approximated by a sufficiently large planar surface positioned between the source and the near-field area of interest. If the total field, that is phase and amplitude, is known on the surface, the field beyond this can be obtained numerically. Methods that utilize the total measured field in the near-field area of a source to determine an equivalent magnetic current source, have been presented by Taaghol and Sarkar (1996), and more recently by Las-Heras et al. (2006). Generally, however, only the RMS values of the field are measured. The methods presented in this paper attempt to The authors would like to thank The Swedish Labour Market Insurance, AFA for their financial support to this project.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 133-143 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836708

COMPEL 27,1

134

recreate the phase information using the measured RMS values on a set of planes in the near field of the source, thereby finding the desired total field (Figure 1). This work is applicable, but not limited, to wireless base station dosimetry, and extends the constant phase approximations made by Gandhi and Lam (2003). The adjoint field method Two different methods have been developed. The first method is based on the adjoint fields, and involves finding equivalent dielectric properties between two parallel planes, e.g. planes 1 and 2 in Figure 1. If the measured electric field amplitudes on plane 1 are used as source with the field components set to be in phase, these properties should yield the correct total field, phase as well as amplitude, on plane 2. The numerical FDTD scheme is used for the calculations, and a gradient-based optimization algorithm is used to find the dielectric properties that give the correct field amplitudes and phase on the planes 2 and 3. Versions of this type of optimization algorithm is used in tomography and can be found in Takenaka et al. (1997), Gustafsson and He (2000) and Fhager and Persson (2005). As a first step towards a 3D method, we use the optimization algorithm in 2D to try to find the equivalent dielectric properties using the field amplitudes on planes 2 and 3, which actually looks like lines in 2D symmetry. In the original (Fhager and Persson, 2005) optimization algorithm the functional to be minimized is: Z TX M X N 


E
m ð1; s; R
n ; tÞ 2 E
meas ðR
n ; tÞ
2 dt; ð1Þ Fð1; sÞ ¼ m 0

m¼1 n¼1

meas where E
m ð1; s; R
n ; tÞ is the calculated field and E
m ðR
n ; tÞ the measured field on

Source

Figure 1. Source in front of planes where field amplitudes were measured

3 2 1

planes 2 and 3. M is the number of sources and N is the number of measurement points on planes 2 and 3. The minimization is done with a conjugate-gradient algorithm. The Fre´chet differential of the functional can be written: F 0 ð1; sÞ ¼ kG1 ð
xÞ; ›1l þ kGs=ksl ð
xÞ; ›sl;

ð2Þ

where the inner product is defined as a surface integral over the area where equivalent dielectric properties are searched for: ðð að
x Þ; bð
x Þ ¼ að
xÞbð
xÞdS: ð3Þ h i S

The gradients can be written as: M Z X G1 ð
xÞ ¼ 2 m¼1

T


~ m ð1; s; x
; tÞ · ›t E
m ð1; s; x
; tÞdt E

ð4Þ

0

M Z X Gs=hsi ð
xÞ ¼ 2hsi

T


~ m ð1; s; x
; tÞ · E
m ð1; s; x
; tÞdt; E

ð5Þ

m¼1 0

where E
m ð1; s; x
; tÞ is the numerically computed E-field in the area where equivalent
~ m ð1; s; x
; tÞ is the solution to the adjoint dielectric properties are searched for and E problem, ksl is a parameter compensating for the different scaling of the gradients. meas We have here modified the algorithm slightly so that the phase of E
m ðR
n ; tÞ is set

to be equal to that of Em ð1; s; Rn ; tÞ at each iteration step of the procedure. The meas amplitudes of E
m ðR
n ; tÞ are obtained from the measurements. The phase angle gradient method The second method developed makes use of the equivalent magnetic surface current density on plane 1:

s ¼ 22n^ £ E; M

ð6Þ

where n^ is a unit vector perpendicular to plane 1 pointing towards the other planes and E
is the total electric field on plane 1, to calculate the electric fields on planes 2 and 3.
s in free space is: The electric vector potential from M ðð 1
s ð
r0 Þ Gð
r; r
0 Þds 0 : ð7Þ M F
¼ 4p S Here, the surface S is plane 1, 1 is the permittivity and Gð
r; r
0 Þ is the Green’s function: 0

e 2jkjr
2
r j Gð
r; r
Þ ¼ ; jr
2 r
0 j 0

ð8Þ

where k is the wavenumber. If the surface S is divided into a square grid with the area DS, grid cell the integral can be approximated as: 1 X
0 ð9Þ F
< Ms ð
r p ÞGð
r; r
0p ÞDS; 4p p

Methods for modeling of complex sources 135

COMPEL 27,1

136

where r
0p represents the point in the middle of the grid cell number p. The electric field
If the can then be calculated using equation (9) and the relation between E
and F. system of coordinates is chosen such that plane 1 is the xy-plane, the resulting expressions are: Ex ¼ 2

›Gð
r; r
0p Þ DS X E x ð
r 0p Þ 2p p ›z

ð10Þ

Ey ¼ 2

›Gð
r; r
0p Þ DS X E y ð
r 0p Þ 2p p ›z

ð11Þ

  ›Gð
r; r
0p Þ ›Gð
r; r
0p Þ DS X 0 0 Ez ¼ þ E y ð
r p Þ E x ð
r p Þ : 2p p ›x ›y

ð12Þ

If plane 1 is chosen large enough and DS small enough, equations (10), (11) and (12) give good approximations for the fields on planes 2 and 3. The field amplitudes are known on plane 1, so the field on the other planes can be regarded as a function of the unknown phase angles of the tangential components of E
on plane 1. After the phase angles on plane 1 have been initiated, e.g. to be zero, the resulting field estimates on planes 2 and 3 can be calculated. To find the correct phase, the initial angles are altered in small steps, so that the field amplitudes

j E i jn ; where n is a
computational grid point, converge to the measured values
E m i n : A functional J of the phase can be defined as:  
m
2 



m

2 
m
2 1X 





J; : ð13Þ j E x jn 2 E x n þ E y n 2
E y
þ j E z jn 2 E z n 2 n n The phase angles are changed in the opposite direction of the phase angle gradients of J, so that J is minimized. To obtain expressions for the gradients, the analytical derivatives of J with respect to the phase angles in the measurement points on plane 1, is calculated. The phase angle gradient method is suitable for parallelization as many iterations are needed for convergence. Results To evaluate the two methods, field values from an analytical formula for an infinitesimal dipole were used instead of the measurements. The frequency of 1 GHz was used. The plane closest to the source was 60 £ 60 cm and the other two planes were both 30 £ 30 cm. The distances from plane 1 to 2 and from plane 2 to 3, in Figure 1, were 5 cm and 6 cm, respectively, in the test case for the phase angle gradient method. In the test case that was used to test the adjoint field method, the distance between the plane closest to the source and the next plane was increased to 17 cm. Results for the adjoint field method Figure 2 shows the 2D test case, that was used for the adjoint field method. Lines, instead of planes, each as long as the width of the corresponding planes in Figure 1 were used in the test case.

Methods for modeling of complex sources

equivalent dielectric properties

137 source

plane 1

plane 2

Figure 2. Test case for the adjoint field method

equivalent source plane

Figure 3 shows the E-field amplitudes on planes 1 and 2, in Figure 2, if the correct total field on the equivalent source plane with correct phase is used. It also shows the obtained results from the developed method using only the RMS values as source together with the calculated dielectric properties of the plate in front of the source. In Figure 4, the correct phase angles and the phase angles that the adjoint field method gives are shown. .

E field amplitude

0.38 0.36 0.34 0.32 0.3 0

10

20

30 40 position (plane 1)

50

60

70

E field amplitude

0.31 0.3 0.29 0.28 0.27 60

correct amplitudes amplitudes with equivalent dielectric properties

70

80

90 100 position (plane 2)

110

120

130

Figure 3. E-field amplitudes

COMPEL 27,1

1.6

phase

1.4

138

1.2 1 0.8 0

10

20

30

40

50

60

70

120

130

position (plane 1) 0.4

phase

0.2 0 – 0.2

Figure 4. Phase angles

– 0.4 60

correct phase phase with equivalent dielectric properties

70

80

90 100 position (plane 2)

110

It can be seen that the calculated dielectric properties give, amplitudes that are similar to the correct amplitudes and phase angles very close to the correct angles. Results for the phase angle gradient method Figure 5 shows the test case, that was used for the phase angle gradient method. The initial values of all the phase angles were set to zero and the phase angle gradient method was used to calculate the phase angles. The system of coordinates was chosen such that plane 1 was the xy-plane. Figure 6 shows the calculated phase angles for the y-component of the E-field on plane 2. In Figure 7, the correct phase angles for Ey on plane 2 are shown. Figures 8 and 9 show the difference between calculated and correct phase angles for the y-component and the x-component of the E-field. It can be seen that the phase angle gradient method gave phase angles that are close to the correct ones. In Figure 9, some points have large differences between the calculated and correct phase angles. These generally occur when the amplitude of Ex is small. Since, the amplitudes are small, the larger differences are not important. It is also reasonable that the differences in such points are larger, as the errors in the phase angles, in points were the amplitudes are small, do not change the functional minimized much. The functional J as function of iteration number is shown in Figure 10 showing excellent convergence. In Figure 11, the summed and weighted phase angle error:

ferror

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2

 2  uP  2 u 2 diff diff

E f þ E f u n j x jn x;n y n y;n u  ¼t

P  2 2 jE xj þ
E y
n

n

n

ð14Þ

Methods for modeling of complex sources 139

1 equivalent source plane

2

3

source

Figure 5. Test case for the phase angle gradient method

Phase angles for Ey from program 1.5

2

phase [rad]

1.5

1

1 0.5 0.5

0 –0.5 30 30

20 20 y [cm]

10

0

10 0

0

x [cm]

as function of iteration number is shown. fdiff x;n is the error in the calculated phase angle for Ex in the measurement point on plane 2 or 3 with number n and fdiff y;n is the corresponding error for Ey. The average is weighted with the amplitudes, as it is more important that errors are small in points where the amplitude is large, than in points with small amplitudes. It can be seen in Figure 11 that the number of iterations can be significantly reduced.

Figure 6. Calculated phase angles for Ey

COMPEL 27,1

Correct phase angles for Ey

1.4 2

140

1.2

phase [rad]

1.5

1

1

0.8

0.5

0.6

0

0.4 0.2

– 0.5 30 30

20 20

10

Figure 7. Correct phase angles for Ey

y [cm]

10 0

0 –0.2

x [cm]

0

Difference phase angles Ey 0.15

phase difference [rad]

0.25 0.2

0.1

0.15 0.1 0.05

0.05 0 – 0.05

0

– 0.1 30

Figure 8. Difference between calculated and correct phase angles for Ey

30

20 20 y [cm]

10

10 0 0

–0.05

x [cm]

The phase angle gradient method was also tested for the frequency 100 MHz, instead of 1 GHz. Figure 12 shows the weighted phase angle error as function of iteration number for that case.

Difference phase angles Ex 0.2 0.1

0.4 phase difference [rad]

Methods for modeling of complex sources

0

0.2 0

141

–0.1

–0.2

–0.2

–0.4 –0.3 –0.6 30 30

20

–0.4

20 y [cm]

10

10 0 0

–0.5 x [cm]

Figure 9. Difference between calculated and correct phase angles for Ex

J

101 100 10–1 10–2 10–3 10–4 10–5

0

1

2

3

4

5 6 iteration

7

8

9

10 × 104

It can be seen that much less iterations were needed for the 100 MHz case, than for the 1 GHz case. The computation time was 4 h 16 min with 20 processors, for the 1 GHz case and 4 min 8 s with 40 processors, for the 100 MHz case. Conclusions The phase angle gradient method is very promising. It gave good results for the test cases and will be tested further. The next step is to test the method with results from

Figure 10. J as function of iteration number

COMPEL 27,1

Weighted phase angle error (rad)

100

142 10–1

Figure 11. Weighted phase angle error as function of iteration number for the 16Hz case

10–2

0

1

2

3

4

5 6 iteration

7

8

9

10 × 104

Weighted phase angle error (rad)

100

10–1

Figure 12. Weighted phase angle error as function of iteration number for the 100 MHz case

10–2

0

0.2

0.4

0.6

0.8

1 1.2 iteration

1.4

1.6

1.8

2 × 104

real measurements. The adjoint field method is functioning well for the 2D test case and a 3D implementation of the method is planed. The developed methods are very useful in situations with an electromagnetic source that is hard or time consuming to make an accurate model of. Both methods have the important advantage that they can be used for many different sources without detailed knowledge of the sources. It therefore shows promise to become a versatile tool for the field dosimetry engineer.

References Fhager, A. and Persson, M. (2005), “Comparison of two image reconstruction algorithms for microwave tomography”, Radio Science, Vol. 40, S3017. Gandhi, O.P. and Lam, M.S. (2003), “An on-site dosimetry system for safety assessment of wireless base stations using spatial harmonic components”, IEEE Transactions on Antennas and Propagation, Vol. 51 No. 4, pp. 840-7. Gustafsson, M. and He, S. (2000), “An optimization approach to two-dimensional time domain electromagnetic inverse problems”, Radio Science, Vol. 35 No. 2, pp. 525-36. Las-Heras, F., Pino, M.R., Loredo, S., Alvarez, Y. and Sarkar, T.K. (2006), “Evaluating near-field radiation patterns of commercial antennas”, IEEE Transactions on Antennas and Propagation, Vol. 54 No. 8, pp. 2198-207. Taaghol, A. and Sarkar, T.K. (1996), “Near-field to near/farfield transformation for arbitrary near-field geometry, utilizing an equivalent magnetic current”, IEEE Transactions on Electromagnetic Compatibility, Vol. 38 No. 3, pp. 536-42. Takenaka, T., Tanaka, T., Harada, H. and He, S. (1997), “FDTD approach to time-domain inverse scattering problem for stratified lossy media”, Microwave and Optical Technology Letters, Vol. 16 No. 5, pp. 292-6. About the authors Markus Johansson received the MSc degree in electrical engineering from The Royal Institute of Technology, Stockholm, Sweden in 2003. He is currently working towards the PhD degree at Chalmers University of Technology in Gothenburg, Sweden. His research interests include computational electromagnetics and dosimetry. Markus Johansson is the corresponding author and can be contacted at: [email protected] Lovisa E. Nord received the MSc degree in engineering physics from Lund Institute of Technology, Lund, Sweden, in 2005. She used to be a PhD student at Chalmers University of Technology in Gothenburg, but interrupted her studies in 2006 and is currently working as a Microwave Engineer at Applied Composites AB ACAB in Linko¨ping, Sweden. Rudolf Kopecky´ received the MSc degree in biomedical engineering from Czech Technical University, Prague, Czech Republic in 2000 and the PhD degree at Chalmers University of Technology, Gothenburg, Sweden in 2006. His research interest includes use of the FDTD method for detailed RF dosimetry in biological tissues. Andreas Fhager received the MSc in engineering physics, 2001, the Licentiate of Technology degree, 2004, and the PhD degree, 2006, in microwave imaging from Chalmers University of Technology, Gothenburgh, Sweden. Currently he is an Assistant Professor at Chalmers working with microwave imaging methods for biomedical applications, e.g. breast cancer detection. Mikael Persson received the MSc and PhD degrees from Chalmers University of Technology, Gothenburg, Sweden, in 1982 and 1987, respectively. He is Professor of electromagnetics at Chalmers University of Technology. He is the author or coauthor of over 100 journal and conference papers in fusion and computational electromagnetics.

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

Methods for modeling of complex sources 143

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

COMPEL 27,1

144

Fundamental investigation of 3D optimal design of open type magnetic circuit producing uniform field Norio Takahashi and Koji Akiyama Department of Electrical and Electronic Engineering, Okayama University, Okayama, Japan, and

Hirokazu Kato and Kanji Kishi Medical School, Okayama University, Okayama, Japan Abstract Purpose – To provide an approach to design the optimal open type magnetic circuit of permanent magnet producing uniform field. Design/methodology/approach – The Biot-Savart’s law and evolution strategy are used for the initial design of permanent magnet configuration. In order to improve the uniformity, the ON/OFF method, which is the topology optimization method, is used for determining the shape of magnetic material which is set around the permanent magnet. Findings – The optimal topology of permanent magnet and shape of magnetic material around it, which can produce nearly uniform field of about 0.15T, is obtained. The obtained uniformity is 3,583 ppm. More work for improving the uniformity is necessary. Originality/value – A new approach for obtaining the optimal shape of open type magnetic circuit which may be used for magnetic resonance imaging is carried out. Keywords Optimization techniques, Electromagnetism, Numerical analysis Paper type Research paper

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 144-153 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836717

1. Introduction The development of an open type magnetic circuit of permanent magnet having uniform field is required for the magnetic resonance imaging (MRI) equipment for clinical practice (Yao et al., 2005) or interventional diagnostic such as a mobile universal surface explorer (Popella and Henneberger, 2001). Traditional magnets for MRI contain two pole pieces with intervening support members so that they all have the limited surgeon access. Then, an open architecture permanent magnet MRI is proposed (Yao et al., 2005), where a patient to be imaged is supposed to be located in one side of a planar or many-sided surface of permanent magnet system. But, it is not easy to get a sufficiently strong and homogeneous magnetic field required at the imaging region. In this paper, firstly, a magnet system is optimized using the novel optimization technique, evolution strategy (ES), combined with the Biot-Savart’s method. Next, the construction of magnetic material surrounding the magnet is optimized using the ON/OFF method combined with the 3D finite element method (Okamoto et al., 2006). A preliminary result of open type magnetic circuit having uniform field is illustrated.

2. Initial design using Biot-Savart’s law and evolution strategy 2.1 Model and method of analysis Firstly, a magnet system composed of three magnets shown in Figure 1 is optimized using the Biot-Savart’s law for analyzing the magnetic field of permanent magnet combined with the ES, so that the flux in the imaging region (called as a target region) becomes uniform. The strength of magnetization of magnet is 1.36T. The flux density B at a point P is given using the Biot-Savart’s law as follows (Abele, 1993): B¼2

1 7P 4p

ððð


  1 M·7 dV r V

Investigation of 3D optimal design 145

ð1Þ

where, 7P is the differential operator in the point P, V is the volume of magnet, r is the distance between the magnet and the point P, M is the magnetization of magnet. The following equation can be obtained from equation (1): 8 9 Bx > > > = < > By ¼ 2 1 > > 4p > ; : Bz >

2 ›I

x

›xP

6 6 ›I x 6 ›yP 6 4 ›I x

›zP

›I y ›xP ›I y ›yP ›I y ›zP

3

9 8 Mx > > > 7> = < ›I z 7 7 M y ›yP 7 > > > 5> ›I z : M z ;

›I z ›xP

ð2Þ

›zP

L4

y

L3

M θ M

M θ x

L 1 L2 (a) x-y plane

z L2

80

L1

x

(b) x-z plane

Figure 1. Model for initial design

COMPEL 27,1

where, Bx, By and Bz are the x-, y- and z-components of flux density. Ix, Iy, Iz are the x-, yand z-components of the calculated integrals in equation (1): ððð
 1 I¼ 7 dv ð3Þ r v

146

By assuming that the range of integral of brick magnet is x1 % x % x2, y1 % y % y2, z1 % z % z2, each component of equation (2) can be given by:    2 X 2 X 2 ›I x X ð yP 2 yj ÞðzP 2 zk Þ ¼ ð21Þiþjþk · 2tan21 ðxP 2 xi Þr ijkP ›x P i¼1 j¼1 k¼1

ð4Þ

   2 X 2 X 2 ›I y X iþjþk 21 ðzP 2 zk ÞðxP 2 xi Þ ¼ ð21Þ · 2tan ð yP 2 yj Þr ijkP ›yP i¼1 j¼1 k¼1

ð5Þ

   2 X 2 X 2 ›I z X ðxP 2 xi Þð yP 2 yj Þ ¼ ð21Þiþjþk · 2tan21 ðzP 2 zk Þr ijkP › zP i¼1 j¼1 k¼1

ð6Þ

2 X 2 X 2 ›I x X ¼ ð21Þiþjþk · ln{r ijkP þ ðzP 2 zk Þ} ›yP i¼1 j¼1 k¼1

ð7Þ

2 X 2 X 2 ›I x X ¼ ð21Þiþjþk · ln{r ijkP þ ð yP 2 yj Þ} › zP i¼1 j¼1 k¼1

ð8Þ

2 X 2 X 2 ›I y X ¼ ð21Þiþjþk · ln{r ijkP þ ðxP 2 xi Þ} › zP i¼1 j¼1 k¼1

ð9Þ

where, xP, yP and zP are the coordinates of point P of which the flux density is calculated. rijkP is the distance between the point P and the corner point (xi, yj, zk) of brick magnet. 2.2 Design goal and obtained result The design goal of open-type MRI is to provide a uniform flux distribution of specified flux density or stronger field in the target region. The functions W1 and W2 in the target region to be minimized are given as: ððð   B2y þ B2x dV ð10Þ W1 ¼ V

W2 ¼

ððð

ðBx 2 Bx0 Þ2 dV

ð11Þ

V

where V is the volume of target region, Bx, By, Bz are the x, y, z-components of flux density and Bx0 is the specified value of the x-components of flux density (¼ 0.15T) in the target region. The objective function W is the linear combination of W1 and W2 as follows: W ¼ k1 W 1 þ k2 W 2

ð12Þ

where k1 and k2 are the weighting coefficients. The lengths L1 , L4 and the angle of magnetization vector of magnet are optimized using the ES. The coefficients k1 and k2 are adjusted at each step so that the order of W1 is the same with that of W2. The obtained dimension of magnet and the angle of magnetization are shown in Figure 2. Table I shows the average flux density Bxave and the uniformity u in the target region. The uniformity u is defined by: u¼

Bx max 2 Bx min Bx ave

ð13Þ

3. Optimization using on/off method 3.1 Model for optimization In order to improve the uniformity of flux distribution, the dimension of permanent magnet system obtained in Section 2 is fixed. The dimension is slightly changed so that the mesh of brick elements of ON/OFF method can be produced. Then, the specified value of Bx0 is changed to 0.15909T. The shape of the magnetic material (design domain, see the gray part in Figure 2) which are set around it is optimized using the ON/OFF method. Figure 3 shows the FEM model. The shape of finite element in the design domain is not changed, but the material of it (iron or air) is changed. The FEM model is composed of edge-based brick elements of first order (123,234 elements, 131,040 nodes, 385,158 edges, 360,924 unknowns). The design domain is subdivided into 16,288 cubes of 2 £ 2 £ 2 mm. The directions of magnetization of magnet 1, 2 and 3 are þ 3.648, 08 and 2 3.648, respectively. The material used in the design domain is the carbon steel S45C. 3.2 ON/OFF method In the ON/OFF method, the region where the topology (shape of magnetic material) should be determined is subdivided into regular mesh of brick elements. The shape of mesh is not changed. Only the material of it is changed. If the objective function W is reduced by assuming the material of an element as an iron, the material of it should be changed to the iron (ON). If W is increased by assuming the material of an element as an iron, the material of it should not be changed and remain air (OFF). The sensitivity analysis is used to determine whether the magnetic material is located in the design domain or not. The sensitivity is accurately calculated by using the adjoint variable method. The equation for FEM is given as:

Investigation of 3D optimal design 147

COMPEL 27,1 y design domain

8

26 44

26 98

3.64°

M

M

4 10

84

10

18

4

148

target region

M 3.64° 10

4

x magnet 1

magnet 2

magnet 3

(a) x-y plane z

design domain

target region 22

26

14

80

4

8

x

magnet 2

Figure 2. Obtained magnet and design domain

magnet 3

(b) x-z plane

HA ¼ G

ð14Þ

where H is the coefficient matrix, A is the column vector of magnetic vector potential, and G is the right-hand side vector. Taking the derivative of equation (14) with design variable pk in an element k:

›H ›A ›G AþH ¼ ›p k ›pk ›pk

ð15Þ

Equation (15) can be rewritten as follows:
 ›A ›G ›H ~ A ¼ H 21 2 ›pk ›pk ›pk

Investigation of 3D optimal design 149

ð16Þ

~ is the value obtained by solving equation (14). If the objective function is where A expressed as the function W( pk, A) of the permeability in design domain and the magnetic vector potential, the sensitivity is given by: dW ›W ›W T ›A ¼ þ dpk ›p k ›A ›p k

Initial Optimal

ð17Þ

Bxmin (T)

Bxmax (T)

Bxave (T)

u

2 0.00780 2 0.15137

2 0.08768 2 0.15233

20.04161 20.15180

1.9195 0.0063

Table I. Flux density and uniformity (initial design)

magnet 2 design domain

z

x

y magnet 3 target region

Figure 3. FEM model

COMPEL 27,1

150

Substituting equation (16) into equations (17) and (18) can be obtained:
 dW ›W ›W T 21 ›G ›H ~ A ¼ þ H 2 dpk ›p k ›A ›pk ›pk

ð18Þ

In order to avoid the calculation of the inverse of H, an adjoint vector l is introduced. The adjoint equation is given by: HT l ¼

›W ›A

ð19Þ

l is obtained by solving equation (19), and is calculated by substituting l into equation (20).
 dW ›W ›H ~ T ›G A ð20Þ ¼ þl 2 dpk ›pk ›pk ›pk Equation (20) suggests that only one extra solution for the adjoint vector is sufficient to determine the sensitivity to all parameters, rather than obtaining each value per parameter, providing a computationally fast method for deriving the gradients. In this analysis, the permeability in an element of design domain is treated as a design variable pk. In the ON/OFF method, the magnetic material is located in the element i if the sensitivity dW/dni is positive. On the other hand, the element i with negative sensitivity is regarded as the air. 3.3 Process of calculation The flow chart of the ON/OFF method is shown in Figure 4. . Step 1: decision of initial topology. The initial topology in the design domain is decided by the material arrangement in each element. . Steps 2 and 7: FEM ! W. The objective function W of the design domain is obtained from the calculated value using FEM. . Step 3: adjoint variable method. Solving the adjoint equation (19), the sensitivity is calculated by using equation (20). . Step 4: sorting of elements in order of sensitivity. Each element is ranked in order of the absolute value of sensitivity. . Step 5: set up of changeable elements. If the sensitivity dW/dnrk with respect to permeability nk in element k is negative, the permeability in the element k should be increased. Then, the magnetic material is located in the element k. On the other hand, if the sensitivity dW/dnrk is positive, the permeability in the element k should be decreased. Then, the air is allocated in the element k. In this step, the element, of which the state is changeable, is selected. . Step 6: change of topology. The topology is modified following the information obtained in step 5.

Investigation of 3D optimal design

START decision of initial topology

step 1

FEM

step 2

W

step 3

adjoint variable method

step 4

sorting of all elements in order of sensitivity

step 5

set up of changeable elements

151

k=k+1 No Yes

Nm = 0

step 6

FEM

step 10

annealing step 9

change of topology

No

W

W (k+1) < W (k)

step 7 step 8

Figure 4. Flow chart of ON/OFF method

Yes END

.

.

Step 8: W(kþ 1) , W(k). If the objective function W (kþ 1) is smaller than W (k), return to step 3, otherwise go to step 9. Step 9: annealing. If the objective function is not improved at all, the changeable elements Nm is decreased using the following equation: Nm ¼ g · Nm

.

ð21Þ

where g is the annealing factor, which is chosen as 0.9. Repeat steps 6-9 until some improvement of the objective function is detected. Step 10: Nm ¼ 0. If the number of changeable elements Nm becomes zero, the computation is terminated. Otherwise, return to step 6.

Using this algorithm mentioned above, a fast convergence characteristic and a good topology can be obtained. 3.4 Results and discussion The initial material in design domain is chosen as the air. The specified value Bx0 is set to 2 0.15909 which is the average value of the flux density in the target region. The optimal topology is shown in Figure 5. Many magnetic bodies are distributed around the magnet. Figure 6 shows the flux distribution. Table II shows the flux density and uniformity of flux in the target region. The uniformity is improved and the specified value of flux density is obtained. The total number of FEM calculations is 356, and the CPU time is about 18.8 h by using PC (CPU: Intel Pentium IV Processor 3.2 GHz, RAM: 2.0 Gbyte).

COMPEL 27,1

152 z

x

y

(a) viewpoint α

z y

x

Figure 5. Optimal result (eighth iterations)

(b) viewpoint β

4. Conclusion One approach of the optimal design of open type magnetic circuit producing uniform field is shown. In this analysis, the dimension of the obtained magnet is slightly changed so that the mesh of brick elements for the ON/OFF method can be produced, but this procedure causes the decrease of accuracy. If the technique of non-conformal mesh (Muramatsu et al., 2002) is applied, this problem can be solved. This is the future subject.

Investigation of 3D optimal design

target region

153

y

x

Figure 6. Flux distribution

Initial Optimal

Bxmin (T)

Bxmax (T)

Bxave (T)

u

2 0.15852 2 0.15877

20.15956 20.15934

20.15909 20.15912

0.006540 0.003583

References Abele, M.G. (1993), Structures of Permanent Magnets, Generation of Uniform Fields, Wiley, New York, NY. Muramatsu, K., Yokoyama, Y. and Takahashi, N. (2002), “3-D magnetic field anlysis using nonconforming mesh with edge elements”, IEEE Trans. on Magn., Vol. 38 No. 2, pp. 433-6. Okamoto, Y., Akiyama, K. and Takahashi, N. (2006), “3-D topology optimization of single-pole-type head by using design sensitivity analysis”, IEEE Trans. on Magn., Vol. 42 No. 4, pp. 1087-90. Popella, H. and Henneberger, G. (2001), “Object-oriented genetic algorithms for two-dimensional design optimization of the magnetic circuit of a mobile magnetic resonance device”, Int. Jour. of Applied Electromagnetics and Mechanics, Vol. 15, pp. 219-23. Yao, Y., Fang, Y., Koh, C.S. and Ni, G. (2005), “A new design method for completely open architecture permanent magnet for MRI”, IEEE Trans. on Magn., Vol. 41 No. 5, pp. 1504-7. Corresponding author Norio Takahashi can be contacted at: [email protected] To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

Table II. Flux density and uniformity (ON/OFF method)

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

COMPEL 27,1

154

Design of a double-sided tubular permanent-magnet linear synchronous generator for wave-energy conversion Danson M. Joseph and Willem A. Cronje School of Electrical and Information Engineering, University of the Witwatersrand, Johannesburg, South Africa Abstract Purpose – The purpose of this paper is to present a double-sided tubular linear machine layout direct-drive applications, with particular focus on wave-energy conversion. The paper documents both the computational and mathematical analysis of this novel machine layout. Design/methodology/approach – The selection and finite-element optimisation of the permanent-magnet array is presented. The machine is then modelled using magnetic circuit theory. By simultaneously solving the system of equations, a demonstrative design is developed and simulated so as to validate the mathematical model and compare the performance of the new layout with a traditional layout. Findings – A surface-mounted magnetic array, with unshaped-poles, is most suitable for the proposed layout. The mathematical model exhibits a suitable level of accuracy for design and analysis purposes. The calculated resultant force differs from the FEA calculation by 1.85 per cent. A higher force-density is exhibited by the proposed layout, when compared with flat layouts, with a reduction of 36.5 per cent in the spatial footprint and magnetic material of the machine. Research limitations/implications – Although the research is focused on the application of wave-energy conversion, the techniques are application-independent. However, certain design decisions should be reviewed for other applications. Practical implications – The practical implementation of such a machine poses many mechanical obstacles. These have been solved in theory, and are being implemented at the time of writing. Originality/value – The combination of a double-sided and a tubular layout has not previously been researched. This research fills that void and provides designers with the technical background and a mathematical model for development of such devices. Keywords Electric machines, Electromagnetism, Waves Paper type Research paper

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 154-169 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836726

1. Introduction Wave-energy provides various benefits over other forms of renewable energy. As a renewable resource, waves provide a continual supply whereas solar or wind energies are periodic or temporal. The nature of a specific coastal area, in terms of the frequency and intensity of off-shore swells, is relatively constant. The seasonal variation of wave intensity conveniently follows the seasonal energy demand. Therefore, wave-energy is the most concentrated and consistent form of renewable energy (Ocean Power Delivery Ltd, 1998). Various studies have been performed on the extraction of energy from ocean waves (Heath, 2003), and the use of linear generators for this purpose (Polinder et al., 2003). A double-sided permanent-magnet linear synchronous machine is presented as a

proposed solution for such direct-drive systems. The selection of the translator configuration, and the optimisation of the magnetic poles are discussed. Having selected an appropriate translator configuration, a mathematical model is developed using magnetic circuit theory. To validate the model, a demonstrative design is developed and simulated using finite-element analysis. The performance of this design is compared to that of a commercial double-sided flat machine designed for the application.

Linear synchronous generator 155

2. Overview of the proposed machine layout Double-sided flat designs increase the air-gap of the machine, linking more flux than single-sided machines. Tubular layouts encapsulate the magnetic field within the machine, suffering less from the fringing effects present in flat layouts. This maximises the magnetic and spatial efficiencies for a given performance. The proposed design is a hybrid of the flat and tubular layouts, as shown in Figure 1. Force production is a function of air-gap surface area. The proposed layout

Tm Tp

rli

ri ro

Notes: Most of the mechanical detail has been left out for clarity. The tubular magnet array passes between the two co-axial stator sections

Figure 1. Cut-away of the proposed double-sided LSG layout (not to scale)

COMPEL 27,1

156

increases the air-gap surface area, without an equally proportional increase in machine volume, by creating two co-axial air-gaps. Thus, a higher force-density is realised. One disadvantage of this layout is the associated complexity. This may be acceptable given the nature of the application and the associated long-term benefits. An example of a platform in which such a device may be present is the Archimedes Wave Swing (AWS) (AWS Ocean Power Ltd, 2004). Its operation is shown in Figure 2. This system is an example of the use of direct-drive linear generators and provides suitable design specifications for a comparative study of the respective generators. Both the system specifications and the generator specifications are used to design a double-sided tubular machine. This machine is compared to the AWS machine in order to compare the two layouts. The AWS has the following dynamic characteristics: F ¼ peak force ¼ 1 MN vs ¼ peak velocity ¼ 2:2 m=s h ¼ strong length ¼ 7 m From these specifications, the designers of the current AWS generator produced a double-sided flat machine with the following relevant dimensions. Each AWS device makes use of two of these machines, each having external permanent-magnet translators (Polinder et al., 2004):

Air-filled chamber moves up under reduced pressure

Air-filled chamber moves down under increased pressure Mobile Permanent Magnet Array

Air-Filled Chamber

Stationary Stator Section

Figure 2. Concept of the AWS

Notes: The air-filled chamber resonates with the frequency of the waves above. The change in water height above the device changes the pressure on the device resulting in reciprocating motion

Linear synchronous generator

L ¼ stator length ¼ 5 m Lmag ¼ magnet array length ¼ 8 m

tp ¼ pole – pitch ¼ 0:1 m l g ¼ air – gap length ¼ 5 mm

157

l stack ¼ stack length ¼ 1 m The rationale behind selecting the above dimensions is to maximise the linkage when the translator is at maximum speed, and to maximise the generated emf by decreasing the pole-pitch and hence increasing the frequency of operation. When the machine is at its lowest speed, the overlap is not at its maximum, but because the frequency is low at these speeds, the power output at these points in the translators stroke is not large enough to justify the cost of further length. The air-gap length is limited to 5 mm for mechanical reasons. 3. Design of the permanent-magnet array In selecting an appropriate translator layout, both the magnetic circuit and the mechanical strength need to be considered. Mechanically, the strength of the entire array is considered; magnetically, the force-capability (FC) and total-force-ripple (TFR) are of importance. 3.1 Permanent-magnet array configurations The three possible magnet array configurations are shown in Figure 3. The axially-aligned buried magnets produce very low-harmonic content with suitable flux levels for use (van Zyl and Landy, 2003). The radially-aligned buried magnets have slightly more harmonics than the former. Both these magnet arrangements possess very large salience. Salience enhances the switched operation of the translator’s ro

ri

τp

τm

z

(a) r

(b) Iron

Permanent Magnet

(c) Flux Path

Figure 3. Permanent-magnet array configurations, shown per pole-pair; a) axially-aligned, buried; b) radially-aligned, buried; c) radially-aligned, surface-mounted

COMPEL 27,1

158

motion, and hence introduces additional cogging forces. In a low-maintenance environment, this is not suitable because it will increase bearing wear and mechanical stress. Another characteristic of these two designs is the relatively low-mechanical strength of the magnet array over long translator lengths. As the sections are stacked vertically along z, rods are needed to run through the magnet and iron blocks to strengthen the stack. However, the length of the stack (translator length) is limited by the tensile strength of these rods. An increase in translator length, and hence weight, is accompanied by an increase in rod thickness, compromising magnetic performance. A surface-mounted magnet array accounts for both of these issues. Firstly, the array is relatively non-salient since the magnets’ permeability is comparable to that of air. Secondly, the mechanical support is a solid tube, gathering strength from its circular structure and its continuous form. Consequently, the most suitable magnet array is a tubular back-iron with surface mounted magnets on both the inner and outer surfaces, as shown in Figure 3(c). 3.2 Optimisation of the surface-mounted magnetic poles Although the layout is novel, the basic structure is tubular, and as such can be considered in a similar fashion to other tubular machines (Wang et al., 2001; Canders et al., 2003). In considering the design of these translators, both the FC and the TFR are of importance. These two outcomes have been studied for unshaped magnets by Wang et al. (2001) where the magnet-pitch to pole-pitch ratio, tm/tp, and radial dimensions were varied to quantify the performance. The conclusions were that increasing the dimensions of the magnets would increase the FC, which is intuitively correct since a greater magnetic volume is present, and that the ratio tm/tp is optimally 0.8 for a minimised TFR, regardless of radial dimensions. A second investigation into the TFR was performed by Canders et al. (2003). This investigation included the possibility of shaped poles. However, for an unshaped pole, Canders et al. suggest that the optimal value of tm/tp is 0.6 for a minimised TFR. Canders et al. consider the harmonic content of the magnetic flux distribution alone. The finite-element simulations presented confirm the analysis because only the harmonic content of the magnetic flux distribution is considered, according to equation (1), were Bv is the component of magnetic flux for the v th harmonic: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 v¼2 Bv TFR ¼ ð1Þ B1 Wang et al. move a step further, integrating this flux distribution into the calculations for the force production. The TFR values from the finite-element analysis done by Wang et al. are based on the simulated forces, and not the magnetic flux distribution. The analysis presented by Wang et al. results in a modified TFR equation, as shown in equation (2): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 v¼2 ðK wv Bv Þ ð2Þ TFR ¼ K w1 B1 Most notably, Wang et al. include the winding factor Kwv for the v th harmonic as calculated by equation (3). This suggests that the stator winding can negate the effects

of harmonics from the excitation system on the force production: K wv ¼

  sinðvp=2mÞ vptc £ sin g £ sinðvp=2gmÞ 2 tp

ð3Þ

where m, number of phases; g, slots/pole/phase; tc, coil pitch (m); tp, pole pitch (m). To determine which of the two models most accurately describes the TFR optimisations from the flux distribution, finite-element analysis has been employed to compare the true TFR from direct force measurements with the analytical solutions presented by the previous authors, using indirect methods involving the measured flux distribution. 3.2.1 Finite-element optimisation of magnetic poles. The tubular dimensions of importance are shown in Figure 4(a). These dimensions have been chosen because the axisymmetric nature of the array allows for this level of abstraction. The radii ro and ri are shown for reference, but are of little consequence because of the axisymmetric nature of the machine (Canders et al., 2003). A section of the finite-elements model is shown in Figure 4(b). A slotless model was used to eliminate slot harmonics which would otherwise result in a flux distribution that would misrepresent the true contribution of the magnet array. Therefore, a current sheet was used to represent the three-phase windings. From these dimensions, simulations were performed using the dimension ratios: . wm/ls varying from 1/8 to 7/8 in steps of 1/8; . wi/wm varying from 0.2 to 1.0 in steps of 0.2; and . tm/tp varying from 0.5 to 0.95 in steps of 0.05.

