[by jan melkebeek, r. belmans] compel - selected (bookos.org)

ISBN 0-86176-897-3 ISSN 0332-1649

COMPELThe International Journal for Computation and Mathematics in Electrical and Electronic Engineering

Selected papers from the international conference onelectrical machines (ICEM) 2002, Bruges, BelgiumGuest Editor: Professor dr. ir. Jan Melkebeek

Co-Editor: Professor R. Belmans

Volume 22 Number 4 2003

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Access this journal online _________________________ 812

Editorial advisory board __________________________ 813

Abstracts and keywords __________________________ 814

Guest editorial ___________________________________ 820

Application of stochastic simulation in theoptimisation process of hydroelectric generatorsE. Schlemmer, W. Harb, J. Schoenauer and F. Mueller ________________ 821

Eddy-current computation on a one pole-pitch modelof a synchronous claw-pole alternatorChristian Kaehler and Gerhard Henneberger ________________________ 834

A closer view on inductance in switched reluctancemotorsIrma Hajdarevic and Hansjorg Kofler______________________________ 847

Spatial linearity of an unbalanced magnetic pull ininduction motors during eccentric rotor motionsA. Tenhunen, T.P. Holopainen and A. Arkkio _______________________ 862

COMPELThe International Journal for Computation andMathematics in Electrical and ElectronicEngineering

Selected papers from the international conference on electricalmachines (ICEM) 2002, Bruges, Belgium

Guest EditorProfessor dr. ir. Jan Melkebeek

Co-EditorProfessor R. Belmans

ISSN 0332-1649

Volume 22Number 42003

CONTENTS

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Calculation of eddy current losses and temperaturerises at the stator end portion of hydro generatorsSt Kunckel, G. Klaus and M. Liese ________________________________ 877

Comparison of two modeling methods for inductionmachine study: application to diagnosisC. Delmotte-Delforge, H. Henao, G. Ekwe, P. Brochet and G-A. Capolino ___ 891

Finite element modeling of the temperaturedistribution in the stator of a synchronous generatorJosef Schoenauer, Erwin Schlemmer and Franz Mueller _______________ 909

Discrete-time modeling of AC motors for high powerAC drives controlS. Poullain, J.L. Thomas and A. Benchaib___________________________ 922

Design of a mass-production low-cost claw-polemotor for an automotive applicationR. Felicetti and I. Ramesohl ______________________________________ 937

A magnetic network approach to the transientanalysis of synchronous machinesM. Andriollo, T. Bertoncelli and A. Di Gerlando _____________________ 953

Numerical magnetic field analysis and signalprocessing for fault diagnostics of electricalmachinesS. Poyhonen, M. Negrea, P. Jover, A. Arkkio and H. Hyotyniemi ________ 969

Thermal modeling and testing of ahigh-speed axial-flux permanent-magnet machineF. Sahin and A.J.A. Vandenput ___________________________________ 982

Current shapes leading to positive effects onacoustic noise of switched reluctance drivesM. Kaiserseder, J. Schmid, W. Amrhein and V. Scheef ________________ 998

Vibrations of magnetic origin of switched reluctancemotorsLieven Vandevelde, Johan J.C. Gyselinck, Francis Bokose andJan A.A. Melkebeek_____________________________________________ 1009

Two-dimensional harmonic balance finite elementmodelling of electrical machines taking motion intoaccountJ. Gyselinck, P. Dular, L. Vandevelde, J. Melkebeek, A.M. Oliveira andP. Kuo-Peng __________________________________________________ 1021

A general analytical model of electrical permanentmagnet machine dedicated to optimal designE. Fitan, F. Messin and B. Nogarede ______________________________ 1037

CONTENTScontinued

Modelling of electromagnetic losses in asynchronousmachinesL. Dupre, M. De Wulf, D. Makaveev, V. Permiakov, A. Pulnikov andJ. Melkebeek___________________________________________________ 1051

Staged modelling: a methodology for developingreal-life electrical systems applied to the transientbehaviour of a permanent magnet servo motorF. Henrotte, I. Podoleanu and K. Hameyer __________________________ 1066

3D h-f finite element formulation for thecomputation of a linear transverse flux actuatorG. Deliege, F. Henrotte, H. Vande Sande and K. Hameyer _____________ 1077

Constrained least-squares method for the estimationof the electrical parameters of an induction motorMaurizio Cirrincione, Marcello Pucci, Giansalvo Cirrincione andGerard-Andre Capolino__________________________________________ 1089

Calculation of eddy current losses in metal parts ofpower transformersErich Schmidt, Peter Hamberger and Walter Seitlinger ________________ 1102

Optimal design of high frequency induction motorswith the aid of finite elementsAtanasi Jornet, Angel Orille, Alberto Perez and Diego Perez ____________ 1115

Analytic calculation of the voltage shape of salientpole synchronous generators including damperwinding and saturation effectsGeorg Traxler-Samek, Alexander Schwery and Erich Schmidt __________ 1126

Numerical modelling of electromagnetic process inelectromechanical systemsVyacheslav A. Kuznetsov and Pascal Brochet ________________________ 1142

A coupled electromagnetic-mechanical-acousticmodel of a DC electric motorMartin Furlan, Andrej Cernigoj and Miha Boltezar___________________ 1155

Current distribution within multi strand windingsfor electrical machines with frequency convertersupplyOliver Drubel__________________________________________________ 1166

New books _______________________________________ 1182

Note from the publisher___________________________ 1183

Awards for Excellence ____________________________ 1184

Erratum__________________________________________ 1186

Index to volume 22, 2003 _________________________ 1187

CONTENTScontinued

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Editorialadvisory board

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COMPEL : The International Journalfor Computation and Mathematics inElectrical and Electronic Engineering

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EDITORIAL ADVISORY BOARD

Professor O. BiroGraz University of Technology, Graz, Austria

Professor J.R. CardosoUniversity of Sao Paulo, Sao Paulo, Brazil

Professor C. ChristopoulosUniversity of Nottingham, Nottingham, UK

Professor J.-L. CoulombLaboratoire d’Electrotechnique de Grenoble,Grenoble, France

Professor X. CuiNorth China Electric Power University, Baoding,Hebei, China

Professor A. DemenkoPoznan University of Technology, Poznan, Poland

Professor E. FreemanImperial College of Science, London, UK

Professor Song-yop HahnSeoul National University, Seoul, Korea

Professor Dr.-Ing K. HameyerKatholieke Universiteit Leuven, Leuven-Heverlee,Belgium

Professor N. IdaUniversity of Akron, Akron, USA

Professor A. JackThe University, Newcastle Upon Tyne, UK

Professor A. KostTechnische Universitat Berlin, Berlin, Germany

Professor T.S. LowNational University of Singapore, Singapore

Professor D. LowtherMcGill University, Ville Saint Laurent, Quebec,Canada

Professor O. MohammedFlorida International University, Florida, USA

Professor G. MolinariUniversity of Genoa, Genoa, Italy

Professor A. RazekLaboratorie de Genie Electrique de Paris - CNRS,Gif sur Yvette, France

Professor G. RubinacciUniversita di Cassino, Cassino, Italy

Professor M. RudanUniversity of Bologna, Bologna, Italy

Professor M. SeverThe Hebrew University, Jerusalem, Israel

Professor J. TegopoulosNational Tech University of Athens, Athens, Greece

Professor W. TrowbridgeVector Fields Ltd, Oxford, UK

Professor T. TsiboukisAristotle University of Thessaloniki, Thessaloniki,Greece

Dr L.R. TurnerArgonne National Laboratory, Argonne, USA

Professor Dr.-Ing T. WeilandTechnische Universitat Darmstadt, Darmstadt,Germany

Professor K. ZakrzewskiPolitechnika Lodzka, Lodz, Poland

Application of stochastic simulationin the optimisation processof hydroelectric generators

E. Schlemmer, W. Harb, J. Schoenauer andF. Mueller

Keywords Stochastic modelling,Optimization, Genetic algorithms,Risk analysis

In this paper, stochastic optimisation and riskestimation techniques are applied to theproblem of hydroelectric generator design.Optimisation results from deterministicsimulations can involve considerable risksdue to unavoidable variations in systemproperties as well as environmental conditions.Therefore, stochastic simulation is used toinclude the effects of parameter scatter andnoise effects in the computer models. Thisallows the evaluation of the scatter inperformance and thus an assessment ofreliability and quality of the simulatedsystem.

Eddy-current computation on a onepole-pitch model of a synchronousclaw-pole alternator

Christian Kaehler and Gerhard Henneberger

Keywords Eddy currents, Finite elements,Alternators

This paper deals with 3D finite-elementcalculation of eddy currents in the claws ofa claw-pole alternator taking the rotationalgeometry movement into account. Twotransient edge-based vector formulations areutilised. The reduction of the model to onlyone pole pitch in combination with a specialboundary pairing in the air gap for theapplied lock-step method is presented.Calculations of varying material conductivityare performed with simplified end windings.The speed characteristics of the eddy currentswith real conductivity and realistic endwindings concludes the paper.

A closer view on inductance in switchedreluctance motors

Irma Hajdarevic and Hansjorg Kofler

Keywords Motors, Inductance,Magnetic fields, Finite element method

Switched reluctance motors are promisingcandidates for a wide variety of drive

applications. The theoretical description ofsuch motors is often reduced to the rathersimple, but clear concept of cI-characteristics.In contrast to this, the machine itself is a realelectrical machine and must therefore beanalysed as is done with conventionalmachines although the experimental machineis constructed as simple as possible. The paperwill first describe some well known basicsconnected to very short machines and flux. Inthe next step, the calculation of stray fluxquantities aided by 3D-FEM is discussed andfinally a comparison of calculation andexperimental measurements is given.

Spatial linearity of an unbalancedmagnetic pull in induction motorsduring eccentric rotor motions

A. Tenhunen, T.P. Holopainen and A. Arkkio

Keywords Motors, Magnetic forces, Rotors

There is an unbalanced magnetic pullbetween the rotor and stator of the cageinduction motor when the rotor is notconcentric with the stator. These forcesdepend on the position and motion of thecentre point of the rotor. In this paper,the linearity of the forces in proportion tothe rotor eccentricity is studied numericallyusing time-stepping finite element analysis.The results show that usually the forces arelinear in proportion to the rotor eccentricity.However, the closed rotor slots may break thespatial linearity at some operation conditionsof the motor.

Calculation of eddy current losses andtemperature rises at the stator endportion of hydro generators

St. Kunckel, G. Klaus and M. Liese

Keywords Eddy currents, Temperature,Generators

This paper deals with a calculation method ofeddy current losses and temperature rises atthe stator end teeth of hydro generators. It canbe used for analysing and evaluating differentdesign variants when optimising the statorcore end portion. The calculation methodsimulates the three-dimensional local coreend field, but uses only a two-dimensionalcalculation model. Amongst all the statorteeth it treats the tooth with the highest axial

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COMPEL : The International Journalfor Computation and Mathematics inElectrical and Electronic EngineeringVol. 22 No. 4, 2003Abstracts and keywords# MCB UP Limited0332-1649

and radial magnetic flux impact. The paperpresents a collection of calculation algorithmsof the method and provides some resultsgained for two different stator core enddesigns.

Comparison of two modeling methodsfor induction machine study:application to diagnosis

C. Delmotte-Delforge, H. Henao, G. Ekwe,P. Brochet and G-A. Capolino

Keywords Diagnosis, Modelling,Induction machines

This paper presents two modeling methodsapplied to induction machine study in order toconstruct a tool for diagnosis purpose. Thefirst method is based on permeance networksusing finite element analysis to calculatemagnetic equivalent circuit parameters. Thesecond method consists of the elaboration ofan electric equivalent circuit obtained fromminimal geometrical knowledge on stator androtor parts of the machine on study. Thesetwo methods are presented and their resultsare compared with respect to the normal androtor broken bar operation. For this study, asimple structure induction machine with threestator coils and six rotor bars has beeninvestigated. The presented results concernstator currents and electromagnetic torque forthe rated speed and the magnitude of thestator current harmonic components havebeen compared.

Finite element modeling of thetemperature distribution in the statorof a synchronous generator

Josef Schoenauer, Erwin Schlemmer andFranz Mueller

Keywords Finite element modelling,Temperature, Thermal modelling, Generators

In this paper, we applied the finite elementmodeling to the stator temperature distributionof a hydroelectric generator. The electricallosses produce a temperature distributionin the stator of a synchronous generator.For the calculation and optimization of thetemperature distribution, a full parameterizedthermal model of the stator was created usingthe finite element method. Now it is possibleto calculate the thermal effects of differentparameter modifications and additionally we

can optimize the heat transfer for the statorwith variant calculations. The most importantbar fitting systems and its thermal efforts areincluded in this thermal stator model. Ourtargets are to decrease the expensive andtime-consuming laboratory measurements inthe future and improve the accuracy of thestandard calculation software. To estimate theaccuracy of the finite element model we buildan additional laboratory model.

Discrete-time modeling of AC motorsfor high power AC drives control

S. Poullain, J.L. Thomas and A. Benchaib

Keywords Time-domain modelling, Motors,Control

This paper proposes a new discrete-timeformulation of state-space model for voltagesource inverter (VSI) fed AC motors,introducing the free evolution of the motorstate and characterized by both thesimplification of torque and flux outputequations and the definition of a predictivereference frame oriented on the rotor freeevolution vector. The potential of theproposed model for high dynamics discrete-time controller synthesis is illustrated throughan application to SM-PMSM.

Design of a mass-production low-costclaw-pole motor for an automotiveapplication

R. Felicetti and I. Ramesohl

Keywords Rotors, Motors,Electrical machines, Mechatronics

This paper describes a thermal and electricalmodel, used at Robert Bosch GmbH for thedesign of an innovative motor for a water-pump. In addition, it offers an example of ahighly integrated mechatronic system. Abonded-ferrite inner rotor has been developedwith an integrated front centrifugal impellerwhich is driven by the magnetic interactionof a rotating field created by claw-poles. Thetwo phase unipolar coil arrangement is fed byan internal circuit using two MOSFETScontrolled by the commutation signal from abipolar Hall-IC. This is the first mass-production example of an electrical machinefor an automotive application where the clawpole topology is used to realise the armature of

Abstracts andkeywords

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the motor (i.e. the rotating field) and not theexcitation field.

A magnetic network approach to thetransient analysis of synchronousmachines

M. Andriollo, T. Bertoncelli andA. Di Gerlando

Keywords Magnetic forces,Synchronous machine, Simulation

The technique for the simulation of thedynamic behaviour of rotating machinespresented in the paper is based on anequivalent circuit representation of the mag-netic configuration. The circuit parametersare obtained by a preliminary automatedsequence of magnetostatic FEM analysesand take into account the local magneticsaturations. The adopted solution technique isbased on an invariant network topologyapproach: its application, presented for theoperation analysis of a low-power synchronousgenerator, allows a great reduction of thecalculation time in comparison with acommercial FEM code for the transientsimulation.

Numerical magnetic field analysis andsignal processing for fault diagnosticsof electrical machines

S. Poyhonen, M. Negrea, P. Jover,A. Arkkio and H. Hyotyniemi

Keywords Condition monitoring,Electromagnetic fields, Electrical machines,Finite element method, Signal processing,Fault analysis

Numerical magnetic field analysis is usedfor predicting the performance of aninduction motor and a slip-ring generatorhaving different faults implemented in theirstructure. Virtual measurement data providedby the numerical magnetic field analysis areanalysed using modern signal processingtechniques to get a reliable indication ofthe fault. Support vector machine basedclassification is applied to fault diagnostics.The stator line current, circulating currentsbetween parallel stator branches and forcesbetween the stator and rotor are compared asmedia of fault detection.

Thermal modeling and testing of ahigh-speed axial-fluxpermanent-magnet machine

F. Sahin and A.J.A. Vandenput

Keywords Flux, Permanent magnets,Thermal analysis, Performance,Electrical machines

This paper gives an overview of the design,manufacturing and testing of a high-speed(16,000 rpm and 30 kW) AFPM synchronousmachine, which is mounted inside, and as anintegral part of, a flywheel. This system willsubsequently be used for transient energystorage and ICE operating point optimizationin an HEV. The paper focuses on the majordesign issues, particularly with regard to thehigh rotational speed, and investigates theloss mechanisms which are apparent therein,e.g. iron losses, rotor losses, and frictionlosses. The paper describes the high-speedtesting facility and includes measured results,which will be compared to calculated values.

Current shapes leading to positiveeffects on acoustic noise of switchedreluctance drives

M. Kaiserseder, J. Schmid, W. Amrhein andV. Scheef

Keywords Torque, Optimization, Motors

A torque ripple minimization technique forswitched reluctance motors is shown in thispaper. Precalculated current shapes areapplied to reduce torque ripple and to raisethe degrees of freedom of the application in thecommutation region. The optimization criteriafor this region can be chosen freely. Therefore,it is possible to take positive effect to somemotor characteristics like power losses,mechanical vibrations or acoustic noise.

Vibrations of magnetic origin ofswitched reluctance motors

Lieven Vandevelde, Johan J.C. Gyselinck,Francis Bokose and Jan A.A. Melkebeek

Keywords Motors, Vibration, Noise,Magnetic forces

Vibrations and acoustic noise are some of thefundamental problems in the design andexploitation of switched reluctance motors

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(SRMs). Adequate experimental and analysismethods may help to resolve these problems.This paper presents a theoretical analysis ofthe magnetic force distribution in SRM and aprocedure for calculating the magnetic forcesand the resulting vibrations based on the 2Dfinite element method. Magnetic field andforce computations and a structural analysisof the stator have been carried out in order tocompute the frequency spectrum of thegeneralized forces and displacements of themost relevant vibration modes. It is shownthat for these vibration modes, the frequencyspectrum can be predicted analytically.The theoretical and the numerical analyseshave been applied to a 6/4 SRM and anexperimental validation is presented.

Two-dimensional harmonic balancefinite element modelling of electricalmachines taking motion into account

J. Gyselinck, P. Dular, L. Vandevelde,J. Melkebeek, A.M. Oliveira and P. Kuo-Peng

Keywords Finite element method,Electrical machines, Harmonics,Magnetic devices

An original and easy-to-implement method totake into account movement (motion) in the2D harmonic balance finite element modellingof electrical machines is presented. The globalharmonic balance system of algebraicequations is derived by applying theGalerkin approach to both the space andtime discretisation. The harmonic basisfunctions, i.e. a cosine and a sine functionfor each nonzero frequency and a constantfunction 1 for the DC component, are used forapproximating the periodic time variation aswell as for weighing the time domainequations in the fundamental period. Inpractice, this requires some elementarymanipulations of the moving band stiffnessmatrix. Magnetic saturation and electricalcircuit coupling are considered in theanalysis as well. As an application example,the noload operation of a permanent-magnetmachine is considered. The voltage andinduction waveforms obtained with theproposed harmonic balance method areshown to converge well to those obtainedwith time stepping.

A general analytical model of electricalpermanent magnet machine dedicatedto optimal design

E. Fitan, F. Messin and B. Nogarede

Keywords Permanent magnets,Electrical machines, Modelling

What is new in this work is the genericcapabilities of the proposed analytical modelof permanent magnet machines associatedwith a novel deterministic global optimizationmethod. That allows to solve some moregeneral inverse problem of designing. Theanalytical approach is powerful to take intoaccount various kinds of constraints(electromagnetical, thermal, etc.). The inverseproblem associated with the optimal design ofactuators could then be formulated as amixed-constrained optimization problem. Inorder to solve these problems, interval Branchand Bound algorithms which have alreadyproved their efficiency, have made it possibleto determine some optimized rotatingmachines.

Modelling of electromagnetic losses inasynchronous machines

L. Dupre, M. De Wulf, D. Makaveev,V. Permiakov, A. Pulnikov and J. Melkebeek

Keywords Hysteresis, Modelling,Electrical machines

This paper deals with the numericalmodelling of electromagnetic losses inelectrical machines, using electromagneticfield computations, combined with advancedmaterial characterisations. The papergradually proceeds to the actual reasons whythe building factor, defined as the ratio of themeasured iron losses in the machine and thelosses obtained under standard conditions,exceeds the value of 1.

Staged modelling: a methodology fordeveloping real-life electrical systemsapplied to the transient behaviour of apermanent magnet servo motor

F. Henrotte, I. Podoleanu and K. Hameyer

Keywords Electrical machines, Modelling,Design

This paper presents a methodology toachieve a global dynamic model of anelectrical system that consists of a battery,


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an inverter, a permanent magnet servo motorand a turbine. The stress is placed on thefact that a classical finite element modelwould not be able to provide a satisfactoryrepresentation of the transient behaviour ofthe whole system. A staged modelling isproposed instead, which succeeds inproviding a complete picture of the systemand relies on numerous finite elementcomputations.

3D h-f finite element formulation forthe computation of a linear transverseflux actuator

G. Deliege, F. Henrotte, H. Vande Sande andK. Hameyer

Keywords Finite element analysis,Permanent magnets, Optimization

A finite element analysis of a permanentmagnet transverse flux linear actuator ispresented. In this application where we needa small model (for optimisation purposes) aswell as a high accuracy on the computedforce, we propose to combine several modelswith different levels of size and complexity, inorder to progressively elaborate an accurate,but nevertheless tractable, model of thesystem.

Constrained least-squares method forthe estimation of the electricalparameters of an induction motor

Maurizio Cirrincione, Marcello Pucci,Giansalvo Cirrincione andGerard-Andre Capolino

Keywords Induction motor, Identification,Estimation

This paper presents for the first time theanalytical solution of the constrainedminimization for the on-line estimation of theelectrical parameters of an induction motor.The method is fully described mathematicallyand its goodness is verified experimentally ona suitably set up test bench. This methodologypermits the almost correct computation of allthe so called K-parameters, which is notalways the case in current literature, thusresulting in the correct estimation of theelectrical parameters.

Calculation of eddy current losses inmetal parts of power transformers

Erich Schmidt, Peter Hamberger andWalter Seitlinger

Keywords Eddy currents, Power losses,Power transformers, Finite element analysis

To maintain quality, performance andcompetitiveness, the eddy current losses inmetal parts of power transformers in therange of 50-200 MVA are investigated in amore detailed form. The finite elementcalculations utilize different modellingstrategies for the current carrying metalparts. Several global and local results arefurther used to obtain simplified calculationapproaches for an inclusion in the initialdesign and the design optimization. Theresults from two finite element approachesusing nodal and edge based formulations willbe compared with measurements.

Optimal design of high frequencyinduction motors with the aid of finiteelements

Atanasi Jornet, Angel Orille, Alberto Perezand Diego Perez

Keywords Finite elements, Induction motor,Power losses

The motors for high-speed operation fed byfrequency converters produce, first, a highamount of hysteresis and eddy losses inboth stator and rotor iron, and secondlya temperature increase of the rotor dueto current distribution in its rotor slots.Conventional calculation using analyticaltools could not calculate precisely therequired parameters in order to obtain anoptimal model to build a prototype that itsproperties confirm that calculated values withthe model. With a finite element methodapplication for magnetic field and heattransfer, the required elements to design anew prototype could be elaborated veryprecisely and it is also a tool to prove thealready existing motors for high speedapplications. This allows us also to designenergy efficient electrical drives according tothe recommendations of the last EEMODSconference held in London in the year 1999with the support of the European committeefor energy saving.

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Analytic calculation of the voltageshape of salient pole synchronousgenerators including damper windingand saturation effects

Georg Traxler-Samek, Alexander Schwery andErich Schmidt

Keywords Synchronous machine,Optimization

A novel analytic method for determiningthe no-load voltage shape of salient polesynchronous generators is presented.The algorithm takes into account the fullinfluence of the damper winding and thesaturation effects in the stator teeth. Maininterest is an easy and very fast calculationmethod, which can be used as a criteria forthe selection of the number of stator windingslots in the initial design calculation or anoptimization process. The analytical resultsobtained are compared with the results oftransient finite element analyses.

Numerical modelling of electromagneticprocess in electromechanical systems

Vyacheslav A. Kuznetsov and Pascal Brochet

Keywords Electrical machines,Numerical analysis

A general approach to the formation ofmagnetic equivalent circuit describing themagnetic process inside the electric machinesis proposed. This formation is based on toothcontour method. Coupling with external andinternal electric circuits of electric machines isemphasized as well as mechanical couplingwith load. The resulting model allows thesimulation of electromechanical converter, butwith the number of element being fewer byseveral orders compared to traditional finiteelement models. Non-linearity such assaturation or electronic switch is taken intoaccount. General equations for the magneticfields and electric circuits of electricalmachines are written using a common basis –the nodal potential method. The wholeprocess is illustrated on the simulation ofa claw poles alternator compared withmeasurements.

A coupled electromagnetic-mechanical-acoustic model of a DC electric motor

Martin Furlan, Andrej Cernigoj andMiha Boltezar

Keywords Electrical machines,Magnetic forces, Noise,Boundary element method

In this article, we present an investigation intothe sound radiation from a permanent-magnetDC electric motor using the finite-element (FE)and boundary-element (BE) models. A three-times-coupled electromagnetic-mechanical-acoustic numerical model was set-up topredict the acoustic field. The first stage wasto calculate the magnetic forces that excite thestructure of the motor by using the FEM. Inthe second stage, the exciting magnetic forceswere applied to the structural model, wherethe harmonic analysis was carried outusing the FEM. The last stage was to modelthe acoustics by using the BEM. In orderto evaluate the numerical model, thecomputational results were compared withthe vibration and acoustic measurements anda reasonable agreement was found.

Current distribution within multi strandwindings for electrical machines withfrequency converter supply

Oliver Drubel

Keywords Variable frequency,Eddy currents, Electrical machines

The current distribution within multi strandwindings is investigated for transient currentand voltage supplies. The difference in lossesbetween transient and sinusoidal waveformsis elaborated. Therefore, a wide range offrequencies as well as different kinds oftransient waveforms has been investigated.The definition of the skin depth is no longersufficient. A new parameter is required fortransients, which is related to time. Thisparameter will be defined and called ‘‘skintime’’. A numerical method is developedbased upon a finite element transientcalculation. The method is applied to thewinding as well as to the core. A comparisonwith measurements verifies the approachdescribed.


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COMPEL : The International Journalfor Computation and Mathematics inElectrical and Electronic EngineeringVol. 22 No. 4, 2003p. 820# MCB UP Limited0332-1649

Guest editorial

In this special issue of COMPEL, 26 papers presented at ICEM2002-Bruges arepublished. The International Conference on Electrical Machines (ICEM) is theonly major international conference devoted entirely to electrical machines.Started in London in 1974, ICEM is now established as a regular biennial event.Following the very successful conferences in Istanbul in 1998 and Helsinki in2000, ICEM2002 was held in Bruges, Belgium, the Venice of the North.

Out of the 502 papers that have been presented at ICEM2002, only 26 paperswere selected.

The selection of these papers was based on their relation to the scope ofCOMPEL on the one hand and on the quality of the paper and of itspresentation at the conference on the other hand.

Together with Professor Ronnie Belmans, my co-chair of ICEM2002,I would like to thank those reviewers of ICEM available for the reviewingand selection of the journal papers. Thanks also to Dr ir. Lieven Vandevelde,Mr Nic Vermeulen and Mr Tony Boone of the Department of Electrical Energy,Systems and Automation of Ghent University for the administrative work incollecting the reviews and corrected papers.

Professor Dr ir. Jan MelkebeekScientific Chair ICEM2002

Application of stochasticsimulation in the optimisation

process of hydroelectricgenerators

E. Schlemmer, W. Harb, J. Schoenauer and F. MuellerVA TECH HYDRO GmbH & Co, Weiz, Austria

Keywords Stochastic modelling, Optimization, Genetic algorithms, Risk analysis

Abstract In this paper, stochastic optimisation and risk estimation techniques are applied to theproblem of hydroelectric generator design. Optimisation results from deterministic simulations caninvolve considerable risks due to unavoidable variations in system properties as well asenvironmental conditions. Therefore, stochastic simulation is used to include the effects ofparameter scatter and noise effects in the computer models. This allows the evaluation of thescatter in performance and thus an assessment of reliability and quality of the simulated system.

1. IntroductionIn the optimisation process of large synchronous generators (Figure 1), manydesign decisions are based on risk estimations. If we consider, for example, ahighly penalised quantity such as a guaranteed efficiency, the designer will tryto avoid any risk to violate this criterion. On the other hand, if a quantity is ofsome technical importance but a slight deviation from this quantity is notassociated with a severe drawback, the same engineer will be more willing toaccept a design that exhibits a certain risk of missing the target value of thisparameter.

Optimisation of generators performed manually is therefore often a processof estimating the effects of some key design variables (such as the width of theair gap or the length of the iron core) on the main target values (such asreactance values, efficiency, losses etc.) and the assessment of admissible riskscaused by unavoidable variations in the design variables. In this context, theexpert knowledge of the design engineer is of extreme importance since he isable to assess situations from his experience. At the present stage, theimplementation of such highly specific expert knowledge into design programsis not state-of-the-art. As a consequence, optimisation procedures whichemploy the use of deterministic models often lack acceptance of decisionmakers since they do not take the real world effects of uncertainty andtolerances into account. Additionally to the effects of keeping models simpleenough to be handled efficiently, which may cancel out effects that could have alarge impact on the system, a great number of noise factors are present whichaffect designs at different stages. The system’s parameters will vary due tomeasurement errors or production variation. The models will exhibit numerical

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Application ofstochastic

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Vol. 22 No. 4, 2003pp. 821-833

q MCB UP Limited0332-1649

DOI 10.1108/03321640310482832

noise and there will be uncertainties in environmental conditions and part wearor deterioration. The conventional method to deal with this uncertainties is theusage of heuristic margins of safety. The system is designed in such a way thatenough space is assured between the expected system response and theallowed value so that the critical limits are exceeded only with a very lowprobability. From the works of Alloto et al. (2000), Cullimore (2000, 2001),Haecker (2000), Marczyk (2000), Reuter and Watermann (1999) and Reuter et al.(2000), several methods for the handling of stochastic simulations are known.

. Sensitivity analysis method, tries to determine the effects of local (small)variations of design parameters.

. Probabilistic-based methods, such as robust design, induce noise on thesystem and vary parameter settings accordingly to mitigate the impact ofuncertainty on the system.

. Interval methods, specify each parameter as a range and try to determinethe response range.

. Fuzzy set theory, tries to exploit human experience by mapping commonsense rules to fuzzy mathematical representations.

Of course, all of these methods have their own shortcomings.. When conducted after deterministic optimisations, sensitivity analyses

often reveal massive robustness problems when parameters deviate fromtheir optimal settings.

Figure 1.CAD drawing of a largehydro-electric generator

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. Probabilistic-based methods often need prohibitively large computationalresources, especially when the simulations themselves are very timeconsuming.

. Interval methods are often overly conservative since they aim principallyat the worst case.

. Sometimes, even for the expert, it is not possible to formulate his humanexperience in terms of fuzzy variables.

Henceforth, our interest will be focused on probabilistic-based techniqueswhere the input parameters are perturbed according to the given distributions.By doing so, the unavoidable manufacturing tolerances and variations inmaterial parameters are simulated by a scatter of points around thedeterministic input values. Figure 2 sketches the principle of a stochasticsimulation. In deterministic simulations, it is assumed that the input variablesat the nominal point are known exactly thus giving exact solutions. Stochasticsimulation gives a better picture of reality by assuming a certain scatter aroundthe nominal point according to known or estimated probability distributions.Consequently, the results are themselves distributed around some expectedvalue and can therefore be used for risk estimation purposes.

We have chosen this approach because a single conventional designcalculation is very time efficient. Therefore, many of them can be executed at

Figure 2.Scheme of a stochastic

simulation


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little cost. Moreover, the enhancement of the existing code for deterministicoptimisation using evolutionary algorithms by a Monte Carlo branch is more orless straightforward.

Within one optimisation step of the evolutionary algorithm, severalsimulations according to the pre-defined distributions of the input parametersare run. The result of these simulations is given as a multi-dimensionaldistribution in the output space. The projection of this distribution into thepre-defined output variables is compared with admissible intervals of theseoutput variables. Straightforwardly, the risk to violate these bounds iscalculated as a function of the weighed number of trials outside the admissibleintervals. In the process of building up the Pareto front in multi-criterionoptimisation, this risk function is an additional criterion for optimisation. As aconsequence, only solutions that exhibit a relatively low sensitivity againstinput parameter variations are chosen for the further conduct of theoptimisation process where these solutions form the genetic pool for the nextgeneration of trials.

Figure 3 gives an overview of the conventional method of handlingguaranteed values, which can be modelled using penalty functions in theoptimisation routines. For example, the values of the sub-transient reactancemust exceed a guaranteed value x 00

dgar. The contrary behaviour is stipulated forthe transient reactance. Of course, for physical reasons, the values of x 0

d and x 00d

are in close relation with each other and cannot be changed independently.

Figure 3.Sketch of a classical riskestimation via penaltyfunctions in the contextof a stochasticsimulation. Thecalculated reactancevalues are separatedfrom the guaranteedvalues by safety margins(sm). Above or below theeconomical intervals,over-engineering (oe)takes place

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In order to minimise the risk of violating these criteria, margins of safety areintroduced. Consequently, no calculated value of the reactance is allowed to liewithin these margins. Adjacent to the respective margins, there is a region ofeconomically feasible reactance values whereas any value surpassing thisregion could be termed as “over-engineered”. If for some reasons, the calculatedvalues of the reactance values lie within the margins of safety, a penaltyfunction can be used to force these values as near as possible to the admissibleregions.

However, the above holds only for the deterministic case. The reactancevalues scattered around the nominal points exhibit a distribution around thecalculated values. This effect can be seen in Figure 4. Although the twosolutions are equal in terms of the margin of safety between the guaranteed andthe calculated value, the right solution is clearly preferred due to riskconsiderations. Whereas on the left, the guaranteed value is exceeded with verylow probability, the solution on the right is rather likely to fail.

2. Problem statementThe problem can be formalised equivalently to a multi-objective minimisationof a partial non-linear and non-differentiable function which is subject tocertain constraints (Zitzler, 1999). A stochastic multi-objective optimisationproblem (SMOP) consists of a set of n parameters (decision variables), a set of kobjective functions, and a set of m constraints. Objective functions andconstraints are functions of the decision variables, where x is the decisionvector, y is the objective vector, X is denoted as the decision space, and Y iscalled the objective space. The feasible set Xf is defined as the set of decisionvectors x that satisfy the constraints e(x).

Figure 4.Difference between

deterministic andstochastic simulation in

assessing the results


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minimise y ¼ fðxÞ ¼ ð f 1ðxÞ; f 2ðxÞ; . . .; f kðxÞÞ

subject to eðxÞ ¼ ðe1ðxÞ; e2ðxÞ; . . .; emðxÞÞ # 0

where x ¼ ðx1; x2; . . .; xnÞ [ X y ¼ ð y1; y2; . . .; ykÞ [ Y

ð1Þ

Furthermore, we split the decision vector into two vectors, namely j and p,where j indicates the control variables which one wishes to influence and p thedesign variables which are not subject to control, either because they are fixedor they are noise variables. We then introduce stochastic functions ui and vi

which map the purely deterministic values of ji and pi to their stochasticcounterparts xi according to the pre-defined probability distributions.

xi ¼ uiðjiÞ or xi ¼ við piÞ ð2Þ

As we deal with the multi-objective case, Pareto optimality has to be introducedin the usual way. For any two objective vectors r and s,

r ¼ s iff ;i [ {1; 2; . . .; k} : ri ¼ si

r $ s iff ;i [ {1; 2; . . .; k} : ri $ si

r . s iff r $ s ^ r – s:

ð3Þ

Additionally, Pareto dominance is defined for any two decision vectors r and ssuch that

r � s ðr dominates sÞ iff fðrÞ . fðsÞ

r X s ðr weakly dominates sÞ iff fðrÞ $ fðsÞ

r < s ðr is indifferent to sÞ iff fðrÞ � fðsÞ ^ fðsÞ � fðrÞ:

ð4Þ

A decision vector x [ Xf is said to be non-dominated regarding a set A # Xf

iff

’6 a [ A : a � x: ð5Þ

The entirety of all Pareto-optimal solutions is called the Pareto-optimal set; thecorresponding objective vectors form the Pareto-optimal front or surface.Therefore A # Xf; the function p(A) gives the set of non-dominated decisionvectors in A:

pðAÞ ¼ {a [ Aja is non-dominated regarding A}: ð6Þ

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The corresponding set of objective vectors f( p(A)) is the non-dominated frontregarding A. Furthermore, the set Xp ¼ pðXfÞ is called the Pareto-optimal setand the set Yp ¼ fðXpÞ is denoted as the Pareto-optimal front.

Stochastic simulation can be incorporated very naturally in the scheme ofPareto optimality by simply adding further constraints ei (x) which denote therisk of violating the pre-defined guaranteed values, similar to the situation inFigure 3. Therefore, in the case of Figure 3, where the guaranteed value isexceeded towards positive values, we define a probability density functiongi (x) which results from the Monte-Carlo-trials of the stochastic simulationsand a user-defined weight-function wið yiÞ ¼ wið f iðxÞÞ:

eiðxÞ ¼

Z 1

hi¼yi;gar

giðxÞwið f iðxÞÞ dhi ð7Þ

Usually, wi weighs the deviation of yi ¼ f iðxÞ according to its distance betweenthe actual value and the guaranteed value. In the case of wi being the identityfunction, we would get the expected value of yi as a risk value. In most cases,gi(x) is given as the ratio of the number of cases exceeding a given interval andthe total number of cases. For the weight function wi, a suitably defined barrieror penalty function (e.g. Figure 3 in the case of safety margins) is defined by theuser.

Obviously, no additional effort is needed for the handling of a constraint ofthe form (7) within the Pareto scheme.

In the actual case, the vector x comprises mainly the following componentswith their respective defining intervals:

. dimensions and number of stator copper strands,

. dimensions of the pole,

. dimensions of the field copper,

. diameter and pitch of the damper bars,

. number of stator slots,

. dimension and number of the ventilation ducts,

. iron length, and

. air gap.

This seems restrictive, but in principle, any variable used in the designprogram can be specified from outside. Additionally, a user-defined scatteraccording to a given probability distribution can be superimposed on allcomponents of x.

The most important constraints are as follows:. transient and sub-transient reactances,. short circuit ratio If,oc/If,sc,


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. temperature rises,

. losses,

. voltage harmonics,

. manufacturing costs, risk and penalty,

. clearances between adjacent field windings,

. mechanical tensions in dovetails and poles, and

. admissible field and current density values.

Dependent variables can be calculated from the input variables using an LL(1)expression parser, i.e. the user can define expressions in a familiar notation inthe text format. The same is true for expressions in the output variables of thedesign program. Although LL-parsing is not a new approach, with this featurea big step towards greater universality is taken.

3. Optimisation methodsThe definition of the general evolutionary algorithms used in this paper hasalready been given (Schlemmer et al., 2000) and need not be repeated here.Moreover, there exists an abundance of literature on the topic, the interestedreader might refer to Fujita et al. (1998), Goldberg (1989), Maekinen et al. (1998),Michalewicz and Fogel (2000), Rechenberg (1994), Zitzler (1999) and thereferences therein. The extensions of the evolutionary algorithms, namelyfuzzy-based taboo search, sharing scheme, meta-optimisation using DOE(Montgomery, 1997) and the application of regular expressions for expressionparsing have already been described elsewhere (Schlemmer et al., 2001).

4. ApplicationsAt the end of the problem statement and the following comments, we havedescribed the decision variables and the respective constraints. One mightargue that we have modelled but a small subspace of the whole problem’scomplexity and even this is heavily restricted by the defining intervals for thedecision variables. However, most heuristic optimisation techniques inengineering work only because there are excellent starting points. In ourcase, this starting point is determined by the so-called “first design program”that embodies quite an impressive body of engineering knowledge and designpractices. Most of the limits stipulated by international and company standardsare already satisfied by the solutions of this program. Additionally, guaranteedvalues and customer demands have to be propagated to the optimiser as can beseen in Figure 5. The optimiser has to control the loop embracing all importantdesign programs, has to prepare input values for these programs and finallyhas to interpret their resultant values for the determination of the next steps inthe optimisation process.

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The optimisation goal for the problem at hand is to find a Pareto-optimalsolution for a hydroelectric generator with respect to minimal costs, losses, andpenalised violations of several geometric and electromagnetic constraints.Additionally, the risk of falling below x 00

dgar and exceeding x 0dgar as well as

violating an interval for the short circuit ratio, If,oc/If,sc, is to be minimised.Figure 6 gives an impression of the evolution of the risk for x 00

d over thegenerations during the optimisation process.

The risk is calculated within an additional Monte Carlo loop where theprobability of violating the limits is given by the ratio of “successful” trials toall trials. For the calculation of the risk value itself, the violations can be

Figure 5.Schematic description ofthe propagation of limits

and guaranteed valuesthroughout the suite of

design programs

Figure 6.Development of the riskassociated with x 00

d overthe conduct of an

optimisation run. Eachbox-plot summarises a

generation of 40individuals. The risk is

reduced until a negligiblequantity is achieved


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weighed by user-defined functions, most frequently employing a suitable normof the distance between the objective value and the respective limit. Figures 7and 8 show the difference between the two weight functions.

In Figure 7, only the number of violations was counted whereas in Figure 8,this number was weighed by the quadratic distance between the reactancevalue and the limit. The two variables air gap width and length of pole werechosen arbitrarily. In this very case, no distinct pattern of low-risk andhigh-risk areas can be found. This shows the need for an automated scheme forrisk evaluation.

Figure 7.Contour plot of notweighed risk ofexceeding theguaranteed value of x 00

d

versus air gap and polelength deviation from therespective nominalvalues. That is, theweight factor in (7) isset to 1

Figure 8.Contour plot of weighedrisk for the samedeviations as above.The risk was weighedproportional to theamount of reactancefalling short of thespecified value

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In Figure 9, the Pareto front of standardised costs, losses and penalty sum ascomputed by a genetic algorithm and an evolution strategy can be seen. Forbetter visibility, the projections of these values have been shown in Figure 10.

Whereas the costs vs losses graph shows the expected hyperbolic shape, thepicture is less clear for the costs vs penalty graph. Since all of these variantsexhibit no risk of violating one of the reactances or of If,oc/If,sc, without furtherinformation, no decision can be made at this point thus leaving the arbitrationto the design engineer.

Figure 9.Pareto front after

concurrent optimisation.For all of the solutions inthe Pareto front, the riskvalue according to (7) is

minimised

Figure 10.Projections of the Pareto

front of Figure 9


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5. ConclusionsIn this paper, we have demonstrated that evolutionary algorithms can copewith the problem of stochastic multi-objective optimisation of hydro-electricgenerators in a straightforward manner. In order to achieve robust designs, it isvery often insufficient to find an optimal configuration in a deterministic way.Therefore, the uncertainty of the input parameters and the noisy environmenthas to be taken into account for the minimisation of the risk of failure. Using asimple Monte-Carlo technique, this risk minimisation problem is reduced toa case where the risk is just another component in the objective vector of amulti-criterion optimisation problem which is solved in the sense of Paretooptimality. Employing this approach, a concurrent optimisation of ahydroelectric generator with respect to costs, losses, and the risk of violatingguaranteed reactance values is carried out.

References

Alotto, P., Molfino, P. and Molinari, G. (2000), “Optimisation of electromagnetic devices withuncertain parameters and tolerances in the design variables”, private communication,Proceedings of OIPE 2000, 25-27 September 2000, Torino.

Cullimore, B.A. (2000), “Reliability engineering and robust design: new methods for thermal/fluidengineering”, C&R White Paper, http://www.crtech.com

Cullimore, B.A. (2001), “Dealing with uncertainties and variations in thermal design”,Proceedings of InterPack ’01, Kuaui, Hawaii, http://www.crtech.com

Fujita, K., Hirokawa, N., Akagi, S. and Kitamura, S. (1998), “Multi-objective optimal design ofautomotive engine using genetic algorithm”, 1998 ASME Design Engineering TechnicalConferences, 13-16 September, Atlanta, GA, http://www.lania.mx

Goldberg, D.E. (1989), Genetic Algorithms in Search, Optimization and Machine Learning,Addison-Wesley Longman, Inc., Reading, MA.

Haecker, J. (2000), “Statistical analysis of manufacturing deviations and classification methodsfor probabilistic aerothermal design of turbine blades”, Thesis, University of Stuttgart.

Maekinen, R.A.E., Neittaanmaeki, P., Periaux, J. and Toivanen, J. (1998), “A genetic algorithm formultiobjective design optimization in aerodynamics and electromagnetics”, ECCOMAS98, Wiley, New York, http://www.lania.mx

Marczyk, J. (2000), “Stochastic multidisciplinary improvement: beyond optimization”, 8thAIAA/NASA/USAF/ ISSMO Symposium. on Multidisciplinary Analysis and Optimization,September 2000, Long Beach, http://www.easi.de/company/publications

Michalewicz, Z. and Fogel, D.B. (2000), How to Solve It: Modern Heuristics, Springer-Verlag,New York.

Montgomery, D.C. (1997), Design and Analysis of Experiments, 5th ed., Wiley, New York.

Rechenberg, I. (1994), Evolutionsstrategie ‘94, Frommann-Holzboog, Stuttgart.

Reuter, R. and Watermann, A. (1999), “Application of uncertainty management to MADYMOoccupant simulations”, 2nd European MADYMO Users Conference, Stuttgart, 1999, http://www.easi.de/company/publications

Reuter, R., Hoffmann, R. and Kamarajan, J. (2000), “Application of stochastic simulation in theautomotive industry”, AMERI-PAM, October 2000, Southfield, Michigan, http://www.easi.de/company/publications

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Schlemmer, E., Harb, W., Kleinhaentz, R., Lichtenecker, G. and Mueller, F. (2000), “Optimisationof large salient pole generators using evolution strategies and genetic algorithms”,Proceedings of ICEM 2000, 28-30 August 2000, Espoo, Finland, pp. 1030-4.

Schlemmer, E., Harb, W., Kleinhaentz, R., Lichtenecker, G. and Mueller, F. (2001), “Multi-criterionoptimisation of electrical machines using evolutionary algorithms and regularexpressions”, Proceedings of HYDRO 2001, 27-30 September 2001, Riva del Garda,Italy, pp. 567-76.

Zitzler, E. (1999), “Evolutionary algorithms for multiobjective optimization: methods andapplications”, Thesis, Swiss Federal Institute of Technology, Zuerich.


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Eddy-current computation ona one pole-pitch model ofa synchronous claw-pole

alternatorChristian Kaehler and Gerhard Henneberger

Department of Electrical Machines (IEM),Aachen Institute of Technology (RWTH), Aachen, Germany

Keywords Eddy currents, Finite elements, Alternators

Abstract This paper deals with 3D finite-element calculation of eddy currents in the claws of aclaw-pole alternator taking the rotational geometry movement into account. Two transientedge-based vector formulations are utilised. The reduction of the model to only one pole pitch incombination with a special boundary pairing in the air gap for the applied lock-step method ispresented. Calculations of varying material conductivity are performed with simplified endwindings. The speed characteristics of the eddy currents with real conductivity and realistic endwindings concludes the paper.

1. IntroductionClaw-pole alternators are used for the generation of electricity in automobiles.There are three basic requirements to them: the output performance must beimproved, the audible noise reduced and the efficiency increased. A descriptionof the magneto-static field calculation, used for output optimisation, and theanalysis of the structural-dynamic and acoustic behaviour can be found inthe work of Kuppers (1996) and Ramesohl (1999).

The efficiency of machines is decreased by different loss mechanisms. In thecase of the claw-pole alternator, these are dominantly the ohmic losses in thecoils and losses caused by the eddy currents in conducting materials. Both canbe broken down into rotor and stator parts. Whereas the ohmic losses can bedirectly calculated in dependence of the coil currents, an analytic description ofthe eddy-current losses is not possible.

Finite-element method (FEM) is used to calculate the eddy currents inconducting materials, which are induced by an alternating magnetic field.A time harmonic approach can be applied if the geometry is not shifting,all material properties are linear and sinusoidal currents are used.

In the case of the claw-pole synchronous machine, the rotor is turning with adefined speed while the direct current is used in the excitation coil of the rotor.



All calculations have been utilised on a claw-pole alternator of the Compact Generator Series ofthe industrial partner Robert Bosch GmbH. For modelling and discretization the commercialprogram ANSYS Version 6.0 has been used.

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In generator mode, the stator coils are driven by three-phase current. All steelmaterials are non-linear. Therefore, a time-stepping algorithm has to beutilised.

In this paper, the applied transient edge-based ~A-approach (Kameari andKoganezawa, 1997) and ~A 2 ~A; ~T-approach (Albertz and Henneberger, 2000)are outlined. The 3D FE model of the claw-pole alternator with one and twopole pitches and also with simplified and realistic end windings are described.Special attention is laid on the meshing strategy required by the pairingalgorithm, which defines the rotational movement. The results at load in thegenerator mode obtained by comparing another model with one pole pitch to amodel with two pole pitches are presented. Computations of the one pole-pitchmodel with varying material conductivity in the claw regions show theapplication range of both the FE formulations. Calculations on the onepole-pitch model with realistic end windings and real-life material conductivityare performed. The characteristic curve of the eddy-current loss over thealternator speed in the generator mode concludes this paper.

2. Theory of the edge-based solverThe applied edge-based solver is part of an object-oriented solver package(Arians et al., 2001). It applies two different FE eddy-current formulations onsimply-connected eddy-current regions.

2.1 ~A 2 ~A; ~T - formulation

The ~A 2 ~A; ~T-approach presented by Albertz and Henneberger (2000) uses twovector potentials, the magnetic vector potential ~A and the electric vectorpotential ~T; to compute the flux density ~B and the current density ~J :

~B ¼ 7 £ ~A; ~J ¼ 7 £ ~T: ð1Þ

The solver separates the model in the eddy-current free regions V1, where thefollowing equation for ~A is solved:Z

V1

7 £ ~ai · n7 £ ~AðtÞ dV1 ¼

ZV1

ð ~ai · ~J0ðtÞ þ 7 £ ~ai · n~BrÞ dV1 ð2Þ

and for eddy-current regions V2, the equations read:ZV2

ð7 £ ~ai · n7 £ ~AðtÞ2 ~ai ·7 £ ~TðtÞÞ dV2 ¼ 0

ZV2

�7 £ ~ai ·

1

s7 £ ~TðtÞ þ 7 £ ~ai ·

›

›t~AðtÞ

�dV2 ¼ 0:

ð3Þ

~J0ðtÞ describes the given coil current density while ~Br defines remanence.The material parameters n and s represent the non-linear reluctivity and

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835

the linear conductivity, respectively. ~ai defines the shape function of an edgeelement (in this solver tetrahedra).

The time-stepping algorithm interpolates the time-dependent variableslinearly as follows:

~AðtÞ ¼ t · ~Anþ1 þ ð1 2 tÞ~An ditto for ~T; ~J0

›

›t~AðtÞ ¼

1

Dtð~Anþ1 2

~AnÞ;

ð4Þ

where n represents the number of the transient step, Dt the time in betweentransient steps and t the relaxation factor.

To improve the convergence behaviour, the current potential is scaled(Kaehler and Henneberger, 2002). The usual periodic and Dirichlet boundaryconditions are used on the model boundaries for ~T and ~A: The boundarycondition between the eddy-current free regions V1 and the eddy-currentregions V2 for the current vector potential reads (Biro and Preis, 2000):

G12 : ~T £ ~n ¼ 0; ð5Þ

where ~n is the normal vector of the boundary region. Since in this applicationall eddy-current regions are continuous and short circuited, equation (5) caneasily be achieved by a Dirichlet condition Ti ¼ 0 on all edges i of theboundary G12.

2.2 ~A - FormulationThe ~A-approach applies only the magnetic vector potential ~A in all regions.Here, the formulation for eddy-current regions reads:

ZV2

7 £ ~ai · n7 £ ~AðtÞ þ s›~AðtÞ

›tdV2 ¼ 0; ð6Þ

while the eddy-current free regions are solved with equation (2). Thetime-stepping algorithm uses equation (4) again. The magnetic flux density ~Band the eddy-current density ~J are computed as follows:

~B ¼ 7 £ ~A; ~J ¼ 2sAnþ1 2 An

Dt: ð7Þ

In this approach, no boundary conditions have to be applied on the boundariesbetween the eddy-current and non-conducting regions. Only the usualconditions for ~A on the model boundaries are used.

Since the resulting global FEM matrix for the ~A 2 ~A; ~T-approach is notsymmetric, it is solved by the SSOR preconditioner and the TFQMR solverof the ITL package (Lumsdaine et al., n.d). The matrix for the ~A-approach is

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symmetric, thus allowing the use of the Cholesky-CG combination (Kameariand Koganezawa, 1997) of the same package.

Saturation effects are computed with an overlaying Newton-Raphsonprocedure for each transient step. The relaxation factor used in betweentransient steps is chosen as t ¼ 2=3 (Galerkin-scheme) (Zienkiewicz andTaylor, 1991).

3. Finite element modelOnly magnetically relevant components of the synchronous claw-polealternator are modelled. The end windings is modelled with straight coils orrealistically. Since the geometry of the alternator is symmetric over two polepitches, a 608 model with periodic boundaries can be utilised (Figure 1(b))(Kaehler and Henneberger, 2002). With antiperiodic boundaries and a specialgeometric regrouping the model can be reduced to one pole pitch or 308(Figure 1(a)). Thus, either with the same calculation time denser meshes andtherefore smaller Peclet numbers (Rodger et al., 1990) can be computed, or withidentical mesh density or Peclet number the number of elements can be halvedand the calculation time nearly quartered.

3.1 Winding headIn order to have geometric identity after a rotor movement of one stator-toothpitch or 108 mechanical, the end windings is simplified as in Figure 1, whereeach coil runs straight through the whole model. The advantage of thissimplification lies in a periodic behaviour of the calculation after 108 and notthe usual 608 when taking the end windings into account. Thus, the settlingtime of calculation can be detected easily.

A model with realistic end windings is shown in Figure 1(c). This model willlater be compared to the model with simplified end windings and used whencalculating the speed characteristics of the eddy-current loss.

Figure 1.Models with translucent

stator regions andsimplified (a,b) and

realistic (c) end windings


837

3.2 Meshing strategy for edge reorderingTo represent the rotational movement, a lock-step method is utilised. In thismethod, no real movement takes place. Instead, boundary conditions are used,which pair edges in each step depending on the rotational angle, while the meshremains stationary.

This edge-grouping routine of the transient solver depends on a specialair-gap discretization. To implement the change of geometry, the FE mesh ofthe alternator is separated into moving elements in the rotor and stationaryelements in the stator. The boundary area of these two meshes is located in themiddle of the air gap. It is meshed identically in both separate meshes.

The boundary mesh is partitioned into equidistant areas in the direction ofmovement. One of these areas (exactly as wide as the step angle) is modelledand meshed in the periodic case (Figure 2(b)). In the antiperiodic case the firstarea is partitioned into four area meshes, which are created by mirroring (step 1and 2 in Figure 2(a)). All other meshed areas are generated by symmetricrotation of the first areas (step 3 in both figures). A zoom on the actualantiperiodic boundary mesh in the middle of the air gap of the claw-polealternator model is shown in Figure 3.

The search function for the pairing of two edges is defined by three vectors.The first ~s1 defines translatory movement in x-, y- or z-direction, the second ~s2

rotational movement around the x-, y- or z-axis and the third ~s3 multiplication ofthe x-, y- and z-value. With these three vectors translatory as well as rotatorymovement can be considered.

In the case of the claw-pole alternator the vectors differ for the two models.In the periodic case (608 or two pole pitches) the search vectors at step n read:

~s1 ¼ n · ~0; ~s2 ¼ n · ð08; 08; 18ÞT; ~s3 ¼ ð1; 1; 1ÞT: ð8Þ

In the antiperiodic case (308 or one pole pitch) the search vectors in periodicregions are identical to (8), while in antiperiodic regions they read:

~s1 ¼ n · ~0; ~s2 ¼ n · ð08; 08; 18ÞT; ~s3 ¼ ð1; 1;21ÞT: ð9Þ

Antiperiodic regions appear when

a · ð2k 2 1Þ , ~s2 · ð0; 0; 1ÞT # a · 2k k [ Z; ð10Þ

Figure 2.Generation of boundaryarea mesh in air gap.(a) Antiperiodic case,(b) Periodic case

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with a ¼ ah 2 al ¼ 308 being the difference of the lower model boundaryal ¼ 758 and the higher boundary ah ¼ 1058:

The corresponding edges are detected by the position of their nodes. Theycan be inserted into the FEM matrix as periodic or antiperiodic boundaryconditions.

Since the edge directions change in the antiperiodic case, due to the negativesign of ~s3 in equation (9), the directions of the fluxes also change automaticallyin these regions for the claw-pole alternator. Thus, all paired edges are insertedas periodic binary constraints into the FEM matrix.

4. Calculations and resultsThe calculations are conducted at constant speed. The mechanical step angleamounts to a ¼ 18; leading to, for example, Dt ¼ 55:556ms in betweentransient steps for a speed of n ¼ 3; 000 rpm: The excitation current isimpressed in the rotor. The three-phase current of the real alternator ingenerator mode is injected in the stator coils. It turns synchronously with therotor.

The calculations on the 308 and the 608 model are compared at low materialconductivity, proving that the use of the smaller antiperiodic model yieldscorrect results. The conductivity is varied up to the real material conductivityof the claws utilising both transient formulations. The computations withsimplified and realistic winding differ only in the average eddy-current loss

Figure 3.Antiperiodic boundary

area mesh of the utilisedmodel


839

value. Finally, the speed characteristic of the average eddy-current loss withrealistic end windings in generator mode is determined.

4.1 Comparison of one and two pole-pitch modelFor the comparison of the one and the two pole-pitch model (Figure 1(a) and (b))a low material conductivity of s ¼ 4:0 £ 102 (Vm)21 and the ~A 2 ~A;~T-approach are selected, since in this calculation the exact value of the eddycurrents is not of interest, but the difference in between the model solutions.

The total eddy-current loss in the claws over the rotation is shown inFigure 4 for both models. Additionally, the relative difference is shown on thesecondary axis. After a short settling time of about 15 time steps, a periodicityof the eddy-current losses of Da ¼ 108 mechanical occurs, as expected for thesimplified end windings. The eddy-current distributions for both the models fora specific time step are shown in Figure 5 on the same scale. The maximumeddy-current values as well as the maximum magnetic flux densities arelocated on the lower flank of the claw (generator effect), the rotor turningmathematically positive.

The calculation of the average energy density �w of the eddy currents over aperiod of the losses leads to Figure 6:

�w ¼Dt

s

XN

n¼1

~J2

n ; ð11Þ

with N being the number of steps in a period and ~Jn the eddy-current densityof that element at step n. Again the maximum is located on the lower flank.

Figure 4.Eddy-current loss vsrotation for materialconductivitys ¼ 4.0 £ 102 (Vm)21

at constant speedn ¼ 3,000 rpm

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The local differences between both the models are caused by the differences inthe mesh density.

The distribution of the average loss in energy can later be used as excitationfor thermal solvers. Since the thermal distribution will only differ by diffusioneffects from the energy distribution, the maxima distributions in Figure 6(a)and (b) can already be compared to hotspot distributions in temperaturemeasurements on the surfaces of the rotor claws.

The global results (relative difference 1 , 1025 per cent) as well as theeddy-current density and energy distributions are close to identical, provingthe eddy-current distribution as well as the magnetic flux density to beantiperiodic in the one pole-pitch model. With this model, the element numbercan be reduced by a factor of two, although the mesh density and the local errorstays identical in all model regions. Thus, the number of unknowns is nearlyhalved and the calculation time about quartered.

Since it has been proven that calculations on one pole-pitch models yieldcorrect results, all further calculations are conducted on these models due totheir shorter calculation time.

Figure 5.Eddy-current

distribution J (A/m2) atstep 25 for conductivitys¼ 4.0 £ 102 (Vm)21

and speed n ¼ 3,000 rpm


841

4.2 Variation of material conductivity in the clawsThe material conductivity in the claws is varied using both transientformulations, beginning with the ~A 2 ~A; ~T-approach for low and using the~A-formulation for high material conductivity.

In order to achieve smooth convergence and to reduce the settling time, thecomputation with the ~A 2 ~A; ~T-approach is started with the materialconductivity of the claws being s ¼ 4:0 £ 102 (Vm)21 or Peclet number Pe <0:001; which is defined as (Rodger et al., 1990):

Pe ¼vlms

2: ð12Þ

In equation (12) v represents the velocity, l the characteristic length of anelement in the direction of movement and m the permeability of the elementmaterial.

The conductivity is subsequently increased to s ¼ 1:0 £ 106 (Vm)21 orPe < 2:5 as depicted in Figure 7. The real conductivity of s ¼ 4:0 £ 106

(Vm)21 ðPe < 10Þ leads to divergence when using the ~A 2 ~A; ~T-formulation.

Figure 6.Average eddy-currentenergy distributionw (W/m3) forconductivitys¼ 4.0 £ 102 (Vm)21

and speed n ¼ 3,000 rpm

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Therefore, the simulation is finished with the ~A-approach which onlyconverges for high material conductivities. Here, the material conductivitiess ¼ 4:0 £ 105 (Vm)21 ðPe < 1Þ and s ¼ 4:0 £ 106 (Vm)21 ðPe < 10Þ arecomputed.

In the ~A-approach, it is essential to start with a static step since otherwisethe eddy currents rise dramatically in the first step. Without static start stepthe solver computes the eddy currents that would build-up, if the machinestarted from no excitation and zero speed to full excitation and full speed in onetime step. The relaxation time would then prolong to about 100 transient steps.With static start step, the computation simulates the spontaneous change fromzero to full conductivity of the claw-pole material. Here, the relaxation time ofabout 15 transient steps is comparable to the relaxation time of the ~A 2 ~A;~T-approach.

The average total eddy-current loss over the material conductivity of thewhole generator is depicted on a logarithmic scale in Figure 7. Both transientformulations show a nearly potential dependency of the eddy currents on thematerial conductivity of the claw.

4.3 Comparison of simplified and realistic winding headThe two models of one pole pitch with simplified and realistic end windings(Figure 1(a) and (c)) yield slight differences in the total eddy-current loss when

computed with the ~A-formulation.Both models are identically meshed in all regions. The different coil topology

is generated by different material definitions in the winding-head regions of thestator mesh. Thus, the differences are only caused by the winding-headdefinition and not by the discretization.

Figure 7.Average eddy-current

loss vs materialconductivity at constant

speed n ¼ 3,000 rpm


843

The periodicity of the simplified end windings is congruent with theassumptions taken in Section 3. As shown in Figure 8, the loss period amountsto the expected 108 mechanical or 10 transient steps. Due to the three-phasecurrent in the end windings of the wave winding in the stator, the periodicity ofthe model with realistic end windings also amounts to 108 mechanical or 10transient steps. In this model, the current distribution in the end windings isalso periodic over 108 mechanical. Therefore, the magnetic flux and theeddy-current distributions follow the same period.

The slight differences are caused by the differing coil-current path in thestator regions. The average eddy-current loss of the model with realistic endwindings is about 2 per cent lower than the loss of the model with simplifiedwindings.

4.4 Speed characteristic of the eddy-current lossThe main aim of the transient calculations lies in determining the eddy currentsof the real claw-pole machine for all working points.

In order to do so, the material conductivity of iron s ¼ 5:0 £ 106 (Vm)21 attemperature T ¼ 1758C is used for the massive steel regions of the claws.Thus, the Peclet number amounts to Pe < 12:5:

The ~A - approach is applied on the 308 model with realistic end windings(Figure 1(c)). The stator currents in generator mode at constant excitation in therotor of I f ¼ 4 A are impressed into the stator coils. The constant alternatorspeed is varied from n ¼ 1;500 to 6;000 rpm as shown in Figure 9. Theresulting average eddy-current energy for n ¼ 6;000 rpm is shown in Figure 10.

The result leads to the characteristic speed curve of the averageeddy-current loss of the synchronous claw-pole alternator in generator mode

Figure 8.Eddy-current loss vsrotation with simplifiedand realistic endwindings for materialconductivitys ¼ 4.0 £ 106 (Vm)21

at constant speedn ¼ 3,000 rpm

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on a half-logarithmic scale is shown in Figure 9. This speed characteristic of theloss can later be used in electric circuit or domain simulations of the alternatorand of the whole automobile.

5. ConclusionsIn this paper, a transient 3D FEM to calculate the eddy currents in the claws ofa synchronous claw-pole alternator is presented taking the rotationalmovement and two edge-based vector formulations into account.

Special attention has been laid on the comparison of a one and a twopole-pitch model and their meshing strategy in the air gap. Both models yieldidentical eddy-current and average energy distributions, which have been

Figure 9.Eddy-current loss vsalternator speed for

material conductivitys ¼ 5.0 £ 106 (Vm)21 in

generator mode

Figure 10.Average eddy-current

energy �w (W/m3) atspeed n¼ 6,000 rpm for

material conductivitys ¼ 5.0 £ 106 (Vm)21


845

depicted in this paper. Thus, by the use of antiperiodic edge grouping, thenumber of unknowns has been reduced by about a factor of two withoutloosing precision.

On the smaller model with one pole pitch, calculations with varying materialconductivity have been performed showing the potential dependency of theeddy-current loss on the conductivity of the claws and the application range ofthe two transient formulations.

Additionally, two models of one pole pitch with simplified and realistic endwindings have been compared. While their periodicity stays identical, theaverage eddy-current loss with realistic end windings is slightly lower.

Last, the speed characteristic in generator mode of the average eddy-currentloss as well as the average eddy-current energy at speed n ¼ 6;000 rpm on therotor claws with realistic material conductivity and end windings has beencalculated and presented.

References

Arians, G., van Riesen, D. and Henneberger, G. (2001), “Innovative object oriented environmentfor designing different finite element solvers with various element types and shapes,”Record of the 13th Compumag Conference on the Computation of Electromagnetic Fields,Evian, France, July 2001, Compumag, Vol. II, pp. II218-19.

Albertz, D. and Henneberger, G. (2000), “On the use of the new edge based ~A;2~A; ~T; formulationfor the calculation of time-harmonic, stationary and transient current field problems”,IEEE Transactions on Magnetics, Vol. 36 No. 4, pp. 818-22.

Biro, O. and Preis, K. (2000), “An edge finite element eddy current formulation using a reducedmagnetic and a current vector potential”, IEEE Transactions on Magnetics, Vol. 36 No. 5,pp. 3128-30.

Kaehler, C. and Henneberger, G. (2002), “Eddy current computation in the claws of a synchronousclaw pole alternator in generator mode”, IEEE Transaction on Magnetics, Vol. 38 No. 3,pp. 1201-4.

Kameari, A. and Koganezawa, K. (1997), “Convergence of ICCG method in FEM using edgeelements without gauge condition”, IEEE Transactions on Magnetics, Vol. 33 No. 2,pp. 1223-6.

Kuppers, S. (1996), “Numerische Verfahren zur Berechnung und Auslegung von Drehstrom-Klauenpolgeneratoren (Numerical methods for the calculation and design of three-phaseclaw-pole alternators)”, PhD thesis, Department of Electrical Machines, Aachen Institute ofTechnology, Shaker verlag, Aachen.

Lumsdaine, A., Siek, J. and Lie-Quan Lee, “The iterative template library - itl”, Available: http://www.lsc.nd.edu/research/itl, [Online]

Ramesohl, I. (1999), “Numerische Gerauschberechnung von Drehstrom-Klauenpolgeneratoren(Numerical acoustic cal-culation of tree-phase claw-pole alternators)”, PhD thesis,Department of Electrical Machines, Aachen Institute of Technology, Shaker verlag, Aachen.

Rodger, D., Leonhard, P.J. and Karaguler, T. (1990), “An optimal formulation for 3D movingconductor eddy current problems with smooth rotor”, IEEE Transactions on Magnetics,Vol. 26, pp. 2359-63.

Zienkiewicz, O.C. and Taylor, R.L. (1991), The Finite Element Method, McGraw-Hill BookCompany, London, New York.

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A closer view on inductance inswitched reluctance motors

Irma HajdarevicDaimlerChrysler, Advanced Propulsion Systems, Stuttgart

Hansjorg KoflerInstitut fur elektrische Maschinen und Antriebstechnik,Fakultat fur Elektrotechnik und Informationstechnik,

Technische Universitat Graz, Graz, Austria

Keywords Motors, Inductance, Magnetic fields, Finite element method

Abstract Switched reluctance motors are promising candidates for a wide variety of driveapplications. The theoretical description of such motors is often reduced to the rather simple, butclear concept of CI-characteristics. In contrast to this, the machine itself is a real electrical machineand must therefore be analysed as is done with conventional machines although the experimentalmachine is constructed as simple as possible. The paper will first describe some well known basicsconnected to very short machines and flux. In the next step, the calculation of stray flux quantitiesaided by 3D-FEM is discussed and finally a comparison of calculation and experimentalmeasurements is given.

1. IntroductionThere is no dispute over the fact that magnetic field and flux are the dominantfactors in a switched reluctance motor. The representation of the motor withsimple CI-characteristics hides the real problems of the motor in connectionwith its magnetic field. If we adopt the usual description of electrical machineswith main and stray inductance or main and stray reactance values we get amuch closer insight into the behaviour of the machine and have parameters athand for modelling the machine in connection with the transient states of itsoperation. If we use the separation of inductance in main and stray inductancein this machine we have to adapt calculation schemes to exploit theseparameters in advance. To back up this experimentally, two machines with anidentical cross section, but with different stack length have been built andmeasured too (Hajdarevic, 2000) (Figure 1).

The methods applied start with conventional calculation schemes, passthrough two-dimensional (2D) FEM calculation to three-dimensional (3D) FEMcalculation. The used FEM code is ANSYSe. The experimental machines arebuilt-up from laminations of ordinary electrical machinery. The magneticproperties of the laminations are known and used throughout the analyticaland numerical calculations. The rotor is also laminated and built-up from the



The authors acknowledge the assistance of the work shop of the institute of electrical Machinesand Drives for their diligent work in preparing the model machines.

Switchedreluctance

motors

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Vol. 22 No. 4, 2003pp. 847-861


DOI 10.1108/03321640310482850

same material type. The six phase windings are split into two branches, onesurrounds the central tooth of the stator pole and the second one surrounds allthe three teeth of a pole. In the break between the poles two phases lay adjacentin such a way that one phase fills half of the break area and the other themirrored other half. One can say that the phase sides lay in parallel positioninside the pole break. No overlapping of the phases is present in the endwinding overhang of the different phases. Linkage in this area is possible onlyfor the two branches of the individual phases (Figures 1, 3, 5, 7, 11, 12, 15, 17).

The principle layout of the machines can be observed best in the meshstructure for the 3D-FEM calculation (Figure 2). For the analytical calculationthe structure is simpler as only gap, teeth, pole faces and yokes aredistinguished.

2. Some practical considerationsThe usual approach for the torque production of switched reluctance motors isa search for the change of magnetic energy when moving the rotor from theposition aligned with the break between the stator poles over the positionaligned with the pole and again to a position aligned with the break betweenthe stator poles. This approach does not produce enough parameters for thetransient operation of the machine. Therefore, we are looking for a moredetailed evaluation of parameters. The voltage equations for two adjacent poleswith the windings (phases) getting involved in the rotation of the rotor in two

Figure 1.Cross-section of SRmachines

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different time periods are shown later. One of these periods is the operation ofthe phase as a stand alone equipment. This means no currents are present inthe complete operation time in the other phases. The second period is governedby the action of two adjacent phases at the same time. One can imagine thatthis operation also has drawbacks on the isolated operation period as thephases will show different entrance conditions to this period in contrast to thereal stand alone operation.

u1 ¼ i1R þdC11

di1

di1

dtþ

dC11

dg

dg

dt2 Ln;l

di2

dtð1Þ

u2 ¼ i2R þdC22

di2

di2

dtþ

dC22

dg

dg

dt2 Ln;l

di1

dtð2Þ

For all this we need a precise knowledge of the magnetic circuit of the motor.In our first attempt to get this insight we rely on the conventional methods.

The different parts of this equivalent circuit are the elements air gapincluding the slot opening effects as well as the break between the poles. Teethand teeth heads, stator yokes make up the stator magnetic resistance and rotorpole zone and rotor bulk body make the rotor magnetic resistance (Figure 3).This modelling of the magnetic circuit in the machine is still insufficient asthere is no realistic account of the length of the machine. In the calculation ofthe air gap magnetic resistance an attempt to cover the finite length of the corestack is used, which includes the small fringe field effects at the end of the stackor stack packages. The effects of the flux in the overhang section of thewindings are not included in the calculation of the magnetic circuit as they do

Figure 2.Mesh structure for the

3D-FEM calculation

Switchedreluctance

motors

849

not contribute from the torque production, but add an inactive additionalreactance which might be considered as an external reactance too (Michaelidesand Pollock, 1994, Williamson and Shaikh, 1992). Aided by the very simpleapproach to the magnetic circuit we can define the distribution of the fluxdensity in the gap. Correct evaluation with respect to the linkage of the twocoils in one pole results in the first step towards the analytical calculation of theCI-characteristics. Figure 4 for simplicity does not include the small amount ofexcitation, which is necessary for the rotor and the stator yoke. But themagnetic path length in the stator yoke is different from the situation in highpole machines, since in this six pole structure of the stator we close the pathover half the circumference to the opposite pole and not to the adjacent as isusual in ordinary electric machinery.

