ontwerp van een energie-efﬁciente¨ design of an energy ...ldupre/phd_hendrik_vansompel.pdf ·...

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Ontwerp van een energie-efficientepermanentemagneetbekrachtigde machine met axiale flux

Design of an Energy Efficient Axial Flux Permanent MagnetMachine

Hendrik Vansompel

Promotoren: prof. dr. ir. Luc Dupre, prof. dr. ir. Peter SergeantProefschrift ingediend tot het behalen van de graad vanDoctor in de Ingenieurswetenschappen: Werktuigkunde-Elektrotechniek

Vakgroep Elektrische Energie, Systemen en AutomatiseringVoorzitter: prof. dr. ir. Jan MelkebeekFaculteit Ingenieurswetenschappen en ArchitectuurAcademiejaar 2012-2013

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ISBN 978-90-8578-616-0NUR 959Wettelijk depot: D/2013/10.500/49

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Design of an Energy Efficient Axial Flux Permanent MagnetMachine

Hendrik Vansompel

Dissertation submitted to obtain the academic degree ofDoctor of Electromechanical Engineering

Publicly defended at Ghent University on 23 August 2013

Supervisors:Prof. dr. ir. Luc Dupre, prof. dr. ir. Peter SergeantElectrical Energy Laboratory (EELAB)Department of Electrical Energy, Systems and Automation (EESA)Faculty of Engineering and ArchitectureGhent UniversitySt.-Pietersnieuwstraat 41B-9000 Ghent, Belgiumhttp://www.eesa.ugent.be

Members of the examining board:Prof. dr. ir. Jan Van Campenhout (chairman) ELIS, Ghent UniversityProf. dr. ir. Hendrik Rogier (secretary) INTEC, Ghent UniversityProf. dr. ir. Luc Dupre (supervisor) EESA, Ghent UniversityProf. dr. ir. Peter Sergeant (supervisor) EESA, Ghent UniversityProf. dr. Elena Lomonova University of Technology EindhovenProf. dr. Malcolm McCulloch University of OxfordProf. dr. ir. Hans Vande Sande University of Antwerp, Atlas Copco Airpower n.v.Prof. dr. ir. Alex Van den Bossche EESA, Ghent University

EN GIN EE RING

AR C H IT E C T U R

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http://www.eesa.ugent.be

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Preface

Comprehensive legislation has been passed in the European Union with the objec-tive to reduce the energy consumption and therefore CO2 emissions. In the Direc-tive 2009/28/EC of 23 April 2009 on renewable energy and Regulation 640/2009/EC of 22 July 2009 with regard toecodesignrequirements for electrical motors, animportant role is drawn towards the development of new electrical motors and gen-erators: energy usage and the efficiency of electrical motors in the industrial envi-ronment, electricity produced from renewable energy sources through hydropowerand wind power in the internal electricity market, and an increasing share of electriccars in production. These legislations require to reach newlevels of efficiency andenergysavings, even under the most demanding conditions. Combining premiumquality materials with advanced technology, the electrical motors and generatorswill be designed to operate highly reliable no matter how demanding the processor application, and to have low life cycleeconomiccosts.

Prior to these legislations, research towards axial flux permanent magnet ma-chines was started at the department of Electrical Energy, Systems and Automationin 2008 by prof. dr. ir. Alex Van den Bossche and prof. dr. ir. Peter Sergeant asa generator in a domestic combined heat and power application. Already at anearly stage, the axial flux permanent magnet machine technology showed superiorproperties such as excellent energy efficiency and high power density. Neverthe-less, more research on this machine was necessary. Therefore, a BOF associationresearch project was requested and granted to prof. dr. ir. Luc Dupre (Ghent Uni-versity) and prof. dr. ir. Peter Sergeant (University College Ghent). This researchproject would focus on the energy efficiency of axial flux permanent magnet ma-chines. In 2010 an international patent on some aspects of the axial flux permanentmagnet machine technology was filed.

I would like to acknowledge the Ghent University Association for theirfinancial support through the BOF association research project 05V00609. Alsospecial thanks to those who where directly involved in this research: prof. dr. ir.Luc Dupre and prof. dr. ir. Peter Sergeant for being my promoters, prof. dr. ir.Alex Van den Bossche for the many splendid ideas, and eng. Damian Kowal forproviding me his preliminary research files.