Linear synchronous generator 159

For each set of relative dimensions, 100 samples were taken as the translator moved through two pole-pitches, and the forces calculated directly using the Maxwell Stress Tensor method. With this number of samples across one pole-pair, a maximum of ro

ls

ri

τp

τm z

z r

wm

wi

(a) Relevant machine dimensions for finite-element optimisation. The axis of symmetry is on the left

r (b) A section of the finite-elements model used for the simulations. The stator sections were extended beyond the ends of the translator so that no forces presented themselves from the salience at the ends of the stator sections. The windings are represented by three phase current sheets on the surfaces of the slotless stators. The axis of symmetry is shown on the left

Figure 4. Details of the finite-element model

COMPEL 27,1

50 harmonics can be accounted for, according to Nyquist’s sampling theorem. However, a conservative value of 25 harmonics have been considered. The TFR was calculated according to equation (4) where Fv is the force component of the v th harmonic. These components are the Fourier coefficients of the respective harmonic:

160 TFR ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 v¼2 F v

ð4Þ

F1

The results were interpolated using a spline interpolation along the values of tm/tp. Since, there were more points simulated for tm/tp than the other ratios, the interpolation error incurred would be minimised, while producing significantly smoother optimisation contours. An example of a typical TFR plot, with both interpolated and non-interpolated results is shown in Figure 5, illustrating the accuracy of this interpolation. The flux pattern for each set of dimensions was recorded, and the TFR calculated indirectly according to equations (1) and (2). This allows for a comparison to be drawn between the direct TFR and the indirect TFR from the analytical solutions provided by Wang et al. and Canders et al., to determine which is the most accurate model. The exact method of integration used by the two authors was not possible for the simulated model. Instead, the flux was measured by considering the flux passing through the stator cores in the z direction. This is a relatively good approximation to the mathematics applied by the two authors as it is the most accurate measure of the linking flux. 3.2.2 Results of the finite-element analysis and calculations. The first analysis employed the Maxwell Stress Tensor method, provided by the FEA package, to

0.6 Original Data Interpolated Data

0.55 0.5 Force Ripple [%]

0.45 0.4 0.35 0.3 0.25 0.2

Figure 5. A typical TFR plot as a function of tm/tp for wm/ls ¼ 0.5 and wi/wm ¼ 0.8

0.15 0.1 0.45

0.5

0.55

0.6

0.65 0.7 τm / τn

0.75

0.8

0.85

0.9

0.95

directly calculate the force production. Figure 6(a) shows the optimal pole dimensions for a minimised TFR. The TFR and FC values for the corresponding dimensions in Figure 6(a) are shown in Figure 6(b) and (c), respectively. If one were optimising for maximum FC, then the value of tm/tp must be at it is maximum since this represents the greatest magnetic volume. Figure 6(d) shows TFR contours for this optimisation. Having measured the flux distribution, the analytical solutions of Wang et al. and Canders et al. were used according to equations (2) and (1), respectively, to calculate the TFR and FC indirectly from the flux distribution. The results of equation (2) are shown in Figure 7. The results of equation (1) are not shown because the optimal value of tm/tp, for minimising the TFR, evaluated to 0.5 for all situations. This suggests that according to equation (1), the optimal values lie below tm/tp ¼ 0.5 because 0.5 is the lowest extreme in the range of values for which simulations were performed. The results of the Maxwell Stress Tensor force calculations in Figure 6(a) would indicate that this is incorrect. 3.2.3 Analysis of the FEA results. Figure 6(a)-(d) shows that there is benefit in using shaped poles. For both shaped and unshaped poles, however, there is a trade-off between the TFR and FC. Figures 6(a) and 7 show that equation (2) most accurately relates the TFR and FC to the magnetic flux distribution. Thus, the winding factor is of significance, and becomes a variable in any optimisations of the magnetic pole’s dimensions. To understand the reason for this dependence on the winding factor, consider the example in Figure 8. The system-wide performance gains from using shaped over unshaped poles be affected by the scale of the device. If a large device is designed, the salient forces from

7/8

6/8

5/8

0.

5/8

w m / ls

15

4/8

6/8

0. 0.2 25 0

4/8 0.05

3/8

w m / ls

0.82

0.84

6 0.8

80

0.90

161

0.

0.88

7/8

Linear synchronous generator

3/8 0.10

2/8

2/8

0.15 0.20

0.6

0.8

0.4

0.6

1/8 1.0

0.8

wi/wm

(a) contours of the optimal value of / for a minimised TFR

(b) %TFR contours for the corresponding dimensions in Figure 6(a) 7/8 0.5

0.4

0.3

4/8

4kN 3kN 2kN 0.8

0.2

5/8

5kN

0.6

0.1

6kN

0.4

7/8

6/8

9 0. .8 0 .7 0 0.6

8kN

7kN

0.2

0.2

wi/wm

6/8 5/8 4/8

3/8

3/8

2/8

2/8

1/8 1.0

0.2

0.4

0.6

0.8

wi/wm

wi/wm

(c) FC contours for the corresponding dimensions in Figure 6(a)

(d) %TFR contours for a maximised FC ( / = 0.95)

1/8 1.0

w m / ls

0.4

w m / ls

0.2

1/8 1.0

Figure 6. Results from the finite-element analysis of the magnetic system calculated using the Maxwell Stress Tensor method

COMPEL 27,1

0

7/8

0

162

0.70

0.75

0.8

5/8 4/8

wm/ls

0.8

5

0.9

6/8

3/8 2/8

Figure 7. Contours of the optimal value of tm/tp for a minimised TFR, calculated from the flux distribution

1/8 0.2

0.4

0.6 wi/wm

0.8

1.0

Notes: The difference between this and Figure 6(a) can be attributed to the approximation in the method of flux measurement

Back Iron

Permanent Magnets

Coil 1

PM Flux Distribution: 3rd Harmonic Fundamental

Coil 2

Figure 8. The winding factor affects the FC and TFR

Notes: In this case, both coils experience a force from the fundamental component of the flux distribution, however, only coil 2 will experience a force component from the presence of the harmonic. For each coil, the relative impact of each harmonic in the flux distribution is different, leading to different TFR and force capabilities

the ends of the stators become negligible relative to the FC of the device. Shaped poles would have a greater effect on the TFR here than in a smaller device, where the salience at the ends of the stator sections would produce significant reluctance forces, negating any gains achieved from employing shaped poles. 4. Development of a mathematical model for design and analysis purposes Design procedures developed for electric machines do not typically link the stator to the rotor/translator until the stator is designed. In this new configuration, the values of the outer and inner radii of the translator iron (ro, ri), not only determine the size of the

stator, but also the energy of the magnets. Therefore, the stator and translator must be designed simultaneously. The machine can be simplified into a magnetic system and an electric system. The specific electric loadings (Qi/o) and magnetic flux (F) couple the two systems. The subscripts of “i” and “o” represent values from the inner and outer sections of the generator, respectively. Having reduced all dimensions to terms involving Qi/o and F the design of a machine for any given pair of outer and inner translator-iron radii (ro, ri), is given by the simultaneous solution to the following system of equations. From these results one could optimise for efficiency, size or cost. 4.1 The electric system The specific electric rating, S (VA) is expressed as: S ¼ 3 £ ðV pho £ I pho þ V phi £ I phi Þ But : V phx ¼ 4:44 £ f K w N phx F [S ¼ 3 £ 4:44f K w FðN pho I pho þ N phi I phi Þ

ð5Þ

where Vphx, phase voltage (Vrms); Iphx, phase current (Arms); Nphx, series-turns per phase; F flux per pole (Wb); Kw, winding factor; f, frequency vs =ð2 £ tp Þ (Hz); vs, translator linear velocity (m/s); tp, system pole-pitch (m). The specific electric rating is not directly useful, but given a system efficiency and a power-factor, this rating can be related to the output force: F £ vs ¼ S £ h £ cos u 

 vs [ Fvs ¼ 3 £ 4:44 h cos uK w F £ ðN pho I pho þ N phi I phi Þ 2 tp F¼

3 £ 4:44h cos uK w F ðN pho I pho þ N phi I phi Þ 2tp

ð6Þ

where F, translator force (N); h, system efficiency (per cent); cos u, power factor. The specific electric loading is defined as the RMS ampere-conductors per unit length of the air-gap’s length (Gieras and Piech, 2000): Qx ¼

2 £ N phx £ 3 £ I phx L

ð7Þ

where L, stator length (m). This then reduces the electric system equation to: F¼

1:11 £ hcosuK w FL ðQo þ Qi Þ tp

ð8Þ

Linear synchronous generator 163

COMPEL 27,1

164

4.2 The magnetic system The magnetic circuit, as shown in Figure 9, is governed by the magnetic analogue of Ohm’s law as expressed by equation (9). The stator slots are reduced to steel sections with an effective air-gap, according to Carter’s coefficients (Gieras and Piech, 2000). It can be assumed, with reasonable accuracy, that the mmf of the circuit is dropped across the air-gap alone. This assumption will be further justified when the core areas are modelled to prevent saturation, and hence any significant effect of these reluctances: f mi þ f mo ¼ FðRi þ Ro Þ þ f bi þ f bo

ð9Þ

where fmx, magnet mmf (AT); fbx, stator back-mmf (AT); Rax, air-gap reluctances (H2 1). 4.2.1 Air-gap reluctances. The air-gap reluctances can be expressed as integrals and simplified as follows: Z ri 2wm dr lnððr i 2 wm Þ=ðr i 2 wm 2 kg l g ÞÞ Rai ¼ ¼ ð10Þ 2pm0 tp m 2 p r t p r i 2wm 2kg l g 0 Rao ¼

Z

ro þwm þkg l g

ro þwm

dr lnððr o þ wm þ kg l g Þ=ðr o þ wm ÞÞ ¼ 2pm0 tp m0 2pr tp

ð11Þ

where kglg, effective air-gap length (m); wm, magnet thickness (m). 4.2.2 Flux per pole. Although some flux flows in the outer circuit alone, passing through the thin iron section of the translator, for design purposes, the inner and outer fluxes are considered identical. For the purpose of consistency, the flux densities will all be defined relative to the density at the inner-most surface of the inner magnets. This is done because it should experience the highest flux-density in the magnets because of the radial design. The demagnetisation curve of NdFeB magnets is relatively linear, allowing for the operating points (Bopx, Hopx) to be chosen according to the required permeance. Given that B ¼ 2 mrm0H represents a line of constant permeance, m ¼ mrm0, the intersection of this line with the demagnetisation curve of the magnets yields the operating point for the given value of permeance. To prevent the demagnetisation of the magnets in operation, the value of mr is typically chosen to 2 x (Rao + Rai)

+

Figure 9. Simplified magnetic circuit per pole-pair

2 x (fmo + fmi)

+ 2 x (fbo + fbi)

− Φ

−

be greater than 2. From this, fmo and fmi are given by wmHopo and wmHopi, respectively. The flux density in the inner air-gap is given by: Bi ¼

tm £ Bopi tp

ð12Þ

This must be checked to see if it is reasonable. A rough guide is to multiply this value by tt/tb, the ratio of a tooth pitch to the pitch of a tooth and its adjacent slot. This result should be maintained below 1.5 T. Finally, the flux-per-pole is given by: F ¼ Bopi £ tm £ 2pðr i 2 wm Þ

ð13Þ

4.2.3 Stator core saturation constraints. The tubular nature of the device limits the depth of the inner stator’s windings and the area of the inner stator’s core. To prevent saturation in the core, and to determine the slot area, equations based on core saturation will be used. The outer stator is not limited, but a similar equation will be used to allow for simultaneous solutions. The flux densities in the cores (Bcore) are chosen to be 1.5 T so as to prevent core saturation. It follows that: Bcore ¼

F Acorex

ð14Þ

given: Acorei ¼ pðr i 2 wm 2 l g 2 Lsp1 Qi Þ2 2 pðr i 2 wm 2 l g 2 Lsp1 Qi 2 Lsp2i Þ2

ð15Þ

Acoreo ¼ pðr o þ wm þ l g þ Lsp1 Qo þ Lsp2o Þ2 2 pðr o þ wm þ l g þ Lsp1 Qo Þ2

ð16Þ

These equations introduce three variables, Lsp1, Lsp2i and Lsp2o, which allow for the depths of the laminations, Lsx, to be written in terms of Qx. This is accomplished by relating the area of the slots to the current in the coils: Astx ¼

N slotx ¼

I phx J

2 £ N phx sl=p=ph £ p £ K w

[ Aslotx ¼

Astx £ N slotx K fill

But : Aslotx ¼ tdx £

tp 2 £ sl=pole

Using the definition of the specific electrical loadings, we can equate the two expressions for slot area and eliminate current and series-turns variables:

Linear synchronous generator 165

COMPEL 27,1

166

[tdx ¼

2Qx K w K fill J

ð17Þ

where Astx, area of one series-turn (m2); Aslotx, area of one slot (m2); Nslotx, number of turns per double layer slot; sl/p/ph, slots per pole-pair per phase; sl/pole, slots per pole; p, number of pole-pairs; tdx, slot depth (m); J, current density (A/m2). The stator lengths are thus defined by equation (18). Lsp1 can be calculated from given parameters, and Lsp2i and Lsp2o are two of the variables to be solved for: Lsx ¼

2 Qx þ Lsp2x K w K fill J

Lsx ¼ Lsp1 Qx þ Lsp2x

ð18Þ

If Qo – Qi then there are too few equations to solve for the relevant variables. To account for this, one introduces rli, being the radius of the inner-most surface of the inner stator’s core, as shown in Figure 1. This then provides the relationship in equation (19). Alternatively, if the outer radius of the machine rlo has a restriction on it, this equation can be replaced by equation (20): r li ¼ r i 2 wm 2 l g 2 Lsp1 Qi 2 Lsp2i

ð19Þ

r lo ¼ r o þ wm þ l g þ Lsp1 Qo þ Lsp2o

ð20Þ

4.2.4 Stator back-mmf. The mmf of a sinusoidally excited system of balanced three-phase currents is given by: f bx ¼

6N phx I phx K w 2tp K w Qx ¼ pp p

ð21Þ

using the definition of the specific electric loadings, Qx, and that L/2p ¼ tp. 4.3 Translator iron thickness The mass of the translator (mt) can be determined analytically and used to quantify the strength of the translator iron. The worse case scenario is if the peak force is exerted on the translator vertically upwards. The total force Ft is then Ft ¼ mtg þ F. A tensile pressure of P MPa can thus be calculated according to equation (22), and compared with the tensile strength of the steel (typ: 400-650 MPa (Wikipedia, 2005)): P¼

Ft 2 r 2i Þ

p ðr 2o

ð22Þ

5. Confirmation of the mathematical model using finite-element analysis To validate the mathematical model, a machine was designed where both stators were contributing as much as possible to the force production, while minimising the difference between ro and ri so that the translator’s mass and magnetic volume were minimised. The specifications are shown in Table I.

Parameter

Parameter definition

F L Lmag tp tm Kw sl/p/ph ro ri wm Lsi Lso Lsp2i Lsp2o NPh Iphi Ipho P

Linear synchronous force Stator length Magnet array length Pole-pitch Magnet-pitch Winding factor Slots/pole/phase Translator outer radius Translator inner radius Magnet thickness Inner stator length Outer stator length Inner core length Outer core length Series turns/phase Inner stator current Outer stator current Translator tensile pressure

Value 0.98 MN 5m 8m 0.1 m 0.08 m 0.933 2 0.26 m 0.25 m 11.7 mm 0.213 m 0.105 m 0.071 m 0.011 m 1492.8 122.4 A 81.7 A 128.3 MPa

Linear synchronous generator 167

Table I. Demonstrative machine specifications

Finite-element analysis was used to confirm the magnetic performance of the machine. A section of the finite-element model is shown in Figure 10. As the generator is just as easily a motor, the simulations used currents to generate a force on the translator. The currents were derived from the solutions to the specific electric loadings Qo and Qi. The simulation results are shown in Figure 11. This demonstrative design produced the force necessary for the application. Table II compares the characteristics of this design with the AWS’ double-sided flat LSG device (Polinder et al., 2003, 2004). The AWS uses significantly more material than the co-axial design and has a higher specific electric loading which can be roughly translated into resistive losses.

z r

Figure 10. Finite-elements model (axis of symmetry shown on left)

COMPEL 27,1 Force on Translator (MN)

1.0

168

Figure 11. Results of FEM simulations showing the total force on the translator and the reluctance and Lorentz components thereof

0.5

0

– 0.5 Total Force Lorentz Component Reluctance Component

–1.0 0

2

Notes: The peak force is 981.5kN which is a 1.85% deviation from the design specification

Parameter Table II. Comparison of linear machine layouts

0.5 1 1.5 Translator Position from Start (Pole-Pitches)

Magnet Volume (m3) LSG footprint (m2) Electric loading (kAT/m)

Double-sided linear machine

AWS’ linear machine

0.240 0.458 128.8

0.384 0.720 141.3

6. Conclusion A double-sided tubular linear synchronous machine layout is presented for direct-drive applications with specific application to wave-energy conversion. The selection of the translator configuration suggests that a surface-mounted magnet array without shaped poles is the most suitable configuration for such a device. The optimisation of the magnetic poles in terms of TFR and FC is dependant on the stator winding factor. The mathematical model developed to aid in the design and analysis of such a device has been validated using finite-element analysis, with an error of 1.85 per cent in the model. The proposed layout produces the required performance with significant benefits in terms of material volume and hence, cost. The example given uses 62.5 per cent of the magnetic material and has a footprint of 63.5 per cent of the AWS’ LSG. The physical validation of this model is currently in progress with the construction of a 5 kN machine.

References AWS Ocean Power Ltd (2004), “AWS ocean energy”, available at: www.waveswing.com (accessed 30 June 2007).

Canders, W., Mosebach, H. and Shi, Z. (2003), “Analytical and numerical investigation of PM excited linear synchronous machines with shaped magnets”, Proceedings on the 4th International Symposium on Linear Drives for Industry Applications, LDIA2003, pp. 465-8. Gieras, J.F. and Piech, Z.J. (2000), Linear Synchronous Motors – Transportation and Automation Systems, CRC Press, New York, NY. Heath, T. (2003), “Realities of wave technology”, available at: www.wavegen.co.uk/pdf/art.1727. pdf (accessed 30 June 2007). Ocean Power Delivery Ltd (1998), The Resource, Ocean Power Delivery Ltd, Edinburgh, available at: www.oceanpd.com/Resource/default.html (accessed 30 June 2007). Polinder, H., Damen, M.E.C. and Gardner, F. (2004), “Linear PM generator system for wave energy conversion in the AWS”, IEEE Transactions on Energy Conversion, Vol. 19, pp. 583-9. Polinder, H., Mecrow, B., Jack, A., Dickinson, P. and Mueller, M. (2003), “Linear generators for direct-drive wave energy conversion”, paper presented at the Electric Machines and Drives Conference IEMDC2003, IEEE International, pp. 798-804. van Zyl, A.W. and Landy, C.F. (2003), “Optimisation of the secondary design for a modified tubular linear synchronous motor”, Proceedings on the 4th International Symposium on Linear Drives for Industry Applications, LDIA2003, pp. 375-8. Wang, J., Jewell, G. and Howe, D. (2001), “Analysis and design optimisation of slotless tubular permanent magnet linear motors”, Proceedings on the 3rd International Symposium on Linear Drives for Industry Applications, LDIA2001, pp. 84-9. Wikipedia (2005), “Tensile strength”, available at: http://en.wikipedia.org/wiki/Tensilestrength (accessed 30 June 2007). About the authors Danson M. Joseph was born in Johannesburg, South Africa. He graduated in 2004 with his BSc (ENG) in Electrical Engineering. He is currently conducting research, on which this paper is based, in preparation for his PhD (ENG) in Electrical Engineering at the University of the Witwatersrand, Johannesburg, South Africa. Danson M. Joseph is the corresponding author and can be contacted at: [email protected] Willem A. Cronje received the B.Ing, M.Ing and D.Ing degrees from the Rand Afrikaans University in Johannesburg, South Africa, in 1985, 1988 and 1993, respectively. He has been working in power electronics and magnetic design since his undergraduate studies, and has developed a deeper interest in magnetic device modelling over the years since completing his D.Ing. He has been co-author of a number of papers in this field. He currently holds a chair in Electrical Machines and Drives at the UUniversity of the Witwatersrand, in Johannesburg, in South Africa. E-mail: wacronje @ieee.org

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

Linear synchronous generator 169

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

COMPEL 27,1

Magnetic shielding of buried high-voltage (HV) cables by conductive metal plates

170

Peter Sergeant, Luc Dupre´ and Jan Melkebeek Department of Electrical Energy, Systems and Automation, Ghent University, Gent, Belgium Abstract Purpose – To study the magnetic shielding of buried high-voltage (HV) cables by adding conductive metal plates on the ground surface above the cables. Design/methodology/approach – The field is calculated with eight rectangular conductive plates above the cables, positioned with their long edge either parallel to the cables or transversal to the cables. Here, the circuit method is used. In this method, the shield is replaced by a grid of straight filaments in which the unknown currents are searched by solving an electrical circuit. Findings – It is observed from the calculation results that it is important to have a perfect electrical connection between adjacent plates. In the area above the shield, an “infinite” contact resistance between neighbouring plates results roughly in double field amplitude compared to the situation with contact resistance zero. The positioning of the rectangular plates (parallel or transversal to the cables) has not much influence on the shielding. The shielding efficiency as a function of the shield size is studied as well. The circuit method is validated by measurements on an experimental setup at reduced scale. Research limitations/implications – The circuit method is applied to conductive objects and not to ferromagnetic objects. Practical implications – As the circuit method is rather fast also for 3D geometries with thin plates, the shielding of HV cables can be evaluated in a computationally more efficient way than by using, e.g. finite elements. Originality/value – The circuit method is already described in the literature. The originality of this paper is the study – by this circuit method – of the effect of several parameters (size of the shield, contact resistance, orientation of the plates) on the shielding efficiency. Keywords High voltage, Magnetism, Circuit theory, Magnetic fields Paper type Research paper

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 170-180 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836735

1. Introduction High-voltage (HV) power cables generate a relatively high-magnetic field in a small area around the cables. The field should comply with the reference levels of the European Government on the limitation of exposure to electromagnetic fields (2004/40/EC (European Parliament, 2004), 1999/519/EC (European Parliament, 1999)). To obtain optimal field reduction with the least possible amount of material, it is necessary to know what the optimal size of the shield is, how the sheets should be positioned and how they should be connected to each other. This work was supported by the Fund of Scientific Research Flanders (FWO) projects G.0322.04 and G.0082.06, by the GOA project BOF 07/GOA/006 and by the IAP project P6/21. The first author is a postdoctoral researcher with the FWO.

The simulations are carried out for the geometry shown in Figures 1 and 2 by using the circuit model explained in Section 2. The numerical values are given in Table I. The three HV cables are at distance D below the ground surface. The current in each of the three phases has amplitude I. In the simulation, the copper shield consists of eight plates of 0.5 £ 4 m. The plates are drawn in thick line in Figure 2 while the grid of the circuit method is shown in thin line.

Magnetic shielding

171 2. Circuit model The circuit model is able to evaluate the shielding performance of any 3D magnetic field source by thin, conductive 3D shields. “Thin” means that the thickness of the z

Measurement points

P2

dp

Ground

zm

Conductive sheets

y x

bp

Figure 1. Geometry in the xz-plane of the three cables and the shield consisting of eight conductive sheets next to each other on the ground surface, and with longest edge parallel to the cables

Ds

D 1

2

d

3

d

1

z (m)

0.5

1m

0 –0.5 –1 1.5 m

–1.5 2

4

1 0

2

–1 y (m)

–2

0 –2

x (m)

Notes: Each plate (thick line) is divided in a 6 × 3 grid of filaments (thin line). The small crosses at the edges of the plates indicate where contact resistance is modelled and the circles at height z = zm = 1 m are the field evaluation points. The HV cables are much longer than shown

Figure 2. Geometry in 3D of the three buried HV cables and the shield consisting of copper plates in transversal configuration (i.e. longest plate edge is orthogonal to the cables)

COMPEL 27,1

172

shield is small compared to the penetration depth. The thickness of the copper plates in the simulations is 3 mm. This means that the circuit method is applicable at 50 Hz as for this frequency, the penetration depth in copper is 9.3 mm. An application example of the method is given in Clairmont and Lordan (1999). The magnetic shield as well as the source conductors – in this case the three power cables – are replaced by a grid of wire filaments (the “mesh”). Each wire has its own resistance, self-inductance and mutual inductance with all other filaments. In each node, the filaments are electrically connected. This results in an electrical network in which the unknown currents can be found by solving Kirchhoff’s laws. Once the currents in all filaments are known, the final step is to calculate in the considered point the magnetic field due to each filament by Biot-Savart’s law. The total field is the superposition of all source and shield current contributions. For each filament, the resistance, the self-inductance and the mutual inductances must be calculated. The resistances are those of bars made from the same material as the shield and with a diameter chosen such that the sheet and the equivalent mesh of filaments contain the same amount of material. Think of the solid sheet being “remelted” into a grid structure. The self induction of the filament with length l and radius r is calculated by the classical formula (Clairmont and Lordan, 1999):
 m0 l 2l ln 2 1 ð1Þ L¼ 4p rg with rg the geometric mean radius of the filament: rg ¼ re2 1/4. For the mutual induction between two straight filaments in 3D space, an exact formula was published in Campbell (1915). The next step in the circuit method is to solve an electrical circuit with nn nodes and nt branches. The calculation of all resistances R and all inductances L results in the matrix [R þ jvL ], which is a nt £ nt matrix. The equations: ½R þ jvL
_I þ V _ ¼V _ t

i

ð2Þ

t

give the relation between the nt voltages over all branches V _ and the currents _I t t in these branches. Underlined symbols represent phasors in the frequency domain. The vector V _ symbolizes the contribution of the source (induced voltages). As no external i currents are supplied in the nodes, the sum of all currents in a node is zero:

Quantity

Table I. Dimensions and phase current used for the simulations and for the experimental setup at reduced scale

D D Ds zm I lp bp dp N

Simulation

Experiment

0.250 m 1.500 m 1.521 m 1.00 m 900 A 4.00 m 0.50 m 3.0 mm 8

0.100 m 0.200 m 0.224 m 0.300 m 5.0 A 0.60 m 0.30 m 3.0 mm 2

Description Distance between cables Depth below surface Distance origin to cable 3 Height of evaluation points Phase current amplitude Plate length Plate width Plate thickness Number of plates

½T
_I ¼ 0

ð3Þ

t

where [T] is a (nn 2 1) £ nt matrix, with one row for each node except the reference node. The element Tij is 1 if branch j starts in node i, 2 1 if branch j arrives in node i T and 0 in all other cases. If equations (2) and (3) are combined with V _ t ¼ ½T
V _ n , then the followin1g equation is obtained with the branch currents and node voltages as unknowns: # " #2 I 3 " _ ½R þ jvL
½2T T
2V _ t i 4 5¼ ð4Þ V _ ½T
½0
0 n Finally, the Biot-Savart law: HðrÞ ¼ _

1 4p

Z

J_ ðsÞ £ ðr 2 sÞ

V

3 jr 2 s j

dv

ð5Þ

yields for every branch the magnetic field caused by the calculated current. The volume integral is for a straight filament reduced to a line integral. The current density J_ is replaced by the current _It ð jÞ for filament j. 3. Simulation results 3.1 Unshielded cables We consider the three cables as shown in Figure 1 and Table I, however without the shield. equation (5) yields the flux density in the point (0, 0, 0) in absence of conductive and/or magnetically permeable materials in the neighbourhood: Bð0; 0; 0Þ ¼ B _ _ ð0; 0; 0Þ þ B _ ð0; 0; 0Þ þ B _ ð0; 0; 0Þ 1

¼

m0 I_ 1 2pDs

2



3



  m0 I_ 3 D m0 I_ 2 D d d 1x þ 1x 2 1z þ 1x þ 1z Ds Ds Ds 2pD 2pDs Ds

ð6Þ

The currents with amplitude I in the cables are assumed to be sinusoidal: I_ 1 ¼ I eð2p=3Þj , I_ 2 ¼ I e0j and I_ 3 ¼ I eð22p=3Þj . 3.2 Refinement of the grid The studied shield consists of eight plates of 4 £ 0.5 m with a total surface of 4 £ 4 m or 16 m2. The word “shield” represents the combination of eight plates. For every plate, the number of filaments along the length is denoted nl and the number of filaments along the plate width is nb. Figure 3 shows the influence of nb and nl on the field amplitude in a point P2 at one meter height above the shield center. For the parallel configuration, the number of filaments along the width nb is important. The choice nb ¼ 1 is unacceptable: the plate is modelled along its width (x-direction) by only one filament that has the same width as the plate itself. Owing to the coarse discretization in x-direction, the y-axis currents cannot choose the position they have in reality. Consequently, the shielding efficiency is underestimated. The too low nb cannot be compensated by a very high nl, as the calculated field for nb ¼ 1 and increasing nl converges to a wrong value. For nb ¼ 3 or higher, the curve converges to

Magnetic shielding

173

COMPEL 27,1

11 Parallel, nb=1 Parallel, nb=3 Parallel, nb=5 Transversal, nb=1 Transversal, nb=3 Transversal, nb=5

10 9 8 7 B ( mT)

174

6 5 4 3 2

Figure 3. Influence of the grid refinement on the calculated magnetic flux density in the point P2(0, 0, 1)

1 0

0

2

4

6

8

10 nl

12

14

16

18

20

Notes: Each copper plate is divided along the length in nl filaments and along the width in nb filaments

an acceptable value for B. In the parallel configuration, firstly the important parameter nb should be chosen sufficiently high. Secondly, the less important parameter nl “tunes” the accuracy. For nl ¼ 6, the field amplitude in the considered point is overestimated by about 20 percent compared to corresponding amplitude in case of a very fine grid. For the value nl ¼ 16 chosen in the simulations, the error is smaller than 10 percent. For the eight plates, the solution of the 544 unknown node potentials and the 1,039 unknown currents in the filaments requires on a 1 GHz PC a computation time of about 4 min per shield and almost 30 s per point where B should be evaluated. In the transversal configuration, the refinement along the length nl has more influence than nb. For the transversal configuration, nb ¼ 3 and nl ¼ 16 is a good choice. The overestimation of the field amplitude in the (worst-case) point P2(0,0,1) is also less than 10 percent, which can be considered as sufficiently accurate for shielding applications. The conclusion can be quoted from Clairmont and Lordan (1999) in order to determine the maximal allowed cell size: as a general rule, it appears that at distances greater than the largest dimension of the cells, the results match measurements very well. For an evaluation distance zm of 1 m, nl should be at least 4 for a 4 m long copper plate. Is extra refinement possible, then refinement in the direction transversal to the cables is more important than refinement parallel to the cables. 3.3 Contact resistance For the parallel configuration (like in Figure 1), the current distribution of Figure 4 is obtained. The line thickness is proportional to the current in the filaments. In the case of a high-contact resistance between the plates (Figure 4(a)), the currents cannot flow from one plate to another. Consequently, the currents flow mainly along the outer edge of every plate because this trajectory encloses the largest surface.

Magnetic shielding

2

1

y (m)

175 0

–1

–2 –2

–1

0

1

2

1

2

x (m) (a) 2

y (m)

1

0

–1

–2 –2

–1

0 x (m) (b)

Notes: The line thickness of a filament is proportional to the amplitude of the induced current in the filament. For clarity, the refinement parameter nl was reduced from 16 to 6 while nb = 3 remains unchanged. The contact resistances between adjacent plates are in (a) high i.e. 0.1 Ω and in (b) low, i.e. 0 Ω

Figure 4. Current distribution in the eight copper plates parallel to the HV cables (the cables and the longest plate edges are vertical in the figure)

COMPEL 27,1

176

The thick lines in the middle of the shield correspond with high currents in adjacent sheets having the same amplitude but opposite direction. Their amplitude is about 26.4 A in the vertical filaments around the point (0, 0, 0) for 900 A phase current in the HV cables. For contact resistance zero, Figure 4(b) shows that these two opposite currents cancel each other, resulting in two thin lines in the middle of the shield. The maximal current is now found along the outer edge of all plates together. The currents choose the outer filaments y ¼ 2 m and y ¼ 2 2 m along the x-axis. To return along the y-axis, many parallel filaments are chosen, which explains why no thick lines occur in vertical direction. The maximal current of 22.1 A is found in the points (0, 2, 0) and (0, 2 2, 0). The currents in filaments with the same thickness in Figure 4(a) and (b) have comparable amplitudes. The conclusions are similar when observing the transversal configuration (like in Figure 2). Figure 5 shows the magnetic flux density norm in several points at z ¼ 1 m above the shield. If the sheets are perfectly connected (resistance Rc ¼ 0), the shielding improves especially in the region above the shield. In the region next to the shield however – at large distance from the cables – the influence of the contact resistance is rather limited. Here, all curves are close to each other except a “reference” curve obtained by finite elements. The shield modelled by 2D finite elements is infinitely long, and this has significant effect on the shielding even at large distance. Consequently, it is necessary to use a 3D method as the 2D finite element model is not suitable for modelling this shielding application. Both for the parallel and the transversal configuration of the plates, an excellent electric contact between adjacent sheets is necessary for good shielding. 3.4 Sheet configuration In the parallel configuration, the rectangular sheets are positioned with their long edge parallel to the cables. In the transversal configuration, the longest edge of the sheets is orthogonal to the cables. For contact resistance Rc ¼ 0, Figure 5 shows that the 20

B0 Parallel, Rc = 100 mΩ Parallel, Rc = 0.2 mΩ Parallel, Rc = 0 mΩ Transversal, Rc = 100 mΩ Transversal, Rc = 0.2 mΩ Transversal, Rc = 0 mΩ 2D Finite Element Method

18 16 14 B (mT)

12 10 8

Figure 5. Magnetic flux density along the x-axis at 1 m above the eight sheets for parallel and transversal configurations and contact resistances Rc

6 4 2 0

0

0.5

1

1.5

2 x (m)

2.5

3

3.5

4

difference is negligible as the two configurations represent the same shield. For Rc ¼ 100 mV – a high resistance – there is no difference either. Only for intermediate Rc ¼ 0.2 mV the parallel configuration is slightly better. We can summarize that the influence of the sheet configuration is negligible. 3.5 Shield dimensions To study the influence of the shield dimensions, the original configuration of eight sheets with a total dimension of 4 £ 4 m is modified without changing the number of sheets. For high-contact resistance, Figure 6 shows that the shielding improves for larger shields. Nevertheless, the field reduction does not exceed a factor two even for the largest shield of 8 £ 8 m. The lack of electrical contact cannot be compensated by increasing the size of the shield. For contact resistance zero, the field is reduced up to six times for the largest shield. The width of the shielded zone is roughly determined by the width of the shield unless the shield is very short (along the y-axis). In the latter case, the length also influences the curve as shown in Figure 7(a), where the shielding is given in the point (0, 0, 1) for shields with different lengths as a function of the width. The maximal field amplitude occurs above the edge of the shield and not above the center, although the unshielded field is maximal above the shield center.