Figure 4 shows the influence of saturation on the shape of the flux densitydistribution very clearly. The smallest shown excitation level produces doublethe flux density in the central region than that is present at the edge of the rotorpole. The peak flux density in the figure is slightly above 1 T at the highestvalue of excitation shown. If we now turn to the unaligned position of the rotor,when the gap opposite to the middle of the stator pole is largest, then one caneasily see that the return path of the magnetic flux is very different from thepath of the aligned position. However already in the discussion of the alignedposition, the return path was omitted as no influence was seen on the shape ofthe flux density in the gap. The large gap even at the edge position of the rotorpole declines the magnetic characteristic to such an extent that tooth region

Figure 3.Equivalent magneticcircuit of the SR machine

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saturation in the stator pole is not present. The shape of the flux density in thisposition is only governed from the distributed excitation in the stator pole andthe varying gap (Figure 5). The flux density distribution is defined with twogaps replacing the actual distribution which is shown later. At the edge of thestator pole the flux density shall drop to zero in the distance of the gap width.

With these assumptions one finds the following flux density distributionsfor some excitation levels as before (Figure 6).

No saturation in the distributions is present. The extremely simplifieddistribution permits the calculation of the CI-characteristic in the unalignedposition and the previous one permits the calculation in the aligned position.The high flux density region in the middle of the distribution is caused by theexcitation of both the windings, the inner and outer one, the flux defined withthis central portion is linked with the inner coil. The total flux expressed by theintegration of the flux density along the complete pole arc is linked to the outercoil. So we have accounted correctly for the linkage between the two coils. Inany case, we use the same effective length of the stator core although we will

Figure 4.Flux density distribution

in the gap of the SRmachine at aligned

position

Switchedreluctance

motors

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find in the course of our examination that we have two different effectivelengths in aligned and unaligned position.

This short introduction of the analytic calculation immediately convincesthe reader that the results will suffer from the necessary simplifications.Therefore when we turn to the calculation with FEM codes, we will find thatsome simplifications can now be abandoned, but still some have to be included.In 2D-FEM code we have to account for the active length of the sheet stack. Wemust not think about stray fluxes in the slots of the machine as the code willautomatically include those fluxes in the calculation result. But we have to beaware of the problems of modelling the conductors inside the slots. Usually, nodistinction is made for individual conductors inside a slot. The current in theslot is distributed uniformly over the cross-section occupied from theconductors. This simplification reduces the number of elements in the code, butomits the special flux linkage of the individual conductors at their distinctposition inside the slot. We return to the problem of the active length that has tobe considered only when 2D-FEM is applied. A closer look on axial cuts of theswitched reluctance motor and on the very first approach of the flux density

Figure 5.The gap in the unalignedposition of the SRmachine

Figure 6.Flux density distributionin the gap of the SRmachine at unalignedposition

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profile at the end of the machine shows that one has to expect a smaller activelength in aligned and a larger one in unaligned position, but in any case thelength is larger than the stack length (Figure 7).

At this point, we turn to the situation outside the stack. The end windings ofthe stator coils of the individual phases can be viewed as stand alone objects.The nested end windings of one phase are linked due to the geometrical layout.One pole will be viewed as part of an ordinary single phase machine with twoslots per phase and pole and in the slots a number of conductors, which in ourcase corresponds with the number of turns of each of the pole coils.Questionable is what magnetic conductance L one has to choose for theexperimental machines. We compare the lay out with the coil heads present insingle phase synchronous machines and will therefore adopt the specificconductance l (0.18) reported for such machines. Similar ideas are used for thecalculation of the slot stray inductance and respective pole break inductance.By all these presented decisions the complete inductance calculation can beperformed first on an analytic base with formulas known since long in thedesign of electric machinery (Schuisky, 1960, Vogt, 1996), secondly in a semi

Figure 7.Flux density profile in an

axial cut of the SRmachine at aligned andunaligned position (zero

of axial coordinate at theonset of constant flux

density level, current inwindings 5.5 A)

Switchedreluctance

motors

853

analytic way we use FEM for the stack length of the machine and the analyticformulas for the end winding region and finally, we may turn over to a 3D-FEMmodel in which all the inductance values can be calculated. In this 3D-FEMcase the number of FEM elements even in this small machine is a severerestriction. So, here again a reduction of the structure to the main feature has totake place and details of the design of the machine must be left behind. Animportant feature of a 3D-FEM model is some lack of isotropic properties of thestator and rotor core. Due to the build-up by sheets we have less magneticconductivity in the direction normal to the stack than we have in the plane ofthe laminations. The relative permeability of the stack in the z-direction istaken into account by an expression which is 1 divided by the differencebetween 1 and the stacking factor (in the case of the model machines between0.92 and 0.94). There should not be a lengthy discussion on other problems in3D-FEM. We want to mention that the distance between the structure of theend winding housing and the end winding in a real machine will be of greatimportance in the calculation. The experimental machines are not verysensitive in this respect as there is no housing present. The next figures showthe front view of the aligned and unaligned 3D models of the machines (AnsysMagnetics, 1994) (Figures 8 and 9).

3. Experimental and calculation resultsIn this section, the formulas, tables and graphs from calculation andexperiment are put together. The analytic CI-characteristics are calculatedwith:

C ¼ 2 leff wcoil

Z arcend

arcstart

Bgap dx þ wcoil

Z arc2

0

Bgap dx

� �ð3Þ

The integration limits arcstart and arcend cover one tooth head width, whereasthe borders 0 and arc1 cover the complete stator pole arc consisting three toothheads and two slot openings as indicated in the Bgap distribution diagrams

Figure 8.View of aligned 3Dmodel of the SR machine

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(Figures 4 and 6). In Figure 4, the calculation must be done for differentexcitation currents as we want to observe saturation. For the unaligned case asingle evaluation would be enough to get the data for a figure similar toFigure 10.

In the short machine, we observe a relative linear dependence of thecalculated difference of measured C and calculated C which says that the strayinductance in the aligned position is unsaturated. In the long machine, we haveto expect another value of stray reactance. This is because the slot strayinductance increases with length, but the coil head reactance remainsunchanged (Figure 11).

The result in this case disturbs the picture, which usually is adopted forstray inductances. In the unsaturated region the calculated difference increaseslinearly, but from the onset of saturation the difference remains constant. Butcloser inspection of the short machines reveals a similar behaviour although

Figure 9.View of unaligned 3D

model of the SR machine

Figure 10.Comparison of measured

and calculatedCI-characteristics in the

short SR-motor

Switchedreluctance

motors

855

not so well defined. The cause of this effect is supposed to be heating of thewindings during the experiments. As the embedded length of conductors is halfthe length in the short machines the effect there is not so prominent. If we turnour attention to the linear extrapolation of the difference calculated for themeasured values and the analytical results for the effective length of themachine we should get a measure for the sum of the stray fluxes in the slotsection and the end winding section. By definition the result in the longmachine although larger should have less than twice the value found in theshort machine. Because of identical construction of both machines the flux inthe winding overhang should be equal. The slot section flux is doubled bytwice the length. One easily recognises that the statement is not fulfilled by thepresentation of the experimental and calculated numbers. When concentratingon the final values at 5 A of excitation we close in on values which can begained from the analytical calculation of stray fluxes. We propose that thisproblem is due to the simplified presentation of the magnetic characteristic ofthe building material. For calculation reasons, the characteristic is reduced insuch a way that no reversal of permeability occurs when passing from low tohigh excitation levels. The analytical as well as the FEM solution is based onthis simplification. The experiment, however, will use the actual permeabilitybehaviour and thus the calculation at low currents must be incorrect (Figure 12).

In flux density regions below 0.5 T the material in reality needs more ampereturns than used for the analytical as well as the FEM calculation. Vice versa theusage of the correct H values will reduce the calculated stray C values infavourable direction to get purely calculated and mixed calculated stray Cvalues to coincide.

Disappointed from these problems, we now try to solve the problem by3D-FEM calculation. This “true” model of reality will certainly produce whatwe need, namely exact values in advance of the production of a motor(Figure 13). We are searching for a solution which allows us to do without

Figure 11.Comparison of measuredand calculatedCI-characteristics in thelong SR-motor

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expensive prototyping and testing. The modelling of the machine forcalculation is presented already in Figures 2, 8 and 9.

In this case, we have rearranged the presentation and switched to theinductance values gained from the different sources. Experimental will be stillthe solid line, the dashed will be the analytic calculation improved by addingalso analytically produced values of stray inductance and finally in dash dotstyle the results of the 3D-FEM calculation are shown (Figure 14).

The measured is the highest value in this graph. 2D and 3D values crosseach other, but are in unsaturated region well below the measured. The 3Dcurve approaches the measured curve at high excitation. The traces do notindicate a systematic mistake in calculation. The correction of the 2Dcalculation says that the analytical gained value of coil head inductance hasbeen added. At this time we have to admit that neither of our calculationmethods delivers the results of the experiment.

Figure 12.Comparison of actual

and calculatedBH-characteristic of the

sheet material

Figure 13.Flux density for

“aligned” position and5.5 A

Switchedreluctance

motors

857

A last topic in connection with stray inductance shall be addressed in thefollowing. Due to the chosen switching arrangement of the poles in theswitched reluctance motor eventually two adjacent poles might be active atthe same time. The linkage of these poles traces back to the fact that the polewindings in the stator lay side by side in the pole break. Under this assumption,we represent the linkage with the inductance value calculated as slot strayinductance of a wide slot. As we already know that these calculations and thestray inductance deducible from experiment do not fit very well we use also amuch larger value namely five times the first value. By this we can study theinfluence of less or more coupled phases in the overlap time (MATLAB, 1993;SIMULINK, 1997) (Figure 15).

The usual view on current in switched reluctance motors concentrates on asingle winding with no interaction with other windings. This makes easy thetheoretical treatment of the current rise and decay in the time the phase is

Figure 14.Comparison of L frommeasurement, 2D(corrected) and 3Dcalculation

Figure 15.Current traces atdifferent speed values inuncoupled phases

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active. It is however, still possible that two phases adjacent to each other comeinto operation in an overlapping time period. What is happening in such a casewill be addressed in the following. We know that in principle, the flux linkageof the two phases comes from the coil sides which lay in the same pole breakside by side. The respective inductance therefore must be closely related to theslot stray inductance in the wide slot which makes up the pole break. InFigures 16 and 17 we will now have a look on the speed values as in Figure 15but with two coupled phases.

The traces show small influence of the overlap on the current decrease. Thecurrents die out in equal time with or without coupling. During the rise time weobserve changes. The peak current by coupling is enhanced. This can be seenbest in the case of an enhanced coupling reactance (dotted lines). Forcomparison, the undisturbed course of the current (full line) is given in thefigures. A deeper investigation at a later time will show the influence of thedescribed current trace changes on the running performance of the motor. Thisimplies the coupling of the electric model and the mechanic model of the motorand eventually also the supply source equivalent circuit. This topic will beaddressed in a future paper.

Figure 16.Current traces in two

adjacent phases at lowspeed

Switchedreluctance

motors

859

4. ConclusionsWhile Figure 14 shows the discrepancy between prediction and measurementone can see that the calculated result approaches gradually the experimentalone at high saturation. This indicates that the air gap used in both calculations(taken from the drawings) is not exactly the same as the gap present in themodels. The narrow gap in the machine is subjected to tolerances due to themanufacturing process and if one looks at the value of such tolerances inobjects of the standard model size 0.15 mm are mentioned. This in fact is largecompared to the gap width and a gap narrower than the drawn is possible. Thiswill increase the values of C and L in the unsaturated region, but will notinfluence the much saturated values. So the observed relations in the curvescan be understood. The deviations found for the stray inductance values withrespect to the slot stray inductance might be due to the large uncertainty of thel-values in the formulas. What remains unsolved is the poor performance ofthe experiments in relation to the coil head stray inductance. One can attributethis to the fact that no data on specific magnetic permeance of coil head inswitched reluctance motors exist. The chosen value of l was simply taken fromthe idea of similarity between the coil heads in single phase machines and thatof the switched reluctance motor. A closer search on the l values may givebetter results. However, in the 3D calculation no special restrictions with

Figure 17.Current traces in twoadjacent phases at highspeed

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respect to the coil heads have been applied and no remarkable improvement ofthe results was found.

References

Ansys Magnetics, User’s Guide for Revision 5.0A (1994), Vol. I, Swanson Analysis Systems, Inc.

Hajdarevic, I. (2000), “Ein Betrag zur Berechnung des geschalteten Reluktanzmotors durchLosung seines zwei- und dreidimensionalen Magnetfeldes”, Dissertation, TU Graz, Graz,Austria.

MATLAB, User’s Guide (1993), The MathWorks, Inc.

Michaelides, A.M. and Pollock, C. (1994), “Effect of end core flux on the performance of theswitched reluctance motor”, IEE Proc.-Electr. Power Appl., Vol. 141, p. 308.

SIMULINK, Using SIMULINK (1997), The MathWorks, Inc.

Williamson, S. and Shaikh, A.A. (1992), “Three-dimensional effects in l/i diagrams for switchedreluctance motors”, Proc. ICEM, p. 489.

Further reading

Schuisky, W. (1960), Berechnung elektrischer Maschinen, Springer-verlag, Wien, Austria.

Vogt Karl (1996), Berechnung elektrischer Maschinen, VCH Verlag, Weinheim, Germany.

Switchedreluctance

motors

861

Spatial linearity of anunbalanced magnetic pull in

induction motors duringeccentric rotor motions

A. TenhunenLaboratory of Electromechanics,

Department of Electrical and Communications Engineering,Helsinki University of Technology, Finland

T.P. HolopainenVTT Industrial Systems, Technical Research Centre of Finland, Finland

A. ArkkioLaboratory of Electromechanics,

Department of Electrical and Communications Engineering,Helsinki University of Technology, Finland

Keywords Motors, Magnetic forces, Rotors

Abstract There is an unbalanced magnetic pull between the rotor and stator of the cage inductionmotor when the rotor is not concentric with the stator. These forces depend on the position andmotion of the centre point of the rotor. In this paper, the linearity of the forces in proportion to therotor eccentricity is studied numerically using time-stepping finite element analysis. The resultsshow that usually the forces are linear in proportion to the rotor eccentricity. However, the closedrotor slots may break the spatial linearity at some operation conditions of the motor.

IntroductionAn electrical motor converts electrical energy into mechanical work. Themagnetic field in the air gap of the machine generates the tangential forcesrequired for the energy conversion, but the field also produces other forcecomponents that may interact with machine structures and excite harmfulvibrations. At low frequencies, the vibration amplitudes may be large enoughto couple the electromagnetic system with the mechanical system. Theelectromechanical interaction changes the vibration characteristics of themachine, e.g. it may induce additional damping or cause rotor dynamicinstability.

The nature of these interaction forces has an effect on the methods requiredto model the electromechanical interaction in the machines. If the forces arelinear in proportion to the rotor displacement, the electromagnetic andmechanical systems can be analysed separately and a highly reducedsimulation model can be used to study the effects of the electromechanicalinteraction.



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Conventionally, the forces acting between the rotor and the stator have beenstudied by analytical means. There are many papers, in which the effects of therotor eccentricity on the unbalanced magnetic pull are studied analytically(Belmans et al., 1987; Ellison and Yang, 1971; Freise and Jordan, 1962; Smithand Dorrell, 1996). The problem with the analytical models is how to evaluatethe equalising currents induced in the windings by the asymmetric fluxdistribution. The effects of saturation and stator and rotor slotting are alsodifficult to model by analytical means.

Numerical field calculation methods have been used rarely for calculatingthe forces due to eccentric rotors (Arkkio and Lindgren, 1994; Tenhunen, 2001).A time-stepping analysis is used for studying the effects of equalising currentsinduced by an eccentric rotor in the parallel circuits of the stator windings onthe forces (DeBortoli et al., 1993).

The references cited above focus on the two special cases of whirling motion,i.e. the static and dynamic eccentricity. However, the whirling motion of therotor can also occur on some other frequencies. Fruchtenicht et al. (1982)developed analytical tools to study the cage induction motor in a more generalwhirling motion.

Arkkio et al. (2000) presented a linear force model using complex variablesfor the electromagnetic forces acting between the rotor and stator:

FðvwÞ ¼ KðvwÞ1ðvwÞ ð1Þ

where F is the total force, K is the frequency response function of the system, 1is relative eccentricity, which is defined as a ratio between the whirling radiusand average air-gap, and vw is the angular frequency of the whirling motion inrelation to the stator. This model fits for non-synchronous whirling motion.They also determined the model parameters for an induction motor bynumerical simulations and verified the results by measurements. Later on, thisforce model was incorporated with a mechanical rotor model and theinteraction phenomena were studied (Holopainen et al., 2002). However, animportant open question is the linearity of the forces in proportion to theeccentricity. The problem can be stated as:

�F ¼ �Fð1; s;U ;vwÞ ¼ 1 �F 0ðs;U ;vwÞ ð2Þ

where s is the slip and U is the line voltage.In this paper, the assumption of linearity is studied in order to establish the

limits of application of this previously developed force model. The spatiallinearity is studied numerically using impulse method (Tenhunen et al., 2002) infinite element analysis to calculate the frequency response functions betweenthe forces and the whirling radius of the rotor at different voltage and slipvalues. The results show that the assumption of linearity is usually valid forsmall values of relative eccentricity. However, the closed rotor slots may breakthe linearity at some operation conditions.

An unbalancedmagnetic pull

863

Analytical studyAt first, the analytical theory of the rotor eccentricity is presented briefly. Therotor eccentricity is considered as a rotor in whirling motion. When the rotor iseccentrically positioned with respect to the stator bore, the air gap length d is afunction of the angular displacement x and time t

dðx; tÞ ¼ d0½1 2 1 cosðx 2 vwt 2 wwÞ� ð3Þ

where d0 is the average air gap length and ww is the phase angle. The air gappermeance L varies inversely with the air gap length (Fruchtenicht et al., 1982)

Lðx; tÞ ¼m0

dðx; tÞ¼X1l¼0

Ll cos½lðx 2 vwt 2 wwÞ� ð4Þ

with the Fourier coefficients

Ll ¼

m0

d

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 12

p for l ¼ 0

2m0

d

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 12

p1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 12

p

1

!l

for l . 0

8>>>>><>>>>>:

ð5Þ

The magnetomotive force is assumed to be sinusoidally distributed in the airgap. The expression for the magnetomotive force is then

Fmðx; tÞ ¼ Fm cosðpx 2 vt 2 wmÞ ð6Þ

The magnetic flux density b(x,t) is a product of magnetomotive force Fm(x, t )and the air gap permeance L(x,t)

bðx; tÞ ¼ Fmðx; tÞLðx; tÞ ð7Þ

We consider only the motors, in which the number of pole pairs is bigger thanone and neglect the homopolar flux by supposing that the integral of the fluxdensity around the rotor is zero. Then, by taking only the first harmonics of theair gap permeance into account, the product in equation (7) gives the fluxdensity distribution in the air gap.

bðx; tÞ ¼Bp cosðpx 2 vt 2 wmÞ þ Bp21 cosððp 2 1Þx 2 ðv2 vwÞt

2 ðwm 2 wwÞÞ þ Bpþ1 cosððp þ 1Þx 2 ðvþ vwÞt 2 ðwm þ wwÞÞ

ð8Þ

where the amplitudes of the flux density harmonics are Bp ¼ FL0 and Bp^1 ¼FLp^1:

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The radial component of the force between the rotor and stator to the z- andy-directions (vertical and horizontal directions with respect to thecross-sectional geometry) is obtained by surface integral over the rotor outersurface

Fz ¼

Z 2p

0

bðx; tÞ2

2m0cosðxÞRl dx

Fy ¼

Z 2p

0

bðx; tÞ2

2m0sinðxÞRl dx

ð9Þ

where l is the length and R is the outer radius of the rotor.As a result, one gets the force vector, which rotates at the whirling frequency

vw. The force vector is presented as a complex form:

F ¼pRl

2m0ðBpBp21 þ BpBpþ1Þ {cosðx 2 vwt 2 wwÞ

þ j sinðx 2 vwt 2 wwÞ}

ð10Þ

The force in equation (10) presents only the radial component of the force. Theforces are usually divided into the radial component in the direction of theshortest air gap and a tangential component perpendicular to the radial one.Fruchtenicht et al. (1982) presented the common expression for theelectromagnetic forces, including also the tangential component of the force.

The amplitude of the force vector depends only on the amplitudes ofthe permeance waves L, because the magnetomotive force Fm is constant.Figure 1 shows the relative values of Fourier coefficients of the permeancewaves L0 and L1.

The radial component of the force is proportional to the product L of theFourier coefficients of the permeance waves L0 and L1. The product written asa series is

L ¼m0

d

�2$1þ

3

413 þ

7

815 þ

51

6417 þ · · ·

%ð11Þ

The first term in equation (11) is the linear part and the rest presents thenon-linear part. Figure 2 shows the product and the relative error done whenthe forces are supposed to be linear in proportion to the displacement (sum ofthe non-linear terms in equation (11)).

Figure 2 shows that according to the analytical theory, the assumption of thespatial linearity is valid for small values of relative eccentricity.


865

This analytical study shows that the unbalanced magnetic pull is linear inproportion to the displacement of the rotor if the radius of the whirling motion,i.e. amplitude of the eccentricity is small. The effects of saturation are not takeninto account in the analytical expression of the forces. The flux densityharmonics created by the rotor eccentricity influence on the saturation and viceversa. That is why the spatial linearity is also studied numerically bytime-stepping finite element analysis in the next section.

Figure 2.Product of the Fouriercoefficients (thick line)and the error of the linearassumption (thin line)

Figure 1.The Fourier coefficientsof the permeance wavesL0, marked by W, andL1, marked by £ as afunction of relativeeccentricity, 1

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Method of numerical studyThe calculation of the magnetic field and operating characteristics ofthe induction motor is based on time-stepping finite-element analysis of themagnetic field. The details of the method are presented by Arkkio (1987). Themagnetic field in the core region of the motor is assumed to be two-dimensional.End-winding impedances are used in circuit equations of the windings to modelthe end effects approximately. The magnetic field and circuit equations arediscretised and solved together as a system of equations. The time-dependenceof the variables is modelled by the Crank-Nicholson method. The method ofanalysis neglects the homopolar flux, but it should properly model the effects ofequalising currents, slotting and saturation.

The method presented by Coulomb (1983) was used for computing theelectromagnetic forces. It is based on the principle of the virtual work, and theforces are obtained as a volume integral computed in an air layer surroundingthe rotor. In the two-dimensional formulation, the computation reduces to asurface integration over the finite elements in the air gap. This method waschosen because it has given accurate results when computing the forces of theelectrical machines and it is verified by measurements (Arkkio et al., 2000). Theforces are calculated at each time step and as a result one gets the forces as afunction of simulation time.

The motion of the rotor is obtained by changing the finite-element mesh inthe air gap. Second order, isoparametric, triangular elements were used.A typical finite-element mesh for the cross-section of the test motors containedabout 10,000 nodes.

The impulse method in the finite element analysis is used to calculate thefrequency response of the electromagnetic forces. The details of the impulsemethod are presented by Tenhunen et al. (2002). The basic idea of the impulsemethod is to move the rotor from its central position for a short period of time toone direction, fixed into the stator coordinate system. This displacementexcitation disturbs the flux density distribution in the air gap, and by doingthis, produces forces between the rotor and the stator. Using spectral analysistechniques, the frequency response functions are determined using theexcitation and response signals. The length T of the displacement pulse, whichis defined as 1ðtÞ ¼ 1ð1 2 cosð2pt=TÞÞ; t1 , t , t2; was 0.01 s and the totalsimulation time was 1 s with constant time-step of 0.05 ms. To increase thespectral resolution, the sample size extended to 2 s was obtained by addingthe zeros to the end of the sample leading to the frequency resolution of 0.5 Hz.The discrete excitation and force signals were transformed into the frequencydomain by the fast Fourier transform without filtering or windowing. Thenumber of sample points was 8,192.

The frequency response function presents the electromagnetic forces perwhirling radius as a function of whirling frequency. Then, if the forces havespatial linearity property, the frequency response is independent of amplitude


867

of the excitation pulse. The effective amplitude of the cosine excitation pulse ishalf of the maximum value.

Three parameters are varied in the analysis: the supply voltage, theamplitude of the displacement pulse, and the slip. The radial and tangentialcomponents of the frequency response of the electromagnetic forces are studiedat range 0-50 Hz of whirling frequency, which is the fourth parameter inequation (2). The limits of linearity are studied by varying the input parametersand comparing the frequency responses.

ResultsTwo machines, 15 and 37 kW four-pole cage induction motors were chosen fortest motors to study the spatial linearity of the electromagnetic forces. Thequarters of the cross-sectional geometry of the motors are shown in Figures 3and 4 and the main parameters of the motors are presented in Table I. The maindifference between these motors is that the 15 kW motor has open and the37 kW motor has closed rotor slots.

The values of the varied parameters in the analysis were the following:the used voltages were 100, 250 V and the rated voltage (380 V for 15 kWmotor and 400 V for 37 kW motor). The used values for the slip weres ¼ 0; 1.6 and 3.2 percent for 15 kW motor and s ¼ 0 and 1.6 percent forthe 37 kW motor. The frequency response functions (FRF) were calculatedusing a displacement pulse with amplitudes 10, 20, 30 and 40 percent ofthe air gap.

Figure 3.The cross-sectionalgeometry of the 15 kWmotor

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At first, the 15 kW motor, which has open rotor slots was studied. Figure 5shows the FRF of the radial component and Figure 6 the FRF of the tangentialcomponent of the force. The voltage was 100 V and the slip s ¼ 0 for all theused displacement pulses.

The radial component of the FRF grows slightly when the amplitude of thedisplacement pulse increases. In Figure 5, the upper most curve is calculated bya 40 percent displacement pulse and the lowest is calculated by a 10 percentpulse. The radial component of FRF by the 40 percent displacement pulse is onan average about 18 percent larger than the corresponding response calculated

Figure 4.The cross-sectional

geometry of the 37 kWmotor

Parameter (kW) 15 37Number of poles 4 4Number of phases 3 3Number of parallel paths 1 1Outer diameter of stator (mm) 235 310Core length (mm) 195 249Inter diameter of stator (mm) 145 200Airgap length (mm) 0.45 0.8Number of stator slots 36 48Number of rotor slots 34 40Connection Delta StarRated voltage (V) 380 400Rated frequency (Hz) 50 50Rated current (A) 28 69Rated power (kW) 15 37

Table I.The main

parameters of thetest motors


869

by the 10 percent pulse. Anyway, the difference in the FRF is less than10 percent for the pulses of 20 and 30 percent.

The tangential component of the FRF seems to be independent of theamplitude of the pulse in this case. All the four responses in Figure 6 are almostequal, the maximum difference between them is less than 2 percent of theamplitudes.

At voltage level 100 V, the maximum value of flux density is 0.69 T, so thereare no saturation effects. At 250 V voltage, the maximum flux density is 1.32 Tand the saturation slightly affects the magnetic field. Instead, at 380 V voltage,the motor is strongly saturated ðb ¼ 2:04 TÞ: Figures 7 and 8 show the FRF

Figure 5.The radial componentsof the FRF function ofthe forces at U ¼ 100 V,s ¼ 0 percent for the15 kW motor. The curvesrepresent all thedisplacement pulses

Figure 6.The tangentialcomponent of the FRF ofthe forces at U ¼ 100 V,s ¼ 0 percent for the15 kW motor. Theoverlapping curvesrepresent all thedisplacement pulses

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at rated operating point U ¼ 380 V and s ¼ 3:2 percent for all the useddisplacement pulses.

The frequency response functions presented in Figures (5)-(8) have a typicalshape for the four pole cage induction motors. The behavior of the FRF as afunction of the pulse amplitude is the same for all the calculated voltage levels.To study the spatial linearity more precisely, the electromagnetic forces arecalculated from the FRF at the whirling frequency of about 10 Hz. The behaviorof the forces as a function of the displacement is similar at reduced voltagelevels, so the results are presented only at the 250 V voltage level. The radial

Figure 7.The radial components

of the FRF of the forcesat U ¼ 380 V, s ¼ 3.2percent for the 15 kW

motor

Figure 8.The tangential

component of the FRF ofthe forces at U ¼ 380 V,

s ¼ 3.2 percent for the15 kW motor


871

and tangential component of the forces are presented as a function of therelative displacement in Figure 9 at 250 V and in Figure 10 at 380 V voltage forslip values s ¼ 0; 1.6 and 3.2 percent.

According to the results presented in Figures 9 and 10, the tangentialcomponent of the force is almost a linear function of the rotor displacement andit is independent of the slip. However, near to synchronous speed, thetangential component of the force depends strongly on the slip, but it is stilla linear function of the rotor displacement. The slip has a visible effect on

Figure 9.The forces as a functionof relative displacementat 250 V and whirlingfrequency 10 Hz for the15 kW motor

Figure 10.The forces as a functionof relative displacementat 380 V and whirlingfrequency 10 Hz for15 kW motor

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the radial components of the forces. The radial component of the force followsthe analytical theory well in the reduced voltages 100 and 250 V.

Table II shows the average difference of absolute value of the total force atwhirling frequency range 0-50 Hz as a function of displacement pulse comparedwith the total force calculated by 10 percent displacement pulse. The differenceis calculated by calculating the difference of the absolute value for each of thestudied whirling frequencies and taking the average of the differences at validfrequency range. The total forces are calculated by varying the values of thevoltage and slip.

According to Table II, the frequency response of the forces is almost thesame for 10 and 20 percent rotor displacements. We can assume that the spatiallinearity is valid for smaller than 10 percent displacements.

Actually, Table II shows the error, which results when using the assumptionof spatial linearity, at different values of relative rotor displacement fordifferent operating characteristics of the motor.

After the analysis of the 15 kW motor, the spatial linearity is studied for the37 kW motor. Figure 11 shows the FRF of the radial component and Figure 12the FRF of the tangential component of the force at 400 V voltage in no loadcondition (s ¼ 0 percent) for all the used displacement pulses.

Figure 11 shows the nonlinear behavior of the forces at this operating pointand used displacements. The smallest pulse gives the largest radial componentof the FRF. The exception is the synchronous speed, at which no equalizingcurrents are induced into the rotor cage and the FRF is independent of the pulse.The closed rotor slots can explain the nonlinear behavior. The eccentricityharmonics in equation (8) open the rotor slots magnetically and the dampingcurrents start to flow in the rotor cage when the amplitude of the displacementincreases. The tangential components of the FRF (Figure 12) are almost equal

s ¼ 0 percent s ¼ 1.6 percent s ¼ 3.2 percent

U ¼ 100 V20 percent 2.2 3.1 4.830 percent 6.4 9.2 14.540 percent 14.4 20.7 33.7



Note: The total forces are calculated with varying values of voltage and slip

Table II.The average

difference (percent)of the total force as a

function ofdisplacement pulsecompared with the

total forcecalculated by

10 percentdisplacement pulse

for the 15 kW motor


873

for all the pulses. However, at whirling frequencies near to the synchronousspeed the 10 percent pulse gave notably lower forces.

Table III shows the average difference of absolute value of the total force atwhirling frequency range 0-50 Hz as a function of displacement pulse comparedwith the total force calculated by 10 percent displacement pulse for 37 kWmotor. The total forces are again calculated with varying values of voltage andslip.

The same effect, which is shown in Figure 11, seems to occur also at allthe studied voltage levels at no load. If the rotor displacements increase,

Figure 11.The radial componentsof the FRF of the forcesat U ¼ 400 V, s ¼ 0percent for the 37 kWmotor

Figure 12.The tangentialcomponent of the FRF ofthe forces at U ¼ 400 V,s ¼ 0 percent for the37 kW motor

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the saturation level in the iron over the rotor slots increases, and more andmore induced damping currents flow in the rotor cage. The induced currents inthe rotor cage damp the harmonics created by the rotor displacement, and bydoing this, damp nonlinearly the forces and break the spatial linearity propertyat relatively low values of the rotor displacement.

At rated load, the force-rotor displacement relation follows approximatelythe analytical expression. The absolute value of the total force increasesslightly when the rotor displacements increase.

DiscussionsThe 15 kW cage induction motor has open rotor slots. The saturation hashardly no effects on the forces. The increase of the load linearises the forcesslighly in proportion to displacement. For the 37 kW test motor, the saturationeffects are more complicated. At no load condition, the harmonics created byrotor displacement open magnetically the rotor slots through the saturationwhen the displacement grows enough. For loaded motors, the fundamentalfield opens the slots and the spatial linearity is valid.

However, the amplitudes of the vibrations are usually very small, just fewpercents of the air gap. Based on this fact, the results obtained indicate that theassumption of the spatial linearity is valid for studying the electromechanicalinteraction. One should be aware of the possibilty that the closed rotor slotsmay cause nonlinearity at some operating points of the motor.

ConclusionsThe linearity of the electromagnetic forces between the rotor and the stator inproportion to the rotor displacement is studied in this paper. At first, thebackground of the study is presented analytically. The spatial linearity is

s ¼ 0 percent s ¼ 1.6 percent

U ¼ 100 V20 percent 216.0 5.430 percent 223.2 17.640 percent 224.2 47.0



Note: The total forces are calculated with varying values of voltage and slip

Table III.The average

difference of thetotal force (percent)

as a function ofdisplacement pulsecompared with the

total forcecalculated by

10 percentdisplacement pulse

for the 37 kW motor


875

studied numerically using time-stepping finite element analysis. The resultsindicate that usually the forces are linear in proportion to the rotordisplacement. However, the closed rotor slots may break the spatial linearity ofthe forces at some operating points of the motor.

References

Arkkio, A. (1987), “Analysis of induction motors based on the numerical solution of the magneticfield and circuit equations”, Acta Polytechnica Scandinavica Electrical Engineering Series,Helsinki, No. 59.

Arkkio, A. and Lindgren, O. (1994), “Unbalanced magnetic pull in a high-speed induction motorwith an eccentric rotor”, Proceedings of ICEM’94, 5-8 September 1994, Paris, France,pp. 53-8.

Arkkio, A., Antila, M., Pokki, K., Simon, A. and Lantto, E. (2000), “Electromagnetic force on awhirling cage rotor”, IEE Proceedings – Electric Power Applications, Vol. 147 No. 2,pp. 353-60.

Belmans, R., Vandenput, A. and Geysen, W. (1987), “Calculation of the flux density and theunbalanced pull in two pole induction machines”, Archiv fur Elektrotechnik, Vol. 70,pp. 151-61.

Coulomb, J.L. (1983), “A methodology for the determination of global electro-mechanicalquantities from a finite element analysis and its application to the evaluation of magneticforces, torques, and stiffness”, IEEE Transactions on Magnetics, Vol. 19 No. 6, pp. 2514-19.

DeBortoli, M.J., Salon, S.J., Burow, D.W. and Slavik, C.J. (1993), “Effects of rotor eccentricity andparallel windings on induction machine behavior: a study using finite element analysis”,IEEE Transactions on Magnetics, Vol. 29 No. 2, pp. 1676-82.

Ellison, A.J. and Yang, S.J. (1971), “Effects of rotor eccentricity on acoustic noise from inductionmachines”, Proceedings of IEE, Vol. 118 No. 1, pp. 174-84.

Freise, W. and Jordan, H. (1962), “Einseitige magnetische Zugkrafte in Drehstrommaschinen”,ETZ-A, Vol. 83 No. 9, pp. 299-303, (Germany).

Fruchtenicht, J., Jordan, H. and Seinsch, H.O. (1982), “Exzentrizitatsfelder als Ursache vonLaufinstabilitaten bei Asynchronmachinen, Parts 1 and 2”, Archiv fur Electrotechik,Vol. 65, pp. 271-92, (Germany).

Holopainen, T.P., Tenhunen, A. and Arkkio, A. (2002), “Electromagnetic circulatory forces androtordynamic instability in electric machines”, Proceedings of 6th International Conferenceon Rotor Dynamics, 30 September-3 October 2002, Sydney, Australia, pp. 456-63.

Smith, A.C. and Dorrell, D.G. (1996), “Calculation and measurement of unbalanced magnetic pullin cage induction motors with eccentric rotors. Part 1: Analytical model”, IEE Proceedings –Electric Power Applications, Vol. 143 No. 3, pp. 193-201.

Tenhunen, A. (2001), “Finite-element calculation of unbalanced magnetic pull and circulatingcurrent between parallel windings in induction motor with non-uniform eccentric rotor”,Proceedings of Electromotion’01, 19-20 June 2001, Bologna, Italy, pp. 19-24.

Tenhunen, A., Holopainen, T.P. and Arkkio, A. (2002), “Impulse method to calculate thefrequency response of the electromagnetic forces on whirling cage rotors”, Proceedings ofCEFC’2002, 16-19 June 2002, Perugia, Italy, p. 109.

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Calculation of eddy currentlosses and temperature rises atthe stator end portion of hydro

generatorsSt. Kunckel, G. Klaus and M. Liese

Department of Electrical Engineering and Information Technology,Institute of Electrical Power Engineering,

Dresden University of Technology, Chair of Electrical Machines,Dresden, Germany

Keywords Eddy currents, Temperature, Generators

Abstract This paper deals with a calculation method of eddy current losses and temperature risesat the stator end teeth of hydro generators. It can be used for analysing and evaluating differentdesign variants when optimising the stator core end portion. The calculation method simulates thethree-dimensional local core end field, but uses only a two-dimensional calculation model. Amongstall the stator teeth it treats the tooth with the highest axial and radial magnetic flux impact. Thepaper presents a collection of calculation algorithms of the method and provides some resultsgained for two different stator core end designs.

1. IntroductionWhen designing the magnetic circuit of an electrical machine generally, themachine’s cross section is only considered. Effects at the stator core ends aretypically neglected although overheating of the teeth at the axial core ends canharm the integrity at least of large machines (Liese et al., 1990). Overheatingcan be based on both undue large eddy current losses in the tooth laminationsand insufficient local cooling, whilst the eddy currents are caused by magneticfluxes penetrating perpendicularly into the laminations at the core ends(Figure 1).

To overcome this situation, a calculation method was developed forpredicting the temperature rises at the core end teeth for various stator core enddesigns. It was transferred into a computer programme tailored for routineapplication by a hydro generator manufacturer. Based on the applicationprofile, a new challenge had to be faced. Obviously, determination of the statorcore end field demands a numerical field calculation. To date this requiresinvolvement of an expert commonly using commercial numerical software.But this time it was considered a prerequisite that later application of thecalculation software should not require any scientific personnel and notime-consuming input data processing for making the generator data processfor the numerical software. The programme module was requested to besuitable for integration into the existing design calculation software of



Calculation ofeddy current

losses

877


Vol. 22 No. 4, 2003pp. 877-890


DOI 10.1108/03321640310482878

the manufacturer and all data needed should be automatically acquired bythe module itself by using the files of the manufacturer’s routine programmeonly.

This demand for integration was particularly challenging, because itrequired short calculation times adapted to the conventional software foriterative treating of numerous design variants in order to identify the best one.Considering the additionally targeted, automated and self-organisingcalculation runs, without the support of an expert, the use of commercialsoftware for numerical field calculation was disregarded. In addition,voluminous three-dimensional numerical field calculations (Fujita et al., 2000)were considered impossible due to the various restrictions although the endtooth field and the eddy current distribution within a tooth are typicalthree-dimensional problems. The circumstances described demand for newsolutions in:

. developing a calculation model which simulates the three-dimensionalcase, but needs a two-dimensional calculation only and thus, indeveloping the appropriate two-dimensional model for adequatesimulation, the three-dimensional condition was a main target;

. using an utmost number of simplifications for shortening the calculationtime without major violation of the correct solutions;

. creating a tailor made numerical software module, which is specialised forthe automated generation of the calculation models and the selforganising calculation procedures comprising a combined field, eddycurrent loss and temperature distribution calculation.

It is felt that this project paved the way for harmonising existing conventionaldesign calculation and recent numerical software commonly operatedseparately.

Figure 1.Imaginary component ofmagnetic field

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The peculiarity of this new calculation approach is in the development andcombination of models for an adequate calculation of the stator end tooth field,eddy currents in the teeth and resulting temperature rises in one package inorder to make the calculation fast enough to be applicable for routinecalculation during the machine’s design stage. Different from othertwo-dimensional and quasi three-dimensional models (Khan et al., 1989;Mercow and Jack, 1992), the new calculation model has a separate highlypermeable magnetic yoke in the axial-radial plane for simulating the peripheralstator flux guided in the stator yoke.

2. Modelling of the magnetic end fieldIn the first step, the two-dimensional model in Figure 2 was developed forsimulating the three-dimensional field problem. It is adapted to thelongitudinal machine section. But, since the field effects in the stator endportion are of interest, only the core length is downscaled in the model. Itsimulates the original three-dimensional field conditions to the best byutilising a separate magnetic interference yoke between the stator and rotoryokes as can be seen on the left side in Figure 2. It allows for similarradial magnetic fluxes in the tooth of the model as in the maximum loadedteeth of the real hydro generator. This applies for both the rated main fluxexcited by the current ampere turn loads Ql and QF of the stator and fieldwinding and the end winding stray flux generated by the same ampereturn loads, but in the end winding region.

Figure 2.Two-dimensionalcalculation model


losses

879

It is obvious that the stray flux penetrating the end laminations of the toothwill be disembarked via radial fluxes in the tooth as is the case in the realmachine. In the latter, the flux loop is closed circumferentially via thestator and rotor yokes whereas this is done by the interference yoke inthe model.

Since the flux conditions within the interference yoke are of no interest, itspermeability can be assumed arbitrarily. It was experienced that with a relativepermeability of approximately 5,000, the same air gap main flux was driven bythe stator and rotor ampere turn loads as in the real machine. For finaladjustment of the model’s air gap induction, the ampere turn loads Ql and QF

on the interference yoke are automatically adjusted by the programme. Thereference induction to be achieved is the maximum air gap induction of the realgenerator. Amongst other data, it is known from the results of the routinedesign calculation programme. With almost all the rated air gap induction, theoriginal flux densities and saturation conditions are achieved in the modeltooth. For others than rated conditions, the programme converts the ampereturn loads accordingly.

The one purpose software for treating the stator core end field only wascompleted by developing algorithms for automatic generation of the gridsystem for discretizing the calculation area in Figure 2. The grid meshesdeveloped are fine in the stator tooth area and coarse in the areas of minorinterest like the artificial interference yoke. The grid generation software isflexible enough for treating all stator end portion variants which may arise inthe course of the iterative optimisation of the generator design.

The programme calculates the vector potentials for all grid nodes utilisingthe finite integral method (FIM), which when compared to others provides aless complicated method regarding both establishing the equation system andhandling the grid system (Eckhardt, 1978; Liese, 2000; Silvester and Ferrari,1983). Finally, it derives the magnetic induction components connected tovector potentials. As an example, Figure 1 shows a plot of a stator core end fieldgained by simulating rated operation conditions of a 780 MVA hydrogenerator. Considering a maximum air gap induction of roughly 1.15 T, theenlarged distances between the flux lines representing the end portion strayflux indicate a maximum stray flux induction of almost 0.35 T entering the endlamination perpendicularly.

3. Modelling of the eddy current pathsThe three-dimensional structure of the calculation domain becomes mostobvious when considering the eddy current flow paths. They are directedperpendicular to the driving axial magnetic flux components and flow withinthe planes given by the insulated laminations, but cannot flow unrestrictedly inthe circumferential direction since they are forced to develop closed loopswithin the tooth cross sections. Thus, it was decided to build-up resistance

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networks, as shown in Figure 3, in several planes along the axial length of theend tooth for calculating the eddy current distribution in the end tooth.

At each plane, the driving mesh voltages are induced by the axial fluxcomponents piercing the tooth laminations gained by the preceding numericalfield calculation. The resistance networks that build-up automatically take intoaccount the electric conductivity and the dimensions of the lamination sheets.The self organising network generation software allows for slits worked intothe laminations for reducing the eddy currents as well as stepped end parcels ofthe stator core.

After calculation of the first set of lumped eddy currents in all calculationplanes, they are fed into a subsequent numerical field calculation run. Thisprovides new vector potentials, induction components and driving meshvoltages for the eddy current networks reflecting the first feedback of the eddycurrents on the end tooth field. In the course of an iterative process comprisingsubsequent eddy current and field calculations, a final eddy currentdistribution in the lumped networks will be achieved enabling thedetermination of the eddy current losses dissipated in the network resistances.

According to Figure 4, determination of the resistances is still based on someideal assumptions neglecting possible skin effects within the laminations.Improvements for taking them into account are developed, but not yetimplemented.

Each mesh cross section is assumed to be penetrated by a constantinduction. The induction rule provides:

2yðEyð2xÞ2 EyðxÞÞ þ 2xðExðyÞ2 Exð2yÞÞ ¼ 2jvB4xy ð1Þ

Figure 3.Eddy current resistance

network of a plane of theend tooth


losses

881

This means that the maximum current density is considered to be linked to theflux that penetrates the whole cross section hb. Each circulation on a smallerrectangular circumference, e.g. with the path lengths 2x and 2y (Figure 4)encloses a smaller flux. Thus, there is no single mesh flux. But it is compatiblewith the calculation model to assume that the mesh fluxes driving theequivalent eddy currents through the lumped equivalent resistances of themeshes in Figure 5 are encircled by loops along the middle between the meshboundaries and the mesh centres, in case the resistances are likewise defined bythe appropriate lengths on this loop. Thus, with the definitions x ¼ 0:5b andy ¼ 0:5h and the symmetry conditions:

Figure 5.Heat source network

Figure 4.Elementary eddy currentmesh

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Eyð2xÞ ¼ EyðxÞ and Exð2yÞ ¼ ExðyÞ ð2Þ

it follows:

2hEy 2b

2

� �þ 2bEx 2

h

2

� �¼ 2jvBbh ð3Þ

which results in:

Gy

k2

b

2

� �h þ

Gx

k2

h

2

� �b ¼ 2

1

2jvBbh ð4Þ

Considering the linear decline of the mesh current densities assumed inFigure 4, it follows:

Gy 2b

2

� �¼ Gx

h

2

� �h

bð5Þ

resulting in:

Gx

h

2

� �¼ Gx max ¼ 2

1

2jvkB

b2h

b2 þ h2ð6Þ

Taking the mean current guided through the upper horizontal lumped meshresistance:

I x m ¼Gx max

2

h

2d ð7Þ

and the voltage drop along this resistance:

Ux m ¼Gx max

k

b

2ð8Þ

one gains the equivalent resistance:

Rx ¼Ux max

I x m

¼b

khdð9Þ

Likewise, one gains for the vertical lumped mesh resistance:

Ry ¼Uy max

I y m

¼h

kbdð10Þ

In case of adjacent meshes, two resistances appear connected in parallel. Theymust be converted into one resulting resistance:


losses

883

Rx1 ¼ Rx and Rx2ðiÞ ¼RxðiÞRxðiþ1Þ

RxðiÞ þ Rxðiþ1Þð11Þ

and:

Ry1 ¼ Ry and Ry2ðiÞ ¼RyðiÞRyðiþ1Þ

RyðiÞ þ Ryðiþ1Þð12Þ

respectively, whereas the mesh voltage is given by:

Um ¼ 2j2pf NBbh

4: ð13Þ

It must be emphasised that the iterative incorporation of the eddy currentsprovides the freedom for using the equivalent lumped eddy current networkemployed using resistances only and no mesh reactances. The magnetic fluxesgenerated by the eddy currents are directly taken into account during theiterative numerical field calculations and must not be pictured in the electricalnetwork by equivalent products of inductivities and the appropriate eddycurrents.

4. Modelling of the temperature distributionAnother self organizing lumped network is build-up automatically forcalculating the temperature rise distribution within the stator end tooth. It isadapted to the longitudinal section of the end tooth, according to Figure 5,taking into account the heat conduction within the whole end tooth, towardsthe stator yoke and the heat transitions into the cooling air at both axial ends ofthe core end parcel and the air gap.

The eddy current losses calculated in the preceding calculation step are fedinto the mesh nodes of the heat source networks. As is the case with the gridsystem generation for the numerical field calculation as well as with thelumped eddy current networks, the algorithms for calculating the resistances ofthe lumped heat source network are also fit for taking into account all thegeometrical features of the stator end portion which may arise from possibledesign modifications.

For simplification of the formulae, all resistances connected in seriesbetween each two nodes are combined to single resulting resistances. Finally,the temperature rises are determined at all nodes of the thermal network byprocessing the related equation system.

The resistances of the thermal network are computed conventionally. Forthe heat conduction resistances, one gains:

Rth C ¼l

lAð14Þ

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where l is the actual conduction length, l the specific thermal conductivity andA the actual conduction cross section within the end tooth.

The heat transition resistances are given by:

Rth T ¼l

aASð15Þ

where a is the heat transition coefficient and A the transition surface.Appropriate resistances must be determined for three heat transition surfaces.The first one concerns the axial transition from the stator core end packagetowards the cooling air in the first radial vent duct of the stator core. To beprecise, this is the only heat transition resistance which can be calculated ratherprecisely based on a heat transition coefficient, a, connected to the well-definedcooling air velocity in the vent. The cooling flow conditions at the outer toothend surface indicated with “Front side” in Figure 5 providing no guidance at allfor the cooling air and in the air gap are much less defined. Considering thealmost uncontrolled coolant flows, these heat transition coefficients could onlybe guessed. The best guess was:

a ¼ 40: ð16Þ

Although the manufacturer of hydro generators felt that this value matchedhis experiences, yet it is basically the weakest data used. Thus, furtherimprovement of accuracy of the calculation would not require furtherimprovement in the field and loss calculation, but a more precise determinationof the heat transition coefficients is hardly achievable.

Different to the coolant at the “Front side” having an almost constant coldair temperature over the whole tooth height, the cooling air moving along thestator cooling vent and the air gap becomes continuously warmer. Thismechanism is taken into account by means of thermal resistances comprisingthe heat capacity of the coolant. They are calculated as follows:

Rth A ¼1

crvAð17Þ

where c is the specific heat capacity, r the density of air, v the cooling airvelocity and A the cooling channel cross section. Knowing the thermalresistances and the losses fed into the nodes of the thermal network thetemperature rises for the nodes are automatically calculated by the programme.For avoiding misunderstandings, it is emphasised that these temperature risessuperpose the “normal temperature rises” caused by the losses of the indirectlycooled stator bars, which are dissipated via the stator teeth and by the ironlosses in the teeth. Thus, even in case the temperature rises treated in the paperwould not be very large, they can become harmful for the machine’s insulationafter adding them to the “normal temperature rises”.


losses

885

5. Programme systemThe numerical algorithms were transformed in a programme using theFORTRAN 77 standard. Once more, it was experienced that more developedprogramming languages could not be used, because they are not standard forindustries relying on programmes developed in the long run. For calculatingthe magnetic field, eddy current losses within the stator core end, the fieldcalculation programme module FIELD was developed. In this programmemodule also the temperature rise is calculated.

The input data needed by FIELD are generated by a preparation programmemodule called DATA. The detailed programme structure is portrayed inFigure 6.

The numerical algorithm are programmed as self organising routines with adynamic data structure. Allowing for treating different machines with differentsizes and modified geometries.

Figure 6.Structure of theprogramme system

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The numerical algorithm is developed as a self organising routine with adynamic data structure. This is necessary for the calculation of differentmachines with different geometric sizes.

6. ResultsTwo stator core end designs were treated. The first calculation case covered astator core end without stepped laminations representing the weakest endportion design (“case 1” of Figure 7).

The second case concerns a design variant with a conventionally steppedstator core end portion according to Figure 7.

Figure 8 depicts the results of the eddy current losses calculation performedfor the two stepping variants for unslitted teeth. As expected, the maximumlosses developed at the outer tooth tip at the very core end. With the steppeddesign variant, the eddy current losses are reduced to a quarter of the previouslosses. In addition, for both designs, different tooth slit configurations wereconsidered, specified in Figure 9, by the relative slit heights hslit per cent of 0, 25,33, 50, 66, 75 and 100 per cent based on the definition:

htooth slit per cent ¼htooth slitv

htooth100 ð18Þ

The hot spot revealed to be always close to the location of maximum losses. Ascan be seen, the stator end temperature rises are considerably influenced by thedesign, i.e. the stepping and slit conditions. In all cases, for relative tooth slitheights above 50 per cent, a further increase in the slit heights resulted in minorreductions of the temperature rises only.

7. ConclusionThis paper presents a new method for calculating the eddy current losses andtemperature rises of the stator end teeth of hydro generators combined with acollection of results for different stator end portion design variants. All resultsgained were considered by the manufacturer to generally comply with his

Figure 7.Stator core end designswith variable heights of

the tooth slit


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887

experiences and the findings on his machines. The new calculation routine istailored for becoming an integrated software module of the manufacturer’sconventional design calculation software. The software developed is able tosimulate the three-dimensional local field of differently designed stator core endportions of hydro generators and the related eddy current loss and temperaturerise distributions. It is designed for very short execution times and automatedapplication.

An input data processing module is an integral part of the software package.Two calculation cases were used for exemplifying the various possibilities

for influencing and reducing the axial stray flux and eddy current loss impactsat the stator core end by stepping and slitting the tooth laminations resulting indifferent temperature rises.

Figure 8.Eddy current losses –teeth unslit

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Figure 9.Temperature rises


losses

889

References

Eckhardt, H. (1978), Numerische Verfahren in der Energietechnik, Teubner Studienskipten,B. G. Teubner Stuttgart, Stuttgart.

Fujita, M., Tokumasu, T. and Yoda, H. et al. (2000), “Magnetic field analysis of stator core endregion of Lagrange turbogenerator”, IEEE Transactions on Magnetics, Vol. 36 No. 4,pp. 1850-3.

Khan, G.K.M., Buckley, G.W. and Bennett, R.B. et al. (1989), “An integrated approach for thecalculation of losses and temperatures in the end-region of large turbine generators”, IEEEProceedings 89 SM 753-5 EC.

Liese, M. (2000), “Numerical field calculation approach without generic discretization errors”,ICEM 2000 Proceedings, 2000 International Conference on Electrical Machines Helsinki,28-30 August, Finland, Vol. III, pp. 1795-9.

Liese, M., Boeer, J. and Eggleston, R.S. (1990), “Upgrading of turbine generators”, InternationalJoint Power Generation Conference, 21-25 October, Boston, Massachusetts, PWR Vol. 10,pp. 119-25.

Mercow, B.C. and Jack, A.G. (1992), “The modelling of segment lamination in three-dimensionaleddy current calculations”, IEEE Transactions on Magnetics, Vol. 28 No. 2, pp. 1122-5.

Silvester, P.P. and Ferrari, R.L. (1983), Finite Elements for Electrical Engineers, CambridgeUniversity Press, Cambridge.

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Comparison of two modelingmethods for induction machinestudy: application to diagnosis

C. Delmotte-DelforgeCREA-INSSET, Saint-Quentin, France

H. Henao and G. EkweCREA-Universite de Picardie Jules Verne, Amiens, France

P. BrochetL2EP-Ecole Centrale de Lille, Villeneuve d’Ascq, France

G-A. CapolinoCREA-Universite de Picardie Jules Verne, Amiens, France

Keywords Diagnosis, Modelling, Induction machines

Abstract This paper presents two modeling methods applied to induction machine study inorder to construct a tool for diagnosis purpose. The first method is based on permeance networksusing finite element analysis to calculate magnetic equivalent circuit parameters. The secondmethod consists of the elaboration of an electric equivalent circuit obtained from minimalgeometrical knowledge on stator and rotor parts of the machine on study. These two methods arepresented and their results are compared with respect to the normal and rotor broken baroperation. For this study, a simple structure induction machine with three stator coils and sixrotor bars has been investigated. The presented results concern stator currents andelectromagnetic torque for the rated speed and the magnitude of the stator current harmoniccomponents have been compared.

IntroductionThe increasing use of electric drives in critical applications requiring highsafety levels (e.g. nuclear power plants, electric vehicles, etc.) has led to thedevelopment of fast and reliable failure detection methods. For the first time,different methods have been based on the observation of signals coming fromnon-invasive sensors (Penman et al., 1986). However, the pertinence of theanalysis depends a lot on accuracy of measurement and discrimination abilitybetween normal and fault states. So, it is necessary to dispose of an importantdatabase that requires many experimental results resulting in a highexperimental cost. Another way to get these results is to use modeling methodsto build the database. This required not only the development of a completeand accurate modeling method for electromagnetic devices, but also the abilityto simulate electrical faults within a reasonable time in order to study a largenumber of situations.



Comparison oftwo modeling

methods

891


Vol. 22 No. 4, 2003pp. 891-908


DOI 10.1108/03321640310482887

From this point of view, two complementary models on the permeancenetwork method (PNM) (Derrhi et al., 1999) and the circuit-oriented model(COM) (Henao et al., 1997) have been developed. These two models takegeometrical specificity of the motor into account, but stay simpler than finiteelement method (FEM), which needs a large amount of computation time(Bangura and Demerdash, 1999).

In this paper, the two methods are presented and their results are comparedwith respect to the normal and rotor broken bar operation. The presented resultsconcern stator and rotor currents for the rated speed operating point, which arecompared with respect to the magnitude of the different harmonic components.

Proposed modelsThe PNMThe magnetic equivalent circuit method is based on the decomposition of theelectromagnetic system into flux tubes. Each tube is characterized by itspermeance and all the permeances are linked together to give a model of themagnetic circuit like in a simple electrical circuit (Derrhi et al., 1999), but thevariables were magnetic flux and magnetic potential.

Induction machines can be decomposed in elementary magnetic circuits,built around a tooth, a slot and the concerned yoke portion. An elementaryequivalent circuit is composed of three permeances: tooth permeance, yokepermeance and leakage permeance (Figure 1). The complete model of themachine is obtained by the association of elementary equivalent circuits for thestator and rotor parts. These two parts are connected together by air-gapmodeling. In order to model the connection between electric and magneticcircuit, magnetomotive force sources are introduced in series with the toothpermeances of the magnetic circuit. These magnetomotive force sources areobtained by applying Ampere’s law around the slots (Figure 1).

All permeance computations are performed with magnetostatic FEMsimulation, resulting in an accurate model. The air-gap is modeled withvariable permeances depending on the rotor position. Permeances are locatedbetween each stator tooth and rotor tooth. During rotational motion, the valueof the permeance increases while the concerned teeth move closer, reaches a

Figure 1.Elementary magneticcircuit

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maximum when they are face to face and decreases while they are movingapart (Figure 2).

Several values of the air-gap permeances are determined for different statorand rotor tooth positions by several FEM simulations. Then, the computedcharacteristic is interpolated by Fourier decomposition (Figure 3).

The induction machine model described earlier leads to the completepermeance network. Accounting for the studied area, the network is quite large.Nevertheless, one can have a good idea of the magnetic equivalent circuit bydrawing an elementary network representing a pair of stator and rotor slotsand the whole air gap (Figure 4).

The system of the resulting differential non-linear equations that describeselectromagnetic and dynamic behaviors of the motor, is solved by a softwareapplication (SiRePCE) using the Matlab environment. This application hasalready been presented (Der, 2000).

COMThis modeling method is based on the machine representation as an electriccircuit with resistors, inductors, capacitors and voltage or current sources(Ekwe et al., 2001). Discrete representation of the electrical characteristics ofstator coils or rotor bars is strongly dependent on resistive and inductive

Figure 2.Air-gap permeance angle

between the ith statortooth and the jth rotor

tooth

Figure 3.Air-gap permeance curvevs angle between the ith

stator tooth and the jthrotor tooth


methods

893

parameters. Isolation between turns effect is associated with capacitiveelements that can be neglected considering the frequency domain of interest(less than 2 kHz). The stator model is obtained from the representation of eachcoil turn. Electrical parameters such as inductances, mutual inductances orleakage inductances are obtained by applying elementary laws to magneticcircuits (Figure 5). The association of elementary circuits according to the statorcoil connections gives the stator model. The principal elements of the equivalentelectric circuit corresponding to the turn m of a stator coil are given by:

. Rsm – conductor resistance

. Lsm ag – air-gap inductance

Figure 4.Equivalent permeancenetwork of the inductionmachine

Figure 5.Equivalent COM for astator coil

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. Lsm s – leakage inductance for the slot

. Lsm ce – leakage inductance for head of coils

. Lsm;1;L

sm;2; . . .Lm;h; . . .;L

sm;n – mutual inductance between the turn m and

the other stator turns, with h – m and n the number of stator turns. Lsr

m;1;Lsrm;2; . . .;L

srm;Q – mutual inductance between the turn m and the rotor

meshes

For the rotor cage (in the case of a squirrel-cage machine), each mesh is formedby two bars and the corresponding portion of end-ring circuit. Each mesh isrepresented by the following elements (Figure 6):

. Rrb; Rr

er – resistance associated to bar and end-ring portions. Lr

ag – air-gap inductance. Lr

s – leakage inductance of slot. Lr

er – leakage inductance of end-ring portions. Lr

k;1;Lrk;2; . . .;L

rk;l ; . . .;L

rk;Q – mutual inductance between the mesh k and

the other rotor meshes, with k – l and Q the number of rotor meshes. Lrs

k;1;Lrsk;2; . . .;L

rsk;n – mutual inductance between the mesh k and the stator

turns

The connection of the different meshes leads to the whole rotor representation.The different circuit elements associated with stator and rotor can be

classified in terms of dependence on rotor position. For constant parameterelements, conductor resistances, self inductances of stator turns, rotor meshes,and mutual inductances between the stator turns and rotor meshes can beconsidered. Mutual inductance between the stator turns and rotor meshesdepends on the relative position between the stator and rotor. For theconsidered time length, which is a function of observed electrical phenomena,the variation of conductor resistance due to temperature is not taken intoaccount. The constant parameter elements can be associated with coupledbranches of linear circuits ðR;LÞ and the variable parameters to control voltagesources depending on induced currents. For the stator turn m, this last effectwith only one electromotive force can be represented as:

emf sm ¼

XQ

k¼1

emf sm;k ¼

XQ

k¼1

d

dtLsr

m;kðuðkÞÞirk

n oð1Þ

where u(k) is the position of the mesh k in the stator reference.For the rotor mesh k, this emf depends on stator currents ði s

1 ; is2 ; . . .; inÞ :

emf rk ¼

Xn

m¼1

emf rk;m ¼

Xn

m¼1

d

dtLrs

k;mðuðkÞÞism

n oð2Þ


methods

895

Figure 6.Equivalent COM for arotor mesh

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In order to simulate this model, a circuit simulation software can be used(PSPICE, EMTP, MATLAB/PSB, . . .).

Test of the proposed methodsThe two methods have been applied to an elementary three-phase machine of2 kW, 230 V/400 V, 50 Hz, 2 poles, 2920RPM with six stator and rotor slots.In the PNM, a relative permeability mr ¼ 2;000 is used to characterize themagnetic material without taking saturation into account. In COM, the ironpermeability is considered as infinite. In the COM, the stator-rotor mutualinductance is obtained by a trapezoidal piecewise representation where thefringe effect in the air-gap is neglected. In the PNM, inductances of head coiland end-ring portion are neglected.

In the simulation, it was considered that the power supply is perfectlysinusoidal. The first simulation results concern the rated speed with themachine operating in with rated power supply without fault. Figure 7 showsthe results obtained from the two methods. The rms value of the statorcurrent fundamental component is 6.9 A for PNM and 6.6 A for COM. InFigure 8, it can be observed that the stator harmonic components are affectedby the number of rotor slots and the rotor slip following the expression (Nandiand Toliyat, 2001):

$ks

N r

pð1 2 sÞ^ 1

%f s ks ¼ 0; 1; 2; 3; 4; . . . ð3Þ

where Nr is the number of rotor slots, p the number of pole pairs, fs is thepower supply frequency, and s is the rotor slip.

The current spectrum observation is limited to a bandwidth of 800 Hz. Inthis band, the frequencies given by expression (3) fits with simulation resultsfor both methods. The magnitude and the frequency of these components arepresented in Table I.

The good agreement between the two methods with two pairs ofcomponents (242-342 Hz and 534-634 Hz) having approximately each one thesame level can be observed. For the first pair of components, the differencebetween the two results is less than 30 per cent and for the second one is lessthan 70 per cent.

The simulation results for the electromagnetic torque are also compared(Figure 9). In Figure 10, the electromagnetic torque spectrum shows that itsmain frequency components are multiples of the corresponding rotor rotationfrequency following the expression:

ktVr

2pkt ¼ 0; 6; 12; 18; . . . ð4Þ


methods

897

Figure 7.Stator current simulationat rated speed for asinusoidal power supply.(a) PNM, (b) COM

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Figure 8.Normalized stator

current spectrum limitedto a bandwidth of 800 Hz(at rated speed). (a) PNM,

(b) COM


methods

899

with Vr the rotor speed. The mean value (0.0 Hz) for PNM is 8.2 N m and forCOM is 8.4 N m. It can be observed that the main frequency components of theelectromagnetic torque are obtained with the same relation approximatelybetween them for both methods too.

Considering now electrical fault simulation, we will examine the case of abroken rotor bar. This fault can be interpreted as an unbalance in the rotor cageand the effect in the machine behavior is the introduction of new frequencycomponents in the stator current (Milimonfared et al., 1999). The newexpression for the stator current frequency components is:�

kbspð1 2 sÞ^ s

�f s kbs ¼ 1; 3; 5; 7; . . . ð5Þ

Figure 11 shows stator current for this case. It can be seen that its magnitude ismodulated, but this is not the only effect in current spectrum as can beobserved in Figure 12 and Table II. The rms value of the stator currentfundamental is in this case is 5.8 A for PNM and 5.3 A for COM.

The predicted frequency components in stator currents, under rotor bar, isgiven by equation (5) are also obtained by both the methods. Particularly, thenew components are located in the sideband of frequencies excited in normaloperation (Table III). For these last frequencies, the location are the same as forthe normal operation, but the magnitude is less.

The torque waveform is also modified by the rotor fault. As the proposedtest motor has a very low number of rotor slots (i.e. six), the effects of onebroken bar is very important. With a “normal” machine, several broken rotorbars are necessary to get the same magnitude of the sideband components(Nandi and Toliyat, 2001).

The effect of broken bars in the electromagnetic torque spectrum can bedetected with the following frequency components (Melero et al., 2000):

kbtVr

2p^ 2sf s kbt ¼ 0; 2; 4; 6; . . . ð6Þ

Table IV presents the relative values of harmonics for electromagnetic torque,showing the apparition of side band harmonics around harmonics obtained

Harmonic components (dB)

Frequency (Hz)

Model 50 242 342 534 634

PNM 0.0 213.0 212.9 226.3 226.1COM 0.0 29.6 210.2 216.0 216.2

Table I.Maximum relativevalues of statorcurrent frequencycomponents(ks¼0,1,2 ands ¼ 2.7 per cent)

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Figure 9.Electromagnetic torque

(at rated speed). (a) PNM,(b) COM


methods

901

Figure 10.Normalizedelectromagnetic torquespectrum limited to abandwidth of 800 Hz(at rated speed). (a) PNM,(b) COM

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Figure 11.Stator current simulation

at rated speed with onebroken rotor bar.

(a) PNM, (b) COM


methods

903

Figure 12.Normalized statorcurrent spectrum limitedto a bandwidth of 800 Hz(one broken rotor bar).(a) PNM, (b) COM

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without fault (Figures 13 and 14). The mean value of torque is 5.8 Nm withPNM and 5.4 Nm with COM.

ConclusionTwo simulation methods applied to induction machines have been presented.In the first step, simulation techniques for a simple induction machinestructure have been studied. The first results show a good correspondencebetween the modeled electromagnetic phenomena and the frequencycomponents of the stator current in both healthy and faulty conditions.The two methods give similar results and allow both the study of faultinfluence and diagnosis in good conditions. Both methods need a minimumknowledge on machine topology such as repartition of coil in slot, diameterof air-gap. PNM also needs dimensional parameters to built magnetostaticfinite elements model in order to determine permeance values. PNM can


Frequency (Hz)

Model 47.3 50 242 244.7 339.3 342 534 536.7 631.3 634

PNM 213.7 0.0 214.9 218.7 224.1 214.8 229.5 227.3 239.6 229.4COM 26.3 0.0 211.4 216.5 216.4 211.4 218.2 224.7 226.7 218.2

Note: kb¼1,5,7,11,13 and s ¼ 2.7 per cent

Table II.Maximum relative

values of statorcurrent frequencycomponents with

one broken bar


Frequency (Hz)

Model 0 292 584

PNM 0.0 4.2 1.7COM 0.0 5.6 2.8

Note: kt¼0,6,12 and Vr¼306 rad/s

Table III.Maximum relative

values ofelectromagnetic

torque frequencycomponents


Frequency (Hz)

Model 2.7 289 292 294 581 584 586

PNM 29.6 28.8 3.8 29.8 217.1 23.4 210.1COM 28.2 21.9 5.8 23.1 21.4 2.5 22.2

Note: kbt¼0,2,4,6,8,10,12 and Vr¼306 rad/s

Table IV.Maximum relative

values ofelectromagnetic

torque frequencycomponents with

one broken bar


methods

905

Figure 13.Electromagnetic torquewith one broken rotorbar (at rated speed).(a) PNM, (b) COM

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Figure 14.Normalized

electromagnetic torquespectrum limited to a

bandwidth of 800 Hz (onebroken rotor bar).(a) PNM, (b) COM


methods

907

easily take into account the saturation phenomena (Derrhi, 2000). The mainbasic differences concern the form of stator-rotor inductance, the inductanceof end-ring and head of coil and the permeability of the magnetic material.The different computation times cannot be compared for the moment asdifferent software have been used, but this will be done in the future whencommon software will be used.

References

Bangura, J.F. and Demerdash, N.A. (1999), “Diagnosis and characterization of effects of brokenbars and connectors in squirrel-cage induction motors by time-stepping coupled finiteelement-state space modelling approach”, IEEE Transactions on Energy Conversion,Vol. 14 No. 4, pp. 1167-76.

Derrhi, M., Delmotte-Delforge, C. and Brochet, P. (1999), “Fault simulation of induction machinesusing a coupled permeance network model”, Proceedings of the IEEE InternationalSymposium on Diagnostics for Electrical Machines, Power Electronics and DriwesSDEMPED’99, Gijon, Spain, pp. 401-6.

Derrhi, M. (2001), “Modelisation of the induction machine by the permeance network method,validated by diagnosis”, PhD Thesis, University of Picardie.

Ekwe, G., Henao, H. and Capolino, G.A. (2001), “A new block-scheme circuit model of three-phaseinduction machine using MATLAB/SIMULINK”, Proceedings of International Symposiumon Diagnostics for Electrical Machines, Power Electronics and Drives (SDEMPED’01),Grado (Italy), pp. 355-60.

Henao, H., Capolino, G.A. and Poloujadoff, M. (1997), “A circuit-oriented model of inductionmachine for diagnostics”, Proceedings of the IEEE International Symposium onDiagnostics for Electrical Machines, Power Electronics and Driwes SDEMPED’97, Carryle Rouet, France, pp. 185-90.

Melero, M.G. et al., (2000), “The ability of on-line tests to detect inter-turn short-circuits in squirrelcage induction motors”, Proceedings of the International Conference in Electrical Machines(ICEM’00), Espoo (Finland), Vol. 2, pp. 771-5.

Milimonfared, J., Kelk, H.M., Nandi, S., Minassians, A.D. and Toliyat, H. (1999), “A novelapproach for broken-rotor-bar detection in cage induction motors”, IEEE Transactions onIndustry Applications, Vol. 35 No. 5, pp. 1000-6.

Nandi, S. and Toliyat, H.A. (2001), “Novel frequency domain based technique to detect incipientstator inter-turn faults in induction machines”, Proceedings of the IEEE IndustryApplications Conference IAS’2000, Rome (Italy), Vol. 1, pp. 367-74.

Penman, J. et al., (1986), “Condition monitoring of electrical drives”, IEE Proceedings, Part B,Vol. 133 No. 3, p. 3.

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908

Finite element modeling of thetemperature distribution in the

stator of a synchronousgenerator

Josef Schoenauer, Erwin Schlemmer and Franz MuellerVA TECH HYDRO GmbH & Co, Elingasse, Austria

Keywords Finite element modelling, Temperature, Thermal modelling, Generators

Abstract In this paper, we applied the finite element modeling to the stator temperaturedistribution of a hydroelectric generator. The electrical losses produce a temperature distribution inthe stator of a synchronous generator. For the calculation and optimization of the temperaturedistribution, a full parameterized thermal model of the stator was created using the finite elementmethod. Now it is possible to calculate the thermal effects of different parameter modifications andadditionally we can optimize the heat transfer for the stator with variant calculations. The mostimportant bar fitting systems and its thermal efforts are included in this thermal stator model. Ourtargets are to decrease the expensive and time-consuming laboratory measurements in the futureand improve the accuracy of the standard calculation software. To estimate the accuracy of thefinite element model we build an additional laboratory model.

1. IntroductionThe electrical losses during the hydroelectric generator operation effectuate aheating of the generator. For the stator bars this means a high temperature rise.An accuracy prediction or calculation of the location and the magnitude of thestator temperature is very important for the design process, because the usefullife from the used insulation materials depends on the magnitude of the statorbar temperature. Therefore, the standardisation DIN VDE 0530 limits thetemperature rise of a stator winding (Figures 1 and 2).