Hendrik Vansompel, 20 May 2013

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Contents

Preface v

Contents x

Summary xi

Samenvatting xv

List of Abbreviations xix

List of Symbols xxi

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Scientific Publications . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Articles in International SCI Journals . . . . . . . . . . . 51.4.2 Publications in the Proceedings of International Conferences 5

2 Introduction to the Axial Flux PM Machine 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Basic Axial Flux PM Machine . . . . . . . . . . . . . . . 82.1.2 Topologies . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Yokeless And Segmented Armature (YASA) Topology . . . . . .112.2.1 From Torus to YASA . . . . . . . . . . . . . . . . . . . . 122.2.2 Structure Proposal . . . . . . . . . . . . . . . . . . . . . 152.2.3 Modular Stator . . . . . . . . . . . . . . . . . . . . . . . 172.2.4 Air Cooling Possibilities . . . . . . . . . . . . . . . . . . 20

2.3 Test Case Machine . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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viii Contents

3 Modelling the Axial Flux PM Machine 293.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Multislice 2D Modelling . . . . . . . . . . . . . . . . . . . . . . 303.3 Analytical multislice 2D Modelling . . . . . . . . . . . . . . . . 32

3.3.1 Magnetic Vector Potential from a Current Sheet . . . . . .333.3.2 Permanent Magnet Magnetic Field . . . . . . . . . . . . . 373.3.3 Armature Magnetic Field and Current Distribution . . .. 393.3.4 Effect of Stator Slotting . . . . . . . . . . . . . . . . . . 423.3.5 Flux Linkage and Back Electromotive Force . . . . . . . . 473.3.6 Electromagnetic Torque and Electric Power Output . . .. 493.3.7 Electric Resistance and Resistive Heating . . . . . . . . .503.3.8 Stator Core Loss . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . 573.4.1 3D Modelling . . . . . . . . . . . . . . . . . . . . . . . . 573.4.2 Multislice 2D Modelling . . . . . . . . . . . . . . . . . . 573.4.3 Mathematical Modelling of Nonlinear Material . . . . . .613.4.4 Maxwell Stress Harmonic Filter Method . . . . . . . . . . 62

3.5 Comparison of Example Results . . . . . . . . . . . . . . . . . . 653.6 Energy Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6.1 Motor Mode . . . . . . . . . . . . . . . . . . . . . . . . 673.6.2 Generator Mode . . . . . . . . . . . . . . . . . . . . . . 683.6.3 Important Remarks on Energy Efficiency . . . . . . . . . 69

3.7 Parameter Optimisation . . . . . . . . . . . . . . . . . . . . . . . 703.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 Comparison of Non Oriented and Grain Oriented Material 854.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2 Magnetic Flux Density in the Stator Teeth . . . . . . . . . . . . .894.3 Anisotropic Material Model Based on the Magnetic Energy. . . . 91

4.3.1 Anisotropic Material Model . . . . . . . . . . . . . . . . 914.3.2 Magnetic Energy . . . . . . . . . . . . . . . . . . . . . . 934.3.3 Epstein Measurement Restriction . . . . . . . . . . . . . 95

4.4 Core Loss Model (Loss Separation) . . . . . . . . . . . . . . . . 954.4.1 Quasi-Static Loss Component . . . . . . . . . . . . . . . 964.4.2 Classical Dynamic Loss Component . . . . . . . . . . . . 964.4.3 Excess Dynamic Loss Component . . . . . . . . . . . . . 964.4.4 Unidirectional Model . . . . . . . . . . . . . . . . . . . . 97

4.5 Simulations and Comparison . . . . . . . . . . . . . . . . . . . . 974.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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Contents ix

5 Eddy Current Loss in the Permanent Magnets 1095.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.2 Analysis of the Air Gap Magnetic Flux Density . . . . . . . . . .112

5.2.1 Influence of Stator Slotting . . . . . . . . . . . . . . . . . 1125.2.2 Influence of Armature Reaction . . . . . . . . . . . . . . 116