Magnetic shielding

177

3.6 Distance between the cables and the shield If the shield is considered to be fixed on the ground surface (z ¼ 0), the distance D shows the depth at which the cables are buried. The quantities that are not mentioned, have the default values of Table I. In Figure 7(b), the influence of this distance is studied. In the curves (1), the position of P2 is changed together with the changing D, in order to obtain a constant distance between the cables and the field evaluation point P2. Consequently, the unshielded field B0 is constant. For increasing D, the shielding seems to improve. For the curves (2), the evaluation point P2 is fixed. The shielded field is mitigated in the most efficient way for small D. It is favourable to put the shield very 20

B0 2 × 2, Rc=0.1 Ω 2 × 2, Rc=0 Ω 2 × 4, Rc=0.1 Ω 2 × 4, Rc=0 Ω 4 × 4, Rc=0.1 Ω 4 × 4, Rc=0 Ω 8 × 4, Rc=0.1 Ω 8 × 4, Rc=0 Ω 8 × 8, Rc=0.1 Ω 8 × 8, Rc=0 Ω

18 16 14 B (mT)

12 10 8 6 4 2 0

0

0.5

1

1.5

2 x (m)

2.5

3

3.5

4

Figure 6. Magnetic flux density along the x-axis at 1 m above the eight sheets in parallel configuration for several dimensions length £ width of the shield and for several contact resistances Rc

COMPEL 27,1 B (mT)

15

178

lp = 0 m lp = 2 m lp = 4 m lp = 8 m

10 5 0

0

2

4 6 Shield width (m)

8

10

(a)

B ( mT)

102 B0, (1) B, (1) B0, (2) B, (2)

101

100

0

0.5

1

1.5

2 D (m)

2.5

3

3.5

4

(b)

Figure 7. Magnetic flux density B with shield or B0 without shield in (0, 0, 1) above a shield as a function of (a) the shield width and length lp; (b) the vertical distance between the shield and the HV cables; (c) the distance between adjacent cables

B ( mT)

60 B0 B

40 20 0

0

0.2

0.4

0.6

0.8

1

Distance cables d hs (m) (c) Notes: For the curves (1), the distance between the cables and P2 was kept constant so that the vertical position of P2 varies; for the curves (2) the point P2 is kept fixed so that the distance between the cables and P2 varies

close above the cables, because few flux lines tend to go around the shield. The latter flux lines reach the region above the shield by using the space besides the shield. For smaller D-values, the shield behaves approximately like an infinitely large plate. For an increasing distance between adjacent cables, Figure 7(c) shows that both the unshielded field and the shielded field in the point P2 increase almost linearly. 4. Experimental validation and conclusions The shielding of HV cables by conductive plates has been experimentally verified on a scale model with two plates and dimensions in Table I. Figure 8 shows that the correspondence between measurements and simulations is good.

0.7 0.6 0.5 B ( mT)

Magnetic shielding

B0, Num B0, Exp Parallel, Num, Rc=1Ω Parallel, Exp, Rc=1Ω Parallel, Num, Rc= 0 Ω Parallel, Exp, Rc= 0 Ω Transversal, Num, Rc=1Ω Transversal, Exp, Rc=1Ω

0.4

179

0.3 0.2 0.1 0

0

0.2

0.4

0.6 x (m)

0.8

1.0

1.2

Notes: The curves show the numerical results (Num); the markers show the experimental data (Exp)

We can conclude that, in order to obtain good shielding with a shield above buried HV cables, the shielding plates should be electrically connected. Moreover, the shield should be sufficiently large and the vertical distance between the cables and the shield should be minimal. References Campbell, G. (1915), “Mutual inductances of circuits composed of straight wires”, Physics Review, Vol. 5 No. 6, pp. 452-8. Clairmont, B.A. and Lordan, R.J. (1999), “3-D modeling of thin conductive sheets for magnetic field shielding: calculations and measurements”, IEEE Transactions on Power Delivery, Vol. 14 No. 4, pp. 1382-91. European Parliament (1999), “Council recommendation 1999/519/EC of 12 July 1999 on the limitation of exposure of the general public to electromagnetic fields (0 Hz to 300 GHz)”, Official Journal of the European Union, L 199, pp. 59-70. European Parliament (2004), “Directive 2004/40/EC of the European parliament and of the council of 29 April 2004 on the minimum health and safety requirements regarding the exposure of workers to the risks arising from physical agents (electromagnetic fields)”, Official Journal of the European Union, L 184, pp. 1-9.

About the authors Peter Sergeant was born in 1978. In 2001, he graduated in electrical and mechanical engineering at the Ghent University, Belgium. In 2006, he received the degree of Doctor in Engineering Sciences from the same university. He joined the Department of Electrical Energy, Systems and Automation, Ghent University in 2001 as Research Assistant. From 2006, he is postdoctoral researcher for the FWO. His main research interests are numerical methods in combination with optimization techniques to design nonlinear electromagnetic systems, in particular actuators and

Figure 8. Magnetic flux density along the x-axis at 0.3 m above the two sheets for several configurations and for several Rc.

COMPEL 27,1

180

magnetic shields. Peter Sergeant is the corresponding author and can be contacted at: [email protected] Luc Dupre´ was born in 1966, graduated in electrical and mechanical engineering in 1989 and received the degree of Doctor in Applied Sciences in 1995, both from the Ghent University, Belgium. He joined the Department of Electrical Energy, Systems and Automation, Ghent University in 1989 as a Research Assistant. From 1996 until 2003, he has been a postdoctoral Researcher for the FWO. Since, 2003, he is a Research Professor at the Ghent University. His research interests mainly concern numerical methods for electromagnetics, especially in electrical machines, modeling, and characterisation of magnetic materials. Jan Melkebeek was born in 1952 and graduated in electrical and mechanical engineering in 1975. He received the Doctor in applied sciences degree in 1980 and the “Doctor Habilitus” in electrical and electronical power technology in 1986, all from the Ghent University, Belgium. He was a visiting Professor at the Universite Nazionale de Rwanda in Bultare, Rwanda, Africa in 1981 and a visiting Assistant Professor at the University of Wisconsin, Madison in 1982. Since, 1987, he has been a Professor in electrical engineering at the Engineering Faculty of the Ghent University. Since, 1993, he has also been the Head of the Department of Electrical Power Engineering. His teaching activities and research interests include electrical machines, power electronics, variable frequency drives, and control systems theory applied to electrical drives.

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

Improved AC-resistance of multiple foil windings by varying foil thickness of successive layers D.C. Pentz and I.W. Hofsajer

AC-resistance of multiple foil windings 181

Industrial Electronics Technology Research Group, University of Johannesburg, Auckland Park, South Africa Abstract Purpose – This paper aims to introduce a technique for optimizing conductor dimensions for construction of helical planar inductor windings with reduced ac- and dc-resistance. The loss reduction is evaluated on simulation and experimental level. Design/methodology/approach – Helical planar windings are currently manufactured by forming each conductive and insulating layer individually. The conductive layers are only interconnected later to form the winding. This process allows greater freedom in selecting the optimum conductor dimensions on a per-layer basis. Methods are proposed for sinusoidal and non-sinusoidal current excitation waveforms and shaped windings are introduced for further loss reduction where conductors are in close proximity to air gaps in magnetic cores. Findings – Traditional optimization of the conductor thickness used for foil wound inductors renders one single value used for each of the respective layers in the winding. This is a result of the manufacturing process involved in making traditional “barrel” windings. This optimization technique has simply been applied in planar inductors for the lack of alternatives up to now. It will be shown that large loss reduction may be achieved by manipulating the conductor dimensions of each layer individually. Originality/value – A new approach to optimization problems verified experimentally offers more efficient inductors. Keywords Eddy currents, Magnetic devices, Optimization techniques, Foil, Electrical resistivity Paper type Research paper

Introduction The ever increasing demand for high power density converters calls for a re-evaluation of component optimization. New material developments have moved the electromagnetic boundaries for magnetic components such as transformers and inductors away even further but thermal problems still form a major stumbling block in further reducing the size of these components. Successful extraction of heat from a component is largely a function of the surface area of the device and becomes increasingly difficult with reduction in component size. Planar magnetic components have a number of advantages amongst which the increased surface area allows for better thermal management. A conventional “barrel”-wound inductor is shown in Figure 1 together with an exploded view of a typical planar inductor showing the main differences in geometry and manufacturing. Loss mechanisms The main sources of thermal losses in magnetic component structures are core and conduction losses, which limit the maximum attainable power density of converters

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 181-195 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836744

COMPEL 27,1

182

Figure 1. (a) Conventional EE-core inductor with “barrel” foil winding; (b) planar EE-core inductor with helical planar winding (a)

(b)

(Strydom and Van Wyk, 2003). Conduction losses are a direct result of increased ac-resistance at high frequencies subsequent to proximity and skin effects in the conductive medium. Work done up to the present moment is based on finding an optimum foil thickness with which the entire winding is constructed (Dowell, 1966; Snelling, 1989; Ferreira, 1994) as shown in Figure 1(a). Alternative methods are proposed in this work with each layer thickness optimized individually for helical planar windings. The initial optimization is performed for sinusoidal current excitation. An existing strategy for non-sinusoidal excitation is adapted to suit the per-layer approach. Simulation and experimental results are offered for a few cases. Another loss mechanism involves eddy currents being induced in conductors that are in close proximity of the air gap in gapped core inductors and can be reduced by shaping the winding in this region to avoid the fringing flux (Pollock and Sullivan, 2005). The advantages offered by the technique used for planar winding construction are again exploited to reduce ac- and dc-losses. These concepts are also investigated on simulation and experimental level. Mathematical model The mathematical model described by Dowell (1966) is used as basis in this work. A winding section is defined as a number of layers in a winding situated between points of zero and maximum mmf. The fields are assumed to be one-dimensional because conductor widths are much larger than their thickness and the return conductors are far enough not to influence the field patterns around the conductors under investigation. The ac resistance of a winding section comprising p number of layers can then be calculated using equation (1) (Dowell, 1966; Snelling, 1989; Ferreira, 1994):

RacTotal

D ¼ Rdc 2




 sinh 2D þ sin 2D 2ð p 2 2 1Þ sinh D 2 sin D þ cosh 2D 2 cos 2D 3 cosh D þ cos D

ð1Þ

where: D¼

Foil thickness h ¼ Skin depth d

ð2Þ

and the skin depth is a function of the operating frequency f, the material conductivity s and permeability m: 1 Skin depth ¼ d ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðpsmr m0 f Þ

ð3Þ

The optimum foil thickness is calculated by solving ðdRac =dhTotal Þ ¼ 0 and used for the construction of the whole winding. This optimum value is a function of the number of turns and the operating frequency. Equation (4) describes the ac-resistance of each individual layer in the winding section and to date it has been assumed that the dc-resistance of each layer will be the same. This does not necessarily have to be the case when constructing a helical planar winding, a feature exploited throughout this work. The ac-resistance of a particular layer is a function not only of the mentioned parameters but also of its unique position in the magnetic field designated by m in equation (4). The new method proposed is based on differentiating equation (4) for each layer of foil in a winding section thus determining the optimum foil thickness hm for each individual layer. A general simplified equation (5) in terms of m can be written and solved to find the minimum and maximum points:
 D sinh D þ sin D 2 sinh D 2 sin D þ ð2m 2 1Þ ð4Þ Racm ¼ Rdcm 2 cosh D 2 cos D cosh D þ cos D mðm 2 1Þ½cosh2 D þ cos2 D
2 ½2mðm 2 1Þ 2 1
cosh D cos D ¼ 0

ð5Þ

The solutions were all obtained using MATLAB. In order to prevent having to solve the equations for each layer a curve fitting technique (Nelder-Mead simplex algorithm) was employed using MATLAB, yielding the following general equation for calculating the coefficient C(m) and the optimal foil thickness for each layer using equations (6) and (7): CðmÞ ¼ 3:0785e21:1056m þ 0:5737e20:0523m

ð6Þ

hopt ðmÞ ¼ CðmÞ £ d

ð7Þ

A cross section of a number of conductors experiencing zero-mmf in the centre and maximum mmf toward the outer conductors is shown in Figure 2. This is typically true for planar windings embedded in an ungapped magnetic core and two winding sections may be optimized using the proposed method, resulting in the windings close to the point of zero-mmf being thicker than the ones experiencing higher magnetic field intensities. Two windings were used in the simulations to follow. The reference winding is optimized using the traditional method and all the layers are 0.125 mm thick. The newly proposed method results in foil dimensions as shown in Figure 2.

AC-resistance of multiple foil windings 183

COMPEL 27,1 0.1mm 0.125mm

184

0.2mm 0.3mm

MMF

0.3mm 0.2mm

Figure 2. Proposed eight-layer planar winding with associated mmf-diagram

0.125mm 0.1mm

The predicted ac-resistance calculated using the one-dimensional solutions proposed by Dowell is shown in Figure 3. A graph representing the percentage difference calculated with FEM-simulations is also included showing good correlation with the one-dimensional predictions. The percentage difference is calculated using the reference winding as the benchmark. The dc-resistance of the new structure reduces by 18 per cent and the ac-resistance is 11.5 per cent lower than that of the reference winding at 100 kHz. The foil thickness used for each layer in the simulations is chosen based on availability of copper foils used to construct the actual windings.

Air core inductors A planar air-core inductor optimized for 100 kHz sinusoidal excitation and constructed according to the new strategy was compared to a similar inductor where the optimal foil thickness (0.133 mm) for the entire winding was calculated and a standard 0.125 mm foil size was used for construction of all the layers. Figure 4 shows a photograph of the reference inductor. A similar winding henceforth referred to as the new winding was constructed using foil dimensions shown in Figure 2. The resulting ac-resistance as a function of frequency was measured with an LCR-meter and is shown in Figure 5. The one-dimensional field assumption is not valid any more for this physical construction and FEM 2D simulations were also done at discrete frequencies to verify the experimental results. Reduction in the dc-resistance amounts to 18 per cent and the ac-resistance at the design frequency of 100 kHz is approximately 12.5 per cent. These results were also confirmed calorimetrically and the results are shown in Figure 6 with good correlation between the expected and measured losses. The differences are attributed to measurement error and the fact that 2D FEM software cannot account for field distribution and subsequent losses around the ends of the windings.

AC-resistance of multiple foil windings

0.3 Ref (1-D)

AC-resistance (ohm)

0.25

New (1-D) 0.2

185

0.15

Design frequency (100kHz)

0.1 0.05 0 104

105

106

5 FEM-magnetic core 1D-prediction

0

% Difference

–5 –10 –15 –20 –25 101

102

103 104 Frequency (Hz)

105

106

Gapped ferrite cores Since, many inductor applications require air gaps in the magnetic circuit to prevent core saturation the performance of the constructed windings was evaluated under these conditions. Figure 7 shows a photograph of the arrangement from the rear end. Spacers are used to align the centre turns with the air gap. The results for these inductors are shown in Figure 8 and an up to 20 per cent reduction in losses at the design frequency (100 kHz) is predicted and verified with LCR-measurement. The excitation levels set up by the impedance analyzer are very low and core losses are generally ignored. Comparative measurements were obtained for two different types of core material and compare well with FEM 2D simulation results. Figure 9 shows the calorimetric results obtained for the two inductors for one of the core sets. It should be remembered that the total losses measured include core losses at higher levels of excitation. To obtain the conductor losses the core losses may be taken from core material data sheets. The data sheet information reflects core losses under

Figure 3. Predicted difference in ac-resistance of reference and new structure

COMPEL 27,1

186

Figure 4. Photograph of planar winding

0.25 Ref(FEM) New(FEM) Ref(Act) New(Act)

AC-resistance (ohm)

0.2 0.15 0.1 0.05 0 101

102

103 104 Frequency (Hz)

105

106

–9 FEM Act

–10 –11 % Difference

–12 –13 –14 –15 –16 –17

Figure 5. LCR-measurements and FEM-simulation for planar air core inductors

–18 –19 101

102

103 104 Frequency (Hz)

105

106

AC-resistance of multiple foil windings

Losses (W)

Air core (Sinusoidal excitation) 4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00

–14.5%

FEM Measured

Figure 6. Calorimetric results (air core)

2

1

187

Inductor

Air gap

Figure 7. Photographs of gapped core inductor assembly

sinusoidal excitation. Experiments have been performed successfully to measure core losses under the same conditions. In a single turn inductor without any air gaps the required flux excursion can be set up with a much lower current and since the amount of copper is minimized the core losses tend to be much higher than the conductor losses. Conductor losses are again predicted using FEM and subtracted from the total losses to reveal the core losses. Measured values compare well to data sheet information. Optimization for non-sinusoidal current excitation An existing method (Hurley et al., 2000) was adapted to optimize each layer individually for non-sinusoidal excitation. This method is based on the original Dowell equations for the total resistance of a p-layer winding section approximated by using

COMPEL 27,1

1 Ref (F44-material)

Rac (Ohm)

0.8

188

New (F44-material) Ref (3F3-material)

0.6

New (3F3-material) 0.4 0.2 0 104

105 Frequency (Hz)

106

10 New vs. Ref (F44) 5

% Difference

0

New vs. Ref(3F3) FEM-2D New vs. Ref

–5 –10 –15 –20

Figure 8. Results for gapped ferrite core inductors

–25

102

104 Frequency (Hz)

106

Sinusoidal excitation 3.00 –18%

Losses (W)

2.50 2.00

–24%

Conductor FEM

1.00

Conductor meas

0.50

Figure 9. Calorimetric results for gapped core inductors

Total Measured

1.50

0.00 1

2 Inductor

series expansions for the trigonometric and hyperbolic functions in the total-resistance equation and only the rms-values of the current and its derivative are used instead of the Fourier coefficients. The following equation is then derived for the optimal foil thickness normalised to the skin depth: sffiffiffiffiffiffiffiffiffiffiffi h 1 vI rms ð8Þ Dopt ¼ ¼ 4 pffiffiffi d c I 0rms where:

c¼

5p 2 2 1 15

ð9Þ

0 Irms is the rms-value of the resulting current, Irms the rms-value of the first derivative of the current, p is the total number of layers in a winding section and v is the fundamental frequency. This work uses the same basic equation as proposed (Hurley et al., 2000). Equation (9) is adapted by performing a series expansion on equation (4), which describes the resistance of each layer in the winding section and yields equation (10):

cm ¼

60m 2 2 60m þ 16 60

ð10Þ

Equations (8) and (10) can now be used to calculate the optimal foil thickness for each of the layers in the winding section. 0 Substituting Irms and Irms for a 100 kHz sinusoidal current waveform yields similar results for a four-layer winding section to those obtained through the direct method using equations (6) and (7). Recall that availability of foils largely dictates the sizes used in the prototypes. For waveforms containing a considerable dc-component it is suggested that the new calculated foil thickness only be used up to the point where the suggested value is lower than the reference foil thickness and the reference thickness used further on. This philosophy will prevent excessive increase in dc-resistance of specific turns causing localized power dissipation. The effect of this technique on the loss-improvement was investigated for several different continuous and discontinuous current waveforms. Improvements of up to 26 per cent were obtained in cases where windings were optimized for discontinuous triangular current waveforms. The additional amount of copper (36 per cent) used in the new winding raised the question about the justifiability of the new technique in terms of cost. A winding shaping technique was considered next to minimize losses due to air gap proximity and to investigate the effects when using different thickness layers. Windings shaped in the region of the air gap The fringing effect in gapped core inductors cause excessive losses in conductors situated in close proximity to the gap. The winding can be shaped in this region and conductors arranged parallel to the flux patterns in order to minimize induced losses (Pollock and Sullivan, 2005). Figure 10 shows the concept of notching the winding in

AC-resistance of multiple foil windings 189

COMPEL 27,1 rMax

190

lg

rNotch

Figure 10. Winding shaped in the region of air gap

the region of the air gap. Carefully consider Figure 1(b) to see how the width of the individual turns are varied beforehand to form the circular indentation once they are stacked. Figure 11 shows a photograph of an actual shaped winding. Circular notching has previously been considered but proposed for barrel foil windings only. In this paper, the technique is applied to planar foil windings. If the foil windings in structures are shaped in this manner the dc-resistance also increases and therefore a compromising optimum design has to be found considering both the dcand the ac-loss components. The width of the centre foils will be reduced in order to avoid the high flux density regions around the air gap. Since, the newly proposed scheme results in thicker foils being used in the centre of the winding where material will be removed to form the indentation it was decided to investigate the matter further. Again different waveforms were considered and numerous simulations performed in order to find the application that best utilizes the advantages offered when using varying foil sizes in the winding section. Trapezoidal current waveforms such as the one shown in Figure 12 has a dc-component and very high harmonic content. An example of a specific simulation will now be discussed. Both the reference and the new winding were optimised for a frequency of 100 kHz, a duty cycle of D ¼ 0.5, I1 ¼ 2.9 A and I2 ¼ 3.7 A. These values were specifically chosen to yield available thickness values in copper foil. The air gaps in the EE-core centre and outer legs are 0.5 mm and the insulation thickness between windings is 0.4 mm. The reference winding comprises of eight turns of 0.5 mm copper foil and the new winding section starts with a 1 mm copper plate in the middle, followed by a 0.625 mm and two 0.5 mm foils arranged similar to the layers shown in Figure 2. Three sets of windings are typically constructed as shown in Figure 13. L1 and L2 are constructed with all the layers formed from the same thickness copper foil. L1 is left unshaped and L2 is shaped

AC-resistance of multiple foil windings 191

Figure 11. Actual shaped inductor winding

i(t) I2

I1

t DT

T

Figure 12. Trapezoidal current waveform

COMPEL 27,1

192

(a) L1

Figure 13. Inductor winding shapes

(b) L2

(c) L3

to its optimum notch radius. The layers in L3 are optimized individually and the winding shaped to its optimum notch radius. The simulation current used is 1 A peak and the simulation is performed at 10 Hz to obtain the dc-resistance and harmonic frequencies up to 1 MHz. The electromagnetic loss for each layer is calculated every time for the sinusoidal excitation at the particular frequency and the total losses calculated using the Fourier coefficients of the chosen current waveform. At first the windings are left unshaped and the loss values obtained are used as a reference set. The windings are now notched incrementally in the simulation and all results are compared to the reference set. The notching process is normally done intuitively in incremental steps until the minimum loss point is obtained. Alternative solutions that converge much quicker are currently being developed. In case of this particular example both the reference winding and the new winding optimal notch radius turned out to be the maximum possible value (rMax) before altering the width of the outer most turns. The notch radius was equal to 2.9 mm for the reference winding and 3.52 mm for the new winding. Consider the graphs in Figure 14 for the results. The fact that the new winding can be notched further than the reference winding results in effectively only 10.8 per cent more copper being used compared to the notched reference winding. The new winding notched to the optimum point however displays a 21.4 per cent improvement when compared to the reference winding notched to its optimum point. Compared to a reference winding which is not notched the shaped reference winding has a loss reduction of 81 per cent and the new

AC-resistance of multiple foil windings

4.5 Ref losses New losses

4

EM-loss (W/m)

3.5 3

193

2.5 2 1.5 1 0.5

0

1

2 Notch radius (mm)

3

4

20 Notched ref vs. orig ref Notched new vs. orig ref

% Improvement

0 –20 –40 –60 –80 –100

0

1

2 Notch radius (mm)

3

4

winding 85 per cent. For the physical windings, parallel foils had to be used in order to obtain the design thickness. It was decided to insulate these parallel layers in the region of the air gap because the lamination effect should further reduce the losses. Simulations were again performed using the laminated layers and the loss prediction for the new winding indeed showed an improvement of 29 per cent (vs 21.4 per cent) with respect to the notched reference. Windings were constructed and the resistance measured at different frequencies using an LCR-meter. The loss calculation through the Fourier coefficients shows an improvement of 24 per cent when comparing the two notched windings and the overall improvement compared to a reference winding which is not notched is 78 per cent (vs 81 per cent simulated) for a notched reference and 81 per cent (vs 86.5 per cent simulated with laminated turns) for a notched winding comprising layers of different thickness. A concern at this point is a slight misalignment of the various layers in the stack resulting in a notch shape which is not

Figure 14. EM-loss comparison

COMPEL 27,1

194

perfectly circular and the manufacturing technique is being altered to obtain a better result. The trapezoidal current waveform is typical of a continuous-mode fly-back converter topology. Generating this current waveform to characterise the behaviour of the inductors designed is the next step towards performing the calorimetric experiments for verification of these results. Since, the core material is nonlinear, superposition cannot be used to obtain the core losses for non-sinusoidal current waveforms meaning that core losses for this experiment will have to be measured. Conclusions The proposed scheme of optimising the foil thickness of individual layers proves to reduce both ac- and dc-resistance. An existing optimisation technique used for non-sinusoidal current profiles was successfully adapted to suit this new method. The improvement in losses was investigated for various waveforms combined with the notching technique described and best results were obtained for trapezoidal waveforms where it is expected that the additional amount of copper used could be justified by the percentage improvement in losses obtained in the process. The manufacturing process has to be altered to achieve better alignment and the results have to be confirmed with higher levels of excitation. References Dowell, P.L. (1966), “Effects of eddy currents in transformer windings”, Proceedings of the IEE, Vol. 113, pp. 11387-94. Ferreira, J.A. (1994), “Improved analytical modeling of conductive losses in magnetic components”, IEEE Transactions on Power Electronics, Vol. 9 No. 1, pp. 127-31. Hurley, W.G., Gath, E. and Breslin, J.G. (2000), “Optimizing the ac resistance of multilayer transformer windings with arbitrary current waveforms”, IEEE Transactions on Power Electronics, Vol. 15 No. 2, pp. 369-76. Pollock, J.D. and Sullivan, C.R. (2005), “Modelling foil winding configurations with low ac and dc resistance”, paper presented at the Power Electronics Specialists Conference (PESC). Snelling, E.C. (1989), Soft Ferrites, Properties and Applications, 2nd ed., Butterworths, London. Strydom, J.T. and Van Wyk, J.D. (2003), “Volumetric limits of planar integrated resonant transformers: a 1 MHz case study”, IEEE Transactions on Power Electronics, Vol. 18 No. 1, pp. 237-47. About the authors D.C. Pentz is a Senior Lecturer in the Department of Electrical and Electronic Engineering – Science at the University of Johannesburg. He received his B.Ing. degree from the University of Pretoria in 1990 and his M.Ing degree from the Rand Afrikaans University in 2001. He is currently busy with a doctoral study with the Industrial Electronics Technology Research Group. D.C. Pentz is the corresponding author and can be contacted at: [email protected]

I.W. Hofsajer was born in Johannesburg, South Africa. He received the B.Ing, M.Ing, and D.Ing degrees in electrical engineering from Rand Afrikaans University, Johannesburg, South Africa, in 1991, 1993, and 1998, respectively. He worked in the field of electromagnetic interference at the South African Atomic Energy Corporation before joining the Faculty of the Rand Afrikaans University. He is currently Associate Professor in the School of Electrical and Information Engineering at the University of Witwatersrand, Johannesburg. His interests include power electronics and electromagnetics. E-mail: [email protected]

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

AC-resistance of multiple foil windings 195

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

COMPEL 27,1

196

Influence of the magnetic model accuracy on the optimal design of a car alternator J. Cros, L. Radaorozandry, J. Figueroa and P. Viarouge

LEEPCI, De´p. de Ge´nie E´lectrique and Informatique, Universite´ Laval, Que´bec, Canada Abstract Purpose – The machine design with optimization method using analytical models is efficient to evaluate a large number of variables because these models are faster to solve. Nevertheless, the validation of the final optimal solution by FE simulation often shows that some specification constraints are not verified. To solve the problem, it is possible to apply a hybrid approach for the design method while combining analytical methods and 3D FE simulations to compensate analytical model errors. The paper addresses this. Design/methodology/approach – Each intermediate optimal solution is evaluated by FE simulation to quantify the analytical model errors. Correction coefficients are derived from this evaluation and another optimization process is performed. With this method, the convergence of the hybrid optimal design process is obtained with a limited number of FE simulations. Findings – This study shows that it is possible to compensate errors of analytical models with a limited number of 3D field calculations during a global optimization design process. The 3D FE software validates the optimal solution but this solution is also a function of the sensitivity of analytical models that is not improved by the correction method. Practical implications – This error compensation of analytical models using FE simulations can be applied for the design of a wide range of electromagnetic devices with optimization methods. Originality/value – This paper presents a correction method that guaranteed the validity of the solution after the optimization process when analyzed with a FE software. Keywords Electric machines, Optimization techniques, Simulation, Alternating current Paper type Research paper

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 196-204 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836753

1. Introduction The electric machine design process can use optimization techniques to maximize cost-performance objectives while respecting the specifications of the application. Such a design approach needs magnetic and electrical design models with adequate sensitivity to the design parameters variations and sufficient accuracy in order to guarantee an efficient convergence of the optimization process towards one true optimal solution. Heavy time consuming FE simulations are usually not applicable during the iterative optimization process in the case of 3D devices. Analytical models are preferable in the case of global optimal design process because they are faster to solve and more versatile in terms of topological parameter variations. But the validation of the final optimal solution by FE simulation often shows that some specification constraints are not verified, because the accuracy of the analytical models is limited. In this paper, a hybrid approach is proposed to solve this problem. The optimization process is always using analytical design models but each intermediate optimal solution is evaluated by FE simulation to quantify the

analytical model errors. Correction coefficients of the analytical models are derived from this evaluation and another optimization process is performed. With this method, the convergence of the hybrid optimal design process is obtained with a limited number of FE simulations. The convergence of this method is verified in the case of the optimal design of a conventional claw-pole Lundell car alternator, which has a 3D structure.

Influence of the magnetic model accuracy 197

2. Specifications and electrical modelling of a Lundell car alternator The design of a conventional claw-pole Lundell car alternator system (Figure 1) with an uncontrolled rectifier connected to a 12 V battery is used to validate the proposed method. The main specifications of this application are the output DC current, torque and efficiency vs speed characteristics shown on Figure 2. These specifications are very tight due to the constraints imposed by the car manufacturers. It has been demonstrated by Figueroa et al. (2005) that the Lundell alternator can be modeled as a three phase non-salient sinusoidal voltage generator with an equivalent circuit composed of a sinusoidal voltage source connected in series with a linear

120

12

100

10

80

8

60

6

40

4 DC output current [A] Efficiency [%] Torque [Nm]

20 0 0

2,000

4,000 Speed [RPM]

6,000

Torque [Nm]

DC output current [A] Efficiency [%]

Figure 1. Rotor and stator of a Lundell car alternator

2 0 8,000

Figure 2. Specifications of the 12 V car alternator system

COMPEL 27,1

198

inductance and a resistance. If the losses and voltage drops in the rectifier are neglected and if the battery is modeled as an ideal voltage source, it is possible to obtain an analytical solution of the steady state current waveforms for any operation speed. This method has the advantage to be very fast for the performance estimation of an alternator (Figueroa et al., 2005). In practice, the parameters of the electrical alternator model are derived from the no-load emf vs rotor field current characteristic E V ð jrotor Þ and the short circuit characteristic I CC ð jrotor Þ. The armature phase resistance is measured at the rated temperature rise. The stator cyclic inductance (L) is derived from these tests by applying equation (1) where v is the electrical pulsation. It must be noticed that this inductance value varies with the rotor field current because magnetic saturation is occurring in the iron yokes:

Lð jrotor Þ ¼

E V ð jrotor Þ v · I CC ð jrotor Þ

ð1Þ

Figure 3 shows a comparative analysis between experimental and simulation results for different field currents which validates the performance of the proposed modeling method. The accuracy of the model highly depends on the parameter identification or computation method. During the alternator design, one can use the same approach to estimate the output DC current vs speed characteristic. But in this case, the electrical parameters must be identified by using FE or analytical techniques with the following method: . Compute the no-load direct axis flux (lsat) for the rated field current ( jrotor) while taking into account the saturation of the magnetic material. . Compute the no-load direct axis flux (llin) for the rated field current ( jrotor) assuming a linear magnetic material.

70

DC bus current [A]

60

Figure 3. Comparative analysis of experimental and simulation results of the alternator output characteristics

50 40 30 J=3A (experience) J=6A (experience) J=3A (analytical model) J=6A (analytical model)

20 10 0 0

1,000

2,000 3,000 Speed [RPM]

4,000

5,000

.

.

Compute the stator winding direct axis inductance (Llin) with a linear magnetic material. Compute the stator winding resistance (R).

Influence of the magnetic model accuracy

The non linear inductance value of the armature is then derived from: Lð jrotor Þ ¼ Llin ·

lsat ð jrotor Þ llin ð jrotor Þ

ð2Þ

199

3. Magnetic modelings Different magnetic models have been used to compute the equivalent circuit parameters of the alternator. The first is based on a magnetic reluctance network that takes account of the machine geometry and the magnetic material saturation. The second is a hybrid model that uses the analytical magnetic field computation in the air gap of an equivalent machine without stator and rotor slotting and a yoke reluctance network. The third method is based on 3D FE modelings. 3.1 Reluctance model of the Lundell alternator The magnetic reluctance network is based on the knowledge of the machine geometry and of the flux circulation paths. Figure 4 shows the equivalent network which can be reduced to two half rotor poles in the case of a structure with one slot per pole and per phase. Re is the air gap reluctance under a half rotor pole. Rt and Ry are the stator reluctances associated to the teeth and the stator yoke. Rcl, Rch and Rco are the rotor reluctances associated to the claws, the cheek and the rotor core. Rl is a reluctance associated to the flux leakage between the claws and the rotor core. The reluctance values are computed from the alternator geometrical dimensions and the B(H) characteristics of the magnetic materials, in the case of a calculation with magnetic saturation. The values of the linear and saturated values (llin) and (lsat) of

ℜy le ℜt 2

ℜt

ℜe

ℜe

ℜcl

ℜl

ℜch

ll

ℜcl

ℜch ℜco

lco 2

N.jrotor

Figure 4. Equivalent magnetic reluctance network

COMPEL 27,1

the no-load flux in the air gap (le) can be derived from the linear and non linear resolution of the equations system (3): 8 ½2 · ðRe þ Rt þ Rcl Þ þ Ry
· l2e 2 Rl · ll ¼ 0 > > < ½2 · Rch þ Rco
· l2co þ Rl · ll ¼ N · jrotor > > : l ¼ l þ 2·l

200

co

e

ð3Þ

l

Equation (4) is used to compute the linear value of the cyclic inductance (Llin):

Llin ¼ 2 · p ·

4 1 · 3 2 · ðRt þ Re þ Rcl Þ þ Ry þ ðRl · ð2 · Rch þ Rco Þ=ðRl þ 2 · Rch þ Rco ÞÞ ð4Þ

The saturated value of the inductance (Lsat) is then derived from equation (2). 3.2 Hybrid alternator model with analytical magnetic field computation in the air gap and yoke reluctances This method neglects the saliency of the stator and rotor slotting. It is then possible to perform the analytical resolution of the magnetic field in the air gap with a similar approach to the one proposed by (Zhu and Howe, 1993): 8 ~ ~ ¼7£A > m0 H > > >
 > < _ ~ ¼ Pð _ an r n þ bn r 2n Þsin nu z^ A ð5Þ n > > > > _ > : _ an ¼ 2m0 an and bn ¼ m0 bn The potential vector is used to facilitate the flux and inductance computations. To compute the flux, it is sufficient to integrate the vector potential on the surface covered by a single stator tooth. The total no-load flux per phase can then be derived from the knowledge of the complete configuration of the stator winding. However, in the case of the car alternator, it is not possible to neglect the stator and rotor reluctances because the magnetic saturation is important and the air gap thickness is small. For these reasons, we have developed a hybrid model by adding a reluctance network to estimate the rotor MMF drop in the magnetic material of the rotor and stator yokes. This hybrid model is used to compute the no-load flux and the armature inductance. In this latter case, we take account of the leakage flux in slots and slot openings and the cyclic inductance is calculated with the complete stator winding configuration. 3.3 3D FE model of the Lundell alternator With the 3D FE modeling, the electrical parameters of the alternator can be directly obtained by applying two methods: a method using the scalar potential and another

one using the vector potential. Both methods are used to estimate the computation errors (Henneron et al., 2004). 4. Accuracy of magnetic models To evaluate the relative accuracies of the different magnetic models, the simulated output performances of a given Lundell car alternator with 36 slots and 12 poles have been compared. We have chosen a structure that respects the hypotheses and the constraints of all modeling methods. For this purpose, the real shape of the rotor structure and claws has been slightly adapted and the geometry of Figure 5 has been adopted. Table I shows the results of a 3D FE simulation with scalar and vector potentials resolutions. The average values of the parameters obtained with the two FE resolution methods are taken as reference values (Henneron et al., 2004). The maximal relative error on every parameter is calculated in relation to these reference values. The maximal relative error is encountered for the linear no-load flux value llin. Table II shows the results obtained with the reluctance and hybrid analytical models of the same alternator. The hybrid analytical model has a better accuracy than the reluctance network. Figure 6 shows the output DC current vs speed characteristics obtained with each set of parameters obtained with the different calculation methods. One observes some important differences between the analytical modeling methods and the 3D FE model. The short-circuit current at high speed is underestimated and the starting generation speed is lower with analytical methods. One can conclude that the lower accuracy of the analytical models limits the performance of a design procedure using optimization techniques.