Additionally, our customers demand often a lower stator temperature rise asin the standards required. This guaranteed temperature rise will be measuredafter commissioning in the interlayer between the upper and lower bar. If theguaranteed temperature rise is violated, an expensive penalty could cause.In this consequence, a continuous improvement of our stator temperaturecalculation is necessary. Particularly, the accuracy of the stator temperaturecalculation in the interlayer is decisively for the risk reduction in the designprocess. For the improvement of the stator temperature calculation we havestarted a research program with the following activities.

1. Creation of a simplified and parameterized finite element (FE) thermalstator model and compare their results with a similar laboratory statormodel.



Finite elementmodeling

909


Vol. 22 No. 4, 2003pp. 909-921


DOI 10.1108/03321640310482896

2. Examination of different stator winding fitting systems and their thermalefforts. After the implementation of the examination results in the finiteelement model we additionally considered the effects of the conventionaland the new U-spacer system.

3. Creation of an advanced finite element thermal stator model withconsideration of the bar strands and their thermal efforts.

4. Comparison of the finite element calculation results with generatormeasurements and standard thermal calculation.

The range of application for our FE model confines to air cooled hydroelectricgenerators with a forward flow ventilation system. The cooling air flows fromthe rotor over the air gap to the radial stator ventilation ducts. Inside theventilation ducts the heat transfer increases the temperature of the cooling air.By using all stator symmetries, we obtained a small FE model.

2. FE modelingFor the temperature field calculation, we used the commercial FE softwareANSYS. With the ANSYS Parametric Design Language (APDL) we found the

Figure 1.CAD drawing of a largehydroelectric generator

Figure 2.Stator winding andframe of a hydroelectricgenerator

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possibility to create a parameterized model. After the input of the major statordimensions the calculation runs automatically. These necessarydeterminations are material properties, winding fitting system, kind ofspacers, characteristic air flow and the electrical losses in stator bar and statorcore. This approach is due to the high number of repetitive calculations thefastest way (Figure 3).

The procedure of the FE modeling consists of the main parts namely,pre-processing, solution and post-processing like in Figure 3. Figure 3 lists themain steps of the creation of the FE stator model. This model creation phasestarts with the determination of the pre-processor with the APDL command\PREP7[3]. Additionally, with ANTYPE we perform a static analysis. The nextstep is to define the generally used parameters, which are valid for differentkinds of stators. Thereafter we define keypoints with the command K, Number,X, Y, Z in the active coordinate system for area description. Near by the X, Y, Zcoordinates are the previously defined parameters. The next step is to connectthe areas with the command A, K1, K2, K3. However, for the area descriptionwe have to define only areas with a homogeneous material, thereby anassignment of the material properties is possible. The 2D 4-node thermal solidPLANE 55 is a plane element with a 2D thermal conduction capability. ThePLANE 55 has four nodes with the single degree of freedom, temperature, ateach node. The third dimension we generate with the command VEXT basedon the previously defined areas. The SOLID 70 is a 3D 8-node thermal solid

Figure 3.Procedure of the FE

modeling with APDL


911

and has a 3D thermal conduction capability. The element has eight nodes withthe temperature as a single degree of freedom. The element is applicable to a 3Dsteady state or transient thermal analysis. Before the 3D extrusion occurs,automatic meshing with AMESH is necessary. This command creates newnodes within areas. For the modeling of the air flow through the statorventilation ducts, we have used the FLUID116 and SURF152 elements. TheFLUID116 (thermal fluid pipe) is a 3D element with the ability to conduct heatand transmit fluid. Heat flow is due to the conduction within the fluid and themass transport of the fluid. The heat transfer by convection needs additionallya connection between FLUID116 and the surface element SURF152. In thiscase, the film coefficient may be related to the fluid flow rate. The SURF152(3D thermal surface effect) is used for various load and surface effectapplications. It is overlaid onto the ventilation duct area. For these bothelements, we need a previous calculation of the ventilation duct air flow rate, airinlet temperature and the film coefficients. The calculation of the heat transfercoefficient in the stator core air gap, area is also necessary. Owing to thecomplex and undefined flow in the air gap, a computational fluid dynamicsexamination was necessary, thereby we could deduce the following generalcalculation formula for the film coefficient in the air gap.

vs ¼u

2þ aw ! w ¼

V

pLStsspð1Þ

Res ¼2sspvs

g

Nu ¼ cðResÞ1

a ¼Nul

2ssp

whereV¼ total air flow rateu ¼ circumferential velocity (outer rotor diameter)p ¼ number of polesa, c, 1 ¼ coefficientsLSt ¼ stator core lengthssp¼ air gap widthRe ¼ Reynolds numberNu ¼ Nußelt numbera ¼ heat transfer coefficient

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For the calculation of the heat transfer coefficients in the radial stator ducts, weuse the standard VDI calculation formulas of a pipe. Depending on the laminaror turbulent flow condition in the duct, we use the following equations for theaveraged Nußelt number. The first equation is applicable for Re , 2;300:

Nu ¼ 0:664Pr1=3

ffiffiffiffiffiffiffiffiffiffiffiRe dh

L

r! Re # 2;300 ð2Þ

Between Re ¼ 2;300 and 10,000, we calculate the averaged Nußelt number asfollows:

Nu ¼ 0:0214ðRe0:8 2 100ÞPr0:4 1 þdh

L

� �2=3" #

ð3Þ

where Pr is the Prandl number, dh the hydraulic length for the rectangularducts, and L the length of the stator ducts.

The composition of a stator bar is not uniform, it consists of strands with avery thin insulation. It is necessary to consider this 0.15 mm thin insulation forthe FE model, because the thermal conductivity of the strand insulationmaterial is less than the thermal conductivity of copper. But the big amount ofstrands makes the APDL program behind the FE model more extensive,therefore we use directional equivalent thermal conductivities for the whole bar(Figure 4).

The calculated equivalent thermal conductivities consider the strandsdepending on the bar coordinate. For the deduction of the equivalent thermalconductivity equations, we dispose the thermal resistances of the differentmaterials according to the coordinate. The following example contains only thecalculation of the equivalent thermal conductivity in the x direction, becausededuction of the other directions is similar. First, we calculate the copper strandarea Aq,cu and the strand insulation area Aq,tiso, which are related to 1 m barlength:

Aq;cu ¼ htl £ 1 m ð4Þ

Figure 4.Components of the

stator bar


913

Aq;tiso ¼ tiso £ 1 m ð5Þ

The heat resistance Rq,cu of the copper-insulation-copper element (serial) andthe heat resistance Rq,tiso of the through-going insulation layer (serial) arecalculated as follows:

Rq;cu ¼ ntbbtl

Aq;culcu

� �þ

ðntbtiso þ tfestÞ

Aq;culisoð6Þ

Rq;tiso ¼ ntbbtl

Aq;tisoliso

� �þ

ntbtiso þ tfestÞ

Aq;tisolisoð7Þ

From the above we obtain the total heat resistance Rq,ges in the x coordinateby parallel combination of the previous heat resistances:

Rq;ges ¼Rq;cuRq;tiso

nthðRq;cu þ Rq;tisoÞð8Þ

After transforming that calculation formula, we calculate the equivalentthermal conductivity in the x coordinate as follows:

lx ¼ntbðbtl þ tisoÞ þ tfest

Rq;gesðhtl þ tisoÞnth £ 1 mð9Þ

wherehtl ¼ strand lengthbtl ¼ strand widthtiso ¼ thickness of the strand insulation (for both sides)lcu ¼ thermal conductivity of copperliso ¼ thermal conductivity of the strand insulationnth¼ number of the strands in the y directionntb¼ number of the strands in the x direction

According to our measured generator in section 4, we calculated the followingequivalent thermal conductivities. The big directional differences of theequivalent thermal conductivities influence the local temperature distributiondecisively.

lx ¼ 4:845 W=mK ly ¼ 2:924 W=mK lz ¼ 341:26 W=mK

For the heat generation of the FE stator model the accurate knowledge of theelectrical stator bar and core losses is necessary. We consider the followinglosses in the FE model:

. ohmic losses in the strand;

. additional ohmic losses in the strand;

. additional ohmic losses by transposed strands;

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914

. additional ohmic losses (end winding);

. core yoke losses;

. core tooth losses;

. load dependent stray losses in the core tooth.

After calculation of the different losses, we summarize these enumerated lossesto different heat generation rates for the bar and core. The larger amount of theheat dissipation goes from the stator bars over the winding fitting system intothe stator core (Figure 10). From the lamination the heat flow leads away byconvection through the ventilation ducts. The insulation of the laminationdecreases the thermal conductivity considerably in the z direction, therefore, wehave to consider this influence in the FE model too. The preliminary calculationof the stator air inlet temperature is also necessary. After leaving the cooler thecold air passes through the fan and the poles. A partial air flow can passadditionally the end winding and the end plate area. For the previous heat up ofthe cooling air we consider the following losses:

. end winding losses;

. end plate losses;

. pole and damper winding losses;

. exciter losses; and

. windage losses.

After the determination of these losses, we calculated the air heat up andcompared it with the measured stator outlet air temperature. The measuredtemperature deviation was less than 3 K. We take this mentioned heatgeneration rates in the FE model and attach it with the APDL command BFE.This command defines an element body load like heat generation rate in athermal analysis. With SFE we specify surface loads on elements like theconvection in the ventilation ducts and in the stator air gap. The expected totalair flow comes from a specific air flow calculation program and was comparedwith the measured total air flow rate. Approximately more than 70 percent ofthe whole generated stator heat leads away from the bar to the core. Thereforean accurate modelling of the winding fitting system is important, the requiredbasics for the thermal characterization were given to us in a lab examination.A description of the used winding fitting systems and their handling in theFE model follows in the next paragraph.

. Elastic winding fitting system. The elastic winding fitting system (EWB)embeds the stator winding bars with a conductive elastomere. The highthermal conductivity of the elastomere in comparison with air enables abetter heat transfer. For the magnitude of the EWB coverage the differentspacer systems are decisive. So the new U-spacer system provides a highcoverage of approximately 70 percent according to the bar wide sides.


915

With the conventional spacer system only a practical coverage of25 percent was possible. For the narrow sides of the bar the EWBcoverage is about 100 percent. The manufacturing deviation of the EWBcoverage is low, therefore we use this standard EWB distribution for theFE model. The thickness of the EWB layer is also a predefined parameter(Figures 5 and 6).

. Slot side ripple spring system. This system involves elastic pre-stressedspring elements, which are inserted in the slot between the bar and core.For the FE model we use an equivalent thermal conductivity, whichcombines the air and the spring layer.

. Slot filler strip system. The filler strips are inserted between the bar andcore and the winding is fastened by impacting the slot wedge underradial pre-stress (top ripple spring). For the FE model we use anequivalent thermal conductivity which combines the air and the striplayer.

After the pre-processing phase we start with the /SOLU and SOLVE commandthe solution phase and for further evaluation of the temperature result we applythe /POST1 post-processor. Figures 5 and 6 show the meshed FE stator modelincluding the different spacer systems.

Figure 5.FE stator model withconventional spacersystem

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3. Comparison with laboratory measurementsFor the assessment of the accuracy of the FE thermal stator model, we buildtwo different laboratory stator models. The difference between the constantlyheated models is only the previously described spacer system. The wholelaboratory model consists of three ventilation ducts and four laminated coreparcels as shown in Figure 7. For the generation of the defined air flow rate wemounted the laboratory model in a suitable channel of our wind tunnel. Wemeasured the air flow rate and for the determination of the core temperature

Figure 6.FE stator model with

U-spacer system

Figure 7.Simplified laboratory

stator model


917

field we used 30 core thermocouples. Additionally, we equipped the model with30 thermocouples for the air temperature distribution in the stator ducts. Forthe comparison of the investigated variants, we heated the bars with a constantload.

The results of the comparison between the FE stator model and thelaboratory stator model are shown in Figures 8 and 9. This comparison wasrealized under the same air inlet conditions and heat generation loads.

Figure 8.Mean stator bartemperature deviation asa function of the mean airvelocity in the core toothduct

Figure 9.Mean core temperaturedeviation as a function ofthe mean air velocity inthe core tooth duct

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In Figure 8, the mean stator bar temperature deviation as a function of themean velocity in the core tooth duct is represented.

The used mean stator bar temperature is an average temperature of anupper and a lower bar according to one slot. Figure 9 shows the mean coretemperature deviation as a function of the mean air velocity in the core toothduct. The percentage of the deviation is less than 4.5 percent over the wholevelocity range. For the stator bar temperature the following described effect isessential. The manufacturing tolerance of the slot width leads to a deviation ofthe expected layer thickness (for example air/elastomere thickness) in the gapbetween the core and bar. Thereby the direct contact area varies between theside ripple spring or filler strip and core. According to Figure 10, the stator corefor the direct heat transfer (convection) to the cooling air has a big influencewith a percentage of more than 70 percent. In comparison to the core the shareof the bar and the wedge are inconsiderable.

4. Application for a hydroelectric generatorAfter the creation of the FE stator model and the comparison with a laboratorymodel we apply the FE thermal model for an accomplished hydroelectricgenerator. For a comparison, we measured the stator bar and stator coretemperatures during the heat run after commissioning. The maindeterminations for the FE calculation are listed as follows:

. identical dimensions;

. elastic winding fitting system with conventional spacer system;

Figure 10.Partitioning of the direct

heat dissipation fromcore, bar and wedge to

the cooling air as afunction of the mean air

velocity in the core toothduct


919

. consideration of the bar strands;

. consideration of the ventilation duct and air gap heat transfer; and

. consideration of the total core and bar losses.

The FE calculation results of the following stator temperature distribution(Figure 11).

The comparison of the FE calculated stator temperatures with the heat runmeasurements and the standard calculation results are shown in Figure 12.

Figure 11.Temperaturedistribution of theinvestigatedhydroelectric generator(FE model calculation)

Figure 12.Radial statortemperature distributionof the investigatedgenerator. Comparisonbetween the FEcalculation, standardcalculation andmeasurements

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For the core yoke the deviation of the calculated and measured temperatures isinconsiderable. The only larger deviation is in the interlayer. Here themeasured temperature is lower than the calculated value. For the interlayer wehave the possibility to improve our standard calculation program with theknowledge of the FE model examinations.

5. ConclusionThe electrical losses in the bar and in the core produce a temperaturedistribution in the stator of a synchronous generator. Using the FE method, wecreated a full parameterized thermal model for the calculation and optimizationof the temperature distribution of the stator. With the FE model we could reachan accurate temperature distribution of the whole stator, as the comparisonswith the laboratory model and generator heat run measurements have shown.This FE model contains the thermal effects of the core, of bar components(strand) and different winding fitting systems. They are the elastic windingfitting system, the side ripple spring system and the slot filler strip system.Therewith it is possible to calculate the thermal effects of parametermodifications and additionally we can optimize the heat transfer for the statorwith variant calculations. Therefore, a continuous improvement of ourstandard calculation programs is practicable. For the automatic generation ofthe parameterized thermal model we used the APDL. It will be possible todecrease the expensive and time-consuming laboratory measurements in thefuture and improve the accuracy of the stator temperature distribution.

Further reading

Driesen, J., Belmans, R. and Hameyer, K. (2001), “Finite element modeling of thermal contactresistances and insulation layers in electrical machines”, IEEE Transactions on IndustryApplications, Vol. 37 No. 1.

Farnleitner, E. (2000), Statorwarmeubergangsmodell: Experimenteller Vergleich vonDistanzsteganordungen und Wicklungseinbauarten, TB E 050 VA Tech Hydro.

Muller, G. and Groth, C. (2000a), “FEM fur Praktiker”, Band 1 Grundlagen, Expert Verlag,Renningen.

Muller, G. and Groth, C. (2000b), “FEM fur Praktiker”, Band 3 Temperaturfelder, Expert Verlag,Renningen.

Rechberger, K. (2000), 3D-FEM Modellrechnungen mit ANSYS zur Nachbildung desStatorerwarmungsmodells, TB E 085 VA Tech Hydro.

Verein Deutscher Ingenieure, VDI-Gesellschaft Verfahrenstechnik und Chemieingenieurwesen(1997), Berechnungsblatter fur den Warmeubergang, Springer Verlag, Berlin Heidelberg.


921

Discrete-time modeling ofAC motors for high power

AC drives controlS. Poullain, J.L. Thomas and A. Benchaib

Power Electronics Research Team, ALSTOM T&D, Massy, France

Keywords Time-domain modelling, Motors, Control

Abstract This paper proposes a new discrete-time formulation of state-space model for voltagesource inverter (VSI) fed AC motors, introducing the free evolution of the motor state andcharacterized by both the simplification of torque and flux output equations and the definition of apredictive reference frame oriented on the rotor free evolution vector. The potential of the proposedmodel for high dynamics discrete-time controller synthesis is illustrated through an application toSM-PMSM.

1. IntroductionThe AC motors (e.g. induction machine or synchronous motor) control laws,used for adjustable speed drives (ASD) industrial systems, are now generallybased on a set of discrete-time controllers implemented in real-time embeddedmicrocomputers. However, the synthesis of these controllers stays in relationwith a continuous-time model of the motor. Considering the synchronization ofthe switching frequency of the VSI actuator with the sampling period d of thedigital controllers, the inverter is equivalent in average value to a sample andhold actuator. In such a case, the torque dynamics is directly in relation withthe switching constraints introduced by the inverter. More particularly, forhigh-power AC-drives, i.e. .1 MW, where the switching frequency is stronglylimited by the power switches constraints, the conventional control approaches,such as field oriented control (FOC) (Blaschke, 1972; Leonhard, 1990; Thomas,1998; Vas, 1990), see a drastic reduction of their torque dynamics.

Then, in the particular case of high-power AC-drives, if high torquedynamics is required, it is necessary to introduce in the machine modeling thediscrete nature of the inverter by assuming a hold stator voltage vector over ad period. The PWM modulation is then considered as a simple transformationof the average stator voltage over the sampling period d in control pulsescompatible with the inverter topology. Then, the most efficient and naturalapproach to control the torque and the flux magnitude is to make controllersynthesis using discrete-time model of the AC motor fed by a stator voltagevector considered as constant over a d period.



The authors would like to express their gratitude particularly to J.C. Alacoque of ALSTOMTransport, for initiating and supporting this work in relation with specific railway applications.

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In this paper, a new discrete-time formulation of AC motors state-spacemodel is presented, introducing the free evolution of the motor state. As theproposed modeling is based on a general formulation of continuous-timePARK’s state-space model, it is relevant for any type of AC machines(i.e. induction or synchronous motors). From both free evolution formulationand specific symmetry properties of input discrete-time matrices, the proposednew discrete-time state-space model is characterized by:

(1) a simplification of the discrete-time torque and flux output equations: thequadratic nonlinear continuous-time equations are transformed intosecond order equations.

(2) the introduction of a new reference frame naturally establishing theprediction property, based on the rotor flux free evolution.

Finally, in order to highlight the potentialities of the proposed discrete-timemodeling, application to AC motor torque and flux discrete-time controllersynthesis is introduced. More particularly, the case-study of a surface-mountedpermanent magnet synchronous motor (SM-PMSM) is illustrated throughsimulation results.

2. Continuous-time model of an AC motor in the (a, b) referenceframe2.1 PARK’s equations state-space representationGiven the PARK’s equations of the motor and considering the stator currentand rotor flux components x ¼ ½isa isb fra frb�

T as state variables andthe stator voltage components u ¼ ½vsa vsb�

T as the control vector, thecontinuous-time state-space model of any AC motor can be defined as thefollowing general form:

_x ¼ AðvÞx þ Bu ð1Þ

with

AðvÞ ¼

a 2bv g xv

bv a 2xv g

l nv m 21v

2nv l 1v m

2666664

3777775

ð2Þ

and

Discrete-timemodeling ofAC motors

923

B ¼

z 0

0 z

h 0

0 h

26664

37775 ð3Þ

with v ¼ npV; where np is the pole-pair number and V is the rotor speed.The value of the different coefficients of matrices A(v) and B will depend on

the type of machine. As an example, for an induction machine, theseparameters are given by:

s ¼ 1 2L2

m

LsLr; Rsr ¼ Rs þ Rr

L2m

L2r

; Tr ¼Lr

Rr

a ¼ 2Rsr

sLs; b ¼ 0; g ¼

1

Tr

Lm

sLsLr

x ¼Lm

sLsLr; l ¼

Lm

Tr; n ¼ h ¼ 0

m ¼ 21

Tr; 1 ¼ 1; z ¼

1

sLs

where s is the leakage factor and Tr is the rotor time constant. Furthermore Rs,Rr are, respectively, the stator and rotor phase resistances, Ls, Lr are the statorand rotor phase inductances and Lm is the mutual inductance.

For a surfaced-mounted permanent magnet synchronous motor, thesecoefficients are defined as:

a ¼ 2Rs

Ls; b ¼ g ¼ l ¼ 0; x ¼

1

Ls

n ¼ m ¼ h ¼ 0 1 ¼ 1 z ¼1

Ls

where Rs is the stator phase resistance and Ls is the stator phase leakageinductance.

2.2 Output equationsAccording to torque tracking control objective of any AC motor (i.e. mechanicalenergy control), the output equation associated with equation (1) can beexpressed in the following quadratic form:

T ¼ KTðfraisb 2 frbisaÞ ¼ KTð ~fr £~isÞ ð4Þ

where KT ¼ npðLm=LrÞ.

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According to magnetic flux regulation objective (i.e. magnetic energycontrol), the output equation associated with equation (1) is dependent on thetype of AC motor.

For an induction machine, the magnetic energy can be regulated through thecontrol of the rotor flux magnitude introducing the following quadraticequation:

k ~frk2¼ ðfrafra þ frbfrbÞ ¼ ~fp · ~fp

� �ð5Þ

For a synchronous motor, the rotor flux level is applied through an additionaldevice (e.g. rotor magnet, rotor induction winding). So, its magnitude has not tobe controlled through stator voltage components. However, introducing statorvoltage limitation, the stator flux magnitude must be considered. With a rotorinduction winding synchronous motor, the stator flux magnitude can beregulated through the control of the induction winding current. For a PMSM,such a stator flux magnitude regulation is not possible. Then, a new magneticenergy quadratic output equation, which is more general than equation (5), isintroduced as follows:

W ¼ KWðfraisa þ frbisbÞ ¼ KW~fr ·~is

� �ð6Þ

where KW ¼ np for a SM-PMSM.Through equation (6), the relative position of rotor flux vector and stator

current vector can be controlled in order to modify the stator flux magnitude.

3. Discrete-time model of an AC motor in the (a, b) reference frame3.1 PARK’s equations state-space representationThe discretization of the continuous-time model (1), using the classical methodand assuming that u(k) is constant over the time period [kd, (k+1)d ], can beexpressed as:

xðk þ 1Þ ¼ FðvÞxðkÞ þ GðvÞuðkÞ ð7Þ

with

FðvÞ ¼ eAðvÞd

GðvÞ ¼ A21ðvÞ½eAðvÞd 2 I �B

(ð8Þ

Considering here the new discrete-time representation introduced by Monacoand Normand-Cyrot (1998), the discrete-time dynamics (7) can be viewed as thecomplementary contribution of both the state free evolution and the controlinput. For notation simplicity, the prediction of any vector ~x at the instant tk+1,is denoted by ~xp. The free evolution part is defined as the one-step prediction of


925

the state/outputs vectors, when the control input is kept to zero. The associatedfree evolution is called ~x0

p. Then, it yields:

~ip ¼ ~i 0p þ G1~vs

~fp ¼ ~f 0p þ G2~vs

8<: ð9Þ

with

G1 ¼al cl

2cl al

" #; G2 ¼

bl 2dl

dl bl

" #ð10Þ

where the coefficients al, bl, cl and dl result from the development order, denotedas l, for l $ 1; of the discrete-time input matrix G. The main property of theapproach proposed here is based on the invariant structure of the matricesG1 and G2. Their symmetry remains the same for all order of the seriesdevelopment used for the matrix exponential eA(v)d computation. Note that ~vs

defines the control input at the instant tk ¼ kd; constant over the time period[kd, (k+1)d ].

3.2 Output equationsThe torque objective is then associated with the following predictive outputequation (using proposed notations):

Tp ¼ KTð ~fp £ ~ipÞ ðIM-PMSMÞ ð11Þ

as the flux control objective can be related to:

k ~fpk2¼ ð ~fp · ~fpÞ ðIMÞ ð12Þ

W p ¼ KWð ~fp ·~ipÞ ðPMSMÞ ð13Þ

Introducing in equations (11)-(13), the stator current and rotor flux discrete-timedynamics (9) and after some computations using specific symmetry propertiesof the Gi matrices, the predictive output equations can be expressed as follows:

Tp ¼ T 0p þ KTð ~c

0p £ ~vsÞ2 KTðaldl 2 blclÞk~vsk

2ð14Þ

with

~c 0p ¼ ½G1�

T ~f 0p 2 ½G2�

T~i 0p ð15Þ

and

T 0p ¼ KT

~f 0p £ ~i 0

p

� �ð16Þ

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926

k ~fpk2¼ k ~f 0

pk2þ 2bl

~f 0p · ~vs

� �2 2dl

~f 0p £ ~vs

� �þ b2

l þ d2l

� �k~vsk

2ð17Þ

W p ¼ W 0p þ KW

~j 0p · ~vs

� �þ KWðalbl 2 cldlÞk~vsk

2ð18Þ

with

~j 0p ¼ ½G1�

T ~f 0p þ ½G2�

T~i 0p ð19Þ

and

W 0p ¼ KW

~f 0p ·~i 0

p

� �ð20Þ

4. Geometric interpretation4.1 Torque circle: CT

The reformulation of equation (14) gives:

DT ¼ ~c 0p £ ~vs

� �2 ðaldl þ blclÞk~vsk

2ð21Þ

with

DT ¼ ðTp 2 T0pÞ=KT

Developing equation (21), it yields:

DT ¼ c 0pavsb 2 c 0

pbvsa 2 ðaldl þ blclÞ v2sa þ v2

sb

� �ð22Þ

Equation (22) represents a conic equation which can be interpreted as a “statorvoltage circle” associated with the torque equation in the (a, b) reference frame.For notation simplicity, it is called Torque circle: CT.

Given lT ¼ 2ðaldl þ blclÞ; the center coordinates, denoted as (mTa,mTb), ofthe circle CT are defined as follows:

~M 0Tp

¼1

2lT

2c0pb

c0pa

24

35 U

mTa

mTb

" #

Then, the circle equation for voltage locus relative to the torque control isgiven by:

ðvsa 2 mTaÞ2 þ ðvsb 2 mTbÞ

2 ¼DT

lTþ k ~M 0

Tpk

2¼ R2

T ð23Þ

where RT defines the radius of the circle CT.


927

4.2 Flux circle: CF

Applying a similar approach to equation (17) or (18), a “stator voltage circle”(called Flux circle: CF) associated with the flux equation in the (a, b) referenceframe can be defined as follows:

ðvsa 2 mFaÞ2 þ ðvsb 2 mFbÞ

2 ¼DF

lFþ k ~M 0

Fpk2¼ R 2

F ð24Þ

where RF defines the radius of the circle CF.For an induction machine, associated with flux output equation (17), the flux

circle parameters are given by:

lF ¼ b2l þ d2

l

DF ¼ k ~fpk22 k ~f 0

pk2

~M 0Fp ¼

1

lF

2blFp0a 2 dlFp0b

dlFp0a 2 dlFp0b

" #U

mFa

mFb

" #

In the particular case of a PMSM, associated with flux output equation (18), theflux circle parameters are given by:

lF ¼ albl 2 cldl

DF ¼ ðW p 2 W 0pÞ=KW

~M 0Fp ¼

21

2lF

j 0pa

j 0pb

24

35 U

mFa

mFb

" #

4.3 Application to discrete-time (a,b) controller synthesisIn this section, the synthesis of torque and flux dead-beat controllers is brieflypresented in order to highlight the potential of the proposed discrete-timemodeling of an AC machine. The desired objective for the controller synthesisis to guarantee in one d period the torque set-point value T*

p and the fluxset-point value F*

p (or W *p). In a first step, a general solution is presented. Then,

the particular case-study of the SM-PMSM is detailed. Results aboutapplication to induction machine are given by Thomas and Poullain (2000a).

4.3.1 General solution. Considering torque and flux control objectives andthen introducing torque circle CT and flux circle CF, the general solution is

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obtained by solving the following system of equations in order to determine thevsa and vsb components:

ðvsa 2 mFaÞ2 þ ðvsb 2 mFbÞ

2 ¼ DFlF

þ k ~M0

Fpk2

ðvsa 2 mTaÞ2 þ ðvsb 2 mTbÞ

2 ¼ DTlT

þ k ~M0

Tpk2

8><>: ð25Þ

Given ST ¼ DT=lT and SF ¼ Df=lF; the torque and flux equations ofsystem (25) can be arranged as follows:

N 2v2sa þ N 1vsa þ N 0 ¼ 0

P1vsa þ P0 ¼ vsb

8<: ð26Þ

where

P1 ¼2ðmTa 2 mFaÞ

ðmTb 2 mFbÞ; P0 ¼

ðSF 2 STÞ

2ðmTb 2 mFbÞ;

N 2 ¼ 1 þ P21; N 0 ¼ P2

0 2 2mFbP0 2 SF;

N 1 ¼ 2P1P0 2 2mF2bP1 2 2mFa

From equation (26), two solutions are mathematically admissible. Then theeffective control, i.e. applied to the motor, can be defined as the solution whichsatisfies the minimal energy condition, obtained here for the minimum modulusof ~vs. In the case of an induction motor used as traction power train fortramway applications, Figure 1 shows the general solution, illustrated by theright side intersection of the two circles, based on motor parameters given inAppendix A and for a given speed.

4.3.2 SM-PMSM case-study. In the case of a SM-PMSM, the coefficients bl, cl,dl of discrete-time input matrices Gi are equal to zero, resulting from b ¼ g ¼l ¼ n ¼ m ¼ h ¼ 0 in continuous-time matrices A(v) and B. As aconsequence, considering equations (14), (15), (18) and (19), the two circleequations of the system (25) are reduced to two straight lines as follows:

al F0pavsa þF0

pbvsb

� �¼ DF

al F0pavsb 2F0

pbvsa

� �¼ DT

8>><>>:

ð27Þ

Then, the vsa and vsb component solution, defining a dead-beat controller, isdirectly given by the inversion of the system (27):


929

vsa

vsb

" #¼

1

k ~f0

pk2

F0pa 2F0

pb

F0pb F0

pa

24

35 Df=al

DT=al

" #ð28Þ

Defining

cosðr0pÞ ¼

F0pa

kf0pk

sinðr0pÞ ¼

F0pb

kf0pk

8>>>>>><>>>>>>:

the equation (28) can be reformulated as follows:

vsa

vsb

" #¼

1

k ~f 0pk

R r0p

� �21 Df=al

DT=al

" #ð29Þ

with the matrix R r0p

� �21

defined as:

Figure 1.Flux/torque circles for aninduction motor

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930

R r0p

� �21

¼

cos r0p 2sin r0

p

sin r0p cos r0

p

24

35

The system (29) highlights a particular rotation matrix, denoted by R�r0

p

�,

oriented on the angular position r0p of the free evolution ~f 0

p of the rotor fluxvector ~fp over the sampling period d.

5. New discrete-time (d,q) reference frame5.1 Discrete-time (d,q) reference frameThe presence of the rotation matrix R r0

p

� �in system (29) is used here to

introduce a new discrete-time (d,q) rotating reference frame, oriented on the

angular position r0p as shown in Figure 2, resulting in f0

p~d¼ k ~f 0

pk and f0p~q ¼ 0.

This new (d,q) reference frame can be defined as a predictive reference frame,offering new potentialities for high-dynamics controller synthesis for ACmotors in a discrete-time context. It can be compared with the standard (d,q)rotating reference frame, shown in Figure 3, associated with the classicalcontinuous-time field-oriented control (FOC).

Figure 2.Discrete-time (d, q)

reference frame

Figure 3.Continuous-time (d, q)

reference frame


931

Furthermore, for a SM-PMSM, as ~f 0p ¼ ~fp, the angular position r0

p coincideswith the angular position rp. Then, the (d,q) reference frame is a directtransposition in a discrete-time context of the classical (d,q) reference frame.

5.2 Discrete-time (d,q) controller synthesis for a SM-PMSMThe interest of the new (d,q) reference frame for discrete-time controllersynthesis is highlighted here through an application to SM-PMSM. Applicationto an induction machine with associated simulation results are presented anddiscussed by Thomas and Poullain (2002a).

5.2.1 Discrete-time (d,q) controller synthesis. Equation (29) becomes:

R r0p

� � vsa

vsb

" #¼

1

k ~f 0pk

Df=al

DT=al

" #ð30Þ

Defining

vs ~d

vs~q

" #¼ R r0

p

� � vsa

vsb

" #ð31Þ

and introducing equation (31) in equation (30), it yields:

vs ~d

vs~q

" #¼

1

k ~f 0pk

Df=al

DT=al

" #ð32Þ

Furthermore, reformulating DF¼ W p2W 0p

� �=KW and DT¼ Tp2T0

p

� �=KT

in the (d,q) reference frame, it comes:

DF

DT

" #¼

Fp ~dip ~d2F0p~d

i 0p~d

Fp ~dip~q2F0p~d

i 0p~q

264

375 ð33Þ

Then, considering that for a SM-PMSM:

Fp ~d¼F0p ~d¼k ~f0

pk

it comes:

DF

DT

" #¼k ~f0

pk

ip ~d2 i 0p ~d

ip~q2 i 0p~q

264

375 ð34Þ

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932

Introducing equation (34) in equation (30), the discrete-time (d, q) controller isgiven by:

vs ~d

vs~q

" #¼

1

al

ip ~d2 i0p ~d

ip~q2 i0p~q

264

375 ð35Þ

This controller can be interpreted as a fully decoupled dead-beat controllerexhibiting a very simple structure. Then, the dead-beat control of torqueset-point Tp* is performed, through ip~q¼ i*p~q, using the vsq component of statorvoltage as the stator flux level is regulated, through ip ~d¼ i*

p~d, using the vsd

component of stator voltage.5.2.2 Simulation results. In order to illustrate the high dynamics

performance of the torque dead-beat controller proposed in equation (35),Figure 4 shows the torque response consecutive to a2TNom to TNom torqueset-point inversion, exhibiting the one sample d torque response. The motorparameters used for the simulation results presented here are given in theAppendix and the operating conditions are defined as follows:

d ¼ 1 ms V ¼ 2Q5 rad=s i*p ~d

¼ 0 A

Figure 4.Dead-beat 2TNom to

TNom torque inversion


933

Figure 5 presents the stator current (d, q) components evolution, showing onesample d inversion of the ipq component as well as the regulation of the ipd

component to a zero set-point.Figure 6 shows the corresponding stator current (a,b) components

evolution, exhibiting the one sample phase shift of both current componentswhen torque inversion occurs.

6. ConclusionsIntroducing the free evolution principle, a new formulation of the discrete-timePARK’s model of any AC motors is presented in this paper. Regarding classicaldiscrete-time modeling, the proposed model introduces simple second orderoutput equations, well-adapted for high dynamics torque and flux controllersynthesis.

From the proposed formulation, a full analogy with the classical fieldoriented concept can be made by introducing a predictive (d, q) reference frame,offering new potentialities for discrete-time controller synthesis.

Furthermore, the proposed approach highlights the interest of using ageometric representation for analysis and synthesis of discrete-time controllers.Then, the geometric representation appears to be a well-adapted powerful toolfor a natural introduction of voltage and/or current limitations.

Moreover, all the presented torque/flux controller synthesis are based on aperfect knowledge of flux components. Obviously, the proposed discrete-time

Figure 5.Stator current (d, q)components evolution

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934

model can be used in the predictive part of an associated flux observer definedin the (a,b) reference frame. In this particular case, the formulation in the (d, q)reference frame is not pertinent.

In order to get full benefit from the proposed discrete-time modeling of ACmotors, voltage and/or current limitations has to be considered and analyzedon the basis of the geometric representation. Then, the synthesis of highdynamics torque and flux discrete-time controllers could be performed,naturally introducing voltage and current limitation constraints.

Furthermore, applying the approach presented here to salient polespermanent magnet synchronous motor would complete the range of applicationto AC machines.

The discrete-time controllers presented in this paper for both inductionmachine and SM-PMSM are protected by patent applications (Thomas andPoullain, 2000b; Thomas et al., 2002).

References

Blaschke, F. (1972), “The principle of field orientation as applied to the new transvectorclosed-loop control system for rotating filed machines”, Siemens Rev., Vol. 39, pp. 217-20.

Leonhard, W. (1990), Control of Electrical Drives, Springer-Verlag, Berlin.

Monaco, S. and Normand-Cyrot, D. (1998). Discrete-time state representations, a new paradigm, ofPerspectives in Control: Theory and Applications, Chapter 4, Springer, Berlin, Vol 1,pp. 191-203.

Figure 6.Stator current (a,b)

components evolution


935

Thomas, J.L. (1998), “Future practical developments in vector control principles”, IEEColloquium, Savoy Place, London.

Thomas, J.L., Alacoque, J.C., Poullain, S. and Benchaib, A. (2002), Procede et dispositif decommande de regulation d’urne machine electrique tournante courant alternatif, enparticulier synchrone, French Patent 0102012.

Thomas, J.L. and Poullain, S. (2000a), Discrete-time field-oriented control for induction motors,31st Annual Power Electronics Specialist Conference, Galway, Ireland.

Thomas, J.L. and Poullain, S., (2000b), Procede de regulation d’une machine tournante etdispositif d’alimentation d’une telle machine, European Patent 1045514A1.

Vas, P. (1990), Vector Control of AC Machines, Oxford University Press, Oxford.

Appendix. Motors parametersInduction motor

Rs ¼ 21:4 mV Ls ¼ 8:34 mH

Rr ¼ 19:2 mV Lr ¼ 8:24 mH

np ¼ 2 Lm ¼ 8:00 mH

SM-PMSMRs ¼ 20:0 mV Ls ¼ 0:75 mH

np ¼ 4 k ~fpk ¼ 0:736 Wb

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936

Design of a mass-productionlow-cost claw-pole motor for an

automotive applicationR. Felicetti and I. Ramesohl

Robert Bosch GmbH, Division: Body Electronics,Technical Centre for Electronic Commutated Motors, Buhl, Germany

Keywords Rotors, Motors, Electrical machines, Mechatronics

Abstract This paper describes a thermal and electrical model, used at Robert Bosch GmbH for thedesign of an innovative motor for a water-pump. In addition, it offers an example of a highlyintegrated mechatronic system. A bonded-ferrite inner rotor has been developed with an integratedfront centrifugal impeller which is driven by the magnetic interaction of a rotating field created byclaw-poles. The two phase unipolar coil arrangement is fed by an internal circuit using twoMOSFETS controlled by the commutation signal from a bipolar Hall-IC. This is the firstmass-production example of an electrical machine for an automotive application where the clawpole topology is used to realise the armature of the motor (i.e. the rotating field) and not theexcitation field.

1. IntroductionThis paper shows several design and calculation tools for a special claw-polemotor.

First, the construction of the motor, with some peculiarities, is presented.The cost-aspects and ambient conditions as an automobile application have tobe taken into consideration. The most important ambient condition is thetemperature of the fluid in which the rotor (Figure 1) runs which is about 1208Cand the temperature of the surrounding air is more than 1008C.

A 3D magnetic field calculation of the claw-pole arrangement is shown inFigure 2 to provide informations about saturation effects in the claws, themagnetic flux density distribution and armature reaction effect.

In Section 2, the electrical principle of the motor is described and in Section 3a novel method of characterising electrical machines in terms of their thermalbehaviour is presented. The methods described are transferable to any type ofelectrical machine and gives the electrical machine designer a fast andcomfortable way to optimise the winding and active magnetic componentsthrough the definition of the electrical loading and current density. The methodpresented, in this paper, has the advantage of providing a good overview of thethermal behaviour of a machine with varying electric load, current density andmain geometrical dimensions. The use of this method is very efficient in thedevelopment stage because it is fast and reliable. For fine optimization, thecoefficients of the model can be adapted through the use of FE-tools or withmeasurement adapted R-C-meshes.



Design of a motor

937


Vol. 22 No. 4, 2003pp. 937-952


DOI 10.1108/03321640310482913

Figure 1.Auxiliary water-pumpmotor. (a) Magneticactive components(prototype), (b) explodedview of all components ofthe pump

Figure 2.No-load flux densitydistribution of half of themagnet and claw polestructure

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938

Section 4 presents an analytically based electromagnetic model of the motorwith a detailed description on how to calculate all relevant components of theequivalent electric circuit.

Thermal and electrical results are compared with measurements andpresented in Section 5. A further section explains the parameter-calculation.

2. The claw-pole motor and driveThe usual application of the claw-pole topology is the production of anexcitation field in synchronous motors and alternators.

The claw-pole arrangement considered here is used to generate the armaturereaction field of the motor (Figure 2). A bonded-ferrite inner rotor magnetisedwith eight poles provides the excitation field. In order to avoid corrosion, nointernal rotor iron yoke was allowed. Therefore, a lateral magnetisation was thebest practical solution. At standstill, the rotor is positioned with its q-axisaligned with the symmetry axis of the eight armature claw-poles. This is thenatural tendency of the rotor to rest in the position of minimum magneticreluctance. The cogging torque of this machine is a helpful functionalphenomenon because it aids the starting torque. The armature field is providedby two bobbin coils, which are positioned in the internal toroidal space of theiron core. This produces good magnetic mutual coupling. The coils areswitched to the battery voltage, depending on the position of the rotor, by twoMOSFETS. The circuit is shown in Figure 3(a). The Hall-IC senses the rotorposition and switches the phases.

The current flows from the battery through the switched phase for a rotorangle of 180 electrical degrees. This generates the electromagnetic torque.To complete one electrical revolution, the first phase is then switched-off andthe other switched-on.

This switching strategy is favoured due to the good magnetic couplingbetween the two phases. The rapid decrease of the current observed in thefirst phase of this topology (Figure 3(b)) is not accomplished by thedissipation of the leakage field energy through the FETs, but through itstransfer from one phase to the other by a good magnetic coupling. Due to this,no significant transient time exists in the commutation. Poor couplingbetween the two coils ðM ! LÞ would lead to high avalanche losses in theFETs. The mechanical design and the quality of the winding processdetermines the coupling factor.

3. Pre-sizing and thermal designTo calculate the size of the motor, depending on the required torque, thefollowing relationship can be applied:

Tem ¼ KBavAR2int

L ð1Þ

Design of a motor

939

where

Tem¼ electromagnetic torque (N m)

K ¼ motor constant (-)

Bav¼ average air gap induction (T)

A ¼ electric loading (A/m)

Rint¼ air gap radius (m)

L ¼ stator length (m)

The maximum allowed electric loading A leads to the smallest possible motorsize and is limited thermally by the winding insulation and permanentmagnets. If the specific machine iron-losses cFe (W/kg) and the permitted

Figure 3.Motor model andsimulated current.

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940

copper over-temperature are fixed in advance, it is possible to predict the valueof A from the following equation:

AðSÞ ¼a*

air ·DTair þ a*water ·DTwater

Rint

Rext

� �KCu

KRKfS

rKbKCuS 2 þ cFegFeKFeð2Þ

where

S¼ current density (A/m2)

r¼ specific electric copper resistance (Vm)

a*air; a

*water ¼ convection/radiation factors (W/m2K)

Kb¼ duty factor (per cent)

KCu¼ copper fill factor (per cent)

KFe¼ iron-copper volume ratio (per cent)

cFe¼ specific iron losses (W/kg)

gFe¼ specific iron mass (kg/m3)

KR¼ radial factor (-)

Kf¼ current form factor (-)

With reference to Figures 4 and 5, it is possible to explain the meaning of eachparameter.

The electric loading and the current density are so defined:

A ¼pWI average

pRintð3Þ

Figure 4.The stator radial

structure

Design of a motor

941

S ¼W I rms

LH KCuð4Þ

where

p¼ pole pairs (-)

W ¼ number of turns/phase (-)

Iaverage ¼ mean current value over a period (A)

Irms¼ effective phase current (A)

L ¼ stator axial length (m)

H ¼ stator radial height (m)

The copper fill and the iron factors are calculated using the followingequations:

KCu ¼2Wp

f2Cu

4

LHð5Þ

KFe ¼AFe

LH¼

ðL þ H 2 2tsÞ

LHð6Þ

where fCu is the copper wire diameter (m), AFe is the iron cross-sectional area(m2), and ts is the steel lamination thickness (m).

It is also possible to describe the heat sources in the motor (copper and ironlosses) by the following equations:

PCu ¼ LHKCu2pRmrS 2KB ð7Þ

Figure 5.The stator cross-section

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942

PFe ¼ cFegFeKFeLH2pRm ð8Þ

where Rm is the average radius of the winding

Rm ¼Rint þ Rout

2ð9Þ

The ratio between A and S is the dimensional length (Vogt, 1996):

A

S¼

KCu

Kf

LH

t, to a length ðmÞ ð10Þ

where t is the tooth pitch (m) and

Kf ¼I rms

I averageð11Þ

It is assumed that the heat-rate flows from the stator to the ambient and waterin the radial direction only.

Depending on the enclosed construction of the motor, the lateral surfaces ofthe stator can be assumed adiabatic. With this assumption, it is possible toexpress the heat transfer coefficients a*

airðW=8C m2Þ and a*waterðW=8C m2Þ

(depending on the stator geometry and thermal properties of the materials andneglecting the temperature drop on the metal parts) as:

a*air ¼

1

aairþ

dout

lairþ

dout

lplastic

� �21

ð12Þ

a*water ¼

1

awaterþ

dint

lairþ

dint

lplastic

� �21

ð13Þ

where

aair¼ convection + radiation air coefficient (W/8C m2)

awater ¼ convection + radiation water coefficient (W/8C m2)

lair¼ thermal air conductivity (W/8C m)

lplastic ¼ thermal plastic conductivity (W/8C m)

At steady state, the power losses are equal to the total heat flow through thestator:

PCu þ PFe ¼ a*air2pRoutL · DTair þ a*

water2pRintL · DTwater ð14Þ

where DTair is the copper over-temperature referred to ambient (8C), andDTwater is the copper over-temperature referred to water (8C)

Design of a motor

943

By substituting equations (7) and (8) in equation (14), taking intoconsideration the relationships (3), (4) and (10), it is possible to deriveequation (2).

It produces the A(S ) curve shown in Figure 6. From this curve, themaximum value of the electric loading Amax and the related current density Scan be determined, which relates to the permitted copper over-temperature.This curve has several properties. The most important is the ratio between Aand S (the slope of a straight through the origin) which is proportional to theslot depth of the motor H (m).

It is possible to express H as a function of S which is represented by h(S ).This curve is also shown in Figure 6 and it can be calculated using thefollowing equation:

hðSÞ ¼AðSÞ

S

Kf

KCuð15Þ

This curve has a maximum value of hth which is named the thermal slot depthof the machine under the specified thermal conditions.

hth ¼ hðSÞjS¼0 ¼a*

air ·DTair þ a*water ·DTwater · Rint

Rout

� �

cFegFeKFeKRð16Þ

This is the maximum permitted radial height of the stator, which is compatiblewith the assumed copper over-temperatures.

Figure 6.A and h characteristicsagainst S

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944

It is interesting to note that the slot depth of the machine at Amax is half ofthe thermal slot depth. It is possible to demonstrate that:

hAmax¼ h SjdA

dS¼0

� �¼

hth

2ð17Þ

These considerations help to formulate the first design of motor by choosingthe minimum rotor volume from equation (1) at Amax and assuming a radialheight of the stator equal to half of its thermal slot depth hth.

4. Electromagnetic modelThe model used for a machine phase winding is an easy model due to thefollowing reasons (Figure 7):

(1) the mutual inductance is involved only during phase commutation,

(2) under normal operating conditions, the iron is not saturated, and

(3) the back e.m.f. is known from the FEM calculation.

During the time period when only one phase is switched to the battery voltage,the electrical and the mechanical conditions can be described by the followingequations:

UDCðtÞ2 R iðtÞ2 LsðtÞdiðtÞdt 2 eðtÞ ¼ 0

TemðtÞ þ T loadðtÞ þ T lossesðtÞ þ TcoggingðtÞ þ I dVðtÞdt

¼ 0

8<: ð18Þ

and

Figure 7.Model for one phase of

the machine

Design of a motor

945

eðtÞ ¼PNi¼1

2 ›CiðtÞdt

¼ 2pWVðtÞPNi¼1

›FiðuðtÞÞ›uðtÞ

TemðtÞ ¼

PNi¼1

eiðtÞiðtÞ

VðtÞ¼ 2pW

PNi¼1

›FiðuðtÞÞ›uðtÞ

iðtÞ

8>>>>>><>>>>>>:

ð19Þ

where

UDC¼ supply voltage (V)

R ¼ phase resistance (V)

Ls¼ leakage inductance (H)

I ¼ moment of inertia (kg m2)

V¼ angular speed (rad/s)

Fi¼ ith harmonic of the flux (Vs)

u ¼ electric angular rotor position (rad)

Tem¼ electromagnetic torque (N m)

Tload ¼ mechanical load (N m)

Tlosses¼ torque due to the magnetic and mechanical losses (N m)

The solution of these differential equations for the steady-state is possible byan analytical closed form calculation, if it is assumed as a nearly constantangular rotor speed V, which agrees well with the reality of the motor (motorspeed ripple ,4 per cent). In this case, equations (18) and (19) can be written inthe following way:

UDCðtÞ2 RiðtÞ2 pVLsdiðuÞdu 2 eðuÞ ¼ 0

�Tem þ �Tload þ �Tlosses ¼ 0

8<: ð20Þ

and

eðuÞ ¼ 2pWVPNi¼1

›FiðuÞ›u

�Tem ¼

1T

R T

0

PNi¼1

eiðtÞiðtÞ dt

V¼ 2pW 1

p

Z p

0

PNi¼1

›FiðuÞ›u

iðuÞ du

8>>>>><>>>>>:

ð21Þ

where �Tx is average value over a period of the x-torque (N m).For the ith flux harmonic, the following expression is assumed:

FiðuÞ ¼ fi sinðiuþ giÞ ð22Þ

and for the ith e.m.f. harmonic:

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946

eiðuÞ ¼ 2pWVifi sin iuþ gi þp

2

� �ð23Þ

Assuming that the reference rotor angular position is that at the switch-on of aphase, it is possible to operate a variable substitution:

u ¼ b2 a2 g1 2p

2ð24Þ

where a is the commutation angle (phase advance switching) (rad).Under this assumption, the phase current is equal to the sum of the two

b-functions, the transient response it(b) and the steady-state response ip(b),respectively:

itðbÞ ¼ Kða;VÞe2

b

pVL

R ð25Þ

ipðbÞ ¼U

R2pWV

XN

i¼1

ifi

ziðVÞsin iðb2a2g1Þ þgi 2 ði21Þ

p

22wiðVÞ

h ið26Þ

where

ziðVÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR 2 þðipVLÞ2

qð27Þ

wiðVÞ ¼ a tan ipVL

R

� �ð28Þ

The phase current over an half period is:

iðbÞ ¼ itðbÞþ ipðbÞ ð29Þ

To find the specific solution of this general integral, it is necessary to fix theinitial condition of i(b).

This can be achieved through energy considerations. Depending on the goodvalue of the magnetic coupling factor ðk ø 1Þ during the commutation almostall of the energy of the leakage field is transferred from one phase to the other.It can be assumed that:

i1ðpÞ ø 2i2ð0Þ ¼ 2i1ð0Þ ð30Þ

or

iðpÞ ø 2ið0Þ ð31Þ

Applying this condition, it is possible to obtain the value of the unknownconstant K(a,V):

Design of a motor

947

Kða;VÞ ¼1

1 þ e2 p

pVLR

22U

Rþ pWV

XN

i¼1

ifi

ziðVÞ

(::

£ sin 2iðaþ g1Þ þ gi 2 ði 2 1Þp

22 wiðVÞ

h ih

þ sin iðp2 a2 g1Þh

þgi þ ði 2 1Þp

22 wiðVÞ

ii)ð32Þ

In Figure 8, the results of a link current simulation for the claw-pole motor ata speed of 3,170 rpm are shown.

5. ConclusionsThe prototype of the claw-pole motor was calculated with the describedanalytical tools. The validity of the thermal analysis tool was confirmed bymeasurements carried out on the prototype auxiliary water-pump. Themeasured copper, iron, ambient and water temperatures are shown in Figure 9.

The measured copper over-temperatures, with respect to the water andambient temperatures given in Figure 9, are very near to the predicted values(Table I):

The analytical model can also simulate with good accuracy theelectromechanical behaviour of the machine. Figure 10 shows the measuredcurrent from the claw-pole motor at a speed of 3,170 rpm. Channel 1 (Ch1) ofFigure 10 shows the shape of the link-current waveform, which agrees wellwith the simulated current waveform given in Figure 8. Channel 2 (Ch2) shows

Figure 8.Link current simulation

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Figure 9.Results from the thermal

measurements

Figure 10.Motor link current

Predicted (8C) Measured (8C)

20 2340 42 Table I.

Design of a motor

949

the measured current (voltage across a precision shunt). The rms values of thesimulated and measured currents agree very well: simulated 1.12 Arms;measured 1.14 Arms.

6. Analytical calculations6.1 The electrical resistance of a phaseIn the area A of Figure 5, ð2 £ W Þ turns of circular wire with diameter f andaverage length is equal to:

la ¼ 2pðRint þ ts þ tpÞ þ ðRout 2 ts 2 DÞ

2ð33Þ

The resistance of a phase is then:

RðW ;TÞ ¼ r0½1 þ a0ðT 2 T0Þ�la

pf 2

4

W

¼ r0½1 þ a0ðT 2 T0Þ�2la

KCuLHW 2

ð34Þ

6.2 The inductance of a motor phaseThe inductance of a phase consists generally of the main self inductance, Lm

and the leakage inductance, Ls. In this motor, it is impossible to find apreferable magnetic path for the main stator flux into the rotor with theabsence of a rotor yoke. It is therefore, possible to consider the stator flux, in thedirection of the rotor, as a part of the leakage flux. The inductance of a phaseconsists then only of the leakage inductances. It is possible to determine threedifferent leakage inductance components.

(1) The leakage inductance in the slot. This component is dependent on themagnetic energy in the winding volume, Lss

. Its calculation is based onthe following equation, which terms the magnetic energy in the windingslot when a phase is excited:

1

2Lss

I 2 ¼

ZV s

1

2m0H

2sðI Þ dV s ð35Þ

It is possible, using equation (33) to demonstrate (Richter, 1967) that:

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Lss¼ 2pm0

W 2

L 2 2ts

H 2c

4þ

H cðRint þ tp þ tsÞ

3

"

þD2

2þ D · ðH c þ Rint þ tp þ tsÞ

#:

¼ m0lssLW 2

ð36Þ

(2) The leakage inductance between the claws. The simplest way to calculatethis leakage inductance Lst

is by using the reluctance between the claws(Figure 11)

Ls t¼

W 2

R¼ NCm0ts

L

d

W 2

cos2ðgÞ¼ m0ls t

LW 2 ð37Þ

(3) The leakage inductance due to the tooth leakage flux. The calculationof Lsth

can be approximated by the use of the principle of magneticsymmetry and Carter’s theory for the air-gap field (Pestarini, 1943). Anadditional condition is required: adjacent claws should quasi lay on aplanar surface. Due to the high number of claws in this machine it ispossible to match this condition quite well (Figure 12).

It results that:

Lsth¼ m0lsth

LW 2 ð38Þ

where

lsthø

5

4ð39Þ

The total leakage inductance of a phase is then:

Ls ¼ Lssþ Lst

þ Lsthð40Þ

Figure 11.Top view of the claws

Design of a motor

951

References

Pestarini, G.M. (1943), Elettromeccanica, Edizioni Italiane, Rome, Vol. 1, pp. 55-7.

Richter, R. (1967), Elektrische Maschinen, Birkhauser Verlag, Basel-Stuttgart, Vol. 1, pp. 265-78.

Vogt, K. (1996), Berechnung Elektrischer Maschinen, VCH-Verlag, Weinheim, p. 434.

Further reading

Smolenski, I. (1980), Electrical Machines, MIR publisher, Moscow, Vol. 2, pp. 34-41.

Figure 12.The symmetry principle

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A magnetic network approachto the transient analysis of

synchronous machinesM. Andriollo, T. Bertoncelli and A. Di Gerlando

Department of Electrical Engineering – Politecnico di Milano,Milano, Italy

Keywords Magnetic forces, Synchronous machine, Simulation

Abstract The technique for the simulation of the dynamic behaviour of rotating machinespresented in the paper is based on an equivalent circuit representation of the magneticconfiguration. The circuit parameters are obtained by a preliminary automated sequence ofmagnetostatic FEM analyses and take into account the local magnetic saturations. The adoptedsolution technique is based on an invariant network topology approach: its application, presentedfor the operation analysis of a low-power synchronous generator, allows a great reduction of thecalculation time in comparison with a commercial FEM code for the transient simulation.

IntroductionFor many years, the most commonly used tool in advanced design and study ofelectromagnetic devices has been represented by the electromagnetic FEManalysis (Kunze et al., 1991; Nabeta et al., 1996; Preston and Sturgess, 1993;Schmidt et al., 2000, 2001).

More recent implementations of the FEM-based codes allow the transientsimulation, taking into account the dependence of both the sources and thegeometrical configuration on time. Nevertheless, the steep application of suchcodes does not allow a deep insight into the electromagnetic behaviour andgenerally results in high calculation times, even if a single configuration has tobe analysed under various operative conditions. Getting worse, theperformance assessment in consideration of parametric variations istime-consuming in proportion to the number of configurations to be examined.

Low-power synchronous generators emphasize such problems, due to therelevant “cross-coupling” effect between the d and q axes m.m.f.s caused by thehigh local saturation in the pole shoe zone.

The alternative approach, presented in this paper, is based on therepresentation of the synchronous machines by an equivalent magnetic circuit(Andriollo et al., 2001), whose various elements accurately characterise thebehaviour of different zones by suitable magnetic permeances (reluctances) andm.m.f.s. Such method, implemented in a code, allows the fast and accurate



Work financed by the Italian National Ministry of Education, University and Research (MIUR),Cofin 1999, Title: “Electromagnetic analysis, modelling and design optimisation of low-powersynchronous generators”.

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Vol. 22 No. 4, 2003pp. 953-968


DOI 10.1108/03321640310482922

calculation of the winding flux linkages, given the rotor position and thewinding currents.

An effective procedure for the transient analysis was then obtained,integrating such code in a step-by-step procedure for the numerical integrationof the voltage differential equations. Several integration techniques wereconsidered, and their performances from the numerical viewpoint wereinvestigated.

In the examples of application, the results obtained by the code based on theproposed technique are compared with the ones related to a commercial FEMcode for the electromagnetic transient analysis (Ansoft Maxwell 2D TransientCode, v.8.0.22, 2001).

Determination of the flux linkagesTo illustrate and test the method, a 2D typical configuration is considered(Figure 1(a)), related to a 2-poles 3-phase synchronous generator with 24 statorslots, with single-layer windings.

It exhibits all the local saturation phenomena, particularly relevant in thelow power machines; in order to evidence the cogging effects due to the statorteeth, an open slot configuration was considered, even if it is not used in actualmachines. Various circuit patterns were examined, related to differentgeometrical subdivisions; a good agreement with the results of thecorresponding FEM analyses was obtained by the circuit representation ofFigure 1(b), related to the partition of Figure 1(a) (Andriollo et al., 2001).

The nodes si, ci (i ¼ 0; . . .; ns 2 1 with ns ¼ 24 number of stator slots) and rj

( j ¼ 0; . . .; nr 2 1 with nr ¼ 8 divisions of the rotor surface) identify,respectively, stator teeth, stator yoke radial sections and rotor boundaries(configuration symmetry was deliberately ignored, to demonstrate theprocedure applicability also to unsymmetrical structures).

Parameters of the magnetic circuitThe configuration is subdivided into different magnetic zones, the permeanceof which (or reluctance) is represented alternatively as:

. a linear parameter, unaffected by the load condition, but generallydependent on the stator-rotor relative position, for the magnetic paths inair (air-gap and leakage);

. a flux-dependent parameter, for paths mainly developing insideferromagnetic branches, and therefore approximately restricted in atime-independent geometrical structure, but affected by the magneticsaturation.

The air-gap permeances are distinguished in:. “mutual” air-gap permeances (li, j), related to the fluxes flowing between

the ith stator tooth edge and the jth rotor boundary edge;

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. air-gap leakage permeances (l‘s,i and l‘r,i), associated to the fluxesbetween ith and ði þ 1Þth stator teeth and between jth and ð j þ 1Þth rotorboundaries, respectively.

An automated sequence of FEM analyses allows to calculate the air-gappermeances. In order to perform these evaluations, an infinite magneticpermeability mFe of the iron core must be provisionally assumed: the air-gappermeances are obtained as the ratio between the fluxes (flowing between theinvolved zone and the adjacent ferromagnetic surfaces) and the difference ofthe magnetic potential impressed by means of suitable probe sources. Theinterpolation of the results obtained by such analyses sequence allows to definethe various permeances as functions of the rotor position a (Figure 2(a)).

According to such scheme, each stator tooth is ideally coupled with eachrotor zone and vice versa for every position a (in practice, the related

Figure 1.Synchronous machine

structure and model: (a)machine structureconsidered for the

analysis (dashed lines:boundaries of the

ferromagnetic branches);(b) equivalent magnetic

network (only somerepresentative elementsare shown and labelled)

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permeance function rapidly decays as the distance between the two zonesincreases). Hence, the network structure, albeit non-planar, is time-invariant,the dependence on a being assigned to the permeance values: this choicegreatly simplifies the analysis, because no topological update is needed duringrotation.

Also the iron core reluctances are evaluated via a suitable sequence ofautomated FEM analyses (Andriollo et al., 2001):

. the flux w in each considered branch is evaluated by a non-linear FEA,applying suited probe m.m.f. sources, taking into account the proper

Figure 2.(a) Air-gap permeancesas functions of the rotorposition a; (b) iron corereluctances as functionsof the proper branchfluxes (uts: stator teeth;uys: stator yoke sections;utr1,3: polar shoeexpansions; up: polarbodies)

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non-linear B(H) relationship for that ferromagnetic branch and assumingmFe !1 in the rest of the iron core;

. the air-gap magnetic voltage drop (MVD) can be determined byevaluating the corresponding air-gap permeance, as previously explained;

. deducting the air-gap MVD contribution from the total MVD, the drop inthe ferromagnetic branch and the corresponding reluctance u iscalculated;

. the previous steps are repeated for different values of the probe m.m.f.sources;

. the interpolation of {u,w) values gives the non-linear function u(w)(Figure 2(b)).

Non-linear magnetic circuit solution algorithmWhile some solvers were previously developed, just for the analysis of no-loadconditions (Di Gerlando et al., 1995), the following method described leads todirect and general formulations; its integration in an efficient code for transientanalysis allows good performances from the point of view of both the accuracyand speed of calculation. The solution of the magnetic network is based onexplicit recursive formulations of the magnetic potentials of the nodes si, ci andsubsequent expressions of the fluxes on iron core branches.

With reference to the circuit of Figure 1(b), let us introduce some furthermagnetic stator and rotor quantities:

Us, i – magnetic potential of the ith stator tooth head;wts, i – flux flowing out the ith stator tooth;wys, i – flux flowing through the stator yoke section between the ith and the

ði þ 1Þth teeth;As, i – ampere turns in the ith stator slot (between the ith and the ði þ 1Þth

teeth);Ur, j – magnetic potential of the jth rotor zone surface;wtr, j – flux flowing out through the jth rotor zone;wpr – flux flowing through the rotor pole;Ar, j – ampere·turns related to rotor lap embraced by the jth and the

ð j þ 1Þth zone.

Calculation of the scalar magnetic potentialsStator. With reference to the generic lap of the ith stator slot, the following ns

recursive equations hold (i ¼ 0; 1; . . .; ns 2 1; i þ 1 is the remainder of thedivision of i þ 1 by ns, so that when i ¼ ns 2 1, i þ 1 ¼ 0; similarly, for i ¼ 0;i 2 1 ¼ ns 2 1):

Us;iþ1

2 U s;i ¼ A*s;i; ð1Þ

with

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957

A*s;i ¼ As;i 2 u

ts;iþ1w

ts;iþ12 uys;iwys;i þ uts;iwts;i ð2Þ

The terms A*s;i represent the slot ampere·turns lessened by the magnetic

potential drops due to teeth and yoke reluctances. The fluxes conservationimplies that:

wts;i ¼ wys;i21

2 wys;i ð3Þ

To solve the set of equation (1), the further equation is introduced:

Xns21

i¼0

ls;iU s;i ¼ 0 ð4Þ

where

ls;i ¼Xnr21

j¼0

li; j

is the ith stator tooth gap permeance; the following relation is obtained:

U s;0 ¼ 2

Xns21

i¼1

ls; iF*s; i21

Xns21

i¼0

ls; i

¼ 2

Xns21

i¼1

ls; iF*s; i21

Ls; rð5Þ

with

F*s; i ¼

Xi

h¼0

A*s; h

and

Ls; r ¼Xns21

i¼0

ls; i ¼Xns21

i¼0

Xnr21

j¼0

li; j;

being Ls, r the global permeance between the stator and rotor.By means of equation (5), the other potentials Us,i are determined by the

recursive relation (1).Rotor. With reference to Figure 1(b), a sequence of rotor magnetic laps

analogous to the stator ones can be recognised, and a set of nr 2 1 recursiveequations like (1) could be therefore obtained. Taking advantage of the

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geometrical and magnetic symmetry, the problem is simplified, since only nr/2equations are required, corresponding to contiguous rotor laps embracing a1808 arc (it results U r; jþnr=2 ¼ 2U r; j). Defining the quantities related to therotor ampere-turns:

A*r;0 ¼ Ar;0 þ utr;1wtr;1 þ upwpr

A*r;1 ¼ Ar;1 þ utr;2wtr;2 2 utr;1wtr;1

A*r;2 ¼ Ar;2 þ utr;3wtr;3 2 utr;2wtr;2

A*r;3 ¼ Ar;3 2 upwpr 2 utr;3wtr;3

ð6Þ

the following equations are obtained:

U r;0 ¼ F*r;3=2 ¼

A*r;0 þ A*

r;1 þ A*r;2 þ A*

r;3

� �

2

U r;1 ¼ U r;0 2 F*r;0

U r;2 ¼ U r;0 2 F*r;1

U r;3 ¼ U r;0 2 F*r;2

ð7Þ

with

F*r; j ¼

Xi

k¼0

A*r;k:

Calculation of fluxesStator. Once the stator and rotor potentials are defined, the flux delivered bythe generic ith stator tooth is given by:

wts;i ¼ 2lts;i21

Us;i21

þ l*s;iU s;i 2 lts;iU s;iþ1

2Xnr21

j¼0

lijU r; j ð8Þ

with l*s;i ¼ ls;i þ lts;i þ l

ts;i21the ith tooth total gap permeance.

Rearranging equation (3), the stator yoke fluxes can be explicited as in thefollowing:

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959

wys;1 ¼ 2cs;1 þ wys;0

wys;2 ¼ 2cs;2 þ wys;0

..

.

wys;ns21 ¼ 2cs;ns21 þ wys;0

ð9Þ

with

cs;i ¼Xi

h¼1

wts;h

Since the net m.m.f. acting inside the stator yoke is null, the sum of the MVDsalong the yoke must be zero, i.e.:

Xns21

i¼0

uys;iwys;i ¼ 0 ð10Þ

Combining equations (9) and (10), wys,0 can be expressed as:

wys;0 ¼

Xns21

i¼1

uys;ics;i

Xns21

i¼1

uys;i

ð11Þ

and so all the other stator yoke fluxes can be determined.Rotor. The flux in the jth rotor zone can be expressed as a function of the

magnetic potentials in the form:

wtr; j ¼ 2ltr; j21

Ur; j21

þ l*r; jU r; j 2 ltr; jU r; jþ1 2

Xns21

i¼0

lijU s; i ð12Þ

with l*r; j ¼ lr; j þ ltr; j þ l

tr; j21the jth zone gap permeance.

The rotor pole flux wpr is then obtained by:

wpr ¼ wtr;1 þ wtr;2 þ wtr;3 ð13Þ

The previous equations define a non-linear system, requiring an iterativeprocedure to be solved. In such application, a fixed-point technique is adopted:let X(n) be the set of values of the magnetic circuit quantities (ampere-turns,magnetic potentials, fluxes) evaluated at the nth step. Entering such values in

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the previous equations, a preliminary updated set Xuðnþ1Þ is obtained; the new

solution vector X ðnþ1Þ is then determined as:

X ðnþ1Þ ¼ bXuðnþ1Þ þ ð1 2 bÞXu

ðnþ1Þ: ð14Þ

A suited choice of the relaxation factor b , 1 is essential to prevent thenumerical instability, limiting at the same time the number of iterations:the higher the saturation, the lower is the instability threshold value of b. Theconvergence estimation is based on the calculation of the flux variation Dw(n)

from the ðn 2 1Þth to the nth iteration (sub-subscripts (n21) and (n)):

DwðnÞ ¼

Xns21

i¼0

wts;ðnÞ

i 2 wts;ðn21Þ

i

� �2

þXnr21

j¼0

wtr;ðnÞ

j 2 wtr;ðn21Þ

j

!2

Xns21

i¼0

w2

ts; iðnÞ

þXnr21

j¼0

w2

tr; jðnÞ

ð15Þ

The iterations are continued until the number n exceeds a maximum nmax orDw(n) goes below a threshold value 1w.

The process is outlined by the flow diagram of Figure 3.

Determination of the flux linkagesThe stator winding flux linkages c1, c2, c3 are determined by multiplying thefluxes wts,i with the linkage coefficients {wp;0; . . .;wp;ns21} of the pth phasewinding, according to the following formulations:

cp ¼Xns21

i¼0

wp;iwts;i with p ¼ 1; 2; 3

wp;i ¼ w0p;i21 2

1

ns

Xns21

i¼0

w0p;i

w0p;0 ¼ 0; w0

p;iþ1 ¼ w0p;i þ gp;i with i ¼ 0; . . .; ns 2 2

ð16Þ

The pth phase connection coefficients gp,i are given by the number ofconductors in the ith slot with a leading + if current flows towards thereader, – if it flows in the opposite direction, 0 if pth phase has no conductorin the slot.

For a symmetrical 3-phase winding, once the phase 1 linkage coefficientsare determined, the corresponding ones for phases 2 and 3 are simply

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obtained by shifting the coefficients to the left of ns/3 and 2ns/3 positions,respectively.

Analogous expressions can be defined for the rotor windings. More directly,the flux linkage of the Ne turns excitation winding is given by:

ce ¼ 2N eðwtr;1 þ wtr;2 þ wtr;3Þ: ð17Þ

Procedure for the transient analysisWith reference to Figure 4, let v ¼ {v1 2 v0; v2 2 v0; v3 2 v0; ve};i ¼ {i1; i2; i3; ie}; c ¼ {c1;c2;c3;ce} be the voltage, current and fluxvectors, respectively (e-subscripted quantities refer to the excitation winding,the other elements to the armature phases). Consider the voltage equations aswritten according to the active bipole representation and express them in thematrix form:

v ¼dc

dt2 Rw · i

vp 2 v0 ¼ Rlip þ Lldip

dtð p ¼ 1; 2; 3Þ; ve ¼ 2V e

ð18Þ

Figure 3.Outline of the procedurefor the fluxes calculation

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with Rw diagonal matrix related to the winding resistances {Ra, Ra, Ra, Re} andVe field supply constant voltage.

The wye midpoint voltage v0 can be determined according to thecharacteristics of the neutral connection: if such connection, as in most cases, isabsent (i.e. i1 þ i2 þ i3 ¼ 0), the sum of the armature winding equations gives:

v0 ¼ pc1 þ c2 þ c3

3

� �¼ pc0 ð19Þ

with c0 the homopolar component of the flux.According to equation (19), equation (18) can be rearranged as:

vp ¼ Rlip þ Lldip

dt¼ 2

d

dtðcp 2 c0Þ2 Raip p ¼ 1; 2; 3

ve ¼ 2V e ¼ 2dce

dt2 Reie

ð20Þ

Defining the equivalent global flux linkages fp ¼ cp 2 c0 þ Llip andincluding the armature winding resistances in the resistance diagonal matrixR with non-null elements {Ra þ Rl ; Ra þ Rl ; Ra þ Rl ; Re}; equation (20)rearranged as:

u ¼df

dtþ R · i with u ¼ {0; 0; 0;V e} and f ¼ {f1;f2;f3;fe} ð21Þ

Referring the subscripts ðk; k þ 1Þ to tk; tkþ1; respectively, and posingakþ1 2 ak ¼ v ·Dt; with rotational speed v and Dt ¼ constant the integrationof equation (21) from tk to tkþ1 ¼ tk þ Dt by the trapezoidal rule yields:

u kþ1 þ u k

2Dt ¼ R ·

i kþ1 þ i k

2Dt þ ðf

kþ12 f

kÞ ð22Þ

Evidencing the occurring (k þ 1Þth state quantities i kþ1; fkþ1

; it results:

Figure 4.Circuit representation of

the armature andexcitation windings (Ra:

armature phaseresistance; Rl, Ll: load

resistance andinductance; Re: excitation

winding resistance)

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i kþ1 þ2

DtR21 ·f

kþ1¼ R21 ·

u kþ1 þ u k

22 i k þ

2

DtR21 ·f

kð23Þ

where all the right-side quantities are known, as soon as the previous kth stateis determined. Due to the non-linear dependence of f

kþ1on i kþ1; equation (23)

has to be solved via an iterative algorithm, according to the following points(where (q), (q21) superscripts denote the values related to the current qthiteration and to the previous one, respectively):

. the flux fðqÞ

kþ1can be expressed by its Taylor series expansion arrested to

the first-order term, starting from the previous estimated values{fðq21Þ

kþ1; iðq21Þ

kþ1 } :

fðqÞ

kþ1¼ fðq21Þ

kþ1þ

›f

›i

��ðq21Þ

kþ1

· iðqÞkþ1 2 iðq21Þkþ1

� �ð24Þ

. substituting equation (24) in equation (23) and collecting iðqÞkþ1 yields:

I þ2

DtR21·

›f

›i

��ðq21Þ

kþ1

!iðqÞkþ1 ¼ 2

2

DtR21· fðq21Þ

kþ12

›f

›i

��ðq21Þ

kþ1

iðq21Þkþ1

!