5.3 Multislice 2D - 2D Coupled Modelling . . . . . . . . . . . . . . . 1175.3.1 Multislice 2D Simulations . . . . . . . . . . . . . . . . . 1185.3.2 Air Gap Magnetic Flux Density Reconstruction . . . . . . 1185.3.3 2D Magnet Eddy Current Calculation . . . . . . . . . . . 1205.3.4 Model Verification . . . . . . . . . . . . . . . . . . . . . 125

5.4 Magnet Segmentation . . . . . . . . . . . . . . . . . . . . . . . . 1285.5 Rotor Back Iron Loss . . . . . . . . . . . . . . . . . . . . . . . . 1305.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6 Influence of the Stator Slot Openings 1376.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.2 Stator Core Losses . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.2.1 Permanent Magnets . . . . . . . . . . . . . . . . . . . . . 1386.2.2 Armature Reaction . . . . . . . . . . . . . . . . . . . . . 1406.2.3 Combination . . . . . . . . . . . . . . . . . . . . . . . . 140

6.3 Analysis of the Air Gap Magnetic Flux Density . . . . . . . . . .1446.3.1 Permeance Function . . . . . . . . . . . . . . . . . . . . 1446.3.2 Eddy Current Losses in the Permanent Magnets . . . . . . 1446.3.3 Cogging . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.4 Evaluation and Comparison . . . . . . . . . . . . . . . . . . . . . 1476.4.1 Stator Core Losses . . . . . . . . . . . . . . . . . . . . . 1476.4.2 Eddy Current Losses in the Permanent Magnets . . . . . . 1496.4.3 Electromagnetic Torque and Power . . . . . . . . . . . . 1496.4.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7 Combined Wye-Delta Connection 1557.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1557.2 Combined Wye-Delta Connection . . . . . . . . . . . . . . . . . 1607.3 Simulations and Comparison with Wye-Connection . . . . . .. . 1647.4 Influence on the Machine’s Losses . . . . . . . . . . . . . . . . . 166

7.4.1 Circulating Current in the Delta-Connected Coils . . .. . 1677.4.2 Stator Core Losses . . . . . . . . . . . . . . . . . . . . . 1697.4.3 Magnet Eddy Current Losses . . . . . . . . . . . . . . . . 1707.4.4 Higher Energy Efficiency . . . . . . . . . . . . . . . . . . 172

7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

8 Losses in VSI-PWM fed machines 1758.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.1.1 Field-Oriented Control . . . . . . . . . . . . . . . . . . . 1768.1.2 Pulse Width Modulation . . . . . . . . . . . . . . . . . . 178

8.2 Simulation of the Current Waveform . . . . . . . . . . . . . . . . 1808.2.1 Machine Model . . . . . . . . . . . . . . . . . . . . . . . 1818.2.2 Simulated Current Waveform . . . . . . . . . . . . . . . 183

8.3 Core Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1858.3.1 Internal Loops . . . . . . . . . . . . . . . . . . . . . . . 1858.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . 187

8.4 Eddy Current Loss in the Permanent Magnets . . . . . . . . . . . 1888.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

9 Concluding Remarks 1959.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1959.2 Energy Efficiency Map . . . . . . . . . . . . . . . . . . . . . . . 1979.3 Drive Train Optimisation . . . . . . . . . . . . . . . . . . . . . . 199

9.3.1 Small Wind Turbines . . . . . . . . . . . . . . . . . . . . 1999.3.2 Electric Vehicles . . . . . . . . . . . . . . . . . . . . . . 201

9.4 Recommendations for Future Research . . . . . . . . . . . . . . . 202Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

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Summary

Electrical motors and drive trains powered by electrical machines are major con-sumers of electric power. They are estimated to account for 43%-46% of all globalelectricity consumption as well as for 69% of all electricity used by industry1, andtheir share is even to increase in the near future with the emerging number of elec-tric vehicles in circulation. As a major part of this electric power is still generatedby traditional fossil-fueled power stations, electrical motors are indirect contribu-tors to greenhouse gas emissions. Therefore, comprehensive legislation has beenpassed in the European Union. In the Regulation 640/2009/ECof 22 July 2009 withregard to ecodesign requirements for electrical motors, the focus is on the energyusage and the energy efficiency of electrical motors in industrial environments.

On the other hand, in Directive 2009/28/EC of 23 April 2009 onrenewableenergy, the shift from traditional power generation to electricity produced by re-newable energy sources such as hydropower and wind power is discussed.