Influence of the magnetic model accuracy 201

B (T) 2

1

Figure 5. Results of 3D FE simulation of a claw pole alternator 0

llin (mWb) lsat (mWb) Llin (m H)

Scalar pot.

Vector pot.

Average values

Maximal relative error (per cent)

41.1 9.51 340

31.4 8.15 286

36.3 8.8 313

13.3 7.7 8.5

Table I. Electrical parameters with 3D FE method

COMPEL 27,1

202

5. Optimization design method and error compensation The global optimization methodology detailed by Cros et al. (2004) that can use different analytical models is applied to the alternator optimal design. It is associated to an error compensation mechanism illustrated by the flowchart of Figure 7. For each intermediate optimal design solution, one compares the electric parameters given by the analytical model to those obtained by a 3D FE computation. If there is a difference between these two sets of parameters, this solution is not acceptable. One must apply some correction factors on the electrical parameters of the analytical model to improve its accuracy as shown on equation (6). Another optimization process is then initiated: 8 > < llin ¼ ka · ðllin ÞAN Llin ¼ kl · ðLlin ÞAN ð6Þ > : lsat ¼ ksat · ðlsat ÞAN (llin)AN, (Llin)AN and (lsat)AN are the electrical parameters computed with the analytical model. ka, kl, ksat are the correction factors derived from equation (7). llin, Llin and lsat are the corrected parameters to be used in the next optimization process. a is a level-headedness coefficient between the new and the old correction factor (0 , a , 1). The correction factors are initialized to 1 for the first intermediate optimization step with the analytical model. After each intermediate optimization step, a 3D FE simulation is performed. The correction factors are then computed with equation (8) by using the FE and analytical model parameters. The final optimal solution is obtained when both set of parameters are identical:

Table II. Electrical parameters with analytical methods

Reluctance model

Maximal relative error (per cent)

Hybrid analytical model

Maximal relative error (per cent)

46.6 11.8 501

28.6 33.6 60.1

40 10.1 406

10.4 14.7 29.6

llin (mWb) lsat (mWb) Llin (m H)

160 140

DC current [A]

120 100 80 60 3D FE scalar potentials 3D FE vector potentials Magnetic reluctance model Analytical 2D field computation

40

Figure 6. Simulated DC current output vs speed characteristics with different magnetic models

20 0 0

2,000

4,000 Speed [RPM]

6,000

8,000

Influence of the magnetic model accuracy

Analytical design iterative process with global optimization New correction factors for analytical models

Optimal solution found

203

3D FE evaluation and validation No

Figure 7. Flowchart of the design method with global optimization and model error correction

Yes Final solution

8 ka ¼ ð1 2 aÞ · kai21 þ a · kai > > < kl ¼ ð1 2 aÞ · kli21 þ a · kli > > : ksat ¼ ð1 2 aÞ · ksati21 þ a · ksati


llin2FE kai ¼ llin2AN




k li ¼

i

Llin2FE Llin2AN



ð7Þ




ksati i

lsat2FE ¼ lsat2AN

 ð8Þ i

The optimal process with error compensation has been applied to the design of an alternator with the specifications of Figure 2 and a fixed total volume. Three iterations with the 3D FE software are sufficient to minimize the analytical model errors and to validate the optimal found solution. Figure 8 shows a comparative analysis between the final optimal results obtained with use of the two analytical corrected models. One can notice that the two optimal structures have the same electrical model parameters but with different number of turns per slot. The performances of these structures are validated with the 3D FE software. The optimal structure obtained with the hybrid analytical model has a lower total Alternator FE analysis

Electrical model

Global parameters

RMS no-load flux: 12.35 mwb

Armature specific loading As = 64318 A/m Field specific loading Ar = 43446 A/m Armature Current density Js = 27.1 A/mm2 Field Current density Jr = 10.2 A/mm2 Armature specific loading As = 58660 A/m Field specific loading Ar = 43687 A/m Armature Current density Js = 29.3 A/mm2 Field Current density Jr = 10.4 A/mm2

Cyclic inductance: 134 µH Reluctance model

Stator resistance: 67 mΩ Number of turns per slot : 12 RMS no-load flux: 12.35 mwb

Hybrid analytical model

Cyclic inductance: 134 µH Stator resistance: 67 mΩ Number of turns per slot : 11

Weight

Ptot = 6.7 kg Pcu = 0.96 kg Pfe = 5.7 kg

Ptot = 6 kg Pcu = 0.86 kg Pfe = 5.2 kg

Figure 8. Comparative analysis of optimal solutions obtained with different analytical models

COMPEL 27,1

204

weight and smaller rotor claw widths than the other one, while using the same total volume. Global design parameters as armature specific loadings and current densities are similar. The correction method improves the accuracy and has an influence on the final result in order to satisfy all constraints and specifications. However, the sensitivity of each analytical model to the design parameters variations has not been improved and this comparative analysis shows that optimal solutions directly depend on the analytical model used. 6. Conclusion This study shows that it is possible to compensate errors of analytical models with a limited number of 3D field calculations during a global optimization design process. The overall convergence of the process is very fast and the 3D FE software validates the solution. However, the optimal solution depends on the sensitivity of the analytical model that is not improved by the correction method. References Cros, J., Figueroa, J. and Viarouge, P. (2004), “Analytical design method of polyphase claw-pole machines”, paper presented at the Annual meeting of IEEE – Industrial Application Society, October 3-7. Figueroa, J., Cros, J. and Viarouge, P. (2005), “Analytical model of a 3-phase rectifier for the design of a car alternator”, Electrimacs’ 2005, April 17-20. Henneron, T., Cle´net, S., Cros, J. and Viarouge, P. (2004), “Evaluation of 3D finite element method to study and design a soft magnetic composite machine”, IEEE Trans. on Magnetics, Vol. 40 No. 2, pp. 786-9. Zhu, Z.Q. and Howe, D. (1993), “Instantaneous magnetic field distribution in brushless permanent magnet DC motors”, IEEE Transactions on Magnetics, Vol. 29 No. 1, pp. 136-58. About the authors J. Cros received the doctor degree from the Institut National Polytechnique of Toulouse in 1992. He joined the Department of Electrical Engineering of Laval University in Quebec in 1995 as Professor. He is working in the research laboratory LEEPCI on electric actuator design with soft magnetic composite material. J. Cros is the corresponding author and can be contacted at: [email protected] L. Radaorozandry was born in Toliara, Madagascar, in 1979. He received the engineering degree from the Ecole Supe´rieure Polytechnique d’Antananarivo in 2003. He is currently working toward the MSc degree at LEEPCI, Laval University. His main interests are power electronics and electrical machines. J. Figueroa was born in Santiago, Chile in 1972. He received the engineering degree from Universidad de Chile in 1998 and the MSc degree from Laval University in 2003. Currently, he is working towards a PhD degree at LEEPCI, Laval University. His main interests are power electronics and electrical machines P. Viarouge received the doctor of engineering degree from the Institut National Polytechnique, Toulouse, France in 1979. Since, he has been a Professor with the Department of Electrical Engineering, Laval University, Quebec. He is working in the research laboratory LEEPCI on power electronics, AC drives and permanent magnet machine design.

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

Modeling of a beam structure with piezoelectric materials: introduction to SSD techniques Romain Corcolle, Erwan Salau¨n, Fre´de´ric Bouillault and Yves Bernard

Modeling of a beam structure

205

Laboratoire de Ge´nie Electrique de Paris,Gif-sur-Yvette, France, and

Claude Richard, Adrien Badel and Daniel Guyomar LGEF, INSA, Villeurbanne, France Abstract Purpose – To provide a model that allows testing and understanding special damping techniques. Design/methodology/approach – The finite element modeling takes into account the piezoelectric coupling. It is used with a non linear electrical circuit. The approach leads to an accurate tool to observe the behavior of the non linear damping techniques such as synchronized switch damping. Findings – The model has been validated by comparison with Ansysw but the CPU time required for the model is around one hundred times shorter. Research limitations/implications – The proposed model is 1D and the assumptions to use it are not verified for all structures. Practical implications – The authors obtain a useful tool for the design of damping structures (for example to find the best localisation of the piezoelectric patches and to test electrical circuits). Originality/value – The model is used for the design and conception of damping as well as for harvesting structures. Keywords Finite element analysis, Modelling, Damping, Electric contacts Paper type Research paper

1. Presentation of device The device presented (Figure 1) is quite elementary but represents rather well real structures. It is constituted of a cantilever beam, piezoelectric material stuck on one face of the beam and electrodes on the piezoelectric material near the clamp. Geometric parameters in 1D are shown in Figure 2. In the case of collocated patches, electrodes are plugged in parallel. 2. Structure damping: SSD techniques It is well-known that adding a resistor across the electrodes of a free vibrating piezoelectric transducer increases the vibration damping of this oscillator. But, the optimum resistor value is frequency dependent, and so, the damping efficiency is narrow banded. Synchronized switch damping (SSD) techniques described below are more efficient (Richard et al., 2000). These techniques are non-active solutions to damp vibrating devices. Most of the time, electrodes are left on a open circuit. Then, electrodes are pluged on a circuit for a very short time. Two different circuits are interesting: short circuit (SSDS for SSD on a short circuit) and inductor (SSDI for SSD on an inductor). The best time to switch

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 205-214 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836762

COMPEL 27,1

Electrode

y

z bp Piezo

206

hp h

Figure 1. Cantilever beam with piezoelectric patches

b f x

Piezoelectric poling xp

+V

e3 hp

e1

h

O +V

Figure 2. Model parameters

l

electrodes on a circuit is at a maximum displacement of the beam extremity (which means a maximum voltage on the electrodes too). In this case, the voltage is 90 degrees out of phase with the motion then enhancing the damping mechanism. 2.1 SSDS When switching on a short circuit, V(t) ¼ 0. Then, when left open, the charge Q(t) stays constant until the next switch. 2.2 SSDI Switching on an inductor allows the inversion of the voltage. In this case, the voltage amplitude is higher than during the SSDS technique, improving damping. 2.3 Experimental results A sinusoidal force is applied to the extremity of the beam. Figure 3 shows the measured voltage and displacement of the extremity of the beam of both SSDS and SSDI techniques. With a sinusoidal external force, voltage is not sinus; SSD techniques are non linear, that is why modeling is a solution.

20

Displacement (mm) - Voltage (V)

Displacement (mm) - Voltage (V)

Modeling of a beam structure

30

25

15 10 5 0 –5 –10 –15

20 10 0

207

–10 –20

–20 –25

0

0.04

0.08

0.12

0.16

0.2

–30

0

0.04

0.08

0.12

0.16

0.2

Time in seconds

Time in seconds

3. Variational principle The most used variational principle for dynamic structures is the Hamilton one. It claims that, for any time interval: Z t2 d ðL þ WÞdt ¼ 0 ð1Þ t1

where L is the Lagrangian (L ¼ J 2 H : kinetic energy J and electric enthalpy H ) and W is the virtual work of external mechanical and electric forces. All variations must vanish at t ¼ t1 and t ¼ t2. The different energies are (Piefort, 2001): "


 # Z Z 1 ›u 1 2 ›u 2 2 ›u3 2 1 t r þ þ r {_u}{_u}dV ð2Þ J¼ dV ¼ 2 › t › t › t V v 2 as the kinetic energy:

8 9 u1 > > > = < > u2 along the ð~e1 ; ~e2 ; ~e3 Þ axisÞ; ð{u} is the displacement field vector : > > > ; : u3 > Z 1 t ð {S}{T} 2t {E}{D}ÞdV ð3Þ H¼ V 2

as the electric enthalpy (where {T} is the strain tensor, {S} the stress tensor, {D} the electric displacement vector and {E} the electric field vector): Z t W¼ {u}{F V }dV þ fQ ð4Þ V

as the virtual work of external forces ({FV} represent body applied forces, Q the charge accumulated on the electrodes and f the electric potential). And so, for arbitrary variations of the displacement field {du} (making variation of the strain {dS}) and of the electric potential df (making variations of the electric field {dE}):

Figure 3. SSDS (left) and SSDI (right) voltage shapes

COMPEL 27,1

208

Z

t 2 Z

ðr t {du_ }{_u} 2t{dS}½c E
{S} þt{dS}t ½e
{E} þt{dE}½e
{S} t1 V  t T t þ {dE}½1
{E} þ {du}{F V }ÞdV þ dfQ dt

0¼

The kinetic term can be integrated: Z Z t2 t r t {du_ }{_u}dt ¼ ½r t {du}{_u}
t21 2 t1

t2

r t {du}{€u}dt

ð5Þ

ð6Þ

t1

and, {du} being equal to zero at t ¼ t1 and t ¼ t2, the first term vanishes. Equation (5) is true for any time interval so the term that have to be time-integrated is equal to zero. Finally: Z ðr t {du}{€u} 2t{dS}½c E
{S} þt{dS}t ½e
{E} þt{dE}½e
{S} ð7Þ V t T t þ {dE}½1
{E} þ {du}{F V }ÞdV þ dfQ ¼ 0 4. Equations First of all, equations of elasticity and piezolectricity are valid in 3D. To develop a 1D (or 2D) model, approximations have to be done. For example, a typical approximation in mechanical problems is plane stress/strain formulations. The strain is linked to the displacement field. According to the geometry of the system, only bending is taken account. The use of Love-Kirchhoff hypothesis leads to this displacement field (Chevalier, 1995): 8 u1 ¼ 2z ››wx > > < u2 ¼ 0 ð8Þ > > : u3 ¼ wðxÞ where w(x) is the displacement of the neutral axis along the ~e3 axis. In fact, this displacement field is a plane stress approximation. And so, with Euler-Bernoulli hypothesis, only S11, T11 and T22 are non equal to zero. The beam behavior law is simply: T 11 ¼

E S 11 1 2 v2

ð9Þ

The piezoelectric behavior laws are not so simple to determine; many formulations exist, linked to the choice of unknowns. Because of the plane stress approximation, the mechanic compliance has to be used. That is why the piezoelectric behavior laws to use can be: ( {S} ¼ ½s E
{T} þt ½d
{E} ð10Þ {D} ¼ ½d
{T} þ ½1 T
{E}

So, equations become in 1D: 8 S 11 ¼ s E11 T 11 þ s E12 T 22 þ d31 E 3 > > < S 22 ¼ 0 ¼ s E12 T 12 þ s E22 T 22 þ d32 E 3 > > : D3 ¼ d31 T 11 þ d32 T 22 þ eT E 3

Modeling of a beam structure ð11Þ

33

209

Eliminating T22 from these equations (and noticing that s E11 ¼ s E22 and d31 ¼ d32) and keeping S11 and E3 as unknowns (in the second member) lead to this piezoelectricity behavior equations: 8 < T 11 ¼ c Eeq S 11 2 eeq E 3 ð12Þ : D3 ¼ eeq S 11 þ 1Seq E 3 The equivalent coefficients are function of s E11 ; s E12 ; d31 and eT33 : Moreover, S11 and E3 are linked to the unknowns to determine w(x) and f: S 11 ¼ 2z

› 2w ›x 2

E3 ¼ 2

f hp

ð13Þ

5. Finite element formulation 5.1 Hermite elements In finite element formulation, the displacement field {u} (only function of w(x) in this case) is related to the corresponding node values {ui} (wi in this case) by the mean of the shape functions [l ]. In a lot of problems, Lagrange elements are used because of their simplicity. But in that case, these elements would not be appropriated. Indeed, Lagrange elements has a C 0-continuity so, the solution w(x) rebuilt from the node values would be continuous. But, in bending structure problems, this is not enough; the derivative of w(x) has to be continuous too: this is the C 1-continuity. That is why Hermite elements must be used (Ida and Bastos, 1997), they are made to ensure C 1-continuity (some other Hermite elements can ensure C 2-continuity too). At each node, the unknown is not only wi anymore, but also the derivative at this node ›xwi. So, in 1D like in this model, one element (two nodes) is linked to four unknowns, and so to four shape functions (l1, l2, l3 and l4). On a Hermite element, the solution w(x) is equal to: 9 8 w1 > > > > > > > > > = < ›x w1 > ð14Þ wðxÞ ¼ ½lðxÞ
> w2 > > > > > > > > ; : ›x w2 > with ½lðxÞ
¼ ½l1 ðxÞ l2 ðxÞ l3 ðxÞ l4 ðxÞ
: So, derivating w(x) is equivalent to derivating li(x). In this case, S11 can be related to the node unknowns:

COMPEL 27,1 S 11

9 8 w1 > > > > > > > > >  2 > = < › w 2 x 1 ›w dl ¼ 2z 2 ¼ 2z ðxÞ > w2 > ›x dx 2 > > > > > > > ; : ›x w2 >

ð15Þ

210 5.2 Matrix assembly In equation (7), the volume integration is equal to the sum of the integration on each element (segments along x). By writing equality of node values of an element with its neighbouring ones, global matrices can be assembled from elementary matrices for each element. So, the global equation system is: 8 < ½M
{U€ i } þ ½K UU
{U i } þ ½K U f
{f} ¼ {F i } ð16Þ : ½K fU
{U i } þ ½K ff
{f} ¼ {Q} 6. System solving 6.1 Alternative formulation It has to be noticed that no damping has been taken account in the equations. Because of experimental results, mechanical damping can be added in a global matrix equation by: 8 < ½M
{U€ i } þ ½C
{U_ i } þ ½K
{U i } ¼ {F i } ð17Þ : with ½C
¼ b½K
This is called b-damping (b dimension in s). In that case with coupling between mechanic and electricity, only the mechanical stiffness matrix [C ] ¼ b[KUU] has to be multiplied. The whole system is now: #8 9 " " #8 9 " #( ) ( ) ½K UU
½K U f
Ui Fi ½C
0 < U_ i = ½M
0 < U€ i = þ þ ð18Þ ¼ ½K fU
½K ff
f Q 0 0 : f_ ; 0 0 : f€ ; Renaming these global matrices, the system is: 8 9 8 9 ( ) ( ) < U€ i = < U_ i = Ui Fi ½M
þ ½C
þ ½K
¼ € _ f :f; :f; Q

ð19Þ

In particular cases, it will be worth to determine Q as an unknown and put f as an external electric force. Mass and damping matrices will be the same, only the stiffness matrix will change: 2 3 ½K UU
2 ½K U f
½K ff
21 ½K fU
½K U f
½K ff
21 5 ð20Þ ½K0
¼ 4 2½K ff
21 ½K fU
½K ff
21

Modeling of a beam structure

And so, the system equation is: ( ) ( ) ( € ) ( _ ) Fi Ui Ui Ui 0 ½M
€ ¼ þ ½C
_ þ ½K
f Q Q Q

ð21Þ

For example, if the electrodes are left open (Q ¼ 0), the equation (19) will be the one to use putting Q ¼ 0 in the external force vector; whereas if the electrodes are short-circuited the equation (21) will be the one to use putting f ¼ 0 in the external force vector. 6.2 Newmark method The Newmark method is used to solve a time-dependent equation. A vector is created at every time step and is function of the solution vector, its velocity and its acceleration at the time before (time (n 2 1)Dt). Thanks to this vector, the new solution vector (time nDt) can be computed, as well as velocity and acceleration. Next time iterations are computed repeating the same process. In the case of a resistor connected to the electrodes, the circuit equation is f ¼ RðdQ=dtÞ: Equation (21) may be used, and the external electric force f may be put in the first member (and forced to zero in the second member), changing the damping matrix: " # ½C
0 ½C
¼ ð22Þ 0 2R It is the same for an inductor f ¼ Lðd2 Q=dt 2 Þ (f may be put in the first member and forced to zero in the second member), changing in that case the mass matrix: " # ½M
0 ½M
¼ ð23Þ 0 2L For a resistor and an inductor plugged in serial, both matrices would have to be changed. With the SSD techniques, the circuit changes every-time. When the electrodes are left open, equation (19) is used (with Q forced to the value it had at the switch off until the next switch on). When switching on short-circuit or inductor, equation (19) is used (with f forced to zero in the external force vector and eventually matrices changing (with L and R)). The difficulty is to switch between two different equations during the computation (and so switch between f and Q as an unknown); when changing equation, new initial conditions have to be given. 7. Model results To evaluate the model, results are compared with a 2D ANSYS model. First of all, the developed model is able to compute 50,000 time steps in 30 s for a 50 nodes beam. With ANSYS, the same computation is about one hour long (150 nodes in a 2D mesh). Every computation is done close to the first modal frequency.

211

voltage [V]

212

For simple circuit plugs, frequency response is suffisant because every unknowns are harmonic. But in the case of SSD techniques, the electric potential f is not harmonic; so, time computation is necessary. Figure 4 shows the beam extremity displacement and the piezoelectric voltage computed with the developed model in the case of an open circuit. ANSYS model plots are not given here because the shapes are exactly the same (with each plugged circuit), only small differences appear on amplitudes and resonance frequencies. These differences will be given in Table I. Next figures show that the damping can be increased thanks to a resistor, but even more with the SSDI technique (Figures 5-7). The voltage is out of phase with the displacement, enhancing the damping (the ideal resistor value at a frequency is the one which brings a 908 phase out). The voltage shape is the one expected. It has to be noticed that the displacement is not anymore a sinus. Damping energy can be evaluated with the area of the cycle strain/stress taken at the middle of the patches. This cycle shows that for the same displacement, the SSDI technique dissipates a lot more energy during a period than the resistor for example. As the shapes were the same between the developed model results and the ANSYS ones, small differences in amplitudes and resonance frequencies have been noticed. Table II presents the maximum amplitude close to the first mode and its frequency for each circuit with the same force applied at the extremity of the beam. This developed model seems to be quite accurate because of the concordance with the ANSYS model results. Amplitudes are very close with less than 1 percent different

Figure 4. Open circuit displacement and voltage

Table I. Results comparison

150

3

100

2

50

1

0

0

–50

–1

–100

–2

–150 0

Model amplitude Model frequency (Hz) ANSYS amplitude ANSYS frequency (Hz)

0.01

0.02 time [s]

displacement [mm]

COMPEL 27,1

–3 0.04

0.03

Open circuit

R

SSDS

SSDI

2.44 mm 63.32 2.12 mm 63.23

0.54 mm 62.78 0.53 mm 62.68

0.27 mm 63.32 0.27 mm 63.23

45.8 mm 63.32 45.8 mm 63.23

10

0.2

0

0

–10

213

– 0.2

–20 0

voltage [V]

Modeling of a beam structure displacement [mm]

0.4

0.01

0.02 time [s]

0.03

15

0.06

10

0.04

5

0.02

0

0

–5

– 0.02

–10

– 0.04

–15

– 0.06 0.04

0

0.01

0.02 time [s]

0.03

Figure 5. Displacement and voltage with a resistor plug

– 0.4 0.04

displacement [mm]

voltage [V]

20

in most cases (only the open circuit has a 10 percent difference). Resonance frequencies are close too, with less than a 0.1 Hz difference with each circuit. To show the SSD techniques efficiency, damping times can be compared. A sinusoidal force is applied until the permanent regim is reached and then canceled. The damping time is equal to the duration to reach 5 percent of the permanent regim amplitude. 8. Conclusion Such a coupled problem has been solved thanks to numerical modeling. Experiment and computed solution (ANSYS and this model) are not so far. Using this model in a optimization process would maybe lead to a smarter use of piezoelectric materials. SSD techniques are very efficient with a very low power comsuption (about 1 mW to control the switch). New techniques are being developed: new damping ones and harvesting ones too.

Figure 6. Displacement and voltage with SSDI

COMPEL 27,1

1

× 106

214

stress T11 [Nm–2]

0.5

0

– 0.5

open circuit resistor

Figure 7. Damping energy for the same displacement

Table II. Damping times

SSDI –1 –6

Damping time

–4

–2 0 2 strain S11 [no dimension]

4

6 × 10– 6

Open circuit

R

SSDS

SSDI

2.95 s

650 ms

320 ms

140 ms

References Chevalier, L. (1995), Me´canique des syste`mes et des milieux de´formables, Ellipses, Paris, pp. 330-62. Ida, N. and Bastos, J.P.A. (1997), Electromagnetics and Calculation of Fields, 2nd ed., Springer, Berlin. Piefort, V. (2001), “Finite element modelling of piezoelectric active structures”, Thesis report. Richard, C., Guyomar, D., Audigier, D. and Bassaler, H. (2000), “Enhanced semi passive damping using continuous switching of a piezoelectric device on an inductor”, paper presented at the 7th SPIE International Symposium on Smart Materials and Structures. Corresponding author Yves Bernard can be contacted at: [email protected]

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

Micromagnetism3D micromagnetismmagnetostatic coupling magnetostatic coupling technique for MR reading heads modeling I. Firastrau

215

Transilvania University of Brasov, Brasov, Romania and Laboratoire SPINTEC, CEA-CNRS-INPG-UJF, Grenoble, France

L.D. Buda-Prejbeanu Laboratoire SPINTEC, CEA-CNRS-INPG-UJF, Grenoble, France

J.C. Toussaint Laboratoire SPINTEC, CEA-CNRS-INPG-UJF, Grenoble, France and Laboratoire Louis Ne´el, CNRS-INPG-UJF, Grenoble, France, and

J-P. Nozie`res Laboratoire SPINTEC, CEA-CNRS-INPG-UJF, Grenoble, France Abstract Purpose – The purpose of this paper is to develop an original approach to simulate the reading process for multitrack shielded magneto-resistive reading (MR) heads. Design/methodology/approach – The shields and the media are of micron size while the sensor has sizes comparable with the characteristic length scales of the magnetic materials which are of the order of nanometer. Because of this large difference of scales between the different parts of the head, the macroscopic shields and the media are described by a boundary element method (BEM) approach, while the sensor is treated by micromagnetism in order to reconstruct the response of shielded multitrack MR head. To select the most favorable approach, several releases were implemented and compared. A technique based on a full-coupling procedure was found to be the most general but too expensive in time. Appling the perfect-imaging method directly into the micromagnetic simulator, the authors succeed in accelerating the computation without loosing accuracy. Findings – Solving by BEM the Poisson equation for the scalar magnetic potential only the surfaces interfaces are discretised, saving thus computation time and memory resources. In addition, for multi-tracks data pattern, the magnetic scalar potential may be estimated with a good approximation by considering a periodic system along the crosstrack direction. By applying the Fourier series expansion for the magnetic charges distribution along the crosstrack direction, the initial BEM 3D problem can be treated as a bi-dimensional one. Originality/value – This macroscopic-microscopic coupling technique allows a full description of the behaviour of the magnetic sensor in its environment, being a useful tool for the design and the optimisation of the multitrack MR reading heads. Keywords Magnetism, Boundary elements, Methods, Magnetic storage Paper type Research paper

1. Introduction The magnetic storage industry is continuously looking for suitable solutions to increase the capacity and to reduce the access time to the information. For tape recording, one possibility consists of writing adjacent track packages, far more space

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 215-223 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836771

COMPEL 27,1

216

efficient since the guard bands are positioned only between packages (O-Mass, n.d.). Read-out, nevertheless, has then to be performed by a single element, as there is no space left to use multiple adjacent read heads. The challenge, in this case, is to define the minimum track width for read, without detrimental cross-talk. In order to investigate the reading process based on a multitrack magneto-resistive (MR) head, we built an original numerical tool based on boundary element method (BEM) (Brebbia, 1978) and micromagnetic modeling (Brown, 1963). In a reading head the MR sensor is inserted between two magnetic shields which enable to separate the fields produced by two adjacent bits written on the media. The shields and the media are of micron size or larger, while the sensor has dimensions comparable with the characteristic lengths of magnetic materials which are of the order of the nanometer (exchange length, magnetic domain wall width). Modeling of such systems represents therefore a complex computational issue since some elements of the head have to be treated macroscopically while the sensor has to be analyzed at the nanometric scale by using a micromagnetic approach. A such microscopicmacroscopic coupling technique seems to be the proper way to analyze the performances of the multitrack heads. 2. Model description In the case of longitudinal magnetic recording, the magnetization lies in the plane of the media, and the information is stored by the presence or by the absence of a magnetic transition (Figure 1). The thin active element of the reading head picks up the magnetic field generated by these magnetic transitions written on the media. This field is usually calculated by 2D models limited to ideal shields having infinite permeability (Yuan and Bertram, 1993). For 3D heads with arbitrar permeability, large-scale finite element method (FEM) computation could be used (Suzuki and Ishikawa, 1998). Because the MR reading heads for disk/tape drives discussed here are systems with very different dimensions, the BEM seems the most adapted. This is because only the interfaces are discretized and not the whole volume, as a common FEM approach does, saving thus a lot of computation time and memory resources. Sensor

y Shield m r

x z

Shield m r

Guard-band

n tio

c

ire

kd ac str

os Cr

Figure 1. A general view of a shielded multitrack MR reading head

s Media Downtrack direction

Magnetic transition

t

T

To evaluate media stray field in the presence of macroscopic shields, characterized by a finite magnetic permeability mr, the Poisson’s equation for the scalar magnetic potential is solved. In addition, for multitrack reading head, the magnetic scalar potential may be estimated with a good approximation by considering a periodic system along the crosstrack direction (Oz). By applying the Fourier series expansion for the magnetic charge distribution along the crosstrack direction, the initial BEM 3D problem can be transformed into a set of bi-dimensional problems, as many as the number of modes of the Fourier expansion. For each of these modes, p, the magnetic scalar potential fp is the solution of a 2D modified Helmholtz type equation: "
# 2p p 2 D2 fp ðrÞ ¼ 2rp ðrÞ ð1Þ T where rp denotes the p coefficient of the Fourier series expansion for the magnetic charge distribution and T the system’s period. To simplify the writing we are limited here only at the volume charges. The uniqueness of the equation (1) solution is assured by the passage and limit conditions: h

fp ðrÞ continous; ;r [ surface domain i ~ fm ðrÞ 2 7 ~ fa ðrÞ · n ¼ sp ðrÞ; ;r [ ›D m > ›D a 7 p p h i ~ fa ðrÞ 2 mr 7 ~ fs ðrÞ · n ¼ 0; ;r [ ›D a > ›D s 7 p p

ð2Þ

fp ðr ! 1Þ ! 0 Here, the index m symbolizes the geometric domain of the magnetic charges, a corresponds to domain of air and s to the shield’s one, respectively. sp denotes the surface magnetic charge density on the boundary ›D a of the air domain, whose normal unit vector is n. The method is conceptually straightforward, but some care had to be taken in order to insure numerical accuracy. The spatial distribution of the magnetostatic field of the media evaluated by BEM serves as an input to a micromagnetic description of the sensor’s response. The magnetic state of the sensor is the result of the competition between several interactions: the magnetocrystalline anisotropy, the exchange interaction, the magnetostatic interaction and Zeeman coupling. Each possible magnetic state corresponds to a local minimum of the total free energy of the ferromagnetic system. These equilibrium states may be reached by integrating the Landau-Lifshitz-Gillbert equation (Fruchart et al., 1998): ð1 þ a 2 Þ

›m ¼ 2g ðm £ m 0 H eff Þ 2 ga½m £ ðm £ m 0 H eff Þ
›t

ð3Þ

where g is the gyromagnetic factor and a is the Gilbert damping constant. m is a unit vector who defines the orientation of the magnetization vector MðrÞ ¼ M s mðrÞ. Its amplitude, Ms is assumed to be constant and represents the spontaneous magnetization of the ferromagnetic material, here the sensor. H eff represents the effective magnetic field derived from the functional derivation of the free energy:

Micromagnetismmagnetostatic coupling 217

COMPEL 27,1

218

H eff ¼ H anis þ H Zeeman þ H ex þ H D

ð4Þ

It involves the anisotropy and Zeeman fields, which are local terms, the exchange field which acts on short distances and a long range term, the demagnetizing field. Because of its non-local character, the demagnetizing field is the most difficult to calculate. The most efficient computation technique is to express it as a convolution of the gradient Green function G, solution of the equation DGðr 2 r 0 Þ ¼ 2dðr 2 r 0 Þ and the magnetic charges distribution inside the sensor r: H D ðrÞ ¼ 2½7r 0 Gðr 2 r 0 Þ £ rðr 0 Þ


ð5Þ

Our approach was developed in two levels. The first one previously reported (Firastrau et al., 2004), called partial coupling, does not take into account the effects of the shields on the micromagnetic state of the sensor. In the equation (1), the rp factor denotes in this case only the media charges distribution along the cross-track direction. Thus, during the reading process, the field issued from the media is computed by the BEM code and after that it is injected into a micromagnetic computation. It appears as a Zeeman field in the H eff expression (4). But, an usual micromagnetic simulation supposes that the ferromagnetic system is isolated. So, the shields act only on the stray field from the media and their direct effect on the sensor is not explicitly included. In reality, the sensor is coupled with the shields, therefore a certain modification of the self demagnetizing field inside the sensor, is expected. These effects are managed by a full coupling technique which adjusts progressively, in a self-consistent manner, the magnetic state of the sensor. In the equation (1), rp symbolizes this time all magnetic charges of the head. The starting point of the iterative procedure is the magnetic state of the sensor found by applying the partial coupling technique. After that, knowing the distribution of the magnetic charges of the head, the BEM approach allows us to evaluate the demagnetizing field inside the head in the presence of the shields. Thus, a better estimation of this field is obtained which enables to correct the magnetic state of the head during the next micromagnetic computation step. The procedure is repeated several times, after each micromagnetic minimization the correction to the demagnetizing field is updated by a BEM computation. The refinement is complete when the changes between two successive micromagnetic minimizations are below a settled tolerance. The major advantage of this full coupling technique is its high degree of generality allowing to deal with shields of various shapes with any value of the magnetic permeability. Unfortunately this technique is expensive in computation time because of iterative refinement of the solution. To solve this problem, for the case of high-permeable shields ðmr ! 1Þ, an alternative technique based on the perfect-imaging method (Yuan and Bertram, 1994) has been implemented. This time, the two shields are supposed to be infinite with respect to the y and z directions. Thus, the magnetic charges distributed inside the sensor are subject of multiple reflections with respect to each free surface of the shields. The magnetic charges of each image contribute to the demagnetizing field of the head. The sensor and its magnetic images indeed form a periodic magnetic system along the downtrack direction (Ox). The period is the double of the shield-to-shield distance denoted g (Figure 2). The form of the demagnetizing field becomes in this case:

þ1 h X

H D ðrÞ ¼ 2

i ~ r 0 Gðr 2 r 0 2 k n Þ £ ðrðr 0 Þ þ r~ðr 0 ÞÞ 7

ð6Þ

n¼21

where k n is a vector defined as k n ¼ 2gni. This periodical feature was directly integrated into our micromagnetic simulator, already customized to deal with periodic systems. Furthermore, by using the asymmetry relation existing between the magnetic charges of the sensor and that of its images r~ð2x; y; zÞ ¼ 2rðx; y; zÞ, only the magnetic system has to be discretized and not the entire gap volume. Thus, once the stray field issued from the media is applied, the magnetic state of the sensor is found by just one micromagnetic minimization. The method appears to be very promising because of the reasonable time of computation achieved.