þ R21·ðukþ1 þ ukÞ2 ik þ2

DtR21 ·f

k

ð25Þ

with I 4 £ 4 identity matrix; finally iðqÞkþ1 is given by:

iðqÞkþ1 ¼ RDt

2þ

›f

›i

��ðq21Þ

kþ1

!21

· fk2 fðq21Þ

kþ1þ

›f

›i

��ðq21Þ

kþ1

· iðq21Þkþ1

þðukþ1 þ ukþ1 2 R · ikÞDt

2

�;

ð26Þ

. the quantities

fðqÞ

kþ1;›f

›i

��ðqÞ

kþ1

can be quickly determined via the method of solution of the magneticnetwork described earlier;

. as starting values ðq ¼ 0Þ; the following ones are assumed (a value has tobe updated from ak to akþ1):

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ið0Þkþ1 ¼ ik

fð0Þkþ1

¼ f ið0Þkþ1;akþ1

� � ›f

›i

��ð0Þ

kþ1

¼›f

›iið0Þkþ1;akþ1

� � ð27Þ

indicating with N the diagonal matrix containing the number of the windingturns and with 1i, a predefined threshold value, the iteration is stopped whenthe relative value of the ampere-turns variation, Di(q) verifies the followingcondition:

DiðqÞ ¼N · iðqÞkþ1 2 N · iðq21Þ

kþ1

N · iðqÞkþ1

, 1i: ð28Þ

Examples of applicationThe described technique was applied to the purely hypothetical configurationsketched in Figure 1(a), in order to compare the results with the ones obtainedby a commercial FEM code for the electromagnetic transient analysis (AnsoftMaxwell 2D Transient Code, v.8.0.22, 2001). A time step Dt ¼ 50ms (angularstep: vDt 1808=p ¼ 0:98) was adopted in the proposed procedure to achieve anadequate angular resolution, while Dt ¼ 138:9ms (angular step: 2.58) in theFEM transient analysis. Mechanical transient was neglected, assuming aconstant rotation speed v ¼ 314:16 s21 (3,000 rpm).

No-load excitation current build-upAs a first case, the excitation current build-up at no load was considered,starting from null initial value and applying a constant voltage V e ¼ 100 V tothe field winding ðN e ¼ 1; 260 turns, Re ¼ 20VÞ: To reach a nearly completesteady state running, more than 8 s are required. Figures 5 and 6 show thecurrent ie and flux linkage ce of the excitation winding, respectively, asfunctions of time, comparing the results of the proposed procedure and FEMtransient analysis. The cogging effect on the excitation current is evidenced inFigure 5(b). The relative difference is about 3 per cent for the current values,and <1.3 per cent with reference to the fluxes.

Simulation of load insertionA sudden load insertion ðRa þ Rl ¼ 3:2VÞ at t ¼ 0 s was simulatedmaintaining V e ¼ 100 V and assuming initial current values {i1; i2; i3; ie} ¼{0; 0; 0; 5 A}: The currents and the flux linkages of the armature windings inthe early instants of the transient are compared with the FEM results inFigure 7, showing a good agreement.

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Figure 5.No-load field current ie:(a) during the entiresimulation (with ripplesmoothing for sake ofclearness); (b) focused for10 ms period to displaythe cogging effect

Figure 6.Build-up of the excitationflux ce at no-load

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It is worth to remark that the calculation time with the FEM code is at least oneorder of magnitude higher than that of the proposed method, the same timeresolution being maintained.

ConclusionsA method for the electromagnetic transient analysis of synchronous machines,based on the solution of an equivalent magnetic circuit, was described. Suchmethod could be extended to a wider class of electromechanical devices withminor modifications of the algorithmic structure.

The examples of application of a code, based on such method, show a goodagreement with the results of a commercial FEM code, allowing on the otherhand a great reduction in the calculation time.

Figure 7.(a) Armature currentsand (b) flux linkagesafter a resistive load

insertion

A magneticnetwork

approach

967

References

Andriollo, M., Di Gerlando, A. and Porzio, D. (2001), “A design oriented magnetic circuit model oflow-power synchronous generators”, Proceedings of the 4th International Symposium onAdvanced Electromechanical Motion Systems – Electromotion ’01, 19-20 June 2001,Bologna, Italy, pp. 535-40.

Ansoft Maxwell 2D Transient Code v.8.0.22 (2001).

Di Gerlando, A., Perini, R. and Vistoli, I. (1995), “A field-circuit approach to the design orientedevaluation of the no-load voltage harmonics of salient pole synchronous generators”,Conference Electric Machines and Drives 95, Durham, UK, pp. 390-4.

Kunze, W., Kuß, H. and Bolter, F-Tth. (1991), “Application of numerical field calculation toselected problems of electrical machines”, Conference SM100, Zurich, CH, pp. 1193-8.

Nabeta, S.I., Foggia, A., Coulomb, J.L. and Reyne, G. (1996), “Finite element simulations ofunbalanced faults in a synchronous machine”, IEEE Transactions on Magnetics, Part 1,Vol. 32 No. 3, pp. 1561-4.

Preston, T.W. and Sturgess, J.P. (1993), “Implementation of the finite-element method intomachine design procedures”, 6th International Conference on Electrical Machines andDrives, pp. 312-17.

Schmidt, E., Grabner, C. and Traxler-Samek, G. (2000), “Reactance calculation of a 500 MVAhydro-generator using a finite element analysis with superelements ”, Electric Machinesand Drives Conference, pp. 838-44

Schmidt, E., Grabner, C. and Traxler-Samek, G. (2001), “Determination of reactances of largehydro-generators using finite elements and domain decomposition”, Canadian Conferenceon Electrical and Computer Engineering, Vol. 2, pp. 811-17.

Further reading

Ostovich, V. (1989), Dynamics of Saturated Machines, Springer-Verlag, New York.

Sturgess, J.P. and Preston, T.W. (1993), “Damper cage design using the finite-element method”,Conference on Electrical Machines and Drives, Oxford, UK, pp. 457-62.

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Numerical magnetic fieldanalysis and signal processing

for fault diagnostics ofelectrical machines

S. PoyhonenDepartment of Automation and Systems Technology,

Control Engineering Laboratory, Helsinki University of Technology,Finland

M. Negrea, P. Jover and A. ArkkioDepartment of Electrical Engineering, Laboratory of Electromechanics,

Helsinki University of Technology, Finland

H. HyotyniemiDepartment of Automation and Systems Technology,

Control Engineering Laboratory, Helsinki University of Technology,Finland

Keywords Condition monitoring, Electromagnetic fields, Electrical machines,Finite element method, Signal processing, Fault analysis

Abstract Numerical magnetic field analysis is used for predicting the performance of aninduction motor and a slip-ring generator having different faults implemented in their structure.Virtual measurement data provided by the numerical magnetic field analysis are analysed usingmodern signal processing techniques to get a reliable indication of the fault. Support vectormachine based classification is applied to fault diagnostics. The stator line current, circulatingcurrents between parallel stator branches and forces between the stator and rotor are compared asmedia of fault detection.

IntroductionCompanies dealing with electrical machinery find condition monitoring anddiagnostics more and more important. The supervision of electrical drivesystems using non-invasive condition monitoring techniques is becoming astate-of-the-art method for improving the reliability of electrical drives in manybranches of the industry.

Typical questions are, how to detect a starting fault, how to distinguish adeteriorating fault from a harmless constructional asymmetry, which are thephysical quantities that best indicate a fault and how to measure them, and howshould the measured signals be processed to get the most reliable diagnosis.

The basis of any reliable diagnostic method is an understanding of theelectric, magnetic and mechanical behavior of the machine in healthy-state andunder fault conditions. The aim of computer simulation of magnetic field



Numericalmagnetic field

analysis

969


Vol. 22 No. 4, 2003pp. 969-981


DOI 10.1108/03321640310482931

distribution and operating characteristics is to foresee the changes of motorperformance due to the changes of parameters as a consequence of differentfaults. Simulation results represent the contribution to the correct evaluation ofthe measured data in diagnostic procedures, which are the important part ofsupervision system based on expert systems and artificial intelligence methods(Filipetti et al., 1995).

Concerning the fault indicators, even though thermal and vibrationmonitoring have been utilized for decades, most of the recent research has beendirected toward electrical monitoring of the motor with emphasis on inspectingthe stator current of the motor because it has been suggested that stator currentmonitoring can provide the same indications without requiring access to themotor (MCS, 1992). In particular, a large amount of research has been directedtowards using the stator current spectrum to sense specific rotor faults(Benbouzid, 1997; Kliman, 1990). In processing plants, within electrical drivesthe vibration monitoring is utilized to detect the mechanical faults of rotatingelectrical machines. The vibration monitoring has been used also to detect theelectromechanical faults like broken or cracked rotor bars in squirrel-cageinduction motors (Muller and Landy, 1996).

Different kinds of artificial intelligence based methods have becomecommon in fault diagnostics and condition monitoring. For example, fuzzylogic and neural networks (NN) have been used in modeling and decisionmaking in diagnostics schemes. Also, numerical classification methods arewidely used in the area of modern fault diagnostics, and in particular, faultdiagnostics of electrical machines. For example, in Alguindigue et al. (1993) andLi et al. (2000), used NN based classifiers in the diagnosis of rolling elementbearings.

Support vector machine (SVM) based classification is a relatively newclassification method, and it is claimed to have better generalization propertiesthan NN based classifiers. Another interesting feature of SVM based classifieris that its performance does not depend on the number of attributes of classifiedentities, i.e. dimension of classified vectors. That is why it is noticed to beespecially efficient in large classification problems. In fault diagnostics process,this property is very useful, because the number of attributes chosen to be thebase of diagnostics is thus not limited.

SVM had been for the first time successfully applied to fault diagnostics ofelectrical machines in Poyhonen et al. (2002). There we used SVM’s to classifyfaults of a 15 kW induction motor from the stator line current. In the presentpaper, the method is extended to classify several faults of a 1.6 MVA slip-ringgenerator and 35 kW cage induction motor. We extend our study concerningthe media for fault detection to the analysis of the circulating currents onparallel branches and forces acting on the rotor.

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Numerical model for fault simulationsElectromagnetic model of a machineThe magnetic field in the core of the machine is assumed to betwo-dimensional. The three-dimensional end-region fields are modeledapproximately using end-winding impedances in the circuit equations of thewindings. The magnetic vector potential A satisfies the equation

7 £ ðn7 £ AÞ ¼ J ð1Þ

where n is the reluctivity of the material and J is the current density. Thecurrent density can be expressed as a function of the vector potential and theelectric scalar potential f

J ¼ 2s›A

›t2 s7f ð2Þ

where s is the conductivity of the material. In the two-dimensional model, thevector potential and the current density have only the z-components

A ¼ Aðx; y; tÞez

J ¼ J ðx; y; tÞez

ð3Þ

The scalar potential f has a constant value on the cross-section of atwo-dimensional conductor, and it is a linear function of the z-coordinate. Thegradient of the scalar potential can be expressed with the aid of the potentialdifference u induced between the ends of the conductor. By substitutingequation (2) in equation (1) the field equation becomes

7 £ ðn7 £ AÞ þ s›A

›t¼

s

‘uez ð4Þ

where ‘ is the length of the conductor. A relation between the total current iand the potential difference u is obtained by integrating the current density(equation (2)) over the cross-section of the conductor

u ¼ Ri þ R

ZS

s›A

›tdS ð5Þ

where R is the dc resistance of the conductor. The circuit equations for thedamping cage are constructed by applying Kirchhoff’s laws and equation (5)for the potential difference.

Time-dependence. A time-dependent field is solved by discretizing the timeat short time intervals Dt and evaluating the field at time instantst1; t2; t3; . . .ðtkþ1 ¼ tk þ DtÞ: In the Crank-Nicholson method, the vectorpotential at time tkþ1 is approximated


analysis

971

Akþ1 ¼1

2

›A

›t

��kþ1

þ›A

›t

��k

� �Dt þ Ak ð6Þ

By adding the field equations written at times tk and tkþ1 together andsubstituting the sum of the derivatives from equation (6), the equation

7 £ ðnkþ17 £ Akþ1Þ þ2s

DtAkþ1 ¼

s

‘ukþ1ez

2 7 £ ðnk7 £ AkÞ22s

DtAk 2

s

‘ukez

� �ð7Þ

is obtained. The time discretization of the potential difference equation (5) gives

1

2ðukþ1 þ ukÞ ¼

1

2Rðikþ1 þ ikÞ þ R

ZS

Akþ1 2 Ak

DtdS ð8Þ

Equations (7) and (8) form the basic system of equations in the time-steppingformulation. Starting from the initial values and successively evaluating thepotentials and currents of the next time-steps, the time variation of thequantities is worked out.

Numerical solution. The construction of the circuit equations and the detailsof the numerical solution of the coupled field and circuit equations have beenpresented by Arkkio (1988). The finite element discretization leads to a largenon-linear system of equations in which the unknown variables are the nodalvalues of the vector potential and the currents or potential differences of thewindings. The equations are solved by the Newton-Raphson method.

Finite element mesh. The magnetic field of a healthy electrical machine isperiodic, typically from one pole pair to the next one. A fault in the machinedisturbs this symmetry, and the whole machine cross section has to bemodeled. For fault detection purposes, we are more interested in qualitativethan exact quantitative results, and the finite element meshes used can berelatively sparse, as long as the geometric symmetry is the same as for thefaulty machine. In this study, triangular first-order finite elements are used,and the finite element meshes typically contain 6,000-8,000 elements.

Motion of the rotor. In a general time-stepping analysis, the equations forrotor and stator fields are written in their own reference frames. The solutionsof the two field equations are matched with each other in the air gap. The rotoris rotated at each time-step by an angle corresponding to the mechanicalangular frequency. The rotation is accomplished by changing the finite elementmesh in the air gap.

Operating characteristics. The magnetic field, the currents and the potentialdifferences of the windings are obtained in the solution of the coupled field and

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circuit equations as discussed earlier. Most of the other machine characteristicscan be derived from these quantities.

Modeling the faultsThe faults studied in the numerical simulations are:

. shorted turn in stator winding (ST)

. shorted coil in stator winding (SC)

. shorted turn in rotor winding (slip-ring generator) (RT)

. shorted coil in rotor winding (slip-ring generator) (RC)

. broken rotor bar (cage induction motor) (BB)

. broken end ring (cage induction motor) (BR)

. static or dynamic rotor eccentricity (SE, DE)

. asymmetry in line voltage (slip-ring generator) (VA)

To model a shorted turn, the sides of this turn in the finite element mesh aresubstituted by conductors in perfect short-circuit. There is no galvanic contactbetween the reduced phase winding and the new, shorted conductors. A shortedcoil is treated in a similar manner.

Static eccentricity is obtained by shifting the rotor by 10 percent of the radialair-gap length, and rotating the rotor around its center point in this newposition. Dynamic eccentricity is obtained by shifting the rotor by 10 percent ofthe air-gap length, but rotating it around the point that is the center point of thestator bore. A 10 percent eccentricity is not yet a real fault, but an eccentricitypossibly growing should be detected at an early stage.

The circuit equations of the cage winding are composed of the potentials ofthe bars inside the rotor core, bar ends outside the core and end ring segmentsconnecting the bars. When modeling a broken bar, the resistance of the bar endoutside the core is increased to a value 100 times the dc resistance of the wholebar. When modeling a broken end ring, the resistance of an end-ring segmentbetween two bars is increased to a value 1,000 times the original resistance ofthe segment.

An asymmetry in the line voltage is not a fault (NF) in the machine, but itmay cause problems if the fault diagnostics algorithms identify the voltageasymmetry as a machine fault. The asymmetric voltage is obtained by adding5 percent of negative phase-sequence voltage to an originally 100 percentpositive phase-sequence supply voltage.

Fault indicatorsThe parameters studied as fault indicators are line currents, circulatingcurrents between parallel stator branches, and force between the stator androtor.


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To compute the stator currents, the parallel branches are treated as separatephases. If there are n parallel branches, the machine is treated as a 3n-phasemachine supplied from a n £ 3-phase voltage source. In the case of a starconnection, all the 3n phases have a common star-point, and a line current isobtained as the sum of n branch currents. A circulating current is half of thedifference of two branch currents.

The electromagnetic force acting between the stator and rotor is computedfrom the air-gap field using the method developed by Coulomb (1983).

SVM for multi-class classificationSVM based classification is a relatively new machine learning method based onthe statistical learning theory presented by Vapnik (1998). In SVM, an optimalhyperplane is determined to maximize the generalization ability of the classifierby mapping the original input space into a high dimensional dot product spacecalled feature space. The mapping is based on the so-called kernel function.The optimal hyperplane is found in the feature space with a learning algorithmfrom optimization theory, which implements a learning bias derived fromstatistical learning theory (Cristianini and Shawe-Taylor, 2000).

A SVM based classifier is a binary classifier. In fault diagnostics of anelectrical machine, there exist several fault classes in addition to healthyoperation. We need a method to deal with a multi-class classification problem.In this study, we use a mixture matrix approach that is suggested by Mayorazand Alpaydin (1999).

SVM is used to build pair-wise classifiers for all considered classes. For ann-class classification problem nðn 2 1Þ=2 pair-wise classifiers cover all pairs ofclasses. Each classifier is trained on a subset of the training set containing onlytraining examples of the two involved classes. Final solution to the n-classproblem is reconstructed from solutions of all two-class problems.

Often, simple majority voting between pair-wise classifiers is applied toreconstruct the n-class solution, but with this approach problems occur, if twoseparate classes get equal amount of votes. Majority voting approach does nottake into account possible redundancy in pair-wise classifiers’ outputs. With amixture matrix approach, the n-class solution is found by linear combination ofpair-wise classifiers’ solutions. The mixture matrix is designed in the trainingphase of classifiers to minimize the mean square error between the correct classdecision and the linear combination of pair-wise classifiers’ outputs.

Preprocessing dataPower spectra estimates of the stator line currents have often been used as amedium of fault detection of electrical machines (Benbouzid, 1997). Maindisadvantage of classical spectral estimation techniques, like FFT, is theimpact of side lobe leakage due to windowing of finite data sets. Windowweighting decreases the effect of side lobes. Further, in order to improve

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the statistical stability of the spectral estimate, averaging by segmenting thedata can be applied. The more segments are used, the more stable the estimateis. However, the signal length limits the number of segments used, but withoverlapping segments the number of segments can be increased. In this study,Welch’s method is used to calculate the power spectra estimates of signals inmotors. The method applies both the window weighting and the averagingover overlapping segments to estimate the power spectrum. In this study, weuse Hanning window sized 500 samples, and number of overlapping samples is250.

Before estimating the power spectra, adaptive predictive filtering is appliedto mitigate the impact of noise (Valiviita et al., 1999). Noise filtering is appliedonly in the case where stator line current is used as a medium of fault detection.

Simulation resultsNumerical analysisTable I gives the main parameters of the cage induction motor and slip-ringgenerator. In the present study, the cage induction motor was fed from afrequency converter. The stator and rotor windings of the slip-ring generatorwere connected to sinusoidal voltage sources. The stator is delta connected andthe rotor is star connected. Figure 1 shows the cross-sectional geometry of theslip-ring machine. Figures 2 and 3 show two examples of simulated circulatingcurrents flowing in the parallel stator branches. The current in Figure 2 iscaused by 10 percent dynamic eccentricity in the slip-ring generator operatingat a capacitive power factor of 0.8 at half load. Figure 3 shows the circulatingcurrent in the cage induction motor with one broken rotor bar and loaded bythe rated torque. The rated currents of the machines are 1,600 and 62 A,respectively.

Figures 4 and 5 show the x-components of the forces acting between thestator and rotor for the slip-ring generator with the 10 percent dynamiceccentricity and for the cage induction motor with a broken rotor bar,respectively.

Induction motor Slip-ring generator

Number of poles 4 4Parallel branches 2 4Stator connection Star DeltaRotor connection – StarRated power (kW) 35 1,600Rated frequency (Hz) 100 50Rated voltage (V) 400 690Rated current (A) 64 1,500

Table I.Main parameters ofthe studied electrical

machines


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975

Figure 1.Cross-sectional geometryof the slip-ring generator

Figure 2.Circulating currentsflowing in parallelbranches of the statorwindings for the slip-ringgenerator with 10 percentdynamic eccentricity

Figure 3.Circulating currentsflowing in parallelbranches of the statorwindings for the cageinduction motor with onebroken rotor bar

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Support vector machine based classificationGenerating the sample set. Measurement error is included to the virtualmeasurement data by adding noise to the signals. Mean value of noise is zeroand variance is 3 percent of the amplitude of the current.

The power spectrum estimates are calculated 80 times from different partsof signals in each fault case to generate a sample set. The support vectorclassifiers are trained and tested separately in different load situations. Half ofthe samples are chosen for training the classifier, and half of the samples areleft for testing the classifier’s generalization ability. An average healthyspectrum from the training set is chosen to be a reference, and all the otherspectra are scaled with it. The difference values from the reference create thesample set.

Classification results. We have six fault classes and the healthy class, son ¼ 7; and we need to design 21 two-class classifiers that are combined togenerate the final classification decision for a spectrum sample. Choosing thekernel function used in building, the SVMs have a considerable influence on theclassification results. There does not exist general rules for choosing the kernelfunction, but the best kernel function depends on the application where SVMsare used. When the cage induction motor was studied, all classifiers weredesigned with a radial basis kernel function width equal to 11 except whenstudying forces as indicators of faults. When the slip-ring machine was studied,

Figure 4.Force acting between the

stator and rotor for theslip-ring generator with

10 percent dynamiceccentricity

Figure 5.Force acting between the

stator and rotor for thecage induction motor

with one broken rotor bar


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977

and also when forces were studied in both machines, a first order polynomialkernel function was used.

In Tables II-IV, fault classification results of a 35 kW inverter-fed cageinduction machine are presented. In Table II, a stator line current has been usedas a medium of fault detection and in Table III, the circulating currents betweenparallel branches have been used. In Table IV, forces on the rotor have beenused as a media of fault detection. In Tables V-VII, fault classification results ofa 1.6 MVA slip-ring generator are presented.

All of the faults from both machines in all load situations are highly moreaccurately detected from the circulating currents between parallel branches orfrom the forces that acts on the rotor than from the stator line current. Whenusing either of these indicators while monitoring the slip-ring machine, evenoperation under an external disturbance, i.e. asymmetry in line voltage, iscorrectly classified to the healthy operation class. Only problems occur inmonitoring the slip-ring machine: shorted rotor coil and shorted rotor turnfaults tend to get mixed up.

Percent NF BB BR ST SC SE DE Total

Full load 20 45 23 100 100 5.0 7.5 43Half load 28 18 20 100 100 38 38 49No load 70 85 70 100 100 50 68 78Total 39 49 38 100 100 31 38 57

Table II.Correct faultclassificationpercentages of acage inductionmotor, stator linecurrent as a mediumof fault detection


Full load 100 100 100 100 100 100 100 100Half load 100 100 100 100 100 100 100 100No load 83 75 100 100 100 100 100 94Total 94 92 100 100 100 100 100 98

Table III.Correct faultclassificationpercentages of acage inductionmotor, circulatingcurrents betweenparallel branches asmedia of faultdetection


Full load 100 100 100 100 100 100 100 100Half load 100 100 100 100 100 100 100 100No load 100 100 100 100 100 100 100 100Total 100 100 100 100 100 100 100 100

Table IV.Correct faultclassificationpercentages of acage inductionmotor, force on therotor as a medium offault detection

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The circulating currents and the force on the rotor are natural fault indicatorsas they are zero for all healthy symmetric machines, but non-zero for faults thatcause asymmetry in the magnetic field distribution of a machine.

The stator line current is also a good indicator of faults in noiseless situation,but in real world applications noise is always present. Shorted turn and shortedcoil in stator are always detected regardless of noise. However, healthyoperation is often misclassified. Thus, stator line current cannot be used asmedium of fault detection without improvements on the classification methodor in noise filtering. Dropping such fault classes out of the classificationstructure that cannot be separated from the healthy operation, we couldconstruct a classifier that is able to correctly detect healthy operation andshorted coil and shorted turn faults from the stator line current. Also otherways of estimating the power spectrum of the current may give someenhancement.

Percent NF NF/VA RC RT ST SC SE DE Total

C 100 100 50 100 100 100 100 100 94O 100 100 100 100 100 100 100 100 100S 100 100 100 100 100 100 100 100 100U 100 100 100 100 100 100 100 100 100Total 100 100 88 100 100 100 100 100 99

Table VI.Correct faultclassificationpercentages

slip-ring generator,circulating currents

between parallelbranches as media

of fault detection


C 100 100 100 45 100 100 100 100 93O 100 100 100 100 100 100 100 100 100S 100 100 100 88 100 100 100 100 98U 100 100 100 100 100 100 100 100 100Total 100 100 100 83 100 100 100 100 98

Notes: C ¼ base speed, power factor 0.8 capacitive; O ¼ 1.12 £ base speed, resistive load;S ¼ base speed, resistive load; U ¼ 0.88 £ base speed, resistive load.

Table VII.Correct faultclassification

percentages of aslip-ring generator,

force on the rotor asa medium of fault

detection


C 40 0 25 73 100 100 15 15 46O 28 0 100 63 100 100 25 35 56S 35 0 35 53 100 100 33 48 50U 38 0 100 100 100 100 15 48 63Total 35 0 65 72 100 100 22 37 54

Table V.Correct faultclassification

percentages ofslip-ring generator,

stator line current asa medium of fault

detection


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Experimental resultsThe 35 kW cage induction motor was fed from an inverter having a switchingfrequency of 3 kHz. A DC generator was used for loading the motor. Thecurrents, voltages, power and supply frequency were measured using a wideband power analyzer. The measurements were carried out for three differentload conditions. Figure 6 presents a schematic view of the measuring set-up.

The current and voltage waveforms were recorded with a transient recorder.Hall sensors (LEM) were used as current transducers, and the voltages weremeasured through an isolation amplifier. The sampling frequency was 40 kHzand a typical number of samples was 20,000. The recording system wascalibrated using the measurements from the power analyzer.

Healthy operation, broken rotor bar operation and operation under inter-turnshort circuit were analyzed. Power spectra estimate samples were calculatedfrom the current, and a SVM classification structure was constructed only forthese three classes. If measurement data were used in both training and testingthe classifier, all test samples were correctly classified in all load situations(Table VIII).

ConclusionsNumerical magnetic field analysis was used to provide virtual measurementdata from healthy and faulty operation of the machines and support vector

Figure 6.Schematic of themeasuring set-up

NF BB ST Total

Full load 100 100 100 100Half load 100 100 100 100No load 100 100 100 100

Table VIII.Correct faultclassificationpercentages fromthe measurementsof a cage inductionmotor, statorcurrent as a mediumof fault detection

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classification was applied to fault diagnostics of a cage induction motor and aslip-ring generator. Stator line current, circulating currents between parallelbranches and forces acting on the machine’s rotors were compared as media offault detection. Circulating currents between parallel branches and forces onrotor were found to be superior indicators of faults compared to the statorcurrent. However, in experimental studies, healthy operation, broken baroperation and operation under inter-turn short circuit were correctly classifiedbased on the stator current, if measurement data were used in both training andtesting the classifier.

References

Alguindigue, I.E., Loskiewicz-Buczak, A. and Uhrig, R. (1993), “Monitoring and diagnosis ofrolling element bearings using artificial neural networks”, IEEE Transactions onIndustrial Electronics, Vol. 40 No. 2, pp. 209-17.

Arkkio, A. (1988), “Analysis of induction motors based on the numerical solution of the magneticfield and circuit equations”, Helsinki, Acta Polytechnica Scandinavica, ElectricalEngineering Series No. 59, ISBN 951-666-250-1, p. 97.

Benbouzid, M.E.H. (1997), “Induction motor diagnostics via stator current monitoring,” Proc.1997 Int. Conf. Maintenance and Reliability, Knoxville, TN, Vol. 1, pp. 1-10.

Coulomb, J.L. (1983), “A methodology for the determination of global electromechanicalquantities from the finite element analysis and its application to the evaluation of magneticforces, torques and stiffness”, IEEE Transactions on Magnetics, Vol. MAG-19 No. 6,pp. 2514-9.

Cristianini, N. and Shawe-Taylor, J. (2000), Support Vector Machines and Other Kernel-BasedLearning Methods, Cambridge University Press, Cambridge.

Filipetti, F., Franceschini, G. and Tassoni, C. (1995), “Neural networks aided on-line diagnosticsof induction motor rotor faults”, IEEE Transactions on Industry Applications, Vol. 31 No. 4.

Kliman, G.B. (1990), “Induction motor fault detection via passive current monitoring,” Proc. 1990Int. Conf. Electrical Machines, Cambridge, MA, Vol. 1, pp. 13-17.

Li, B., Chow, M.-Y., Tipsuwan, Y. and Hung, J.C. (2000), “Neural-network-based motor rollingbearing fault diagnosis”, IEEE Transactions on Industrial Electronics, Vol. 47 No. 5,pp. 1060-9.

Mayoraz, E. and Alpaydin, E. (1999), “Support vector machines for multi-class classification”,IWANN 1999, Vol. 2, pp. 833-42.

MCS (1992), “Methods of motor current signature analysis”, Mach. Power Syst., Vol. 20,pp. 463-74.

Muller, H. and Landy, C.F. (1996), “Vibration monitoring for broken rotor bars in squirell cageinduction motors with interbar currents”, SPEEDAM 96, 5-7 June, Capri, Italy, Vol. A2.

Poyhonen, S., Negrea, M., Arkkio, A., Hyotyniemi, H. and Koivo, H. (2002), “Support vectorclassification for fault diagnostics of an electrical machine,” Proceedings ICSP’02, 26-30August, Beijing, China.

Valiviita, S., Ovaska, S.J. and Vainio, O. (1999), “Polynomial predictive filtering in controlinstrumentation: a review”, IEEE Transactions on Industrial Electronics, Vol. 46 No. 5,pp. 876-88.

Vapnik, V.N. (1998), Statistical Learning Theory, Wiley, New York.


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Thermal modeling and testingof a high-speed axial-flux

permanent-magnet machineF. Sahin

Philips CFT, Mass Products and Technologies, Eindhoven,The Netherlands

A.J.A. VandenputEindhoven University of Technology, Eindhoven, The Netherlands

Keywords Flux, Permanent magnets, Thermal analysis, Performance, Electrical machines

Abstract This paper gives an overview of the design, manufacturing and testing of a high-speed(16,000 rpm and 30 kW) AFPM synchronous machine, which is mounted inside, and as anintegral part of, a flywheel. This system will subsequently be used for transient energy storage andICE operating point optimization in an HEV. The paper focuses on the major design issues,particularly with regard to the high rotational speed, and investigates the loss mechanisms whichare apparent therein, e.g. iron losses, rotor losses, and friction losses. The paper describes thehigh-speed testing facility and includes measured results, which will be compared to calculatedvalues.

1. IntroductionIt has been shown that the use of electric vehicles (EVs) and hybrid electricvehicles (HEVs) reduces total CO2 emissions. Consequently, the improvementof such vehicle drives has become a major topic, and the HEVs are especiallyfavorable in the short term considering the difficulties introduced by batteriesand the current lack of an EV charging infrastructure. The use of a novel HEVdrive system (Cadee and van Rooij, 1997) where the layout is shown in Figure 1,is being investigated. This drive system has many advantages including anincreased speed, and a faster acceleration of an HEV given a particular batterysize, while minimizing fuel usage. Also, by channeling power into and out ofthe flywheel, the peak current demand on the battery system is reduced, amajor concern in high-speed driving and during regenerative braking sincesuch peak currents drastically reduce battery life.

The inner city and highway driving speeds of the flywheel are specified tobe 7,000 and 16,000 rpm, respectively, at a torque level of 18 N m. The totaldrive system is highly demanding in terms of electrical machine efficiency andfurthermore the HEV layout specifies that the machine must be designed smallenough to fit in a cylindrical volume of 240 mm diameter with 150 mm height.

These specifications, among others, lead to one major implication in terms ofelectromagnetic requirements, namely a high torque density, which is the mainjustification of the suitability of an AFPM machine in a HEV application.



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In addition to the shape related advantages and other issues of suitability forthe application, extensive study demonstrated the torque density advantage ofthe axial-flux machine in comparison with the radial-flux type (Sahin, 2001).

2. Design aspectsThere are numerous alternatives for the design of AFPM machines assummarized by Zhang et al. (1996), such as internal rotor (Platt, 1989),internal stator ( Jensen et al., 1992), or multidisk type, slotless or slotted stators,and rotors with interior or surface-mounted permanent magnets. For thisapplication, slotted stators and surface-mounted permanent magnets areproposed by Sahin and Vandenput (1999). Due to its shape and compactness anAFPM machine is proposed to be mounted inside the flywheel.

Single stator-two rotor type AFPM machines have lots of advantages suchas lower end-winding lengths as explained by Jensen et al. (1992). On the otherhand, taking into account the size and shape constraints, the best machine forthis particular application is found as the two stator-one rotor type. As shownin Figure 2 permanent magnets are placed on the rotor which is an integral partof the flywheel, and the stators are fixed to the housing. The outside diameter,inside-to-outside diameter ratio, number of slots, magnet span, magnetskewing, and stator offsetting affect the performance of the machine to greateror lesser extents, and need to be carefully chosen to obtain an optimal design.These choices are discussed in detail by Sahin and Vandenput (1999).

Figure 1.Hybrid drive system

Figure 2.AFPM machine flywheel

arrangement

Thermalmodeling and

testing

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The number of slots and the winding configuration of the stators determine themagnitude of the mmf harmonic components (Sahin and Vandenput, 1999). TheAFPM machine was designed with 24 slots and 5/6 short-pitched statorwindings, which together reduce both the harmonic content and the length ofthe end windings. The machine is designed almost torque ripple free bychoosing a 1508 magnet span (plus a small span to compensate for fringing)and with one slot pitch magnet skewing. These methods (distribution, shortpitching of the windings, and magnet skewing) decrease the output torque ofthe machine. Nevertheless, they are advantageous in decreasing the lossescaused by the higher-order winding and slot harmonics. This fact becomesmore important at higher speeds where the rotor losses become a major concernas discussed by Sahin (2001).

A high-speed rotating rotor with a flywheel tends to generate extremely highair friction losses. Under these circumstances, air pressure reduction inside themachine is inevitable. Reduced air pressure naturally decreases the heattransfer rate from the rotor to the stator, which results in a rapid rise of themagnet and rotor temperature in a very short time. The high speed andevacuation based thermal problems are also aggravated with the highfrequency related eddy-current losses occurring in the rotor magnets and rotorsteel. These losses constitute further heating sources for the magnets. Therecan be two solutions offered to this problem: either by the lamination of themagnets together with the usage of a low-loss material for the rotor steel or byeliminating the causes of rotor losses as much as possible during the designprocess. In this study, considering the potential mechanical problems that itcould lead to, the first solution is left aside.

An extensive space harmonic analysis of possible structures was carried outand the design variables were evaluated in terms of their space harmoniccontributions. This study helped to choose a good combination of the designparameters, which ultimately resulted in a design with low space harmoniccontent and consequently low torque ripple. Accordingly, the magnitude of therotor losses was suppressed. Although the major higher-order harmoniccomponents could be suppressed by design, they have not been fullyeliminated, and it has been shown that the 11th and 13th ordered componentsare still the dominating rotor loss producing mechanisms. Of course, inaddition to the losses induced by the time harmonic components caused by thePWM drive scheme, there are other losses in the rotor which eventually resultin heat. Under this situation, it is clear that the level of armature excitationshould be modest by means of a dominant magnet flux density, whichconstitutes a trade off. Due to the nature of the application, as low as possibleno-load losses are preferred. Since the rotor is integrated into a flywheel,no-load losses always exist and keep on reducing the overall system efficiency.Reduction of the no-load losses can only be made possible via the reduction of

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the stator core losses, since the air friction losses are suppressed by means ofreduced air pressure.

In order to decrease stator core losses, the magnet excitation may be keptvery low and a very low flux density may be permitted in the stator cores. Yet,this contradicts with the previous concept, which demands a compromise.In this design, both of these contradicting conditions were tried to be satisfied,by keeping the magnet excitation dominant (around 0.73 T), and designing thestator cores with lower maximum flux density levels (1.2 T), with the cost of arelatively (to the extent permitted by the volume constraint) thicker statorback-iron. Considering the ultimate condition that a very low-loss steel is usedfor the stators, core losses can easily be minimized with this design. This is dueto the fact that the machine was designed with only four poles, and themaximum frequency was kept around 533 Hz. With modern thin low-loss steellaminations, and with a relatively low level of core flux density, an acceptablelow amount of core losses could be achieved at this frequency.

3. LossesThe design is based on efficiency optimization, which depends on the properestimation of the losses. This estimation is also important to analyze thethermal behavior of the machine which is a critical aspect in high-speedmachines. The AFPM machine mainly suffers from the following losses.

3.1 Copper lossesThe I 2R losses cover a great deal of the total losses. The high-frequencyeddy-current losses in the windings are around 0.4 per cent of the total copperlosses at rated speed since the stator coils are made of small-diameterconductors. In an AFPM machine the major part of the copper losses aregenerated in the end windings rather than in the slots. Therefore, the endwinding design deserves special attention. Using short pitched windings thelength of the end windings reduces. On the other hand, the longer coil spansshould correspond to the inner-end windings. The end windings are shown inFigure 3.

3.2 Core lossesDue to the difficulty of getting a laminated toroidal stator core made of thinsilicon, isotrophic steel, the M-4 grain-oriented silicon steel is chosen for theprototype motor. The core loss data, which are available only in the easydirection of magnetization, are used to fit the Steinmetz equation (Miller andHendershot, 1995), that describes the specific loss in W/kg as

pco ¼ 0:014492 B1:8 f þ 0:00004219 B2 f 2: ð1Þ

For a fine calculation of the core losses, the machine is divided into severalregions. Since the magnitude of the flux density varies at different axial

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cross-sections of the machine (between the inside and the outside diameter), atseveral axial cross-sections of the machine finite elements analysis (FEA) isconducted at different rotor positions over a pole pitch. The resultanttime-varying flux density waveforms and their higher frequency harmoniccomponents are obtained for various parts of the machine such as tooth,tooth-tops and the stator yoke at various axial cross-sections. From this, a fineestimation of the core losses is made. In order to take the anisotropic effects ofM-4 steel in the axial direction into account, the calculated losses for the teethare multiplied with a coefficient based on the measurements of Moses (1992).

3.3 Rotor lossesFor high-speed permanent-magnet motor applications, rotor losses generatedby induced eddy currents may amount to a considerable part of the total losses.The eddy currents are mainly induced in permanent magnets, which are highlyconductive, and also in the rotor steel. The major causes of eddy currents canbe categorized into the following three groups:

. No-load rotor eddy-current losses caused by the existence of slots.With the introduction of tooth-tops, the magnitude of the loss caused byslots can be made very small.

. On-load rotor eddy-current losses induced by the major mmf windingharmonics. In the current design the major winding harmonics are the11th and the 13th. The loss contribution of the higher order windingharmonics is relatively smaller.

. On-load eddy-current losses induced by the time harmonics of the phasecurrents due to PWM modulation.

There is no trivial way to remove the heat generated in the magnets and thus,the estimation of the rotor eddy-current losses is particularly important.Especially in the case of vacuuming where the rotor to frame and rotor to

Figure 3.End windings

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stators convection resistances are relatively large, rotor heat removal becomesa major problem. Excessive heat may result in the demagnetization of themagnets and rotor destruction. For our prototype, the reduction of the rotorlosses was not chosen as a major objective, and therefore the simplest rotorconstruction was realized. A substantial reduction of the rotor losses to anegligible level can be realized by the choice of a proper low-loss material forthe rotor complemented by a proper lamination of the magnets.

Using

P ¼

ZV

s ~E2 dV ¼

ZV

~J2

sdV ; ð2Þ

and considering the time average over a period T, the power loss equation for amagnet is

P ¼1

T

Z T

0

Z Lm

0

Z Li

0

Z þt=2

2t=2

J 2ðx; tÞ

sdx dy dz dt; ð3Þ

where Lm, Li and t are the thickness, length and width of the magnet,respectively.

Eddy-current loss in the magnets and the rotor steel may be calculated usingFE-AC analysis. The analysis is repeated for every space harmonic component(up to order 49), with the combination of simulated time harmonic componentsof the current waveform. It should be noted that for a certain space harmonic,the magnet width t is equivalent to the pole pitch of that space harmoniccomponent. Figure 4 shows the flux plot for the 11th order space harmonic andthe fundamental of the current as a representative example. The thin surfacecurrent density layer is defined between the stator and the airgap and the effect

Figure 4.Flux lines for the 11th

space harmoniccomponent of the

winding obtained fromFE-AC analysis at

6,240 Hz

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of the slotting is neglected. The frequency is 12 £ 520 Hz in the examplerepresenting 15,600 rpm. The loss at full load due to 11th space harmoniccomponent is found as 27 W in the magnet and 1 W in the rotor iron.

The analytical method, which uses the positional magnet flux densitywaveforms obtained from static FE solutions (Henneberger, 1998), can also beused to calculate rotor losses. Using this type of solution the contributions ofstator slotting can be found through the flux density variation in the magnet atno-load. The rotor losses caused by slotting is found to be only 10 W at15,600 rpm due to the use of tooth-tops.

3.4 Mechanical lossesFriction losses in the air space of high-speed machines largely contribute to thetotal losses. Especially considering the fact that the circumferential speed of theAFPM machine without flywheel is 217 m/s and with flywheel it is around330 m/s, which is many times higher than the standard 50 Hz machines, theheat created by air friction is not tolerable. Hence, the machine is designed to bevacuumed (100 mbar). In order to calculate the loss contribution of the airfriction, the methods recommended by Saari (1998) are used. The windagelosses under normal and reduced air pressure conditions are shown in Figure 5.

3.5 EfficiencyThe estimated efficiency map under reduced air pressure condition is shown inFigure 6.

4. Thermal analysisAn accurate estimation of the thermal behavior of an electrical machine isimportant considering the fact that safe operating conditions and overloadingcapabilities are dependent on the temperature rise. Temperature limits exist forpermanent magnets, winding insulations, and the glue used to attach themagnets. Besides, the winding resistances, consequently I 2R losses, and thepermanent-magnet flux are temperature dependent.

Figure 5.Air friction losses undernormal and reduced airpressure

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Having a lot of variable factors, unknown loss contributions with theircomplicated three-dimensional distributions, it is obvious that an exactdetermination of the machine’s thermal behavior is impossible. Nevertheless, inthe case of a high-speed machine knowing the order of magnitude of thetemperature rises allows the designer to choose a suitable cooling strategy andsuitable materials in order to increase the machine’s capability.

The thermal equivalent circuit is an analogy of the electric circuit in whichthe loss sources, thermal resistances and thermal capacitances are representedas current sources, resistances and capacitances, respectively. All machineelements are described by node sources, having an average surfacetemperature with respect to the ambient and a thermal capacitance. Theelements are connected to each other by conduction or convection resistances(Henneberger, 1998). Assuming constant thermal parameters and neglectingradiation, the linear differential equation for each node becomes

Pi ¼ Ci

dvi

dtþXn

j¼0

1

Rij

ðvi 2 vjÞ ð4Þ

where Pi, Ci, Rij and vi are heat loss in node i, thermal capacitance to ambient,thermal resistance between nodes i and j, and temperature in node i,respectively.

The location of the nodes and the power loss sources in a scaled quartermodel of the machine are shown in Figure 7.

It should be noted that, to each loss source a current source is connected andeach node has a capacitance to the ambient. The calculation of the thermalconvection resistances to the ambient air, end-cap air, airgap and cooling watercan be found in the work of Henneberger and Ben Yahia (1995) and Saari(1998).

Figure 6.Estimated efficiency mapat 100 mbar air pressure

condition

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5. Development5.1 Mechanical designThe safe operating conditions of the machine are set by means of mechanicalconstraints. The machine with its currents and permanent magnets representsa magnetic system which creates magnetic related forces. With rotation andcentrifugal forces added the overall mechanical system creates displacementswhich result in vibrations. The major magnetic forces which should beconsidered for the axial-flux machine, are the axial attractive forces betweenthe rotor permanent magnets and the stator cores, and the centrifugal forcesacting on the rotor.

Figure 7.Thermal equivalentcircuit

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With a high centrifugal force and also considering the resultant axialattractive forces acting on a magnet, it is calculated that a glue shouldwithstand a force density around 5 N/mm2. The tests conducted on thecommercially available glues at several temperatures up to 1208C showed thatthe protection of the magnets against such high centrifugal forces is notpossible only by means of a glue. The problem is solved by means of acombination of the following two measures:

(1) The nickel coating of the magnet is slightly grinded without causingoxidation of the magnet to increase the adhesive force of the glue.

(2) The magnets are ordered with ramped shaped corners. After they areglued on the rotor iron, a glass-fibre rim is placed around the rotormagnets (between the magnets and the iron part) as shown in Figure 8.

Modal analysis technique, which is commonly used in mechanical engineering,is used to obtain the eigen-frequencies of the rotor structure. The modal shapesof the rotor and related displacements or deformation shapes are computed atthe eigen-frequencies of the structure. The most important issue here of courseis that all of these frequencies are much higher than the operating frequencies.With reduced air pressure inside the machine, the frame can considerablycontribute to the vibration thus reducing reliability. Therefore, thedisplacements of the frame are computed and used for the determination ofits thickness. In order to minimize the displacement, the frame is thickened onboth sides and the end windings of the stator are filled with epoxy. A maximumdisplacement around 0.01 mm is achieved, while complete evacuation inside themachine is assumed.

Stress analysis of the rotor is conducted for the condition where the flywheelis assumed to be attached to the rotor. The resulting stress values(at 16,000 rpm) are shown in Figure 9. In this case the maximum stress valuelies around 1 GPa.

Figure 8.Glass-fibre rim around

the rotor magnets

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5.2 ManufacturingThe most binding difficulty is faced during the search for proper statorlaminations. The AFPM machine’s stator is a toroid, which should be made ofa very long and thin lamination segment. Due to the difficulty of getting alaminated toroidal stator core made of thin silicon and isotrophic steel fora reasonable price, the M-4 grain-oriented silicon steel is chosen for theprototype machine.

Another difficulty faced during the material search phase was related withthe glass-fiber rim around the magnets. Since the product bought did not haveenough strength, another one was prepared.

Slots are carved in the laminated stator cores with spark erosion technique.Since it is a very expensive method, punching is also advisable in massproduction. Yet with punching, making slot tops with any desirable shape isnot possible. In order to have a cost effective design, rectangular or round slotsshould be preferred in that case.

The stators are hand wound while paying special attention to make the endwindings shorter and the amount of copper in the three phases is the same onboth the stator units. A slot filling factor of 0.56 is achieved and the maximumcurrent density in the slot is determined as 6 A/mm2. The prototype parts(frame, stator and rotor) are shown in Figure 10.

6. PerformanceThe only way to truly verify the above loss analysis was to measure the loss inthe machine. This section describes the method of measuring the machine loss.

6.1 Machine test facilityThere are two methods used to measure the loss in electric motors: measuringthe loss directly, and measuring the difference between power input and poweroutput. The first method is very accurate when calorimetric methods are used.McLeod et al. (1998) describes the method, the errors that creep intocalorimetric measurements and how to avoid them, and summarizes past

Figure 9.Centrifugal stresses inthe (half) rotor with theattached flywheel at16,000 rpm

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research on the topic. However, this method is complicated and expensive toset-up.

The second method suffers from accuracy constraints. Measuring the powerinto an electric machine is problematic when an inverter is used sinceaccurately measuring PWM voltage waveforms is difficult (Domijan et al.,1998). Furthermore, loss measurement accuracy is poor since one is measuringthe difference between the two large numbers. Nevertheless, this method isoften used and was used for our tests.

The test bed consists of a 4-quadrant, field oriented, current-regulated, VSIfed 3,000 rpm servo motor drive, coupled to the machine under test through abelt drive transmission. The pulleys and belts can be changed depending on themachine to be tested. Figure 11 shows the set-up.

As mentioned earlier the accuracy of Pin2Pout measurements can be low. Tominimize the measuring errors, a LEM Norma D6000 power analyser is used tomeasure the electrical power into the machine, and a GIF 20,000 rpm, 20 N mtorque transducer is installed. Both measurement devices have an accuracy ofbetter than 0.1 per cent.

6.2 Machine measurementsAfter the manufacturing of the machine, the first step in the process ofmeasurements is the measurement of all phase resistances and inductances. It isnot only beneficial for determining the machine parameters, but also it is acheck for manufacturing. Unexpected differences between the predicted andmeasured phase resistances and inductances, or unexpected differencesbetween phases may imply a construction error. The phase inductances weremeasured by means of an impedance analyzer. The measured inductances atseveral rotor positions were equal. This test can be seen as a check on themagnets and airgaps. Under normal circumstances, the inductance should notvary with respect to rotor position for surface-mounted PM machines.

Figure 10.AFPM machineprototype parts

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The back-emf test is important since the relationship between the producedtorque and the back-emf is obvious. From this test, the machine’s back-emfconstant can also be derived. By driving the AFPM test machine with theinduction machine at several speeds, the phase-emf (rms) values were recordedand compared with the predicted ones. A line-emf wave form recorded by

Figure 11.Machine testing facility

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means of a digital scope at the speed of 9,065 rpm is plotted in comparison withthe predicted waveform as shown in Figure 12. Except the small differencecaused by the airgap lengths, the waveforms coincide. Small hatches on themeasured emf waveform are caused by the low precision of the magnetproduction. The magnets have small dimensional differences between eachother.

While doing the back-emf measurement test, the no-load losses of themachine were also measured by means of the torque and speed sensors.Measured losses are compared with the predictions and extreme differences aredetected. As it was already predicted during the first stage of the project, theonly possible explanation is that the unpredictable (hysteresis) losscomponents are appearing in the stator cores because of the anisotropy ofthe grain-oriented M-4 steel. Figure 13 shows the measured efficiency andphase current values with respect to varying torque at 5,000 rpm as arepresentative example. A linear relation between the current and the torque isobserved.

Figure 12.Measured and predicted

line-emf waveforms at9,065 rpm

Figure 13.Measured phase current

and efficiency withrespect to torque at

5,000 rpm

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Taking into account the amount of core losses observed from the no-load test,the experimental efficiency results and the other loss components are found tocoincide with the predictions for the greater part. This means that if a properstator steel was used instead of the M-4 steel, the desired efficiency levelswould have been easily reached. For example, at 7,000 rpm and 13.84 N m, thetotal loss is measured as 947 W (output power Pout ¼ 10:15 kW; input powerP in ¼ 11:097 kW; power factor cosf ¼ 0:988; efficiency h ¼ 91:47 per cent). Ifthe unexpected loss of 460 W under reduced air pressure (100 mbar) did notexist, the efficiency value of 95.6 per cent could have been achieved.

It can be concluded from the conducted tests that the thermal analysis of theAFPM machine by means of an equivalent circuit gives sufficient informationabout the thermal behavior of the machine. The differences between themeasured and the predicted temperatures are not more than 128C. In theexample shown in Figure 14, a case with water cooling (4 l/min water flow rate)and reduced air pressure (100 mbar) is investigated. The phase currents are setto 30 A rms, and the speed is 7,000 rpm. Steady-state temperatures are reachedin this case. Stator winding temperatures indicated in the figure are measuredfrom a thermocouple which is placed in a slot. The measured and the predictedsteady-state winding temperatures converged to the same level.

7. ConclusionsThe paper has aimed to summarize the critical aspects concerning the design,manufacturing and testing of a high-speed axial-flux permanent-magnetmachine which will be applied in a hybrid electric vehicle application. Thehigh-speed design aspects were given. Analysis of the losses and the thermalbehavior of the machine are included. Mechanical constraints and the aspectsof manufacturing are summarized. The machine’s test bench is described andmeasurements of loss are shown.

Figure 14.Measured temperaturerises of the magnet,windings and stator yokein comparison with thepredicted results at 30 Aphase current, 7,000 rpmspeed, 4 l/min water flowrate at 100 mbar

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References

Cadee, T.P.M. and van Rooij, J.H.M. (1997), “Hybrid driving system”, Patent number: EP 0 665 B1.

Domijan, A., Carkowski, D. and Johnson, J.H. (1998), “Nonsinusoidal electrical measurementaccuracy in adjustable-speed motors and drives”, IEEE Transactions on IndustryApplications, Vol. 34 No. 6, pp. 1225-33.

Henneberger, S. (1998), “Design and development of a permanent magnet synchronous motor fora hybrid electric vehicle drive”, PhD thesis, Catholic University Leuven.

Henneberger, G. and Ben Yahia (1995), “Calculation and identification of a thermal equivalentcircuit of a water cooled induction motor for electric vehicle applications”, IEEInternational Conference on Electrical Machines and Drives, pp. 6-10.

Jensen, C.C., Profumo, F. and Lipo, T.A. (1992), “A low loss permanent-magnet brushless DCmotor utilizing tape wound amorphous iron”, IEEE Transactions on Industry Applications,Vol. 28 No. 3, pp. 646-51.

McLeod, P., Bradley, K.J., Ferrah, A., Clare, J.C., Wheeler, P. and Sewell, P. (1998), “High precisioncalorimetry for the measurement of the efficiency of induction machines”, IEEE-IndustryApplications Society Annual Meeting, pp. 304-11.

Miller, T.J.E. and Hendershot, J.R. (1995), Design of brushless permanent magnet motors, OxfordUniversity Press, Oxford.

Moses, A.J. (1992), “Problems in modelling anisotrophy in electrical steels”, International Journalof Applied Electromagnetics in Materials No. 3, pp. 193-7.

Platt, D. (1989), “Permanent magnet synchronous motor with axial flux geometry”, IEEETransactions on Magnetics, Vol. 25 No. 4, pp. 3076-9.

Saari, J. (1998), “Thermal analysis of high-speed induction machines”, PhD thesis, ActaPolytechnica Scandinavica.

Sahin, F. (2001), “Design and development of an high-speed axial-flux permanent-magnetmachine”, PhD thesis, Eindhoven University of Technology.

Sahin, F. and Vandenput, A.J.A. (1999), “Design considerations of the flywheel-mountedaxial-flux permanent-magnet machine for a hybrid electric vehicle”, 8th EuropeanConference on Power Electronics and Applications, CD.

Zhang, Z., Profumo, F. and Tenconi, A. (1996), “Axial flux wheel machines for electric vehicles”,Electric Machines and Power Systems, Vol. 24, pp. 883-96.

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Current shapes leading topositive effects on acoustic

noise of switched reluctancedrives

M. Kaiserseder, J. Schmid and W. AmrheinInstitute of Automatic Control and Electrical Drives, Department ofElectrical Control and Power Electronics, Johannes Kepler University

Linz, Austria

V. ScheefRobert Bosch GmbH, Department FV/FLP, Gerlingen, Germany

Keywords Torque, Optimization, Motors

Abstract A torque ripple minimization technique for switched reluctance motors is shown in thispaper. Precalculated current shapes are applied to reduce torque ripple and to raise the degrees offreedom of the application in the commutation region. The optimization criteria for this region canbe chosen freely. Therefore, it is possible to take positive effect to some motor characteristics likepower losses, mechanical vibrations or acoustic noise.

1. IntroductionThe advantages of switched reluctance drives like high mechanical andthermal robustness (no permanent magnets, no rotor coils), simple mechanicalstructure, simple power electronics compared to AC and DC drives, hightorque/mass ratio and low production costs have made them an attractivesolution for industrial applications. The control strategy used in the past – thecurrent was kept constant during conduction period – was very simple, but thenon-linearities between phase current, rotor position and torque caused a veryhigh ripple in the torque profile. Due to the high ratio of harmonics in thefrequency spectrum of radial force, disagreeable mechanical vibrations aregenerated leading to bad acoustic behavior. The increasing availability ofhigh-power fast switches, integrated circuits and high performance digitalsignal processors has recently motivated research in advanced techniques forminimizing these problems.

Finding the source of acoustic noise within electric motors is an old topic. Soa survey of noise sources within the switched reluctance motors (SRM) waspublished by Hendershot (1993). Theoretical analysis of vibrations of SRM’s(Colby et al., 1996; Pillay and Cai, 1999, Vandevelde, et al., 1999) showingcritical mode shapes like the oval mode for 6/4 SRM and finite element analysisof periodic excitation of natural vibration modes are the basis for all acousticalnoise minimization methods. Zero voltage loop conduction, voltage smoothing



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or intelligent random modulation of PWM frequency (Blaabjerg et al., 1994) areused to reduce effects caused by current switching. Also specialized motordesign like reluctance shark-machines can be taken into account (Rasmussenet al., 2000).

Other control strategies, using precalculated torque sharing functions, werestudied not only to reduce acoustical noise, but also to reduce torque ripple(Choi et al., 2000; Kim and Ha, 1996; Kjaer et al., 1996; Stankovic et al., 1996).Another method, reducing torque ripple, deals with feedback linearizationcontrol (Haiqing et al., 1996; Ilic-Spong et al., 1987; Rossi et al., 1994; ) andconverts the non-linear characteristics of the system into an equivalent linearform. Simulations of open loop strategies show that it is possible to produceripple free torque, but an unprecise model of the drive leads to less performance.

Most methods used to reduce torque ripple of a SRM suffer from excessivememory requirement and high complexity. Furthermore, a well known motormodel is needed. The main advantage using open loop strategies withprecalculated current shapes is the possibility to take positive effect to featureslike mechanical vibration, acoustic noise or copper losses.

2. Purposed open loop control methodConventional torque sharing function methods lead to the purposed approachthat works on drives in which at least two phases are used to produce torque atany given time. Precalculated current shapes are applied to reduce torqueripple and to raise the degrees of freedom of the application in the commutationregion. To raise efficiency only phase current, which generates positive torqueis permitted. The presented work shows the minimization of the gradient inradial force as optimization criteria for acoustic noise and mechanicalvibrations in the overdetermined commutation region. The algorithm causesindependent turn on and turn off angles for current conduction at differenttorque levels. Saturation effects during commutation were also taken intoaccount.

Structural finite element analysis is used to predict the mode shapes andfrequencies of excited vibrations. A DSPACE signal processor board is used forexperimental implementation. Because of the varying turn on and turn offangles the precalculated and optimized current shapes have to be stored intwo-dimensional lookup tables. A simple PI controller is used to control thedefault precalculated and optimized current shapes, torque is impressed. Phasecurrent and rotor position are taken as inputs for the current controller. Themanipulated output is the voltage at the stator coil.

2.1 SRM modelThe presented algorithm has been tested on a 6/4 SRM and on a 18/12 SRM butonly the results of the 6/4 motor will be shown in this paper. The basic set ofequations for the electrical model of a SRM is

Switchedreluctance drives

999

vi 2 iiri 2dCi

dt¼ 0 ð1Þ

where the phase voltage vi and the phase current ii can be measured and theflux linkage Ci is calculated by the finite element program. The subscript iindicates the phase number ði ¼ 1. . .mÞ:

The mechanical behavior of the SRM is described by the following equations

Xm

i¼1

Ti 2 TL 2 Jdv

dt¼ 0 ð2Þ

and

dw

dt2 v ¼ 0: ð3Þ

The moment of inertia, J, depends on the drive design and the phase torque Ti isa non-linear function of phase currents (ii, ij) and rotor position, w.

2.2 Basic conceptAccording to equation (1), the major source of acoustic noise in SRM’s is theradial force acting on the stator poles. The shape of this force is a non-linearfunction of phase current ii and rotor position w (Figure 1). Radial force is

Figure 1.Normalized radial forceis a non-linear function ofnormalized phase currentii and rotor position w

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a result of the finite element calculation. A characteristic value for the acousticnoise is the sound power

W ð4Þ ¼ r0cs

Zn

v2rms dS: ð4Þ

For comparison under the same boundary conditions the terms of air densityr0, sound velocity, c, and radiation loss factor, s, are the same. Provided thatthe surface does not change, comparison of acoustic noise can be found by thevariation of sound power W(v). Under the mentioned simplifications and onlyfor comparison, sound power can be set proportional to the mean square rmsvalues of some (n) housing points velocities v2

rms like

W ð4Þ <Pn

1 v2rms

n: ð5Þ

Due to the steep slope of radial force at the end of the conduction period, a highratio of harmonics in the frequency spectrum of this force excites noisymechanical vibrations. For a given speed, excitation of critical natural vibrationmodes can be avoided by minimizing a default order of harmonics. The chosenoptimization (minimization of the gradient of radial force) reduces the numberand the level of harmonics and therefore, it will reduce the acoustical noise overa wide range of speed.

The given task, minimization of the gradient of the radial force andreduction of torque ripple, leads to an optimization problem, where the requiredobjective function

�� ›Frad

›x¼ min ð6Þ

has to be solved subject to

Xm

i¼1

Ti ¼ const ð7Þ

The region of torque generation is now split into two sections (Figure 2): section(A) where only one phase generates torque and section (B) where torque isproduced by two phases.

2.3 OptimizationIn section (A), defined between wstart_SectA and wend_SectB, there is nooptimization needed to find current waveforms for constant torque because thecontour lines of the torque shape (Figure 3) define the shape of the phasecurrents.


1001

Figure 2.Normalized currentconduction andnormalized motor torque

Figure 3.Normalized currentshapes defined by theircontour lines for differenttorque levels

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Section (B) starts at rotor position wstart_SectB and ends at wend_SectB. In thissection, there are an infinite amount of combinations of current pairs for theconducted phases to reach the default torque level.

The set of equations that has to be solved in section (A) is the same as inequations (6) and (7), but it is under determined. Using the Lagrange multiplierl a solution can be found.

Optimizing equations (6) and (7) leads to a more dimensional optimizationproblem

QF1rði1ðwÞÞ

›wþ

F2rði2ðwÞÞ

›w¼ min ð8Þ

TðiiðwÞ; ijðwÞ;wÞ ¼ const ð9Þ

w ¼ ½wstart_SectB. . .wend_SectB�: ð10Þ

By reducing the size of dimensions, it is possible to eliminate the rotor positionw from the equations. The valuation factor Q sets a higher priority to the fallingedge F1rði1ðwÞÞ=›w of the radial force. Boundary values for this equation are

F1rðwstart_SectBÞ ¼ F1rðwend_SectAÞ ð11Þ

F1rðwend_SectBÞ ¼ 0: ð12Þ

The systems (8) and (9) decoupled from rotor position w can now be written as

f ðiiw ; ijwÞ ¼ QF1rwði1wÞ

›wþ

F2rwði2wÞ

›w! min ð13Þ

Tw ¼ gðiiw ; ijwÞ ¼ const ¼ TL ð14Þ

for every rotor position w.To find a solution for the above-problem, Lagrange deals with the fact, that

the gradients at the extreme points must have the same orientation. TheLagrange multiplier, l is introduced to take account of the different length ofthe gradients and to make the Lagrange equation

Lði1; . . .im;l1;lkÞ ¼ f ðiÞ þXk

i¼1

liðci 2 giðiÞÞ ð15Þ

solvable.


1003

The m+k solutions of the optimization problem are

›mþkLði1; . . .im; l1; . . .; lkÞ

›i1; . . .; ›im; ›l1; . . .; ›lk

¼ 0: ð16Þ

li describes the variety of the optimum, changing the constant in the auxiliarycondition.

There is also a numeric method leading to solutions for the aboveoptimization problem, but this algorithm will not be described in detail.Because of the huge amount of data this numeric algorithm can find thesolution under less time exposure. Another advantage of the numeric algorithmis the use of the calculated finite element calculation torque data whereas norestrictions for the optimization criteria are given.

3. Comparison of simulated and experimental resultsExperimental results indicate that the effect of stator yoke saturation (Figure 4)in commutation regions may not be neglected.

The resulting torque in commutation section is not equal to the arithmeticsum of the phase torque values of the conducted phases. Taking this fact intoaccount the acoustic optimization results in an almost ripple free torque profile.

For implementation, a DSPACE signal processor board is used. Therefore,the precalculated current shapes are stored in the two-dimensional lookuptables. The output of the used current controller is the mean value of a pulse

Figure 4.Effect of stator yokesaturation on normalizedmotor torque

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width modulated (PWM) voltage. This voltage is proportional to the voltageacross the stator coil vphase. This soft chopping technique (Miller, 1996) leadsalso to less acoustic noise. The current controller takes the phase current androtor position as inputs to calculate the manipulated variable, vphase.

The structure of the algorithm may cause different turn on and turn offangles for current conduction (Figure 5) at different torque levels.

To see how the optimization works, a comparison between two simulatedand optimized current waveforms is given in Figure 6.

Both optimization criteria have been verified in the experiment (Figure 7).Due to the mechanical asymmetry of the SRM, there is still a small ripple in thetorque profile.

This method can be used for torque control with feedforward in certainapplications. For the smaller offset of the torque control loop a controller withhigher dynamic can be designed.

For the shown point of operation the acoustic level of the structure bornesound could be reduced by 2.8 dB at a speed of 1,000 rpm (Figure 8). At differentspeeds of 500-3,000 rpm and the same motor torque level, the reduction ofthe acoustic level of structure borne sound was between 2.8 and 3.4 dB.

Further measurements for other motor torque levels show similar resultsand will not be presented in this paper.

Figure 5.Normalized computed

current shapes fordifferent torque levels


1005

Figure 6.Comparison of simulatednormalized torqueperformance for differentoptimized currentwaveforms. (A) Lowgradient in radial force,and (B) low copper losses

Figure 7.Comparison of measurednormalized torqueperformance for differentoptimized currentwaveforms. (A) Lowgradient in radial force,and (B) low copper losses

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4. ConclusionsThe conventional torque sharing functions have led to the work of this paper.There is no angular restriction in the torque producing region and theoptimization criteria for a default constant torque can be chosen freely. Due tothe fact that only 66 percent of conduction period is optimized for a minimumgradient of the radial force a reduction in mechanical vibration can be reached.There is a reduction of structure borne sound of about 3 dB over a wide speedrange. Controlling precalculated current shapes leads to an open loop structurefor the motor torque. The memory requirement should not be higher than thefeedback linearization controls or real time calculating systems.

References

Blaabjerg, F., Pedersen, J.K., Nielsen, P., Andersen, L. and Kjaer, P.C. (1994), “Investigation andreduction of acoustical noise from switched reluctance drives in current and voltagecontrol”, Proc. ICEM ’94, Vol. 3, pp. 589-94.

Choi, C., Lee, D. and Park, K. (2000), “A new torque sharing function method for ripple free torquecontrol of a switched reluctance motor”, Proc. ISIM, pp. 199-204.

Colby, R.S., Mottier, F.M. and Miller, T.J.E. (1996), “Vibration modes and acoustic noise in afour-phase switched reluctance drive”, IEEE Trans. Ind. App., Vol. 32, pp. 1264-357.

Haiqing, Y., Panda, S.K. and Chii, L.Y. (1996), “Performance comparison of feedback linearizationcontrol with PI control of four-quadrant operation of switched reluctance motor”, Proc.IEEE APEC’96, pp. 956-62.

Figure 8.Comparison of measured

displacement of ahousing point and the

terzband frequencyanalysis for different

optimized currentwaveforms. (A) Low

gradient in radial force,and (B) low copper losses


1007

Hendershot, J.R. (1993), “Causes and sources of audible noise in electrical motors” Proc. 22ndIncremental Motion Control Systems and Devices Symposium, pp. 259-70.

Ilic-Spong, M., Marino, R., Peresada, S. and Taylor, D. (1987), “Feedback linearizing control ofswitched reluctance motors”, IEEE Trans. Automat. Contr., Vol. AC-32, pp. 371-9.

Kim, C.H. and Ha, I.J. (1996), “A new approach to feedback-linearization control of variablereluctance motors for direct-drive applications”, IEEE Trans. Control Systems Technology,Vol. 4 No. 4, pp. 348-62.

Kjaer, P.C., Gribble, J.J. and Miller, T.J.E. (1996), “High-grade control of switched reluctancemachines”, Conf. Rec. IEEE IAS Annu. Meeting, pp. 92-100.

Miller, T.J.E. (1996), Switched Reluctance Motors and their Control, Magna Physics Publishingand Claredon Press, Oxford.

Pillay, P. and Cai, W. (1999), “An investigation into vibration in switched reluctance motors”,IEEE Trans. on Ind. App., Vol. 35, pp. 589-1596.

Rasmussen, P.O., Blaaberg, F., Pedersen, J.K. and Jensen, F. (2000), “Switched reluctance-sharkmachines – more torque and less acoustic noise”, Proc. IAS 2000, Vol. 1, pp. 93-8.

Rossi, C., Tonielli, C. and Tonielli, A. (1994), “Feedback linearizing and sliding mode control of avariable reluctance motor”, Int. J. Control, Vol. 60 No. 4, pp. 543-68.

Stankovic, A.M., Tadmor, G. and Coric, Z.J. (1996), “Low torque ripple control of current-fedswitched reluctance motors”, Conf. Rec. IEEE IAS Annu. Meeting, pp. 84-91.

Vandevelde, L., Gyselinck, J.J.C. and Melkebeek, J.A.A. (1999), “Theoretical and numericalanalysis of vibrations of magnetic origin of switched reluctance motors”, Proc.COMPUMAG 99, pp. 58-9.

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Vibrations of magnetic originof switched reluctance motors

Lieven VandeveldeElectrical Energy Laboratory (EELAB), Department of Electrical Energy,

Systems and Automation (EESA), Ghent University, Belgium

Johan J.C. GyselinckDepartment of Electrical Engineering (ELAP), Montefiore Institute,

University of Liege, Belgium

Francis Bokose and Jan A.A. MelkebeekElectrical Energy Laboratory (EELAB), Department of Electrical Energy,

Systems and Automation (EESA), Ghent University, BelgiumKeywords Motors, Vibration, Noise, Magnetic forces

Abstract Vibrations and acoustic noise are some of the fundamental problems in the design andexploitation of switched reluctance motors (SRMs). Adequate experimental and analysis methodsmay help to resolve these problems. This paper presents a theoretical analysis of the magnetic forcedistribution in SRM and a procedure for calculating the magnetic forces and the resultingvibrations based on the 2D finite element method. Magnetic field and force computations and astructural analysis of the stator have been carried out in order to compute the frequency spectrumof the generalized forces and displacements of the most relevant vibration modes. It is shown thatfor these vibration modes, the frequency spectrum can be predicted analytically. The theoreticaland the numerical analyses have been applied to a 6/4 SRM and an experimental validation ispresented.

1. IntroductionThe main sources of vibration and noise in switched reluctance motors (SRMs)have been highlighted in earlier publications (Cameron et al., 1992; Pillay andCai, 1999). Experiments have been carried out on existing SRMs of variousconfigurations to identify the factors which affect the production of vibrationand noise (Cameron et al., 1992; Wu and Pollock, 1995). It is an established factthat the main contributors to vibration and noise in SRMs are the sources ofmagnetic origin. All mechanical components of the machine contribute tovibration and noise in varying degrees and are supposed to be modelled.However, the deformation of the stator lamination stack due to magnetic forcesis the main source of vibrations and noise in SRMs (Cameron et al., 1992; Colbyet al., 1996; Wu and Pollock, 1995). These sources are affected by the control



This research has been carried out with the financial support of the Fund for Scientific Research –Flanders (Belgium) (F.W.O.-Vlaanderen) and in the frame of the Interuniversity Attraction Poles(IAP P4/20-P5/34) supported by the Belgian government. L. Vandevelde is a Postdoctoral Fellowof the Fund for Scientific Research – Flanders.

Vibrations ofmagnetic origin

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Vol. 22 No. 4, 2003pp. 1009-1020


DOI 10.1108/03321640310482968

strategy used and are characterized by complex wave patterns. The oddharmonics of these waves excite more vibrations and noise than the evenharmonics (Neves et al., 1998). It is due to this complexity that proper analyticalas well as 2D/3D finite element (FE) electromagnetic and structural analysis arerequired. However, in a comparative FE analysis of modes and frequenciesof 2D and 3D SRM stator structures, it was shown that 2D FE analysis ofvibrations and noise in the stator structures is adequate, and that the error isonly about 15 percent (Neves et al., 1998).

The aim of this paper is to contribute to both the theoretical and thenumerical analysis of vibrations of magnetic origin of the SRM.

First, the exciting magnetic forces are studied with respect to theirdistribution in time and space by determining the time harmonic and spatialorders of the forces. This analysis includes the effects of possible anomalies(asymmetries) of the SRM, the converter and the load.