Both legislations require to reach new levels of efficiency and energy savingseven under the most demanding conditions. Combining premium quality materialswith advanced technology, the electrical motors and generators will be designed tooperate highly reliable no matter how demanding the processor application, and tohave low life cycle economical costs.

Regarding to the electrical motor technology, one particular electrical machinetopology shows superior advantages: the axial flux permanent magnet machine.Axial flux permanent magnet machines have an excellent flexibility at a variety ofrotational speeds, which makes them perfectly suitable forhigh-speed-low-torqueas well as low-speed-high-torque applications. Moreover,axial flux permanentmagnet machines are very compact; the axial length of the machine is much smallercompared to radial machines, which is very often crucial forbuilt-in applications.The slim and light-weight structure results in a machine with a relatively highpower density. Finally, but very important in the focus of this work, axial fluxpermanent magnet machines have a good energy efficiency.

Although numerous variants on axial flux permanent magnet topologies exist,this work focuses on the topology that is in literature oftenreferred to as the yoke-

1Energy efficiency policy opportunities for electrical motor driven systems, International EnergyAgency, 2011 Working Paper in the Energy Efficiency Series, by Paul Waide and Conrad U. Brunner,OECD/IEA 2011

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xii Summary

less and segmented armature or segmented armature torus. The absence of a statoryoke enforces the compactness and corresponding power density. Simultaneously,the yokeless and segmented armature topology introduces a modular constructionfor the stator: the different stator core elements are manufactured individually, pro-vided with a tooth coil winding, and finally arranged together into a stator. As thewinding process takes place out of the machine, easy windinghandling as well asgood conductor filling factors are obtained, to meet the demands on economicalmanufacturing cost and energy efficiency.

A benefit of axial flux permanent technology in general, and the yokeless andsegmented armature topology in specific, is the good energy efficiency. Never-theless the aim of this work is to perform more elaborated research in order toincrease the energy efficiency. Therefore this work introduces premium qualitymaterials combined with new concepts and advanced technologies. Proper eval-uation of the introduced energy efficiency increasing features requires the devel-opment of appropriate mathematical models. As for the past decades research onelectrical machines was focussing on radial machines only,the mathematical mod-elling of axial flux variants is discussed extensively in this work. A multislice-2Dmodelling technique, in which only 2D mathematical calculations are involved tomodel the inherent 3D structure of the axial flux machine topology, is introduced.Although the majority of the calculations in this work are performed with finiteelement analysis, an analytical model is introduced as it gives a good insight in theruling physics whereas finite element analysis rather behaves like a black box.

As the axial component of the magnetic flux density in most parts of the sta-tor core elements dominates, the non oriented low grade laminated silicon steel inthe stator core elements is replaced by a grain oriented highgrade material. Ad-ditional development of mathematical tools to model the nonlinear anisotropicbehaviour and losses in the stator core material were necessary in order to allowproper comparison of both material grades. The advantageous low losses and ex-cellent permeability of the grain oriented material in the rolling (axial) directionresults in a machine with strongly reduced stator core losses and a slight increaseof the electromagnetic torque. Despite the higher economical cost of high gradematerial, the surcharge can be justified by the significantlybetter energy efficiency.

A neodymium alloy (NdFeB), a type of rare-earth magnetic material, is intro-duced as permanent magnet material because of its excellentenergy density. AsNdFeB has a good electric conductivity and the permanent magnets are surfacemounted, variations in the air gap magnetic flux density result in induced eddycurrents in the permanent magnets. These variations in the air gap magnetic fluxdensity are partly caused by the stator slot openings and partly by the armature re-action. The induction of eddy currents in the permanent magnets should be limitedas they deteriorate the energy efficiency, but even worse, may result in local irre-versible demagnetisation of the permanent magnet materialif temperature exceedsa certain level. To model these eddy currents, a 2D frequencydomain based modelthat takes the skin depth into account was developed. Finally, segmentation of the

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permanent magnets with the purpose to limit the eddy currents, was introduced andevaluated mathematically.