Micromagnetismmagnetostatic coupling 219

3. Application The three micromagnetic-magnetostatic coupling approaches described before are applied here to study the reading process with a multitrack MR reading head. In this example, the head has to read a crosstrack data pattern consisting of identical transitions, situated at 40 nm distance from the bottom of the head. The tracks large of t ¼ 1 mm are separated by guard bands of s ¼ 250 nm. The thickness of the magnetic medium is 200 nm and the magnetization is supposed to vary linearly between two magnetic domains in a region of 50 nm. The sensor with a size of 15 nm £ 1 mm £ 2 mm, made of permalloy ðm0 M s ¼ 1:07 T; Aex ¼ 2 £ 10211 J=m; uniaxial magnetocrystalline anisotropy with m0 H K ¼ 2:14 mT along the z-axis), is divided into 3 £ 160 £ 200 cells. It is placed between two magnetic shields characterized by a constant magnetic permeability mr and a gap width of 200 nm. The shields are height of 2 mm and width of 1 mm. To insure a linear response of the head an uniform bias field of 15.4 mT is applied along the y direction. Periodic boundary conditions are imposed along the z direction and the micromagnetic simulations are made with the Gilbert damping constant set to one. The equilibrium state of the ferromagnetic system is found when the angle between the magnetization and the effective field decreases below 102 6 rad. To estimate the head signal response on the excitation of one track transition, the leads assumed to be infinitely thin, are located on the sensor surface at distances equal to the track width t. The output signal of the head is proportional to the mean value of g

2g Shield

µr→∞

2g

y x

µr→∞

2g Shield

Note: The head is considered to be uniformly magnetized and the magnetic charges are indicated by their sign

Figure 2. Scheme for the perfect imaging method in the case of the AMR multitrack head

COMPEL 27,1

220

the square of the angle between the leads’ current, which flows here along the z direction and the sensor magnetization distribution (Bertram, 1994). Two different values mr ¼ 50 and mr ¼ 3,500 for the magnetic permeability of the shields were considered in the simulations. As Figure 3(a) shows, the amplitude of the response signal and its pulse width at half-amplitude (PW50) diminishes when the mr increases. In fact, more permeable the shields are, smaller the magnetic flux generated by the media in the sensor is. The magnetic moments inside the sensor are indeed less aligned with the Oy direction with increasing the shield permeability and the mz magnetization component, responsible for the sensor response, is then less important. We remark also that the shape of the response curves recorded by the partial (Figure 3(a)) and by the full (Figure 3(b)) coupling techniques are rather similar. Instead the signal amplitude on the full coupling is approximately half of one of the partial coupling. This significant difference is due to direct magnetostatic interactions between the sensor and the shields. In the partial coupling approach, the stray field of the media is calculated in the region of the sensor, by taking into account the shields 0.55

µr = 50 µr = 3500

signal~

0.50 0.45 0.40 0.35

partial coupling

0.30 0.25 –500

0.24

signal~

500

µr = 50 µr = 3500 perfect imaging

0.22

Figure 3. (a) Simulated MR signal in the partial coupling approach for a weak (mr ¼ 50), respectively, strong (mr ¼ 3500) shields permeability value; (b) comparison between simulated MR signal obtained in the full coupling and the perfect imaging approaches

–250 0 250 downtrack position x0(nm) (a)

0.20 0.18 0.16

full coupling

0.14 0.12

–500

–250 0 250 downtrack position x0(nm) (b)

500

but not the sensor. In contrast, in the full coupling approach, the magnetostatic interactions of the sensor on the magnetic shields are taken into account. Thus, the magnetic flux is better canalized by the sensor and tends to accentuate the rotation of the magnetic moments along the y direction in comparison with the partial coupling approach (Figure 4(a) and (b)). This effect is also emphasized by the curves shown on the Figure 4(c), which correspond to the distribution of the magnetization (mz component) along the y direction, in the middle of the sensor obtained with the partial and the full coupling approach, respectively. In spite of its generality and high accuracy, the full coupling technique is too time consuming to be used for head optimization, especially when the number of the tracks inside a track-package is large (16, 32 or higher). Hopefully, the response curve obtained by applying the perfect-imaging method is almost superposed with the curve obtained by the full coupling technique (Figure 3(b)). The slight difference observed is due to the fact that in the perfect imaging approach the shields are supposed to be infinitely long along the y and z directions, which is not the case in the full coupling one.

Micromagnetismmagnetostatic coupling 221

y

→ Hbias

Ku z x (a)

(b)

1.0 partial coupling full coupling 0.8

mz

0.6 0.4 0.2 0.0

0

200

400

600

800

1000

y(nm) (c) Note: The graphic representations are made for shields with magnetic permeability of 50 and for the downtrack position of 2,5nm

Figure 4. Micromagnetic configurations of the sensor calculated by the: (a) partial; (b) full coupling technique; and (c) profile of the mz component of the magnetization along the y direction on the partial/full coupling approach

COMPEL 27,1

222

In addition, comparatively to the full coupling approach, the perfect-imaging method provides around 60 percent gain in term of computation time. In conclusion, for shields with a strong magnetic permeability the perfect imaging coupling method gives a good approximation of the signal response of the MR head, with reasonable computation time. Moreover, for weak magnetic permeability shields, the response head calculation can be accelerated by considering the partial coupling as a first estimation in the iterative process of full coupling method. Thus, for a qualitative but rapid estimation of the response curve of the head we can use the partial coupling technique, while for an accurate calculation the full coupling approaches are more adapted. 4. Conclusions By coupling the BEM and the micromagnetic approach, we fully describe the behavior of a magnetic sensor in its environment. Three 3D micromagnetism-magnetostatic coupling techniques are proposed. An approach based on a full-coupling procedure was found to be the most general but expensive in time. Appling the perfect-imaging method directly into the micromagnetic simulator, we succeed to accelerate the computation without loosing in accuracy. Problems such as cross-talk between adjacent tracks could be addressed. This numerical approach is a useful tool to design optimized multitrack MR reading heads. References Bertram, H.N. (1994), Theorie of Magnetic Recording, Cambridge University Press, Cambridge. Brebbia, C.A. (1978), The Boundary Element Method for Engineers, Pentech Press, London. Brown, J.F. (1963), Micromagnetics, Wiley, New York, NY. Firastrau, I., Buda, L.D., Toussaint, J-Ch. and Nozie`res, J.P. (2004), “Boundary element method and micromagnetism coupling for the magneto-restive heads modeling”, J. Magn. Magn. Mater., Vol. 738, pp. 272-6. Fruchart, O., Nozie`res, J.P., Kevorkian, B., Toussaint, J-Ch., Givord, D., Rousseaux, F., Decanini, D. and Carcenac, F. (1998), “High coercivity in ultrathin epitaxial micrometer-sized particles with in-plane magnetisation: experiment and numerical simulations”, Phys. Rev., Vol. 57, pp. 2596-606. O-Mass (n.d.), private comunication. Suzuki, Y. and Ishikawa, C. (1998), “A new method of calculating the medium field and demagnetizing field for MR heads”, IEEE Trans. Magn., Vol. 34 No. 4, pp. 1513-5. Yuan, S.W. and Bertram, H.N. (1993), “Magnetoresistives heads for ultra high density recording”, IEEE Trans. Magn., Vol. 29 No. 6, pp. 3811-6. Yuan, S.W. and Bertram, H.N. (1994), “Cross-track characteristics of shielded MR heads”, IEEE Trans. Magn., Vol. 30 No. 2, pp. 381-7. About the authors I. Firastrau was born on July 15, 1978. She graduated from the Transilvania University of Brasov in 2000, where she studied Physics and Mathematics. She received her PhD in Physics from the Institut National Politechnique de Grenoble in 2004. Her thesis topic involved the modelling of the reading process using magnetic heads. Her expertise was completed by two

years of post-doc in Spintec Laboratory in the spin electronics area. I. Firastrau is the corresponding author and can be contacted at: [email protected] L.D. Buda-Prejbeanu is an Assistant Professor at Institut National Politechnique de Grenoble (France) since 2005 (www.enspg.inpg.fr/). As a researcher, she is working in Spintec Laboratory from CEA Grenoble (France) (www.spintec.fr). Her work concerns the nanoscaled magnetic systems. Mainly, she is developing micromagnetic solvers to study the magnetisation dynamic under external excitations. These simulations serve to design/optimise magnetic applications (reading MR heads, magnetic oscillators, magnetic media). J.C. Toussaint was born in 1965. Since 2003, he has been a Full Professor of Solid State Physics and Applied Mathematics for Numerical Simulation at Institut Politechnique de Grenoble (France) (www.enspg.inpg.fr/), a researcher at the Neel Institut department NANOsciences CNRS Grenoble (France) (www.spintec.fr). His research consists in developing numerical solvers for micromagnetism and in applying them to the modeling of magnetisation configurations of nanoscaled systems in statics and in dynamics.

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

Micromagnetismmagnetostatic coupling 223

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

COMPEL 27,1

224

Analysis of the stray magnetic field created by faulty electrical machines V.P. Bui, O. Chadebec, L-L. Rouve and J-L. Coulomb Grenoble Electrical Engineering Laboratory, Domaine Universitaire, Saint Martin d’He`res, France Abstract Purpose – This paper aims to compute the magnetic stray field created by faulty electrical machines. Design/methodology/approach – This paper proposes two approaches to compute the magnetic stray field created by faulty electrical machines. The first one presents a homogenized FEM method. The second one is based on a combination of an analytical expression for the magnetic field in the machine air gap with an integral method. Findings – The studies show good agreement and demonstrate the reliability of the approach. Originality/value – Two models developed in this paper originally used to compute the stray magnetic field of electrical machines. They can contribute to develop new tools for fault monitoring. Keywords Electric machines, Finite element analysis, Electrical faults, Magnetic fields, Measurement Paper type Technical paper

1. Introduction The analysis of stray magnetic field created by electrical machines is known as a method for fault monitoring (Selvaggi et al., 2004; Henao et al., 2003). However, this task is not without complexity, the main difficulties being the need of a 3D model for computing the stray magnetic field and a high ratio between the internal and the external magnetic fields (106). An accurate formulation to compute the stray field has been proposed (Froidurot et al., 2002). It is based on a standard FEM computation with infinite box and then a post-processing with magnetic moments method. A first magnetostatic scalar FEM resolution is achieved. Once the scalar potential is obtained in the whole machine, the magnetization M in ferromagnetic parts Vmag is computed. Then it is possible to compute the stray field thanks to a volume integral equation which links the magnetization M to the outside magnetic induction B: ! ððð ~ · ~r ~ M M m0 ~ ~0 3 5 ~r 2 3 dV þ B ð1Þ B¼ r r 4p V mag

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 224-234 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836780

where r is the distance between the point where the magnetic induction is computed and the integration point, B0 is the induction created by winding. This integral post-processing approach is called moment method has been successfully validated by measurements (Froidurot et al., 2002). However, this 3D modeling is very memory consuming. In the case of faulty machines, geometry or physical symmetries are lost. Thus, the number of unknowns becomes too much significant to allow the modeling of the whole machine.

Therefore, the goal of this paper is to propose new models to evaluate the influence of faulty operation modes on the stray magnetic field of a synchronous machine, while keeping an acceptable number of unknowns. The first one carried out with homogenized FEM approach. This method is already known, but is originally adapted to the stray magnetic field computation. The second one is an original use of a classical integral method combined with an analytical expression of the induction in the machine air gap.

Analysis of the stray magnetic field 225

2. Proposed models 2.1 Homogenized FEM model The basic idea of this model is to replace the tooth and slot area by an equivalent homogenized continuous materials with characteristic B(H). The main assumption lies in an idealized distribution of induction flux inside the machine. The fields lines are tangential (according to a circular way) in the yoke and become radial in a zone located in the vicinity of the air gap and delimited, on both sides, by the end of the slots of the stator and the rotor. In the absence of a significant induced reaction (during no-load operation mode), this assumption is realistic because it does not modify the total reluctance associated inductor flux. Another assumption of our model concerns the linear materials law. To schematize the behavior of the induction flux in teeth, let us consider the Figure 1, in which Bt and Bs are average induction values, respectively, in the teeth and the slots. Let us suppose that the average values of the excitation field in the iron and in the vacuum are both equals to the given value H. The average induction B on this tooth and slot area is thus given by the expression: BðHÞ ¼

m0 £ mr £ H £ Lt þ m0 £ H £ Ls Lt þ Ls

ð2Þ

where mr is the permeability of the ferromagnetic material, Lt and Ls present the widths of tooth and slot. Thus, equation (2) leads to a homogenized equivalent permeability for each tooth and slot area. Moreover, as the magnetic flux density is mainly forced to remain radial, the permeability is then anisotropic. The radial m1, tangential m2, vertical m3 components of this permeability are calculated by: Ls

Lt

Bs

Bt

B

Figure 1. Simplified tooth and slot, associated induction in the electric machine

COMPEL 27,1

m1 ¼ m3 ¼

m2 ¼

226

mr £ Lt þ Ls Lt þ Ls

ð3Þ

Lt þ Ls Ls

ð4Þ

As the consequence, we only need a coarse mesh in each tooth and slot area, in comparison with standard FEM modeling where to take into account the real slot geometry leads to a very fine mesh. The number of unknowns is largely decreased, which allows modeling the complete machine without symmetry. In particular, several fault operation modes can be modeled (rotor short-circuits, for example). The stray magnetic field is computed by a post-processing with magnetic moments method. However, this kind of 3D FEM modeling is still difficult and heavy to manage (geometry description and mesh, need of CPU memory, problem of cancellation error). This is why, a simpler model is proposed below. 2.2 Air gap surface model This model is based on a combination between an analytical expression of the magnetic field in the machine air gap and an integral method. In a first assumption, we consider that the materials law is linear and the slot geometry is not taken into account. The model is then carried out by considering two remarks as follows. 2.2.1 Field due to magnetizations. Denote V a ferromagnetic body that channels a flux of induction (high-magnetic permeability). The field created by this device can be computed by the well known expression: ððð ~ 0 ÞÞdV ~ gðMðx ð5Þ BðxÞ ¼ m0 V

where x is the measurement point, x0 is the integration point, g is the mathematical expression of the field created by a dipole. In such a case, it is usual to consider that in this volume, M has a free divergence. The volume V creates exactly the same field as a surface S with a charge distribution q ¼ M · n, where n is the normal going outside from the volume. More precise explanations of this representation can be found (Durand, 1968). Then, the field can be computed by: ðð ~ 0Þ · n ~ ~ ðx 0 ÞÞdS f ðMðx ð6Þ BðxÞ ¼ m0 S

where f is the mathematical expression of the field created by a charge. This approach is called Coulombian representation of a ferromagnetic body. In the example shown in Figure 2, volume V channels perfectly the induction flux. Charges located on all the faces where M is perpendicular to the normal vanish. Only two opposite charges remain at the top and bottom of the volume. 2.2.2 Field due to the windings. Let us now consider a single loop C in which flows a current I. The magnetic field created by this current is the same as the one created by

every surface S delimited by C with a dipole distribution equal to I· n, where n is the normal to the surface (Durand, 1968). It is easy to see that such a surface dipole distribution is equivalent to the line current. To this end, we consider a network of small loop currents. The contributions of all currents cancel each other except on the line C. If the sizes of the small loops tend to 0, we get a distribution of current vortex equivalent to dipoles. This result is known as the Ampere equivalence (Figure 3). Let us now consider an electrical machine made of windings, an air gap, a stator and rotor. Under the equivalences above, the stray magnetic field is created (Durand, 1968): . inductor coils in which currents flow; and . charge surface density M · n located on surfaces which delimits the interface between ferromagnetic materials and air regions (M is the magnetization of the materials and n the external normal going from the materials to the air).

Analysis of the stray magnetic field 227

In an electrical machine, three main surfaces must be considered. The first one is the external boundary (stator/air interface), and the two others deals with the air gap (air gap/stator and air gap/rotor) which are closed to each other. These two last surfaces carry nearly same charges distribution, opposite signs (the induction in the air gap is mainly radial and continuous so associated magnetizations are the same and both normal have opposite directions). If we neglect the slot geometry, both surfaces can be replaced by an average one of the air gap carrying a distribution of normal dipoles Tmag. In most part of electrical machine, this dipole distribution is easy to determine, the field B in the air gap being well known and directly linked to M. Many analytical expressions already exist which depend, for example, on the poles number and the induction in the air gap. So, for a 2p-pole electrical machine with a sinusoidal air gap induction, this distribution created by the ferromagnetic materials can be represented: q+ = + M.n ++ ++ n

V S

M n ––

– – q = – M.n –

Note: Both magnetic fields computed are the same

Figure 2. Representation of a magnetized volume and its equivalent surface charge distribution

loop C

I surface S

Figure 3. Equivalence between the current flowing in a winding and a distribution of normal dipoles

COMPEL 27,1

228

T mag ðu; tÞ ¼ e £

mr 2 1 £ Bairgap_ max £ cosð pu 2 vs tÞ m0 mr

ð7Þ

where e is the thickness of the air gap, mr is the relative permeability of the material, u is the mechanical position of the rotor and vs is the electrical pulsation and t is the time. We can now apply the equivalence with rotor windings, lines C being the coils and the surface S being the average surface of the air gap. In most part of an electrical machine, the sum of the different I · n dipoles created by all the windings can be considered, in a first approximation, as sinusoidal. So, the field created by the windings can be represented by a distribution of normal dipoles located on the cylindrical average air gap surface. Its expression is: T wind ðu; tÞ ¼ 2At_ max £ cosð pu 2 vs tÞ

ð8Þ

where At_max is the maximum number of Ampere-turn for one pole. Let us notice that with such a distribution, end-winding effect is taken into account in the model. Thus, a distribution of normal dipoles T is obtained on the cylindrical surface in the air gap which is the sum of Tmag or Twind and takes into account both ferromagnetic and windings effects. Note that Twind and Tmag have opposite directions. In others words, Twind creates the source field and Tmag represents the shielding effect of the ferromagnetic materials. It remains now to determine the charge distribution on the external surface. We consider the machine as a whole ferromagnetic cylinder with an internal field source created by the distribution T described previously (its influence can be expressed with a simple numerical integration on the air gap surface). Then, we can use a classical global integral equation to obtain external charges q (Durand, 1968): ðð ~r · n ~ mr þ 1 1 ~ 0 ðPÞÞ ~ ·B qðPÞ 2 qðM Þ 3 dS ¼ 2ðn ð9Þ r 2p S mr 2 1 where r is the vector between a point P of the surface and M the integration point, B0 is the induction created by internal normal dipoles distribution T. By meshing the surface and using a point-matching approach, we get a full linear system resolved with LU decomposition. Then, charges distribution q is obtained and the stray magnetic field can be computed. Once the values of normal dipoles T and externals charges q at each time step is obtained, the stray magnetic field created in the air can be calculated thanks to a relation: ! ðð ðð ~ · ~r ~ ~r T T m m0 0 ~ ~ 3 5 r 2 3 dS þ q dS ð10Þ B¼ r r 4p Sair_gap 4p Sexternal r 3 As a consequence, this model is thus easy to implement by considering two cylindrical surfaces (air gap and external surfaces) meshed into rectangular elements. The air gap one is carried out with uniform normal dipoles distribution. Note that in this model, the mesh does not have any motion. The computation of the integral equations (9) and (10) is provided by Gauss numerical integration method. One of the main advantages of this approach is the simplicity of the machine description. In particular, lots of references provide analytical expressions of the field in

the air gap of electric machine in faulty operation modes (Timar, 1989). For example, these harmonic expressions concern rotor short-circuits and rotor dynamic eccentricity:
 vs t Tðu; tÞ ¼ k £ cosð pu 2 vs tÞ þ lcc £ cos u 2 ð11Þ p



 vs t Tðu; tÞ ¼ k £ cosð pu 2 vs tÞ £ 1 þ lde £ sin u 2 p

ð12Þ

where lcc and lde represent the constants linked to the fault amplitude of short-circuits and eccentricity, respectively. 3. Analysis of stray magnetic field In this section, the models are applied to a no-load operation mode of a complex four-poles synchronous generator working at low frequency. In a first step, the stray magnetic field of the healthy machine is computed thanks to our both models and the results are compared to those obtained by our reference. This reference is a standard FEM modeling (Figure 4(a)) in Flux3D software. A post-processing magnetic moments method to compute the stray magnetic field is used. Once both models are validated, different rotor faults (short-circuit windings, dynamic eccentricity) are introduced and their influence on the stray magnetic field are analyzed. 3.1 Modelling We have modeled in the first place a machine without iron housing. The model based on homogenized FEM approach has been carried out (Figure 4(b)) in Flux3D. Different regions correspond to different permeability values and directions. To avoid the field singularity around conductors crossing the material, the rotor windings are set in the machine air gap. Because of the FE numerical noise in the air region, stray magnetic field is also calculated by a post-processing magnetic moments method. Let us notice that the number of elements is highly reduced. We have also modeled the machine with our integral approach (Figure 4(c)). The internal cylinder is the air gap surface with normal dipoles distribution. The external surface represents the charge distribution obtained after the resolution of the linear system. 3.2 3D stray magnetic field computation The FEM homogenized model, the air gap surface model and the standard FEM model have been compared. The stray magnetic field obtained on the sensors outside the healthy machine (Figure 4(c)) are presented and show a very good adequacy (Figure 5) for all modeling, which enables us to validate our approaches. In this case the error is less than 10 percent, in some other cases still better than 20 percent. 3.3 Low-frequency analysis of stray magnetic field The stray magnetic field created by healthy studied machine can be approximated as follow (Wikswo and Swinney, 1985).

Analysis of the stray magnetic field 229

COMPEL 27,1

230

(a)

(b)

External surface with charges distribution

Sensors postions

Figure 4. (a) 3D standard FEM modeling; (b) homogenized FEM modeling of the whole machine; (c) boundary integral modeling of the whole machine

Normal dipoles distribution Charges value (Am) –600

–400

–200

0

200

400

(c)

Its stray magnetic field can be approximated by the rotation of a quadrupole (four-poles electrical machine) at the rotating pulsation vs/2 (Figure 6). In this case, the spectrum of this field will be constituted by a spectral line at synchronous frequency fs. If we introduce a rotor short-circuits of one pole machine, the machine loses its symmetry. The stray magnetic field can be approximated by the sum of two more simple representations (Figure 7): (1) one quadrupole, representative of the healthy machine (magnetic signature in 1/r 4); and (2) a more complex term with a very rich harmonic decomposition. If we introduce a dynamic eccentricity in this machine, its symmetry is lost. The stray magnetic field can be simply approximated by the sum of three distributions (Figure 8):

× 10–4 Reference (FEM with fine mesh) Surface Air-gap model Homogenised FEM

Magnetic induction (Tesla)

5 4

Analysis of the stray magnetic field 231

3 2 1 0 0.2

0.25

0.3

0.35 0.4 0.45 0.5 Distance to axis (meter)

0.55

0.6

0.65

Figure 5. Comparison of stray magnetic field obtained with our models and standard FEM

ws/2

Figure 6. Representation of a healthy machine

ws/2

ws/2

Note: This representation is the sum a quadrupole and a more complex term

ws/2

ws/2

Figure 7. Representation with a fault in the rotor windings

ws/2

Note: This representation is the sum a dipolar term, a quadrupole one and an octopole one

Figure 8. Representation with a rotor dynamic eccentricity

COMPEL 27,1

232

(1) the rotation of a quadrupole, representative of the healthy machine (magnetic signature in 1/r 4); (2) the rotation of two dipoles (magnetic signature in 1/r 3); and (3) the rotation of an octopole (magnetic signature in 1/r 5) which will be largely attenuated because of its strong decreasing law. Thanks to our models and the qualitative approximations above, faulty machines can thus be studied with an acceptable number of elements. The first fault considered is a one-ninth rotor short-circuits of one pole. The model used is the homogenized one. By analyzing the radial component of the stray magnetic field on one magnetic sensor outside the machine, we obtain the following low-frequency spectrum (Figure 9). Added to the synchronous frequency ( fs ¼ 5 Hz), a new spectral component appears at the rotation speed (2.5 Hz). The air gap surface model has also used to predict the magnetic stray field created by faulty machine with a dynamic eccentricity. The low-frequency spectrum is obtained. Added to the synchronous frequency ( fs ¼ 5 Hz) on Figure 10, two new spectral components appear at 2.5 and 7.5 Hz. 4. Conclusions In this paper, two new models are presented to compute the 3D stray magnetic field created by electric machines. The first one presents a well known homogenized technique, but originally used to compute the stray magnetic field. The second is a new, very light and easy to implement one. These two models permit also to precisely predict the frequency contents and the associated amplitudes of the stray magnetic field created by faulty machines. These models can easily be used to develop new tools methods for fault monitoring.

Healthy machine Short circuit rotor winding machine

90 80

dB

70

Figure 9. Low frequency spectrum obtained by the homogenized FEM model – comparison between healthy machine and rotor short-circuit machine

60 50 40

1

2

3

4

6 5 Frequency (Hz)

7

8

9

Analysis of the stray magnetic field

Healthy machine Rotor dynamic eccentricity machine

110 100

dB

90

233 80 70 60

1

2

3

4

6 5 Frequency (Hz)

7

8

9

Note: Comparison between healthy machine and a machine with an eccentricity

References Durand, E. (1968), Magne´tostatique, Masson et Cie, Paris. Froidurot, B., Rouve, L-L., Foggia, A., Bongiraud, J-P. and Meunier, G. (2002), “Magnetic discretion of naval propulsion machines”, IEEE Trans. Magn, Vol. 38 No. 2, pp. 1185-8. Henao, H., Demian, C. and Capolino, G-A. (2003), “A frequency-domain detection of stator winding faults in induction machine using an external flux sensor”, IEEE Transactions on Industry Applications, Vol. 39 No. 5. Selvaggi, J.P., Salon, S., Kwon, O-M. and Chari, M.V.K. (2004), “Calculating the external magnetic field from permanent magnets in permanent-magnet motors-an alternative method”, IEEE Trans. Magn, Vol. 40 No. 5, pp. 3278-85. Timar, P.L. (1989), Noise and Vibration of Electrical Machines, Elsevier Science, New York, NY. Wikswo, J.P. Jr and Swinney, K.R. (1985), “Scalar multipole expansions and their dipole equivalents”, Journal of Applied Physics, Vol. 57 No. 9, pp. 4301-8.

About the authors V.P. Bui was born in Danang, Vietnam, in 1980. He graduated in electrical engineering from HoChiMinh City University of Technology, Vietnam, in 2003. He is currently a PhD student at the Grenoble Electrical Engineering Laboratory (G2Elab), France. He studies the magnetic field computation methods, faults diagnosis of the electrical machines. O. Chadebec was born in Sens, France, in 1973. He graduated in electrical engineering from the Ecole Nationale Supe´rieure d’Inge´nieurs Electriciens de Grenoble (ENSIEG), France, in 1997. He received his PhD in electrical engineering in 2001 from the Institut National Polytechnique de Grenoble (INPG), France. At the present time, he is Research Associate CNRS in the team “Modeling and CAD in electromagnetism” of the Grenoble Electrical Engineering Laboratory (G2Elab), France. His current interests are magnetic field computation methods, inverse problems and low magnetic measurements. O. Chadebec is the corresponding author and can be contacted at: [email protected]

Figure 10. Low-frequency spectrum obtained by the “air gap” surface model

COMPEL 27,1

234

L-L. Rouve is currently a Research Engineer at the INPG, in the team “ERT CMF” of the Grenoble Electrical Engineering Laboratory (G2Elab), France. Her current interests are low-magnetic field measurements, electrical machines, inverse problems and identification. J-L. Coulomb was born in Nıˆmes, France, in 1949. He graduated in electrical engineering from the (ENSIEG), France, in 1972. He received his PhD in 1975 and the DSc in 1981 from Institut National Polytechnique de Grenoble (INPG), France. He is currently a Professor at the INPG. His activity of research is in the team “Modeling and CAD in electromagnetism” of the Grenoble Electrical Engineering Laboratory (G2Elab), France, relates to the field computation, the finite elements, the mesh generation, optimization and inverse problems, and the CAD software.

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

Analysis of the structure-dynamic behaviour of an induction machine with balancing kerfs C. Schlensok

Structuredynamic behaviour 235

Bosch Rexroth AG, Lohr am Main, Germany, and

K. Hameyer Institute of Electrical Machines, RWTH Aachen University, Aachen, Germany Abstract Purpose – To present results of research closely linked with real life applications. It resumes work of a period of about two years. Design/methodology/approach – Applying the finite-element method (FEM) the impact of balancing kerfs in the bars of squirrel-cage rotors of a small scale, mass series induction machine (IM) is studied. For the analysis and design optimization of the IM both, 2D electromagnetic, multi-slice and 3D structure-dynamic models are considered. Introducing and applying a novel 2D-3D force-transformation scheme, all possible balancing variants of the IM are studied in terms of electromagnetic and mechanical behaviour. Findings – The obtained results lead to a significant improvement of the studied IM. In fact, it is found, that the method of balancing the rotor by carving the rotor bars results in higher unbalanced pull rather than reducing it. This is due to electromagnetic unbalance caused by balancing. Hence, the IM is no longer balanced in series production. This again leads to a major economic benefit. Research limitations/implications – Using the FEM for simulation of structure dynamic problems is often limited to how the boundary layers are handled. In real life materials are not “connected” but glued or clamped. Therefore, the behaviour can only be adopted by manipulating the material parameters derived from iterative parameter adoption by measurement. Practical implications – Owing to the findings the IM is no longer balanced in series production, leading to a significant reduction of costs. In general, the applied methods can be used for the analysis and optimization of any kind of manufacturing or tolerance problem of electrical machines such as various kinds of eccentricity, punching kerfs, broken bars, magnetization errors in permanent-magnet machines, etc. Originality/value – This contribution gives a close insight of how to study the impact of manufacturing and tolerance problems of electric machinery, applying the method to an IM with balancing kerfs. Keywords Electric machines, Acoustics, Vibration, Finite element analysis, Induction machine, Harmonic analysis Paper type Research paper

1. Introduction 1.1 General approach For the analysis of the impact of manufacturing and tolerance problems in electrical machines a two-step chain of numeric models is applied as shown in Figure 1. With the finite-element method (FEM) electromagnetic and mechanical models are computed. The electromagnetic model consists of stator, winding, rotor, and air-gap and is excited by currents in the stator winding. Depending on the speed the rotor is moved in each simulation-time step applying the moving-band method (De Gersem et al., 2006).

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 235-245 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836799

COMPEL 27,1

The electromagnetic model provides the flux-density distribution from which the net-force, the torque, and the surface-force density are derived. The latter is used as ex-citation for the succeeding mechanical model. This model consists of all mechanical parts as described later in the paper. It provides the deformation of the machine from which the body-sound is derived.

236

1.2 Concrete problem Balancing rotating machine parts, for example, rotors of electrical machines, can be performed by various principles (Lingener, 1992). Depending on the mechanical part, its size, and the installation space either mass is added on the opposite side of the rotor’s unbalance (e.g. balanced wheel rims on cars) or mass is removed at its location. This method is preferred for rotors of electrical machines. Here, an induction machine (IM) with squirrel-cage rotor is analysed balanced by removing mass. As Figure 2 shows, the mass is milled out of the iron of the rotor lamination on both ends of the IM. In the studied machine four neighbouring rotor teeth are affected on each end different from the prototype shown. The maximum tolerated length and width of the kerfs are shown in Figure 3. From the longitudinal section Akerf of the kerf and the quadrangle Aquad the modified, effective length is derived: ~leff ¼ Akerf ¼ 12:74 mm < 17% · l Fe : Aquad

Figure 1. Computational two-step chain

Figure 2. Rotor of IM with balancing kerfs

ð1Þ

geometry material currents speed

Electromagnetic model

flux density forces torque

geometry material forces

Mechanical model

deformation body sound

balancing kerfs

2. Electromagnetic simulation In order to estimate the worst case configuration the maximum size of the balancing kerfs is considered for the electromagnetic, 2D finite-element model. Figure 4 shows the section of the rotor containing the kerfs. As a reference a model without kerfs is applied. Both models are simulated with a transient solver (van Riesen et al., 2004). The model is simulated at nominal speed n ¼ 1,200 min2 1 and at stator frequency of f1 ¼ 48.96 Hz. For each of the N ¼ 2,056 time steps simulated the flux-density distribution is provided. From this the torque and the surface-force density on the stator teeth are derived (Jordan, 1950; Schlensok and Henneberger, 2004; Timar, 1989). The analysis of the air-gap flux-density shows that the “balanced” model generates extra pole-pair numbers (Figure 5). The four-pole IM with NS ¼ 36 slots in the stator and NR ¼ 26 in the rotor now has odd numbers showing the dynamic eccentric impact of the balancing kerfs to the machine’s behaviour. Nevertheless, the most significant occurring pole-pair numbers n are: v¼p¼2

ð2Þ

v ¼ N R ^ p ¼ 24; 28

ð3Þ

v ¼ N S ^ p ¼ 34; 38:

ð4Þ

Structuredynamic behaviour 237

For the analysis of the torque the results of the two models are combined, since the balancing kerfs only affect an axial portion of the rotor (about 17 per cent of the 1.6

0.7

19

1.6

90° tooth

slot tooth (b) Cross section.

(a) Longitudinal section.

Figure 3. Maximum measures of the balancing kerf

kerf slot slot

rotor

stator

Figure 4. FE-model of IM with maximum worst-case balancing kerfs

COMPEL 27,1

0.8

B(v)additional B(v)reference

0.7

238

Bradial in T

0.6 0.5 0.4 0.3 0.2

Figure 5. Pole-pair numbers of the air-gap flux-density of both models

0.1 0.0

0

4

8

12

16 20 24 pole-pair number v

28

32

36

40

armature length on each end of the rotor). Therefore, the torque is averaged. 66 per cent are taken from the reference model and each 17 per cent of a model with kerfs depending on their location on the circumference. There are 13 models, since the kerfs on both ends can be shifted against each other. For the calculation of the torque in fact a modified multi-slice models (Gyselinck et al., 2001) is applied as Figure 6 shows in brief. The axial length of the rotor lz is assigned to i ¼ 1 . . . 5 slices with the individual axial length of lz · hi according to a weight of gi. In order to exclude other effects skewing has been neglected in the FEM-model. The multi-slice model results in a rather small impact to the torque behaviour decreasing the average torque by about 1.4%. However, balancing results for the studied IM in high-net forces of about F ¼ 23 N acting onto the bearings. Therefore, the analysed IM is no longer balanced. The resulting un-balanced fugal forces are smaller than the net forces produced by balancing. Owing to symmetry the net force of the reference model can be neglected. As the analysis of the surface-force density s on the stator teeth shows the balancing kerfs have some local effect to the force excitation of the IM. Figure 7 shows the surface-force density-excitation for the same stator teeth and the same time step for both models. It is stated that the kerfs generate some extra force peaks and raise the amplitude of already existing harmonics. Since, the kerfs rotate with the rotor this results in a modulation of the surface-force density-spectrum shown in Figure 8 (Jordan, 1950; Schlensok and Henneberger, 2004). Next to this, the s is raised for the regions of the stator teeth in the range of the kerfs throughout the spectrum.

Figure 6. Multi-slice model for torque and structure-dynamic simulation

lz 0.033

0.17

0.594

0.17

0.033

hi

–0.3625

–0.286

0.0

0.286

0.3625

gi

reference

kerfs

reference

kerfs

reference

Kraftdichte in N/m2

Kraftdichte in N/m2

1.00E+06

1.00E+06

9.00E+05

9.00E+05

8.00E+05

8.00E+05

7.00E+05

7.00E+05

6.00E+05

6.00E+05

5.00E+05

5.00E+05

4.00E+05

4.00E+05

3.00E+05

3.00E+05

2.00E+05

2.00E+05

1.00E+05

1.00E+05

0.00E+00

0.00E+00

Structuredynamic behaviour 239 Figure 7. Surface-force density for both models at same time step

32

sreference ( f )

28

skerfs ( f )

s in N / cm2

24 20 16 12 8 4 0

0

100

420

250 620 f in Hz

940

1040 1140

Figure 8. Spectrum of the surface-force density modulated with rotor speed

3. Structure-dynamic simulation In the next step, the mechanical, structure-dynamic behaviour of the IM is studied to estimate the impact of the balancing kerfs. Therefore, a mechanical model is applied (Schlensok et al., 2006) which consists of all mechanical parts of the IM, such as stator, rotor, shaft, housing, bearings, and housing caps (Figure 9). The IM is mounted on the front plate. Hence, the rear part of the machine can oscillate freely. For the analysis of the vibrations of the IM the surface-force density derived from the electromagnetic model is transformed to the frequency domain and than for each stator winding

short circuit rings

bearing

rotor end shield

shaft stator

case

mounting plate

Figure 9. Exploded view of the structure-dynamic model of the IM with squirrel-cage rotor

COMPEL 27,1

240

regarded frequency to the mechanical model. Significant frequencies are the slot harmonics of stator and rotor. The rotor-slot harmonic is modulated with the double stator frequency. Finally, the rotor speed has to be considered due to the dynamic-eccentric effect of the balancing kerfs. The studied frequencies are: f ¼ 20, 98, 422, 520, 618, 720, 942, 1,040, and 1,138 Hz. For these frequencies the force excitations of four different electromagnetic models are taken into account: reference model, model with kerfs only on front end of rotor, model with kerfs on rear end, and model with kerfs on both ends. The force transformation applies the 2D multi-slice model from the electromagnetic simulation described above (Figure 6). With a novel method the forces are transformed from the slices to the ranges of the stator of the mechanical model assigned to the corresponding slice. By this, higher resolutions in the frequency and space domains are reached in comparison to an equivalent 3D, electromagnetic model. Figure 10 shows the procedure of transformation. For 3D, mechanical elements which show an overlapping of force assignment the value of s is averaged. This is the case for element 6. The force of slice 2 is weighted with a and of slice 3 with b. A 3D, electromagnetic FE-model would afford by far more elements and would result in an unreasonable computing duration if the same resolution of the spectrum should be reached (Df ¼ 2 Hz at a cut-off frequency of fco ¼ 1,200 Hz). As own studies have shown the applied novel method results in more accurate results of the structure-dynamic model when compared to measurements of the body-sound level. After simulation of the mechanical model by applying the 13 different balanced and the reference model in the electromagnetic force calculation, the deformation is analysed in three different ways. 3.1 Deviation of the deformation In a first step, the deviation of the deformation is analysed. This principle subtracts the deformation values of all nodes of the model considering the results of a balanced and the reference model. Therefore, positive values show regions of higher deformation of the balanced model and negative values consider higher values for the reference model. Figure 11 shows exemplarily the difference of the deformation for f ¼ 1,040 Hz of housing and stator. Here, balancing kerfs on both ends of the rotor are compared to the reference model. Similar to a dynamic-eccentric rotor the balancing kerfs result in higher deformation of the complete structure. Especially, the rear (free) end of the IM oscillates stronger (dark regions in figure). Independent of which balanced model is regarded, this effect can be stated. 1

2

3

4

5

structure-dynamic model 1 element layer

Figure 10. Principle of force transformation

6

7

8

9

a b

electromagnetic Multi–Slice Model slice 1

slice 2

slice 3

...