Further, a 2D FE procedure for calculating the vibrations is presented, whichis based on a novel force calculation method (Vandevelde et al., 1998) and on themodal superposition technique ( Javadi et al., 1995). By considering thevibrations as rotating waves, the frequency spectrum and the direction ofrotation of the waves can be predicted on the basis of the theoretical analysis.

The theoretical and the numerical analyses have been applied to a 6/4 SRMand the results are compared with vibration measurements at different loadconditions.

2. Theoretical analysisFor the theoretical analysis of the magnetic forces acting on the stator of aswitched reluctance motor and the ensuing deformation, we divide the statorcross section into Ns sectors, i ¼ 0; . . .; N s 2 1; with Ns the number of statorpoles, as shown in Figure 1 for a 6/4 SRM. We further consider a point P0 in thezeroth sector and the corresponding points – i.e. with the same relativeposition – in the other sectors. The magnetic force density �fm and thedisplacement �u in this set of Ns points can be resolved into a series ofcomponents with time harmonic orders lk and spatial orders kk. For example,the radial magnetic force f i

r in the points Pi ði ¼ 0; . . .; N s 2 1Þ can be writtenas

f ir ðr0; u0; tÞ ¼

k

XR �Frkðu0; r0Þe

j lk2pf rott2kk2pNs

i� ��

ð1Þ

where frot is the rotor speed (in hertz), r0 and u0 are the polar coordinates of thepoint P0 in the zeroth sector. The direction of increasing i corresponds with thedirection of rotation, which is chosen anticlockwise.

Two components with time harmonic and spatial orders ðlk; kkÞ and ðll ; klÞ;respectively, are equivalent if

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ðlk;kkÞ ¼ ^ðll; kl þ mNsÞ ðm : integerÞ: ð2Þ

Therefore, the following constraints can be imposed on the orders:

lk $ 0 ð3Þ

if lk ¼ 0 : 0 # kk #N s

2

if lk . 0 : 2 N s

2 þ 1 # kk #N s

2

8<: ð4Þ

In this case, the sign of the spatial order kk determines the direction of rotationof the component, viz the same direction as the rotor for positive orders and theopposite direction for negative orders.

In the following paragraphs, the orders ðlk;kkÞ of the force components willbe discussed in the case of a 6/4 SRM.

2.1 Main force componentsFirst, we consider an idealized SRM drive, i.e. we assume that the SRM and itspower electronic supply are completely symmetrical and that the load torque isconstant, whereas the speed may fluctuate due to the electromagnetic torque

Figure 1.Cross section of a 6/4

SRM


1011

ripple of the SRM. Under these conditions, the working principle of the 6/4 SRMleads to the following periodicities in time and space of both radial andtangential magnetic force densities fr and fu:

f iðr0; u0; tÞ ¼ f iþ1 r0; u0; t 21

12frot

ð5Þ

f iðr0; u0; tÞ ¼ f i r0; u0; t 21

4frot

ð6Þ

On the basis of these periodicity conditions (5) and (6), it can be shown easilythat the time and space harmonic orders, lk and kk, are bound by the followingrestrictions:

lk ¼ 4ð3m 2 nÞ

kk ¼ 2n

(ðm; n : integerÞ ð7Þ

We remark that the spatial orders are even and that the time harmonics arequadruples, i.e. the fundamental frequency of the forces is the drive frequencyf 0 ¼ 4f rot:

According to equations (7) and (4), with N s ¼ 6; the main magnetic forcecomponents, i.e. for an idealized 6/4 SRM drive, are determined by:

kk ¼ 0 : lk ¼ 4ð3mÞ ¼ 0; 12; 24; . . .

kk ¼ 2 : lk ¼ 4ð3m 2 1Þ ¼ 8; 20; 32; . . .

kk ¼ 22 : lk ¼ 4ð3m þ 1Þ ¼ 4; 16; 28; . . .

ð8Þ

2.2 Side bandsLoad torque ripples and possible asymmetries of the SRM and the convertermay cause additional components or “side bands” to the main components(equation (8)).

As the fundamental frequency of the phase currents is the drive frequencyf0 ¼ 4frot and as each phase winding A, B and C consists of a pair of coilswound around diametrically opposed poles and connected in series,asymmetries of the converter are characterized by the following ordersðl0k; k

0kÞ :

l0k ¼ 4s1

k0k ¼ 2s2

8<: ðs1;s2 : integerÞ ð9Þ

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The orders of asymmetries of the stator geometry and of static rotoreccentricity (i.e. eccentricity of the shaft and the stator) are given by:

l0k ¼ 0

k 0k ¼ 1s

8<: ð1s : integerÞ ð10Þ

In a reference frame fixed to the stator, the asymmetries of the rotor geometry,such as dynamic rotor eccentricity (i.e. eccentricity of the shaft and the rotorcore), feature the following orders:

l0k ¼ 1r

k 0k ¼ 1r

8<: ð1r : integerÞ ð11Þ

As the torque corresponds to tangential force components of zeroth spatialorder, load torque ripples are characterized by the following orders:

l0k ¼ t

k 0k ¼ 0

8<: ð12Þ

2.3 General expression of the time harmonic and spatial ordersThe magnetic forces in an actual SRM are found by “modulating” the forces inan idealized SRM with the anomalies mentioned earlier. The general expressionof the time harmonic and spatial orders of the magnetic force distribution isthus found by adding the orders ðl0k,k

0kÞ (equations (9)-(12)) to the orders of the

main components ðlk; kkÞ (equation (8)):

lk ¼ 4ð3m 2 nÞ þ 4s1 þ 1r þ t

kk ¼ 2n þ 2s2 þ 1s þ 1r

(ð13Þ

Analogously, the vibrations of magnetic origin are characterized by the ordersðlk; kkÞ (equation (13)). In this respect, we remark that mechanical asymmetriesof the SRM, which may cause a discrepancy between the orders of the excitingforce component and those of the resulting vibration component(s), are alreadyincluded in equation (13), viz. via the term 1s.

The expression of the orders ðlk; kkÞ (equation (13)) can be used forpredicting the frequency spectrum of the vibrations (and thus the noise) ofSRMs and for determining the origin of experimentally observed or computedfrequency components.


1013

3. Numerical analysis3.1 Magnetic force calculationFor calculating the magnetic force distribution, an original method, presentedby the authors in previous papers (Vandevelde and Melkebeek, 2001;Vandevelde et al., 1998), is used. The method is based on the separation of theforces in a magnetized elastic material into magnetic long-range andshort-range forces or stresses. The latter represent mechanical stresses aswell as magnetic short-range forces, i.e. local interactions (magnetostriction).

As shown in Figure 2, the long-range magnetic force acting on a part V1

inside the surface S1 of a magnetic body is defined as the force acting on itwhen separated from the rest of the magnetic material (V2) by means of animaginary gap of infinitesimal width. This way, the short-range forces (i.e. theinteractions on a microscopic scale) between the material in V1 and V2 vanish.The long-range force �F on the material in V1 can be calculated by integratingthe Maxwell stresses �fs over the surface S0

1. On the basis of this definition, thedeformation due to the long-range magnetic forces can be calculated by meansof the following (fictitious) magnetic force density �f *

m (N/m3) (Vandevelde andMelkebeek, 2001; Vandevelde et al., 1998):

�f *m ¼ �7·T *

m; ð14Þ

T *m ¼ �B �H 2

m0

2H 2I þ

m0

10M 2I þ

m0

5�M �M; ð15Þ

which features a singularity at material boundaries, viz. a surface force density�T

*m (N/m2) given by

�T*m ¼

m0

2M 2

n �n 2 �n ·m0

10M 2I þ

m0

5�M �M

� ; ð16Þ

where �n and Mn are the outward normal unit vector on the surface and thenormal component of the magnetisation, respectively.

In 2D FE calculations, by using first order triangular elements, the nodalforces are easily obtained by computing the stresses �n·T *

m on both sides of theelement edges.

By way of illustration, in Figure 3, some results of a static 2D FE simulationof a 6/4 SRM are shown, viz. the flux pattern, the force distribution (nodalforces), and the ensuing deformation (with a magnification of 2,000). The force

Figure 2.Definition of long-rangemagnetic forces

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distribution, shown in Figure 3(c), clearly consists of both volume and surfaceforces. The volume forces are (approximately) directed towards the curvaturecenter of the flux lines, as if trying to deform the magnetic material such thatthe magnetic path length shortens.

3.2 Structural analysis and equations of motionVibrations and the resulting acoustic noise can be reduced if coincidencesbetween the exciting force frequency harmonics and the natural frequencies ofthe machine structure are avoided. Often it is impossible to avoid suchcoincidences at all operating points (Verma and Balan, 1996).

From the perspective of noise and vibrations, the stators of SRMs can bemodeled as a system consisting of a number of masses linked by springs and

Figure 3.Some results of a static

2D FE calculation


1015

damping elements, in order to facilitate analytical solution of the dynamicbehavior of the structure. The vibration model should transform thesedistributed parameters into discrete forms, enabling differential equations ofmotion to be constructed. Alternatively, modal analysis, which is a process offorcing the structure to vibrate mainly at a priori determined resonantfrequencies and vibration modes, can also describe effectively the dynamics ofthe vibrating structure (Cai and Pillay, 2001; Hong et al., 2002; Tang, 1997).

Prediction of the natural modes and frequencies of SRM mechanicalstructures is very important at the design stage. Analytical models ofsimplified structures can be formulated relating the resonant frequencies to thegeometrical parameters and material properties of the machine (Blaabjerg et al.,1999). An approximate formulation for the frequency of the fundamental modecan be achieved by modeling the stator lamination stack as a uniformcylindrical shell and using the energy conservation principle in terms of kineticand potential energies to deduce the frequency (Colby et al., 1996). For relativeaccuracy, structural FE analysis for determining the natural modes andfrequencies is deployed.

For the analysis of vibrations generated by the magnetic forces, a structuralanalysis of the SRM stator has been carried out by means of the 2D FE method.Figure 4 (a)-(e) shows some of the computed mode shapes, viz. those which canbe characterized as a 0th, 2nd and 4th order radial displacement, respectively.The outer stator boundary in undeformed state is shown as dashed line.

If mechanical damping is not considered, the equations of motion in thefrequency domain are written as follows:

2v2kmi �qik þ ki �qik ¼ �fik ð17Þ

where mi and ki are the generalized mass and stiffness, respectively, and �qik and�fik are the complex values of the kth time harmonic (pulsation vk) of thegeneralized displacement and force, respectively, of the ith mode.

3.3 Dynamic FE simulationsThe procedure for calculating the modal vibrations in a SRM can besummarized as follows. First, a series of calculations of the magnetic field,yielding the magnetic force distribution based on the magnetic stress tensor

T *m (equation (15)), are carried out, where the input consists of the (measured)

phase currents or voltages as a function of the rotor position. Owing to theperiodicity condition (5), only a twelfth of a rotor revolution (1/12frot) has to besimulated for a 6/4 SRM. On the basis of the mode shapes, computed by meansof a structural FE analysis of the SRM, and the nodal magnetic forces, thegeneralized forces fi(t), and further by means of a Fourier analysis, the forcecomponents �fik are calculated. Finally, the time harmonics of the modaldisplacements are computed by using equation (17).

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Figure 4.Computed mode shapes,natural frequencies and

modal displacements of a1.2 kW 6/4 SRM( frot ¼23.33 Hz,

f0¼93.33 Hz)


1017

Both FE simulations and measurements have been carried out on a 1.2 kW 6/4SRM at a speed of 1,400 rpm, i.e. f rot ¼ 23:33 Hz; f 0 ¼ 93:33 Hz; and operatingin single pulse mode. For the simulations, an idealized 6/4 SRM drive, asdefined earlier, and constant speed were assumed. Figure 4(f )-( j) shows thecalculated modal displacements for the modes depicted in Figure 4(a)-(e). Byconsidering the nonzero order modes in pairs, e.g. the pair 2/20, two modaldisplacements with the same amplitude and with a phase shift of ^908 result ina rotating wave. For instance, a phase lag (lead) of 908 of the modaldisplacement of mode 20 with respect to mode 2 results in a wave rotating in thesame (opposite) direction as the rotor, which corresponds to a spatial order kk of2 (22). The frequency spectrum and direction of rotation of these waves can bepredicted on the basis of equation (8) for an idealized SRM or equation (13)in general. The computed frequency spectrum and phase shifts of modes 2 and20, as shown in Figure 4(g)-(h), correspond to the predicted frequency spectrumfor spatial orders 2 and 22 (equation (8)). Analogously, the computed resultsfor mode 0 (standing wave) and modes 4/40, shown in Figure 4(f ), (i) and ( j),respectively, agree with the theoretical results (8), taking into account that,according to equation (2), spatial orders 4 and 24 are equivalent to orders 22and 2, respectively.

4. Experimental verificationVibration measurements have been carried out on the 6/4 SRM at a speed of1,400 rpm ð f rot ¼ 23:33 HzÞ and at two load conditions set by controlling thephase currents and the load torque, viz. at a set current I set ¼ 2:0 and 7.5 A,respectively.

For a set current I set ¼ 2:0 A; the torque, which corresponds to forcecomponents of zeroth spatial order ðkk ¼ 0Þ; is very low (less than 1 Nm).Figure 5 shows that in this case, the frequency spectrum of the vibrations ismainly determined by the time harmonic orders lk corresponding to spatialorders kk ¼ 2 and 22, given by equation (8).

By increasing the load torque and the set current to approximately 8 Nm and7.5 A, respectively, also the frequencies corresponding to kk ¼ 0; viz. multiplesof 12frot, appear in the vibration spectrum, as shown in Figure 6. Furthermore,some side bands with a frequency step of frot, which are thus due to rotorasymmetries or load torque ripples, are clearly observed.

5. ConclusionsIn this paper, a procedure for calculating the vibrations of magnetic origin inswitched reluctance motors together with an analytical method for predictingthe frequency spectra of the vibration modes have been presented. Themodeling method assumed an idealized 6/4 SRM drive and a constant speed. Byconsidering a pair of modes of nonzero order, it was observed that two modaldisplacements with the same amplitude and a phase shift of ^908, result in a

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Figure 5.Measured vibrations

(1,400 rpm, Iset¼2.0 A)

Figure 6.Measured vibrations

(1,400 rpm, Iset¼7.5 A)


1019

rotating wave. The frequency spectrum and direction of rotation of this wave ispredicted using the analytical method. Experimental verification throughvibration measurements, conforms the validity of the devised method.Computed vibration mode shapes, resonance frequencies and modaldisplacements as well as measured vibrations at different setups aregraphically displayed.

References

Blaabjerg, F., Rasmussen, P.O. and Pedersen, J.K. (1999), “Multidisciplinary design of electricaldrives”, Proc. International Intelligent Motion Conference, pp. 201-10.

Cai, W. and Pillay, P. (2001), “Resonance frequencies and mode shapes of switched reluctancemotors”, IEEE Transactions on Energy Conversion, Vol. 16 No. 1, pp. 43-8.

Cameron, D.E., Lang, J.H. and Umans, S.D. (1992), “The origin and reduction of acoustic noise indoubly salient variable-reluctance motors”, IEEE Transactions on Industry Applications,Vol. 28 No. 6, pp. 1250-5.

Colby, R.S., Mottier, F.M. and Miller, T.J.E. (1996), “Vibration modes and acoustic noise in afour-phase switched reluctance motor”, IEEE Transactions on Industry Applications,Vol. 32 No. 6, pp. 1357-64.

Hong, J.P., Ha, K.H. and Lee, J. (2002), “Stator pole and yoke design for vibration reduction ofswitched reluctance motor”, IEEE Transactions on Magnetics, Vol. 38 No. 2, pp. 929-32.

Javadi, H., Lefevre, Y., Clenet, S. and Lajoie-Mazenc, M. (1995), “Electro-magneto-mechanicalcharacterizations of the vibration of magnetic origin of electrical machines”, IEEETransactions on Magnetics, Vol. 31 No. 3, pp. 1892-5.

Neves, C.G.C., Carlson, R. and Sadowski, N. (1998), “Vibrational behavior of switched reluctancemotors by simulation and experimental procedures”, IEEE Transactions on Magnetics,Vol. 34 No. 5, pp. 3158-61.

Pillay, P. and Cai, W. (1999), “An investigation into vibration in switched reluctance motor”,IEEE Transactions on Industry Applications, Vol. 35 No. 3, pp. 589-96.

Tang, Y. (1997), “Characterization, numerical analysis and design of switched reluctancemotors”, IEEE Transactions on Industry Applications, Vol. 33 No. 6, pp. 1544-52.

Vandevelde, L. and Melkebeek, J.A.A. (2001), “Magnetic forces and magnetostriction inferromagnetic material”, COMPEL, Vol. 20 No. 1, pp. 32-50.

Vandevelde, L., Gyselinck, J.J.C. and Melkebeek, J.A.A. (1998), “Long-range magnetic force anddeformation calculation using the 2D finite element method”, IEEE Transactions onMagnetics, Vol. 34 No. 5, pp. 3540-3.

Verma, S.P. and Balan, A. (1996), “Vibration model for stators of electrical machinesincorporating the damping effects”, Proc. ELECTRIMACS’96, pp. 755-61.

Wu, C.-Y. and Pollock, C. (1995), “Analysis and reduction of vibration and acoustic noise in aswitched reluctance drive”, IEEE Transactions on Industry Applications, Vol. 31 No. 1,pp. 91-8.

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Two-dimensional harmonicbalance finite elementmodelling of electrical

machines taking motion intoaccount

J. Gyselinck and P. DularDepartment of Electrical Engineering, University of Liege,

Institut Montefiore, Liege, Belgium

L. Vandevelde and J. MelkebeekDepartment of Electrical Energy, Electrical Energy Laboratory,Systems and Automation, Ghent University, Ghent, Belgium

A.M. Oliveira and P. Kuo-PengFederal University of Santa Catarina, GRUCAD/EEL/CTC, SC, Brazil

Keywords Finite element method, Electrical machines, Harmonics, Magnetic devices

Abstract An original and easy-to-implement method to take into account movement (motion) inthe 2D harmonic balance finite element modelling of electrical machines is presented. The globalharmonic balance system of algebraic equations is derived by applying the Galerkin approach toboth the space and time discretisation. The harmonic basis functions, i.e. a cosine and a sinefunction for each nonzero frequency and a constant function 1 for the DC component, are used forapproximating the periodic time variation as well as for weighing the time domain equations in thefundamental period. In practice, this requires some elementary manipulations of the moving bandstiffness matrix. Magnetic saturation and electrical circuit coupling are considered in the analysisas well. As an application example, the noload operation of a permanent-magnet machine isconsidered. The voltage and induction waveforms obtained with the proposed harmonic balancemethod are shown to converge well to those obtained with time stepping.

IntroductionThe steady-state finite element (FE) analysis of electrical machines can becarried out either in the time domain or the frequency domain. The firstapproach, also referred to as time stepping, is mostly followed, in spite of thevery large number of time steps to be carried out. Indeed, the time step Dt to be



The research was carried out in the frame of the Inter-University Attraction Poles forfundamental research funded by the Belgian State. P. Dular is a Research Associate with theBelgian Fund for Scientific Research (F.N.R.S.). L. Vandevelde is a Postdoctoral Fellow of theFund for Scientific Research – Flanders (F.W.O.–Vlaanderen).

Two-dimensionalfinite element

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Vol. 22 No. 4, 2003pp. 1021-1036


DOI 10.1108/03321640310482977

adopted, which depends on the largest relevant frequency of the system understudy, may be very small and in some cases the transient phenomenon that hasto be stepped through before reaching quasi steady-state may decay veryslowly.

The frequency domain or harmonic balance (HB) approach (Lu et al., 1991;Yamada and Bessho, 1988) is not very popular as it has some evidentdisadvantages. It consists of approximating the periodic time variation of themagnetic fields by a truncated Fourier series and results in a single, but verylarge system of nonlinear algebraic equations. Its resolution may be veryexpensive as both the number of unknowns and the bandwidth of the systemmatrix increase with the number of considered frequencies.

Furthermore, it is difficult to take saturation and movement into account.Some of the authors have treated the first aspect, the magnetic nonlinearity, ina recent publication (Gyselinck et al., 2002a). The nonlinear HB equations arestraightforwardly solved by means of the Newton-Raphson method. In thepresent paper, the authors focus on the motional aspect, and extend the methodproposed by Gyselinck et al. (2002a) to problems with periodic movement.In particular, a 2D FE model of a rotating electrical machine having a movingband is considered. The approach can be equally adopted for an arbitrary timeperiodic movement (e.g. linear machines) if a hybrid FE – boundary elementmodel is used (Gyselinck et al., 2002b).

It should be noted that sofar the modelling of electrical machines in thefrequency domain has been given little attention in the literature, unlike its timedomain counterpart. However, one particular case of the frequency domaincalculations has been and still is extensively carried out. It concerns theharmonic simulation of static devices and induction machines (Yahiaoui andBouillault, 1985), in which only one frequency is considered. In inductionmachines, the slip frequency in the rotor is simply affected by multiplying theconductivity of the rotor bars by the slip. The saturation is commonly takeninto account by means of an equivalent bh-curve, the choice of which may affectconsiderably the accuracy (Luomi et al., 1986). Obviously, the single-frequencyapproach can only produce a (rough) estimate of the fundamental components.A more involved approach consists of, e.g. considering a multi-harmonic statorand rotor model separately and identifying the airgap fields of thecorresponding time and spatial order (De Gersem and Hameyer, 2002).

Depending on the type of electrical machine, not all harmonics of thefundamental frequency are present in the stator and the rotor. In order toreduce the computational cost, this should be taken into account in a practicalharmonic balance finite element (HBFE) method implementation. The rotatingfield theory of, for example, synchronous machines (Breahna, 1999), allows todetermine the spectrum of the magnetic field, the currents and the voltages inthe stator and the rotor.

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In the following sections, the extension of the HBFE method to rotatingelectrical machines will be elaborated and applied to a permanent-magnetmachine.

Outline of the method2D magnetostatic problemWe consider a classical 2D magnetostatic problem (Ida and Bastos, 1997). In adomain V in the xy-plane, the given current density j ¼ jðx; yÞ 1 z is directedalong the z-axis. The magnetic field h and the magnetic induction b; thez-components of which vanish, are to be calculated. The constitutive law h ¼ n b;with n the reluctivity, and conditions on the boundary of V are supplied.

Permanent magnets can be included in the analysis as well. The constitutivelaw h ¼ n ðb 2 brÞ; where br is the remanent induction, amounts to anequivalent current density curl ðn brÞ in the permanent-magnet domains and toa current layer on their boundary. In the case of a uniform magnetisation(constant nbr), only the latter is nonzero.

The magnetostatic field problem is mostly formulated in terms ofthe magnetic vector potential aðx; yÞ; which can be chosen along the z-axis:a ¼ aðx; yÞ 1z: From b ¼ curl a ¼ 1z £ grad a; it follows that the magneticGauss law, div b ¼ 0; automatically holds. Remains to satisfy Ampere’s lawcurl h ¼ j; which, expressed in terms of the magnetic vector potential, reads:

curl ðn curl aÞ ¼ j or div ðn grad aÞ ¼ 2j: ð1Þ

A discretisation of the domain V in, for example, first order triangular elementsallows to approximate the potential aðx; yÞ as

aðx; yÞ ¼X#n

l¼1

al alðx; yÞ; ð2Þ

where alðx; yÞ is the piecewise linear basis function that is associated with thelth node in the FE mesh. The total number of nodes is denoted by #n.

Following the Galerkin approach, equation (1) is weakly imposed in V byweighing it with all #n interpolation functions akðx; yÞ :

ZV

ðdivðn grad aÞ þ jÞak dV ¼ 0: ð3Þ

Considering equation (2), partial integration of (3) produces a system of #nalgebraic equations in terms of the #n unknown coefficients al. If the problem islinear (considering a constant reluctivity n), the system of equations is linear aswell. It can be written as follows:

SA ¼ J ; ð4Þ


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where A is the column matrix in which the #n unknowns are assembled, andwhere the elements of the square stiffness matrix S and the column matrix Jare given by

Skl ¼

ZV

n gradak · gradal dV; ð5Þ

J k ¼

ZV

jðx; yÞak dV: ð6Þ

For the sake of brevity, the boundary conditions and permanent magnets arenot explicitly considered in the present analysis.

Rotating electrical machinesFor modelling rotating electrical machines, the FE domain V is commonly splitup into three complementary subdomains: a “stator” Vs, a “rotor” Vr and a thinring Vmb between stator and rotor, the so-called moving band (Figure 1).

As the angular position of the rotor, denoted by u, varies in time, the FEmesh in the stator and the rotor can remain identical with regard to therespective reference frames xy and x 0y 0; while the moving band has to beremeshed. For the sake of convenience, the stator and the rotor are connectedby a single layer of elements in the moving band. As the rotor position changes,the discretisation of the moving band changes in a discontinuous way: theelements are deformed and, intermittently, the topology of the moving bandmesh is (locally) modified in order to maintain elements of good aspect ratio.As there are no nodes situated inside the moving band, the total number ofnodes #n does not vary with the rotor angle u(t), and equals #ns þ #nr; where#ns and #nr are the number of nodes in the stator and the rotor, respectively.

When written in terms of the proper coordinates (i.e. either in the stator orthe rotor reference frame), the interpolation functions are u-independent, andthe interpolation of the magnetic vector potential can be expressed as

aðx; y; tÞ ¼X#ns

l¼1

aslðtÞ aslðx; yÞ in Vs; ð7Þ

Figure 1.2D FE model consistingof a stator Vs, a rotor Vr

and a moving band Vmb

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aðx 0; y 0; t Þ ¼X#nr

l¼1

arlðtÞ arlðx0; y 0Þ in Vr: ð8Þ

In the moving band Vmb, the interpolation is inherently u-dependent:

a ¼s l

Xas lðtÞ as lðx; y; uÞ þ

rl

XarlðtÞ arlðx

0; y 0; uÞ; ð9Þ

where only the basis functions asl and arl associated with the nodes on theouter and the inner boundary, respectively, of the moving band are to beconsidered.

The system of algebraic equations (4) can be partitioned accordingly:

Ssss þ Smb

ss Smbsr

Smbrs Sr

rr þ Smbrr

24

35 As

Ar

" #¼

J s

J r

" #; ð10Þ

where the subscripts indicate the concerned degrees of freedom and thesuperscript (s, r or mb) indicates the subdomain that produces the block in thestiffness matrix.

When ignoring saturation, the diagonal blocks Ssss and Sr

rr; due to thestator and the rotor, respectively, are time-invariant. The blocks Smb

ss ; Smbrr and

Smbsr ¼ S

mbrs

T

; due to the moving band, depend on the rotor position u(t).

Harmonic BalanceLet us now consider a time periodic problem. The current and/orpermanent-magnet excitation, Js(t) and Jr(t), and the rotor position u(t)(modulo 2p) vary periodically in time, with fundamental frequency f and periodT ¼ 1=f :

The HB method consists of approximating the periodic time variation ofAs(t) and Ar(t) by a truncated Fourier series. The corresponding time basisfunctions H(t) are

ffiffiffi2

pcosð2pkftÞ and 2

ffiffiffi2

psinð2pkftÞ for each nonzero

frequency kf, and a constant function 1 for the DC component. A different set offrequencies may be adopted in the stator and the rotor.

The harmonic time discretisation of As(t) and Ar(t) can thus be written as

AsðtÞ ¼X#hs

l¼1

AðlÞs H slðtÞ; ð11Þ

ArðtÞ ¼X#hr

l¼1

AðlÞr H rlðtÞ; ð12Þ


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where #hs is the number of harmonic basis functions considered in the statorand the moving band and #hr is the number of functions in the rotor and themoving band. In the moving band, both sets of frequencies apply.

The harmonic basis functions are orthonormal in the stator and the rotor,respectively, e.g. for those defined in the stator:

1

T

Z T

0

H skðtÞH slðtÞ dt ¼ dk;l: ð13Þ

Adopting the Galerkin approach, the HB system of algebraic equations can beobtained by using the harmonic basis function as test function as well. Theweighing of equation (4) in the fundamental period [0,T ] with a basis functionH(t) can be written as

1

T

Z T

0

ðSA 2 J ÞHðtÞ dt ¼ 0: ð14Þ

Considering the partitioned system (10) and the harmonic functions that aredefined in the stator and the moving band on the one hand, and in the rotor andthe moving band on the other hand, equation (14) can be elaborated further to

1

T

Z T

0

Ssss þ S

mbss

As þ S

mbsr Ar 2 J s

Hsk dt ¼ 0; ð15Þ

1

T

Z T

0

Smbsr As þ Sr

rr þ Smbrr

Ar 2 J r

Hrk dt ¼ 0: ð16Þ

The time discretisation equations (11) and (12) thus leads to a system of#ns#hs þ #nr#hr linear equations in term of an equal number of unknowns.It can be written as follows:

SsssH þ S

mbssH S

mbsrH

SmbrsH S

rrrH þ S

mbrrH

24

35 AsH

ArH

" #¼

J sH

J rH

" #: ð17Þ

The harmonic components of As(t) and Ar(t) are assembled in AsH and ArH

as follows:

AsH ¼

Að1Þs

..

.

Að#hsÞs

266664

377775 and ArH ¼

Að1Þr

..

.

Að#hrÞr

266664

377775; ð18Þ

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and the matrices SsssH etc. can be partitioned into blocks S

sðk;lÞssH etc., where (k,l)

refers to the pair of harmonic functions concerned, i.e. H skðtÞH slðtÞ;H rkðtÞH rlðtÞ or H rkðtÞH slðtÞ:

From the orthonormality of the basis functions (13), it follows that thematrices S

sssH and S

rrrH have a diagonal block structure. Indeed, the blocks of

e.g. SsssH are given by

Ssðk;lÞssH ¼

1

T

Z T

0

Ssss H skðtÞH slðtÞ dt ¼ dk;lSss: ð19Þ

The matrices produced by the moving band generally have a full blockstructure, e.g. the blocks of S

mbsrH are given by

Smbðk;lÞsrH ¼

1

T

Z T

0

Smbsr ðuðtÞÞH skðtÞH rlðtÞ dt: ð20Þ

Different harmonics in the stator (having the same or different frequency) arethus coupled through the nodes situated on the outer moving band contour.The same holds for the rotor harmonics and the nodes on the inner movingband contour. The stator and the rotor harmonics are coupled through all thenodes situated on the outer and inner contour of the moving band.

Practical aspects and extension to nonlinear problemsIn practice, the integrals (20) over the fundamental period [0,T ] areapproximated by a sum, considering a finite number of discrete timeinstants ti. For each corresponding rotor position u(ti), the moving band ismeshed and its static stiffness matrix is calculated. Then the contribution to theglobal HB system matrix is effected, considering all the relevant pairs ofharmonic basis functions.

Saturation can be taken into account as proposed by Gyselinck et al. (2002a).The system of nonlinear algebraic equations can be solved straightforwardlyby means of the Newton-Raphson (NR) method. Hereto, for each NR iterationand for each element situated in a nonlinear region of the FE domain, thedifferential reluctivity tensor ›h=›b multiplied by each relevant pair ofharmonic basis functions has to be integrated over [0,T ]. Saturation causes allharmonics considered in the nonlinear region to be coupled.

The constant contribution of the moving band (where the reluctivity isconstant, n ¼ n0) and of the linear media in the stator and the rotor to theJacobian matrices can be calculated and stored before starting the iterative NRprocess.

Electrical circuit coupling can be easily considered (Lombard and Meunier,1992). If the electrical circuit is linear, the electrical coupling does not require aspecial HB treatment (Lu et al., 1991). However, for nonlinear inductivecomponents and resistive components (e.g. diodes), the differential inductance


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(flux derived with respect to current) and the differential resistance (voltagedrop derived with respect to current) have to be processed in a similar way asthe differential reluctivity tensor (Gyselinck et al., 2002c).

Both the time domain as the HB systems of algebraic equation can be solvedby means of GMRES with ILU preconditioning, after renumbering with thereverse Cuthill McKee algorithm (Saad, 1996; SPARSKIT, 2003). As the fill-in(average number of nonzero entries per row) increases with the number ofconsidered frequencies, it is important for the GMRES convergence and thecomputational cost (computation time and storage requirements) to set thefill-in parameters of the preconditioning to an appropriate value.

An efficient, adaptive HBFE approach consists of performing severalcalculations, starting from the minimal spectrum, which is then graduallyexpanded until sufficient convergence of the obtained waveforms is observed.The iterative NR procedure is advantageously initialised with the previoussolution (if available).

Application example2D FE model of permanent-magnet machineThe proposed HB method is applied to an eight-pole three-phasepermanent-magnet motor (Oliveira et al., 2002). The stator windings are starconnected. The commercial motor has 24 skewed stator slots (one slot per poleand per phase) and the permanent magnets are mounted in a slightlyasymmetric way. As a result, the distortion of the e.m.f. waveforms isconsiderably reduced.

In this paper, we consider an especially assembled motor that has straightstator slots and magnets mounted (nearly) symmetrically. In the following, themeasured noload voltage waveform will be compared to the one obtained withthe 2D FE model using both the time and the frequency domain approach.Induction waveforms in the stator and the rotor will be shown as well.

By imposing anti-periodicity conditions, only one pole has to be modelled.The FE discretisation is shown in Figure 2. The FE mesh has 2,934 first ordertriangles (1,880 in stator, 924 in rotor and 130 in the moving band), leading to#n ¼ 1; 450 degrees of freedom in a time-stepping simulation. The airgap issplit up into three layers (see zoom in Figure 2), the middle of which is themoving band. The mean airgap radius and the minimum airgap width are 25.7and 0.55 mm, respectively. The axial length of the stator and the rotor corestack is 40 mm. The bh-curve used for the stator and the rotor iron is shown inFigure 3.

Time-stepping simulationA time-stepping simulation at 750 rpm is carried out ð f ¼ 50 Hz; T ¼ 20 msÞ:As it concerns a static problem with a constant excitation, there is no transient.One period is time-stepped with Dt ¼ T=360:

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In Figure 4, the obtained line-to-line voltage waveform is shown together withthe measured one. A good agreement is observed. The spectrum of thecalculated line-to-line voltage and phase voltage is shown in Figure 5. Thephase voltage contains important 3rd, 5th, 7th and 9th harmonics. The 3kharmonics do not appear in the line-to-line voltage.

HB simulationsBesides the fundamental, f ¼ 50 Hz; component in the stator and the DCcomponent in the rotor, the magnetic field has ð2k þ 1Þ harmonics in the statorand 6k harmonics in the rotor.

Five HB calculations with increasing spectrum are carried out. They aredenoted as HB 1, HB 5, HB 7, HB 11 and HB 13, and comprise the followingharmonics:

Figure 2.2D FE model of one poleof the permanent-magnet

machine

Figure 3.bh-curve of the stator and

rotor iron


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HB 1: 0, 1

HB 5: 0, 1, 3, 5

HB 7: 0, 1, 3, 5, 6, 7

HB 11: 0, 1, 3, 5, 6, 7, 9, 11

HB 13: 0, 1, 3, 5, 6, 7, 9, 11, 12, 13

In HB 13, e.g. six nonzero frequencies are considered in the stator, while in therotor, the DC component and two nonzero frequencies are considered. Thisresults in a total of 12 þ 5 ¼ 17 harmonic basis functions. For evaluating theHB stiffness matrix of the moving band numerically, 360 time instants ti in[0,T ] and rotor positions u(ti) in [08,908] are considered.

Some of the obtained harmonic components of the flux pattern are shown inFigure 6.

Figure 4.Measured and calculatednoload line-to-linevoltage

Figure 5.Spectrum of line-to-linevoltage and phasevoltage (the amplitude ofthe fundamental 50 Hzcomponent of the phaseand line-to-line voltage is38.5 V and 66.6 V,respectively)

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The waveform of the phase voltage obtained with the five HB calculations andwith the time-stepping simulation are shown in Figure 7. A good convergenceof the HB waveforms to the time stepping one is observed. Note also thatthe distortion of the moving band elements may cause some noise in thetime-stepping voltage waveform, as can be clearly seen in the zoom in Figure 7.

The relative amplitude of the frequency components of the voltage obtainedwith the HB calculations (where the amplitudes obtained with time-steppingserve as reference) is shown as a function of the spectrum (HB 1-HB 13) inFigure 8. Notice that the error of the fundamental component diminishes fromonly 1.2 per cent with HB1 to less than 0.04 per cent with HB 13. HB 13produces an equally good precision for the 3f and 5f components (error of 0.3and 0.05 per cent, respectively). The minimum error of the 7f and 9fcomponents is about 5 per cent.

Figure 6.Harmonic components of

the flux pattern in therotor and the stator


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Figure 7.Waveform of noloadphase voltage obtainedwith time-stepping andwith the five HBcalculations (samelegend for zoom below)

Figure 8.Relative amplitude of thefrequency componentsf ¼ 50 Hz-11f ¼ 550 Hzof the voltage obtainedwith the HB calculations(where the amplitudesobtained withtime-stepping serve asreference) as a functionof the spectrum(HB 1-HB 13)

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The calculated waveforms of the radial induction in two points in a stator toothare shown in Figure 9. It concerns the stator tooth that is aligned with themagnet in Figure 2 and following, and two points on its axis of symmetry, oneclose to the airgap (at r ¼ 26:1 mm) and the other further from the airgap(at r ¼ 30 mm). Again an excellent agreement of the time-stepping results andthe HB results can be observed.

In Figure 10, the radial induction waveforms in two points in the rotor aredepicted. The two points are situated on the symmetry axis of the permanentmagnet, one close to the airgap (at r ¼ 25 mm) and the other further from theairgap (at r ¼ 20 mm). As the harmonics are multiples of six, only one sixth ofa fundamental period, [0,T/6], is shown.

Calculation timesAll calculations have been carried out on a Pentium III 750 MHz. Theapproximate calculation times supplied hereafter should give an indication of

Figure 9.Time-stepping and HB

waveforms of the radialinduction in two points

in a stator tooth: a pointclose to the airgap (up)

and a point further fromthe airgap (below)


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computational efficiency of the HB approach compared to the time-steppingapproach.

The time-stepping simulation (one period, 360 steps, in average 3 NRiterations per time step) has taken 360 s. The HB calculations HB 1, HB 5, HB 7,HB 11 and HB 13 have taken 20, 67, 153, 310 and 590 s, respectively. Thenumber of NR iterations was 7, 5, 4, 4 and 4, respectively. For HB 5-HB 13, theprevious solution (i.e. HB 1-HB 11) has been used as initial solution for the NRiterative process. Without this initialisation, the number of NR iterations waseither 7 or 8.

If only the fundamental component f and some lower order harmonics (3f, 5f)are of interest, the HB approach is certainly more efficient than the time domainapproach.

ConclusionsAn original method in the 2D HBFE modelling of electrical machines takingmotion into account has been presented. The HB system of algebraic equations

Figure 10.Time-stepping and HBwaveforms of the radialinduction in two pointsin the permanentmagnet: a point close tothe airgap (up) and apoint far from the airgap(below)

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has been straightforwardly derived by applying the Galerkin approach to boththe space and time discretisation. The method can be easily implemented as itonly requires some elementary manipulations of the moving band stiffnessmatrix. Magnetic saturation and electrical circuit coupling are considered in theHBFE analysis as well.

The proposed method has been successfully applied to a permanent-magnetmachine. The HB waveforms of the noload voltage converge well to theone obtained with time-stepping. The computational cost of the HBcalculations is favourable provided the number of frequency considered isnot too large.

References

De Gersem, H. and Hameyer, K. (2002), “Air-Gap flux splitting for the time-harmonicfinite-element simulation of single-phase induction machines”, IEEE Trans. Magn., Vol. 38No. 2, pp. 1221-4.

Gyselinck, J., Dular, P., Geuzaine, C. and Legros, W. (2002a), “Harmonic balance finite elementmodelling of electromagnetic devices: a novel approach”, IEEE Trans. Magn., Vol. 38 No. 6,pp. 521-4.

Gyselinck, J., Geuzaine, C., Dular, P. and Legros, W. (2002b), “Multi-harmonic modelling ofmotional magnetic field problems using a hybrid finite element – boundary elementdiscretisation”, Proceedings of the Second International Conference on AdvancedComputational Methods in Engineering (ACOMEN02), Liege, Belgium, 28-31 May, onCDROM.

Gyselinck, J., Dular, P., Geuzaine, C. and Legros, W. (2002c), “Two-dimensional harmonic balancefinite element modelling of electromagnetic devices coupled to nonlinear circuits”,Proceedings of the XVII Symposium Electromagnetic Phenomena in nonlinear Circuits(EPNC2002), 1-3 July, Leuven, Belgium, 11-4 (accepted for COMPEL).

Ida, N. and Bastos, J.P.A. (1997), Electromagnetics and calculation of fields, Springer-Verlag,New York, NY.

Lombard, P. and Meunier, G. (1992), “A general method for electric and magnetic coupledproblem in 2D and magnetodynamic domain”, IEEE Trans. Magn., Vol. 28 No. 2,pp. 1291-4.

Lu, J., Yamada, S. and Bessho, K. (1991), “Harmonic balance finite element method takingaccount of external circuits and motion”, IEEE Trans. Magn., Vol. 27 No. 5,pp. 4024-7.

Luomi, J., Niemenmaa, A. and Arkkio, A. (1986), “On the use of effective reluctivities in magneticfield analysis of induction motors fed from sinusoidal voltage source”, Proceedings of theInternational Conference on Electrical Machines (ICEM86), 8-10 September, Munchen,Germany, pp. 706-9.

Oliveira, A.M., Kuo-Peng, P., Sadowski, N., Andrade, M.S. and Bastos, J.P.A. (2002), “A non-apriori approach to analyze electrical machines modeled by FEM connected to staticconverters”, IEEE Trans. Magn., Vol. 38 No. 2, pp. 933-6.

Saad, Y. (1996), Iterative Methods for Sparse Linear Systems, PWS Publishing Company.

SPARSKIT: a basic tool-kit for sparse matrix computations (2003), http://www.cs.umn.edu/Research/arpa/ SPARSKIT/sparskit.html


1035

Yahiaoui, A. and Bouillault, F. (1985), “Saturation effect on the electromagnetic behaviour of aninduction machine”, IEEE Trans. Magn., Vol. 31 No. 3, pp. 2036-9.

Yamada, S. and Bessho, K. (1988), “Harmonic field calculation by the combination of finiteelement analysis and harmonic balance method”, IEEE Trans. Magn., Vol. 24 No. 6,pp. 2588-90.

Further reading

Breahna, R., Viarouge, P., I. Kamwa, I. and Cros, J. (1999), “Space and time harmonics insynchronous machines”, Proceedings of ELECTRIMACS’99, September 14-16, Lisbon,Portugal, Vol. 3, pp. 45-50.

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A general analytical model ofelectrical permanent magnetmachine dedicated to optimal

designE. Fitan, F. Messine and B. Nogarede

Electroactive Machines and Mechanisms – EM 2 group,Laboratoire d’Electrotechnique et d’Electronique Industrielle,

Unite Mixte de Recherche INPT-ENSEEIHT/CNRS, Toulouse, France

Keywords Permanent magnets, Electrical machines, Modelling

Abstract What is new in this work is the generic capabilities of the proposed analytical model ofpermanent magnet machines associated with a novel deterministic global optimization method.That allows to solve some more general inverse problem of designing. The analytical approach ispowerful to take into account various kinds of constraints (electromagnetical, thermal, etc.). Theinverse problem associated with the optimal design of actuators could then be formulated as amixed-constrained optimization problem. In order to solve these problems, interval Branch andBound algorithms which have already proved their efficiency, have made it possible to determinesome optimized rotating machines.

1. IntroductionNowadays, a designer uses some numerical tools which enable him to take adecision, before the expensive phase of prototype making. A rational way to dothis is to develop an analytical model and then to solve some interestingassociated optimization problems. Nevertheless, the optimal conception ofelectromechanical actuators subjected to a specific schedule of conditionsrepresents a delicate problem. Generally, what the designer has to do first is tochoose the type of structure, then to determine its dimensions, in order tosatisfy the targets which have been defined. In any case, such a proceedingdoes not allow to know if the best structure of the actuator has been chosen,deliberately ignoring the others. Even in the case of conventional machines,which are fairly well known, traditional and empiric methods are no longersatisfactory. Consequently, the choice of the type of structure and its dimensioncannot and must not be treated separately.

After a short introduction of the inverse problem associated with the optimaldesign of the electromechanical actuators, a new analytical model combiningdifferent kinds of rotating machines with permanent magnets will bepresented. Our approach will then be validated on the determination ofmachines which minimize the magnet volume or/and the global volume. Then,all the solutions will be verified by numerical simulations.



A generalanalytical model

1037


Vol. 22 No. 4, 2003pp. 1037-1050


DOI 10.1108/03321640310482986

2. Towards a formulation based on the inverse problem ofoptimal designGenerally, starting from the actuator of a given structure (type of motor, usedmaterials) and known dimensions, some characteristic values such as thetorque, the flux are sought for. In order to achieve this research, Maxwell’sequations are directly solved by numerical methods (finite elements method forexample). In that way, a motor can be constructed by gradually modifyingsome parameters in order to obtain the required performances: iterations of thedirect problem of dimensioning.

Another method of dimensioning consists of setting the problem in terms ofoptimization. Starting from the analytical model of a given structure (obtainedafter making hypotheses from Maxwell’s equations) the dimensions of thestructure enabling to optimize some best criteria are sought for. Then, we speakof an optimal dimensioning problem (Messine et al., 1998).

An even more interesting approach would be the one allowing to definesimultaneously the dimensions and the choice of a structure answering thetargets aimed by the best designer. Consequently, it is necessary to develop ananalytical model joining several structures (for example with slots, withoutslot) and to have an associated resolution method. It is then an optimal designproblem (Messine et al., 2002). This problem can then be formulated as anoptimization problem in which new variables with generic characteristics havebeen introduced, allowing to define more general models; this is amixed-constrained optimization problem.

The different problems stated above are represented in Figure 1.In order to find the optimal design of a class of actuators, a deterministic

global optimization algorithm based on interval analysis (Messine, 1997;Ratschek and Rokne, 1998) has been developed. These methods have alreadybeen validated for design problems (Messine et al., 1998, 2002). Someextensions of such algorithms to deal with mixed problem (integer, real,boolean and categorical variables) have been discussed by Messine et al. (2001,2002). The fact that the global optimum is precisely enclosed using such analgorithm, becomes essential to compare with efficiency all the possiblesolutions due to the different structures.

3. Description of our general analytical modelThe conception of general models is based on the identification of genericparameters, defined as common to different machines (the bore diameter, themechanical air gap,. . .) and of the specific parameters of each one (for example,slots). So, only a design by analytical model is possible, taking more or lessstages of simplifications into account. This modelization, aimed at designing isbased on the magnetic and thermal effects, which are usually adopted.

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3.1 Frame-work of the model and assumptionsThe model considered is intended for rotating machines with surface mountedpermanent magnets. Generally speaking, there are different possibilities ofdesign and feeding for electrical machines, considered here to define a firstgeneral analytical model.

. Rotor configuration. For concentric rotor machines there are twosolutions, either a configuration with an internal rotor (the most commonone where the rotor is held by the stator) or a configuration with areversed rotor where the rotor is external to the stator. Discoidal machinesare not considered here.

. The kind of armature. Two sorts of armature are considered with orwithout slots. In the first case, conductors are wound in slots and thearmature teeth are crossed by most of the magnetic field. In the secondcase, conductors are directly crossed by the magnetic field.

. The kind of waveform. Only rectangular or sinusoidal current waveformsare considered.

So as to obtain a simplified model, some hypotheses about physicalcharacteristics of magnetical materials and considered structures have beendone. Magnetic induction in the mechanical airgap is supposed to be purelyradial. The respective relative permeabilities of the magnets and magnetical

Figure 1.Different kinds of

problems


1039

circuits (yoke and teeth) are fixed as unit and infinity. For machines with slots,the spouts are not integrated. Moreover, the geometrical forms of slot, wedge(for slotless machines) and magnet are supposed to be radial.

The so considered model enables us to re-unit different height structures,shown in Figure 2, working with generic parameters defined later.

To work out this model, our research is based only on a case of firstapproximation (with a view to pre-dimensioning) where our attention isfocussed on expressions relative to the electomagnetic torque and to physicsquantity as the thermic flux and some characteristic values of flux density, etc.

First, there are parameters with generic character allowing to definedimensions or part of dimensions of the so considered actuators:

. D: the bore diameter,

. L: the useful length,

. g: the mechanical air gap,

. la: the thickness of permanent magnet,

. E: the winding thickness,

. C: the thickness of yoke,

. b: the polar arc factor,

. p: the number of pole pairs.

Then, specific parameters, which are used to define dimensions of slot andtooth have to be introduced:

Figure 2.Considered structures ofpermanent magnetmachines

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. a: the length of slot evaluated at the middle of slot,

. d: the length of tooth evaluated at the middle of slot.

In order to integrate different types of structures to the model, a new kind ofvariables, named categorical variable has been introduced (Messine et al., 2001):

. for which scheduling would not, a priori, give a significance (choice of therotoric configuration),

. that allows to allocate some characteristic values to some parameters. Forexample, if M represents the remanent polarization of permanentmagnets, value of 0.6 could be given for a plasto-magnet or value of 0.9 fora rare-earth magnet. These values could be fixed as calculation is inprogress.

In this model, categorical variables are defined as follows (from which eachelementary model could be found):

sr ¼ 21 for an inverse rotoric configuration or sr ¼ 1 for an internal rotoricconfiguration,

se ¼ 0; when considering slotless machine and se ¼ 1; when slot machinesare considered,

sf ¼ 21 for a sinusoidal waveform or sf ¼ 1 for a rectangular waveform.Remark. The choice of the categorical variable values is arbitrary. Here

these ones are proposed because they are interesting for the following generalmodel.

3.2 Magnetical modelConsidering the laws on electromagnetism and studies already made aboutmodelization (Nogarede, 1990, 2001; Slemon and Liu, 1992), a first analyticalgeneral model with a view to conception is proposed. The electromagnetictorque can be found from an energetic calculation. The torque produced iswritten as follows:

Tem ¼ DL�

D þ srð1 2 seÞE�KTBeA ð1Þ

where KT represents the torque coefficient, which depends on the kind ofwaveform (taken into account with the categorical sf variable) and the kind ofarmature (considered with the magnetic leakage coefficient Kf).

KT ¼p

2

ð1 þ sfÞ

2ð1 2 K fÞ

ffiffiffib

pþ

ð1 2 sfÞ

2

ffiffiffi2

p

2

" #ð2Þ

A is the current electric loading. This parameter works with the armaturecategorical se variable. kt is the fitting factor and J the current areal density,both considered as generic parameters.


1041

A ¼ k tEJa

sed þ að3Þ

Be represents the no-load magnetic radial flux density to the bore diameterneighbourhood, the value of which depends on the rotoric configuration (withthe categorical st variable) and the kind of armature (se). M represents themagnetic polarization that can take different values according to the kind ofpermanent magnet considered. M value will be calculated by using thecategorical variable sm.

Be ¼2MðsmÞla

srD ln Dþ 2 Esrð12seÞD2 2srðla þ gÞ

� � ð4Þ

From the flux conservation law, by supposing interpolar leakages equal to zero,the flux density in the yoke B c mainly depends on the kind of waveform andstructure:

Bc ¼D

2pC

ð1 þ sfÞ

2bp

2þ

ð1 2 sfÞ

2

Be ð5Þ

Only for slotless machines with rectangular waveform, a semi-empiricmagnetic leakage coefficient K f must be introduced:

Kf ¼ 1:5pðE þ eÞb

Dð1 2 seÞ

ð1 þ sfÞ

2ð6Þ

When considering machines with slots, it is necessary to deal with the fluxdensity inside the teeth Bt. Like the yoke flux density value, the tooth fluxdensity value will depend on the used materials. An analytical relation can beobtained from the flux conservation law:

Bt ¼d þ a

dBe ð7Þ

When considering machines with slots, there are two relations which must betaken into account about the number of slot Ne; the first is a geometrical onewith:

Ne ¼pðD þ srEÞ

d þ að8Þ

the second is either an electrical one, where q is the number of phases, m thenumber of slots per pole and phase:

Ne ¼ 2pqm ð9Þ

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The yoke and the teeth here are built with the same material which is supposedto be linear to a certain value BM(smt). It can take different values according tothe kind of material considered with the categorical smt variable.

Bc # BMðsmtÞ

Bt # BMðsmtÞ

(ð10Þ

A geometrical relation must be added to the model, to be able to build theactuator. It mainly depends on the rotoric configuration.

D

22 C 2

1 2 sr

2

� E 2

1 þ sr

2

� ð g þ laÞ $ 0 ð11Þ

3.3 Thermal modelIn order to consider the loss of joules and their evacuation, a thermal modelproves indispensable. Here, a general thermal analytical model is proposedconsidering an electric circuit approach (Bertin, 1999; Semail, 1998), in acontinuous rating, considering only the loss of joules due to the electricalwindings (Figure 3). The model is based on a tooth pitch for slots machines(Powell et al., 2000) and a polar pitch for slotless machines, elaborated fromFourier’s law (thermal conduction, only considered in a radial direction) andNewton’s law (thermal convection). The thermal radiation is not considered inthis paper.

The expressions of all thermal resistances allowing thermal convectionmodelization depends on the rotoric configuration sr.

RJ is the thermal resistance that modelizes heat flow in windings. Here, onlyone kind of electrical conductor is considered: copper whose thermalconductivity is lcu.

Figure 3.Considered thermal

model


1043

RJ ¼ ktsr

lcuaeLln 1 þ sr

2E

D

� ð12Þ

When considering thermal conduction, the ae slot coefficient (radian) allows tocalculate the winding area. Moreover, for thermal convection, ae is used to takeinto account the winding area abutting with the mechanical air gap.

ae ¼2a

D þ srEse þ b

p

pð1 2 seÞ ð13Þ

Rd is the thermal resistance that modelizes heat flow in teeth (for machineswith slot) and in wedges (for slotless machines). Slots and wedges areconstructed with materials the characteristics of which are very different (heremainly the magnetical one). In order to deal with these properties, thecategorical variables smt and se are introduced to choose the value of thethermal conductivity lmd.

Rd ¼sr

lmdðse;smtÞad

2 Lln 1 þ sr

2E

D

� ð14Þ

The ad coefficient (radian) allows to calculate the area of tooth or wedge, or totake into account the area abutting with the mechanical air gap.

ad ¼2d

D þ srEse þ ð1 2 bÞ

p

pð1 2 seÞ ð15Þ

Rc is the thermal resistance that modelizes heat flow through the yoke. Thevalue of the thermal yoke conductivity lmd depends on the used materials smt.

Rc ¼sr

lmcðsmtÞacLln 1 þ sr

2C

D þ sr2E

� ð16Þ

The ac coefficient (radian) is nothing but polar pitch or tooth pitch, according tothe considered armature se.

ac ¼ 2a þ d

D þ srEse þ

p

pð1 2 seÞ ð17Þ

In the modelization of convection phenomena, the previous differentcoefficients (ae, ad, ac) are bound to be used.

Re permits to modelize the heat flow in the neighbourhood of the air gap andwindings.

Re ¼2

DaeLheð18Þ

where he represents the characteristic coefficient of convection inside the airgap.

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R1 allows to modelize heat flow in the neighbourhood of the air gap andteeth or wedge.

R1 ¼4

DadLheð19Þ

Rg modelizes heat flow between the yoke and the ambient medium.

R g ¼2

ðD þ 2srðE þ CÞÞacLhað20Þ

where ha represents the characteristic coefficient of convection of the ambientmedium. Since the model considered is based on a polar or a tooth pitch, arelation between the thermal flux fTh and the loss of joules PJ can beelaborated.

fTh ¼PJ

2pð1 þ ðqm 2 1ÞseÞð21Þ

From Fourier’s law and the equivalent electric circuit in Figure 3, the followingrelation can be expressed:

fTh ¼ðT J 2 TeÞ

Req1þ

ðT J 2 TcÞ

Req2

1

1 þ RJ1

Req1þ 1

Req2

� � ð22Þ

where Req1 and Req2 are the equivalent thermal resistances of the consideredcircuit.

Req1 ¼ReðRd þ R1Þ

2Re þ Rd þ R1

ð23Þ

Req2 ¼ Rc þ Rg ð24Þ

TJ, Te, Tc, respectively, represent the value of the temperature inside windings,the air gap and the ambient medium.

The loss of joules PJ is calculated with the Ohm’s law. Winding overhang isnot taken into account.

PJ ¼ rCup JL½D þ srð1 2 seÞE�A ð25Þ

where rCu indicates the resistivity of the considered electrical material (herecopper).

3.4 Relationships with interesting volumesThe different volumes of the machines can be elaborated from simplegeometrical relations. Here, categorical variables are introduced to deal withthe different kinds of structures. The magnet volume Vm is defined as follows:


1045

Vm ¼ bpLla½D 2 srð2g þ laÞ� ð26Þ

The global volume Vg (the external volume) is given by:

Vg ¼pL

4

1 þ sr

2ðD þ 2ðE þ CÞÞ2 þ

1 2 sr

2ðD þ 2ðg þ la þ CÞÞ2

ð27Þ

These two equations are the criteria for the following considered optimizationproblems.

4. Applications and validationsA rigorous conception of machines is achieved through the use of a rationalmethod combined with an appropriate analytical model. Once the globalsolution has been obtained, it is interesting to know whether it proves a rightanswer to the intended physical targets of the model (in terms of torque,flux,. . .). Thus, the solutions can be verified by using a numerical finite elementmethod.

4.1 Problem contextThe model proposed so far has allowed to join height configurations of thepermanent magnet machines, which have distinct geometries and feedings. Wethought it would be interesting to consider other characteristics, linked to thematerials used in the design of electrical machines. Indeed when an actuator isconstructed several materials can be considered according to their magnetical,mechanical, etc., and thermal properties.

In our case, four kinds of materials are considered: permanent magnets (sm),electric and magnetic (smt) conductors, wedges. Each kind of materials can beconstructed with different compounds as presented in Table I. Consequently,the model permits to take 32 different structures into account.

The problem which appears, is to define the structure in terms of dimensionsand compositions enabling to answer the following best targets:

. to minimize the permanent magnet volume (Vm),

. to minimize the global volume (Vg), and

. then to minimize both simultaneously (Multi)

Material Categorical variable Compound M (T) BM (T) l (W/m/K)

Permanent magnet sm Plasto 0.6 2 2NdFeB 0.9 2 2

Magnetical conductor smt Powder 2 1.2 80Stamping 2 1.5 70

Electrical conductor 2 Copper 2 2 386Wedge 2 Aluminum 2 2 204

Table I.Materials

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for a three-phase machine ðq ¼ 3Þ; producing a 10 Nm electromagnetic torque.For the resolution of the problems, the following parameters are introduced andfixed:

. the form factor l, defined as the ratio of the bore diameter to the usefullength ðl ¼ D=LÞ; such as l ¼ 2;

. the use of slot space (t), ratio of tooth to the tooth pitch ðt ¼ d=ðd þ aÞÞ;with t ¼ 0:6;

. the polar arc factor b ¼ 0:85; the fitting factor kt ¼ 0:7; the mechanicalair gap g ¼ 1 mm; the number of slots per pole and phase m ¼ 1;

. the temperature of the air gap Te ¼ 258C; of the ambient medium Ta ¼208C and of the conductors T J ¼ 758C; the resistivity of copper rCu ¼0:018 £ 1026 Vm;

. the characteristic coefficients of convection ha ¼ 4 and he ¼ 12:

The other parameters will be situated between the two minimal and maximalvalues specified in Table II, defining the domain of interest of theseproblems.

Remark 1. The effects due to demagnetisation have been ignored in thefirst approach, because for one thing, NdFeB permanent magnets present ahigh coercive force (about 1,000 kA/m), here used with a low current electricloading (about 20 kA/m), and for another thing, these permanent magnets work

Bounds min Vm min Vg min Multi

D (mm) [60, 200] 98.65 102 112.4la (mm) [3, 50] 3.20 8.2 3.05E (mm) [3, 50] 25.5 17 16.1C (mm) [3, 50] 44.50 7.0 6.1p [1, 6] 1 6 6L (mm) [20, 200] 49.33 51.0 56.2d (mm) [4, 50] 39.0 4.50 5.0a (mm) [2, 50] 26.0 3.0 3.36J (A m22 [3, 6] £ 106 3.01 3.05 3.02A (A m21) [1, 10] £ 104 2.15 1.45 1.36Be (T) [0.1, 0.9] 0.66 0.87 0.7Bc (T) [0.9, 1.5] 0.97 1.40 1.45Bt (T) [0.9, 1.5] 1.10 1.45 1.17sr {21, 1} 1 21 21se {0, 1} 1 1 1sf {21, 1} 1 1 1sm {pl., Nd.} Nd. Nd. Nd.smt {pd., st.} Pd. st. st.Vm £1025(m 3) 3.94 12.5 5.38Vg £1023(m 3) 2.20 0.73 0.77

Table II.Optimization results


1047

at low ambient temperature in this application (about 508C), which is much lessthan the usual level (about 1308C).

4.2 Analytical resultsThe results of the proposed minimizations are given in Table II. The kind ofmachine which minimizes the permanent magnet volume, considering theprevious schedule of conditions, is an actuator with slots. In fact, slotlessmachines constucted with a bigger air gap, the thickness of which is mainlydue to one of the electrical conductors, generally needing magnet volume from1.5 to 2 times superior to the one of machines with slots. Consequently, the typeof permanent magnet is the one presenting the highest magnetic polarization,in this case rare-earth type magnet (NdFeB). The number of pole pairs tends toits minimum value ðp ¼ 1Þ; for which an explanation can be given fromequations (5) and (26). In order to minimize the magnet volume, when b is fixedto a specific value, it seems logical to look for the minimum bore diametervalue, obtained with relation (5) when the number of pole pairs and the fluxdensity in the yoke have lower values. The magnetical material used to buildthe machine is a soft magnetic powder. However, one almost identical solutioncan be calculated with stampings. Both these solutions are justified sincevalues of the flux density in the yoke and teeth are less than 1.2 T, magneticalmaterial supposed to be linear to this value for powder. The choice of aninternal rotor configuration is logical allowing to obtain a value of the borediameter lower than with a reversed rotor machine. As for the kind ofwaveform, it is a rectangular waveform, a result which was not necessarilyexpected.

In the case of the minimization of the external volume, a machine with slotsis obtained again, built with rare-earth type magnet. That produces for mainconsequence to minimize the magnetic active parts and especially to reducethe thickness of yoke and consequently, the external diameter. From relation(5), it can be shown that the thickness of yoke becomes minimum when thenumber of pole pairs and the flux density in the yoke have high values. So, thechosen material is the one that can stand the highest flux density. The choice ofan external rotor configuration confirms the acquired experience for machinesof small dimensions, even if their thermal aspect can seem restrictive.

One can notice that these two results are very different when theirdevelopment (rotor configuration ,materials,. . .) as well as their dimensionsare considered. Then, an interesting problem is to minimize simultaneouslythese two criteria, so as to affirm one of these solutions or to meet a new kindof actuator. This multi-criteria problem is written as shown in Messine et al.(2002), dealing with weight factors, here the global minimum of each criterium:

Multi ¼Vm

min Vmþ

Vg

min Vg:

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The results of this minimization is similar to the external volume minimization.In fact, the solution which minimizes Vg, is obtained with a lower magnetvolume, whereas when Vm is minimized the global volume takes an importantvalue.

The value obtained for the criterium Multi is 2.43, which is not so far fromthe absolute Pareto-Optimal solution 2. A small difference of about 5 per cent isnoticed for the criterium Vg, whereas the second criterium Vm is greater than45 per cent.

4.3 VerificationsOptimal solutions which have been calculated are verified with numerical finiteelements method: EFCAD. In this case only numerical values of theelectromagnetic torque are taken into account.

With regard to the solution allowing to minimize the magnet volume, thenumerical torque value is 9.1 Nm. Consequently, the calculation error due to theanalytical model is about 9 per cent.

Concerning the solution that minimizes the external volume, the numericalvalue obtained for the torque is worth 8.9 Nm. The difference between the tworesults is then about 11 per cent. The same ratio has been obtained for thedifference between the optimal and numerical solutions of the multi-criteriaproblem (Figure 4).

These computational errors are justified, due to the hypotheses which havebeen speculated, especially not integrating spouts and not taking into accountthe whole of the magnetic leakage.

Figure 4.Multicriteria simulation


1049

5. ConclusionIn this paper, a rational method has been elaborated to help the designer tocreate rotating machines before the expensive stage of prototype making. So asto deal with such a method, a general analytical model of permanent magnetmachines has been defined dealing with generic parameters, and built on thebasis of height types of structures. This general model is based on magneticaland thermal aspects, from which assumptions have been speculated. Thedesign inverse problem is set in terms of optimization and has been solved witha deterministic global method: interval Branch and Bounds algorithm (Messine,1997; Ratschek and Rokne, 1998). The global solutions that have beencalculated have then been verified with numerical simulations which validatethe model arround 10 per cent.

References

Bertin, Y. (1999), “Refroidissement des machines electriques tournantes”, Techniques del’ingenieur, D3460.

Messine, F. (1997), “Methodes d’optimisation globale basees sur l’analyse d’intervalle pour laresolution de problemes avec contraintes”, PhD thesis, Institut National Polytechnique deToulouse, France.

Messine, F., Fitan, E. and Nogarede, B. (2002), “The inverse problem associated to the optimaldesign of electromagnetic actuators: application to rotating machines with magneticseffects”, in Michielsen, B. and Decavele, F. (Eds) Proceeding JEE, Vol. 1, pp. 323-8

Messine, F., Monturet, V. and Nogarede, B. (2001), “An interval branch and bound methoddedicated to the optimal design of piezoelectric actuators”, Mathematics and Computers inSciences and Engineering, WSES Press, pp. 174-80.

Messine, F., Nogarede, B. and Lagouanelle, J-L. (1998), “Optimal design of electromagneticactuators: a new method based on global optimization”, IEEE Transaction on Magnetics,Vol. 34 No. 1, pp. 299-308.

Nogarede, B. (1990), “Etude de moteurs sans encoches a aimants permanents de forte puissance abasse vitesse”, PhD thesis, Institut National Polytechnique de Toulouse, France.

Nogarede, B. (2001), “Machines Tournantes: principes et constitutions”, Techniques del’ingenieur, D3411.

Powell, D.J., Schofield, N. and Howe, D. (2000), “Thermal modelling of a permanent magnetbrushless machine for application in a electro-hydraulic actuator”, Proceeding ICEM,Vol. 1, pp. 1987-91.

Ratschek, H. and Rokne, J. (1998), “New Computer Methods for Global Optimization”, WestSussex: Ellis Horwood.

Semail, E. (1998), Physique du Genie Electrique, Lavoisier Tec & Doc, France.

Slemon, G.R. and Liu, X. (1992), “Modeling and design optimization of permanent magnetmotors”, Electric Machines and Power Systems, pp. 71-92.

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Modelling of electromagneticlosses in asynchronous

machinesL. Dupre, M. De Wulf, D. Makaveev, V. Permiakov,

A. Pulnikov and J. MelkebeekDepartment of Electrical Energy, Systems and Automation,

Ghent University, Ghent, Belgium

Keywords Hysteresis, Modelling, Electrical machines

Abstract This paper deals with the numerical modelling of electromagnetic losses in electricalmachines, using electromagnetic field computations, combined with advanced materialcharacterisations. The paper gradually proceeds to the actual reasons why the building factor,defined as the ratio of the measured iron losses in the machine and the losses obtained understandard conditions, exceeds the value of 1.

1. IntroductionDuring the last decade, there has been a vast amount of research worldwide togain a deeper insight into the magnetic behaviour of electrical steel sheets, usedin rotating electrical machines. The standard method for characterisation andqualification of electrical steel is based on unidirectional sinusoidal fluxexcitation at 50 or 60 Hz. Generally, the measured iron loss in electricalmachines exceeds the value which is obtained by multiplying the weight of thesteel in the machine by the loss density specified by standard qualificationmethods. This difference is taken into account by the building factor (Nakata,1984), defined as the ratio of the measured iron losses in the machine and thelosses obtained under standard conditions. Among the reasons for this buildingfactor to exceed the value of 1, one should mention: non-uniform fluxdistribution due to local saturation, harmonics in flux patterns (Cester et al.,1997), local rotational magnetisation, short-circuit between neighbouringlaminations (Marion-Pera et al., 1994), changes in the magnetic characteristicsdue to the mechanical and thermal treatment of the material during theconstruction of the machine, etc. The analysis of the local flux patterns and ofthe local iron losses in electrical machines has been strongly stimulatedand facilitated by the impressive progress in advanced numerical methods.The results from this analysis can lead to interesting suggestions for theimprovement of the efficiency of the electrical machine. We provide in this



The work is carried out in the frame of the GOA project 99-200/4 funded by the researchfoundation of the Ghent University and by the IWT/STWW-project 980357.

Modelling ofelectromagnetic

losses

1051


Vol. 22 No. 4, 2003pp. 1051-1065


DOI 10.1108/03321640310482995

paper a deeper insight into the evaluation of the electromagnetic losses inasynchronous machines.

2. Magnetic and mechanical conditions2.1 Local flux patternsLocal flux patterns are calculated by the electromagnetic field computationsperformed by finite element (FE) techniques in which the material propertiesare described by a single valued magnetisation curve obtained fromEpstein-frame measurements. In the paper, we consider an 18 kW 4 poleasynchronous machine. The stator has a single layer winding with four slotsper pole and per phase. The rotor has 40 unskewed closed slots. The starconnected 18 kW induction machine has been tested at no-load. Two supplieshave been chosen for the motor. The first test category was carried out with asinusoidal supply and the second one with a voltage source inverter. For bothtypes of supply, FE computations were performed in order to study the localfluxpatterns in the machine. The B-loci in the points indicated in Figure 1 arepresented in Figures 2 and 3 for the sinusoidal and inverter supply,respectively. The induction in point 1 varies along one space direction andshows higher harmonics due to non-linear effects (in particular 3rd and 5thharmonics) and slot effects (19th and 21st harmonics). The induction in theyoke (points 3 and 4) and near the tooth yoke interface (point 2) is clearlyrotational, showing slot harmonics to a minor extend. Figure 4 shows theagreement between the computed flux patterns in the machine and the fluxpatterns measured using pick up coils positioned in the machine as shown inFigure 1. When the material characteristics in the field computations aremodified in a realistic way, the fluxpatterns in the machine are almost the same

Figure 1.Cross-section of theasynchronous machinestator

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Figure 2.B-loci sinus supply

Figure 3.B-loci inverter supply


losses

1053

for a certain voltage supply. For the machine under consideration, the materialcharacteristics of the iron core have almost no influence on the flux patternsinside the machine. The flux patterns are mainly defined by the machinegeometry and the supply.

2.2 Alterations brought on by the production processThe performance of the electrical machines may be affected by mechanicalstresses in the steel resulting from mechanical or thermal manufacturingprocesses. Indeed, these processes alter the magnetic properties near thecutting edges. Understanding, quantifying and modelling these alterations isstill a difficult task. In the work of Dupre et al. (1998), modification of themagnetic properties, such as permeability and electromagnetic losses due topunching, laser cutting or erosion was observed on stacked ring cores. Themodifications of the magnetic properties due to local plastic strains andinternal stresses may be identified by studying the evolution of the Vickersmicro-hardness over the tooth width (Ossart et al., 2000). The hardness is verylarge close to the edges, where plastic strains are large, and rapidly decreasesas the distance to the edge increases, reflecting the local effect of stresses.During assembling, the producer usually fixes the core by means ofcompression. It should be noted here that the value of compression rarelyexceeds 8 MPa. This is far below the yield stress of electrical steels, but hasconsiderable impact on anisotropy and may cause additional losses.

Figure 4.B(t)-waveform measuredwith pickup coils as inFigure 1 and calculatedfrom 2D FEM

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3. Electromagnetic lossesBased on the computed field maps inside the machine, the local magneticexcitation conditions for the magnetic material can be derived. Models whichconsider complex magnetic conditions, including the effect of mechanicalstress, must be derived in order to calculate the electromagnetic behaviour andthe corresponding electromagnetic losses.

3.1 Unidirectional flux excitation3.1.1 Statistical loss theory. In the statistical loss theory (Bertotti, 1988), it isstated that the iron losses in laminated materials can be separated into thehysteresis loss Ph ¼ W h f ; the classical loss (“eddy current loss”) Pc ¼ Wc fand the excess loss Pe ¼ We f ; the latter due to extra induced currents becauseof the domain structure. In the framework of the standard loss model and itsextension to non-sinusoidal induction, the concept of loss separation is crucialfor the accurate prediction of the energy losses. Let us consider the case wherethe periodic time dependence of the induction Ba(t) can be described by meansof the Fourier series BaðtÞ ¼ SnBpn sinð2pft þ fnÞ: In the usual case ofsymmetric hysteresis loops, only odd harmonics are considered. The theorydeveloped by Fiorillo and Novikov (1990) shows that a theoretical expressionfor the total energy loss per unit volume is given by

P tð f Þ ¼ W h f þsp2 d 2 f 2

6 n

Xn2 B2

pn

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisGSV 0

p Z 1=f

0 n

X2p nf cos ð2p nf Bpn þ fnÞ

��3=2

dt ð1Þ

where s is the electrical conductivity, d is the lamination thickness, S is thelamination cross-sectional area, G ¼ 0:1357 and the parameter V0 is the wellknown internal field depending on microstructure. It should be mentioned thatthe hysteresis loss component Wh is independent of distortion, provided thereare no local minima in the induction waveform. It is nonetheless important topoint out that also in this case the formulation of the classical and excess lossesremains valid. An objective limitation of this approach, intrinsic to the physicalmodelling of losses, is the requirement of complete flux penetration in thelamination cross-section.