Structuredynamic behaviour

Deviation-Deformation in m 2.50E-09 2.00E-09

241

1.50E-09 1.00E-09 5.00E-10 0.00E+00 –5.00E-10 –1.00E-09 –1.50E-09 –2.00E-09

Figure 11. Example: deviation of the deformation: f ¼ 1,040 Hz

–2.50E-09

3.2 Body-sound index The second analysis scheme is the level of the body-sound index LBSI. The LBSI is an integral value of the deformation of an entire body for instance the housing of the IM. For the nodes of all elements p of the body the normal component of the velocity of deformation ~vp is summed up. The sum is related to the reference values S0 ¼ 1 m2 and h2U 0 ¼ 25 £ 10216 m 2 =s 2 and the level is calculated: PN el R LS ð f Þ ¼ 10 log

p¼1 S p

2 j~vp · n~ p j dS

S 0 · h2U 0

! :

ð5Þ

n~ p is the normal vector of the element p, f is the frequency and Nel the number of elements. Therefore, the LBSI allows for the global evaluation, analysis, and comparison of a body’s entire deformation. The LBSI is a pure computational value and can therefore not be compared to any measurement. The body sound of the housing can be transmitted to other parts of the mechanical construction, i.e. the mounting. Therefore, the LBSI of the housing is studied. Figure 12 shows the results for four models, i.e. the reference model, a model with kerfs on the front and on the rear end of the rotor, and a model each with kerfs on one of both ends. As the studies show, the models with kerfs on both ends of the rotor show a weak dependence on the relative location between the kerfs on each end. Therefore, studying only the model with kerfs on both ends is sufficient.

COMPEL 27,1

120 105

242

LBSI in dB

90 75 60 45 reference kerfs front kerfs rear kerfs both

30 15

Figure 12. Body-sound index of the IM’s housing

0

20 98

422 520 618 720 f in Hz

942 1040 1138

Except for f ¼ 98, 422, and 720 Hz the balancing kerfs rise the level of the body index sound of the housing. Especially the higher frequencies studied at f ¼ 1,040 and 1,138 Hz increase by up to 35 dB. For rotor speed at f ¼ 20 Hz the level is increased up to 60 dB. But being very low for the reference model this order keeps smallest in the spectrum also for the balanced models. Figure 13 shows the difference in the LBSI of the balanced models compared to the reference model. 3.3 Body-sound level Finally, the body-sound level LBS allows for local analysis of the vibration. The LBS is a local value. Therefore, a global analysis as for the LBSI is not possible. Figure 14 shows the analysis lines and the positioning of the acceleration sensor used for measurements. Along the analysis lines the deformation is sampled and the body-sound level is calculated by: LS ¼ 20 · log

a dB: 1ðmm=S2 Þ

ð6Þ

a is the acceleration of the specific node at the regarded frequency f. As an example the radial component of the LS is studied for both analysis lines on the housing for 70 kerfs front kerfs rear kerfs both

60

∆LSm in dB

50 40 30 20 10

Figure 13. Difference of body-sound index of the IM’s housing

0 –10

20 100

420 520 620 720 f in Hz

940 1040 1140

counting direction line of nodes position of sensor

Structuredynamic behaviour 243

1st nodes

Figure 14. Analysis lines and positioning of sensor for the body-sound level

B–side

A–side (driving side)

f ¼ 422 Hz for models with kerfs on either end of the rotor (Figure 15). The LS oscillates strongly on the front end of the IM. This is due to the mounting which is performed by fixing the edges of the drilling holes on the front end of the mounting plate in the housing. This results in the four minima shown in Figure 15. The six maxima stem from the coupling of stator and housing by six spiral-steel springs. The stator’s deformation is transmitted just by the six spiral spring pins which fix the stator in the housing (Figure 9). On the freely oscillating rear end of the IM the LS is higher and is rather independent on the mounting and the coupling of stator and housing. The analysis of the LS at the location of the acceleration sensor shows a similar result as for the body-sound index. Figure 16 exemplarily sums up the results for all variants studied for the radial component of the body-sound level. The levels for higher

90

LS,rad in dB

85 80 75 kerfs on front, front line kerfs on rear, front line kerfs on front, rear line kerfs on rear, rear line

70 65 60

0

60

120

180 angle α [°]

240

300

360

Figure 15. Body-sound level along analysis lines for f ¼ 422 Hz

COMPEL 27,1

50 LS,bal. - LS,ref. in dB

244

kerfs on front end kerfs on rear end both ends

60

40 30 20 10 0

Figure 16. Body-sound level at sensor location

–10 20 100

420 520 620 720 f in Hz

940 1040 1140

orders and for rotor speed increase strongly especially for the case of balancing kerfs on both ends of the rotor. Next to these orders the 31 harmonic at f ¼ 618 Hz increases in the case of balancing kerfs significantly. 4. Conclusions The presented studies and simulation results show the potential of the structure-dynamic simulation chain for improvement and verification of the design of electrical machines. Here, the impact of balancing kerfs on the rotor of an IM is analysed. In general, the applied methods can be used for the analysis and optimization of any kind of manufacturing or tolerance problem of electrical machines such as various kinds of eccentricity, punching kerfs, broken bars, magnetization errors in permanent-magnet machines, and others. As a result of the studies presented in this paper the analysed IM is no longer balanced. The resulting un-balanced fugal forces are smaller than the net forces produced by balancing. Without balancing the IM induces less body-sound to the mechanical system of the application and the IM is hardly excited at rotor speed. For rotor speed (first order) the balancing kerfs result in strongly increasing body sound, i.e. vibration and mechanical stress. At frequencies above f ¼ 1,000 Hz the body sound rises significantly as well. This frequency range is critical since the human ear is very sensible here. Hence, the IM will be experienced more noisy. From this result, the IM is no longer balanced in series production. This leads to a major economic benefit. References De Gersem, H., Ion, M., Wilke, M., Weiland, T. and De-menko, A. (2006), “Trigonometric interpolation at sliding surfaces and in moving bands of electrical machine models”, COMPEL, Vol. 25 No. 1, pp. 31-42. Gyselinck, J.J.C., Vandevelde, L. and Melkebeek, J.A.A. (2001), “Multi-slice FE modelling of electrical machines with skewed slots – the skew discretization error”, IEEE Trans.-Mag., Vol. 37 No. 5, pp. 3233-7. Jordan, H. (1950), Gera¨uscharme Elektromotoren, Verlag W. Girardet, Essen. Lingener, A. (1992), Auswuchten – Theorie und Praxis, Verlag Technik GmbH, Berlin.

Schlensok, C. and Henneberger, G. (2004), “Comparison of static, dynamic, and static-dynamic eccentricity in induction machines with squirrel-cage rotors using 2D-transient FEM”, COMPEL, Vol. 23 No. 4, pp. 1070-9. Schlensok, C., van Riesen, D., Ku¨est, T. and Henneberger, G. (2006), “Acoustic simulation of an induction machine with squirrel-cage rotor”, COMPEL, Vol. 25 No. 2, pp. 475-86. Timar, P.L. (1989), Noise and Vibration of Electrical Machines, Elsevier, Amsterdam. van Riesen, D., Monzel, C., Kaehler, C., Schlensok, C. and Henneberger, G. (2004), “iMOOSE – an open source environment for finite-element calculations”, IEEE Trans.-Mag., Vol. 40 No. 2, pp. 1390-3. About the authors C. Schlensok received the MSc degree in Electrical Engineering in 2000 from the Faculty of Electrical Engineering and Information Technology at RWTH Aachen University Germany. From 2001 to 2006, he has been researcher at the Institute of Electrical Machines at RWTH Aachen University. In 2005, he obtained the PhD degree at the Faculty of Electrical Engineering and Information Technology at RWTH Aachen University with a thesis on numeric simulation and optimisation of induction machines. With the beginning of 2007, he moved to industry to Bosch Rexroth AG in Lohr am Main, Germany. He is now responsible for the design, calculation, and simulation of induction and synchronous machines as well as torque and linear motors. He is member of the developer team for the finite-element software iMoose (www.imoose.de). C. Schlensok is the corresponding author and can be contacted at: Christoph. [email protected] K. Hameyer (Senior MIEEE, Fellow IET) received the MSc degree in electrical engineering from the University of Hannover, Germany. He received the PhD degree from University of Technology Berlin, Germany. After his university studies, he worked with the Robert Bosch GmbH in Stuttgart, Germany, as a Design Engineer for permanent magnet servo motors and board net components. In 1988, he became a Member of the Staff at the University of Technology Berlin, Germany. From November to December 1992, he was a Visiting Professor at the COPPE Universidade Fderal do Rio de Janeiro, Brazil, teaching electrical machine design. In the frame of collaboration with the TU Berlin, he was in June 1993 a Visiting Professor at the Universite´ de Batna, Algeria. Beginning in 1993, he was a Scientific Consultant working on several industrial projects. Currently, he is a Guest Professor at the University of Maribor in Slovenia, the Korean University of Technology and Education (KUTE) in South-Korea. He was awarded his Dr habil. from the Faculty of Electrical Engineering of the Technical University of Poznan, Poland, and was awarded the title of Dr h.c. from the Faculty of Electrical Engineering of the Technical University of Cluj Napoca, Romania. Until February 2004, he was a Full Professor for numerical field computations and electrical machines with the K.U. Leuven, Belgium. Currently he is the Director of the “Institute of Electrical Machines” and holder of the Chair “Electromagnetic Energy Conversion” of the RWTH Aachen University, Germany (www.iem.rwth-aachen.de/). Since 2007 he is dean of the faculty of electrical engineering and information technology at RWTH Aachen, Germany. His research interests are numerical field computation, the design of electrical machines, in particular permanent magnet excited machines, induction machines and numerical optimisation strategies. He is author of more than 100 journal publications, more than 200 international conference publications and author of four books.

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

Structuredynamic behaviour 245

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

COMPEL 27,1

Wound magnetic core consequences on false residual currents

246

B. Colin G2Elab INPG UJF CNRS UMR 5269, Saint Martin d’He`res, France and Schneider Electric, Grenoble, France

A. Kedous-Lebouc and C. Chillet G2Elab INPG UJF CNRS UMR 5269, Saint Martin d’He`res, France, and

P. Mas Schneider Electric, Grenoble, France Abstract Purpose – Geometric or magnetic anomalies in the wound magnetic core of a residual current circuit breaker can be responsible of its abnormal tripping. The purpose of this paper is to analyse the core shape contribution to false residual currents (FRCs). Design/methodology/approach – To study precisely the core shape contribution, FEM simulations are investigated. First a 2D multilayer geometry is described thanks to linear regions. Then an apparent anisotropic bulk core is developed and validated in 2D and in 3D. Findings – The air gaps between the magnetic layers develop a shielding effect responsible of the core high sensibility to primary conductors eccentricity. This effect can be easily represented using an anisotropic bulk core model. Research limitations/implications – The anisotropic material model is basic and has known limitations. Future research should see the development of a new model. Originality/value – FRCs can considerably disturb operation of residual current device. This paper provides new hypothesis on the origin of theses currents and proposes an anisotropic magnetic material model that simplifies FRC study. Keywords Simulation, Finite element analysis, Electric current Paper type Research paper

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 246-255 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836807

1. Introduction Residual current device (RCD) must ensure user safety against electrical shocks without disturbing normal operations of electrical installations. It must detect residual current in installations and trigger the opening of the installation circuit breaker when this current exceeds a given threshold (typically 30 mA). A single phase RCD is made of (Figure 1): . a high-permeability magnetic wound core placed around the primary conductors; and . a secondary coil wound around the core which picks up the flux created by the conductors. The secondary winding measures the sum of the magnetic flux in the core (Figure 1). If a residual current circulates in the primary supply, the measured sum will be

Wound magnetic core consequences

Sec. Coil Core

247 Conductors

Figure 1. Description of the device

proportional to this current and, if it reaches a given threshold, the associated circuit breaker will open. Even under normal conditions, a false residual current (FRC) can be detected by the secondary winding and can lead to abnormal tripping. The FRC is a consequence of a magnetic dissymmetry of the device. If a magnetic or geometric anomaly disturbs the flux circulation, the detected secondary coil flux will be altered, and then a residual term will appear and eventually lead to a tripping (Pearse et al., 1995). The current standards (IEC, IEC standards 60947-2, 61008 & 61009) impose circuit breakers to be immune to nuisance tripping under six times the nominal current (several hundreds Amperes). It is strongly suspected that the geometry of the core has a notable impact on the FRC. It has been previously shown that the position of the ribbon ends had an influence on FRC (Colin et al., 2006). This paper concentrates on the multilayer core modelling and its effects.

2. 2D simulation of a multilayer magnetic core 2.1 Description of the model The physical magnetic core is made of a 20 mm thin high-permeability magnetic material ribbon wound in 70 (approx.) layers, separated the ones from the others by 3 mm air layers. The ribbon is wound in a spiral manner that has an influence on FRCs. Paper (Colin et al., 2006) shows that the effect of the spiral shape can be separated of others FRC sources. Thus, in this paper, the real spiral shape was replaced by concentric layers. About 70 layers are far too much to model: a simplified core, with only 30 layers (at constant core volume), has been simulated in order to understand the physical phenomena taking place in the core. To avoid a costly modelling, air layers were replaced by linear regions with the same magnetic properties as the equivalent vacuum (Figure 2). The thickness of the air layers was kept at 3 mm. The core is meshed using rectangle shape elements (Figure 3) and uses isotropic non linear B(H) law for the magnetic material. The simplified model parameters are given in Table I.

COMPEL 27,1

3 µm vacuum properties Econd

248 em

Rext Rint

Figure 2. 2D core model

Dcond

0

B(T)

Figure 3. Detail of the core mesh/shielding effect

Table I. Simplified model parameters

1.3

Parameter

Description

Dimension

Rint Rext Nm ea em Ec Econd Dcond

Internal radius External radius Nb of iron material layers Airgap Iron material layer thickness Conductors eccentricity Conductors spacing Conductors diameter

18.18 mm 19.35 mm 30 3 mm 40 mm Variable 15 mm 10 mm

In this first model, the primary conductors have been given a 1mm eccentricity. The magnetic dissymmetry created is sufficient to generate FRC. The primary conductors are supplied with equal and opposite 1,000 A currents. The 1,000 turns secondary winding is represented by two conductive layers, placed inside and outside the core. An electrical circuit connects the coils with a 26 V resistance.

Wound magnetic core consequences 249

2.2 Shielding effect The air layers create a phenomenon of shielding: the magnetic flux, following the less reluctive path, circulates in the first inner layers of the core. Thus, in a 2D problem, theses layers are quickly saturated whereas the external layers do not receive magnetic flux (Figure 3). This internal shielding effect amplifies the effect of conductor eccentricity: because of the eccentricity, more flux circulates in one half of the core. The air layers prevent the correct distribution of the additional flux thus creating a magnetic dissymmetry in the core, which leads to FRCs. This is shown on Figure 4: the simulated FRCs increase with the thickness of the air gap, for a given conductor eccentricity. The previous results show the importance of a correct model of the shielding effect. Unfortunately, modelling the core layers requires a very large number of elements. This limits 2D simulations to a low number of layers and makes nearly impossible to build a 3D model of the core. This shielding effect is not restricted to FRC study: applications as current sensors, core shielding, and more generally any wound core-based application can be concerned by shielding effect problems. 3. 2D apparent anisotropic bulk magnetic material 3.1 2D anisotropic material A bulk anisotropic core model was developed in order to reduce the number of elements needed and to allow shielding effect to be taken into account. The objective is to simulate the high-magnetic reluctance in the radial direction using an equivalent B(H) law. In the other direction, along the ribbon, the model uses the intrinsic non linear B(H) law. The principle of the anisotropic core used in a vector potential formulation is described in Figure 5: for each node of the model, the induction field Bxy is projected on the radial (across the air layers) and orthoradial (along the ribbon) directions where 50

FRC (mA)

40 30 20 10 0 0

2

4

6 Airgap (µm)

8

10

12

Figure 4. Impact of air gap on FRC

COMPEL 27,1

Bx, By Br, Bθ Hθ (Bθ)

Hr (Br)

250

Low permeability law

Hr, Hθ

Figure 5. 2D anisotropic B(H) model

Hx, Hy

an inverted magnetic law (Mac Gregor model) calculates the magnetic field H and the permeance matrix yru. Finally, the magnetic field in the Cartesian reference system Hxy is re-built and the yxy matrix calculated. Considering the reluctance of the multilayer core in the radial direction, the equivalent relative permeability mrequ of the anisotropic core was calculated:

mrequ ¼

N m £ em ððN m £ em Þ=mr Þ þ ðN m 2 1Þea

ð1Þ

where mr is the relative permeability of the material (in our case close to 40,000). With the parameters given in Table I, the equivalent permeability is close to 16. For the original 70 layers core, it would be 6.65. 3.2 Simulation: main problems, validations and main limitations This method has been successfully implemented under FLUXe using user subroutines. To achieve that, many problems, especially convergence ones, have been overcome, thanks to delicate adjustments of the solver parameters (Solver GMRES, precision 1 £ 102 8, tolerance 1 £ 102 9, Kyrlov Space 120, 300 Newton Raphson iterations) (CEDRAT, n.d.). However, convergence problems still remain to be solved, especially for high currents and when a secondary winding is simulated. A comparison between the results obtained with the bulk anisotropic model and the isotropic multilayer model is shown on Figure 6. It validates the proposed approach: the induction obtained with the anisotropic model perfectly follows the stair-like distribution of the isotropic multilayer core induction. B (T)

1.5 Anisotropic bulk core model Isotropic multilayer core model

1

Figure 6. Induction profile along the core radius, comparison between the anisotropic bulk core and the isotropic multilayer core

0.5 Radius (mm) 0 18

18.5

19

19.5

This method works only if the two magnetic directions of the simulated anisotropic material are sufficiently different: if the two directions are equivalent (isotropic core), the method will, in some cases, give a false induction. This problem can be illustrated by the following example. For the considered isotropic core, on a node, the H field is such that Hr ¼ Hu, and Hr high enough to saturate the material. In this case, the correct induction on this node should be: pffiffiffi Bru ¼ Bð 2 £ H r Þ

Wound magnetic core consequences 251

But, on each axis, the calculated B field is: Br ¼ Bu ¼ BðH r Þ So: pffiffiffi Bru ¼ 2 £ BðH r Þ If the material is non-linear, the error can be up to 40 per cent. 3.3 Applications of the bulk anisotropic material model The previously simulated and tested simplified core only counted 30 layers. With the anisotropic material, it is now possible to simulate the flux distribution in an equivalent 70 layers core, corresponding to the real core. Logically, the flux concentrates preferentially in the first layers (Figure 7). Using this equivalent 70 layers core, the core sensitivity to primary conductor eccentricity, a major source of FRC, can now be estimated more precisely. The primary conductors represented on Figure 2 have been shifted horizontally of 0.5, 1 and 1.5 mm, to represent a misplacement of the core. The shielding effect generates high FRC, shown on Figure 8. The impact of conductor eccentricity increases almost exponentially with the primary current. For a 1 mm eccentricity, the usual 30 mA threshold is reached at 1,000 A. Even if the conductors are perfectly centered, cores can be subject to this phenomenon: wound core are easily flattened and a 1 mm deformation on a 40 mm diameter core is reachable. Physical measurement on the cores have been launched to confirm theses results.

30 equivalent layers (µr = 16) at 1400 A

B (T)

1.5

70 equivalent layers (µr = 6.75) at 1400 A

1

0.5 Radius (mm) 0 18

18.5

19

19.5

Figure 7. Comparison between 30 and 70 equivalent layers cores

COMPEL 27,1

252

4. 3D apparent anisotropic bulk magnetic material 4.1 3D anisotropic model A 3D version of the anisotropic core model has been developed. This model uses a third magnetic law, independent of the two firsts (radial and orthoradial laws) along the Z-axis. The 3D simulations designed use scalar potential formulation. Since, the previous 2D anisotropic model used a vector formulation, it was not possible to adapt it by simply adding a third magnetic law. The principle of the 3D anisotropic core used in a scalar potential formulation is shown in Figure 9. Designed for scalar potential formulations, it is not possible to use the 3D model in a 2D simulation. To test and compare this model, a special 2D/3D simulation had to be carried out: a 3D core using the 3D anisotropic bulk model, a few millimeters high, bordered by Dirichlet face regions (Figure 10).

FRC (mA)

140 120 100

0.5 mm excentricity 1 mm excentricity 1.5 mm excentricity

80 60 40

Figure 8. FRC created by conductor eccentricity

Primary current (A)

20 0 0

500

1,000

1,500

Hx, Hy, Hz

Hr, Hθ Br (Hr) Low permeability law

Figure 9. 3D anisotropic B(H) model

Br, Bθ Bx, By, Bz

Dirichlet conditions

Figure 10. 3D simulation used to test the 3D anisotropic bulk core model

Bθ (Hθ)

Primary conductors

Core using 3D anisotropic bulk core model

BZ (HZ)

In this model, the magnetic fields created by the conductors are constrained in a 2D plan: 3D edge effects are suppressed by the lateral Dirichlet boundaries. The fields measured in the core are therefore comparable to those measured in the previously tested 2D cores, allowing the 3D model validation (Figure 11). 4.2 3D simulations This anisotropic bulk core model allows original 3D examinations of the flux distribution in a wound core. The core described in Table I has been simulated with and without the anisotropic model, for the same 500 A current (Figures 12 and 13). To simplify the simulations, the X, Y plan is a symmetry plan: only a half of the core is considered. The shielding effect is clearly visible on Figure 13: the induction in the first layer (inside) is very high and decreases abruptly. However, the induction rises in the external layers (Figure 14): the core seems to attract leakage flux from above. The shielding effect keeps the leakage flux in the lasts layers and prevents its distribution

Wound magnetic core consequences 253

1.5 B (T)

2D Model (vec. form.) 3D Model (Scal. form.)

1

0.5

Figure 11. Comparison between 2D and 2D/3D models: induction in the core

Radius (mm) 0 18

18.5

0

0.2

0.4

19

0.6

0.8

19.5

1

Figure 12. Induction, isotropic 3D bulk core, 500 A

COMPEL 27,1

254

B

Z Y X

Figure 13. Induction, anisotropic 3D bulk core, 500 A

A

B (T)

1.5

Induction, isotropic core Induction, anisotropic core

1

Figure 14. Induction comparison between isotropic and anisotropic cores, along X-axis, 500 A

0.5 Radius (mm) 0 18

18.5

19

19.5

in the core. The same phenomenon can be noticed for the inside layers: the shielding effect concentrates the “inner” leakage flux in the first layer. It can be noticed that the use of the anisotropic model does not change, in this case, the global flux in the core: the integration of the two curves of the Figure 14 shows very close results (6 per cent). 5. Conclusion FRCs can be generated by magnetic and geometric asymmetries in the wound core and need very accurate modeling in order to be evaluated. To avoid the expensive simulation of a multilayer core, a bulk anisotropic core model has been created for 2D and 3D applications. These new tools allow original studies, such as internal shielding effect in 2D or 3D cores, or precise estimations of conductor eccentricity impact on FRCs. References CEDRAT (n.d.), Flux Software Manual, Flux 9 – 2D/3D applications – User’s guide. Colin, B. et al., (2006), “Effects of magnetic core geometry on false detection in residual current sensor”, JMMM, Vol. 304 No. 2, pp. e804-6.

Pearse, J. et al. (1995), “Method and apparatus for detecting magnetic anomalies in a differential transformer core”, US Patent 5,446,383. About the authors B. Colin is a Electrical Engineer from ENSEM (Ecole Nationale Supe´rieure d’Electricite´ et de Me´canique), Nancy – France. He has received his PhD degree from the Institut National Polytechnique de Grenoble (INPG), Grenoble – France in 2007. His PhD has been focused on FRCs in circuit breakers and has been carried out at G2Elab (Grenoble Electrical Engineering Laboratory – former LEG) in strong collaboration with Schneider Electric. He now works with Schneider Electric. A. Kedous-Lebouc, received her Electrical Engineer and PhD degrees from the “Institut National Polytechnique de Grenoble” in 1982 and 1985. She is a Senior CNRS Researcher at G2Elab (Grenoble Electrical Engineering Laboratory – former LEG). Her main activity interests are soft magnetic materials and their integration in electrical engineering applications: non conventional characterization, magnetic behavior modeling and use in electromagnetic devices. Since, 2002, she is also involved in a new research theme on “New giant magnetocaloric effect materials and applications in magnetic refrigeration around room temperature”. Her activity as a whole is presented in more than 100 international journal publications and conferences. A. Kedous-Lebouc is the corresponding author and can be contacted at: [email protected] C. Chillet received the PhD degree in electrical engineering from the Institut National Polytechnique de Grenoble (INPG), Grenoble – France, in 1988, for his work on a disc-type permanent magnet synchronous motor. In 1989, he was appointed Researcher at the Centre National de Recherche Scientifique (CNRS) and has worked at the Grenoble Electrical Engineering Laboratory (G2Elab) since. His works concern the study and design of electromagnetic actuators and sensors, especially permanent-magnet motors and devices. P. Mas, received his PhD degree from the “Institut National Polytechnique de Grenoble” in 1987, for his work on integrated magneto-resistive sensors, carried out in the CEA LETI laboratory. He was then appointed as research engineer at Merlin Gerin – Schneider Electric Group. He has worked on research and development projects in the field of magnetism, sensors and actuators. He is currently magnetics senior expert.

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

Wound magnetic core consequences 255

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

COMPEL 27,1

256

Limits and rules of use of a dynamic flux tube model Marie-Ange Raulet, Fabien Sixdenier, Benjamin Guinand, Laurent Morel and Rene´ Goyet Universite´ de Lyon, France, Universite´ Lyon, France and CNRS UMR5005 AMPERE, Villeurbanne, France Abstract Purpose – The purpose of this paper is to analyze the main assumption of a dynamic flux tube model and to define its rules of use. Design/methodology/approach – The studied dynamic model lumps together all dynamic effects in the circuit by considering a single dynamic parameter. A physical meaning of this parameter as well as rules of use of the model are elaborated from analyses performed on several samples. A systematic comparison between experimental and calculated results allows to argue the conclusions. Findings – The model gives accurate results when a weak heterogeneity of magnetic data exists, nevertheless, the saturation phenomenon enlarges the validity domain. By considering the losses separation assumption, the model allows to obtain separately an estimation of losses due to classical eddy currents and due to the wall motion effects. Research limitations/implications – The estimation of the model’s parameter value is still empiric, a work is in progress on this subject. Practical implications – The model’s implementation in a flux tubes network allows to simulate the dynamic behaviour of industrial actuators having massive cores. Originality/value – A physical interpretation of the parameter associated to the dynamic flux tube model is given. Rules of use of the model are also defined. Keywords Eddy currents, Motion, Flux Paper type Research paper

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 256-265 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836816

1. Introduction Nowadays, the miniaturization of electromagnetic devices leads to increase the fundamental supply frequency, moreover, these systems are usually fed by static converters. Thus, the magnetic materials of these devices are hardly stressed due to the fast dynamic working conditions. The design of these devices requires simulation tools, which need to take into account accurately both the description of the geometry of the system and dynamical material laws. A 3D field calculation including a dynamic realistic material law would lead to a prohibitive calculation time and numerical difficulties; at present time, it hardly seems possible with standard computers. Some authors (Gyselinck et al., 2000; Sadowski et al., 2002) consider dynamical effects due to the material in a 2D field calculation considering a modified law of the material. Our laboratory has developed a magnetic dynamical flux tube model (Marthouret et al., 1995). This model points out the dynamical behaviour of the material and considers a simple geometry (flux tube with a constant cross-section). The association of different flux tubes allows to simulate a real industrial device. This model has

already been effectively used to represent different industrial devices. Nevertheless, the main assumption of the model which is to lump together the different dynamical effects has not yet been tested. The purpose of this paper, is to analyse in details the main assumption of this model and to define its validity domain and its rules of use.

Dynamic flux tube model

2. Description of the dynamical flux tube model The model allows to obtain the dynamic behaviour of a magnetic circuit with a constant cross-section, where the anisotropy of the material is neglected. The different dynamic effects developed in the circuit (wall motion, macroscopic eddy currents) are considered in this model by a single representation. The dynamic behaviour of the circuit is described by a first order differential equation which can be represented by a bloc-diagram in Figure 1. The quantity Ba is the average flux density in the cross-section, Hdyn, is the excitation field applied at the surface, and Hstat (Ba), is a fictitious static excitation field value which corresponds to a given value of Ba. The magnetization history of the material can be taken into account by considering an hysteretic static model for Hstat (Ba). The parameter g is a dynamic behavioural parameter. Its value has to be fitted by comparing simulated and measured dynamic loops. This model presents several advantages: . it requires only one parameter g (apart those required to model the static hysteresis), supposed to be independant of the waveform and velocity of the excitation); . it is a time domain model; . the calculation time is short; . it is reversible Ba (Hdyn) or Hdyn (Ba); and . it can be easily introduced in many kinds of softwares (circuit type, design, simulation . . .)

257

The main assumption of this model is to consider the homogeneity of the phenomena in the cross-section of the flux tube. This model has been tested for different materials and devices. For the representation of ferrite components, this model allows to obtain accurate results, in that the assumption of homogeneity holds with good approximation. For other materials, where the conductivity leads to a field diffusion across the section, the model provides quite satisfying results. 3. Validity of the main assumption The dynamic flux tube model lumps together all dynamic effects developed in the circuit. The value of the dynamic parameter g, determines the width of the dynamic Hdyn – Hstat

Hdyn

1 g Hstat Hstat (Ba)

dBa dt

Ba ∫

Figure 1. Dynamic bloc-diagram of the flux tube model

COMPEL 27,1

258

loop of the material. This value depends on the thickness of the circuit, the resistivity r of the material, the wall motions, the static hysteresis phenomenon and the permeability of the material. In the aim to simplify the problem, we limit our study to simple geometries: we consider torus samples where the anisotropy phenomenon is negligible (Gyselinck et al., 2000). Consider the formula (1) associated to the dynamic flux tube model: H dyn 2 H stat ðBa Þ ¼ g:

dBa dt

ð1Þ

This formula can be compared with the expression (2) defined in the case of a magnetic lamination if the skin effect, saturation and edge effects are negligible. H tot ¼ H stat ðBa Þ þ

s · d2 dBa · 12 dt

ð2Þ

where Htot ¼ excitation field at the lamination surface when eddy currents are induced in the cross section, d ¼ thickness of the lamination, s ¼ conductivity of the material, Ba ¼ averaged magnetic flux density over the thickness. In both cases, the homogeneity of magnetic data assumption is assumed. An analogy between equations (1) and (2) allows to estimate the value of the parameterg:

g¼

s · d2 12

ð3Þ

In the aim to validate on the one hand, the estimation of g and on the other hand the lumped model, we carry out successively tests on four samples numbered 1-4. 3.1 Sample tests no. 1 We consider a toroidal sample made of NiFe(50/50) main characteristics are reported in Table I. The value of r is given by the manufacturer and mr is estimated considering the static characteristic Hstat (Ba) in its linear part. Owing to the large thickness of the torus, the weak resistivity and its high-relative permeability, eddy currents are not negligible in this sample. A sinusoidal excitation field H is imposed at the surface of the sample. Table II regroups for different working frequencies (25, 50, 100 and 150 Hz): . the skin depth d: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ð4Þ d¼ * £mo mn £ f

Table I. Sample data

Dout (mm) 18.8

Din (mm)

Thickness (mm)

r (V.m)

mr

9.9

1.1

48 £ 102 8

100,000

.

.

the relative error 11 between the areas of the simulated and measured loops (which is representative of the losses); the quadratic error 12 defined by the formula (5). rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Xn ð5Þ 12 ¼ ð Bsimulated ði Þ 2 Bmeasured ði ÞÞ2 i¼1 n

We observe that the quadratic error 12 increases with the frequency. This result agrees with the indication given by the skin depth value: in fact when d becomes comparable to the size of the section, the assumption of homogeneity does not hold. However, we point out that the classical formula (4) has been obtained for linear materials and with a semi-infinite plane conductor: hence it provides a very approximate result with our samples (for instance see (Ida, 1995) for analytical computations of d with different geometries). Local information inside the cross section is not available. So as to obtain this information, we use a numerical tool based on the magnetic field diffusion (Raulet et al., 2004). Figure 2 shows the simulated excitation field H, normalized with respect of its intensity at the surface, as a function of its relative position in the thickness of the lamination (H (0) ¼ field at the surface, H (100 per cent) ¼ field in the middle of the thickness) for different frequencies. One observes that even at 25 Hz the excitation field H is not uniform; however, the skin effect is less important than the prediction obtained by equation (4). In the same way, Figure 3 shows the normalized flux density as a function of the relative geometric position through the thickness. One sees that the saturation f (Hz)

Relative Amplitude of the applied excitation field (%)

d (mm) 11 (per cent) 12 (per cent)

25

50

100

150

0.22 19 14.5

0.16 4.2 22.7

0.11 23 29.8

0.09 36.7 37.2

Dynamic flux tube model

259

Table II. Sample no. 1 results

100 90 80 70 60 50 40 30 25 Hz 50 Hz 100 Hz 150Hz

20 10 0

0

20 40 60 80 relative distance from the surface (%)

100

Figure 2. Normalized amplitude of the excitation field H versus the relative thickness

COMPEL 27,1

Relative Amplitude of the flux density (%)

100

260

Figure 3. Normalized amplitude of the flux density B versus the relative thickness

90 80 70 60 50 40 25 Hz 50 Hz 100 Hz 150Hz

30 20 10

0

20 40 60 80 relative distance from the surface (%)

100

phenomenon tends to homogeneize the flux density through the thickness of the lamination. Hence, the validity of the flux tube model is enlarged. A last observation concerns the comparison between both errors e 1 and e 2 carried out for 50 Hz frequency operation where a discrepancy appears. Figure 4 shows the simulated and measured loops for this frequency. Both loops have nearly the same area

1 0.8

simulation measure

Averaged Flux density B (T)

0.6 0.4 0.2 0 – 0.2 – 0.4 – 0.6

Figure 4. Measured and simulated loops

– 0.8 –50

0 Excitation field H (A/m)

50

but are hardly different. These different results bring out several preliminary conclusions: . If this model provides accurate results on the estimation of the area of the loop, we must be more careful concerning the estimation of the waveform when a great heterogeneity exits in the cross section. . The skin depth provides information about the validity domain of the tube flux model . The saturation phenomenon tends to homogeneize the flux density distribution, and thus enlarges the limits of use of our model.

Dynamic flux tube model

261

3.2 Sample test no. 2 The second sample is a stack of rings (thickness 0.2 mm) of SiFe(3 per cent). The material resistivity given by the manufacturer is 48 £ 102 8 V.m, its relative permeability mr estimated from the linear part of the quasi-static characteristic is about 6,000. Conversely, to the previous sample, few eddy currents can be induced in the circuit (thin ring): by using equation (4), one sees that the skin effect can be neglected until at least 1,500 Hz. First, we analyse the results provided by the model considering the value of g ¼ 6.94 £ 102 3 computed by equation (3). We carry out some 300 Hz to 2,400 Hz simulations with an imposed excitation field H at the surface. Figure 5 shows the quadratic error 12 computed for each frequency. An important error is observed, and cannot be explained by the skin effect, which is negligible until at least 1,500 Hz frequency. Therefore, we carried out other simulations by taking the value of g which minimizes the quadratic error 12 at the frequency of 800 Hz. The obtained value of g is 35 γ = 0.00694

2400Hz

γ = 0.013

30

25 1200Hz ε2 (%)

20 300Hz

400Hz

800Hz

1

2

3 case

15

10

5

0

4

5

Figure 5. Quadratic error

COMPEL 27,1

262

now 0.013. The quadratic error 12 is reduced for all the considered frequencies (not only at 800 Hz), as shown in Figure 5. Hence, the formula (3) is no more applicable. By considering the assumption of magnetic losses separation (Bertotti, 1998), the excitation field Hdyn at the surface of the sample can be decomposed by the sum of different terms: Hstat (Ba) due to the static law of the material, Hedd due to the eddy currents, and Hexc due to the effects of wall motions. H dyn ¼ H stat ðBa Þ þ H edd þ H exc

ð6Þ

We now compare equations (1)-(6), by using formula (2), so as to obtain the following expression for Hexc:   s · d 2 dBa ð7Þ H dyn ¼ g 2 · 12 dt The dynamic flux tube model lumps different dynamic effects, which are represented by a sole formulation. The representation of dynamical effects associated with the wall motions has a similar formulation as those linked to eddy currents. This result has already been validated in previous works (Tenant and Rousseau, 1995) to simulate ferrite circuits. In this kind of material, dynamic effects due to the wall motions are dominant, and the flux tube model gives accurate results. The assumption of magnetic losses separation together with the identification of the parameter g allows to specify the different energy dissipations (W/kg) associated, respectively, with eddy currents and to wall motions: I s · d 2 dBa · dBa ð8Þ P edd ¼ 12 dt

P exc

 I s ·d2 dBa · dBa ¼ g2 12 dt

ð9Þ

Many simulations using the dynamical flux tube model have been carried out by using the same value of g ¼ 0.013. The contribution of each kind of losses cannot be measured separately. So as to obtain such a comparison, we use the results obtained with the diffusion model (diff) (Raulet et al., 2004). Different simulations have been carried out by imposing a sinusoidal excitation field H at the surface of the sample, for the range of frequencies (800-2,400 Hz). The results are regrouped in Table III. PT are

f (Hz) Model

Table III. Sample no. 2 results

PT (mW/kg) Physt (mW/kg) Pedd (mW/kg) Pexc (mW/kg)

800

1,200

2,400

DFTM

Diff

DFTM

Diff

DFTM

Diff

0.63 0.32 0.16 0.16

0.58 0.38 0.14 0.06

0.75 0.32 0.21 0.21

0.7 0.4 0.2 0.1

0.96 0.25 0.35 0.35

0.89 0.37 0.32 0.2

the total losses, Physt are the static losses, Pedd are the losses due to eddy currents and Pexc are the excess losses. These results lead to different investigations: . Physt are important in this range of frequencies, thus an accurate static hystersis model Hstat (Ba) must be used. . By considering (8) and (9) formulas, the power losses due to eddy currents and to the wall motions are proportional to the coefficients s · d 2 =12 and (g 2 ðs · d 2 Þ=12). . The dynamic flux tube model allows to obtain accurate results, until the skin effect is not dominant.

Dynamic flux tube model

263

3.3 Sample tests no. 3 and no. 4 These tests have been carried out on rings made of CrNiFe. The resistivity is r ¼ 94 £ 102 8 V.m, the relative permeability related to the linear part of the static first magnetization is about 50,000. The manufacturer produces two series of rings with different thicknesses: d ¼ 0.2 and 0.6 mm. Manufacturing precautions are taken into account in the aim to ensure the same magnetic properties of both ring series. So as to avoid the static hysteresis phenomenon, we carry out simulations limited to the first magnetization. For the previous sample, we obtain the value of g which minimizes the quadratic error 12 for the frequency 50 Hz. The obtained results are regrouped in Table IV. The optimal value of g depends upon the thickness of the ring. This result agrees with the formula (3). Conversely, we observe that the parameter g 2 ðs · d2 Þ=ð12Þ seems to be constant. This parameter is characteristic of dynamic effects due to wall motions. This result is coherent, in that wall motions are linked to the structure of the material, and are independent of the geometry of the sample.

4. Conclusion The limits of a dynamic flux tube model is analysed by considering tests (simulations þ measurements) on four different samples. This model lumps dynamic effects, which are represented by a same formulation. It gives accurate results when the skin effect is negligible, and when a weak heterogeneity of magnetic data exists. Nevertheless, the saturation phenomenon enlarges the validity domain of the model. The value of g can be decomposed into two terms: the first one is linked to eddy currents, and is given by the classic expression depending of s and d equation (3).

d (mm) 12 (per cent) g s · d 2/12 g 2 (s · d 2)/(12) 2 · d/d

0.2

0.6

1.77 0.05 0.00354 0.00146 3.08

4.37 0.033 0.0319 0.0011 1

Table IV. Sample nos 3 and 4 results

COMPEL 27,1

The second one seems to be a constant value, depending upon the structure of the material, and not upon the geometry. This decomposition based on the assumption of losses contributions, allows to obtain separately an estimation of the losses linked, respectively, to the static hysteresis, to eddy currents and to wall motion effects. The estimation of g is still empiric; we are working on this subject.

264 References Bertotti, G. (1998), “General properties of power losses in soft ferromagnetic materials”, IEEE Trans. on Magn., Vol. 24, pp. 621-30. Gyselinck, J.J.C., Vandevelde, L., Makaveev, D. and Melkebeek, J.A.A. (2000), “Calculation of no load losses in an induction motor using an inverse vector Preisach model and eddycurrent loss model”, IEEE Trans. on Magn., Vol. 36, pp. 856-60. Ida, N. (1995), Numerical Modeling for Electromagnetic Non-destructive Evaluation, Chapman & Hall, London. Marthouret, F., Masson, J.P. and Fraisse, H. (1995), “Modelling of a non-linear conductive magnetic circuit”, IEEE Trans. on Magn., Vol. 31, pp. 4065-7. Raulet, M.A., Ducharne, B., Masson, J.P. and Bayada, G. (2004), “The magnetic field diffusion equation including dynamic hysteresis: a linear formulation of the problem”, IEEE Trans. on Magn., Vol. 40 No. 2, pp. 872-5. Sadowski, N., Batistela, N.J., Bastos, J.P.A. and Lajoie-Mazenc, M. (2002), “An inverse Jiles-Atherton model to take into account hysteresis and hysteresis in time-stepping finite element calculations”, IEEE Trans. on Magn., Vol. 38, pp. 797-800. Tenant, P. and Rousseau, J.J. (1995), “Dynamic model for soft ferrite”, Power Electronics Specialists Conference Proceedings, Vol. 2, pp. 1070-6. About the authors Marie-Ange Raulet is a full-time Lecturer at the University Claude Bernard Lyon 1 and a researcher at the Ampere Laboratory. Her researches concern the elaboration of accurate magnetic laws and their implementation in field calculations. Marie-Ange Raulet is the corresponding author and can be contacted at: [email protected]

Fabien Sixdenier is a full-time Lecturer at the University Claude Bernard Lyon 1 and a researcher at the Ampere Laboratory. His researches concern magnetic materials and their applications, especially introducing accurate magnetic material laws into actuators modelling. E-mail: [email protected]

Benjamin Guinand received the MS degree in Electrical Engineering from the National Polytechnic Institute of Grenoble (INPG), France, in 2003. He is currently working toward the PhD degree at the Claude Bernard Lyon 1 University, Ampere Laboratory. His researches concern magnetic materials and their applications, especially the study of Barkhausen Noise.

Laurent Morel is a full-time Lecturer at the IUT of University Claude Bernard Lyon 1 and a Researcher at the Ampere Laboratory. His researches concern magnetic materials and their applications, high-speed machine, three-axis magnetic sensor for earth magnetic field and their instrumentations. E-mail: [email protected]

Rene´ Goyet has been Master of Conferences at the University of Paris6 during 30 years and is now at the University Claude Bernard Lyon 1. He has been a Teacher in Electronics, Electrotechnics and Automatics. Besides, he carried out research in driving motors and magnetic modeling for Electrical Engineering. He has authored many technical publications and directed many PhD students.

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

Dynamic flux tube model

265

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

COMPEL 27,1

Finite element formalism for micromagnetism H. Szambolics Institut Ne´el, CNRS-INPG-UJF, Grenoble, France

266

L.D. Buda-Prejbeanu Laboratoire SPINTEC, CEA-CNRS-INPG-UJF, Grenoble, France and Institut National Polytechnique de Grenoble, Grenoble, France

J.C. Toussaint Institut Ne´el, CNRS-INPG-UJF, Grenoble, France and Laboratoire SPINTEC, CEA-CNRS-INPG-UJF, Grenoble, France and Institut National Polytechnique de Grenoble, Grenoble, France, and

O. Fruchart Institut Ne´el, CNRS-INPG-UJF, Grenoble, France Abstract Purpose – The aim of this work is to present the details of the finite element approach that was developed for solving the Landau-Lifschitz-Gilbert (LLG) equations in order to be able to treat problems involving complex geometries. Design/methodology/approach – There are several possibilities to solve the complex LLG equations numerically. The method is based on a Galerkin-type finite element approach. The authors start with the dynamic LLG equations, the associated boundary condition and the constraint on the magnetization norm. They derive the weak form required by the finite element method. This weak form is afterwards integrated on the domain of calculus. Findings – The authors compared the results obtained with our finite element approach with the ones obtained by a finite difference method. The results being in very good agreement, it can be stated that the approach is well adapted for 2D micromagnetic systems. Research limitations/implications – The future work implies the generalization of the method to 3D systems. To optimize the approach spatial transformations for the treatment of the magnetostatic problem will be implemented. Originality/value – The paper presents a special way of solving the LLG equations. The time integration a backward Euler method has been used, the time derivative being calculated as a function of the solutions at times n and n þ 1. The presence of the constraint on the magnetization norm induced a special two-step procedure for the calculation of the magnetization at instant n þ 1. Keywords Simulation, Finite element analysis, Electromagnetism Paper type Research paper

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 266-276 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836825

1. Introduction Thanks to high-resolution fabrication techniques (lithography, patterning, self-assembly), submicron magnetic systems are now routinely fabricated with different materials and precisely controlled sizes and shapes. The shape can be dots, with various forms, rings, wires and rods, antidotes, etc. (Li et al., 2001; Jubert et al., 2001). To understand in detail what happens inside such a magnetic system, with both ultimate time and space resolution, accurate experimental studies and micromagnetic modeling must be combined.

Nowadays most of the micromagnetic softwares are based on the finite difference (FD) approximation, meaning that the magnetic body is divided into regular orthorhombic cells. From a numerical point of view, the implementation of these algorithms is straightforward and due to the periodic discretization, the use of fast Fourier transforms is possible, thus the computation time is significantly reduced (Toussaint et al., 2002). Furthermore, specific integration schemes were developed to integrate the Landau-Lifschitz-Gilbert (LLG) equation describing the magnetization dynamics (Brown, 1963) which conserve implicitly the magnitude of magnetization. Unfortunately, the algorithms based on FD are intrinsically affected by the roughness of the grid at surfaces. Thus, only the systems bounded exclusively by planar surfaces parallel to some axes of the grid, can be in principle reliably computed (Garcı´a-Cervera et al., 2003). To overcome these numerical difficulties, an alternative is the finite element method (FEM) (Braess, 2001), which uses an irregular mesh. The characteristic of FEM is that it is based on the projection of the micromagnetic equations on so-called test functions. Thus, the mathematical background is more complex than in the case of the FD approach, where the physical quantities are estimated locally. Up to now the micromagnetic calculations using irregular mesh, presented by physicists are not based on a projective form of the LLG equation. We show here the steps we followed when building up our FEM approach and the results obtained for two 2D magnetic test cases. 2. Weak form for micromagnetism The principle of micromagnetics is to approximate the magnetization distribution inside a magnetic system with a continuous medium (Brown, 1963). This requires that the variations of the magnetization vector MðrÞ ¼ M s mðrÞ occur on a length scale large enough to approximate the direction angles of neighboring atomic spins with a continuous function. The spontaneous magnetization Ms denotes the mean magnetic moment per unit volume and is assumed to be constant, only the orientation of the magnetization vector m(r) may change in time and space. The magnetization distribution corresponding to an equilibrium state of the ferromagnet is obtained by minimizing the total free energy Etot of the system with respect to m(r). In the continuous medium approximation, Etot is the sum of exchange interactions, magnetocrystalline energy, the Zeeman contribution due to applied field and the dipolar interactions. In the simplest case of uniaxial magnetocrystalline anisotropy of second order, it can be written as: Z Z 2 Aex ½7m
dV þ K 1 ½1 2 ðu K · mÞ2
dV E tot ½m
¼ V V Z Z ð1Þ 1 2 m0 M s m · H app dV 2 m0 M s m · H m dV V V2 This integral expression depends on the material parameters which are Aex the exchange constant, K1 and u K for the anisotropy and Ms. H m is the magnetostatic field, solution of Maxwell’s equations. It coincides with the demagnetizing field produced inside the ferromagnet by the magnetization itself. While the terms describing the anisotropy, the exchange interactions, and the Zeeman coupling are local terms, the demagnetizing field depends on the magnetization distribution over the entire material, thus it remains the most difficult term to compute.

Finite element formalism for micromagnetism 267

COMPEL 27,1

The method adopted here to relax the magnetic configuration towards an equilibrium state consists in integrating the LLG dynamic equations (Brown, 1963):
 ›m ›m ¼ 2m0 g ðm £ H eff Þ þ a m £ ð2Þ ›t ›t

268

Here g0 is the gyromagnetic factor, a is the damping constant and H eff is the effective field obtained by variational derivation of the total energy Etot with respect to m(r). According to equation (1), the effective field is the sum of four fields: exchange field Hex, anisotropy field Hani, magnetostatic field Hm and applied field Happ. The LLG equation respects implicitly the condition imposed on the norm of the magnetization: 1 2 m2 ¼ 0

ð3Þ

Solving these equations by using FEM, as explained below, means deriving their weak form and integrating this by using the most suitable integration method (Braess, 2001). 2.1 Weak form of the magnetostatic equations The evaluation of H m is the most difficult issue because of its long-range character. Let us consider two of Maxwell’s equations in magnetostatics: 7·B ¼ 0

7 £ Hm ¼ 0

ð4Þ

the magnetization being related to the magnetic induction B and to the demagnetizing field H m by: ð5Þ B ¼ m0 ðH m þ MÞ When working with the magnetic vector potential, the starting point is the solenoidal nature of the B vector. This means that it is possible to write it as the curl of a vectorial quantity, called magnetic vector potential A, and the magnetostatic field becomes: Hm ¼

1 1 B2M¼ 7£A2M m0 m0

ð6Þ

The potential A is assumed to vanish at infinity: Aðjrj ! 1Þ ! 0

ð7Þ

To derive its weak form, the magnetostatic equation (6) is multiplied by a vector weighting function v and then integrated over the whole space V: Z v · 7 £ H m dV ¼ 0 ð8Þ V

By using: divðv £ H m Þ ¼ H m · 7 £ v 2 v · 7 £ H m

ð9Þ

and the continuity condition of the tangential component of Hm at free surfaces, the derivation orders are being equilibrated and the weak formulation reads as: Z Z 7 £ v · 7 £ AdV ¼ m0 M · 7 £ vdV ð10Þ V

V

Owing to the invariance of the studied system along the Oz direction only the z component of the vector potential A must be considered, and since v 5 (0, 0, v) the final form of the weak formulation is: Z ›x vð›x Az þ m0 M y Þ þ ›y vð›y Az 2 m0 M x Þ ¼ 0 ð11Þ For the treatment of the condition at infinity (7), a spatial transformation is used. This converts the infinite exterior that must be considered for this problem into a finite domain, so the “open boundary problem” becomes a “closed boundary problem” (Brunotte et al., 1992). The 2D system is thus modified in order to apply the transformation: the upper and the lower semi-infinite regions are converted in two finite domains bounded by straight lines: 2 Y1 , Y # 2Y0 and Y0 # Y , Y1, as shown in Figure 1. The capital letters refer to the coordinates in the transformed domains, while the coordinates in small letters indicate the real space. Since, Hm is expected to decay exponentially at long distances away from the magnetic system, a natural choice for the transformation to be used is:  
Y 1 2 jY j T0   y ¼ TðY Þ ¼ sgnðY Þ · Y 0 2 ð12Þ Log Y1 2 Y0  2p

Finite element formalism for micromagnetism 269

2.2 Weak form of the micromagnetic equations To obtain the weak form of the LLG equations we project these onto vector test functions w: 
 Z ›m ›m 2a m£ w· dV ›t ›t Vm Z ð13Þ w · ½m £ ðH ex þ H ani þ H m þ H app Þ
dV ¼ 2m0 g Vm

In this case, the integration is done only on the magnetic volume Vm. y Y∞ Y0 x

Magnetic system

–Y0 –Y∞ Note: The upper and the lower semi-infinite regions are replaced by two finite domains coloured in grey

Figure 1. 2D test case

COMPEL 27,1

Only the term that contains the exchange field, H ex ¼ 2Aex Dm=ðm0 M s Þ, needs to be transformed. For this term, we need to equilibrate the orders of derivatives of the unknowns and of the test functions. In Cartesian coordinates, this integrand can be rewritten as follows: ðm £ H ex Þ · w ¼

270

2Aex X Dml ½ðm £ e l Þ · w
m0 M s l¼{x;y;z}

ð14Þ

By using the divergence theorem, after integration on the magnetic volume, the weak form for the exchange term is obtained: Z X ts 7ml · n · ½ðm £ e l Þ · w
dS ðm £ DmÞ · wdV ¼ Vm

l¼{x;y;z}

X Z

2

l¼{x;y;z}

ð15Þ 7ml · 7½ðm £ e l Þ · w
dV

Vm

The micromagnetic theory imposes that no exchange torque acts at the free surface. This implies a Neumann condition 7ml · n ¼ 0 on S, known as the Brown condition, so the surface integral from equation (15) vanishes. Finally, after considering also the constraint (3), equation (13) transforms into the weak form:
 Z ›m ›m 2am£ w· dV ›t ›t Vm  Z
2Aex X ›m ›w m£ dV 2 gm0 · ¼g M S l¼{x;y;z} Vm ›x l ›xl ð16Þ Z  w · ½m £ ðH ani þ H app þ H m Þ
dV Vm

þ

Z

2

mð1 2 m ÞdV þ Vm

Z

lw · Vm

›ð1 2 m 2 Þ dV ›m

Here, l is the Lagrange multiplier used for the treatment of the constraint and m is the corresponding test function. Another alternative weak form has been proposed by Alouges, where the test functions w at each mesh node belong to the tangential plane to m (Alouges and Jaisson, 2006). The comparison between the two methods is in progress. 2.3 Finite element discretization The expression (16) corresponds to an ideal weak form for the LLG equations. In this ideal case, after finite element discretization, the Lagrange multiplier l and the vector field m are written as a sum of basis functions {wi} weighted by a set of fitting coefficients. The test functions m and w are calculated using the same functions {wi}. We have used as basis functions second order Lagrange polynomials.

In practice, the constraint on the magnetization norm is applied only at the mesh nodes and only the magnetization and the test functions w are interpolated: m¼

N X

wj m j ¼

j¼1

N X

wj

j¼1

X

m j;q e q

w ¼ wi e p

ð17Þ

q¼{x;y;z}

where N is the number of nodes and p [ {x; y; z}. The temporal scheme is based on the Euler’s backward time integration method where the time derivative is estimated as a FD of the solutions at times n and n þ 1:

›m m nþ1 2 m n ¼ ›t Dt

ð18Þ

and the exchange term is also calculated at time n þ 1. The other terms in the right hand member of the equation (16) are evaluated at time n. Owing to the constraint (3), a two step procedure has to be implemented. ~ nþ1 at time step n þ 1 is firstly determined An estimation of the magnetization m without taking into account the influence of the constraint. Then, a correction dm due to the constraint on the magnetization norm is calculated. The magnetization at time n þ 1 is finally obtained by summing up the two contributions: ~ nþ1 þ dm m nþ1 ¼ m

ð19Þ

Inserting the interpolated expressions (17) and the time derivative of the magnetization equation (18) into the weak form, the following matrix equation is obtained, from which ~ nþ1 is calculated: the solution m n ~ nþ1 ðM ij;pq þ Dij;pq þ DtK ij;pq Þm j;q ¼ ðM ij;pq þ Dij;pq Þmj;q

ð20Þ

where: M ij;pq ¼

Z Z

wi wj dpq dV Vm

wi wj aðe p £ e q Þ · m n dV Z 2Aex X › wi › wj ¼g ½ðe p £ e q Þ · m n
dV M s l¼x;y;z Vm ›xl ›xl

Dij;pq ¼

Vm

K ij;pq

Finite element formalism for micromagnetism

ð21Þ

with implicit summation over j [ {1; . . . ; N} and p [ {x; y; z}. By introducing the constraint term in the equation the following set of equations is obtained: 8 < ðM ij;pq þ Dij;pq þ DtK ij;pq Þdmj;q þ H Tij;p lj ¼ 0 ð22Þ : H ij;q dmj;q ¼ Gi ðm ~ nþ1 Þ where lj represents a Lagrange multiplier which expresses the presence of the constraint and H ij;q is the associated Jacobian matrix:

271

COMPEL 27,1

272

~ nþ1 H ij;q ¼ 2dij m j;q

Gi ¼ 1 2

X q

~ nþ1 m i;q

2

ð23Þ

To simplify the expressions used up to now the equation set (22) is rewritten by using the matrix notation: ( ðM þ D þ DtKÞdm þ H T L ¼ 0 ð24Þ ~ nþ1 Þ H dm ¼ Gðm The general solution of equation (22) may be written as follows:

dm ¼ Nul u þ m d

ð25Þ

where Nul is the matrix that collects all the vectors of Ker(H), and therefore satisfies H Nul ¼ 0. The Lagrange multiplier can be eliminated from the first equation in equation (22) by multiplying it with NulT. One finally obtains: ( NulT ðM þ D þ DtKÞðNul u þ m d Þ ¼ 0 ð26Þ ~ nþ1 Þ H m d ¼ Gðm The method for solving (26) consists in determining firstly md and then u. From the reduced singular value decomposition of H: H ¼ U r S r V Tr

ð27Þ

where Sr is a positive definite matrix, the expression for md is obtained: T ~ nþ1 Þ m d ¼ V r S 21 r U r Gðm

ð28Þ

It is now possible to determine u from equation (26) by replacing md with equation (28): T u ¼ 2K 21 eff Nul ðM þ D þ DtKÞm d

ð29Þ

K eff ¼ NulT ðM þ D þ DtKÞNul

ð30Þ

where Keff denotes:

3. Test cases We present here an application of our finite element approach to magnetic thin films with a perpendicular anisotropy of moderate strength. To this category belong FePd
The equilibrium magnetization configuration of alloys, Co/Pt multilayers or Coð1010Þ. such systems consists of a periodic modulation of the perpendicular component of the magnetization leading to parallel stripe domains (Toussaint et al., 2002). This kind of configuration is well adapted to 2D micromagnetic simulations since the magnetization is nearly invariant along the stripes’ direction (Oz-axis) and is periodic in the other in-plane direction (Ox-axis). Owing to these features, the simulations are done for only one period of the system of length L ¼ 200 nm and thickness h ¼ 40 nm. A schematic representation of the model system is shown in Figure 2.

To test our approach the relaxation process to equilibrium calculated by FEM is compared with the one obtained by a FD approach implemented in the GL_FFT software (by J.C. Toussaint, q Lab. Louis Ne´el). As initial magnetization configuration a sinusoidal profile has been chosen. The magnetization distribution is shown in the Figure 3(a). By monitoring the time evolution of the total energy, shown in Figure 4, we verify, if the time integration scheme describes a dissipation process towards equilibrium. A small energy gap around 1 percent is observed at equilibrium between the FD and FEM calculations. For such physical systems our FEM approach is thus validated. The residual gap can be attributed to the different ways to evaluate the total energy: FD uses local estimations of the magnetization vector and the effective field, whereas in FEM the energy expression (1) is applied to the magnetization field interpolated on each element. We apply now our approach to test cases which, in principle, can be dealt only with FEM. Such a system is a thin film with artificial periodic constrictions (Figure 5). The material parameters considered for this system are the same as before. The dimension of the geometry is: length L ¼ 200 nm and full thickness h ¼ 65 nm. The starting

y h

x z L 0

Finite element formalism for micromagnetism 273

Figure 2. Schematic representation of the stripe structure in a thin film

1

mz

(a)

(b) Notes: For both (a) and (b) the mx and my magnetization components are represented by arrows and the mz component by a grey scale. Material parameters: Aex=2.10 –11 J/m, µ0Ms=1T and K1=105 J/m3

Figure 3. (a) The initial magnetic configuration; (b) the equilibrium magnetization distribution calculated by FEM

COMPEL 27,1

x 105 GLFFT FEM-Az

–1.2

274

Etot(J/m3)

–1.4 –1.6 –1.8 –2 –2.2

0

0.4

0.8 time (a.u.)

Figure 5. Magnetization distribution calculated by FEM at several times in arbitrary units

2×10– 4

1×10– 4

5×10–5

5×10– 6

3×10– 6

0

Figure 4. Time evolution of the total energy density

Note: The same grey scale is used as previously

1.2

1.6

magnetization configuration is similar to that considered for the regular geometry, but is phase shifted with 458C. As expected, the domain walls drift during the equilibration process and finally they are located on the constrictions. The domain walls are pinned on the constrictions, minimizing the energy of the system. This result proves the feasibility of our numerical approach in analyzing systems with complex shape.

Finite element formalism for micromagnetism

4. Conclusions The weak form of the LLG equations presented here seems to be well adapted to deal with 2D micromagnetic systems. Its generalization to 3D systems is in progress and requires for magnetostatics a special treatment of the open boundary based on spherical shell transformations (Brunotte et al., 1992).

275

References Alouges, F. and Jaisson, P. (2006), “Convergence of a finite elements discretization for Landau-Lifschitz equations”, Mathematical Models and Methods in Applied Sciences, Vol. 16 No. 2, pp. 299-313. Braess, D. (2001), Finite Elements, 2nd ed., Cambridge University Press, Cambridge. Brown, W.F. (1963), Micromagnetics, Wiley, New York, NY. Brunotte, X., Meunier, G. and Imhoff, J-F. (1992), “Finite modeling of unbounded problems using transformations”, IEEE Transactions on Magnetics, Vol. 28, pp. 1663-6. Garcı´a-Cervera, C.J., Gimbutas, Z. and Weinan, E. (2003), “Accurate numerical methods for micromagnetics simulations with general geometries”, Journal of Computational Physics, Vol. 184, pp. 37-52. Jubert, P.O., Fruchart, O. and Meyer, C. (2001), “Self-assembled growth of faceted epitaxial Fe (110) islands on Mo (110)/Al2O3”, Physical Review B, Vol. 64, pp. 115419-28. Li, S., Peyrade, D., Natali, M., Lebib, A., Chen, Y., Ebels, U., Buda, L.D. and Ounadjela, K. (2001), “Flux closure structures in cobalt rings”, Physical Review Letters, Vol. 86, pp. 1102-5. Toussaint, J.C., Marty, A., Vukadinovic, N., Ben Youssef, J. and Labrune, M. (2002), “A new technique for ferromagnetic resonance calculations”, Computational Materials Science, Vol. 24, pp. 175-80. About the authors H. Szambolics was born on 12 March 1982. She is graduated at the “Mihai Eminescu” National College in 1999. Then she studied physics and mathematics at the Babes-Bolyai University, Cluj-Napoca. Since, 2005, she has lived in France. At the moment, she is a PhD student in Department of Nanoscience, CNRS Grenoble. H. Szambolics is the corresponding author and can be contacted at: [email protected] Since 2005 L.D. Buda-Prejbeanu, has been an Assistant Professor at Institut National Politechnique de Grenoble (France) (www.enspg.inpg.fr/). As researcher he is working in Spintec Laboratory from CEA Grenoble (France) (www.spintec.fr). His work concerns the nanoscaled magnetic systems. Manly he is developing micromagnetic solvers to study the magnetisation dynamic under external excitations. These simulations serve to design/optimise magnetic applications (reading MR heads, magnetic oscillators, magnetic media). J.C. Toussaint was born in 1965. Since, 2003, he has been a Full Professor of solid state physics and applied mathematics for numerical simulation at Institut Politechnique de Grenoble (France) (www.enspg.inpg.fr) and a researcher at the Neel Institut department NANOsciences CNRS Grenoble (France) (www.spintec.fr). His research consists in developing numerical solvers

COMPEL 27,1

276

for micromagnetism and in applying them to the modeling of magnetisation configurations of nanoscaled systems in statics and in dynamics. O. Fruchart has studied physics in Grenoble and then Paris in Ecole Normale Supe´rieure and Orsay. He graduated in 1998 in Laboratoire Louis Ne´el (CNRS-UJF-INPG), Grenoble, under the supervision of D. Givord. The topic of his PhD was magnetization reversal in ultrathin and submicron-sized ferromagnetic epitaxial dots. His activity broadened to epitaxial self-organization of nanomagnets during his stay at the MPI-Halle in the group of J. Kirschner. Since, that time he has been working back in Grenoble as a permanent Scientist of CNRS, using epitaxial growth for investigating various topics ranging from magnetic domains to magnetic anisotropy in the very nanometer range. He habilitated in 2003 (INPG) and is now working in the research group “Micro- and Nano-magnetism” at the Institut Ne´el (CNRS-UJF-INPG), Grenoble.

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

A 3D electric vector potential formulation for dynamic hysteresis and losses O. Maloberti

3D electric vector potential

277

Laboratoire de Ge´nie Electrique de Grenoble, Saint Martin d’He`res, France and Schneider-Electric Corporate Research Center, Grenoble, France

V. Mazauric Schneider-Electric Corporate Research Center, Grenoble, France, and

G. Meunier and A. Kedous-Lebouc Laboratoire de Ge´nie Electrique de Grenoble, Saint Martin d’He`res, France Abstract Purpose – The purpose of this paper is to introduce the dynamic hysteresis and losses of soft magnetic materials in numerical computation of high-sensitive devices. Design/methodology/approach – So as to do this, the authors propose to lump all the microscopic dynamic effects due to averaging and smoothing techniques that lead to the definition of a dynamic field as proposed by other contributions. In this paper, the method to implement the modified field diffusion process in finite element computations is investigated, explained, detailed and put to the test. Findings – In order to take microscopic magnetization reversal processes and eddy currents that damp the field at the mesoscopic scale, the authors have been led to define a new dynamic property L representative of the magnetic structure and its easiness to change. It is involved in an additional term in both the magnetic behaviour law and the bulk and surface coupling formulations describing the physical problem in iron and at the borders. Research limitations/implications – This model can only be used for macroscopic pieces for which each dimension is bigger than at least four times the characteristic length of magnetic domains. Originality/value – The originality of the paper comes from the need to investigate the possibility to predict iron losses and the corresponding dynamic hysteresis during the processing computation of power electrical devices such as accurate sensors and high-sensitive actuators of earth leakage circuit breaker for example. Keywords Modelling, Eddy currents, Magnetism, Finite element analysis Paper type Research paper

1. Introduction In most soft magnetic materials, dynamic magnetization reversal processes, mainly the motion of Bloch domain walls (Williams et al., 1950), induce complex and incalculable microscopic eddy currents. These last and related magnetic field damping are at the origin of the dynamic hysteresis and associated iron losses (Pry and Bean, 1958). It is however possible to statistically smooth and average the material dynamic behavior (Russakof, 1970), its computation with A-V magnetic vector potential formulations The authors thank ADEME and ANRT for their support.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 277-287 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836834

COMPEL 27,1

278

becomes then natural (Maloberti et al., 2007). With T-F electric vector potential formulations (Chari and Salon, 2000), we are led to change the potential variables T and F. In Section 2, we describe the bulk and coupling formulations dedicated to such materials; then in Section 3, we show and discuss the numerical results obtained with a 3D test case, an inductance; Section 4 is devoted to a high-sensitive actuator for earth leakage circuit breaking applications. Finally, in Section 5, we conclude about the limitations and usefulness of the proposed modeling and methods. 2. Fields formulation and coupling between regions 2.1 Description of a dedicated bulk formulation Considering both classical and microscopic eddy currents, smoothing them (Russakof, 1970) (, . . operator on Figure 1), we can define a convenient property L (Maloberti et al., 2006), that models the magnetic structure dynamics and the magnetic field H damping (equation (3)) (Raulet et al., 2004) due to motion of walls and surrounding microscopic eddy currents (Figure 1). All complexity due to local space variations of microscopic currents loops and due to magnetic objects characteristics (walls surface, density, mobility, . . . etc.) are lumped in the property L. It leads to a frequency-increasing amount of dissipated energy that we should take into account. Enhanced induced Joules losses and magnetization power are then included in an electromagnetic functional PF (Mazauric, 2004): ðð ððð 2 PF ¼ 2 OðE £ HÞd x þ ðB · ›t HÞd3 x

¼

ððð




 sE 2 þ ›t ðH · BÞ d3 x

ð1Þ

E is the electric field, B is the flux density. For small enough time-variations, they check: E ¼ s 21 curlðHÞ

ð2Þ

  B ¼ m H 2 sL 2 › t B

ð3Þ

m is the magnetic permeability, s is the electric conductivity. Performing a variational principle onto the power functional PF (equation (1)) (Mazauric, 2004), we derive a dedicated formulation with delayed electric vector TM and magnetic scalar FM potentials such that:   ð4Þ H 2 sL2 ›t B ¼ ðT M 2 gradF M Þ 8 < divð›t BÞ ¼ 0 : curlðEÞ þ › B ¼ 0 t

ð5Þ

Maxwell (b,h,e,j)

3D electric vector potential

~ 100 µm

b–µ0h +Js –Js

x

279

σe2

≈Λ

σE2

___________

=

σe2

x

h

____ H=

≈Λ

h

µ−1B

____ = µ−1 b x

Notes: We keep the information of magnetic structures (±Js magnetic polarization) and smooth all the microscopic fields (small letters) towards the mesoscopic scale ( operator)

We then propose to write and detail this bulk formulation with the finite element method (FEM) in order to use it for any macroscopic soft geometry including the dynamic hysteresis of its material. Using classical orthogonal shape functions ap and gp ðgpj ¼ ap uj Þ, we can project the unknown variables on a mesh and build a system to determine them at each node p. So equation (5) and the Coulomb gauge (Chari and Salon, 2000) give the following volume residues (equation (6)) (surface ones vanishing because shape functions equal zero on the borders), that we should make as close to zero as possible:

Figure 1. Method that deals with eddy currents at the microscopic and mesoscopic scales (1D Diagram)

COMPEL 27,1

280

0 R¼@

R 1p

1

A R 2q ððð ¼ ð2gradap · ›t BÞd3 x  ððð

curlgqj · s 21 E þ curlgqj s 21 divT M þ gqj ›t B d3 x

ð6Þ

We then give the corresponding Jacobians (Chari and Salon, 2000) and build the matrix equation (7): 00 1 0 1 1 0 1 R 1p R 1p M 11 M 12 B@ C @ A A A 2@ @ A ¼ 2Rtþudt;k R R M 21 M 22 2q 2q tþudt;k

ððð 

tþdt;kþ1

tþdt;k

 md · gradaq d3 x M 11 ð p; qÞ ¼ gradap · dt ððð   md · gqj d3 x M 12 ð p; q; j Þ ¼ 2gradap · dt ððð   md · gradap d3 x M 21 ðq; j; pÞ ¼ 2gqj · dt ! ððð sL 2 m d 21 curlgqj · s uþ M 22 ðq; j; p; i Þ ¼ · curlgpi dt ! md 21 · gpi d3 x þdivgqj · us · divgpi þ gqj · dt

ð7Þ

ð0 , u , 1Þ md is the differential magnetic permeability, ð›B ¼ md ›ðH 2 sL2 ›t BÞÞ. 2.2 Implementation of fields coupling When the problem studied contains several regions with at least one magnetic material, we must comply with some continuity conditions on edges (n is a unit vector normal to the surface enclosing the considered volume): n · dð›t BÞ ¼ 0

ð8Þ

2n £ ðn £ dðEÞÞ ¼ 0

ð9Þ

n · dðcurlðHÞÞ ¼ 0

ð10Þ

2n £ ðn £ dðHÞÞ ¼ 0

ð11Þ

Equations (8) and (10) are due to the flux (equation (5)) and free charges conservation principles, whereas relationships between equations (9) and (11)

correspond to the continuity of the driving fields (equation (2)) enforced by the outer sources. Since, unknown potential variables are often different from one region to another, and with T-F formulations, the conditions (10) and (11) will need specific formulations for the borders in between. In fact equations (8) and (9) are automatically solved due to the previous bulk formulation. In the following, we describe the coupling formulation for the neighboring areas. The Maxwell-Ampe`re equation is enforced by writing the magnetic field H ¼ T-gradF, (Chari and Salon, 2000). Our delayed potentials TM and FM are linked to the total potentials T and F as follow:   T ¼ 1 þ sL 2 m d › t T M

ð12Þ

  grad F ¼ 1 þ sL2 md ›t grad F M

ð13Þ

The conditions (10) and (11) become then equations (14) and (15):     n £ T 2 1 þ sL 2 m d › t T M ¼ 0

ð14Þ

    n £ grad F 2 1 þ sL2 md ›t grad F M

ð15Þ

Between iron and air, this can be done by superimposing n ^ T ¼ 0 on both side and by solving the second equation (15) with a specific surface coupling formulation. The aim is to minimize the following quantity equation (16): ðð

     2 Ckmd k n £ n £ grad F 2 1 þ sL2 md ›t grad F M d2 x

ð16Þ

(C is a regulating constant in comparison to R1 and R2) first with respect to F keeping FM constant; it gives the first surface residue R3 (equation (17)): ðð 

 R 3q ¼ Ckmd k n £ n £ grad aq n      2 £ n £ grad F 2 1 þ sL md ›t grad F M d2 x

ð17Þ

secondly with respect to FM, keeping F constant; it gives the second surface residual R4 equation (18): R4p ¼ 2R3p The corresponding Jacobians are:

ð18Þ

3D electric vector potential

281

COMPEL 27,1

282

ðð

 þCkmd kðn £ ðn £ grad ap ÞÞu · ðn £ ðn £ grad aq ÞÞ d2 x ! ðð sL 2 m d 2Ckmd kðn £ ðn £ grad ap ÞÞ u þ M 32 ð p; qÞ ¼ dt  · ðn £ ðn £ grad aq ÞÞ d2 x ðð
 2Ckmd kðn £ ðn £ grad aq ÞÞu · ðn £ ðn £ grad ap ÞÞ d2 x M 41 ðq; pÞ ¼ ! ðð sL2 md þCkmd kðn £ ðn £ grad aq ÞÞ u þ M 42 ðq; pÞ ¼ dt !