3.1.2 Advanced characterisation. A material hysteresis model whichprovides a description of the time-dependent relation between the magneticfield strength and the magnetic induction in ferromagnetic materials (e.g. SiFe)is required for an accurate analysis of the material behaviour under complexmagnetic field conditions.

In the case of unidirectional magnetisation, one of the most widely usedmodels for magnetic hysteresis is the scalar Preisach model (Bertotti, 1998).


losses

1055

The classical Preisach model is very accurate in describing the quasi-staticmagnetic properties of electrical steel sheets under arbitrary unidirectionalexcitations. In this model, each Preisach dipole has a non-symmetric hysteresisloop defined by the switching fields a and b ðb # aÞ: Depending on the historyof the magnetic field H, the magnetisation f of the dipole takes the value 21 or+1. Then, the magnetisation M(t) is obtained from

M ðtÞ ¼ M revðH ðtÞÞ þ1

2

Z 1

21

da

Z a

21

dbfða;b; tÞPða;bÞ ð2Þ

where P (a, b) is the Preisach distribution function (PDF). The identification ofthe PDF P(a, b) can be performed in different ways (De Wulf et al., 2000;Everett, 1955; Mayergoyz, 1991).

On the other hand, “black box” input-output mathematical models, based onartificial neural networks (ANN), are emerging as a powerful modelling tooland could form an alternative to hysteresis modelling techniques like thePreisach model. Feed forward neural networks (FFNN) have the property toapproximate any non-linear function of an arbitrary number of input andoutput parameters, using standard algorithms. A FFNN hysteresis modeloffers the same accuracy as the classical scalar Preisach model (Makaveev et al.,2001). Here, the neural network topology and input parameters are chosenbased on the properties of the Preisach model.

Based on the loss separation property of SiFe alloys also dynamic hysteresis,i.e. the relation between the time dependent Ba(t) and Hs(t) with

BaðtÞ ¼2

d

Z d=2

0

Bðz; tÞ dz; z-axis orthogonal to lamination ð3Þ

and with Hs(t) denoting the magnetic field at the surface of the lamination, canbe treated conveniently. The quasi-static field Hhyst(t) and the dynamic fieldHdynðtÞ ¼ H cðtÞ þ H excðtÞ; (Figure 5), can be associated with the correspondingloss components for each time point t, leading to the following expression forthe instantaneous power loss Ptot(t) at time t:

P totðtÞ ¼ PhystðtÞ þ PdynðtÞ ¼dBaðtÞ

dtðHhyst þ HdynÞ ð4Þ

The quasi-static and dynamic contributions can thus be treated independently.The first model for dynamic hysteresis starts from the BaHhyst-relation,

obtained from the Preisach model, describing the quasi-static (hysteresis)behaviour of the material. To obtain the BaHs-loop, the total field at the surfaceof the lamination Hs(t) is obtained at each time point as the sum of thehysteresis field Hhyst and the dynamic field Hdyn ¼ H c þ H exc; (Chevalier et al.,1999). Due to the dependency of V0 on the induction Ba, the dynamic field Hdyn

depends on Ba and dBa/dt, but not on the history of the magnetic field.

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Therefore, for this method, it is sufficient to reconstruct the function HdynðtÞ ;HdynðBa; dBa=dtÞ: For the purpose of identification of the function Hdyn onlyBaHs-cycles, measured on Epstein strips at different frequencies, are comparedwith the quasi-static cycle, similar as in Figure 5, (Chevalier et al., 1999). Usingneural networks, equation (4) suggests combining two FFNN. One determinesthe quasi-static BHhyst-loop and the other one yields the dynamic field Hdyn(t),for a given Ba(t) and dBa=dt as input (Figures 6 and 7).

3.2 Circular or elliptical flux excitationsThe case of two-dimensional hysteresis phenomena in the lamination plane isthe subject of interest. The field and induction vectors are restricted to vary inthis plane and are denoted by �HðtÞ ¼ jHðtÞjexpð jfH ðtÞÞ and �BðtÞ ¼jBðtÞjexpð jfBðtÞÞ:

3.2.1 Statistical loss theory. Irregular two-dimensional flux loci appears inmachine cores. The case where one or both components of the two-dimensionalpolarization of non-sinusoidal functions is considered, i.e. BxðtÞ ¼SnBxn sinð2pnft þ fxnÞ (Bxp peak value in the x-direction), ByðtÞ ¼SnByn sinð2pnft þ fynÞ (Byp ¼ peak value in the y-direction). The x-axis ischosen along the main axis of the locus. It is possible to separately predict theunidirectional losses for these two distorted components, see equation (1). It isshown by Appino et al. (1997) that the hysteresis, classical and excess lossunder 2D distorted flux conditions are given by:

Figure 5.Static and dynamic

effects defining the staticand the dynamic

magnetic field


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Figure 6.Measured (Epsteinframe) and calculated(neural networks)magnetisation loopsunder distortedunidirectional fluxexcitation

Figure 7.Measured (Epsteinframe) and calculated(Preisach model)magnetisation loopsunder distortedunidirectional fluxexcitation

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W 2D;disth ¼ W 1D

xhðBxpÞ þ W 1Dyh ðBypÞðR

chðBxpÞ2 1Þ ð5Þ

W 2D;distc ¼

sp2d 2f

6 n

Xn2B2

xn þm

Xm2B2

ym

!ð6Þ

W 2D;diste ¼ W 1D

xe ðBxpÞ þ W 1Dye ðBypÞðR

ceðBxpÞ2 1Þ ð7Þ

where Wx,h, Wx,e and Wy,h, Wy,e are the hysteresis and excess losses underunidirectional excitation of the x and y component, respectively. R c

h and R ce are

material dependent functions.3.2.2 Advanced characterisation. The classical Preisach model provides

accurate results for quasi-static unidirectional (scalar) magnetization, while itsextensions to vector magnetization, proposed by Mayergoyz (1991), stillcontain certain limitations. The Mayergoyz model is computationallyintensive and not valid for circular magnetization at high induction levels,as it does not predict the correct dependence of the losses on the induction.In this section, the FFNN technique is extended to the modelling oftwo-dimensional vector hysteresis. Consider an isotropic vector hysteresissystem with input Bk and output Hk at the time point k. As Bk is the availableparameter to be used as input of the hystersis model during finite elementcalculations, this presentation is convenient. When the magnetization patternsare restricted to circles and ellipses, the magnetic memory state, and theoutput Hk, can be derived from the known input parameters Bk (amplitudejBkj and phase fB

k ) at each time point k, the maximum induction magnitudejBjmax and the axis ratio a ¼ jBjmin=jBjmax (with jBjmin the minimuminduction magnitude), for the corresponding magnetization pattern. Theoutput Hk can be presented conveniently by its amplitude jHkj and the phaseshift uk between Hk and Bk. A FFNN with four inputs would thus yield anaccurate prediction of the relation between Bk and Hk for quasi-static ellipticalmagnetization patterns, for axis ratios a ranging from a ¼ 0 (unidirectionalmagnetization) to a ¼ 1 (circular magnetization), in isotropic materials(Makaveev et al., 2002):

ðj �Hj; uÞ ¼ FFNNðj �Bj;fB; j �Bjmax; aÞ ð8Þ

Dynamic hysteresis or the relation between �BðtÞ and �HtotðtÞ is again treatedbased on the loss separation property of SiFe steels. The quasi-static field�HhystðtÞ and the dynamic field �HdynðtÞ; can be associated with the

corresponding loss components for each time point t, leading to thefollowing expression for the instantaneous power loss Ptot(t) at time t:


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P tot ¼ Physt þ Pdyn ¼d �B

dt�Htot ¼

d �B

dtð �Hstat þ �HdynÞ ð9Þ

The quasi-static and dynamic contributions can thus be treated independentlyfrom each other, analogous to the unidirectional case (Figure 8).

3.3 Influence of mechanical stressThe dependence of magnetic properties on mechanical stress during elastic andplastic deformation can be evaluated by means of recently constructedequipment (Lo Bue et al., 2000; Permiakov et al., 2002). In general, the shape ofthe magnetization loops is changing during the application of stress, Figure 9.The coercive field is increasing while the remanent induction and thepermeability are decreasing. Figure 10 shows the hysteresis losses as afunction of the tensile stress for the magnetization levels of 0.7, 1.0 and 1.2 Tand 50 Hz sinusoidal magnetic flux, while Figure 11 shows the variation of therelative permeability Bp=m0Hp for different values of the compressive andtensile stress.

4. Application of the asynchronous machine at no load4.1 Separation of lossesThe motor, discussed in section 2, was fed by an autotransformer with variablesinusoidal voltage. Eleven voltage levels are applied between 60 and 450 V.Measurements are made after thermal stabilisation. The mechanical losses

Figure 8.Correspondence betweenthe measured andcalculated loops underelliptical flux, usingFFNN

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Figure 9.Variation of

magnetisation loopsunder mechanical stress

Figure 10.Variation of hysteresis

loss as a function ofmechanical stress


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(windage and friction losses) are firstly estimated at 158 W by a powermeasurement at low voltage. The Joule stator losses are calculated by 3 Rstator

I 2, the stator resistances being calculated at the average temperature of thewindings, I is the average current. The iron losses are obtained as the differencebetween the input power and the joule losses in the stator windings and themechanical losses (definition of IEC34-2 standard).

4.2 Local loss mapsKnowing the local excitation the material is subjected to, the local losses insidethe machine of section 2 were determined. The joint contribution of the locallosses was compared with the global losses, measured on the test machine.

Starting from the flux patterns, obtained from the numerical 2D fieldcalculations of section 2, and using the numerical techniques (statistical losstheory, Preisach modelling or FFNN) described in section 3, the localmagnetization behaviour and electromagnetic losses in the stator can becalculated. The obtained local iron losses in the four points of Figure 1 for thetwo supply conditions (sinusoidal supply at 400 V, and invertor supply atfundamental 400 V) are given in Table I. For both supplies and for the fourpoints considered in the stator, three types of calculations are considered.For type 1, we consider only the fundamental of the B-waveform, which isobtained by projecting the local B-locus on the longest axis (reduced sinusoidalunidirectional fluxpattern). Type 2 includes the harmonics of the B-waveform,which is obtained by projecting the local B-locus on the longest axis

Figure 11.Variation of relativepermeability due tomechanical stress(negative value:compression; positivevalue: tension)

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(reduced unidirectional fluxpattern). Type 3 considers the local B-locusincluding harmonics and rotational effects.

It is observed that by considering the local losses in each finite elementtriangle in the stator, the numerical results do not give a good correspondencewith the measured iron losses, e.g. for the sinusoidal nominal supply onemeasures 418 W while 207 W was obtained by computing; for the invertorsupply 640 W was obtained by measurements while the computation results in281 W. Therefore, taking into account higher harmonics and rotational effects,the stator may not be taken to be representative for the evaluation of the ironlosses in the machines.

Consequently, the measured iron losses, obtained as described in section4.1A, even under no load conditions, do not consist of the stator iron losses. Theso called “measured iron losses” may not be located in the stator iron. Lossescould occur in the rotor cage (rotor joule losses), in the stator frame and in therotor iron.

The currents in a squirrel cage of an induction motor cause joule losses.Several computations for the different voltage operating conditions wereperformed and the rotor joule losses in the squirrel cage of the machine neverexceeded 10 W. The flux in the stator frame is pulsating with the samefrequency as the supply. So losses occur in this conductive part of the motor.The frequencies of the flux pulsations in the rotor laminations are above 1 kHz.Because of the high frequencies we can suppose that the most important lossesare the eddy current losses. Considering the losses in the rotor core we obtain abetter correspondence between measured and calculated iron losses: sinusoidalsupply: 418 W measured versus 342 W computed; invertor supply 640 Wmeasured versus 591 W computed. The discrepancy between the results of theloss separation technique, the Preisach model and the FFNN model is small incomparison with the discrepancy between computed and measured total ironlosses (Figure 12).

5. ConclusionsIn this paper, numerical techniques based on the statistical loss theory, thePreisach theory and FFNN have been presented to evaluate the iron losses in

Sinus supply Invertor supplymJ/kg Type 1 Type 2 Type 3 Type 1 Type 2 Type 3

pt1 35.8 39.5 41.5 34.8 49.3 52.1pt2 31.5 35.9 52.1 30.2 45.8 70.4pt3 42.7 42.9 45.6 41.2 53.2 56.6pt4 42.6 42.9 46.1 41.2 49.7 53.6

Table I.Overview loss

density in the fourpoints of Figure 1


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the rotating electrical machines. These techniques result in a precisedescription of the magnetic response of the electrical steel, subjected tounidirectional and rotational field conditions with arbitrary waveform andfrequency. These techniques were used to evaluate the local iron losses inelectrical machines starting from the local flux patterns, obtained from the 2Dnumerical field calculations and from the experiments using pick up coils.Comparing the calculated iron losses with the measured iron losses in electricalmachines, one may conclude that a non-negligible discrepancy is observedwhich must be related to the influence of mechanical stresses in electricalmachines on the magnetic properties and the change (deterioration) of thematerial quality throughout the machine construction.

References

Appino, C., Fiorillo, F. and Rietto, A.M. (1997), “The energy loss components under alternating,elliptical and circular flux in non-oriented alloys”, Proc. 5th International Workshop on 2DMagnetisation Problems, Grenoble, France, pp. 55-61.

Bertotti, G. (1988), “General properties of power losses in soft ferromagnetic materials”, IEEETrans. Magn., Vol. 24 No. 1, pp. 621-30.

Bertotti, G. (1998), Hysteresis in Magnetism, Academic Press, Boston.

Cester, C., Kedous-Lebouc, A. and Cornut, B. (1997), “Iron loss under practical working conditionsof a PWM powered induction motor”, IEEE Trans. Magn., Vol. 33 No. 5, pp. 3766-8.

Chevalier, T., Kedous-Lebouc, A., Cornut, B. and Cester, C. (1999), “Estimation of magnetic loss inan induction motor fed with sinusoidal supply using a finite element software and a newapproach to dynamic hysteresis”, IEEE Trans. Magn., Vol. 35 No. 5, pp. 3400-2.

Figure 12.The measured iron lossesas a function of appliedvoltage at no-loadconditions

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De Wulf, M., Vandevelde, L., Maes, J., Dupre, L. and Melkebeek, J. (2000), “Computation ofPreisach distribution function based on a measured everett map”, IEEE Trans. Magn.,Vol. 36 No. 5, pp. 3141-3.

Dupre, L., Van Keer, R. and Melkebeek, J. (1998), “A study of the influence of laser cutting andpunching on the electromagnetic behaviour of electrical steel sheets”, 4th InternationalWorkshop Electric and Magnetic Fields, Marseille, France, pp. 195-200.

Everett, D. (1955), “A general approach to hysteresis-Part 4: an alternative formulation of thedomain model”, Trans. Faraday Soc., Vol. 51, pp. 1551-7.

Fiorillo, F. and Novikov, A. (1990), “An improved approach to power losses in magneticlaminations under non-sinusoidal induction waveform”, IEEE Trans. Magn., Vol. 26 No. 5,pp. 2904-10.

LoBue, M., Sasso, C., Basso, V., Fiorillo, F. and Bertotti, G. (2000), “Power losses andmagnetization process in Fe-Si non-oriented steels under tensile and compressive stress”,J. Magn. Magn. Mat., Vol. 215, pp. 124-6.

Makaveev, D., Dupre, L., De Wulf, M. and Melkebeek, J. (2001), “Modelling of quasi-staticmagnetic hysteresis with feed-forward neural networks”, J. Appl. Phys., Vol. 89 No. 11,pp. 6737-9.

Makaveev, D., Dupre, L. and Melkebeek, J. (2002), “Neural network based approach to dynamichysteresis for circular and elliptical magnetization in electrical steel sheet”, IEEE Trans.Magn., Vol. 38 No. 5, pp. 3189-91.

Marion-Pera, M.C., Kedous-Lebouc, A., Cornut, B. and Brissonneau, P. (1994), “Evaluation ofinterlaminar losses in magnetic cores”, J. Magn. Magn. Mat., Vol. 133 Nos. 1-3, pp. 156-8.

Mayergoyz, I. (1991), Mathematical Models of Hysteresis, Springer, Berlin.

Nakata, T. (1984), “Numerical analysis of flux and loss distributions in electrical machinery”,IEEE Trans. Magn., Vol. 20 No. 5, pp. 1750-5.

Ossart, F., Hug, E., Hubert, O., Buvat, C. and Billardon, R. (2000), “Effect of punching on electricalsteels: experimental and numerical coupled analysis”, IEEE Trans. Magn., Vol. 36 No. 5,pp. 3137-40.

Permiakov, V., Dupre, L., Makaveev, D. and Melkebeek, J. (2002), “Dependence of power losses ontensile stress for Fe-Si non-oriented steel up to destruction”, J. Appl. Phys., Vol. 91 No. 10,pp. 7854-6.


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Staged modellingA methodology for developing real-life

electrical systems applied to the transientbehaviour of a permanent magnet servo

motorF. Henrotte, I. Podoleanu and K. Hameyer

Department of Electrical Engineering (ESAT/ELECTA),Katholieke Universiteit Leuven, Leuven, Belgium

Keywords Electrical machines, Modelling, Design

Abstract This paper presents a methodology to achieve a global dynamic model of an electricalsystem that consists of a battery, an inverter, a permanent magnet servo motor and a turbine. Thestress is placed on the fact that a classical finite element model would not be able to provide asatisfactory representation of the transient behaviour of the whole system. A staged modelling isproposed instead, which succeeds in providing a complete picture of the system and relies onnumerous finite element computations.

1. IntroductionOne has to analyse the transient behaviour of the system consisting of a:

. battery,

. three-phase inverter bridge,

. permanent magnet (PM) servo motor and

. turbine (mechanical load).

As this system is foreseen for mass production, all elements are kept as simpleand cheap as possible, e.g. very simple inverter with minimum control,standard magnetic iron (high iron losses), low DC supply voltage.Consequently, the system under consideration is characterised by a stronginterrelation between:

. the different components (e.g. the waveshapes of the inverted currentsdepend on the back-emf level, and consequently on the speed, whereas,the torque (and consequently the speed again) depends critically on thosewaveshapes);

. the different physical fields involved (e.g. temperature dependency andmagnetic saturation of the cores);



This paper presents research results of the Belgian programme on Interuniversity Poles ofAttraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming.

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. the different time scales (on the one hand, the small time constant thatcharacterises the electromagnetic phenomena in the motor and thecommutations in the inverter bridge; on the other hand, the large timescale that characterises all mechanical and thermal phenomena).

The goal of the analysis is to dimension the motor (geometry, magnets andcoils) in order to fulfill a set of technical specifications concerning the dynamicsof the system (ramp-up time, acceleration, . . .) and its thermal robustness. It canbe noted that the specifications concern non-magnetic properties of the system,whereas the design parameters are related to the electro-magnetic functioningof the motor. This separation advocates clearly to adopting a global modellingapproach, i.e. a model that involves not only the motor, as is usually the case,but also the supply (inverter) and the load (turbine).

In this context, a classical finite element (FE) analysis is not the mostappropriate approach. Indeed, even if one manages to solve all equationstogether (magneto-dynamic, thermal, power electronics and mechanical), themodel would nevertheless be so heavy and slow that it would be of no practicaluse for design, because designing means exploring thoroughly a large domainof parameters. On the other hand, a simplified analysis based on an analyticalmodel of the motor would fail to provide a sufficiently accurate description ofthe system, because of the critical importance of losses and saturation in thisapplication.

This paper presents a methodology which, by doing a limited number ofsimplifications, allows to combine the conciseness of analytical models with theaccuracy of discrete models (FE models, time-stepping in circuit equations, . . .).The provided model is exactly what is needed at the design stage.

2. Towards staged modelling2.1 FE modellingThe reasons for the success of the FE method are probably its wideapplicability and flexibility, as well as the fact that the discretisation errorcan be theoretically made as small as wished. In view of the developmentof computers and numerical techniques in the recent years, one might feelfree to imagine that there exist no limit to the power of representation ofFE models and that all the various aspects of any system could be takenat once into consideration. Therefore, FE modelling is sometimes seen asthe end of empirical modelling, which is the kind of modelling that consistsof making simplifications and approximations, in carefully selecting therepresentative quantities of the system and finding out the relationshipsbetween them.

But the major trends in FE modelling of electrical devices(i.e. multiphysics, coupling with electronic circuits, 3D geometries, accuratematerial modelling, etc.) have remained unchanged for quite a time now

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(Hameyer and Belmans, 1999). The development and validation of large FEmodels of complex electrical systems has become a common academicexercise, which requires months and which is nowadays a typical PhD thesissubject. Experience however shows that, even if such ambitious projects turnout to be successful at the end, the developed models lack generality, andthey are very heavy and hardly reusable. Moreover, their accuracy cannot becontrolled, unless by experimental means in a limited set of particularsituations.

2.2 Design of electrical systemsLet us consider the problem from another point of view. Engineers think interms of a limited set of variables, even in the presence of complex systems. Thepurpose of design is to build or to modify a system in order to make theinter-relations between those variables match predefined technicalspecifications. Now, the model is the exploratory tool that should help indoing so, i.e. a thing one may interrogate at will instead of doing measurements.

But at R&D stage, designers are still essentially concerned with qualitativequestions and they must usually work on the basis of incomplete, not yet fixedor inaccurate data sets.

On the other hand, in order to be able to play a significant role in a moderndesign process, the model has to account for all important aspects of thesystem, including interactions with its input and output systems. It must alsobe flexible and open to gradual enrichment and improvements as thedevelopment progresses. Finally, its computation should require a reasonableamount of time and its accuracy should always stay under control. These areprinciples of what we call staged modelling: an intellectual construction, basedon well-argued simplifications, and intended to help answering pre-definedtechnical questions. Although FE computations play an important role inbuilding staged models, they are not themselves the kind of model we arelooking for.

The staged modelling of the system under consideration in this paper isdescribed in the following sections. A set of preliminary computations are firstpresented in Section 3. The “over one period stationary” analysis is thenpresented in Section 4 and the transient analysis in Section 5. As this isessentially a paper about methodology, rather than an application paper, thedata and parameters might be incomplete at some places. The authorsapologize for that.

3. Preliminary computations3.1 The flux plotThe key point in the realisation of a staged model is to define carefully theinterface quantities between the different components of the system. Between

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the inverter and the motor, the interfacing quantities are the electricalquantities associated with the three stator phases. In particular, to represent theback-emf of the phase (Miller, 1989), we will pay attention to the so-called fluxplot

fðu; I Þ ¼

ZSph

bðu; I Þ · ~n dS ð1Þ

which gives the total flux embraced by one stator phase as a function of theangular position of the rotor, u and the current in that phase, I. If the motorworks in an equilibrate state, the flux plot of one phase is sufficient, as the fluxplot for the other phases are obtained by a simple angle shift. If the machineworks off saturation, the dependence on I can be neglected. The flux plotgathers in one table all the needed information concerning the motor, seen fromthe point of view of the inverter.

3.2 Core losses parametersIn a PM motor, core losses are mostly located in the stator core. At high speeds,they may override copper losses and therefore need to be estimated carefullyover a wide range of frequencies. A difficulty in the calculation of core losses isthat the magnetic flux density varies not only in time, but also varies widely inspace (Hendershot and Miller, 1994). One may distinguish hysteresis losses,which are proportional to the frequency and eddy current losses, which areproportional to the square of the frequency. One will therefore assume that thecore losses can be expressed accurately by the formula

QcoreðI ;vÞ ¼ C1ðI Þvþ C2ðI Þv2 ð2Þ

where the frequency independent constants C1(I ) and C2(I ) are expressed by

C1ðI Þ ¼

Zstator

Z 2p

0

h ›ubðu; I Þ du

� �; ð3Þ

C2ðI Þ ¼

Zstator

Ceddy

2p

Z 2p

0

j›ubðu; I Þj2

du

� �: ð4Þ

This assumption is one of the simplifications of the staged modelling.One sees that the flux plot (1) as well as the core losses parameters (3) and (4)

depend on b(u, I ) and ›ub(u, I ). They can therefore be computed beforehand fora given geometry of the motor, by postprocessing adequately a series of 2Dstatic FE computations, and then stored into tables.

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4. The “over one period stationary” analysisBecause of the big difference in scale between one electrical period and the timeneeded by the motor to reach its nominal speed, a classical transient analysiswould require several ten thousands of time steps. The transient analysis istherefore split up into two parts. The first part consists of calculating “over oneperiod” the stationary current waveshapes for a fixed speed, v, a fixedcommutation angle, a and fixed temperatures. The second part is the transientdynamic and thermal analysis. This splitting is one of the simplifications of thestaged modelling.

During the “over one period stationary” analysis, the inductances,resistances and back-emfs of the phases, the extra voltages and resistancesdue to the electronic components and the switching strategy of the inverterbridge are all taken into consideration for the computation of the waveshapesof the phase currents and voltages.

One starts from the relation between the electric quantities associated witheach stator phase:

V j ¼ RjI j þ ›tfj; j ¼ A;B;C ð5Þ

where fj is given by equation (1). The total resistance of the phase, Rj iscomputed analytically. It may be augmented by extra resistance due to theelectronic components. The total phase voltage, Vj is equal to the batteryvoltage, Vbattery distributed over the different phases, according to the particulartopology at each instant of time of the inverter and to the connection type (here aWye-connection) (Hendershot and Miller, 1994). One writes

V j ¼ lV battery ð6Þ

where l is the coefficient taken from Table I.The table has six sections, which correspond to the six periods between

the switching-on of two successive phases. For instance, let us consider thelast group of four lines in the table. In the considered period, the phase Ahas to be switched on ð0 !þ1Þ and the phase C has to be switched offðþ1 ! 0Þ, the phase B remaining in the same state ð21 !21Þ.Immediately after sending the switch-on signal to phase A, all threephases are conducting. The inverter topologies corresponding to the lasttwo lines of the group are then used. In both of them, the voltage ispositive in the switched on phase (i.e. phase A), the voltage is negative inthe switched off phase (i.e. phase C ), and the voltage is either positive ornegative in phase B, which allows to control the current in that phase.After a while, the current in phase C reaches zero and the phase ceases tobe conducting. From that instant on, the first two lines of the group areused, the first one to let the current decrease (freewheeling) and the secondone to make it increase in the loop formed by the phases A and B.

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Starting from equation (5), the following explicit time-stepping scheme can bewritten:

I jðuþ DuÞ ¼ I jðuÞ þV j 2 v›ufj 2 RjI jðuÞ

v›Ifj

Du ð7Þ

and applied until the achievement of a stationary waveshape Ij (u) over oneperiod t.

Figures 1 and 2 show examples of the computed stationary waveshapes ofthe phase currents and of the voltages, which are the different terms ofequation (5).

Phase A Phase B Phase C

+1 ! +1 21 ! 0 0 ! 210 Off 0

1/2 Off 21/221/3 2/3 21/3

1/3 1/3 22/3

+1 ! 0 0 ! +1 21 ! 21

Off 0 0Off 1/2 21/2

22/3 1/3 1/32 1/3 2/3 21/3

0 ! 21 +1 ! +1 21 ! 00 0 Off

21/2 1/2 Off21/3 21/3 2/32 2/3 1/3 1/3

21 ! 21 +1 ! 0 0 ! +10 Off 0

21/2 Off 1/21/3 22/3 1/3

2 1/3 21/3 2/3

21 ! 0 0 ! 21 +1 ! +1Off 0 0Off 21/2 1/22/3 21/3 21/31/3 22/3 1/3

0 ! +1 21 ! 21 +1 ! 0]0 0 Off

1/2 21/2 Off1/3 1/3 22/32/3 21/3 21/3

Table I.Coefficient for thedistribution of the

battery voltage overthe different phases

(Wye-connection)

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The mean torque is now computed by

Tðv;aÞ ¼3

2p

Z 2p

0

2 ›ufðu2 a; 0ÞI ðuÞ du2QcoreðI ;vÞ

v: ð8Þ

with Qcore given by equation (2). This first part of the staged model can then beseen as a procedure that gives the torque for given speed, commutation angleand temperatures.

Figure 1.Stationary waveshapesof the phase currents (inall phases) and of thevoltages (in one phase) atlow speed(v ¼ 1 £ 103 rad/s) andwith a¼0

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5. The transient analysisThe transient analysis of the system can now be described. Actually, twotransient analysis are carried over in parallel. One for the temperatures and onefor the speed.

5.1 The thermal analysisTemperature has a strong influence on the behaviour of PM motors (throughthe temperature dependency of the magnet strengths and the electricalconductivity). Knowing the different losses (joule losses in the stator coils,stator iron losses and power dissipated by the power electronics components)and their location in the different regions of the system, an elementary transient

Figure 2.Stationary waveshapes

of the phase currents (inall phases) and of the

voltages (in one phase) ata higher speed

(v ¼ 2.5 £ 103 rad/s) andwith a¼558

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thermal model can be built-up in order to estimate the temperature level of thedifferent parts of the machine. (Hamdi, 1994). The thermal model (Figure 3)associates one node with each region of the model (e.g. coils, yoke, cooling fluid,power electronics components, ambient air around the motor, etc.). Each node isassociated with a temperature TI, a heat source PI and a thermal capacity CI.The nodes are linked with branches. Each branch is associated with a heat fluxQIJ and a thermal conductivity KIJ. All these parameters are estimatedanalytically or by means of dedicated thermal FE analysis.

One can now write evolutionary equations for the temperatures:

›tðCI TI Þ þJ

XQIJ ¼ PI ; ð9Þ

QIJ ¼ 2KIJ ðTJ 2 TI Þ: ð10Þ

5.2 The dynamic analysisThe second part of the transient analysis is the transient dynamic analysis ofthe speed of the motor, according to the ordinary differential equation in time

J›tvþ f ðvÞ ¼ Tðv;aÞ ð11Þ

where J is the inertia of the rotating part, f(v) is the reaction torque exerted bythe turbine and friction forces and where T(v,a) is given by equation (8).

Different strategies of flux weakening, i.e. modification of a as a function ofv, can now be analysed easily (Figures 4 and 5).

5.3 Combination with the “over one period stationary” analysisAt each time step of the transient analysis, one “over one period stationary”analysis is done with the current values of the speed v, of the commutationangle a and of the temperatures TI (which affect the values of all resistancesand of the flux plot). The outcome of the “over one period stationary” analysisis the value of the torque T(v,a) and the values of all losses PI in the differentregions of the system. With these values, the speed v and the temperatures TI

are updated by the explicit schemes described in the previous sections.

Figure 3.Sketch of a thermalmodel with six nodes

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6. ConclusionsBuilding the staged model has implied doing simplifications. In counterpart,the staged model has the big advantage of providing a faithful dynamicdescription of the overall system at a reasonable computational cost. As thedevelopment goes along, the staged model is open to gradual enrichment andimprovements, due to further FE investigations aiming at determining itsdifferent parameters or at estimating the influence of the simplifications thathave been done.

Figure 4.Plot of the torque vs

speed characteristic, withand without flux

weakening

Figure 5.Plot of the torque as a

function of the speed vand the commutation

angle a. Flux weakeningconsists of selecting, at

each speed v, thecommutation angle a

that maximises thetorque

Staged modelling

1075

References

Hamdi, E.S. (1994), Design of Small Electrical Machines, Wiley, Chichester, England.

Hameyer, K. and Belmans, R. (1999), Numerical Modelling and Design of Electrical Machines andDevices, WIT Press, Southampton, UK.

Hendershot, J.R. and Miller, T.J.E. (1994), Design of Brushless Permanent-Magnet Motors,Clarendon Press, Oxford, USA.

Miller, T.J.E. (1989), Brushless Permanent-Magnet and Reluctance Motor Drives, Clarendon Press,Oxford, USA.

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3D h-f finite elementformulation for the

computation of a lineartransverse flux actuator

G. Deliege, F. Henrotte, H. Vande Sande and K. HameyerKatholieke Universiteit Leuven, Leuven, Belgium

Keywords Finite element analysis, Permanent magnets, Optimization

Abstract A finite element analysis of a permanent magnet transverse flux linear actuator ispresented. In this application where we need a small model (for optimisation purposes) as well as ahigh accuracy on the computed force, we propose to combine several models with different levels ofsize and complexity, in order to progressively elaborate an accurate, but nevertheless tractable,model of the system.

1. IntroductionOne of the typical outcomes of the numerical modelling of the dynamics ofactuators is the estimation of the magnetic force exerted on moving parts. Inthe conception and the design of such devices, the geometry of the magneticcores (and consequently of the airgap) is a fundamental issue. Optimizing thegeometry so as to match technical constraints requires numerous numericalcomputations of the system, which makes it highly desirable to dispose of anumerical model of small size. On the other hand, the exerted force depends onthe distribution of the magnetic field around the moving part. It has always athree-dimensional (3D) component, of which the relative importance dependson the geometry as well. The aim of this paper is to design a finite elementmodel of a linear transverse flux actuator, which allows the computation ofthe force with a given accuracy while minimizing the number of unknowns.The geometry and the main characteristics of the actuator are described.A two-dimensional (2D) model and a simplified 3D model limited to a regionaround the mover are presented. A dual approach in both scalar and vectorpotential formulations is proposed. Two methods to compute the magneticforce on the mover are presented.

2. Description of the motorThe permanent magnet transverse flux linear actuator under consideration aimsat fast and accurate positioning. It has been described in previous papers



The authors are grateful to the Belgian “Fonds voor Wetenschappelijk Onderzoek Vlaanderen”for its financial support of this work and the Belgian Ministry of Scientific Research for grantingthe IUAP No. P4/20 on Coupled Problems in Electromagnetic Systems.

3D h-f finiteelement

formulation

1077


Vol. 22 No. 4, 2003pp. 1077-1088


DOI 10.1108/03321640310483011

(Deliege et al., 2002; Vande Sande et al., 2001). The actuator consists of twoindependent motors facing each other (Figure 1). The stators can be seen as longC-cores with toothed lower and upper plates. A coil is wound around eachvertical core. The teeth of the two stators are shifted in space by a quarter ofa pole pitch, so that the reluctance forces in the direction of the movement, i.e. thex-axis, cancel each other out (Deliege et al., 2002; Vande Sande et al., 2001). Themovers are made of blocks of iron and high energy magnets magnetized in the xdirection alternately. A block of non-magnetic material is sandwiched betweenthe movers in order to avoid flux passing from one mover to the other. Themovers are therefore mechanically connected, but magnetically independent,and only one motor needs to be modelled (Figure 2). The magnet and iron blocksforming the mover have the same dimensions as the stator teeth in the x and zdirections. The pole pitch is equal to four times the block length. The position ofthe mover is measured with respect to a reference position for which the firstblock of the mover is aligned with a stator tooth.

3. Finite element modelsIn this application where we need a small model (for optimisation purposes) aswell as well as a high accuracy on the computed force, we propose to combineseveral models with different levels of size and complexity, in order toprogressively elaborate an accurate, but nevertheless tractable, model of thesystem.

Figure 1.Geometry of the overalltransverse flux linearactuator

Figure 2.Full 3D finite elementmodel of the actuator

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3.1 Airgap centred 3D modelWhen dealing with 3D models, it is important to use unknowns sparingly. As theaccuracy of the computed force depends mainly on the accuracy of the computedmagnetic field in the airgap around the mover, we can advantageously leavethe vertical core and the coil outside the model. Therefore, we have defined anairgap-centred 3D model, which focuses on the airgap field and devote amaximum of the available unknowns to the description of the field around themover (Figure 3). The airgap centred 3D model is connected to a simple magneticcircuit that accounts for the vertical core and the coil, in order to get in total acomplete rigorous model of the system.

3.2 2D modelThe 3D effects occurring around the mover cannot be taken into account by a2D model. However, the 2D approximation has definite advantages whencompared to a full 3D approach: the geometry and the control of the quality ofthe mesh are easier and faster; and the computation time is significantly lower.Therefore, the design of a 2D model is generally a preliminary step whichallows the designer to perform many computations to determine the overallbehaviour of the system, and to investigate the influence of the parameters, at areduced computation cost. The 2D model is a slice of the motor in the x-y plane(Figure 4). Two regions are added, above and under the stator teeth, torepresent the part of the stator around which the coil is wounded. The problemis solved with both scalar and vector potential formulations as explained in thefollowing section.

4. Finite element dual formulationsIn this application, dual analysis is used to determine which refinement isnecessary in the airgap to have the force on the mover computed with a givenaccuracy. This question can be answered satisfactorily in the context of a2D analysis, because one may assume that the smallness of the characteristicmesh size needed to obtain a given accuracy will not depend crucially on the3D effect. The costly dual analysis is therefore carried over with the simplified2D model, in order to find out how fine the mesh in the airgap must beto reach the desired accuracy and how coarser it can be at other places.

Figure 3.(a) Full 3D model and

(b) focused 3D model ofthe actuator


formulation

1079

The convergence of the force computed by the 2D dual formulations asa function of the total number of nodes of the mesh is shown in Figure 5.The values obtained with the dual approach give a valuable control on theaccuracy. On the basis of this curve, a relation can be found between thecharacteristic length of the elements of the mesh and the accuracy of the globalquantities (force, energy). This relation helps designing the 3D model, bygiving an approximation of the size and the distribution of the elements in the3D mesh in order to reach a given accuracy. To illustrate the relativecomputation cost of the models, a 3D mesh of more than 500,000 nodes isnecessary to obtain the same accuracy as a 2D mesh of 20,000 nodes.

Figure 4.2D finite element model

Figure 5.Convergence of the forcecomputed in 2D with aand h-f formulations;flux w ¼ 0.002 Wb

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The problem is magnetostatic and the Maxwell equations (1) and (2) must besolved.

div b ¼ 0 ð1Þ

curl h ¼ j ð2Þ

4.1 Scalar potential formulationThe magnetic field h is decomposed into the sum of the gradient of the scalarmagnetic potential f, and the source field h s which must verify curl h s ¼ j; sothat equation (2) is automatically fulfilled.

h ¼ hs 2 gradf ð3Þ

The constitutive law is

b ¼ m0ðh þ mÞ ð4Þ

m ¼

0 in air

mðbÞ or mðhÞ in iron

hc in permanent magnets

8><>: ð5Þ

where hc is a constant depending on the type of permanent magnet. Replacingequation (3) in (4), and then in equation (1), we obtain

divðm0ðhs 2 gradfþ mÞÞ ¼ 0 ð6Þ

of which the weak form isZV

gradf0 · ðm0ðhs 2 gradfþ mÞÞ dV

2

ZGL<GR

f0b · n dG ¼ 0; ;f0 [ F0hðVÞ ð7Þ

with

F0hðVÞ ¼ {f [ L2ðVÞ; gradf [ L2ðVÞ; fjGU<GD

¼ 0}: ð8Þ

4.2 Vector potential formulationThe magnetic induction is expressed as b ¼ curl a; in order to fulfillequation (1). Equation (2) expressed in terms of the vector potential a becomes:

curl� 1

m0curl a 2 m

�¼ j ð9Þ

The weak form is


formulation

1081

ZV

curla0 ·1

m0curla2m

� dV2

ZGU<GD

ða0^hÞ ·ndG¼ 0; ;a0 [F1hðVÞ ð10Þ

with

F1hðVÞ ¼ {a[L2ðVÞ; curla[L2ðVÞ;n^ajGL<GR

¼ 0}: ð11Þ

4.3 Boundary conditionsThe boundary conditions for the 2D model (Figure 4) are set according toTables I and II.

5. Force computation5.1 Direct differentiation of energy and coenergyAn accurate computation of the force profile is one of the goals of this finiteelement analysis. The calculation of electromagnetic forces by directdifferentiation of the magnetic energy or coenergy is simple, easy tounderstand and perfectly rigorous; but it is generally disregarded because itrequires to solve the system several times. However, this drawback vanishes ifone is interested in the force, not at one particular position, but over a range ofpositions. The total magnetic coenergy F and magnetic energy C of the systemare given by

FðhÞ ¼

ZV

1

2m0ðh þ mÞ2 dV ð12Þ

CðbÞ ¼

ZV

1

2m0b · b 2 m · b dV ð13Þ

Making use of equation (4), one checks easily that expressions (12) and (13)verify the relation

Fix I Fix w

GU f¼ I 2RGU

m0ðhs 2 gradfþ mÞ · n dG ¼ wGD f¼ 0 f ¼ 0GL<GR b · n¼ 0 b · n ¼ 0

Table I.Boundaryconditions for theh-f formulation

Fix I Fix w

GU< GD h^ n ¼ 0 h ^ n ¼ 0GL aZ ¼ 0 aZ ¼ 0

GR

RGR

1m0

curl a 2 m� �

^ n dG ¼ I aZ ¼ w

Table II.Boundaryconditions for the aformulation

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CðhÞ þFðbÞ ¼

ZV

b · h dV: ð14Þ

In soft magnetic materials, h c vanishes and we obtain the classical expressionsof energy and coenergy. In permanent magnets, we have a situationrepresented in Figure 6 for a given working point (b, h) such that b · h and F arenegative. Notice that in that case, F and C are not equivalent.

The coenergy F(h) is computed when the scalar potential formulation isused, whereas the energy C(b) is computed when the vector potentialformulation is used. As the problem is solved for a set of successive positions ofthe mover, we can easily compute the value of the component of the force in thedirection of motion at any point xi, with a second order approximation of thederivative of the coenergy or the energy:

FxðxiÞ ¼Cðxiþ1Þ2Cðxi21Þ

xiþ1 2 xi21¼ 2

Fðxiþ1Þ2Fðxi21Þ

xiþ1 2 xi21ð15Þ

5.2 Eggshell methodOn the other hand, if one is interested in the transient analysis of the actuator, amethod that gives directly the force at each position is needed. There are twopossibilities, the virtual work principle and the Maxwell stress tensor.

The virtual work method is fairly general, but neither easy to understandnor to implement. Moreover, the generality of this costly method is not fullyexploited in this particular case where, instead of the local values at nodes of

Figure 6.Energy and coenergy in

the magnet


formulation

1083

the magnetic force, the resultant force exerted on a rigid body is sought after. Itis therefore worth finding a more dedicated and efficient method. Thetechnique we have adopted, i.e. the eggshell method (Henrotte et al., 2002),stems from the application of the Maxwell stress tensor. The latter is preciselyvalid for the computation of the magnetic forces exerted on rigid bodies placedin air. A direct application of the Maxwell stress tensor requires however tointegrate over a surface an expression of which the computation requiresinformation coming from outside the surface (i.e. the normal gradient of thepotential). This makes it necessary to find out, for each integration point inthe surface, the finite element to which it belongs. The eggshell method is aparticular application of the Maxwell stress tensor that avoids thisdisadvantage. It consists of averaging the Maxwell stress tensor over acontinuous class of concentric closed surfaces, which fill up an eggshell shapedregion VB placed around the moving body (Figure 7). Arkkio’s famous formulafor torque computation in electrical machines (Arkkio, 1987) results from theapplication of the same principle in the airgap of an electrical machine,assuming a rigid body rotation of the rotor.

The parallelepipedic eggshell shown in Figure 7 can in that way beconsidered as filled up by a class of parallelepipedic surfaces enclosed insideeach other. The normal to all those surfaces make up a unit vector field n that isuniform over each of the six walls of the eggshell. It can usually be definedanalytically. For rigid body translations, the eggshell method formulae are:

F ¼

ZVB

m0

dhðh · nÞ2

1

2nðh · hÞ

� dVB ð16Þ

F ¼

ZVB

1

m0dbðb · nÞ2

1

2nðb · bÞ

� dVB ð17Þ

where d is the thickness of the eggshell.Figure 8 shows a comparison between the two methods used for force

computation. They give similar results in 2D as well as in 3D, but the method

Figure 7.Eggshell region VB

enclosing a layer of airand the mover(grey volume)

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based on the direct differentiation of energy suffers from the loss of accuracydue to the numerical differentiation.

In total, the eggshell method is more interesting because it requires only theintegration over a smaller volume (VB instead of the complete domain) and allthe needed information is contained in that small volume. We have also found itto be more accurate in the 3D case.

6. ResultsA first set of calculations has been done with the 3D and 2D models, using theh -f formulation. The 2D and 3D meshes contain 7,000 and 87,000 nodes,respectively. The mover displaces over one pole pitch and for each position, thecoil current takes the five values 2200, 2100, 0, 100 and 200 A.

One sees in general that the 3D effect influences sensibly the energy andcoenergy of the system (Figures 9 and 10), as well as the developed force(Figures 11 and 12), but the 2D model captures the most important features ofthe behaviour of the device. The 3D effect of the force is not a simplemultiplication factor, but it depends on the position of the mover. It can beestimated from Figure 8.

The shape of the curves, the relative influence of the coil currents, are allqualitatively well described by the 2D model, at a much lower computationalcost and with a much better continuity. They give a justification that one canpursue as far as possible the geometrical optimisation with the 2D model. Itsuggests indeed that the optimum configuration found by the 2D analysiswill not be very different from the real optimum. The slight irregularities ofthe curves representing the force profiles computed in 3D (Figure 11) show that

Figure 8.Comparison of the forces

computed with theeggshell method and the

differentiation ofcoenergy, in 2D and 3D

(h-f)


formulation

1085

the 3D mesh is too coarse, even if the number of nodes is ten times greater thanin the 2D mesh and despite the fact that, due to the presence of the eggshell,there are several layers of elements in the airgap. This accuracy problem isobviously attributable to a lack of accuracy of the solution itself, and not to themethod used to compute the force, since the results obtained by the

Figure 9.3D (h-f): coenergy of thesystem as the coil currentranges from 2200 to200 A

Figure 10.2D (h-f): coenergy of thesystem as the coil currentranges from 2200 to200 A

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differentiation of the coenergy and the eggshell method confirm each other(Figure 8).

7. ConclusionA linear transverse flux actuator has been described. A 2D and a simplified 3Dfinite element model of the machine have been proposed in order to reduce

Figure 11.3D (h-f): force along X

computed with theeggshell method, as the

coil current ranges from2200 to 200 A

Figure 12.2D (h-f): force along X

computed with theeggshell method, as the

coil current ranges from2200 to 200 A


formulation

1087

the computation time, with a view to the optimisation of the force on the mover.The dual h-f scalar potential and a vector potential formulations in thepresence of permanent magnet materials have been reminded. Two methods tocompute the force have been described, and their respective advantages havebeen pointed out. The finite element models have then been used to computethe force acting on the mover, as a function of its position and the coil current. Ithas been shown that the 2D analysis is unable to describe all the 3D effects, andto accurately evaluate the amplitude of the force and the energy or coenergy ofthe system. However, it can describe the important features of the behaviour ofthe device, and give an idea of the sensitivity of the global quantities to avariation of parameters, at a much lower computation cost. In addition, the 2Dformulations are in some cases easier to implement: the vector potentialformulation, for instance, does not require to build a spanning-tree for gauging.Therefore, if it cannot substitute for a 3D analysis, it nevertheless constitutes avaluable preliminary and complementary step.

References

Arkkio (1987), “Analysis of induction motors based on the numerical solution of the magneticfield and circuit equations”, Acta Polytechnica Scandinavica, p. 56.

Deliege, G., Vande Sande, H., Hameyer, K. and Aerts, W. (2002), “3D finite element computationof a linear transverse flux actuator”, Proceedings of the International Conference on PowerElectronics, Machines and Drives (PEMD2002), April 2002, Bath, UK, pp. 315-9.

Henrotte, F., Deliege, G. and Hameyer, K. (2002), “The eggshell method for the computation ofelectromagnetic forces on rigid bodies in 2D and 3D”, Proceedings of IEEE Conference onElectromagnetic Field Computation (CEFC2002), 16-19 June 2002, Perugia, Italy, pp. 16-19.

Vande Sande, H., Deliege, G., Hameyer, K., Van Reusel, H., Aerts, W. and De Coninck, H. (2001),“Design of a linear transverse flux actuator for fast positioning”, Proceedings of the XIIIthConference on the Computation of Electromagnetic Fields (COMPUMAG2001), July 2001,Evian, France, pp. 54-5.

Further reading

Blissenbach, R., Schafer, U., Hackmann, W. and Henneberger, G. (2000), “Development of atransverse flux traction motor in a direct drive system”, Proceedings of the InternationalConference on Electrical Machines (ICEM00), August 2002, Helsinki, Finland.

Dular, P., Geuzaine, C., Genon, A. and Legros, W. (1999), “An evolutive software environment forteaching finite element methods in electromagnetism”, IEEE Transactions on Magnetics,Vol. 35 No. 3, pp. 1682-5.

Laithwaite, E.R. and Bolton, H.R. (n.d.), “Linear motors with transverse flux”, Proceedings IEE,Vol. 118 No. 12.

Webb, J.P. and Forghani, B. (1989), “A single scalar potential formulation for 3D magnetostaticsusing edge elements”, IEEE Transactions on Magnetics, Vol. 25 No. 11, pp. 4126-8.

Weh, H. and Jiang, J. (1988), “Berechnungsgrundlagen fur transversalfluss maschinen”, ElectricalEngineering: Archiv fur Elektrotechnik, Vol. 71, Springer, Berlin.

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Constrained least-squaresmethod for the estimation of

the electrical parameters of aninduction motor

Maurizio Cirrincione and Marcello PucciI.S.S.I.A-C.N.R., Palermo, Italy

Giansalvo Cirrincione and Gerard-Andre CapolinoDepartement du Genie Electrique-IUP-GEEI, Universite de Picardie,

Amiens, France

Keywords Induction motor, Identification, Estimation

Abstract This paper presents for the first time the analytical solution of the constrainedminimization for the on-line estimation of the electrical parameters of an induction motor. Themethod is fully described mathematically and its goodness is verified experimentally on a suitablyset up test bench. This methodology permits the almost correct computation of all the so calledK-parameters, which is not always the case in current literature, thus resulting in the correctestimation of the electrical parameters.

List of symbolsusg ¼ space vector of the stator voltages

expressed in the general referenceframe

urg ¼ space vector of the rotor voltagesexpressed in the general referenceframe

isg ¼ space vector of the stator currentsexpressed in the general referenceframe

irg ¼ space vector of the rotor currentsexpressed in the general referenceframe

csg ¼ space vector of the stator flux-linkagesexpressed in the general referenceframe

crg ¼ space vector of the rotor flux-linkagesexpressed in the general referenceframe

Ls ¼ stator inductanceLr ¼ rotor inductanceLm ¼ magnetising inductanceRs ¼ resistance of a stator phase windingRr ¼ resistance of a rotor phase windingvr ¼ angular rotor speed (in electrical

angles per second)Ts ¼ stator time constant ¼ Ls/Rs

b0 ¼ Rr/Lr¼ inverse of the rotor timeconstant Tr

s ¼ total leakage factor ¼ 1 2 L2m=ðLsLrÞ

IntroductionAt present, induction motors are widely used in a great deal of industrialapplications for their well-known advantages in comparison with other kinds ofmotors. However, when these motors are used in high-dynamical performancedrives, then the problem of the instantaneous knowledge of their parameters isnecessary, e.g. in field oriented control (FOC) and direct-torque control (DTC)where the real-time computation of the rotor or stator flux-linkage is necessary.



Constrainedleast-squares

method

1089


Vol. 22 No. 4, 2003pp. 1089-1101


DOI 10.1108/03321640310483020

Several techniques have been proposed for correct flux estimation (extendedluenberger observer¼ELO, extended kalman filter ¼ EKF, model referenceadaptive systems ¼ MRAS) (Vas, 1994, 1996), but the drawback of all thesemethods is the amount of necessary computations.

However for on-line estimation of slowly-varying parameters, as in the caseof induction motors, the method of least-squares (LS) is certainly one of themost suitable for both its simplicity and the low computational burden itrequires. In particular, a LS approach, which estimates the electricalparameters on the basis of stator currents and voltages and the rotationalspeed has been developed by Cirrincione and Pucci (2001, 2002), Moons and DeMoor (1995), Pucci and Cirrincione (2001), Ribeiro and Jacobina (2000), Ribeiroet al. (1997), Stephan and Bodson (1994) and Velez-Reyes et al. (1989). Thesemethods have always presented some difficulties because auxiliary variablesand constraints are introduced for retrieving the physical parameters andunfortunately not all of them are easy to detect. In general, only numericalmethods are used for the identification without taking into considerationneither the so called K2 problem, that is the difficulty in estimating one of theparameters with resulting bad estimation of the electrical parameters, nor theconstraint. Stephan and Bodson (1994), developed a more complete method toovercome this problem, also trying to consider the constraints, but at theexpense of the computational burden, without giving, however, any proof ofconvergence. In the work of Ribeiro and Jacobina (2000) the K2 problem causesthe parameter estimation procedure to be split into two phases.

This paper, after underpinning that the parameter estimation of theinduction motor is a LS constrained minimisation, gives an analytical solutionto this problem and verifies this method experimentally with a suitable set upof test bench.

The equation of the induction machine for the application of LSThe employment of any LS technique for real-time identification of inductionmachines requires the mathematical model of the machine itself to berearranged. It is well-known that the induction motor model can be describedby the following stator and rotor space-vector voltage equations in a generalreference frame, which rotates at a given speed vg (Vas, 1994)

usg ¼ Rsisg þdcsg

dtþ jvgcsg

urg ¼ Rrirg þdcrg

dtþ jðvg 2 vrÞcrg

csg ¼ Lsisg þ Lmirg

crg ¼ Lrirg þ Lmisg

ð1Þ

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In the above equations, the different electrical parameters could be constant ortime and space varying depending on the preliminary assumptions. Byeliminating the space vector of the rotor currents, which are not measurable,and expressing the direct and quadrature components in the stator referenceframe, the following matrix equations is obtained under the assumption ofslowly varying speed ðdvr=dt ø 0Þ with the following parameters, calledK-parameters:

K1 ¼1

sTsþ

b0

s; K2 ¼

b0

sTs; K3 ¼

1

sTs; K4 ¼

1

sLs; K5 ¼

b0

sLsð2Þ

disD

dtisD vrisQ 2 dusD

dtþ vrusQ

� �2usD

disQ

dtisQ 2vrisD 2

dusQ

dt2 vrusD

� �2usQ

0BB@

1CCA

K1

K2

K31

K4

K5

0BBBBBBBB@

1CCCCCCCCA

¼

2 d2isD

dt 2 2 vrdisQ

dt

2d2isQ

dt 2 þ vrdisD

dt

0B@

1CA ð3Þ

The relationship that exists between the K-parameters is as follows:

K2K4 ¼ K31K5 ð4Þ

From the K-parameters not all the five electrical parameters (Rs, Rr, Ls, Lr, Lm)can be retrieved as no rotor measurements are available. In fact, theK-parameters determine only four independent electrical parameters that areRs, Rr, s, b0 ¼ 1=Tr in the following way:

Tr ¼K4

K5

s ¼K5

K4ðK1 2 K31Þ

Ls ¼K1 2 K31

K5

Rs ¼K2

K5¼

K31

K4

ð5Þ


method

1091

where the subscripts sD and sQ refer, respectively, to the component along theD-axis (direct axis) and the Q-axis (quadrature axis) of the stationary referenceframe fixed to the stator. The K-parameters to be estimated are 5. It should beremarked that once the parameters are estimated, the rotor flux linkage can bereconstructed by using any flux model (Vas, 1994) and consequently, the torquecan be computed. This allows the estimation of both the load torque and theinertia by using the mechanical equations. Since this computation isstraightforward and already presented by Stephan and Bodson (1994), it isnot dealt with here.

The matrix equation (3) together with the constraint can be written in thefollowing form:

Ax < b

f ðxÞ ¼ 0

(ð6Þ

where f(.) is the constraint function (4). The first equation in (6) can be solvedfor the K-parameters both in steady-state and transient in real-time by using anordinary least-squares (OLS) method, because the main cause of error, themodelling error, is present in the observation vector. The drawback, as pointedout by Cirrincione and Pucci (2001, 2002), Moons and De Moor (1995), Pucci andCirrincione (2001), Ribeiro and Jacobina (2000), Ribeiro et al. (1997), Stephanand Bodson (1994) and Velez-Reyes et al. (1989), is that a suitable signalprocessing system must be designed which employs voltage and currentsensors, analog low-pass filters, digital low-pass filters and differentiatorfilters.

By inspection of the equation (3), it is easy to recognise that the secondcolumn of the matrix has values which are lower than those of the othercolumns. This means two things:

(1) a gradient descent method is unsuitable as it is too slow to converge;

(2) if a constrained minimisation is used it is intuitive to realise that thedifference between the true value, that is the solution satisfying theconstraints, and the 2-norm solution, that is the unconstrained minimum,can be quite apart from each other along the direction of minimumgradient (K2).

This makes the estimation of K2 critical, which is also confirmed by Cirrincioneand Pucci (2001, 2002), Moons and De Moor (1995), Pucci and Cirrincione (2001),Ribeiro and Jacobina (2000), Ribeiro et al. (1997), Stephan and Bodson (1994)and Velez-Reyes et al. (1989) (K2 problem). Two paths can then be followed toovercome this difficulty. The former employs an unconstrained minimisationwhich in a way, takes into account the constraint. This is the method followedby Stephan and Bodson (1994), but no proof about the convergence of thealgorithm has been shown. Also an LS constrained minimisation has been

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suggested by Moons and De Moor (1995), but too many constraints are present,due to the high order of the differential equations derived from the equation (3),which makes the method unsuitable for real world applications. This paperinstead presents a constrained analytical minimisation thus overcoming the K2

problem. This method is fully explained in the following.

The constrained minimization: analytical solutionMethodologyIt is known that in parameter estimation, if the stator reference frame is chosen,an equation as

Ax ¼ b

appears, where A is an m £ 5 matrix, x is the vector of the five unknowns andb is the m £ 1 observation vector. The x vector is computed by an ordinary LSalgorithm, which means that the following function error should be minimised

E ¼ kAx 2 bk22 ¼ ðb 2 AxÞTðb 2 AxÞ ¼ bTb 2 2PTx þ xTRx ð7aÞ

where R ¼ ATA is the autocorrelation matrix ð5 £ 5Þ; while PT ¼ bTA is themutual correlation vector ðP2R 5£1Þ: However, between the x components thefollowing constraint exists

x2x4 ¼ x3x5 ð7bÞ

This means that a constrained LS minimization should be performed. It isnecessary to reduce equation (7(b)) into a canonical form with a suitablereference frame and then accordingly modify the constraint. As explained inthe following, three translations and one rotation are necessary to achieve thisgoal.

(1) Translation. The expression (7(a)) can be easily simplified into

z1 ¼ 22PTx þ xTRx ð8Þ

where z1 ¼ E 2 bTb(2) Rotation. Let y ¼ VTx and V the matrix whose columns are the

normalized eigenvectors of R. Then, by applying this rotation, it follows:

z1 ¼ 22PTVy þ yTLRy ð9Þ

where LR ¼ VTRV is a diagonal matrix formed by the eigenvalues of R,which are all real.

(3) Translation. Let us now translate y by a vector h, i.e.:

y ¼ y þ h ð10Þ


method

1093

The purpose is to determine the value of h so that no first-order term appears inequation (9). The method of the squares completion can be adopted. Indeed bysubstituting equation (10) in equation (9) and eliminating the first order terms,it results:

h ¼ L21R VTP ð11Þ

then equation (9) can be written as:

z1 ¼ 22PTVh þ hTLRh þ yTLRy

4. Translation. From above it results:

z ¼ yTLRy ð12Þ

because:

z1 þ 2PTVh 2 hTLRh ¼ yTLRy ð13Þ

Now the constraint must be rewritten considering that x ¼ Vy ¼ Vðy þ hÞ.Let eT

i ¼ ð0; . . .; 1; . . .; 0Þ be the unit vector whose components are all null savefor the ith component which is one.

Then,

xi ¼ eTi Vðy þ hÞ with i ¼ 1; . . .; 5

The constraint can be rewritten:

ðy þ hÞTVTe2eT4 Vðy þ hÞ ¼ ðy þ hÞTVTe3e

T5 Vðy þ hÞ

which means that the constraint is as follows:

yTv2 þ hTv2

�yTv4 þ hTv4

�¼ yTv3 þ hTv2

�yTv5 þ hTv5

�ð14Þ

where vi is the ith column of VT ði ¼ 1; . . .; 5Þ:Let us define fi and f as follows:

f i ¼ yTvi þ hTvi

�i ¼ 1; . . .; 5: ð15Þ

f ¼ f 2 f 4 2 f 3 f 5 ð16Þ

Then the gradient of f (grady f ; from now on abbreviated as grad f ) can bewritten as

grad f ¼ f 4ðgrad f 2Þ þ f 2ðgrad f 4Þ2 f 5ðgrad f 3Þ2 f 3ðgrad f 5Þ ð17Þ

But grad f i ¼ vi and grad z ¼ 2LRy:Let a be the lagrangian multiplier and Cð y;aÞ the total cost function to be

minimized, defined as:

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1094

Cð y;aÞ ¼ yTLRy þ a f ð18Þ

so:

grad C ¼ 2LRy þ a grad f ð19Þ

Then equation (17) can be developed in the following way:

grad f ¼ ðW þ WTÞy þ ðW þ WTÞh ð20Þ

where

W ¼ v2vT4 2 v3v

T5 ð21Þ

Then to find the minimum of the cost function it is necessary that grad C ¼ 0;i.e.

2LRy þ aðW þ WTÞy þ aðW þ WTÞh ¼ 0

f 2 f 4 ¼ f 3 f 5

(ð22Þ

With this regard, the constraint (14) can be rewritten as follows:

1

2yTðW þ WTÞ y þ hTðW þ WTÞ y þ

1

2hTðW þ WTÞh ¼ 0 ð23Þ

equation (7) is then written as:

2LRy þ a ðW þ WTÞy þ aðW þ WTÞh ¼ 0

12 yTðW þ WTÞy þ hTðW þ WTÞy þ 1

2 hTðW þ WTÞh ¼ 0

8<: ð24a; bÞ

From equation (24(a)) it results:

y ¼ 2a ½2LR þ a ðW þ WTÞ�21ðW þ WTÞh ð25Þ

By substituting equation (25) into equation (24(b)) a non-linear scalar equationwith the scalar unknown a, which can be easily solved for by a numericaltechnique (e.g. the Newton’s method or the secant method).

Description of the test benchThe LS methodology for the estimation of the electrical parameters of aninduction motor has been tested in simulation and experimentally. A test benchhas been built for this purpose. The test bench consists of (Cirrincione et al.,2002):

. one three-phase induction motor with rated values shown in Table I;


method

1095

. one electronic power converter (three-phase diode rectifier and VSIinverter composed of 3 IGBT modules without any control system) ofrated power 7.5 kVA.

. one electronic card with voltage sensors (model LEM LV 25-P) andcurrent sensors (model LEM LA 55-P) for monitoring the instantaneousvalues of the stator phase voltages and currents;

. one voltage sensor (Model LEM CV3-1000) for monitoring theinstantaneous value of the DC link voltage;

. one electronic card with analogue 4th order low-pass Bessel filters andcut-off frequency of 800 Hz;

. one incremental encoder (model RS 256-499, 2500 pulses per round);

. one DSPACE card (model DS1103) with a floating-point DSP.

The VSI inverter is driven by an asynchronous vector modulation technique(switching frequency¼5 kHz) implemented by a software on the DSPACE cardand the DC link voltage sensor permits to take into account the instantaneousvalue of the DC link voltage for the modulation. Figure 1 shows the schematicsof the built test bench.

Special care has been taken for the signal processing of the signals becauseanti-aliasing filters, low-pass filters and differentiators are necessary. Moredetails on the test-bench can be found in Cirrincione et al. (2002).

The estimation algorithm needs the signals of the stator voltages andcurrents, their derivatives (up to the second order for the current and the firstorder for the voltage).

Since the motor can be supplied both by the electric grid and by a VSIinverter, filters for stator voltage and current signals are needed. The presenceof filters, however, causes distortion and time delays of the processed signalswhich, therefore, at each time instant, should be synchronised with one anotherwith respect to the dynamic equation of the induction machine.

Rated power, Prated (kW) 2.2Rated voltage, Urated (V) 220Rated frequency, frated (Hz) 50Pole-pairs 2Stator resistance, Rs (V) 3.88Stator inductance, Ls (H) 252£1023

Rotor resistance Rr (V) 1.87Rotor inductance Lr (H) 252£1023

Three-phase magnetising inductance, Lm (H) 236£1023

Moment of inertia, J (Kg m2) 0.0266

Table I.Parameters of theinduction motor

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Simulation and experimental resultsThe capability of the analytical solution to estimate, by exploiting a speedtransient of the machine, all the electrical parameters of the induction motor(rotor time constant, stator resistance, stator inductance, global leakage factor)has been verified in simulation and experimentally, by using the test benchdescribed earlier. The model of the induction motor for the simulation has thesame parameters of the real motor, as shown in Table 1. Simulations have beenperformed in Matlabw – Simulinkw environment. In particular the followingtest has been performed. The motor, both in simulation and in the experimentalapplication, has been supplied by the VSI inverter, controlled by means of thevector modulation, to which a reference sinusoidal voltage of 220 V and 50 Hzhas been given. The entire speed transient from zero speed to steady-statespeed has been exploited to estimate all the four electrical parameters of theinduction motor.

Figure 2 shows the rotor speed, the isD, isQ stator currents during the start-upof the motor with no load in the simulated test. Figure 3 shows thecorresponding estimation of the parameters, obtained with the constrainedanalytical solution.

Figure 4 shows the rotor speed, the isD, isQ stator currents during the start-upof the motor with no load in the experimental test. Figure 5 shows thecorresponding estimation of the parameters, obtained with the constrainedanalytical solution.

Figure 1.Schematic of the test

bench


method

1097

Figure 2.Rotor speed, isD, isQ

waveforms (simulationresults)

Figure 3.Real and estimatedelectrical parameters ofthe motor (simulationresults)

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Figure 4.Rotor speed, isD, isQ

waveforms(experimental results)

Figure 5.Real and estimated

electrical parameters ofthe motor (experimental

results)


method

1099

The values of the electrical parameters of the real machine are measured withthe usual no-load and locked-rotor tests and have been considered as the truevalues.

Tables II and III show the true and estimated values of the K-parameters ofthe induction motor under test, obtained with the constrained analyticalsolution in the simulation and experimental tests, respectively.

The graphs show that the estimated electrical parameters converge quicklyand smoothly to true ones. Moreover, the tables highlight the correct estimationof all the K-parameters, including K2.

ConclusionsThe original contribution of this paper is the presentation of an analyticalsolution for the on-line identification of an induction motor by a constrainedminimisation of a LS error surface. This solution has been obtained by suitablereference frame transformations. This method permits the quick on-line andsimultaneous computation of all the four electrical parameters of the machineby exploiting a speed transient of the machine.

Therefore, for the first time the so called K2 problem is overcome, whichcauses a bad parameter estimation in the current literature in which thenumerical methods are usually employed where either the constraint is nottaken into account or the procedure is split up into two parts. The goodness ofthis method has been verified experimentally by using a test bench suitably setup, which is moreover flexible enough to permit the experimental verificationof several identification procedures.

The future work will focus on a deeper insight into the analytical solution inorder to obtain the Lagrangian coefficient, a, in a closed form and to study the

True values Estimated values Error per cent

K1 189.0088 189.4606 0.2390K2 946.4266 941.3659 20.5347K31 127.5398 128.0074 0.3666K4 32.8711 32.9568 0.2609K5 243.9244 242.3646 20.6395

Table II.Steady-stateestimatedK-parameters(simulation results)

True values Estimated values Error per cent

K1 189.0088 212.2766 12.3104K2 946.4266 995.9520 5.2329K31 127.5398 147.4474 15.6089K4 32.8711 41.2711 25.5544K5 243.9244 278.7709 14.2858

Table III.Steady-stateestimatedK-parameters(experimentalresults)

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sensitivity of the solution to variations of the values of the data matrix and theobservation vector.

References

Cirrincione, M. and Pucci, M. (2001), “A direct-torque control of an AC drive based on arecursive-least-squares (RLS) method”, IEEE SDEMPED 01, Grado (Go), Italy.

Cirrincione, M. and Pucci, M. (2002), “Experimental verification of a technique for the real-timeidentification of induction motors based on the recursive least-squares”, IEEE AMC 02,Maribor, Slovenia.

Cirrincione, M., Pucci, M. and Vitale, G. (2002), “A test-bench for experiments on control,identification and EMC of high performance adjustable speed drives with inductionmotors”, SPEEDAM 02, Ravello, Italy.

Moons, C. and De Moor, B. (1995), “Parameter identification of induction motor drives”,Automatica, Vol. 31.

Pucci, M. and Cirrincione, M. (2001), “Estimation of the electrical parameters of an inductionmotor in saturated and unsaturated conditions by use of the least-squares method”, IEEEACEMP 01, Kusadasi, Turkey.

Ribeiro, L.A. and Jacobina, C.B. (2000), “Real-time estimation of the electrical parameters of aninduction machine using sinusoidal PWM voltage waveforms”, IEEE Transactions onIndustry Applications, Vol. 36 No. 3.

Ribeiro, L.A. de S., Jacobina, C.B. and Lima, A.M.N. (1997), “The influence of the slip and thespeed in the parameter estimation of induction machines”, IEEE PESC 97.

Stephan, J. and Bodson, M. (1994), “Real-time estimation of the parameters and fluxes ofinduction motors”, IEEE Transactions on Industry Applications, Vol. 30 No. 3.

Vas, P. (1994), Vector Control of AC Drives, A Space-Vector Approach, Cambridge UniversityPress, Cambridge.

Vas, P. (1996), Parameter Estimation and Condition Monitoring, Cambridge University Press,Cambridge.

Velez-Reyes, M., Minami, K. and Verghese, G.C. (1989), “Recursive speed and parameterestimation for induction machines”, IEEE IAS 89.


method

1101

Calculation of eddy currentlosses in metal parts of power

transformersErich Schmidt

Institute of Electrical Drives and Machines,Vienna University of Technology, Vienna, Austria

Peter HambergerVA TECH EBG, Transformatoren GmbH & Co., Linz, Austria

Walter SeitlingerVA TECH Peebles, Transformers Ltd, Leith, Edinburgh, UK

Keywords Eddy currents, Power losses, Power transformers, Finite element analysis

Abstract To maintain quality, performance and competitiveness, the eddy current losses in metalparts of power transformers in the range of 50-200 MVA are investigated in a more detailedform. The finite element calculations utilize different modelling strategies for the current carryingmetal parts. Several global and local results are further used to obtain simplified calculationapproaches for an inclusion in the initial design and the design optimization. The results from twofinite element approaches using nodal and edge based formulations will be compared withmeasurements.

1. IntroductionThe design of large power transformers requires the determination of strayload losses due to the leakage magnetic flux as accurately as possible (Bolteand Lipinski, 1983; Szabados et al., 1987; Valkovic, 1980). Apart from the coreand winding losses, the losses in the steel tank wall and the core clampingconstruction represent the largest components of stray load losses.

The distribution of the leakage magnetic flux strongly depends on thegeometry and material properties of core and steel tank wall. Usually, a tankwall shielding by means of sheets of high electric conductivity as well as highmagnetic permeability will be used to reduce the additional tank losses. Inthose cases, the arrangement of the shielding has a lot of degrees of freedom.Thus, optimization techniques with numerical field calculations are usedtogether as an approach towards an optimal design (Hameyer and Belmans,1999; Reece and Preston, 2000).

The detailed geometry of core, core clamping construction, tank wall andshielding requires 3D numerical calculation methods. Since these calculationscan be very time-consuming due to the nonlinearity of the magnetic materials,simplified calculation approaches are necessary with the initial design,



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the design review or the design optimization. Nevertheless, the 3D finiteelement analysis serves as the reference for the simplified calculation methods.

2. Representation of eddy currents in ferromagnetic materialThe calculation of eddy current losses in magnetic steel is a problem difficult tosolve with classical methods, which have chiefly been developed fornon-magnetic conductive materials. Due to the nonlinear magnetic materialproperties, a closed solution is impossible and numerical methods areconsiderably difficult to apply (Biro et al., 1997). Several finite elementformulations are known for non-ferromagnetic and even ferromagnetic eddycurrent carrying sheets (Biro et al., 1997; Brauer et al., 2000; Hollaus and Biro,2000). Mainly, they describe the required formulations for rather thin sheets aswell as laminations and neglect the influence of saturation in the ferromagneticmaterial. When considering the eddy current losses in the nonlinear tank wall,the described methods serve as a basis of modified modelling and calculationmethods.

It is possible to obtain numerically stable numerical analyses withsophisticated material models. Such investigations have indicated that withsome simplifications regarding the nonlinear magnetic characteristic, a simplemethod for the calculation of eddy currents and their losses in theferromagnetic material can be developed. With this simplification, therelationship between the magnetic excitation H and magnetic flux density Bwithin the material is assumed to be a step function as shown in Figure 1.

This means that in principle, the flux density in the material can only be thesaturation flux density BS, either positive or negative. With this assumptionand with a given constant conductivity of the material, the following facts canbe derived for alternating magnetic fields in the power frequency range.

. The magnetic flux in the material keeps staying at the surface. Thissurface flux generates a saturated layer with the thickness d of this layerdepending only on the peak value of the total amount of flux entering thematerial as shown in Figure 2.