M 31 ð p; qÞ ¼




ð19Þ

· ðn £ ðn £ grad ap ÞÞ d2 x Then the global system, including both volume (R1 and R2) and coupling (R3 and R4) contribution, can be implemented. 3. Numerical results 3.1 Test case We can put the formulation TM-FM to the test. The simple problem of an inductance has been chosen (Figure 2). We automatically take one fields coupling at the interface iron-air into account and compute the power absorbed PL (complex (equation (20)) or real (equation (21)) within the iron core due to the FEM (Chari and Salon, 2000). 3.2 Harmonic results We use an adapted magneto-dynamic application. Active (real part of PL) and reactive (imaginary part of PL) powers (equation (20)) are plotted onto Figure 3. The iron power lost corresponds to the real component. As expected, it increases with the frequency f and the increase of magnetic characteristic length L ( , decrease of the number of walls): ðð 1 P ð v Þ ¼ 2 O ðE _ðx; vÞ £ H _* ðx; vÞÞd2 x _L 2 ððð ð20Þ 1 * ðx; vÞ þ H ð sE ¼ ðx; vÞE ðx; vÞð jvB ðx; vÞÞ* Þd3 x _ _ _ _ 2

v is the angle velocity defined by v ¼ 2pf.

Figure 2. Geometry of the test case studied, an inductance (left); distribution of diffusing total eddy currents and the corresponding skin effect (right)

3D electric vector potential

70,000

Absorbed Power Density (W/m^3)

60,000 50,000 40,000

283

30,000 20,000 10,000 0 –10,000 –20,000

50 10 0 15 0 20 0 25 0 30 0 35 0 40 0 45 0 50 0 55 0 60 0 65 0 70 0 75 0 80 0 85 0 90 0 95 1, 0 00 1, 0 05 0

–30,000 Frequency (Hz) Notes: ∆ : real and imaginary parts with Λ = 0 mm,

: same with Λ = 100 mm

Figure 3. Computation of harmonic power density

3.3 Transient results Using an adapted transient magnetic application, the total power absorbed (equation (21)) has been computed and plotted on Figure 4: ðð ððð
 2 sEðx; tÞ2 þ Hðx; tÞ›t Bðx; tÞ d3 x ð21Þ PL ðtÞ ¼ 2 OðEðx; tÞ £ Hðx; tÞÞd x ¼ 35,000

Absorbed Power Density (W/m^3)

30,000 25,000 20,000 15,000 10,000 5,000 0 –5,000 Time (two periods) Notes: ∆ : f = 50 Hz and 500Hz with Λ = 0 mm,

: same with Λ = 100 mm

Figure 4. Computation of transient power density

COMPEL 27,1

For harmonic fields, the following connections are satisfied: Z   v 2p=v PL ðt Þdt ¼ Re P_L ðvÞ 2p 0

ð22Þ

    PLmax 2 PLmin ¼ 2*P_L ðvÞ

ð23Þ

   hpi PL t ¼ 0 ¼ Re P_L ðvÞ 2 Q_L ðvÞ v

ð24Þ

284

Q_L is defined as P_L without complex conjugation. 3.4 Discussions The iron losses can be computed with a classical formulation coupled to a complex permeability or a dedicated one involving an additional dynamic property L (Figure 3). This last is based on a physical description of magnetization reversal processes and a variational energy principle. It permits a simple a priori transient introduction of the phenomena (Figure 4), improving both computation time and convergence criteria. Experimental identifications of L can be found elsewhere (Maloberti et al., 2006). 4. An appliance: the R5 relay The R5 relay is an actuator destined to earth leakage circuit breaking applications (Figure 5, left). Without any current in the coil; the magnet, stronger than the springs, maintains the pallet against it and the magnetic circuit is closed. When a fault current is to be detected, it creates an opposite flux, which when higher than a certain threshold makes the pallet open (the motion being done by the springs). We plan to study the impact of the domain walls motion onto the threshold and power lost forecasting. The size of the device is near 2 cm3 (, 15 £ 15 £ 7.5 mm3). The coil has got 65 turns and the fault current intensity is 100 mA. We first compute the flux density diffusing into the motionless geometry (pallet slightly open, Figure 5 right).

Coil

Pallet Shutter release flux

0.3

0.2

Figure 5. Geometry of the R5 relay (left); distribution of the diffusing flux density and the corresponding skin effect (right)

0.1

Magnet

Magnetic circuit

Figure 6 shows that the flux responsible for the opening depends on both the frequency and the material dynamic property. If then the leakage current is too “fast” (fault harmonics or high-characteristic rising frequency) the material without enough walls may delay and damp too much the signal to make it usable in the circuit breaking. This information loss is naturally linked to a certain amount of power absorbed and lost within the magnetic circuit. In Figure 7, the quicker the signal is the bigger the power absorbed and the smaller the reactive power are. The walls dynamics seem to decrease the electromechanical conversion performances of the relay and it is in accordance with the results of Figure 6. We need further studies, implementations and simulations with the actual transient motion to complete the analysis.

3D electric vector potential

285

5. Conclusions From material properties and energy considerations, we propose a mesoscopic electromagnetic formulation dedicated to large soft magnetic materials. It is compatible with finite element T-F Formulations and takes iron power losses within the quasi-static approximation into account. It might easily be implemented for transient problems in order to forecast the static and dynamic hysteresis also.

0.12

Shutter release flux density (Wb/m^2)

0.10 0.08 0.06 0.04 0.02 0.00 –0.02 –0.04 –0.06 0

100

200

300

400

500

Frequency (Hz) Notes: ∆: Λ = 50 mm, : Λ = 100 mm, imaginary part of the flux

: Λ = 300 mm. Plain markers for real part and empty ones for

600

Figure 6. Shutter release mean flux density as a function of the frequency

COMPEL 27,1

1.20E-03 1.00E-03

286

Absorbed power (W)

8.00E-04 6.00E-04 4.00E-04 2.00E-04 0.00E+00 – 2.00E-04 – 4.00E-04 0

100

Figure 7. Total absorbed power as a function of the frequency Notes: ∆: Λ = 50 mm, : Λ = 100 mm,

200

300 Frequency (Hz)

400

500

600

: Λ = 300 mm. Plain markers for the real part (losses) and empty ones for the imaginary part (reactive power)

References Chari, M.V.K. and Salon, S.J. (2000), Numerical Methods in Electromagnetism, Academic Press, San Diego, CA, pp. 583-9. Maloberti, O., Kedous-Lebouc, A., Geoffroy, O., Meunier, G. and Mazauric, V. (2006), “Field diffusion-like representation and experimental identification of a magnetic dynamic property”, JMMM Journal, Vol. 304, pp. e507-9. Maloberti, O., Meunier, G., Mazauric, V. and Kedous-Lebouc, A. (2007), “A magnetic vector potential formulation to deal with dynamic hysteresis and induced joule losses within two-dimensional models”, IEEE Transactions on Magnetics, Vol. 43 No. 4. Mazauric, V. (2004), “From thermostatistics to Maxwell equations: a variational approach of electromagnetism”, IEEE Transactions on Magnetics, Vol. 40 No. 2, pp. 945-8. Pry, R. and Bean, C. (1958), “Calculation of the energy loss in magnetic sheet materials using a domain model”, Journal of Applied Physics, Vol. 29 No. 3, pp. 532-3. Raulet, M-A., Ducharne, B., Masson, J-P. and Bayada, G. (2004), “The magnetic field diffusion equation including dynamic hysteresis”, IEEE Transactions on Magnetics, Vol. 42 No. 2, pp. 872-5. Russakof, G. (1970), “A derivation of the macroscopic Maxwell equations”, American Journal of Physics, Vol. 80 No. 6, pp. 1090-4. Williams, H., Shockley, W. and Kittel, C. (1950), “Studies of the propagation velocity of a ferromagnetic domain boundary”, Physical Review, Vol. 80 No. 6, pp. 1090-4. About the authors O. Maloberti was born in 1977 in Gonesse (France). After obtaining his bachelor degree in sciences in Dakar (1995), he moved in France. First in the Fermat Preparatory School of

Toulouse, where he studied maths, physics and engineering sciences. In 1997, he succeeded in entering the High School ENS Cachan. There, he got his first degree in mechanics (1998) and decided to re-orient himself towards advanced studies in physics. So he passed his first and second degree (1999-2000). Afterwards, he began a master degree in optoelectronics (2002). He worked on materials science and electrical engineering for his PhD in the G2Elab laboratory (2003-2006). He is still working today on interactions between materials and electromagnetic fields, and on the design of field sources for the experimental tools and the applications. His fields of interests are interactions between fields or waves and materials. O. Maloberti is the corresponding author and can be contacted at: [email protected] V. Mazauric received a Dipl.-Ing. degree in Electrical Engineering from the National Polytechnic Institute of Grenoble (France) in 1986, a master degree in theoretical physics in 1987 and a master degree in pure mathematics in 1989. In 1987, he joined the French Agency for Aerospace Research and received a PhD in Solid State Physics in 1991. From 1992 to 1994, he was with the Centre for Extreme Materials (Osaka, Japan) to perform research on phase transitions and critical phenomena. Since, 1995, he is with Schneider Electric and is now Principal Scientist in electromagnetism. He is fellow of the Japanese Society for the promotion of science and expert-evaluator at the 7th Framework Program for Research and Development (European Community). G. Meunier is Dipl.-Ing. In Electrical Engineering from the National Polytechnic Institute of Grenoble (INPG), France, in 1977. He received his PhD from the INPG in 1981. He joined the CNRS in 1982 in the Power Electrical Engineering Laboratory of Grenoble where he is presently CNRS Research Director (laboratory G2Elab). His researches are devoted to numerical modelling of electromagnetic phenomena. In the laboratory, he was successively responsible of the “Modelling and CAD team” from 1990 to 1998 and Associate Director from 1998 to 2002. He is presently responsible of the Power Electrical Engineering Master of Research and Doctoral Department. E-mail: [email protected] A. Kedous-Lebouc, received her Electrical Engineer and PhD degrees from the “Institut National Polytechnique de Grenoble” in 1982 and 1985. She is a Senior CNRS Researcher at G2Elab (Grenoble Electrical Engineering Laboratory – former LEG). Her main activity interests are soft magnetic materials and their integration in electrical engineering applications: non-conventional characterization, magnetic behavior modeling and use in electromagnetic devices. Since, 2002, she is also involved in a new research theme on “New giant magnetocaloric effect materials and applications in magnetic refrigeration around room temperature”. Her activity as a whole is presented in more than 100 international journal publications and conferences. E-mail: [email protected]

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

3D electric vector potential

287

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

COMPEL 27,1

288

New discretisation scheme based on splines for volume integral method Application to eddy current testing of tubes Christophe Reboud and Denis Pre´mel CEA, LIST (Laboratoire des Systemes it des Technologies), Gif-sur-Yvette, France

Dominique Lesselier Laboratory of Signals and Systems, De´partement de Recherche en Electromagne´tisme, Gif-sur-Yvette, France, and

Bernard Bisiaux VM France-CEV: Vallourec Research Center, Aulnoye Aymeries, France Abstract Purpose – A numerical model dedicated to external eddy current inspection of tubes has been developed using the volume integral method (VIM). The purpose of this paper is to suggest new discretization schemes based on non-uniform B-splines for the solution of the state equation with the method of moments (MoM). Design/methodology/approach – VIM is a semi-analytical approach providing fast and accurate results for the simulation of eddy current testing (ECT) of pieces with canonical geometries. The state equation derived with this formalism is solved using the Galerkin variant of the well-known MoM. Findings – This paper shows that an accuracy improvement is achieved in MoM by using B-splines with degree 1 or 2 as projection functions in MoM instead of pulse functions. Moreover, comparisons between simulation results show that, for all ECT configurations tested, the use of degree 1 B-splines is sufficient to get this improvement. Originality/value – The use of B-splines functions has already been proposed for MoM in the literature, but not in the framework of the Galerkin variant of MoM. This paper also shows quantitative comparisons between experiment and simulation as well as a study of the minimal degree required to get an accuracy improvement in MoM. Keywords Eddy currents, Numerical analysis, Electromagnetism Paper type Research paper

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 288-297 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836843

1. Introduction Eddy current testing (ECT) is widely used for quality improvement during manufacturing in steel industry. CEA and the Vallourec Research Center have collaborated in order to develop a 3D electromagnetic model for the simulation of ECT configurations: external probes testing a nonmagnetic metallic tube affected by a 3D volumetric flaw. The developments corresponding to these ECT configurations are integrated to the CIVA platform dedicated to non-destructive testing (Reboud et al., 2007). In the case of canonical geometries, the dyadic Green’s formalism (Chew, 1995) associated to the volume integral method (VIM) performs accurate results in a short

computation time. The keystone of the forward problem is the estimation of the internal electric field resulting from the interaction between the flaw and the primary field due to the emitting part of the probe. This interaction is governed by a state equation involving a fictitious volumetric current density. The main problem considered in this paper is how to establish a good representation of this current density using a suitable basis of functions, in order to efficiently solve the numerical problem. Previous works use a numerical scheme which consists in expanding the current volumetric density on pulse or Dirac functions. This choice does not force the solution to be continuous and differentiable. So, other works propose schemes based on uniform splines bases of higher order (Sabbagh, 1993; Bowler, 1991). Unfortunately uniform bases cannot describe a discontinuity of the fictitious current density at the edge of the flaw. To overcome this drawback, we propose a new scheme based on tensor products of non-uniform splines. This paper is organised as follows. The VIM approach is first presented, then the discretisation scheme introduced in method of moments (MoM) is detailed. Finally, the numerical model is validated with experimental data and the accuracy of the new discretisation schemes is illustrated.

New discretisation scheme 289

2. Presentation of the VIM approach Let us consider the following ECT configuration: an external probe testing an infinite, isotropic, linear and non-magnetic conducting tube (permeability m0, conductivity s0) affected by a 3D flaw of bounded support V. A typical ECT configuration is shown in Figure 1. The flaw is represented in the cylindrical coordinates system r ¼ (r, w, z) of the tube by the conductivity s(r), which differs from s0 in V. We define the conductivity contrast function f ðrÞ ¼ ðs0 2 sðrÞÞ=s0 , which can be discontinuous at the edge of V. Let E(r) be the total electric field in V resulting from the interaction between the flaw and the primary electric field Ep(r) emitted by the probe. A fictitious current density denoted by JðrÞ ¼ s0 f ðrÞEðrÞ is introduced in the formalism, and the following state equation is derived:

JðrÞ ¼ J p ðrÞ þ ivm0 s0 f ðrÞ

Z

G ee ðr; r 0 Þ · Jðr 0 Þdr 0 ;

ð1Þ

V

Longitudinal notch Transversal notch

Figure 1. Typical ECT configuration used for nondestructive testing of tubes

COMPEL 27,1

290

where r 0 is the location of a point source, G ee is the electric-electric Green’s dyad for ðr; r 0 Þ [ V2 and J p ðrÞ ¼ s0 f ðrÞE p ðrÞ. The Green’s dyad is calculated in the stratified medium constituted by three regions: air inside the tube, the tube wall, and air outside the tube (Xiang and Lu, 1996): G ee ðr; r 0 Þ

r^ r^ ¼ 2 dðr 2 r 0 Þ k singularity

1 Z 1 1 X 0 0 þ 2 G ee ðr; r 0 ; n; kz Þejnðw2w Þ ejkz ðz2z Þ dkz ; ð2Þ 4p n¼21 21

where d(r –r 0 ) is the Dirac distribution expressed in cylindrical coordinates and: 0

1

1

0

0

B r^ r^ ¼ B @0

0

C 0C A:

0

0

0

The dyad G ee is expressed in equation (2) as an inverse Fourier transform and an inverse Fourier series with respect to the variables z-z0 and w – w0 , respectively. The singularity of G ee , arising when r ¼ r 0 is extracted from the spectral expression. In a second step, the impedance variation dZ of the probe is calculated with an observation equation. In the case of the ECT configuration shown in Figure 1, it combines the primary electric fields (Ep1, Ep2) created by the two identical bobbins functioning in differential mode and driven by an exciting current of magnitude I: Z 1 dZ ¼ 2 ½E p1 ðrÞ 2 E p2 ðrÞ
· JðrÞdr; ð3Þ I V The preliminary calculation of these fields is detailed in Reboud et al. (2007) and Pre´mel et al. (2004) for several probe geometries. Because of the presence of the unknown current density J inside and outside of the integral, equation (1) must be solved numerically. We choose to use the Galerkin variant of MoM (Harrington, 1987), as described in detail in the next section. 3. Discretisation scheme 3.1 Discrete expressions obtained using MoM The unknown J(r) is expressed with N ¼ nr £ nw £ nz basis functions ðBijk ðr 0 ÞÞði;j;kÞ[Nnr £Nnw £Nnz in R3 : Jðr 0 Þ ¼

nw X nz nr X X

J ijk Bijk ðr 0 Þ:

ð4Þ

i¼1 j¼1 k¼1

According to the Galerkin variant of MoM, equation (1) is tested with a set of functions T uvw ðrÞ ¼ Buvw ðrÞ. This leads to the calculation of terms:

J p;uvw ¼ aijk;uvw ¼ b ijk;uvw ¼

Z R

3

Z 3

Z

R

R3

J p ðrÞT uvw ðrÞdr;

ð5Þ

Bijk ðrÞT uvw ðrÞdr;

ð6Þ

f ðrÞT uvw ðrÞ

Z

G ee ðr; r 0 ÞBijk ðr 0 Þdr 0 dr;

ð7Þ

V

and to the following system of discrete equations: ;ðu; v; wÞ [ Nn £ Nnw £ Nnz ; J p;uvw ¼

ð8Þ

X ½aijk;uvw I 3 þ ivm0 s0 b ijk;uvw
J ijk ; ijk

where Jp,uvw and Jijk are 3 £ 1 vectors, bijk,uvw is a 3 £ 3 matrix, and I3 is the 3 £ 3 identity matrix. This system is rewritten in a more classical form: ½J
¼ ðA þ BÞ21 ½J p
;

ð9Þ

where [J] and [Jp] are 3nr nw nz £ 1 vectors, A and B are 3nr nw nz £ 3nr nw nz matrices. The choice of test and basis functions has a great influence on the accuracy and therefore on the convergence of the numerical solution. 3.2 Choice of basis and test functions Test and basis functions used in MoM are defined as tensor products of 1D functions F ri ðrÞ, F wj ðwÞ and F zk ðzÞ : ;r [ R3 ; T ijk ðrÞ ¼ Bijk ðrÞ

¼

nw X nz nr X X

F ri ðrÞF wj ðwÞF zk ðzÞ:

ð10Þ

i¼1 j¼1 k¼1

In the existing version of the model (Pre´mel et al., 2004) the basis and test functions are pulse functions, i.e. polynomials of degree 0. This choice leads to several important simplifications from a numerical point of view. Indeed, matrices A and B defined in equations (6) and (7), respectively, have simplified expressions in this case: aijk;uvw ¼ adijk;uvw ; a [ R; b ijk;uvw ¼

Z

f ðrÞ R3

Z

G ee ðr; r 0 Þdr 0 dr:

ð11Þ ð12Þ

V

Symbol dijk,uvw in equation (11) stands for the Kronecker symbol. Moreover, equation (12) and the symmetry property of dyad G ee : G ee ðr; r 0 Þ ¼ G ee ðr; r 0 ÞT ;

ð13Þ

New discretisation scheme 291

COMPEL 27,1

292

give symmetry properties to the matrix B. Such a numerical scheme provides us with good results for most ECT configurations, but in some critical cases, involving for instance long longitudinal flaws with very thin opening, the convergence of simulation results is not easily reached due to the limitation of the computer memory allowed. A convergence improvement may be achieved by using higher order expansion functions, since pulse functions do not take into account the continuity and the differentiability of the unknown current density J(r) which is numerically evaluated. Uniform B-splines (de Boor, 2001) have been proposed (Sabbagh, 1993; Bowler, 1991) but approximations of J(r) obtained using these bases oscillate at the edge of region V. This problem is illustrated for a 1D case (V ¼ [2 5,5]) in Figure 2: a function J(z) that is discontinuous at the edges of V is approximated with two kinds of bases of B-splines with degree 2. Using the non-uniform basis shown in Figure 2(a), one can see in Figure 2(c) that the approximation Japprox(z) of J(z) is also discontinuous at z ¼ 2 5 and at z ¼ 5. However, if Japprox(z) is constructed from the uniform basis shown in Figure 2(b), it oscillates at the discontinuities locations z ¼ 2 5 and z ¼ 5, as shown in Figure 2(d).

1 z Fk(Z)

0.5 0 –5

0 Z (a)

5

–5

0 Z (b)

5

1 z Fk(Z)

0.5 0

Figure 2. (a) Basis of non-uniform B-splines of degree 2 and range z [ [25,5]; (b) same basis with uniform B-splines; (c) approximation of a function J(z) using the non-uniform basis; (d) approximation of J(z) using the uniform basis

3 2 1 0

J(Z) Japprox(Z)

–5

0 Z (c)

5

J(Z) Japprox(Z)

3 2 1 0 –5

0 Z (d)

5

Here, we propose to use non-uniform B-splines of degree 1 or 2 along the z and w directions in the 3D case for the definition of functions Bijk and Tuvw in equation (10). Then, the solution obtained with equations (9) and (4) is continuous (B-splines of degree 1) and differentiable (B-splines of degree 2) in the flawed region V and may be discontinuous at the edge of the flaw. This can be seen as a generalization of the existing discretisation scheme, since pulse functions are B-splines of degree 0. Starting from the existing ECT model using only pulse functions, a new discretisation scheme using B-splines of degree 1 and 2 for the functions F wj ðwÞ and F zk ðzÞ and pulse functions (B-splines of degree 0) for F ri ðrÞ has been developed. Hence, the function J is still piecewise constant along the r direction since pulse functions are still used along that direction. B-splines of higher degree have been used in a first step along the z and w directions because of the spectral expression of G ee with respect to these variables, see equation (2). Examples of the new 2D expansion functions obtained along the w and z directions are shown in Figure 3.

New discretisation scheme 293

4. Simulation results The convergence of the different schemes with respect to the mesh size are compared for two ECT configurations, shown in Figure 1, involving 3D flaws. The encircling probe testing the tube is constituted by two identical coils connected in differential mode and separated by a gap of 2 mm. The inner and outer radii of the coils are, respectively, equal to 14 and 16 mm, their height is equal to 2 mm, their driving current is 1 A, and the operating frequency is 120 kHz. The tube is non-magnetic and its inner radius and thickness are 9.84 and 1.27 mm, respectively. Its conductivity is 106 S.m2 1. In the first configuration, the flaw is a through-wall transversal notch with a length of 0.1 mm along the z direction and an angular opening of 828. In the second configuration, the flaw is a longitudinal notch with a depth of 0.69 mm, a length of 10 mm along the z direction and a width of 0.1 mm. Figure 4 shows the normalized magnitude and phase of the impedance variation (Figure 6) due to the transversal notch, calculated for different mesh sizes with the three discretisation schemes. For this flaw all schemes have the same behaviour and a quick convergence is observed. From a physical point of view, the currents induced in the tube by the encircling probes are mainly directed along the w direction, so the perturbation caused by this flaw is quite weak. The variations of the current density J

1

1

0.5

0.5

0 2π

0 2π ϕ

π

6 0

5 4

Z (a)

ϕ

π

6 0

5 4

Z (b)

Figure 3. Example of 2D expansion functions obtained for the range [4, 6] £ [0, 2p ] with non-uniform B-splines along the z and w directions, respectively; (a) B-splines of degree 1 (triangles); (b) B-splines of degree 2

COMPEL 27,1

294

Magnitude 1.04

u.a

1.02

1 10,5,5

10,6,6

u.a

10,8,8

1

degree 0 degree 1 degree 2

0.98 10,5,5

10,9,9

Phase

1.02

Figure 4. Schemes convergence for the ECT configuration involving a transversal notch

10,7,7

10,6,6

10,7,7

10,8,8

10,9,9

Mesh size nr,nφ,nz

are consequently smooth and are well described with pulse functions, in spite of the small size of the flaw along the z-axis. However, in the case of the longitudinal notch, the flaw generates an important perturbation of the induced currents, so the density J varies quite much along the w axis in region V, all the more that its size along that direction is very small. Hence, these variations of J are better described using higher-order expansion functions, and one can see in Figure 5 that the scheme using pulse functions exclusively converges much slower than the two other schemes. For the finest mesh used in this case ðnr £ nw £ nz ¼ 5 £ 7 £ 18Þ; the difference between results obtained using the existing scheme (degree 0) and the new ones are about 14 per cent for the magnitude and 2 per cent for the phase. Schemes using degree 1 and 2 functions seem to yield convergence results for this mesh unlike the existing one obtained from pulse functions. Simulation results obtained using degree 1 and 2 B-splines are very close to each other in every ECT configuration tested so far. This result is interesting because the computation time of the dyad G ee increases with the degree of the B-splines used. Thus, using B-splines of degree 1 seems to be sufficient in order to get an improvement of the convergence. 5. Comparison with experimental results Let us consider an experimental validation of the numerical model in the case of the two ECT configurations described in Section 3. Experimental data are compared with simulation results obtained with the three schemes previously presented for the finest

mesh ðnr £ nw £ nz ¼ 5 £ 7 £ 18Þ. Simulated and experimental signals are calibrated so that signals shown in Figure 6(a), corresponding to the transversal notch, have a phase of 108 and a magnitude of 1 V. This flaw is chosen for calibration since all discretisation schemes converge to the same limit in this configuration, see Figure 4. One can observe that simulated ECT signals in Figure 6(c) and (d), corresponding to the schemes using functions of degree 1 and 2, are closer to the experimental one than the signal shown in Figure 6(b) with the scheme using pulse functions only. Quantitative difference of magnitude (in per cent) and phase (in 8) between experiment and simulation, given in Table I, show that an improvement of the model convergence is achieved through the increase of the degree of the expansion functions used in MoM.

New discretisation scheme 295

6. Conclusion and future works A new discretisation scheme based on B-splines in the w, z directions has been proposed for solving the state equation with Galerkin variant of MoM. Non-uniform B-splines are used in order to model accurately the discontinuities of the fictitious current density J. Comparisons between simulations and experimental data show a good improvement of the convergence of the model when higher order B-splines of degree 1 and 2 are used in MoM. This improvement allows the simulation of more complex ECT configurations, and opens up several perspectives. Up to now, pulse functions have been used in the r direction. Another discretisation scheme with higher-order B-splines functions in the r, w and z directions is currently under implementation. We expect to achieve a better convergence with this scheme than with the existing ones, especially for the ECT simulation of through-wall defects.

Magnitude 1 0.9 u.a 0.8 0.7 0.6 5,5,5 5,5,6 5,5,7 5,6,8 5,6,9 5,6,10 5,7,11 5,7,12 5,7,13 5,7,14 5,7,16 5,7,18

Phase 1 u.a 0.95

degree 0 degree 1 degree 2

0.9 5,5,5 5,5,6 5,5,7 5,6,8 5,6,9 5,6,10 5,7,11 5,7,12 5,7,13 5,7,14 5,7,16 5,7,18 Mesh size nr,nφ,nz

Figure 5. Convergence of schemes for the ECT configuration involving a longitudinal notch

COMPEL 27,1

0.1 Imag[dZ] (V)

0

Calibration (1 V,10˚)

Transversal notch: Experiment Simulation

–0.1 –0.4 –0.3 –0.2 –0.1

296

0 (a)

0.1

0.2

0.3

0.4

0.1 Imag[dZ] (V)

Longitudinal notch: Experiment Degree 0 (pulses)

0 –0.1 –0.5

0 (b)

0.5

0.1 Imag[dZ] (V)

0 –0.1 –0.5

Figure 6. Comparison between simulation results and experimental data for both flaws and the three discretisation schemes. The mesh used for simulation is the finest mesh tested in this case: ðnr £ nw £ nz ¼ 5 £ 7 £ 18Þ

0 (c)

0.5

0.1 Imag[dZ] (V)

0 –0.1 –0.5

Longitudinal notch: Experiment Degree 2

0 Real[dZ] (V)

0.5

(d)

Degree of B-splines along w, z 0 1 2

Table I.

Longitudinal notch: Experiment Degree 1 (triangles)

Magnitude difference (per cent)

Phase difference (8)

218.09 26.38 26.13

25 24 24

Notes: Comparisons between experiment and simulated signals obtained with the three discretisation schemes in the case of the longitudinal notch. The mesh size used for simulation is ðnr £ nw £ nz ¼ 5 £ 7 £ 18Þ

References Bowler, J. (1991), “Three dimensional eddy current probe-flaw calculation using volume elements”, Electrosoft, Vol. 2 Nos 2/3, pp. 142-56. Chew, W.C. (1995), Waves and Fields in Inhomogeneous Media, 2nd ed., IEEE Press, Piscataway, NJ. de Boor, C. (2001), A Practical Guide to Splines, Springer-Verlag, New York, NY.

Harrington, R. (1987), “The method of moments in electromagnetics”, Journal of Electromagnetic Waves and Applications, Vol. 1 No. 3, pp. 181-200. ´ Premel, D., Pichenot, G. and Sollier, T. (2004), “Development of a 3D electromagnetic model for eddy current tubing inspection”, International Journal of Applied Electromagnetics and Mechanics, Vol. 19, pp. 521-5. Reboud, C., Pre´mel, D., Pichenot, G., Lesselier, D. and Bisiaux, B. (2007), “Development and validation of a 3D model dedicated to eddy current non destructive testing of tubes by encircling probes”, International Journal of Applied Electromagnetics and Mechanics, Vol. 25 No. 1, pp. 313-17. Sabbagh, H. (1993), “Splines and their reciprocal-bases in volume integral equations”, IEEE Transactions on Magnetics, Vol. 29 No. 6, pp. 4142-52. Xiang, Z. and Lu, Y. (1996), “Electromagnetic dyadic Green’s function in cylindrically multilayered media”, IEEE Transactions on Microwave Theory and Techniques, Vol. 44 No. 4, pp. 614-21. About the authors Christophe Reboud obtained an engineering degree and a MSc in signal processing at Ecole Centrale de Nantes (2003), and a PhD degree in Electrical Engineering (2006) at Universite´ Paris Sud 11. He is currently working at CEA LIST in electromagnetic modelling for the simulation software CIVA dedicated to non-destructive testing. Christophe Reboud is the corresponding author and can be contacted at: [email protected] Denis Pre´mel worked from 1994 to 2002 at the “Ecole Normale Supe´rieure de Cachan” on an Associate Professor position. He is currently in charge at the Commissariat a` l’Energie Atomique (CEA) of the development of numerical models dedicated to ECNDT and inverse scattering resulting in new functionalities within the CIVA platform. E-mail: [email protected] Dominique Lesselier is Directeur de recherche at the Centre National de la Recherche Scientifique (CNRS) since October 1988. As Director, since January 2006, of the Groupement de Recherche CNRS known as GDR Ondes, he is also managing a large network of laboratories and researchers involved in the science of waves. E-mail: [email protected] Bernard Bisiaux is the Head of the NDT Department at the Vallourec Research Center (CEV) since 1990 and is working for Vallourec since 1975. He is in charge of research and development in NDT: Ultrasonic testing, Eddy Current testing and Electromagnetic testing. He is NDT Level III in UT, ET, PT, RT, MT. E-mail: [email protected]

To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints

New discretisation scheme 297

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

COMPEL 27,1

298

Hybridization of volumetric and surface models for the computation of the T/R EC probe response due to a thin opening flaw Le´a Maurice and Denis Pre´mel CEA LIST Saclay, Gif-sur-Yvette, France

Jo´zsef Pa´vo´ Budapest University of Technology and Economics, Budapest, Hungary

Dominique Lesselier Laboratory of Signals and Systems, Department of Research in Electromagnetics, and

Alain Nicolas Centrale Lyon, Ecully, France Abstract Purpose – The purpose of this paper is to describe the development of simulation tools dedicated to eddy current non destructive testing (ECNDT) on planar structures implying planar defects. Two integral approaches using the Green dyadic formalism are considered. Design/methodology/approach – The surface integral model (SIM) is dedicated to ideal cracks, whereas the volume integral method is adapted to general volumetric defects. Findings – The authors observed that SIM provides satisfactory results, except in some critical transmitting/receiving (T/R) configurations. This led us to propose a hybrid method based on the combination of the two previous ones. Originality/value – This method enables to simulate ECNDT on planar structures implying defects with a small opening using T/R probes. Keywords Eddy currents, Tests and testing, Surface defects, Integral equations Paper type Research paper

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 1, 2008 pp. 298-306 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640810836852

1. Introduction Eddy current interactions with planar defects, i.e. defects which have an opening width much smaller than the length of the flaw, have been studied by many authors in the last two decades. A useful approach consists in an idealization of such defects through the definition of an ideal thin crack: assuming that the thickness tends toward zero, the thin crack acts as an impenetrable barrier to the electric current (Bowler, 1994). In the case of a transverse planar flaw located in the (x, y) plane, the spatial variable x along the thickness of the flaw disappears, and the flaw is represented by a current surface dipole density, referred to as p( y, z), which is a scalar quantity depending on two

spatial variables corresponding to the dimensions of the surface of the ideal flaw. The surface dipole density is solution of an integral equation on the surface of the crack, which involves an hypersingular kernel when it is evaluated in the spatial domain (Beltrame and Burais, 2002a, b). In an alternative way, the kernel can be evaluated in the spectral domain (Pa´vo´ and Miya, 1994), and the use of a global approximation of p( y, z) (Pa´vo´ and Lesselier, 2006) may overcome some numerical difficulties coming from specific boundary conditions to be satisfied by p( y, z) (Bowler et al., 1997). This numerical approach leads to the implementation of a fast numerical model which can advantageously be integrated in an iterative procedure for probe design or for performing some parametric studies (Deng et al., 2005). While testing this model on some transmitting/receiving (T/R) configurations, we nevertheless noticed unsuitable results. This led us to compare them to results yielded by the volume integral method (VIM), whose efficiency had already been demonstrated for a great number of NDT configurations (Buvat et al., 2005; Pre´mel et al., 2004). In a first part, this paper gives a review of surface integral model (SIM) and VIM, and a third model is proposed, set up thanks to a combination of SIM and VIM. In a second part, we carry out experiments on T/R probes to evaluate the respective performances of each model, regarding the accuracy of results as well as the computational time and the computational load required. 2. Description of the models Let us consider a conducting non-magnetic slab of conductivity s0. It is tested with an air-core probe functioning either in absolute or in T/R mode, and featuring, respectively, one or several coils. We operate in a time-harmonic regime, with an excitation assumed to vary in time as the real part of exp(2 ivt). The driving current in the transmitting coil (respectively in the receiving coil) has a magnitude of I T (respectively I R) and v stands for the angular frequency. 2.1 Surface integral model (SIM) This model is based on the idealization of the perfectly non-conducting planar flaw, as defined in the introduction. The ideal crack is represented by an equivalent surface distribution: the dipole surface density p(rs) defined by (Bowler, 1994): 2 Eþ t ðr s Þ 2 E t ðr s Þ ¼ 2

1 7t pðr s Þ s0

ð1Þ

2 where rs ¼ ( y, z) is the variable describing the surface of the flaw, E þ t ðr s Þ and E t ðr s Þ are the tangential components of E(rs), the electric field in the vicinity of the flaw, on both sides of the flaw. ft is the differential operator once the normal derivate is removed. Denoting E P the primary field due to the emitting part of the probe, Sf the surface of the flaw, n the unit vector normal to Sf, m0 the permeability of the vacuum, the dipole surface density is determined by solving:

E P ðr 0 Þ · n ¼ 2 lim ivm0

Z

r s !r 0 [S f

Sf

G nn ðr s jr 0s Þpðr 0s Þdr 0s

ð2Þ

Hybridization of volumetric and surface models 299

COMPEL 27,1

 s jr 0 Þ · n, and Gðr  s ; r 0 Þ is the electric-electric dyadic Green’s with G 0nn ðr s jr 0s Þ ¼ n · Gðr s s function (Chew, 1995). The probe response is given by (Bowler, 1994): Z I T · I R DZ ¼ 2 E R ðr s Þ · n pðr s Þdr s ð3Þ Sf

300

R

where E is the electric field which would be due to the receiving coil assumed to operate in the source mode. The transmitting and the receiving coils are identical in the case of an absolute probe, so I T ¼ I R and E R ¼ E P, thus retrieving the usual formula of impedance (Bowler, 1994). This model has been experimentally validated (Pa´vo´ et al., 2004), and a global decomposition of p(rs) is used (Pa´vo´ and Lesselier, 2006) for the case of rectangularly-shaped flaws, written: pð y; zÞ ¼

My X Mz X

mz y p my ;mz f m y ð yÞ f z ðzÞ

ð4Þ

my ¼1 mz ¼1

The approximating functions are defined on the whole surface of the crack, and their explicit formulation, e.g. for a surface breaking flaw, is: rffiffiffi rffiffiffi      2 my b 2 2mz 2 1 my mz · sin · cos f y ð yÞ ¼ p yþ pz f z ðzÞ ¼ b 2 a 2a b mz th th y where f m y and f z are, respectively, the my and mz mode functions in the y and z direction. b and a, respectively, stand for the length of the flaw along the y dimension, z and the depth of the flaw, along the z dimension. Three other definitions of the f m z functions are obtained by applying the boundary conditions on the Fourier modes for the three other types of defects, namely: flaws opening at the bottom of the plate, embedded cracks, and through-wall cracks (Pa´vo´ and Lesselier, 2006). The main advantage of this global approximation is that it enables to take into account the boundary conditions on p(rs), namely (Bowler et al., 1997): 8 on embedded edges