Figure 1.Simplified magnetic

characteristic offerromagnetic material


losses

1103

. A variation in time of the total amount of flux F takes place by themovement of a surface, which separates regions with positive saturationflux density from regions with negative saturation flux density. Thismovement always starts at the material surface and is directed towardsthe inside of the material as shown in Figure 3.

. With v as the angular frequency of the grid, the velocity of the surfacemovement as given by

vS ¼vd

2cos

vt 2 kp

2; 0 , vt 2 kp , p k ¼ 0;^1;^2; . . .; ð1Þ

Figure 2.Magnetic flux at thesurface of ferromagneticmaterial

Figure 3.Magnetic flux densityprofile in saturated layerat various points in timeduring a powerfrequency cycle

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represents the sinusoidal variation of the total flux with time. Wheneverthe total flux F has achieved its peak value and starts to decrease, a newseparating border again changing the sign starts at the material surface.

. Eddy current flows only in this saturated layer of thickness d, parallel tothe surface and perpendicular to the direction of the flux density. But, thevariation of the eddy current density distribution with time is slightlymore complex than for the magnetic flux.

. First, eddy current flows only in the region between the material surfaceand moving border. Secondly, the magnitude of the eddy current densityis proportional to the speed of the border movement, which in generalvaries with time during a power frequency cycle.

It is rather simple to arrive at formulas for the movement of the separatingborder and the eddy current density in the material and finally for eddy currentlosses per area. They are based on either the assumption of a sinusoidal flux Fentering the relevant magnetic part or assuming that the magnetic excitation Hat the surface (equivalent to the total current in the material) is the driver of theeddy currents. It turns out, that due to the nonlinear material characteristic asinusoidal wave shape of the flux results in a non-sinusoidal wave shape of theeddy current, and vice versa. Nevertheless, useful relationships can be derivedas follows:

. The thickness d of the saturated layer is proportional to the peak amountof flux F entering the material.

. The peak value of the total eddy current is proportional to d 2. This isbecause the peak current density as well as the thickness of the eddycurrent layer are both proportional to the peak flux F.

. Consequently, the eddy current losses are proportional to d 3 as

pS ¼2

3pv2sd3B2

S: ð2Þ

This means that the eddy current losses in the ferromagnetic material increaseswith the power 3 with regard to penetrating flux F, but only with the power 1.5with regard to an exciting magnetic field H.

In general, eddy current losses in ferromagnetic material are generated in arather thin layer immediately at the surface. In most practical applications, thethickness of the saturated layer is limited to a few millimetres, otherwise thelosses would cause excessive heat. Finite element methods, in order todetermine eddy current losses, would need to use very fine meshing in such acase, which makes this method rather difficult to apply as well as time andresource consuming. The behaviour described earlier can be used to generatecomparable simple finite elements which nevertheless represent the actualbehaviour quite accurately. The important properties can be concentrated insurface elements, which deal with the fundamental harmonic of flux and


losses

1105

current densities, respectively. The fundamental characteristics of theseelements are as follows.

. Magnetic flux entering the surface element will generate a flux within thesurface.

. Magnetic flux in the surface will generate a current flow in the surfaceperpendicular to the flux direction and with a phase shift dependent onthe effective resistance of the current path. Thereby, the correspondingfundamental harmonics are related as given by

linear case : JS ¼ffiffiffi2

p HS

dsinðvt þ wÞ; tanw ¼ 1:0; ð3aÞ

step � function : JS ¼8ffiffiffi5

p

3p

HS

dsinðvt þ wÞ; tanw ¼ 0:5: ð3bÞ

. The effective resistance is proportional to the inverse of the peak flux andto the specific resistivity of the magnetic material. Thus, it is possible torepresent the mutual influence of flux and eddy currents for complexgeometries including ferromagnetic material in a very efficient way.

3. Finite element modellingThe various finite element calculations are carried out for a three-limb powertransformer with its data as listed in Table I. The coil windings of thistransformer consist of a two layer low voltage winding and a two layer highvoltage winding with two tapping windings.

Contrary to axial-symmetric and 2D analyses as proposed by Reece andPreston (2000), full 3D approaches are used throughout all calculations. Theyare done with the intent on detailed investigations of asymmetry effects in thisthree-limb transformer and their influence on the eddy current losses inparticular in the tank wall. One of the used finite element models for the powertransformer is shown in Figure 4. Due to an assumed symmetry of the core andtank wall arrangement, the model includes one quarter of the transformer.

3.1 Finite element formulationsThe depicted model with 3D hexa- and pentahedral volume elements in theeddy current carrying parts is solved with the EMAS solver (Brauer and

Rated power 160 MVARated high voltage 165 kVRated low voltage 67 kVRated frequency 50 HzShort circuit impedance 0.12Cooling system ONAN/ONAF

Table I.Main characteristicsof the concernedthree-limb powertransformer

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MacNeal, 1994). This solver is based on a nodal A-c formulation with amagnetic vector potential A, a time integrated electric scalar potential c and anincorporated Coulomb gauge (Biro and Preis, 1989).

Additionally, an A formulation based on the work of Biro et al. (1995, 1996)with second order tetrahedral edge elements is used. In these analyses, the eddycurrent carrying regions of the tank wall are represented with surface elementsas proposed by Biro et al. (1997). With these surface elements, the suggestionsmentioned earlier are introduced additionally. As shown in Table II with linearanalyses, a much faster and more numerically stable solution is obtained fromthe edge-based A formulation with 2D surface elements in comparison tothe analysis of the nodal-based A-c formulation with 3D volume elements.This is also true with the various nonlinear calculations.

Due to symmetry and uniqueness, the following boundary conditions areapplied with both finite element formulations:

Figure 4.Active parts of the finite

element model of thepower transformer

containing core, low andhigh voltage windings,

core clamping plate andsteel tank wall (partly

shown)

Formulation Elements Nodes Equations CPU time (s)

Nodal-based A-c with 3D volume elements 122,500 116,200 330,000 8,760Edge-based A with 2D surface elements 73,400 18,100 480,000 3,300

Table II.Comparison of theused finite element

analyses


losses

1107

. a Neumann boundary condition B £ n ¼ 0 is used at the horizontal xymid-plane.

. a Dirichlet boundary condition B · n ¼ 0 is used at the vertical xzmid-plane.

. Dirichlet boundary conditions B · n ¼ 0 are modelled at all outerboundary planes of the tank wall.

3.2 Material modellingThe above-mentioned finite element formulations utilize anisotropicpermeabilities of the non-conducting core and anisotropic conductivities ofthe coil windings as described by Schmidt and Ojak (1996).

Due to the current excitation with impressed current densities and negligibleeddy currents inside the coil windings, any anisotropic coil conductivity isgiven as

½s� ¼

0 0 0

0 sC 0

0 0 0

2664

3775 ð4Þ

with regard to a local cylindrical coordinate system with each of the three limbs.Regarding Figure 4, the anisotropic core permeability can be written as

½m� ¼

mxx 0 0

0 myy 0

0 0 mzz

2664

3775 ð5Þ

The different magnetization directions of limbs and yokes are considered withappropriate functions mxx and mzz in dependence of the local saturation. Thecore lamination is represented by an effective permeability obtained from thestacking factor kF as

myy ¼ m0mF

mF ð1 2 kFÞ þ m0 kF; ð6Þ

where an averaged value mF obtained from mxx and mzz is used.Both the nonlinear permeabilities mxx and mzz of the core as well as the

nonlinear permeabilities of the core clamping plates are modelled using theaveraged magnetic coenergy (Biro et al., 1998),

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mcoðBÞ ¼4

tmaxjHðtÞj

2

4

T

Z T=4

0

dt

Z HðtÞ

Hð0Þ

B · dH ð7Þ

4. Analysis results4.1 Total leakage coefficientAccording to the flux linkages of the high voltage (HV) and the low voltage(LV) winding as

CHV ¼ LHH iHV þ MHL iLV; ð8aÞ

CLV ¼ MHL iHV þ LLL iLV; ð8bÞ

the total leakage coefficient is given as

s ¼ 1 2M 2

HL

LHH LLLð9Þ

Figure 5 shows the total leakage coefficient (9) for opposite rated MMFs withinHV and LV windings including both tapping windings. As shown, the leakage

Figure 5.Leakage coefficient s

versus relativepermeability of the steel

tank wall


losses

1109

coefficient distinctly depends on the permeability of the tank wall. Therefore,the modelling of the nonlinear permeabilities is of great importance. This isespecially significant for accurate values of the eddy currents and the powerlosses in the case of the volume elements representing the core clamping plateas well as the surface elements representing the tank wall.

4.2 Power losses in the tank wallTable III shows the eddy current losses in the tank wall obtained from anexcitation with opposite rated MMFs within HV and LV windings includingboth tapping windings. Thereby, region 1 is along the winding height, region 2means the height between windings and yoke, regions 3 and 4 correspond tothe height of the yoke, regions 5 and 6 represent the upper chamfer of the tankwall and finally region 7 is above the yoke. Figure 6 shows the distribution ofthe power loss density at the inner tank wall planes with regard to thisexcitation.The larger values of the left part of phase 1 and the right part of phase 3 are dueto the much greater tank wall planes in these regions. The asymmetries of thelisted values between the regions are caused by the three-limb constructionbecause the finite element mesh is reflected and shifted in the centered regionsof the three phases. These calculated values are in good accordance with thetotal eddy current losses of 19 kW obtained from measurements.

4.3 Flux density at the tank wallIt is practically impossible to measure the eddy current losses in the tank wallaccurately. This is due to the fact that these losses are only a small part of thetotal load losses of such a power transformer.

For this reason and also for the investigation of the tank wall shieldingefficiency, the magnetic flux carried within the shielding sheets has been

Phase 1 Phase 2 Phase 3Left Right Left Right Left Right

Region 1 3,398 1,089 1,100 1,097 1,077 3,402Region 2 254 107 135 124 128 294Region 3 102 125 121 118 127 95Region 4 423 238 234 205 235 397Region 5 205 142 166 158 142 193Region 6 95 575 621 614 571 94Region 7 219 287 297 296 263 203

Summary 4,696 2,563 2,674 2,612 2,543 4,678

Table III.Eddy current losses(W) in various tankwall regions

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measured. The measuring positions along the tank wall at the xy mid-plane areshown in Figure 7. The peak to peak voltages Ua as listed in Table IV areobtained from auxiliary coils around the sheets in the case of HV coil currentsof iHV ¼ 0:50 with regard to the rated current and nominal frequency off N ¼ 50 Hz: The auxiliary coils are built of W a ¼ 10 windings having a crosssection of Qa ¼ 0:00272 m2: Thus, the average values of the local magnetic fluxdensity Ba as given in Table IV are obtained from

Ba ¼U a

4pf N

1

W aQa

1

iHV: ð10Þ

Comparison of the results from measurements to the calculated results asshown in Figure 8 gives a good agreement with the exception of the outer

Figure 6.Power loss density at the

inner tank wall planesfor rated current

excitation of the HV andLV windings

Figure 7.Positions of the tank

shielding along the tankwall at the xy mid-plane


losses

1111

regions of phase 3. This is due to the asymmetry of the tank wall caused by thetapping switch which is not included in the finite element model.

5. ConclusionsWith the aim of an improvement of initial design and design optimizationcalculation methods, the eddy current losses in the core clamping plates and

Figure 8.Magnetic flux density atthe inner tank wallplanes for rated currentexcitation of the HV andLV windings

Shielding part Voltage (V) Flux density (T)

Phase 1 11 17.2 2.0012 16.2 1.9013 19.0 2.2514 20.2 2.3515 20.1 2.35

Phase 2 21 16.1 1.9022 17.2 2.0023 17.2 2.0024 16.1 1.90

Phase 3 32 19.1 2.2533 19.0 2.2534 14.7 1.7035 12.6 1.50

Table IV.Measured voltageand determined fluxdensity at variouspositions of the tankshielding

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steel tank wall of power transformers in the range of 50-200 MVA areinvestigated with 3D finite element analyses. These calculations are carried outwith the intent on detailed investigations of asymmetries in the three-limbtransformers and their influence on the eddy current losses, in particular, in thetank wall.

The finite element analyses utilize two different formulations. The classicalnodal A-c formulation with a magnetic vector potential A and a timeintegrated electric scalar potential c uses 3D hexa- and pentahedral volumeelements in the eddy current carrying core clamping plate and the tank wall.On the other hand, an A formulation with second order tetrahedral edgeelements is applied which uses 2D surface elements in the eddy currentcarrying tank wall.

Both finite element solutions serve as a reference for the development ofsimplified calculation methods which can be included in the initial designprocess and the design optimization of large power transformers.

References

Biro, O. and Preis, K. (1989), “On the use of the magnetic vector potential in the finite elementanalysis of three-dimensional eddy currents”, IEEE Transactions on Magnetics, Vol. 25No. 4.

Biro, O., Paoli, G. and Buchgraber, G. (1998), “Complex representation in nonlinear time harmoniceddy current problems”, IEEE Transactions on Magnetics, Vol. 34 No. 5.

Biro, O., Preis, K. and Richter, K. (1995), “Various FEM formulations for the calculation oftransient 3D eddy currents in nonlinear media”, IEEE Transactions on Magnetics, Vol. 31No. 3.

Biro, O., Preis, K. and Richter, K. (1996), “On the use of the magnetic vector potential in the nodaland edge finite element analysis of 3D magnetostatic problems”, IEEE Transactions onMagnetics, Vol. 32 No. 3.

Biro, O., Bardi, I., Preis, K., Renhart, W. and Richter, K.R. (1997), “A finite element formulation foreddy current carrying ferromagnetic thin sheets”, IEEE Transactions on Magnetics, Vol. 33No. 2.

Bolte, E. and Lipinski, W. (1983), “Berechnung von Wicklungsstreufluss und Kesselabschirmungbei Gross- und Verteilungstransformatoren”, etz Archiv, Vol. 5 No. 9.

Brauer, J.R. and MacNeal, B.E. (1994), MSC/EMAS User Manual, MacNeal-SchwendlerCorporation, Los Angeles.

Brauer, J.R., Cendes, Z.J., Beihoff, B.C. and Phillips, K.P. (2000), “Laminated steel eddy-currentloss versus frequency computed using finite elements”, IEEE Transactions on Magnetics,Vol. 36 No. 4.

Hameyer, K. and Belmans, R. (1999), Numerical Modelling and Design of Electrical Machines andDevices, WIT Press, Southampton.

Hollaus, K. and Biro, O. (2000), “A FEM formulation to treat 3D eddy currents in laminations”,IEEE Transactions on Magnetics, Vol. 36 No. 4.

Reece, A.B.J. and Preston, T.W. (2000), Finite Element Methods in Electrical Power Engineering,Oxford University Press Inc., New York.


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Schmidt, E. and Ojak, S. (1996), “3D MSC/EMAS simulation of a three phase power transformerby means of anisotropic material properties”, Proceedings of the 1996 MSC World UsersConference, Newport Beach, CA, USA.

Szabados, B., El-Nahas, I., El-Sobki, M.S., Findlay, R.D. and Poloujadoff, M. (1987), “A newapproach to determine eddy current losses in the tank walls of a power transformer”, IEEETransactions on Power Delivery, Vol. 2 No. 3.

Valkovic, Z. (1980), “Calculation of the losses in three-phase transformer tanks”, IEE Proceedings,Vol. 127 No. 1, Part C.

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Optimal design of highfrequency induction motors

with the aid of finite elementsAtanasi Jornet

AEG Fabrica de motores, S.A., Test Laboratory, Terrassa, Spain

Angel Orille, Alberto Perez and Diego PerezElectrical Engineering Department, Industrial Engineering TechnicalHigh College, Polytechnic University of Catalonia, Barcelona, Spain

Keywords Finite elements, Induction motor, Power losses

Abstract The motors for high-speed operation fed by frequency converters produce, first, a highamount of hysteresis and eddy losses in both stator and rotor iron, and secondly a temperatureincrease of the rotor due to current distribution in its rotor slots. Conventional calculation usinganalytical tools could not calculate precisely the required parameters in order to obtain an optimalmodel to build a prototype that its properties confirm that calculated values with the model. With afinite element method application for magnetic field and heat transfer, the required elements todesign a new prototype could be elaborated very precisely and it is also a tool to prove the alreadyexisting motors for high speed applications. This allows us also to design energy efficient electricaldrives according to the recommendations of the last EEMODS conference held in London in theyear 1999 with the support of the European committee for energy saving.

1. IntroductionHigh-speed drives are usually used for industrial applications such as textileand CNC machining. This technology has been developed very quickly becauseof the introduction of high speed drives that could work at rated frequency upto 400 Hz. The speed limitation because of the size of the motors, therequirement of bypass application (the motors could be started also at 50 Hzsupply) results in taking into account of these three models described in Table I.

Nowadays conventional programs, are very good to calculate motors at50 Hz operation, in a few seconds the motor is calculated in a easy way. But forhigh speed motors, these conventional tools could not easily take into accountthe phenomenon of high frequencies. The new tools that are used for someyears are the finite elements. The disadvantage of this tool is that it takeslonger duration to do a calculation and is not very easy.

2. Finite element analysisThe finite element analysis of the three defined models, are summarized in twosections (Table I).

2.1 Model 1In this section, the motors size 63-132 with the following special design areincluded (Table II):



High frequencyinduction motors

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Vol. 22 No. 4, 2003pp. 1115-1125


DOI 10.1108/03321640310483048

. high speed bearings and grease;

. external fan cooling IEC 416;

. low iron losses lamination; and

. special balancing2.1.1 Core losses. Performing an integral over the block of the total core

surface, we can evaluate total hysteresis and laminated eddy current losses fordifferent frequencies supply. An example of flux density plot of a 2 poles motoris shown in Figures 1 and 2. The values are summarized in Table III.

Representing total iron losses vs rotor’s and core’s temperatures, it denotes aclear dependence of the temperature with total iron losses. At 200 Hz frequencysupply, both temperature and iron losses increase enormously. To improve theefficiency, a low-loss quality and thinner lamination have to be used (Figure 3).

2.1.2 Rotor losses. The rotor current density due to skin effect of the firstmodulation harmonic with a rated frequency of 300 Hz supply does not have avery high value, as shown in Table IV (Figure 4).

The harmonics stator current to calculate resistive rotor losses is obtainedby means of FFT analysis of the supplied motor current by the frequencyconverter. Figures 5 and 6 represent losses distribution at 50 and 300 Hz ratedfrequency supply.

For the first model, it is remarkable that hysteresis losses in comparisonwith the rotor losses for a frequency range from 50 to 300 Hz, increase higher.Any improvement of the efficiency could be achieved by the reduction of thisiron losses.

2.2 Models 2 and 3In these models the motors with sizes 160-250 and 280-355 are included, asdescribed in Table I, with the following special design (Table V):

. high speed bearings and grease;

Motor size (IEC 60072) Frequency range (Hz) Maximum speed (1/min)

63-132 0-400 24.000160-250 0-100 6.000280-355 0-60 3.600

Table I.Usual speed rangefor high frequencyinduction motors

Rated power (kW) 5.5UN (V) 400Frequency (Hz) 300IN (A) 12Number of poles 2n (1/min) 17.650

Table II.High speed testedmotor for Model 1

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Figure 1.Magnetic flux density at

no-load condition

Figure 2.Magnetic flux density at300 Hz frequency supply

at rated slip condition

Frequency(Hz)

Fundamental ironlosses (W)

Harmonic ironlosses (W)

Total ironlosses (W)

50 17.52 21.22 38.74150 62.59 86.14 148.73200 93.01 129.83 222.84250 129.74 181.01 310.75300 172.00 239.68 411.68

Table III.Total iron losses at

different frequenciessupply


1117

. external fan cooling IEC 416;

. low iron losses lamination;

. special slot rotor design; and

. special balancing

The next explanation to the iron losses and rotor losses are applicable for bothmodels 2 and 3.

2.2.1 Core losses. These motors are usually fed up to 100 Hz because of theirmechanical limitations. Therefore, the iron losses are not the main cause of theadditional heating indirect fig. citation (Figures 13 and 14).

2.2.2 Rotor losses. Motors designed for sinus supplied operation, when fedby frequency converter a high increase of current density takes place in

Figure 3.Total iron losses vsmotor’s maintemperatures

Frequency(Hz)

Fundamental resistiveslosses (W)

Harmonic resistivelosses (W)

Ratio harmonic and resistivelosses (per cent)

50 17.03 0.48 2.83150 12.07 0.73 6.02200 15.91 0.19 1.11250 12.20 0.82 6.75300 20.20 0.16 0.76

Table IV.Resistive rotorlosses at differentfundamentalfrequency supply

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the upper side of the rotor to slot due to current frequency harmonics of PWMmodulation (Figures 7 and 8). For this output range, current THD is higher thanin motors of model 1, because it is not easy to achieve higher modulationfrequencies because of switches losses.

The current density in rotor slot, has a significant value. This factproduces an increase of joule rotor losses from 16 to 30 per cent (Figures 9and 10).

With the special rotor slot design, the motor fed by frequency convertercould be optimised to keep the same efficiency as it was fed by sinussupply. This solution has some disadvantages, for instance, it requires abypass application. It means that the motor could work when fed by both

Figure 4.Rotor’s current density at

300 Hz rated frequency

Figure 5.Losses distribution at50 Hz rated frequency


1119

frequency converter and sinus supply. If no special considerations aretaken, the starting current would be 12 times higher than the rated current.

Another disadvantage to consider, is that manufacturers are obliged toproduce two different rotor design with the consequent cost increase. Withfinite element analysis of rotor slot, a reduction of joule rotor losses and abypass operation could be achieved as shown in Figure 11.

Figure 6.Losses distribution at300 Hz rated frequency

Rated power (kW) 90UN (V) 400Frequency (Hz) 50IN (A) 159Number of poles 6n (1/min) 988

Table V.High speed testedmotor for model 3

Figure 7.Current density of themodulation frequency

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With this tool, as shown in Figure 12, with a change of the air-aluminium ratio,joule rotor losses could be highly reduced.

A high reduction of Joule rotor losses could be achieved, if the slot was onlyconsidered for frequency converter application (Figures 13 and 14).

Different slot configurations and their main ratings studied are summarisedin Table VI.

3. Thermal modelAs we have a tool to calculate accurately each motor losses, it is thinkable toobtain thermal models of the motor for the complete frequency range. To obtainthe thermal admittance some tests have been conducted at different frequencies

Figure 8.Current density along

slot’s length

Figure 9.Losses distribution for

50 Hz sinus supply


1121

of the selected motor, size 112 (rated output 5.5 kW at 300 Hz). In Figure 15there is a record of the temperature values in different points of the motor forthese frequencies. The test has been conducted for a constant torque andventilation. Above 200 Hz temperatures of both the rotor and the statorincreases very quickly. This phenomenon is produced by an enormous increase

Figure 10.Losses distribution for50 Hz frequencyconverter supplyincrease of the rotorlosses

Figure 11.Special rotor’s slotdesign to optimizeperformance withfrequency convertersupply

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Figure 12.Losses as a function of

ratio h1/h2

Figure 13.Special rotor’s slotdesign to optimizeperformance with

frequency convertersupply


1123

of the iron and mechanical losses. Figure 16 shows the calculated iron lossesand the core temperature versus frequency. It is easy to appreciate thementioned effect.

4. ConclusionsFor high-speed applications with converter fed motors it is not enough to deratethe output of the motor. A special design is required taking into account therequired control range, bypass and the frequency converter signal. Theefficiency of the electrical drive has not to be reduced.

Figure 14.Losses as a function ofratio h1/h2

Rotor’s slot design

RatingsDouble cage

open slotDouble cage

specialSingle

open slotSingle

special slotSingle

closed slot

Pn (kW) 90 90 90 90 90U (V) 400 400 400 400 400In (A) 161.26 161.26 155.85 155.85 152.9Cosf 0.8519 0.8519 0.8849 0.8849 0.9028n (1/min) 990.9 990.9 991.3 991.3 991.5f (Hz) 50 50 50 50 50Mn (Nm) 867.39 867.39 867.04 867.04 866.87I0 /In (p.u.) 0.254 0.254 0.259 0.259 0.261Ia/In (p.u.) 6.724 6.724 7.763 7.763 8.347Ma/Mn (p.u.) 2.588 2.588 1.924 1.924 1.638Mk/Mn (p.u.) 2.598 2.598 2.98 2.98 3.123Fundamental rotorJoule losses (kW) 0.527 0.529 0.436 0.399 0.581Modulation harmonicRotor Joule losses (kW) 1.092 0.214 1 0.122 0.176

Table VI.Different ratings foreach rotor’s slotdesign

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Further reading

Budig, P.K. (2001), Stromrichter-gespeiste Drehstromantriebe. Theorie und Betriebsverhalten vonAsynchronantireben, VDE Verlag.

Fischer, R. (1995), Elektrische Maschinen, Carl Hanser Verlag Munchen, Wien.

Meeker, D. (2001), Finite Element Method, URL source http://femm.berlios.de/dmeeker/

Mohan, N. and Undeland, T.M. (1989), Power Electronics: Converter, Applications and Design,Wiley, New York.

Siemens. (1989), Temperaturfeld und Warmefluß beikleineren oberflachengekuhltenDrehstrommotoren mit Kafiglaufer, Elektrotechnische Zeitschrift ETZ-A.

Silvester, P. and Ferrari, R.L. (1983), Finite Elements for Electrical Engineers, CambridgeUniversity Press, Cambridge.

Vaske, P. and Riggert, J.H. (1974), Elektrische Maschinen und Umformer, B. G. Teubner, Stuggart.

Figure 15.Thermal motor model

Figure 16.Temperature rise as afunction of frequency


1125

Analytic calculation of thevoltage shape of salient pole

synchronous generatorsincluding damper winding and

saturation effectsGeorg Traxler-Samek and Alexander Schwery

ALSTOM (Switzerland) Ltd, Hydro Generator Technology Center, Birr,Switzerland

Erich SchmidtInstitute of Electrical Drives and Machines,

Vienna University of Technology, Vienna, Austria

Keywords Synchronous machine, Optimization

Abstract A novel analytic method for determining the no-load voltage shape of salient polesynchronous generators is presented. The algorithm takes into account the full influence of thedamper winding and the saturation effects in the stator teeth. Main interest is an easy and veryfast calculation method, which can be used as a criteria for the selection of the number of statorwinding slots in the initial design calculation or an optimization process. The analytical resultsobtained are compared with the results of transient finite element analyses.

1. IntroductionSeveral numerical methods can be applied to calculate the airgap magnetic fluxdensity and the voltage shape of salient pole synchronous generators. Simondand Neidhofer (1980) determined the voltage harmonics with finite differencecalculations in combination with an analytical approximation of the stator toothripple, but without the influence of the damper winding. The impact of thedamper winding slot pitch on the airgap magnetic flux density distribution isshown in Rocha and De Arruda Penteado (1991). Additionally, the calculation ofthe voltage shape and the telephone influence factor (TIF) is described in Rochaet al. (1997). Finally, as proposed in Ide et al. (1992) and Kim and Sykulski (2001),finite element analyses including the rotor movement and subsequent Fourierseries expansions of the flux linkage and the phase voltage can be used.

A nonlinear transient finite element analysis can take into account the fullinfluence of the damper winding, the slot opening distribution due to thefractional slot stator winding and the saturation effects in the stator teeth. Theutilization of optimized solving strategies such as a domain decompositionalgorithm can reduce the calculation time (Schmidt, 2001). Nevertheless,



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the transient finite element analysis remains very time-consuming, especiallyin the case of a fractional slot stator winding.

To easily determine the number of stator slots during the first design phase,an analytic method with a fast algorithm has to be applied. In this stage, theengineer needs a simple and reliable yes-no-criteria.

Derived from the ideas mentioned earlier, this paper presents an analyticalmethod for calculating the magnetic flux density in the airgap and the voltageshape under no-load conditions including damper winding and stator teethsaturation effects. Contrary to the airgap permeance function utilized in Rochaand De Arruda Penteado (1991), any pole shoe shape can be considered.

The analytical method presented separates the calculation into three mainelements. First, the rotor magnetic flux density distribution in the airgap withoutthe influence of the stator winding slots is concerned. Second, the stator slotrelative permeance function taking into account the tooth ripple is introduced.Finally, the influence of damper winding currents is included in the calculation.

2. Coordinate transformationThe synchronous speed of the generator at the stator bore diameter is definedaccording to

vS ¼vtp

pð1Þ

where v is the angular frequency of the grid and tp is the pole pitch length. Asdepicted in Figure 1, a stator fixed coordinate system x [ CS and a rotor fixedcoordinate system j [ CR will be used for all calculations. The coordinatetransformation is done with

x ¼ jþ vSt ð2Þ

Figure 1.Stator and rotor fixed

coordinate systems

Analyticcalculation of the

voltage shape

1127

3. Airgap permeance functionThe airgap permeance function LR(j ) takes into account the variable airgapwidth due to the saliency in direct and quadrature axis and the pole shoeshape. Thereby, a non-slotted stator will be considered. With the assumptionof all flux tubes of the magnetic flux density in radial direction, the airgappermeance function is directly obtained from the radial airgap width functiond(j ) as

LRðj Þ ¼ m01

dðj Þð3Þ

Both the radial airgap width as well as the airgap permeance function arereferenced to the fixed rotor coordinate system (j ).

Figure 2 depicts the original geometry of the pole. Figure 3 shows theairgap function d(j ) and the permeance function LR(j ) for the depictedpole shoe with one arc. As shown in Figure 3, a linear approximation ismade in the pole gap region between the two adjacent poles. Thus, theairgap width function in the pole shoe region is applicable for any poleshoe shape.

In general, the airgap function is represented by a discretized functiondi ¼ dðjiÞ; ji ¼ j0 þ iDj; with equidistant support points ji. Consequently, theairgap permeance function LR(j ) will be represented by discrete values LR

iaccording to the support points ji. The Fourier series expansion of LR

i is carriedout with piecewise linear interpolations between the support points.

Figure 2.Geometry of the polewith a shifted damperwinding

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4. Stator slot permeance functionThe relative stator slot permeance function lS(x) describes the influence of thestator winding slot openings with regard to the stator bore diameter assuminga constant airgap. Introducing the stator slot pitch tS, the relative stator slotpermeance function with respect to a fixed stator coordinate system (x ) isrepresented by the Fourier series expansion

lSðxÞ ¼X1

k¼21

lk exp jk2px

tS

� �: ð4Þ

Thus, a stator without any slots is represented by lSðxÞ ¼ l0 ¼ 1 (Freeman,1962).

The Fourier coefficients lk depend on the ratio bS/tS of slot opening width toslot pitch, the ratio d0/tS of airgap width to slot pitch and the stator teethmagnetization Bt. They result from various finite element calculations withdifferent values of the influence quantities and in dependence of the saturationlevel caused by the flux entering the slot pitch.

With modern finite element tools, automated calculations with a variation ofgeometry parameters can be performed (CEDRAT, 2000). The basic geometryof one slot for such calculations is shown in Figure 4. The slot pitch tS is fixedto 100 mm, the slot opening bS and the minimum airgap width d0 varies in therange of bS ¼ 20 . . . 60 mm and d0=bS ¼ 0:125 . . . 2: Therefore, the totalmagnetic flux varies between F0

S ¼ 0:05 . . . 0:50 Vs=m assuming an axiallength of the 2D finite element model of 1 m.

Figure 3.Airgap width d(j )(straight line) and

permeance functionLR(j ) (dashed line)


voltage shape

1129

The resulting magnetic flux density distribution B(x ) along the slot pitchx [ ½0; tS� at the stator bore position is converted to the stator slot permeancefunction as defined by

lSðxÞ ¼BðxÞ

1tS

R tS

0 BðxÞ dx; ð5Þ

which confirms the Fourier coefficient l0 ¼ 1: Figure 5 depicts the calculatedpermeance function (5) for a fixed ratio d0=bS ¼ 0:4 in the range of bS=tS ¼0:2 . . . 0:6 for the non-saturated case of F0

S ¼ 0:05 Vs=m: According to thesedependencies, Figures 6 and 7 depict the decisive Fourier coefficients (4) for thelinear and saturated magnetization.

5. Airgap magnetic flux densityThe distribution of the field winding MMF is approximated by a trapezoidfunction V R(j ). Therefore, the airgap magnetic flux density withoutconsideration of the stator slot openings is obtained from the rotorpermeance function (3) as

BRðj Þ ¼ LRðj ÞV Rðj Þ ð6Þ

Figure 4.Finite element model forthe parametriccalculation of the statorslot permeance functionlS(x)

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1130

Figure 5.Stator slot permeance

function l S(x) assuminga fixed ratio d0/bS¼0.4and linear stator teeth

magnetization withF 0

S¼0.05 Vs/m

Figure 6.Fourier coefficients

lk ¼ l*2k; k¼ 1, 2, 3 of

the stator slot permeancefunction assuming

l0¼ 1, ratiod0/tS¼ 0.266, linear

stator teethmagnetization with

F 0S ¼ 0.05 Vs/m


voltage shape

1131

and will be represented by the Fourier series expansion

BRðj Þ ¼X1n¼21

Bn exp jnpj

tp

� �ð7Þ

Figure 8 shows a comparison between the analytically calculated pole fieldcurve and the curve obtained from a finite element analysis without the effects

Figure 7.Fourier coefficientslk ¼ l*

2k; k¼1, 2, 3 ofthe stator slot permeancefunction assumingl0¼ 1, ratiod0/tS¼ 0.266, saturatedstator teethmagnetization withF0

S ¼ 0.10 Vs/m

Figure 8.Magnetic flux density inthe airgap, analyticvalues (straight line) andvalues from the finiteelement analysis(dashed line)

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of the stator slots according to the reference magnitude of B0 ¼ 1 T: Figures 9and 10 show the corresponding harmonics from Fourier series expansions. Allfigures show the close agreement of both analysis methods.

The above magnetic flux density distribution is invariant to the position ofthe non-slotted stator. The transformation to the stator fixed referencecoordinate system according to equations (1) and (2) yields

Figure 9.Harmonics of the

magnetic flux density inthe airgap, analytic

calculated values

Figure 10.Harmonics of the

magnetic flux density inthe airgap, values from a

finite element analysis


voltage shape

1133

BS0ðx; tÞ ¼

X1n¼21

Bn exp jnpx

tp2 jnvt

� �: ð8Þ

The influence of the stator teeth is taken into account with the relative statorslot permeance function (4) as

BSðx; tÞ ¼X1n¼21

BnlSðxÞ exp jn

px

tp2 jnvt

� �: ð9Þ

Introducing the number of stator slots per pole and phase as q ¼ tp=ðmtSÞ;where m is the number of stator winding phases, the magnetic flux densitydistribution can be rewritten in the form

BSðx; tÞ ¼X1n¼21

X1k¼21

Bnlk exp jðnþ 2kmqÞpx

tp2 jnvt

� �: ð10Þ

This equation represents the distribution of the airgap magnetic flux densitywithout the existence of a damper winding.

6. Effects of the damper windingThe damper winding is considered with a network containing all damper barswith resistances, self and mutual as well as stray inductances as shown inFigure 11.

All harmonic components of the airgap magnetic flux density (10) aretransformed to the rotor fixed coordinate system according to equations (1) and(2) resulting in

BRðj; tÞ ¼X1

n¼21

X1k¼21

Bn l0k exp jðnþ 2kmqÞ

pj

tpþ j2kmqvt

� �ð11Þ

Figure 11.Network of a singledamper bar including theend ring reactances

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Thereby, a correction of the stator permeance coefficients (4) due to the airgapis introduced as

l0k ¼1

cosh 2kpd0

tS

lk: ð12Þ

Each field component Bn, modulated with the stator tooth ripple k – 0; runs incircumferential direction relative to the damper winding bars with a relativespeed given by

vn;k ¼ 22kmq

nþ 2kmq

vtp

pð13Þ

in the case of nþ 2kmq – 0: These harmonic components are introduced in thedamper bar network as external voltages according to

Un;k ¼ lvn;kBn ð14Þ

with an angular frequency as

vDk ¼ 2kmqv: ð15Þ

The case nþ 2kmq ¼ 0 generates equally phased voltages in all damper bars.Consequently, there are no damper currents due to this combination ofharmonics.

Each harmonic component k defines a different frequency as shown inequation (15). Thus, the external voltages of harmonic numbers n can besuperposed in the network. The magnitude and the phase angle of the voltageinduction depends both on the damper bar location and wavelength.

To obtain the current distribution of the damper bars, the damper networkis solved separately for the reaction on each harmonic component k.The magnetomotive force V R

Dðj; tÞ due to the calculated bar currents generateswith equation (3) the airgap magnetic flux density

BRDðj; tÞ ¼ LRðjÞV R

Dðj; tÞ; ð16Þ

which can be expressed by the Fourier series expansion

BRDðj; tÞ ¼

X1i¼21

BD;i exp jipj

tpþ jvDkt

� �: ð17Þ

This distribution is back-transformed to the stator fixed coordinate systemwith equation (15), yielding


voltage shape

1135

B SD ðx; tÞ ¼

X1i¼21

BD;i exp jipx

tp2 jði 2 2kmqÞvt

� �: ð18Þ

Due to the damper current reaction, each exciting harmonic of the airgapmagnetic flux density generates a lot of new harmonic components which canhave a big impact on the voltage shape harmonics.

7. Calculation of the voltage shapeGenerally, a magnetic flux density wave in the airgap with any wavelengthharmonic n 0 and any frequency harmonic m0 can be assumed in the stator fixedcoordinate system as

BSgðx; tÞ ¼ Bg exp jn 0 px

tp2 jm0vt

� �: ð19Þ

Furthermore, let us consider a stator winding coil with the first bar located atx ¼ x0 and the second bar according to the winding pitch Y1 at x ¼ x0 þ tp ys;where ys ¼ Y 1=ðmqÞ: Thus, the induced voltage of this coil can beobtained from

Uiðx0; tÞ ¼ 2d

dtl

Z x0þtpys

x0

Bg exp jn 0 px

tp2 jm0vt

� �dx

� �: ð20Þ

As there is a singularity problem for n 0 ¼ 0 in equation (20), which has to betreated separately. In this case, the voltage of the coil concerned is calculatedfrom

Uiðx0; tÞjn 0¼0 ¼ 2d

dtl

Z x0þtpys

x0

Bg exp ð2jm0vtÞ dx

� �: ð21Þ

The stator winding voltage as the sum of all winding conductor voltagesis summarized in accordance to the winding diagram of the machine insteadof using the well known winding factors. In particular, this allows theconsideration even of special stator winding arrangements.

By comparing equations (10) and (19), each airgap flux density harmonicwave (n, k) is represented by

n 0 ¼ nþ 2kmq; m0 ¼ n: ð22Þ

Consequently, all harmonic numbers k which are modulated with the n thharmonic of the pole field curve generate the same frequency (nv) in the statorwinding. Each frequency harmonic in the voltage has to exist as a wave

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harmonic in the pole field curve with the non-slotted stator. As with equation(20), the resulting magnitude in the stator winding voltage depends on the ratiom0/n 0. Therefore, an amplification factor is defined as

uðnÞ ¼m0

n 0¼

n

nþ 2kmq; ð23Þ

which results in uðnÞ ¼ 1 for a non-slotted stator and can otherwise result inhigh values with j2kmqj near jnj and kn , 0:

The generator terminal voltage results from the contribution of all airgapharmonics (10) and all harmonics generated by the damper winding fromequation (18). From the obtained harmonics, any distortion factor like the THFcan be calculated in accordance with the IEC standard (IEC, 1999).

8. Calculation resultsThe following example shows the calculated voltage shape of a 35 MVAhydro-generator with an unsymmetric arrangement of the damper winding.The results of the analytic calculation are compared to those obtained fromtransient finite element analyses.

Figures 12 and 13 depict the calculated line-to-line voltage harmonicswithout an influence of the damper winding reaction. Figures 14 and 15 depictthe voltage harmonics with the full influence of the damper winding for anunshifted damper cage. On the contrary, Figures 16 and 17 show the effect of

Figure 12.Voltage harmonics

without damper windingreaction, fundamental

harmonic ofU1¼100 per cent,

analytical calculatedvalues


voltage shape

1137

the shifted damper cage. It can easily be seen that the unsymmetricarrangement of the damper winding reduces the voltage harmonics.

9. ConclusionsThe results of the presented analytical method are in good accordance with theresults of a transient finite element analysis. Deviations occur frominaccuracies in the analytical calculation of the pole field curve and using

Figure 13.Voltage harmonicswithout damper windingreaction, fundamentalharmonic ofU1¼100 per cent, valuesfrom a finite elementanalysis

Figure 14.Voltage harmonics witha classical damperwinding, fundamentalharmonic ofU1¼100 per cent,analytically calculatedvalues

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the tooth ripple permeance Fourier coefficients taken out of the sample curves.On the other hand, there may also be some inexactitudes of the finite elementcalculations, in particular, due to the Fast Fourier algorithm used with thetransient analysis.

The main problem in the analytical calculation of the pole field curve arisesfrom the falling edges in the pole gap due to the saliency of the machine. Small

Figure 15.Voltage harmonics with

a classical damperwinding, fundamental

harmonic ofU1¼100 per cent, values

from a finite elementanalysis

Figure 16.Voltage harmonics with

a shifted damperwinding, fundamental

harmonic ofU1¼100 per cent,

analytically calculatedvalues


voltage shape

1139

deviations in the pole field curve characteristic compared to finite elementresults can cause higher deviations in the high-order harmonics, for examplewith ordinal numbers near the tooth ripple.

Nevertheless, the main aim of the presented analytical method is a quickprediction of the voltage shape quality in the initial design calculation of hydrogenerators. The engineer needs a simple criteria for the selection of the numberof stator slots, design of the pole shoe shape and arrangement of the damperwinding.

References

CEDRAT (2000), Flux 2D, Version 7.5, User Manual, September 2000, CEDRAT, France.

Freeman, E.M. (1962), “The calculation of harmonics due to slotting in the flux-density waveformof a dynamo-electric machine”, Proceedings of IEE, Vol. 109, Part C, No. 16.

Ide, K., Takahashi, M., Sato, M., Tsuji, E. and Nishizawa, H. (1992), “Higher harmonicscalculation of synchronous generators on the basis of magnetic field analysis consideringrotor movement”, IEEE Transactions on Magnetics, Vol. 28 No. 2.

IEC 60034-1 (1999), “Rotating electrical machines – Part 1: rating and performance”, 10.2. ed.

Kim, C.E. and Sykulski, J.K. (2001), “Harmonic analysis of output voltage in synchronousgenerator using finite element method taking account of the movement”, Proceedings ofthe 13th Conference on the Computation of Electromagnetic Fields, COMPUMAG, Evian,France.

Rocha, E.J.J. and De Arruda Penteado, A. (1991), “The influence of damper winding slot pitchupon losses and synchronizing torque”, Proceedings of the International Conference on theEvolution and Modern Aspects of Synchronous Machines, Zurich, Switzerland.

Figure 17.Voltage harmonics witha shifted damperwinding, fundamentalharmonic ofU1¼100 per cent, valuesfrom a finite elementanalysis

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Rocha, E.J.J., Uemori, M.K. and De Arruda Penteado, A. (1997), “The damper winding influenceupon salient pole synchronous generator electromotive force”, Proceedings of the IEEEInternational Electric Machines and Drives Conference, IEMDC, Milwaukee, WI, USA.

Schmidt, E. (2001), “Electromagnetic finite element analysis of electrical machines using domaindecomposition and floating potentials”, Proceedings of the 13th Conference on theComputation of Electromagnetic Fields, COMPUMAG, Evian, France.

Simond, J.J. and Neidhofer, G. (1980), “Verfahren zur genauen Berechnung von Kurvenform undSchwingungsgehalt der Spannung bei Schenkelpolmaschinen”, Brown BoveriMitteilungen, pp. 2-80.


voltage shape

1141

Numerical modelling ofelectromagnetic process inelectromechanical systems

Vyacheslav A. KuznetsovMoscow Power Engineering Institute (MPEI), Moscow, Russia

Pascal BrochetEcole Centrale de Lille, France

Keywords Electrical machines, Numerical analysis

Abstract A general approach to the formation of magnetic equivalent circuit describing themagnetic process inside the electric machines is proposed. This formation is based on tooth contourmethod. Coupling with external and internal electric circuits of electric machines is emphasized aswell as mechanical coupling with load. The resulting model allows the simulation ofelectromechanical converter, but with the number of element being fewer by several orderscompared to traditional finite element models. Non-linearity such as saturation or electronic switchis taken into account. General equations for the magnetic fields and electric circuits of electricalmachines are written using a common basis – the nodal potential method. The whole process isillustrated on the simulation of a claw poles alternator compared with measurements.

1. IntroductionThe development of new methods for investigating electromechanical systems,which consist of different electromechanical converters (ECs), electric andmechanical elements and semiconductor devices, and methods of representingmagnetic fields and processes in ECs are among the most important problemsof the present-day electro mechanics. Historically, these two problems havebeen separated from one another to a considerable extent and different methodshave been used to solve them. This can be explained by both the diversity ofthe phenomena that occur in these systems and the modest capability of thecomputing technique which has been used over for many decades. Thesituation has changed during the last 20-25 years. New mathematical methodsfor analysing phenomena and processes in electromechanical devices andsystems have been developed and have become available. The most importantevent was the appearance of new powerful universal and functionally orientedcomputers, which enabled optimisation problems to be showed for anyobjective function and behaviour of the electromechanical systems and theirelements to be analysed both under nominal and extreme operating conditions.



The authors thank the Conseil Regional du Nord Pas de Calais (France) and FEDER for theirfinancial support.

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An electrical machine is one of the most complicated parts of any EC and canbe itself considered as a complex system characterized by magnetic, electrical,wave, thermal, mechanical air- and hydrodynamic phenomena. For thesereasons, the main attention of scientific researcher has been focused on thedevelopment of reliable models and methods of simulations of electricmachines.

For some types of electric machines characterized by a high non-linearity ofthe magnetic materials and induced currents, it is desirable to use models thatretain information on the existing magnetic field. In particular, it is veryimportant for electric machines with non-symmetrical magnetic systems,supplied with non-symmetrical currents and voltages, whose magnetic fieldsdepend strongly on the mutual position of a rotor and stator. Models of thislevel should be suitable for investigating the phenomena in theelectromechanical systems with simultaneous representation of the magneticfields. It would be a qualitatively new stage in the development of thesimulation. By solving the complete set of equations, one can analyse thesteady state and transient processes in any machines taking into account allthe influencing factors. However, the realization of this principal possibilitymeets insurmountable obstacles. Because of their complexity, the complete setof equations cannot be solved analytically or numerically. Moreover, thecalculation of the magnetic field in the interior of the machine is, in a way,over-informative, i.e. it gives redundant information. The existing numericalmethods (finite element, finite difference) are super-universal and they do notexploit the specific features of the interior structure of electric machines, whichactually, allow a considerable reduction in the number of elements whenmodelling the interior space, thereby greatly reducing the computing time.

2. Model reductionThe specific features of electric machines, which can help one to decrease thenumber of discrete elements of the model, are as follows:

. a relatively small air gap between the stator and rotor, except in the caseof permanent magnet machine;

. usually a high permeability of iron compared with vacuum;

. definite periodicity of the tooth-slot discrete structures;

. the relatively weak influence of the magnetic field in the end-windingzones on the process of energy conversion.

The first two features of the magnetic systems of electric machines make itrelatively easy to predict the magnetic field in the air gap between the statorand rotor teeth, particularly in the region of mutual overlap zone, even if thecores are rather highly saturated. The numerous analytical and numericalanalysis of the magnetic field for a permeability of the iron greater than 10m0

show that the lines of force in this case are practically orthogonal to the surface


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of the ferromagnetic cores. Strong orthogonality of the lines on the boundarysurfaces occurs if the iron permeability is infinite. This enables one to simplifythe calculation of magnetic fields, while dispensing with an excessivelydetailed representation of the interior space of the electric machine, as is done,for example, in finite element models.

3. The concept of constructing an equivalent circuit of themagnetic-field space of electric machineAt the first stage, the mutual permeances in the air space between the separateparts of the core surfaces, the ferromagnetic bodies of the electric machine, arebeing determined. The permeability of the ferromagnetic bodies at this stage isassumed to be infinite, so the permeances are calculated only for thenon-magnetic region. The surfaces of the ferromagnetic bodies are divided intoparts, taking into account the periodicity of the special structure and supposednature of the magnetic field in the air gaps of the machine. These parts areusually the tooth crowns, the sidewalls of the slot, pole shoes, etc. Figure 1shows the result of dividing the core surfaces into parts denoted from a to y andthe permeances for magnetic fluxes which exist between the excited part, a (thethick line) and all other parts.

The “excited” part has an arbitrarily specified scalar potential, all the otherparts have a zero potential. This potential distribution along the boundarycorresponds to the so-called special boundary conditions (SBC). In this case, thefluxes in the non-magnetic region occur only from the excited part, a, to thenon-excited parts, b, c, g, h, . . . s, r, etc. (Hecquet and Brochet, 1998; Ostovic,1989). Because of the small air gap and high iron permeability, many of thepermeances between part a and remote parts, t, u, v, w, x, y, k, l, m, etc. havepractically zero values and the magnetic field is concentrated in a very narrowzone near part a. The fluxes between the “excited” part and all other parts canbe calculated by any available method, analytical or numerical, depending onthe form of region and the capability of the computer and the programming.Further, for the same mutual position of the cores, another part, for example, r,is considered to be “excited” as shown in Figure 2 and using the same

Figure 1.Permeances betweenthe parts of cores whilepart a is excited

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procedure, the magnetic fluxes and permeances are determined. The scalarpotential of part a in this case, naturally, is assumed to be zero. Thus, a definite(unit) scalar potential is assigned alternatively to each part and the magneticfluxes and permeances are calculated for the SBC.

The next set of calculations is carried out for some change in the mutualposition of the cores. The angle of rotation g can be chosen based on the methodused, computing resources and computing time available. Analytical andnumerical-analytical methods for determining magnetic permeances can berecommended.

The results of calculations can be stored in the form of the numerical data orgraphical and analytical relations with appropriate approximation support.Some plots are shown in Figure 3, whence it can be seen that there are somepermeances which depend only slightly on the angle of rotation of the rotor andcan even be assumed to be constant without appreciable error (Lcg, Ldf). Someof the permeances are so small that they can be ignored (Leh, Lek etc.). The

Figure 2.Permeances between theparts of cores while part

r is excited

Figure 3.Permeances of branches

a-b, c-g, d-f depending onthe angle g between the

cores when SBC areapplied


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permeances between the tooth crowns are most dependent on the angularposition of the rotor (Lah in Figure 3 and Lah, Lrh in Figures 2 and 3). Theperiodicity of the tooth structure causes the tooth permeance curves to besimilar. For example, Lab and Lah in Figure 3 are absolutely identical and aremerely shifted with respect to one another.

This dramatically reduces the computing time needed to calculate theangular dependences of the tooth permeances. In practice, only 1-3 curves needto be calculated for synchronous machines and only one for induction motors.

It should also be noted that these angular dependences can be calculatedquite independent of the analysis of the magnetic equivalent circuit at thepreliminary stage. Hence, the computational costs involved in thesecalculations will, in no way, increase the overall cost of the design ofthe magnetic circuit.

The results of determining the air-gap permeances can be presented asthe superposition of the local solutions corresponding to Figures 1 and 2 etc.The whole interior space of the machine in this case is represented as a set ofdiscrete elements, determined by convolution of the continuous space. Theresultant non-planar discrete circuit is obtained without the use of the frozenprinciple, which leads to traditional planar equivalent circuit of the magneticsystem (ECMS) and introduces considerable systematic error. In our case, theconvolution of the space is carried out absolutely correctly. The error ofthis operation is determined by the unavoidable discrete representation of theslotted ferromagnetic surfaces.

It is not essential to introduce the concept of the tooth contours, since thesurfaces can be discretized as finely as required. However, an increase in thenumber of parts along the ferromagnetic surfaces does not always lead toincreased accuracy. As can be seen from the pattern of magnetic lines, thesurfaces of the ferromagnetic bodies even under severe saturation conditionscan be considered as lines of equal magnetic scalar potential. This considerablysimplifies the discretization of the interior space of the ferromagnetic bodyusing a limited number of non-linear permeances to represent the magnetizingcharacteristic of the corresponding part of the magnetic circuit of the stator androtor. Computation and experiments show that usually the teeth of the majorityof machines can be accurately represented by 3-4 non-linear permeances, andthe part of the yoke between the two teeth can be represented quite accuratelyby only one permeance. These permeances depend on:

. the magnetic properties of the materials;

. the dimensions of the chosen parts of the ferromagnetic bodies in thecross-section of the machine;

. the axial structure of cores, i.e. axial length, radial ducts, stack factor. . .

This fact is fundamental and distinguishes this method of setting up a modelfrom the models based on the finite element method. Since the permeances of

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the air-gap region in the no planar scheme must be calculated taking intoaccount the axial heterogeneity of the cores (if such exists), the discreteequivalent circuit of the interior space of an electrical machine is not atwo-dimensional version of an ECMS. It is, in essence, an artificial equivalentcircuit representing three-dimensional structure of the magnetic cores withonly a few elements (compared with the similar calculation scheme used in theFEM, for example) and capable of representing the mutual displacement ofthe stator and rotor very economically. When using strict planar schemes withthe FEM and FDM, the meshes must be readjusted for each mutualdisplacement of the cores.

A simplified fragment of the equivalent circuit of the magnetic system of asynchronous machine for half of the pole pitch is shown in Figure 4. The

Figure 4.A simplified fragment ofthe equivalent circuit of

the magnetic system of asynchronous machine for

half of the pole pitch


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non-linear permeances of the air-gap (numbers from 22 to 30) depend onthe angular position of the rotor. The non-linear permeances represent thesaturable parts of the magnetic system: the stator yoke (1-3), the rotor yoke (45),the stator teeth (3, 10, 16, 5, 11, 17 etc.), the rotor pole (34, 30, 41), the edge of thepole shoe (36, 37, 38, 39), and the tooth crowns of the pole shoe (31, 32, 33). Thelinear permeances represent the paths of the stator slot leakage (7, 13, 19, 8, 14,20 etc.) and the rotor inter-pole leakage (42, 43, 44). No special measures areneeded to represent differential leakage. The only field component which is notrepresented is the end-winding fluxes. Usually, this has no appreciableinfluence on the main processes of energy conversion. The end-windingleakage can be taken into account to a certain approximation when setting upthe set of differential equations of the electrical circuits.

The proposed method of constructing an ECMS may be used not only tostudy the electrical machines, but also other types of electromechanical energyconverters (EECs), for example, electrical apparatus with magnetic cores andwindings. The particular feature of the equivalent circuits considered, as hasalready been mentioned, is the non-planar form of the ECMS in the air-gapregions.

4. The concept of constructing a magnetic equivalent circuit of ECThe magnetic equivalent circuit is a discrete circuit reproducing the wholevolume inside the converter, where all particularities of its structure, materialproperties and movement can be found. Particularities due to the windinglocated inside the machine can be taken into account by considering themagnetic shells (Smolenski and Kuznetsov, 2000).

The currents in the windings are the sources of magnetic field in the ECMS.The actual cross-sections of the stator and rotor windings are depicted bycrosses in the stator slots and points in the rotor poles. To ascertain where thesources of the magnetic field (i.e. MMFs) should be placed in the branches ofECMS, the windings of the machine should be represented in the form of a setof magnetic shells with conventional magnetic charges on their surfaces. Anymagnetic shell can be considered as a body, which is in effect “pulled” over thetransverse cross-sections of the coils with equal, but opposite currents. Thelatter condition is strictly necessary. The magnetic shells rest on (“are pulledover”) the actual cross-sections of the coils, both sides of each coil being placedin the slots of the machine in the case of a drum-type winding. Any drum-typewinding can be represented by an equivalent ring winding as shown Figure 5,in which the “lower” sides coincide with the actual cross-sections placed in theslots (with the same current directions). The “upper” sides of this ring windingare placed on the outer perimeter of the stator core and carry the same currents2iB; j and 2iB; jþ1 as in the “lower” sides in the slot.

A magnetic shell is a double charged layer. It is known that if this doublelayer is penetrated (conventionally), the magnetic scalar potential will change

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abruptly. The magnetic scalar potential of any arbitrary point inside the spaceof the machine, created by the current in any turn of the winding, does notdepend on the form of the magnetic shell stretched over this turn. The magneticscalar potential in any arbitrary point is determined solely by the product of thecurrent, which flows in this turn and the solid angle, at which the turn with thecurrent is seen from this arbitrary point. The magnetic flux and flux linkage ofthe turn are also independent of the form of the shell. This fact enables us torepresent any type of machine winding as a combination of magnetic shells,that is very convenient for calculations. For the version shown in Figure 5, aring-type winding is used. There are two coil sides in the slot considered. Eachside has a cross-section and carry currents iB, j and iB; jþ1; which flow indifferent branches of the winding. The branches can be of the same or ofdifferent phases. The number of phases in a slot is not restricted. The othersides of the coils used to form the magnetic shells, when using the ring windingform, carry currents 2iB; j and 2 iB; jþ1 and, as mentioned earlier, are placed onthe periphery of the stator core.

If the machine has a drum winding, each coil is represented by two systemsof magnetic shells placed in different slots. In the case of a ring winding, themagnetic shells essentially coincide with the actual turns of the machinewinding.

Figure 5.Placement of EMF

sources of the magneticshells with currents iB,j

and iB;jþ1 in theequivalent magnetic

circuit


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The magnetic shells carrying currents iB, j and iB; jþ1; shown in Figure 5,intersect the branches of the ECMS A2B2 (branch “K 2 1”), A3B3 (branch “K”)and A4B4 (branch “K þ 1”), produce in these branches step changes in themagnetic scalar potential f K21; j; f K; j and f K; jþ1; f Kþ1; j and f Kþ1; jþ1;respectively. These MMFs are equal to the MMFs of those parts of themagnetic shells, which are “under” the mentioned branches of the ECMS. Forexample, MMF f K21; j in the ðK 2 1Þth branch A2B2 is equal to the product ofthe current iB, j and the number of turns carrying this current, which is “cut off”by the branch ðK 2 1Þ; i.e. A2B2. This number of turns is “under” branchðK 2 1Þ and can be denoted by wK21; j: Then

f K21; j ¼ wK21; jiB; j

In branch K, i.e. A3B3, step changes of a potential will be created by twomagnetic shells with currents iB, j and iB; jþ1:

f K; j ¼ wK; jiB; j;

f K; jþ1 ¼ wK; jþ1iB; jþ1

After introducing additional characteristics of the branch of ECMS in the form^wk; j; where k ¼ 1; 2; . . .;m and j ¼ 1; 2; . . .; n (here m is the total number ofbranches of the ECMS and n is the number of branches of the machine), theECMS acquires the character of an electromagnetic equivalent circuit (EMEC),which can be used for the simultaneous representation of the magnetic fieldsand electromagnetic processes in a machine along with the processes in theelectric circuits comprising the machine windings. This is fundamentally newin the practice of modelling magnetic fields and electromagnetic processes inECs. Thus, the new method differs in principle from the methods of modellingmagnetic fields using, for example, the FDM and FEM.

The vector f of the MMFs of the branches of the ECMS can be expressed inthe form

�f ¼ ½W � · �iB;

where �iB is the column matrix of the currents flowing in the winding branchesof the converter and [W ] is a matrix which transforms the currents of thebranches of the electric circuit into the m.m.f.s of the branches of the ECMS,i.e. the magnetic equivalent circuit.

The matrix [W ] can be called the structural matrix of the winding. It has anumber of columns equal to the number of branches of the electric circuit and anumber of rows equal to the number of branches of the ECMS. The elementwk, j, situated in the kth row and jth column, is equal to the number of turnscarrying current iB, j, which embrace the kth branch of the magnetic circuit.This number has its own sign.

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(Due to limitation of space, only an approximate method for determining wk,j

is described. In fact, the permeances, which are used to calculate the magneticfluxes and flux linkages are close to one another only if the number of branchesof the ECMS per slot is sufficiently large.)

Rigorous analytical and numerical computations have shown that alreadyfor 3-4 branches of the ECMS per slot, the discretization error for thepermeances used to calculate the magnetic fluxes and flux linkages even forsaturated conditions does not exceed a few per cent.

5. The general system of equations describing the magnetic stateand electrical processes in the EECWhen the EMEC is set up, the magnetic fluxes in discrete elements of the EECand electric quantities in the branches of the EEC windings can be determinedby a single universal method. The non-planar form of the ECMS and theelectric circuit comprising the windings of the EEC give purely nodal methodsan advantage; these operate with easily imagined quantities, such as the nodalpotentials and fluxes (currents) of the branches.

The procedure for setting up the equations used in nodal methods is simpleand convenient for computerization. Some properties of the nodal equationsensure rapid convergence of numerical methods, their mathematical apparatusis compact and can be used to solve both magnetic and electric circuits. Thesymmetrical coefficient matrices used in the nodal methods have positiveelements on the principle diagonal and negative or zero off-diagonal elements.The spectrum of these matrices is real and positive, which ensures that themore popular iterative computational methods used to analyse non-linearsystems converge.

A circuit with lumped elements can be described by topological methods.The full description must contain the following information:

. how the branches, i.e. the parts of the circuit between the two nodes, areconnected to one another;

. the sources of MMF and EMF connected in the branches;

. the dependence of the permeance and conductance (linear or non-linear) ofthe branches on the magnetic fluxes or electric currents and the appliedMMF (EMF) or voltage.

The equations describing the magnetic fluxes in discrete elements of the ECMSand the processes in electric circuit, in accordance with the topological methodsare combined into one system:

½A� · �F ¼ 0; ð1Þ

�F ¼ ½Lð �F; gÞ� · ð½A�t �wþ �f Þ; ð2Þ


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�f ¼ ½W � · �iB; ð3Þ

CB ¼ ½W �t �F; ð4Þ

d

dtCB ¼ ½Ae�

t �we þ �eext þ �er 2 ½RB� · �iB þ ½LB:ext�d�iBdt

þ ½CB:ext�21

Z�iB dt þ �ucð0Þ

ð5Þ

½Ae� · iB ¼ 0 ð6Þ

where w and we are the magnetic and electric potentials, cB the flux linkage ofthe winding, iB the currents in the electric circuit including the winding, f theMMF, F the fluxes in the branch of the magnetic circuit, [A ] and [Ae] are themagnetic and electric incidence matrix, [L(F,g)] is the permeances diagonalmatrix, and other notations are obvious.

This system consists of six equations and has six unknown vectors: �w is thevector of nodal potentials of the ECMS; �F is the vector of magnetic fluxes of thebranches of the EMEC; �CB is the vector of the flux linkages of the branches ofthe converter windings; �f is the vector of MMFs of the branches of ECMS; �we isthe vector of nodal potentials of the electric circuit; �iB is the vector of thecurrents in the branches of the electric circuit including the currents in theconverter windings.

The electromagnetic torque M can be found using previously calculatedderivatives of the permeances of the air zones with respect to the angle ofrotation g of the rotor

M ¼ 21

2�U

t

L

›ðLðgÞÞ

›g�UL ð7Þ

where �UL is the vector of MMF across the air-gap branches of the ECMS. Thisvector is determined after determining the nodal potentials �w

�UL ¼ ½A�t �wþ ½W � · �iB ð8Þ

Equations (1)-(8) constitute the complete system of equations of an EECdescribing the states of electric, magnetic and mechanical equilibrium, takingaccount of any non-linearity of magnetic, electric or mechanical origin.

6. A step-by-step simulation of a claw poles alternatorThis type of machine known as Lundell alternator is a very popular device, butits modelling meets many difficulties, one of them being the axial orientation offluxes in the stator core. Air-gap permeances have been determined using a 3Dfinite element package with the mesh shown in Figure 6.

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A part of the 3D permeance network is represented in Figure 7.Simulation results compare very well with measured EMF when the

machine is feeding a resistance through a six-diodes rectifier, as shown inFigure 8. Comparisons have been made on typical harmonics (Hecquet andBrochet, 1998). Many other interesting quantities are available owing to thissimple and efficient universal method. For example, in Figure 9, the evolutionof radial forces applied to a stator tooth are determined for different speeds andfor different stator currents.

Figure 6.Finite element mesh of

the claw-pole alternator

Figure 7.A part of the

3D permeance network; ahalf claw and two

stator teeth


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7. ConclusionThe general numerical modelling of electromagnetic process described hereallow fast and accurate simulations of electrical machine in various load cases.

References

Hecquet, M. and Brochet, P. (1998), “Time variation of forces in a synchronous machine usingelectric coupled network model”, IEEE Trans. on Magnetics, Vol. 34 No. 5, pp. 3656-9.

Ostovic, V. (1989), Dynamic of Saturated Electric Machines, Springer-Verlag, Berlin.

Smolenski, A.V.I. and Kuznetsov, V.A. (2000), “Universal numerical method for simulatingelectromechanical converters and systems”, Elektrichestvo, Vol. 2000 No. 7, pp. 24-34.

Figure 8.EMF on pole pitch, loadcurrent ¼ 16 A

Figure 9.Radial forces applied to astator tooth

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A coupledelectromagnetic-mechanical-

acoustic model of a DC electricmotor

Martin Furlan and Andrej CernigojR&D, ISKRA Avtoelektrika d.d., Sempeter pri Gorici, Slovenia

Miha BoltezarFaculty of Mechanical Engineering, University of Ljubljana, Ljubljana,

SloveniaKeywords Electrical machines, Magnetic forces, Noise, Boundary element method

Abstract In this article, we present an investigation into the sound radiation from apermanent-magnet DC electric motor using the finite-element (FE) and boundary-element (BE)models. A three-times-coupled electromagnetic-mechanical-acoustic numerical model was set-up topredict the acoustic field. The first stage was to calculate the magnetic forces that excite thestructure of the motor by using the FEM. In the second stage, the exciting magnetic forces wereapplied to the structural model, where the harmonic analysis was carried out using the FEM. Thelast stage was to model the acoustics by using the BEM. In order to evaluate the numerical model,the computational results were compared with the vibration and acoustic measurements and areasonable agreement was found.

1. IntroductionThe demand for a quiet environment has influenced almost every producer ofnoisy machinery, including the manufacturers of direct current (DC) electricmotors. Consequently, acoustic noise has become one of the most importantfactors that influence the acceptability of a DC electric motor for customers in avariety of industrial applications.

Improved computational facilities have recently resulted in the possibility ofa more quantitative description of the noise from electric machinery, especiallywhen the magnetic noise dominates. The first step includes the modelling of themagnetic field and the subsequent computation of the magnetic forces thatrepresent the main excitation for the structure of the electric motor. Mostresearch seems to be concentrated on induction machines (Cho and Kim, 1998;Ishibashi et al., 1998; Jang and Lieu, 1991; Lai and Wang, 1999; Wang and Lai,1999; Verdyck and Belmans, 1994), where two-dimensional (2D) modelsdominate. Just a few papers deal with the problem of magnetic force generationin the DC electrical motors.

The second step is the calculation of the structural dynamic response due tothe magnetic forces. Most authors who deal with the structural dynamics use



Model of a DCelectric motor

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Vol. 22 No. 4, 2003pp. 1155-1165


DOI 10.1108/03321640310483075

2D models of induction machines, similar to the approaches in the first step(Ishibashi et al., 1998). The reason for the limited use of three-dimensional (3D)models is probably the result of the complexity of the real structures of theelectric machines and the fact that the results obtained with a 2D analysis ( Jangand Lieu, 1991) give relatively good vibration predictions. Nevertheless, someauthors (Wang and Lai, 1999) showed a successful example of the 3D-structuredynamic modelling, which was proved by comparing with the experimentalresults. They also extended their research to acoustic modelling (Lai and Wang,1999).

The last step in the modelling of electric-machine noise is the calculation ofthe acoustic field. The computation of both the magnetic forces and thestructural dynamic response are usually performed with the finite-elementmethod (FEM). In contrast, the FEM is rarely applied to acoustic analysis,especially when dealing with the external problem, and at high frequencies, dueto the enormous number of degrees of freedom (DOF). Some years back, thesame thing could also be said for the boundary-element method (BEM). Butbecause the BEM is computationally less demanding than the FEM, andbecause of the rapid developments in computational power, it is now possibleto apply the BEM in acoustic modelling. As an even more effective method forpredicting the acoustic field of electric machines, some authors use thestatistical energy analysis (SEA) technique (Lai and Wang, 1999).

The aim of this paper is to show a step-by-step approach for predicting theacoustic noise radiated from the DC electric motors. The analysis wasperformed on a 0.6 kW permanent-magnet DC electric motor, intended to drivean electro-hydraulic power-steering system. The results obtained using withthe FEM/BEM are compared with the experimental data.

2. Method of numerical analysisThe prediction of magnetic noise is normally based on a three-times-coupledelectromagnetic-mechanical-acoustic numerical model. Figure 1 shows atypical model of this kind. To calculate the magnetic forces that excite thestructure of the motor an electromagnetic model needs to be developed, andthe FEM is one of the most frequently used techniques to describe it. Theelectromagnetic model itself is built-up using the geometrical characteristicsand the material data of the investigated electric motor, while the input to themodel represents the operating conditions. Using this information, we cancalculate the magnitude and frequencies of the resulting magnetic forces andapply them to the structural model. Like the electromagnetic model, the FEMwas found to be the most effective numerical tool for the structural model, too.The results of the vibrations, represented by velocities on the exterior surfaceof the electric motor, are used as an input for the acoustic model. In general,acoustic modelling can be done with the FEM or BEM. However, for noise

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scattering into free space, BEM is more convenient because only the exteriorsurface of the electric motor needs to be discretised.

2.1 Magnetic forces calculationFor the magnetic force calculation in the investigated DC electric motor, two 3DFEM models were built-up; the only difference between the models was in therotor-skewing angle. One rotor had no skewing and the other had a skewing ofone rotor slot. Because of the skewing it was impossible to apply symmetry tothe FEM model and therefore, it was necessary to model the whole motor. Thedynamic magnetic field produced as a result of the rotation of the rotor wasapproximated as a time series of magneto-static fields or quasi-static fields.

To allow the rotation of the rotor in the FEM model, the rotor and the statorwere meshed separately and later joined together with the constraint equationsin the position of the air gap. Linear commutation was applied to estimate thecurrent loads in the commutation zones.

The magnetostatic field was solved by the scalar potential formulation andthe non-linear material properties of the magnetic materials were included inthe model. The number of nodes was 52,793 for the rotor and 29,273 for thestator. Figure 2 shows both the FEM models, with the skewed rotor and thenon-skewed rotor.

The magnetic traction acting on the rotor and the stator with the permanentmagnets (Figure 3) was calculated by the Maxwell stress method. Theequation:

F ¼1

m

ZS

Tn dS ð1Þ

with the Maxwell stress tensor

Figure 1.Prediction of magnetic

noise in electric machines


1157

T ¼

B2x 2

12 jBj

2BxBy BxBz

ByBx B2y 2

12 jBj

2ByBz

BzBx BzBy B2z 2

12 jBj

2

66666664

77777775 ð2Þ

was used to calculate the magnetic force acting on a specific part of the electricmotor from the distribution of the flux density B on a closed envelope surface Saround that part with the normal n.

The resulting, calculated magnetic forces and moments acting on eachmagnet and on the rotor were transformed into a cylindrical coordinate system(r, w, z) that could be simply applied to the structural dynamic model of theinvestigated motor.

Analysing the resulting magnetic forces and the moments acting on eachmagnet we found that they had the same amplitude and phase during therotation. Figure 4 shows the resulting radial magnetic force acting on

Figure 2.3D FEM electromagneticmodels, skewed rotor(on the left) andnon-skewed rotor (on theright)

Figure 3.The magnetic forcesacting on the magnet(on the left) and on therotor (on the right) for themodel with the skewedrotor

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the magnet during the rotation of one rotor slot by one tenth of the rotor-slotpitch, as the main source of the magnetic noise. The forces and the momentswere calculated for different motor currents. After examining the character ofthe radial magnetic force acting on the magnet (Figure 4) we found that theforce could be split into a static and a dynamic component. The staticcomponent represents the average value of the magnetic traction action on themagnet with an approximate value of 75 N, which is almost the same for bothFEM models with the skewed and the non-skewed rotors. In contrast, for thedynamic component we found that it was strongly influenced by the motorcurrent as well as by the rotor geometry.

Figure 4.The resulting radial

magnetic force acting onthe magnet during the

rotation for the FEMmodel with skewed rotor

(on the top) andnon-skewed rotor (on the

bottom)


1159

The variation in the resulting magnetic force Fi(t) during the rotation for onerotor slot can be decomposed by the discrete Fourier transform (DFT) methodas follows:

FiðtÞ ¼X1

n¼�1

Fi;neinvt ð3Þ

where Fi,n represents the amplitude corresponding to the frequency componentnv with the direction index i (i ¼ r; w, z). Figure 4 shows the amplitudes of themagnetic forces’ harmonics. Only the first five harmonics, which result fromthe rotation defined by one tenth of the rotor-slot pitch, are calculated. As therotor has 20 slots, these harmonics are the 20th, 40th, 60th, 80th and 100th.

The harmonic decomposition in general, as well as for the resulting radialmagnetic force acting on the magnet (Figure 5) gives a more qualitativedescription of the dynamic component of the magnetic force. Nevertheless,this force represents the source of the vibrations on the electrical machine,and consequently, the magnetic noise. Concentrating on the harmoniccontents of the presented, resulting radial magnetic force and its comparisonbetween the two different FEM models (Figure 5) we found that the 20th and40th harmonics dominate. The values of the 20th harmonic for both the FEMmodels, with the skewed and non-skewed rotors, are very alike and linearlyincrease with the motor’s current. The only significant difference between theharmonic contents of the resulting radial magnetic force acting on themagnet for the presented FEM models occurs at the 40th harmonic. Itsvalues tend to be independent of the motor current for the motor with thenon-skewed rotor, while for the motor with the skewed rotor a linear increaseis observed.

As the experimental investigation of noise and vibrations was conducted fora constant rotation speed of 3,200 rpm, the exciting magnetic forces werecalculated for the corresponding loading conditions. For the motor with theskewed rotor a current of 5 A was applied, and for the motor with thenon-skewed rotor the current was 12 A.

2.2 Structural dynamic responseTo calculate the structural dynamic response of the investigated electric motora 3D FEM model was developed. After considering the construction of themotor, the structural FEM model was built-up from three parts. The statorwas modelled with linear isotropic thin-shell elements. The rotor structure wasapproximated by beam elements representing a shaft, while the armature wasmodelled as added mass to the beam elements. More simplifications of thegeometry were necessary for the end shield, which was modelled with bothshell and beam elements (Figure 6).

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Figure 5.The decomposed

resulting radial magneticforce acting on the

magnet for the FEMmodel with the skewed

rotor (on the top) and thenon-skewed rotor (on the

bottom)

Figure 6.3D solid model (on theleft), and 3D structural

FEM model (on the right)


1161

The most difficult task is to join all the three structures into a single-structureFEM model. The bearings, which represent the connection between the rotorand the front-end shield on one side and between the rotor and the stator on theother side, were modelled with stiffness elements. As the exact values of thebearing stiffness were not known, they were simplified as almost rigid. Threescrew connections between the stator and the end shield were implemented bycoupling nodes at the screw positions.

For verifying the structural response of the whole model, as with theindividual parts, an experimental modal analysis (EMA) was applied to the realstructure. At a later stage, the EMA results were used to improve the structuralFEM model.

The structural response of both motors was calculated using a harmonicanalysis, where the linear behaviour of the structure is considered, governed bythe equation:

K þ ivmC 2 v2mM

� �Xm ¼ Fm ð4Þ

where Xm is the displacement vector, Fm is the force vector and vm is thefrequency of the harmonic excitation, and K, M and C are the stiffness, themass and the damping matrixes, respectively. The analysis includes the firstfour harmonics of the magnetic excitation forces that have the greatestcontribution to the total motor noise, and a constant damping ratio of0.5 per cent is applied. As both motors operate at a rotation speed of 3,200 rpm,the excitation frequency of the 20th harmonic was 1,067 Hz.

Figure 7 shows the mechanically deformed motor resulting from the firstfour harmonics of the magnetic forces. The deformed shape at a particularharmonic is almost identical for both motors, for the skewed and non-skewedrotor only the amplitude of the deformation is different. The results of themeasured and calculated displacements for the collating point are presented inTable I.

2.3 Acoustic fieldThe acoustic field, described by the sound pressure p at an arbitrary point ofthe exterior medium E that involves a surface S with the harmonic vibrationassumed on it, is governed by the Helmholtz wave equation (Kirkup, 1998)

72p þ k2p ¼ 0 ð5Þ

where 72 is the Laplace operator. Knowing the normal velocities on the exteriorsurface S, the surface pressure can be calculated using a BEM formulation(Kirkup, 1998)

HpS ¼ Bvn ð6Þ

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where matrixes H and B describe the discretised acoustical problem, anddepend only on the excitation frequency and the structure shape, but not onthe structural behaviour. pS and vn are the pressure nodal surface vector andthe normal velocity nodal surface vector, respectively. Knowledge of thesepressures and normal velocity values allows us to set-up a discretedescription of the pressure pf at any field point in the exterior medium E(Kirkup, 1998):

pf ¼ hTf pS þ bT

f vn ð7Þ

Figure 7.Vibration response due

to the magnetic forces ofthe motor with the

non-skewed rotor

20th harmonic,1,067 Hz




p (mPa) x (m) p (mPa) x (m) p (mPa) x (m) p (mPa) x (m)

Non-skewedMeas. 7.8 1.8 £ 1028 8.9 2.6 £ 1029 5.7 2.2 £ 1029 4.9 9.6 £ 10210

Calc. 0.44 1.9 £ 1029 25.1 2.2 £ 1028 4.4 1.2 £ 1028 2.5 4.6 £ 1029

SkewedMeas. 3.5 1.30 £ 1029 1.6 8.7 £ 10211 1.5 3.6 £ 10210 – –Calc. 1.1 1.7 £ 1028 1.7 1.1 £ 1028 1.2 1.3 £ 1028 0.1 7.9 £ 1029

Table I.Calculated and

measured results ofsound pressure and

vibrationdisplacement


1163

where hTf and bT

f are the influence coefficient vectors, which depend on theposition of the field point, the excitation frequency and the structure shape.

In this case, the acoustic field of the investigated electric motor is calculatedwith the BEM, where the outer surface of the motor is discretised with 1,411linear, triangular, boundary elements (BEs) (Figure 8). The BE mesh is createdover the structural shell elements sharing the same nodes. This means that thenodal velocities from the structural analysis can be easily extracted to the BEMmodel. To ensure the accuracy of the BEM calculation the maximum size of theelement was less than one sixth of the sound wavelength at the maximuminvestigated frequency. Consequently, the initial mesh in the structuralanalysis can be considered to be fine enough for the subsequent acousticanalysis.

Table I contains the results of the sound pressure level, both measured andcalculated, at a distance of 10 cm from the motor’s surface.

3. Noise and vibration characteristics of the motorsFrom the vibration and sound pressure spectra (Figure 9), measured for themotor with the non-skewed rotor, peaks at the rotor-slot harmonics could beobserved. This phenomenon shows that the magnetic noise, which is related tothe rotor-slot harmonics, strongly dominates. A similar situation can also befound for the spectra of the motor with the skewed rotor, except that therotor-slot harmonics’ dominance is reduced.

4. ConclusionsThe presented investigation shows a step-by-step procedure for predicting theacoustic field of a DC electric motor when the magnetic noise dominates. Theresults obtained with both the structural and acoustic models give reasonableagreement with the measurements. For a more accurate analysis, however,improvements need to be made to the structural model.

Figure 8.3D BEM mesh (on theleft) and acoustic field –instantaneous pressure(on the right)

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References

Cho, D.H. and Kim, K.J. (1998), “Modeling of electromagnetic excitation forces of small inductionmotor for vibration and noise analysis”, IEE Proceedings: Electric Power Applications,Vol. 145 No. 3, pp. 199-205.

Ishibashi, F., Noda, S. and Mochizuki, M. (1998), “Numerical simulation of electromagneticvibration of small induction motors”, IEE Proceedings Electric Power Applications, Vol. 145No. 6, pp. 528-34.

Jang, G.H. and Lieu, D.K. (1991), “The effect of magnet geometry on electric motor vibration”,IEEE Transactions on Magnetics, Vol. 27 No. 6, pp. 5202-4.

Kirkup, S.M. (1998), “The boundary element method in acoustics: a development in Fortran”,Integrated Sound Software, Hebden Bridge.

Lai, J.C.S. and Wang, C. (1999), “Prediction of noise radiated from induction motor”, SixthInternational Congress on Sound and Vibration, pp. 2449-55.

Verdyck, D. and Belmans, R.J.M. (1994), “An acoustic model for a permanent magnet machine:modal shapes and magnetic forces”, IEEE Transactions on Industry Applications, Vol. 30No. 6, pp. 1625-31.

Wang, C. and Lai, J.C.S. (1999), “Vibration analysis of an induction motor”, Journal of Sound andVibration.

Figure 9.Vibration (upper) and

sound pressure (lower)linear spectra for the

motor with thenon-skewed rotor


1165

Current distribution withinmulti strand windings forelectrical machines with

frequency converter supplyOliver Drubel

ALSTOM (Switzerland) Ltd, Turbogenerator business, Birr, Switzerland

Keywords Variable frequency, Eddy currents, Electrical machines

Abstract The current distribution within multi strand windings is investigated for transientcurrent and voltage supplies. The difference in losses between transient and sinusoidal waveformsis elaborated. Therefore, a wide range of frequencies as well as different kinds of transientwaveforms has been investigated. The definition of the skin depth is no longer sufficient. A newparameter is required for transients, which is related to time. This parameter will be defined andcalled “skin time”. A numerical method is developed based upon a finite element transientcalculation. The method is applied to the winding as well as to the core. A comparison withmeasurements verifies the approach described.

IntroductionVariable speed machines are penetrating the market more and more evenfor machines up to 351 MVA, (Kleiner et al., 2001; Rechberger et al., 2000).Frequency converters are used even for turbogenerators of several hundredMVA output for start up and supply for the field winding. This technology isused mainly for large air-cooled generators up to 500 MVA, which arebecoming the preferred technology for gas turbines because of their operationaland maintenance advantages. Although the accumulated power for thisapplication of converters is about 2 GW each year, the main application offrequency converters is for asynchronous machines.

The basic topics investigated in this publication are not restricted to onespecial kind of machine and can be applied to large induction machines as wellas to synchronous machines.

The calculation of losses and equivalent resistances of frequency converterdrives has to take the transient waveform of voltage and current into account,see Figure 1.

The estimation of these equivalent resistances by a calculation, which takesharmonics into account, is not achievable without the use of a computer. Thecurrent density distribution within multi strand windings with transposition iscalculated for different kinds of waveforms. Losses are derived from thecurrent distribution and parameters are defined to compare the differenttransient waveforms with each other. Based on the standard definition of theskin depth a similar parameter is introduced and called skin time. The skin



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time defines how long the maximum current would have to be applied to theminimum skin depth to create the same losses as the transient.

This concept of skin depth and skin time is applied to different waveforms.The way in which this method can also be applied to non-linear material isshown and the work by Drubel and Stoll (2001) is consolidated.

Space and timeThe skin depth is in general defined for sinusoidal current shapes in linearmaterial with one-dimensional geometric properties. The electromagneticquantities depend on time and space according to Figure 2.

The distribution of the flux density is given by equation (1):

Bðy; tÞ ¼ Be2y=ds sinðvt 2 2py=dsÞ ð1Þ

The skin depth for periodic functions and non-linear material was investigatedby Oberretl (1959). It has been analysed in-depth and is not the subject of thispublication.

Three main approaches for the skin depth are generally applied:

(1) at the skin depth the amplitude of the current density as well as the fluxdensity is only 1/e of its maximum value at the surface of the material(Figure 3);

(2) this definition is equal to a decrease in power density of 1/e 2;

(3) the effective dc current flowing in the area of the skin depth would causethe same losses as the real time dependent current causes in a semiinfinite slab as a mean value over the complete time interval (Figure 4).

As soon as transient, non-periodic functions are considered these threeapproaches are no longer sufficient.

Therefore, a new quantity dT is defined:

W el ¼ Rac1

2I 2

maxdT; ð2Þ

where Wel is the loss energy, Rac is the equivalent resistance due to space skindepth, and Imax is the maximum current.

Figure 1.Line current without the

basic harmonic duringrunning up of a 300 MVA

air-cooled turbogenerator

Currentdistribution

1167

The new quantity is a kind of skin depth in time. The losses no longer occurover a complete time period, but over the skin time dT. The energy, which isproduced by the losses, is of main interest. From this energy an adiabaticestimation for the increase in temperature can be derived. In order to estimatethe material volume, which has to be applied for the temperature calculation,the skin depth in space is used. Within this skin depth the losses reduce to 1/e 2.

Figure 2.Flux density over spaceand time for sinusoidalcurrent within a massiveconductor

Figure 3.Definition of skin depthdue to decrease incurrent density

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The losses, which occur during an exponential decrease are only 1/e 2 of themaximum losses after dT. This constant is equal to the time constant T in thecase of an exponential current function, which is flowing through a bar whereno eddy currents occur.

iðtÞ ¼ Imaxe2t=T ð3Þ

PðtÞ ¼ RI 2maxe22t=T ð4Þ

W el ¼ R1

2I 2

maxT ð5Þ

Whereas the skin depth in time is just the time constant, T, if eddy currents areneglected, it is also quite a helpful parameter for the cases where eddy currentsoccur. The equivalent resistance Rac,sT can be estimated using the followingequations (6) and (7):

W el ¼ Rac;sT1

2I 2

maxT ð6Þ

Rac;sT ¼ RacdT

Tð7Þ

Figure 4.Definition of the skin

depth according to losses

Currentdistribution

1169

Due to the fact that the time constants within any complex system are notknown in advance, the appropriate resistance has to be determined iteratively.

It is a helpful parameter to estimate the current and energy, which is withinsome parasitic impulses, especially for shaft voltage calculations and othertransient parasitic phenomenon with small time constants.

The determination of the appropriate skin depth and time is shown with thefollowing example (Figures 5 and 6). One strand of 3 mm is supplied with acurrent of exponential time decrease of 10ms.

The skin time is about five times the time constant T ¼ 10ms:The skin depth is determined from the loss energy density over the depth y

according to Figure 6.The losses are partly due to the mean current density and circulating current

density. The circulating current causes the high losses on the air gap side andincreases the losses in the bottom region. In between the losses are fairly low.

With the skin depth and skin time it is possible to determine the loss energyof the current decrease according to the following equation (8):

W el

bl¼ I 2

max

1

2dsgdT ¼ 0:00028mW=m2: ð8Þ

Investigated geometry and time functionsIn order to describe the current waveform within electrical machines, which areconnected to a static frequency converter, the following three different basic

Figure 5.Loss density and lossenergy density used toderive the skin time dT

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1170

time functions can be applied. In general, the current waveform is acombination of all the three.

. Sinusoidal

iðtÞ ¼ I sinðvtÞ; ð9Þ

. Linear increase

iðtÞ ¼ Ct þ I 0; ð10Þ

. Exponential decrease

iðtÞ ¼ Imaxe2t=T : ð11Þ

A sinusoidal current occurs, for example, after running up a generator. It is thenormal operating condition and is not included in this investigation.

The subject of this investigation is mainly transient exponential functions.These occur for instance during LCI operation or in VRM drives (Colby et al.,1993; De Doncker et al., 1994; Hava et al., 1992). During LCI operation thecurrents are switched from one phase to the other phase (Figure 7).

The linear current shape is applied to investigate the parasitic effects due toswitching. The current shape is given here in Figure 1. This investigation is notlimited to the strands of a winding, but is also applied to non-linear material.Capacitive coupling causes currents over the stator core to ground. It is one of

Figure 6.Skin depth from bottom

and air gap side, totalskin depth ds¼1.55 mm

Currentdistribution

1171

the important reasons for the newly occurring kind of shaft voltages (Chan,1995; Hausberg and Seinsch, 2000).

Two geometries were chosen for the investigation. These are fundamentalcases within electrical machines. On the one hand, one strand is investigatedfor different dimensions and on the other hand, a multi strand conductor in aslot is investigated. The first example describes the condition within the strandat the bottom of a slot. The second one will be found in stator slots oflarge asynchronous or synchronous machines as well as in the rotor of asynchronous machine. The transposition is assumed to be so optimised, thatthe same current flow occurs in each conductor.

Mathematical modelAll cases have been investigated using a transient finite element numericalcalculation. The method has been verified by an analytical model of a semiinfinite slab and sinusoidal currents.

The finite element model is based upon Maxwells equations:

rot ~E ¼ 2›~B

›t; ð12Þ

rot ~H ¼ J : ð13Þ

With the equations of material:

~J ¼ g~E; ð14Þ

and

Figure 7.Phase current during LCIoperation (Colby et al.,1993)

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~B ¼ m · ~H: ð15Þ

The eddy current equation:

D~A ¼ 2mg›~A

›tð16Þ

with

~B ¼ rot ~A: ð17Þ

is derived from these basic equations (12)-(15).In addition to this differential equation in each strand, a current i(t) is

impressed according to equations (9)-(11). In this area the electrical fieldstrength consists not only of the rotational part, but also of a scalar part:

~E ¼ 2›~A

›tþ ~Ei: ð18Þ

The impressed field strength ~Ei is constant over each strand. It is unknown andhas to be derived by the following equation for each strand.

Ig ~Ei 2

›~A

›t

!d~S ¼ iiðtÞ; ð19Þ

where S is the cross-section of copper strand.Whereas equation (16) can be solved analytically for some simple geometries

and time functions, the application of numerical methods is inevitable for morecomplicated time functions and non-linear material.

The non-linear characteristic of the iron is modelled by spline functionsaccording to Schwarz (1986).

Equations (16) and (18) are solved by the application of the numerical finiteelement method. The investigated geometry is one-dimensional.

Að y; tÞ ¼Xm

n21

anð yÞAnðtÞ; ð20Þ

with

anð yÞ ¼ 2yn21

yn 2 yn21þ

1

yn 2 yn21y y # yn

¼ynþ1

ynþ1 2 yn2

1

ynþ1 2 yny y . yn;

ð21Þ

and

Currentdistribution

1173

wð y; tÞ ¼Xm

n21

anð yÞwnðtÞ; ð22Þ

with

E i ¼ 2gradðwÞ; ð23Þ

where m is the no. of nodes, and n is the node n.With greens theorem equation (16) together with equation (18) can be

transformed to:

ZV

wn DAn þ mg›An

›t

� �dy ¼ 2

ZV

gradðwnÞ gradðAnÞ dy

þ

ZV

wnmg›An

›tdy ¼ 2

ZV

wng grad wn

� dy

ð24Þ

This equation is derived for each element form function vn. Additionallyequation (19) is solved for each strand. The impressed electrical field strength isconstant for each strand. With this condition, the algebraic equation system iscompleted and solved for each time step. The calculation procedure is iterativeif non-linear material is considered.

Calculation resultsThree main regions are investigated, which are considered as potential currentpaths during high frequency peaks and exponential decrease. Within staticfrequency converters transients occur due to parasitic phenomena of thethyristors, but also due to the switching itself.

(1) These transients cause losses within normal copper strands of the stator.

(2) They cause losses within the rotor copper.

(3) They cause losses in the iron of the stator core.

The investigation of 1 and 2 is very similar, but it has to be differentiatedbetween the case where the eddy currents are induced by a strand internalsource or by an external source. The current within each strand itself inducescirculating currents in addition to the circulating currents induced by the fluxcaused by current flow within the other strands in the same slot.

Figure 8 shows the current density for both cases. The magnetic fieldstrength at the air gap side, y¼0, of the strand is the same for both cases.

The eddy currents within the strand are determined by the waveform of themagnetic field strength at the surface of the strand. At the air gap side thecurrent density distribution in a strand due to the eddy currents induced by its

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1174

own flow, is the same for the case where the current flow is in an adjacentstrand. It is different at the bottom, where no current is induced by the adjacentstrands.

If the time constant is in the range of 10-100ms, the eddy currents aredominant for multi strand bars with strands depth of some millimetres(Figure 9).

Figure 9 shows the loss energy density for a bar, which consists of 10strands. The loss energy density increases nearly by the square of the distanceto bar no. 10. This shows the dominant influence of the eddy currents. This isconfirmed by the eddy current distribution in Figure 10.

The current density increases in proportion to the distance from strandno. 10. The distribution is determined by the field strength increase of thestrands, which are deeper in the slot. The current density for t ¼ 0 is 667 A/m2.This part of the density is not shown in Figure 10. Only the eddy currentdensity is shown. If the time constant is in the range of some 10 ms the eddycurrents are no longer dominant. Therefore strands of some millimetres depthare sufficient for 50 or 60 Hz applications.

In Figures 9 and 10 the current and loss distribution within a multi strandbar have been investigated for an exponential current decrease. The eddycurrent distribution for a linear current increase and decrease according toFigure 11 is shown in Figure 12.

The current density distribution during the increase and decrease of thecurrent is quite similar to the one for the exponential decrease. It is quite

Figure 8.Current density

distribution after 20msdue to exponential

current decrease withinthe regarded and

adjacent strand

Currentdistribution

1175

Figure 9.Loss energy densitydistribution in a bar with10 strands and a timeconstant of 10ms

Figure 10.Eddy current densitydistribution in a bar with10 strands and a timeconstant of 10ms

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Figure 11.Linear current increase

and decrease

Figure 12.Eddy current

distribution within thestrands of a bar for a

current waveformaccording to Figure 11

Currentdistribution

1177

different at the current peak. Here the current density is relatively small and ashort decrease in the losses occurs.

A typical current distribution for non-linear material is given in Figure 13for a bar with three strands.

Equivalent circuit parametersWhereas several effects of a thyristor’s parasitic phenomenon are oftendescribed, calculations or models to calculate these effects are often missing(Chan, 1995; Hausberg and Seinsch, 2000; Torlay, 1999). Only a fewpublications exist which realise a model of these parasitic effects and comparethem with the measurements (Ammann et al., 1988). Due to the very complexstructure of shaft voltage phenomena, equivalent circuit diagrams are appliedwhich require the estimation of its parameters. One parameter is an appropriateequivalent resistance for the current path. Within Table I an overview ofappropriate parameters is given therefore for a strand with inherent currentflow and in Table II with dominant induced currents due to the current inadjacent strands.

The skin time for a bar with i strands can be derived by the followingequation (25), if eddy currents are dominant:

dT;bar ¼ dT;1 þdT;2

i

Xi

k¼1

ðk 2 1Þ2 ð25Þ

Figure 13.Current densitydistribution fornon-linear material

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The skin time dT,1 is given in Table I, whereas the skin time dT,2 is given inTable II.

Comparison with measurementsAny calculation model can only be regarded as valid if it has been verified bymeasurements. The switching between thyristors of a 15 MW gas turbinestarting device has been chosen as an example. The normal measurementequipment within plants is not applicable for those high frequencies where theskin time becomes important. Special equipment like air-cored Rogowsky coilshave been installed in order to get reasonable results.

The measured thyristor voltage has been impressed in the model and thecalculated stator terminal current is compared with measurements in Figure 14.

The measured and calculated currents fit very well for the first 40ms afterthe thyristor was switched. After this period the dominant frequency decreasesand the applied calculation model needed to be adjusted. The measured andcalculated curves were adjusted to the reference value shown in Figure 14.

ConclusionsA parameter, which allows the influence of the eddy currents upon the timeconstant to be described is a valuable support to modelling transients using

Strand depth (mm) 1.5Time constant (ms) 10 100 1000ds (mm) 1.5 1.5 1.5dT (ms) 33 99 980b/lRacsT (mV) 0.040 0.012 0.012

Strand depth (mm) 2Time constant (ms) 10 100 1000ds (mm) 1.55 1.82 2.0dT (ms) 49 99 990b/lRacsT (mV) 0.057 0.010 0.009

Strand depth (mm) 3Time constant (ms) 10 100 1000ds (mm) 1.7 3 3dT (ms) 90 150 1000b/lRacsT (mV) 0.095 0.009 0.006

Table I.Equivalent

resistance and skintime for different

geometries andinherent current

flow

Strand depth (mm) 2Time constant (ms) 10 100 1000ds (mm) 1.28 1.36 1.33dT (ms) 111 32 4b/lRacsT (mV) 0.155 0.004 0.00006

Table II.Equivalent

resistance and skintime for different

geometries andcurrent flow inadjacent strand

Currentdistribution

1179

equivalent circuits. Especially within frequency converter drives thisparameter allows the exponential current decrease and increase to beunderstood and simulated better and parasitic phenomena often with veryshort time constants to be investigated. Especially the commutation betweentwo switching thyristors, as in pulse width modulated or LCI converters, is inthe range of some 10ms and requires eddy currents to be taken into accountduring current circuit simulations. The defined skin time is a parameter whichfulfils these requirements. An overview of the parameter for different timeconstants and strand dimensions, which are often used, allows its practicalapplication.

The losses are dominated by induced eddy currents within the strands of abar if the time constants are not more than some 100ms. Here the results givencan be applied to any number of strands.

References

Ammann, C., Reichert, K., Joho, R. and Posedel, Z.F. (1988), “Shaft voltages in generators withstatic excitation systems - problems and solutions”, IEEE Trans. on Energy Conversion,Vol. 3 No. 2, pp. 409-19.

Chan, S. (1995), Bearing current, EMI and soft switching in induction motor drives DissertationUniv. Winsconsin, Madison.

Colby, R.S., Otto, M.D. and Boys, J.T. (1993), “An analysis of LCI synchronous motor drives finiteDC link inductance”, IEE Proceedings B, Vol. 140 No. 6, pp. 379-86.

De Doncker, R.W., Demirci, O. and Temple, V.A. (1994), “Characteristics of GTO’s andHigh-Voltage MCT’s in High-Power Soft Switching Converters”, IEEE Trans. on IndustryApplications, Vol. 30 No. 6, pp. 1548-56.

Drubel, O. and Stoll, R.L. (2001), “Comparison between analytical and numerical methods ofcalculating tooth ripple losses in salient pole synchronous machines”, IEEE EnergyConversion, Vol. 16 No. 1, pp. 61-7.

Figure 14.Comparison betweencalculated and measuredstator terminal current at798 rpm

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Hausberg, V. and Seinsch, H.O. (2000), “Shaft voltages and circulating bearing currents ofconverter supplied induction machines”, Electrical Engineering 82, Vol. 87 No. 6,pp. 313-26.

Hava, A.M., Blasko, V. and Lipo, T. (1992), “A Modified C-Dump Converter forVariable-Reluctance Machines”, IEEE on Industry Applications, Vol. 28 No. 5, pp. 1017-21.

Kleiner, F., Ponick, B. and De Wit, B. (2001), “Choosing electric turbocompressor drivers”, IEEEIAS, Vol. 7 No. 4, pp. 45-52.

Oberretl, K. (1959), “Die magnetische Rotorjochspannung bei Asynchronmaschinen”,Elektrotechnik und Maschinenbau, Vol. 76 No. 19, pp. 449-54.

Rechberger, K., Mueller, F. and Kofler, H. (2000), “Single phase sudden short circuits in variablespeed generators“, ICEM 2000, pp. 903-7.

Schwarz, H.R. (1986), Numerische Mathematik, Teubner, Verlag Stuttgart.

Torlay, J.E. (1999), “Etude des courants et tensions d’arbre et de phases dans les grandsalternateurs”, Disstation Inst. Nat. Polytechnique de Grenoble.

Currentdistribution

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New booksLinear synchronous motors: transportation and automationsystemsJacek F. Gieras and Zbigniew J. PiechCRC Press, 2000/328 ppISBN: 0-8493-1859-9

The book provides a comprehensive treatment of all types of brushless linearsynchronous motors (LSMs) – their construction, electromagnetic effects, control,and applications. It explains the physics of hard magnetic materials, discussespermanent magnet and superconducting excitation systems, and addressesapplications in factory automation and high-speed transportation systems. LSMshave better performance and higher power density than their inductioncounterparts, and they can operate with a larger mechanical clearance betweentheir stationary and moving parts.

In chapter 1, ‘‘Topology and Selection’’, the authors discuss topology ofLSMs including stepping and switched reluctance motors, calculations offorces and linear motor traction. Seven numerical examples illustrate theselection of linear motors and their applications to a variety ofelectromechanical traction drives. Chapter 2, ‘‘Materials and Construction’’,describes materials for magnetic circuits of LSMs, permanent magnets (PMs),principles of superconductivity, laminated stacks of LSMs, slotted and slotlessarmature windings, electromagnetic, PM and superconducting excitationsystems. In chapter 3, ‘‘Theory of LSMs’’, methods of calculation of thrust forPM and superconducting LMs are derived and equations for basic parametersof LSMs and linear hybrid stepping motors are given. A case study ofperformance calculation for small flat surface PM LSMs is included. Chapter 4,‘‘Motion Control’’, deals with dynamic performance of LSMs, their methods ofcontrol and precision linear positioning. Chapter 5 describes optical andmagnetic sensors for LSMs. In chapter 6, ‘‘High Speed Maglev Transport’’,details of German ‘‘Transrapid’’ maglev system, Japanese Yamanashi MaglevTest Track, Swissmetro project and Japanese underwater marine express, allwith LSMs, are analyzed. Chapter 7, ‘‘Building and Factory TransportationSystems’’, is devoted to modern linear motor driven roped and ropelesselevators and factory horizontal transportation systems. Chapter 8, ‘‘IndustrialAutomation Systems’’, discusses automation of manufacturing processes withLSMs, in particular, casting processes, machining processes, welding andthermal cutting, surface treatment and finishing, material handling and testing.In Appendix A, ‘‘Magnetic Circuits with Permanent Magnets’’, a circuitalapproach to the calculation of PM systems is described. Appendix B,‘‘Calculations of Permeances’’, presents basic formulae for permanence of PMmagnetic circuits. Appendix C, ‘‘Performance Calculations for PM LSMs’’,shows electromagnetic calculations of a small flat LSM with buried PMs.

Note from thepublisher

1183

COMPEL : The International Journalfor Computation and Mathematics inElectrical and Electronic Engineering

Vol. 22 No. 4, 2003p. 1183


Note from the publisher

Professor Jan Sykulski wins the Leading Editor Award at theLiterati Awards for Excellence 2003We were delighted to present Professor Jan Sykulski, Editor of COMPEL,with a Leading Editor Award at this year’s Literati Awards for Excellence(see www.emeraldinsight.com/literaticlub.htm for further details).

Leading editorships are awarded for (amongst other things):

. the successful development of a journal;

. effective management of the editorial advisory board and the reviewprocess;

. identification of good quality special issues;

. delivering quality copy which satisfies editorial objectives.

COMPEL has gone from strength to strength under the editorship of ProfessorSykulski, and we congratulate him on his efforts and achievements with thejournal. We look forward to continuing our successful working relationship.

Literati Club

Awards for Excellence

Nikolaos V. Kantartzis

Theodoros K. Katsibas

Christos S. Antonopoulosand

Theodoros D. TsiboukisAristotle University of Thessaloniki, Thessaloniki, Greece

are the recipients of the journal’s Outstanding Paper Award for Excellence for their paper

‘‘Unified higher-order curvilinear FDTD-PMLs for 3D electromagnetics andadvectiver acoustics’’

which appeared in COMPEL, Vol. 21 No. 3, 2002

Nikolaos V. Kantartzis was born in Kavala, Greece, on 26 April 1971. He received the Diploma and theDr Eng. Degree in electrical and computer engineering from the Department of Electrical and ComputerEngineering, Aristotle University of Thessaloniki, Greece, in 1994 and 1999, respectively. Since 1999,he has been with the Department of Electrical and Computer Engineering, where he is currently apost doctoral fellow. His main research interests include electromagnetic field analysis andcomputational electromagnetics (FDTD and higher-order methods, FEM, Vector Finite Elements, ABCs),acoustics and EMC problems. He has authored more than 20 refereed journal papers. He is, also,the recipient of the URSI Young Scientist Award, 1999 and Emerald-Compel Best Paper Award, 2001.Dr Kantartzis is a member of the technical chamber of Greece.

Theodoros K. Katsibas was born in Trikala, Greece, on 11 January 1975. He received his Diplomadegree in electrical and computer engineering from Aristotle University of Thessaloniki, Greece,in 1999. He is currently a PhD student at the Department of Electrical and Computer Engineering,Aristotle University of Thessaloniki. His current research is focused on computational electromagneticsand acoustics as well as propagation and scattering of acoustic waves. He is, also, the recipient ofthe Emerald-COMPEL Best Paper Award, 2001. Mr Katsibas is a member of the technical chamberof Greece.

Christos S. Antonopoulos was born in Alexandroupolis, Greece on 20 July 1958. He received theDiploma and the Dr degree in electrical engineering from the Department of Electrical Engineering,Faculty of Engineering, Aristotle University of Thessaloniki, Greece, in 1981 and in 1990, respectively.His doctoral thesis deals with electromagnetic field calculations in stratified media using the T-�method and a boundary element technique. Now his scientific interests include problems dealing withthe Electromagnetic Field Theory, Analytical and Numerical Methods in ElectromagnetismElectromagnetic Compatibility, and Acoustic Propagation. He is currently an Associate Professor withthe Electrical and Computer Engineering Department, Faculty of Technology, Aristotle University ofThessaloniki, Greece.

Theodoros D. Tsiboukis was born in Larissa, Greece, on 25 February, 1948. He received the Diplomadegree in electrical and mechanical engineering from the National Technical University of Athens,Greece, in 1997 and the Dr Eng. Degree from the Aristotle University of Thessaloniki, Thessaloniki,Greece, in 1981. During the academic year 1981-1982, he was a visiting research fellow at the ElectricalEngineering Department of the University of Southampton, England. Since 1982, he has been workingat the Department of Electrical and Computer Engineering of the Aristotle University of Thessaloniki,where he is now a Professor. His main research interests include electromagnetic field analysis byenergy methods, computational electromagnetics (FEM, BEM, Vector Finite Elements, MoM, FDTD,ABCs), inverse problems, EMC problems, and adaptive meshing in FEM analysis. He is the author of sixbooks and has authored or co-authored more than 85 refereed journal articles. He has also organisedand chaired conference sessions and was awarded a number of distinctions. Professor Tsiboukis is amember of various societies, associations, chambers and institutions.

Literati Club

COMPEL22,4

1186


Erratum

Modelling and simulation with neural and fuzzy-neural networks ofswitched circuitsOwing to an error in the production of the above article by Yakup Demir andAysegul Ucar, published in COMPEL, Vol. 22 No. 2, pp. 253-72, the authors’proof amendments had not been incorporated.

Emerald sincerely apologises to the authors and readers for this error.The correct article may be accessed on the Emerald Web site at:

www.emeraldinsight.com/compel.htm

Index

1187


Vol. 22 No. 4, 2003pp. 1187-1200


Index to Compel:The International Journal

for Computation andMathematics in Electrical

and Electronic Engineering,volume 22, 2003

AuthorsALEXEEVSKI, D., see BRAUER, H.

ALIFEROV, A., see LUPI, S.

ALLELLA, F., CHIODO, E. and LAURIA, D., Probabilistic approach to series compensationdesign, No. 2, pp. 372–387.

ALOTTO, P., MAGELE, C., RENHART, W., STEINER, G. and WEBER, A., Robust targetfunctions in electromagnetic design, No. 3, pp. 549–560.

AMRHEIN, W. see KAISERSEDER, M.

ANDRIOLLO, M., BERTONCELLI, T. and DI GERLANDO, A., A magnetic network approachto the transient analysis of synchronous machines, No. 4, pp. 953–968.

ANILE, A.M., SPINELLA, S. and RINAUDO, S., Stochastic response surface method andtolerance analysis in microelectronics, No. 2, pp. 314–327.

ARKKIO, A., see POYHONEN, S.

ARKKIO, A., see TENHUNEN, A.

ASAI, S., see COLLI, F.

BAAKE, E., NACKE, B., BERNIER, F., VOGT, M., MUHLBAUER, A. and BLUM, M.,Experimental and numerical investigations of the temperature field and melt flow in theinduction furnace with cold crucible, No. 1, pp. 88–97.

BAAKE, E., NACKE, B., UMBRASHKO, A. and JAKOVICS, A., Turbulent flow dynamics, heattransfer and mass exchange in the melt of induction furnaces, No. 1, pp. 39–47.

BABINI, A., BORSARI, R., DUGHIERO, F., FONTANINI, A. and FORZAN, M., 3D FEMmodels for numerical simulation of induction sealing of packaging material, No. 1,pp. 170–180.

BALLESTRA, L. and SALERI, F., Numerical solutions of a viscous-hydrodynamic model forsemiconductors: the supersonic case, No. 2, pp. 205–230.

BARGLIK, J., see DOLEZEL, I.

BAUMGARTNER, U., GRUMER, M., JAINDL, M., KOSTINGER, A., MAGELE, CH., PREIS, K.,REINBACHER, M. and VOLLER, S., e-Courseware authoring tools for teachingelectrodynamics, No. 3, pp. 603–615.

BELMANS, R., see DE GERSEM, H.

BENCHAIB, A., see POULLAIN, S.

BEREZA, J., see ZGRAJA, J.

BERNIER, F., see BAAKE, E.

COMPEL22,4

1188

BERTONCELLI, T., see ANDRIOLLO, M.

BINGHAM, C.M., see SEWELL, H.I.

BIRO, O., see PAVO, J.,

BIRO, O., see WEIß, B.

BLUM, M., see BAAKE, E.

BOKOSE, F., see VANDEVELDE, L.

BOLTEZAR, M., see FURLAN, M.

BORSARI, R., see BABINI, A.

BOSSAVIT, A., Extrusion, contraction: their discretization via Whitney forms, No. 3,pp. 470–480.

BRANDSTATTER, B., HOLLER, G. and WATZENIG, D., Reconstruction of inhomogeneitiesin fluids by means of capacitance tomography, No. 3, pp. 508–519.

BRANDSTATTER, B., see SCHWEIGHOFER, B.

BRAUER, H., ZIOLKOWSKI, M., DANNEMANN, M., KUILEKOV, M. and ALEXEEVSKI, D.,Forward simulations for free boundary reconstruction in magnetic fluid dynamics, No. 3,pp. 674–688.

BROCHET, P., see DELMOTTE-DELFORGE, C.

BROCHET, P., see KUZNETSOV, V.A.

BUCHAU, A., HAFLA, W., GROH, F. and RUCKER, W.M., Improved grouping scheme andmeshing strategies for the fast multipole method, No. 3, pp. 495–507.

BURAIS, N., see SIAUVE, N.

CAPOLINO, G-A., see DELMOTTE-DELFORGE, C.

CAPOLINO, G.A., see CIRRINCIONE, M.

CAVALIERE, V., CIOFFI, M., FORMISANO, A. and MARTONE, R., Robust design of highfield magnets through Monte Carlo analysis, No. 3, pp. 589–602.

CERNIGOJ, A., see FURLAN, M.

CHIODO, E., see ALLELLA, F.

CIOFFI, M., see CAVALIERE, V.

CIRRINCIONE, G., see CIRRINCIONE, M.

CIRRINCIONE, M., PUCCI, M., CIRRINCIONE, G. and CAPOLINO, G.A., Constrained least-squares method for the estimation of the electrical parameters of an induction motor,No. 4, pp. 1089–1101.

COLLI, F., FABBRI, M., NEGRINI, F., ASAI, S. and SASSA, K., Removal of SiC inclusions inmolten aluminium using a 12 T static magnetic field, No. 1, pp. 58–67.

COSTA, M.C., see COULOMB, J-L.

COULOMB, J-L., KOBETSKI, A., COSTA, M.C., MARECHAL, Y. and JONSSON, U.,Comparison of radial basis function approximation techniques, No. 3, pp. 616–629.

CUI, X., see ZHANG, B.

DANNEMANN, M., see BRAUER, H.

DAVEY, K.R., Use and analysis of null flux coils, No. 2, pp. 304–313.

Index

1189

DE GERSEM, H., BELMANS, R. and HAMEYER, K., Floating potential constraints and field-circuit couplings for electrostatic and electrokinetic finite element models, No. 1,pp. 20–29.

DE SALVADOR FERREIRA, G., see WEINZIERL, D.

DE WULF, M., see DUPRE, L.

DEDEK, L., DEDKOVA, J. and VALSA, J., Optimization of perfectly matched layer for 2DPoisson’s equation with antisymmetrical or symmetrical boundary conditions, No. 3,pp. 520–534.

DEDKOVA, J., see DEDEK, L.

DELFINO, F., PROCOPIO, R., ROSSI, M. and NERVI, M., A full-Maxwell algorithm for thefield-to-multiconductor line-coupling problem, No. 3, pp. 789–805.

DELIEGE, G., HENROTTE, F., VANDE SANDE, H. and HAMEYER, K., 3D h-ø finite elementformulation for the computation of a linear transverse flux actuator, No. 4, pp. 1077–1088.

DELMOTTE-DELFORGE, C., HENAO, H., EKWE, G., BROCHET, P. and CAPOLINO, G-A.,Comparison of two modeling methods for induction machine study: application todiagnosis, No. 4, pp. 891–908.

DEMIR, Y. and UCAR, A., Modelling and simulation with neural and fuzzy-neural networks ofswitched circuits, No. 2, pp. 253–272.

DI BARBA, P., DUGHIERO, F., LUPI, S. and SAVINI, A., Optimal shape design of devices andsystems for induction-heating: methodologies and applications, No. 1, pp. 111–122.

DI GERLANDO, A., see ANDRIOLLO, M.

DOLEZEL, I., BARGLIK, J., SAJDAK, C., SKOPEK, M. and ULRYCH, B., Modelling ofinduction heating and consequent hardening of long prismatic bodies, No. 1, pp. 79–87.

DRUBEL, O., Current distribution within multi strand windings for electrical machines withfrequency converter supply, No. 4, pp. 1166–1181.

DUGHIERO, F., LUPI, S., MUHLBAUER, A. and NIKANOROV, A., TFH – transverse fluxinduction heating of non-ferrous and precious metal strips: results of a EU researchproject, No. 1, pp. 134–148.

DUGHIERO, F., see BABINI, A.

DUGHIERO, F., see DI BARBA, P.

DULAR, P., see GYSELINCK, J.

DULAR, P., see GYSELINCK, J.

DULAR, P., see SABARIEGO, R.V.

DUPRE, L., DE WULF, M., MAKAVEEV, D., PERMIAKOV, V., PULNIKOV, A. andMELKEBEEK, J., Modelling of electromagnetic losses in asynchronous machines, No. 4,pp. 1051–1065.

EKWE, G., see DELMOTTE-DELFORGE, C.

ERTL, M., KALTENBACHER, M., MOCK, R. and LERCH, R., Numerical analysis of fastswitching electromagnetic valves, No. 3, pp. 715–729.

FABBRI, M., GALANTE, F., NEGRINI, F., TAKEUCHI, E. and TOH, T., Influence of theelectro-magnetic stirring on the boundary layer of a molten steel pool, No. 1, pp. 10–19.

FABBRI, M., see COLLI, F.

COMPEL22,4

1190

FELICETTI, R. and RAMESOHL, I., Design of a mass-production low-cost claw-pole motor foran automotive application, No. 4, pp. 937–952.

FIRETEANU, V. and TUDORACHE, T., Numerical simulations of continuous inductionheating of magnetic billets and sheets, No. 1, pp. 68–78.

FIRETEANU, V., GERI, A., TUDORACHE, T. and VECA, G.M., Transverse flux inductionheating: comparison between numerical models and experimental validation, No. 1,pp. 98–110.

FIRETEANU, V., see PAYA, B.

FITAN, E., MESSIN, F. and NOGAREDE, B., A general analytical model of electricalpermanent magnet machine dedicated to optimal design, No. 4, pp. 1037–1050.

FONTANINI, A., see BABINI, A.

FORMISANO, A. and MARTONE, R., Optimisation of magnetic sensors for currentreconstruction, No. 3, pp. 535–548.

FORMISANO, A., see CAVALIERE, V.

FORZAN, M., see BABINI, A.

FURLAN, M., CERNIGOJ, A. and BOLTEZAR, M., A coupled electromagnetic-mechanical-acoustic model of a DC electric motor, No. 4, pp. 1155–1165.

GALANTE, F., see FABBRI, M.

GEBHARDT, M., see RAUSCH, M.

GERI, A., see FIRETEANU, V.

GEUZAINE, C., see SABARIEGO, R.V.

GOLDSTEIN, R., see NEMKOV, V.

GRENIER, D., see GYSELINCK, J.

GROH, F., see BUCHAU, A.

GRUMER, M., see BAUMGARTNER, U.

GUERIN, C., see PAYA, B.

GYIMOTHY, S., see PAVO, J.,

GYSELINCK, J., DULAR, P., LEGROS, W. and GRENIER, D., Hybrid magnetic equivalentcircuit – finite element modelling of transformer fed electrical machines, No. 3,pp. 643–658.

GYSELINCK, J., DULAR, P., VANDEVELDE, L., MELKEBEEK, J., OLIVEIRA, A.M. andKUO-PENG, P., Two-dimensional harmonic balance finite element modelling of electricalmachines taking motion into account, No. 4, pp. 1021–1036.

GYSELINCK, J., see SABARIEGO, R.V.

GYSELINCK, J.J.C., see VANDEVELDE, L.

HAFLA, W., see BUCHAU, A.

HAJDAREVIC, I. and KOFLER, H., A closer view on inductance in switched reluctance motors,No. 4, pp. 847–861.

HAMBERGER, P., see SCHMIDT, E.

HAMEYER, K., see DE GERSEM, H.

Index

1191

HAMEYER, K., see DELIEGE, G.

HAMEYER, K., see HENROTTE, F.

HARB, W., see SCHLEMMER, E.

HENAO, H., see DELMOTTE-DELFORGE, C.

HENNEBERGER, G., see KAEHLER, C.

HENROTTE, F., PODOLEANU, I. and HAMEYER, K., Staged modelling: a methodology fordeveloping real-life electrical systems applied to the transient behaviour of a permanentmagnet servo motor, No. 4, pp. 1066–1076.

HENROTTE, F., see DELIEGE, G.

HOLLER, G., see BRANDSTATTER, B.

HOLOPAINEN, T.P., see TENHUNEN, A.

HYOTYNIEMI, H., see POYHONEN, S.

IVANYI, A., see SZABO, Z.

IVANYI, A., see KUCZMANN, M.

JAGIELA, M., see LUKANISZYN, M.

JAGIELA, M., see LUKANISZYN, M.

JAINDL, M., see BAUMGARTNER, U.

JAKOVICS, A., see BAAKE, E.

JONSSON, U., see COULOMB, J-L.

JORNET, A., ORILLE, A., PEREZ, A. and PEREZ, D., Optimal design of high frequencyinduction motors with the aid of finite elements, No. 4, pp. 1115–1125.

JOVER, P., see POYHONEN, S.

KOSTINGER, A., see BAUMGARTNER, U.

KAEHLER, C. and HENNEBERGER, G., Eddy-current computation on a one pole-pitch modelof a synchronous claw-pole alternator, No. 4, pp. 834–846.

KAISERSEDER, M., SCHMID, J., AMRHEIN, W. and SCHEEF, V., Current shapes leading topositive effects on acoustic noise of switched reluctance drives, No. 4, pp. 998–1008.

KALTENBACHER, M., see ERTL, M.

KALTENBACHER, M., see RAUSCH, M.

KETTUNEN, L., see SUURINIEMI, S.

KLAUS, G., see KUNCKEL, ST.

KOBETSKI, A., see COULOMB, J-L.

KOFLER, H., see HAJDAREVIC, I.

KOST, A., see WEINZIERL, D.

KRAUZE, A., see MUHLBAUER, A.

KRAUZE, A., see MUHLBAUER, A.

KRZYSZTOF, D., see MARIAN, P.

COMPEL22,4

1192

KUCZMANN, M. and IVANYI, A., Vector hysteresis model based on neural network, No. 3,pp. 730–743.

KUILEKOV, M., see BRAUER, H.

KUNCKEL, ST., KLAUS, G. and LIESE, M., Calculation of eddy current losses and temperaturerises at the stator end portion of hydro generators, No. 4, pp. 877–890.

KUO-PENG, P., see GYSELINCK, J.

KUZNETSOV, V.A. and BROCHET, P., Numerical modelling of electromagnetic process inelectromechanical systems, No. 4, pp. 1142–1154.

LANDES, H., see RAUSCH, M.

LAURIA, D., see ALLELLA, F.

LEGROS, W., see SABARIEGO, R.V.

LEGROS, W., see GYSELINCK, J.

LERCH, R., see ERTL, M.

LI, L., see ZHANG, B.

LIESE, M., see KUNCKEL, ST.

LO, K.L., see YUEN, Y.S.C.

LOWTHER, D.A., Automating the design of low frequency electromagnetic devices – asensitive issue, No. 3, pp. 630–642.

LUKANISZYN, M., JAGIELA, M. and WROBEL, R., Electromechanical properties of a disc-type salient-pole brushless DC motor with different pole numbers, No. 2, pp. 285–303.

LUKANISZYN, M., WROBEL, R. and JAGIELA, M., Field-circuit analysis of constructionmodifications of a torus-type PMDC motor, No. 2, pp. 337–355.

LUPI, S. and ALIFEROV, A., Electromagnetic phenomena in resistance heating of curvilinearcylindrical work pieces, No. 1, pp. 149–157.

LUPI, S., see DI BARBA, P.

LUPI, S., see DUGHIERO, F.

MAGELE, C., see ALOTTO, P.

MAGELE, CH., see BAUMGARTNER, U.

MAKAVEEV, D., see DUPRE, L.

MARECHAL, Y., see COULOMB, J-L.

MARIAN, P. and KRZYSZTOF, D., New approach to the optimisation of three-phase three-wiresystems with sinusoidal voltage sources and nonlinear loads, No. 2, pp. 356–371.

MARTONE, R., see CAVALIERE, V.

MARTONE, R., see FORMISANO, A.

MASMOUDI, A., Stability analysis of the doubly fed machine with emphasis on saturationeffects, No. 2, pp. 410–423.

MELKEBEEK, J., see GYSELINCK, J.

MELKEBEEK, J., see DUPRE, L.

MELKEBEEK, J.A.A., see VANDEVELDE, L.

MESSIN, F., see FITAN, E.

Index

1193

MOCK, R., see ERTL, M.

MUELLER, F., see SCHLEMMER, E.

MUELLER, F., see SCHOENAUER, J.

MUHLBAUER, A., MUIZNIEKS, A., RATNIEKS, G., KRAUZE, A., RAMING, G. andWETZEL, T., Mathematical modelling of the industrial growth of large silicon crystalsby CZ and FZ process, No. 1, pp. 158–169.

MUHLBAUER, A., MUIZNIEKS, A., RATNIEKS, G., KRAUZE, A., RAMING, G. andWETZEL, T., Using of EM fields during industrial CZ and FZ large silicon crystalgrowth, No. 1, pp. 123–133.

MUHLBAUER, A., see BAAKE, E.

MUHLBAUER, A., see DUGHIERO, F.

MUIZNIEKS, A., see MUHLBAUER, A.

MUIZNIEKS, A., see MUHLBAUER, A.

NACKE, B., see BAAKE, E.

NACKE, B., see BAAKE, E.

NEGREA, M., see POYHONEN, S.

NEGRINI, F., see COLLI, F.

NEGRINI, F., see FABBRI, M.

NEMKOV, V. and GOLDSTEIN, R., Computer simulation for fundamental study and practicalsolutions to induction heating problems, No. 1, pp. 181–191.

NERVI, M., see DELFINO, F.

NICOLAS, A., see SIAUVE, N.

NICOLAS, L., see SIAUVE, N.

NIKANOROV, A., see DUGHIERO, F.

NOGAREDE, B., see FITAN, E.

OLIVEIRA, A.M., see GYSELINCK, J.

ORILLE, A., see JORNET, A.

PAVO, J., SEBESTYEN, I., GYIMOTHY, S. and BIRO, O., Approximate prediction of losses intransformer plates, No. 3, pp. 689–702.

PAYA, B., FIRETEANU, V., SPAHIU, A. and GUERIN, C., 3D magneto-thermal computationsof electromagnetic induction phenomena, No. 3, pp. 744–755.

PEREZ, A., see JORNET, A.

PEREZ, D., see JORNET, A.

PERMIAKOV, V., see DUPRE, L.

PETERSON, W., Fixed-point technique in computing nonlinear eddy current problems, No. 2,pp. 231–252.

PODOLEANU, I., see HENROTTE, F.

POULLAIN, S., THOMAS, J.L. and BENCHAIB, A., Discrete-time modeling of AC motors forhigh power AC drives control, No. 4, pp. 922–936.

COMPEL22,4

1194

POYHONEN, S., NEGREA, M., JOVER, P., ARKKIO, A. and HYOTYNIEMI, H., Numericalmagnetic field analysis and signal processing for fault diagnostics of electrical machines,No. 4, pp. 969–981.

PREIS, K., see BAUMGARTNER, U.

PROCOPIO, R., see DELFINO, F.

PUCCI, M., see CIRRINCIONE, M.

PULNIKOV, A., see DUPRE, L.

RAIZER, A., see WEINZIERL, D.

RAMESOHL, I., see FELICETTI, R.

RAMING, G., see MUHLBAUER, A.

RAMING, G., see MUHLBAUER, A.

RATNIEKS, G., see MUHLBAUER, A.

RATNIEKS, G., see MUHLBAUER, A.

RAUSCH, M., GEBHARDT, M., KALTENBACHER, M. and LANDES, H., Magnetomechanicalfield computations of a clinical magnetic resonance imaging (MRI) scanner, No. 3,pp. 576–588.

REINBACHER, M., see BAUMGARTNER, U.

RENHART, W., see ALOTTO, P.

RINAUDO, S., see ANILE, A.M.

ROSSI, M., see DELFINO, F.

RUCKER, W.M., see BUCHAU, A.

RUSEK, J., Category, slot harmonics and the torque of induction machines, No. 2, pp. 388–409.

SABARIEGO, R.V., GYSELINCK, J., GEUZAINE, C., DULAR, P. and LEGROS, W., Applicationof the fast multipole method to the 2D finite element-boundary element analysis ofelectromechanical devices, No. 3, pp. 659–673.

SAHIN, F. and VANDENPUT, A.J.A., Thermal modeling and testing of a high-speed axial-fluxpermanent-magnet machine, No. 4, pp. 982–997.

SAJDAK, C., see DOLEZEL, I.

SALERI, F., see BALLESTRA, L.

SASSA, K., see COLLI, F.

SAVINI, A., see DI BARBA, P.

SCHEEF, V., see KAISERSEDER, M.

SCHLEMMER, E., HARB, W., SCHOENAUER, J. and MUELLER, F., Application of stochasticsimulation in the optimisation process of hydroelectric generators, No. 4, pp. 821–833.

SCHLEMMER, E., see SCHOENAUER, J.

SCHMID, J. see KAISERSEDER, M.

SCHMIDT, E., HAMBERGER, P. and SEITLINGER, W., Calculation of eddy current losses inmetal parts of power transformers, No. 4, pp. 1102–1114.

SCHMIDT, E., see TRAXLER-SAMEK, G.

Index

1195

SCHOENAUER, J., SCHLEMMER, E. and MUELLER, F., Finite element modeling of thetemperature distribution in the stator of a synchronous generator, No. 4, pp. 909–921.

SCHOENAUER, J., see SCHLEMMER, E.

SCHWEIGHOFER, B. and BRANDSTATTER, B., An accurate model for a lead- acid cellsuitable for real- time environments applying control volume method, No. 3, pp. 703–714.

SCHWERY, A., see TRAXLER-SAMEK, G.

SCORRETTI, R., see SIAUVE, N.

SEBESTYEN, I., see PAVO, J.,

SEITLINGER, W., see SCHMIDT, E.

SEWELL, H.I., STONE, D.A. and BINGHAM, C.M., Dynamic load impedance matching forinduction heater systems, No. 1, pp. 30–38.

SIAUVE, N., SCORRETTI, R., BURAIS, N., NICOLAS, L. and NICOLAS, A., Electromagneticfields and human body: a new challenge for the electromagnetic field computation, No. 3,pp. 457–469.

SKOPEK, M., see DOLEZEL, I.

SPAHIU, A., see PAYA, B.

SPINELLA, S., see ANILE, A.M.

STEINER, G., see ALOTTO, P.

STONE, D.A., see SEWELL, H.I.

SUURINIEMI, S. and KETTUNEN, L., Trade-off between information and computability:a technique for automated topological computations, No. 3, pp. 481–494.

SZABO, Z. and IVANYI, A., Adjustment with magnetic field, No. 3, pp. 561–575.

TAKACS, J., Fourier analysis of hysteretic distortions, No. 2, pp. 273–284.

TAKEUCHI, E., see FABBRI, M.

TENHUNEN, A., HOLOPAINEN, T.P. and ARKKIO, A., Spatial linearity of an unbalancedmagnetic pull in induction motors during eccentric rotor motions, No. 4, pp. 862–876.

THOMAS, J.L., see POULLAIN, S.

TOH, T., see FABBRI, M.

TRAXLER-SAMEK, G., SCHWERY, A. and SCHMIDT, E., Analytic calculation of the voltageshape of salient pole synchronous generators including damper winding and saturationeffects, No. 4, pp. 1126–1141.

TUDORACHE, T., see FIRETEANU, V.

TUDORACHE, T., see FIRETEANU, V.

UCAR, A., see DEMIR, Y.

ULRYCH, B., see DOLEZEL, I.

UMBRASHKO, A., see BAAKE, E.

VALSA, J., see DEDEK, L.

VANDE SANDE, H.V., see DELIEGE, G.

VANDENPUT, A.J.A.., see SAHIN, F.

COMPEL22,4

1196

VANDEVELDE, L., GYSELINCK, J.J.C., BOKOSE, F. and MELKEBEEK, J.A.A., Vibrations ofmagnetic origin of switched reluctance motors, No. 4, pp. 1009–1020.

VANDEVELDE, L., see GYSELINCK, J.

VECA, G.M., see FIRETEANU V.

VOGT, M., see BAAKE, E.

VOLLER, S., see BAUMGARTNER, U.

WATZENIG, D., see BRANDSTATTER, B.

WEBER, A., see ALOTTO, P.

WEIß, B. and BIRO, O., Multigrid for transient 3D eddy current analysis, No. 3, pp. 779–788.

WEINZIERL, D., RAIZER, A., KOST, A. and DE SALVADOR FERREIRA, G., Simulation of amode stirred chamber excited by wires using the TLM method, No. 3, pp. 770–778.

WETZEL, T., see MUHLBAUER, A.

WETZEL, T., see MUHLBAUER, A.

WŁODARSKA, J., see WŁODARSKI, Z.

WŁODARSKI, Z. and WŁODARSKA, J., Evaluation of hysteresis loss using variable pinningparameter, No. 2, pp. 328–336.

WROBEL, R., see LUKANISZYN, M.

WROBEL, R., see LUKANISZYN, M.

YIN, H., see ZHANG, B.

YUEN, Y.S.C. and LO, K.L., Simulations of bilateral energy markets using MATLAB, No. 2,pp. 424–443.

ZGRAJA, J. and BEREZA, J., Computer simulation of induction heating system with seriesinverter, No. 1, pp. 48–57.

ZHANG, B., CUI, X., ZHAO, Z., YIN, H. and LI, L., An electromagnetic approach to analyze theperformance of the substation’s grounding grid in high frequency domain, No. 3,pp. 756–769.

ZHAO, Z., see ZHANG, B.

ZIOLKOWSKI, M., see BRAUER, H.

Titles

3D FEM models for numerical simulation of induction sealing of packaging material, BABINI,A., BORSARI, R., DUGHIERO, F., FONTANINI, A. and FORZAN, M., No. 1, pp. 170–180.

3D h-� finite element formulation for the computation of a linear transverse flux actuator,DELIEGE, G., HENROTTE, F., VANDE SANDE, H. and HAMEYER, K., No. 4,pp. 1077–1088.

3D magneto-thermal computations of electromagnetic induction phenomena, PAYA, B.,FIRETEANU, V., SPAHIU, A. and GUERIN, C., No. 3, pp. 744–755.

(An) accurate model for a lead- acid cell suitable for real- time environments applying controlvolume method, SCHWEIGHOFER, B. and BRANDSTATTER, B., No. 3, pp. 703–714.

Adjustment with magnetic field, SZABO, Z. and IVANYI, A., No. 3, pp. 561–575.

Index

1197

Analytic calculation of the voltage shape of salient pole synchronous generators includingdamper winding and saturation effects, TRAXLER-SAMEK, G., SCHWERY, A. andSCHMIDT, E., No. 4, pp. 1126–1141.

Application of stochastic simulation in the optimisation process of hydroelectricgenerators, SCHLEMMER, E., HARB, W., SCHOENAUER, J. and MUELLER, F., No. 4,pp. 821–833.

Application of the fast multipole method to the 2D finite element-boundary element analysis ofelectromechanical devices, SABARIEGO, R.V., GYSELINCK, J., GEUZAINE, C., DULAR,P. and LEGROS, W., No. 3, pp. 659–673.

Approximate prediction of losses in transformer plates, PAVO, J., SEBESTYEN, I.,GYIMOTHY, S. and BIRO, O., No. 3, pp. 689–702.

Automating the design of low frequency electromagnetic devices – a sensitive issue,LOWTHER, D.A., No. 3, pp. 630–642.

Calculation of eddy current losses and temperature rises at the stator end portion of hydrogenerators, KUNCKEL, ST., KLAUS, G. and LIESE, M., No. 4, pp. 877–890.

Calculation of eddy current losses in metal parts of power transformers, SCHMIDT, E.,HAMBERGER, P. and SEITLINGER, W., No. 4, pp. 1102–1114.

Category, slot harmonics and the torque of induction machines, RUSEK, J., No. 2, pp. 388–409.

(A) closer view on inductance in switched reluctance motors, HAJDAREVIC, I. and KOFLER,H., No. 4, pp. 847–861.

Comparison of radial basis function approximation techniques, COULOMB, J-L.,KOBETSKI, A., COSTA, M.C., MARECHAL, Y. and JONSSON, U., No. 3, pp. 616–629.

Comparison of two modeling methods for induction machine study: application to diagnosis,DELMOTTE-DELFORGE, C., HENAO, H., EKWE, G., BROCHET, P. andCAPOLINO, G-A. No. 4, pp. 891–908.

Computer simulation for fundamental study and practical solutions to induction heatingproblems, NEMKOV, V. and GOLDSTEIN, R., No. 1, pp. 181–191.

Computer simulation of induction heating system with series inverter, ZGRAJA, J. andBEREZA, J., No. 1, pp. 48–57.

Constrained least-squares method for the estimation of the electrical parameters of an inductionmotor, CIRRINCIONE, M., PUCCI, M., CIRRINCIONE, G. and CAPOLINO, G-H., No. 4,pp. 1089–1101.

(A) coupled electromagnetic-mechanical-acoustic model of a DC electric motor, FURLAN, M.,CERNIGOJ, A. and BOLTEZAR, M., No. 4, pp. 1155–1165.

Current distribution within multi strand windings for electrical machines with frequencyconverter supply, DRUBEL, O., No. 4, pp. 1166–1181.

Current shapes leading to positive effects on acoustic noise of switched reluctance drives,KAISERSEDER, M., SCHMID, J., AMRHEIN, W. and SCHEEF, V., No. 4, pp. 998–1008.

Design of a mass-production low-cost claw-pole motor for an automotive application,FELICETTI, R. and RAMESOHL, I., No. 4, pp. 937–952.

Discrete-time modeling of AC motors for high power AC drives control, POULLAIN, S.,THOMAS, J.L. and BENCHAIB, A., No. 4, pp. 922–936.

Dynamic load impedance matching for induction heater systems, SEWELL, H.I., STONE, D.A.and BINGHAM, C.M., No. 1, pp. 30–38.

COMPEL22,4

1198

e-Courseware authoring tools for teaching electrodynamics, BAUMGARTNER, U., GRUMER,M. JAINDL, M., KOSTINGER, A., MAGELE, CH., PREIS, K., REINBACHER, M. andVOLLER, S., No. 3, pp. 603–615.

Eddy-current computation on a one pole-pitch model of a synchronous claw-pole alternator,KAEHLER, C. and HENNEBERGER, G., No. 4, pp. 834–846.

(An) electromagnetic approach to analyze the performance of the substation’s grounding grid inhigh frequency domain, ZHANG, B., CUI, X., ZHAO, Z., YIN, H. and LI, L., No. 3,pp. 756–769.

Electromagnetic fields and human body: a new challenge for the electromagnetic fieldcomputation, SIAUVE, N., SCORRETTI, R., BURAIS, N., NICOLAS, L. and NICOLAS,A., No. 3, pp. 457–469.

Electromagnetic phenomena in resistance heating of curvilinear cylindrical work pieces,LUPI, S. and ALIFEROV, A., No. 1, pp. 149–157.

Electromechanical properties of a disc-type salient-pole brushless DC motor with different polenumbers, LUKANISZYN, M., JAGIELA, M. and WROBEL, R., No. 2, pp. 285–303.

Evaluation of hysteresis loss using variable pinning parameter, WŁODARSKI, Z. andWŁODARSKA, J., No. 2, pp. 328–336.

Experimental and numerical investigations of the temperature field and melt flow in theinduction furnace with cold crucible, BAAKE, E., NACKE, B., BERNIER, F., VOGT, M.,MUHLBAUER, A. and BLUM, M., No. 1, pp. 88–97.

Extrusion, contraction: their discretization via Whitney forms, BOSSAVIT, A., No. 3,pp. 470–480.

Field-circuit analysis of construction modifications of a torus-type PMDC motor,LUKANISZYN, M., WROBEL, R. and JAGIELA, M., No. 2, pp. 337–355.

Finite element modeling of the temperature distribution in the stator of a synchronousgenerator, SCHOENAUER, J., SCHLEMMER, E. and MUELLER, F., No. 4, pp. 909–921.

Fixed-point technique in computing nonlinear eddy current problems, PETERSON, W., No. 2,pp. 231–252.

Floating potential constraints and field-circuit couplings for electrostatic and electrokineticfinite element models, DE GERSEM, H., BELMANS, R. and HAMEYER, K., No. 1,pp. 20–29.

Forward simulations for free boundary reconstruction in magnetic fluid dynamics,BRAUER, H., ZIOLKOWSKI, M., DANNEMANN, M., KUILEKOV, M. andALEXEEVSKI, D., No. 3, pp. 674–688.

Fourier analysis of hysteretic distortions, TAKACS, J., No. 2, pp. 273–284.

(A) full-Maxwell algorithm for the field-to-multiconductor line-coupling problem, DELFINO, F.,PROCOPIO, R., ROSSI, M. and NERVI, M., No. 3, pp. 789–805.

(A) general analytical model of electrical permanent magnet machine dedicated to optimaldesign, FITAN, E., MESSIN, F. and NOGAREDE, B., No. 4, pp. 1037–1050.

Hybrid magnetic equivalent circuit – finite element modelling of transformer fed electricalmachines, GYSELINCK, J., DULAR, P., LEGROS, W. and GRENIER, D., No. 3,pp. 643–658.

Improved grouping scheme and meshing strategies for the fast multipole method, BUCHAU,A., HAFLA, W., GROH, F. and RUCKER, W.M., No. 3, pp. 495–507.

Index

1199

Influence of the electro-magnetic stirring on the boundary layer of a molten steel pool, FABBRI,M., GALANTE, F., NEGRINI, F., TAKEUCHI, E. and TOH, T., No. 1, pp. 10–19.

(A) magnetic network approach to the transient analysis of synchronous machines,ANDRIOLLO, M., BERTONCELLI, T. and DI GERLANDO, A., No. 4, pp. 953–968.

Magnetomechanical field computations of a clinical magnetic resonance imaging (MRI)scanner, RAUSCH, M., GEBHARDT, M., KALTENBACHER, M. and LANDES, H., No. 3,pp. 576–588.

Mathematical modelling of the industrial growth of large silicon crystals by CZ and FZ process,MUHLBAUER, A., MUIZNIEKS, A., RATNIEKS, G., KRAUZE, A., RAMING, G. andWETZEL, T., No. 1, pp. 158–169.

Modelling and simulation with neural and fuzzy-neural networks of switched circuits, DEMIR,Y. and UCAR, A., No. 2, pp. 253–272.

Modelling of electromagnetic losses in asynchronous machines, DUPRE, L., DE WULF, M.,MAKAVEEV, D., PERMIAKOV, V., PULNIKOV, A. and MELKEBEEK, J. No. 4,pp. 1051–1065.

Modelling of induction heating and consequent hardening of long prismatic bodies, DOLEZEL,I., BARGLIK, J., SAJDAK, C., SKOPEK, M. and ULRYCH, B., No. 1, pp. 79–87.

Multigrid for transient 3D eddy current analysis, WEIß, B. and BIRO, O., No. 3, pp. 779–788.

New approach to the optimisation of three-phase three-wire systems with sinusoidal voltagesources and nonlinear loads, MARIAN, P. and KRZYSZTOF, D., No. 2, pp. 356–371.

Numerical analysis of fast switching electromagnetic valves, ERTL, M., KALTENBACHER,M., MOCK, R. and LERCH, R., No. 3, pp. 715–729.

Numerical magnetic field analysis and signal processing for fault diagnostics of electricalmachines, POYHONEN, S., NEGREA, M., JOVER, P., ARKKIO, A. and HYOTYNIEMI,H., No. 4, pp. 969–981.

Numerical modelling of electromagnetic process in electromechanical systems, KUZNETSOV,V.A. and BROCHET, P., No. 4, pp. 1142–1154.

Numerical simulations of continuous induction heating of magnetic billets and sheets,FIRETEANU, V. and TUDORACHE, T., No. 1, pp. 68–78.

Numerical solutions of a viscous-hydrodynamic model for semiconductors: the supersonic case,BALLESTRA, L. and SALERI, F., No. 2, pp. 205–230.

Optimal design of high frequency induction motors with the aid of finite elements, JORNET, A.,ORILLE, A., PEREZ, A. and PEREZ, D., No. 4, pp. 1115–1125.

Optimal shape design of devices and systems for induction-heating: methodologiesand applications, DI BARBA, P., DUGHIERO, F., LUPI, S. and SAVINI, A., No. 1,pp. 111–122.

Optimisation of magnetic sensors for current reconstruction, FORMISANO, A. andMARTONE, R., No. 3, pp. 535–548.

Optimization of perfectly matched layer for 2D Poisson’s equation with antisymmetrical orsymmetrical boundary conditions, DEDEK, L., DEDKOVA, J. and VALSA, J., No. 3,pp. 520–534.

Probabilistic approach to series compensation design, ALLELLA, F., CHIODO, E. andLAURIA, D., No. 2, pp. 372–387.

Reconstruction of inhomogeneities in fluids by means of capacitance tomography,BRANDSTATTER, B., HOLLER, G. and WATZENIG, D., No. 3, pp. 508–519.

COMPEL22,4

1200

Removal of SiC inclusions in molten aluminium using a 12 T static magnetic field, COLLI, F.,FABBRI, M., NEGRINI, F., ASAI, S. and SASSA, K., No. 1, pp. 58–67.

Robust design of high field magnets through Monte Carlo analysis, CAVALIERE, V., CIOFFI,M., FORMISANO, A. and MARTONE, R., No. 3, pp. 589–602.

Robust target functions in electromagnetic design, ALOTTO, P., MAGELE, C., RENHART, W.,STEINER, G. and WEBER, A., No. 3, pp. 549–560.

Simulation of a mode stirred chamber excited by wires using the TLM method, WEINZIERL,D., RAIZER, A., KOST, A. and DE SALVADOR FERREIRA, G., No. 3, pp. 770–778.

Simulations of bilateral energy markets using MATLAB, YUEN, Y.S.C. and LO, K.L., No. 2,pp. 424–443.

Spatial linearity of an unbalanced magnetic pull in induction motors during eccentric rotormotions, TENHUNEN, A., HOLOPAINEN, T.P. and ARKKIO, A., No. 4, pp. 862–876.

Stability analysis of the doubly fed machine with emphasis on saturation effects,MASMOUDI, A., No. 2, pp. 410–423.

Staged modelling: a methodology for developing real-life electrical systems applied to thetransient behaviour of a permanent magnet servo motor, HENROTTE, F.,PODOLEANU, I. and HAMEYER, K. No. 4, pp. 1066–1076.

Stochastic response surface method and tolerance analysis in microelectronics, ANILE, A.M.,SPINELLA, S. and RINAUDO, S., No. 2, pp. 314–327.

TFH – transverse flux induction heating of non-ferrous and precious metal strips: results of aEU research project, DUGHIERO, F., LUPI, S., MUHLBAUER, A. and NIKANOROV, A.,No. 1, pp. 134–148.

Thermal modeling and testing of a high-speed axial-flux permanent-magnet machine,SAHIN, F. and VANDENPUT, A.J.A., No. 4, pp. 982–997.

Trade-off between information and computability: a technique for automated topologicalcomputations, SUURINIEMI, S. and KETTUNEN, L., No. 3, pp. 481–494.

Transverse flux induction heating: comparison between numerical models and experimentalvalidation, FIRETEANU, V., GERI, A., TUDORACHE, T. and VECA, G.M., No. 1,pp. 98–110.

Turbulent flow dynamics, heat transfer and mass exchange in the melt of induction furnaces,BAAKE, E., NACKE, B., UMBRASHKO, A. and JAKOVICS, A., No. 1, pp. 39–47.

Two-dimensional harmonic balance finite element modelling of electrical machines takingmotion into account, GYSELINCK, J. DULAR, P., VANDEVELDE, L., MELKEBEEK, J.,OLIVEIRA, A.M. and KUO-PENG, P., No. 4, pp. 1021–1036.

Use and analysis of null flux coils, DAVEY, K.R., No. 2, pp. 304–313.

Using of EM fields during industrial CZ and FZ large silicon crystal growth, MUHLBAUER,A., MUIZNIEKS, A., RATNIEKS, G., KRAUZE, A., RAMING, G. and WETZEL, T., No. 1,pp. 123–133.

Vector hysteresis model based on neural network, KUCZMANN, M. and IVANYI, A., No. 3,pp. 730–743.

Vibrations of magnetic origin of switched reluctance motors, VANDEVELDE, L., GYSELINCK,J.J.C., BOKOSE, F. and MELKEBEEK, J.A.A., No. 4, pp. 1009–1020.

[by jan melkebeek, r. belmans] compel - selected (bookos.org)

Documents

eddycurrent computation

modeling methods

thermal modeling

computation andmathematics

contentsthe current

finite element modeling

belgiumguest editorprofessor

magnetic network approach