Although an optimisation of a limited set of geometrical parameters was per-formed to find a preliminary machine design, other parameters have a significantimpact on the power losses in the machine as well. One of theseparameters is thewidth of the slot openings near the air gap. This parameter has a major influenceon the power losses in the stator core and the permanent magnets. Moreover, theeffect of a parameter variation has a converse impact on bothpower losses: an in-crease in the slot openings results in a reduction of the power losses in the statorcore elements, but increases the eddy current losses in the permanent magnets andvice versa. Moreover it was observed that the slot openings contributeto an un-equal flux density level over the different laminations in the stator core. Insight inthe impact of the slot openings variation on both power losses is provided, and anillustrative case is discussed in which the impact of the slot openings variation onboth power losses is examined quantitatively.

As the fundamental components of the back electromotive forces of the differ-ent tooth coil windings assigned to one phase in the machine have a slightly dif-ferent phase angle, the fundamental winding factor differssignificantly from unityfor specific slot and pole numbers. As low fundamental winding factors result ina poor electromagnetic torque output given a specified armature current density, anovel combined wye-delta connection is introduced. In thiscombined wye-deltaconnection, the phase shifts between the different tooth coil windings are partiallycompensated by the electric phase between the wye and delta connected coils. Im-plementation of the combined wye-delta connection resultsin a significant increaseof the electromagnetic torque output for the same armature current density. Al-though this combined wye-delta connection increases the machine’s torque output,the high harmonic content including zero sequence components in the back elec-tromotive forces results in a circulating current in the delta connected tooth coilwindings. The advantages of a higher electromagnetic torque by the higher wind-ing factor is compared to the additional losses caused by thecirculating current. Itis observed that the increase in the torque output of the machine clearly surpassesthe additional losses.

In most current high-performance electric powered drive trains, the electricalmachines are fed by variable frequency drives. A variable frequency drive allows tocontrol the torque and speed by variation of the motor input frequency and voltage.To vary the voltage (and frequency), most variable frequency drives use pulse widthmodulation. As an illustrative example in this work, the axial flux PM machine isfed by a voltage source inverter using pulse width modulation in which a fieldoriented control strategy is implemented. Although the carrier frequency in thepulse width modulation is chosen sufficiently high, the on and off times of theswitches that determine the voltage waveform, are still visible in the phase currentas well. Although the ripple in the current waveform remainslimited, the additionalharmonic content in the armature reaction increases the losses in the permanent

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xiv Summary

magnets significantly. Moreover the current ripple influences the magnetic fluxdensity in the stator core and therefore the core power losses. Generally a higherfrequency of the carrier waveform results in lower losses inthe machine.

Finally, two applications are discussed in which the axial flux permanent mag-net machines are very suitable: small wind turbines and electric vehicles.

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Samenvatting

Elektrische motoren en elektrisch aangedreven eenheden vormen een zeer belang-rijke groep van elektriciteitsverbruikers. Hun aandeel wordt ingeschat op 43%-46% van het totale elektriciteitsverbruik, en ze zijn goed voor 69% van het elek-triciteitsverbruik in industriele omgevingen2. Het aandeel van elektriciteit ver-bruikt door elektrische motoren wordt verwacht verder te stijgen door de toenamevan elektrische voertuigen in roulatie. Omdat een groot deel van de huidig gepro-duceerde elektriciteit afkomstig is uit centrales die werken op klassieke fossielebrandstoffen, hebben elektrische motoren een indirecte bijdrage tot de productievan broeikasgassen. Daartoe heeft de Europese Unie enkele uitgebreide wettek-sten opgesteld. In verordening 640/2009/EC van 22 juli 2009met betrekking tot deecologische vereisten voor elektrische motoren, ligt de klemtoon op het energiever-bruik en de energie-efficientie van elektrische motoren inindustriele toepassingen.

Anderzijds wordt in richtlijn 2009/28/EC van 23 april 2009 rond hernieuwbareenergie, de transitie van klassieke centrales naar elektrische energie uit hernieuw-bare bronnen zoals waterkracht en windenergie uiteengezet.

Beide maatregelen vereisen het halen van nieuwe standaarden van energie-efficientie en het besparen van energie, zelfs onder moeilijke werkingsom-standigheden. Door de combinatie van superieure materialen en geavanceerdetechnologieen, kunnen nieuwe elektrische motoren en generatoren ontworpenworden, die ongeacht de eisen van het proces of toepassing, betrouwbaar eneconomisch aanvaardbaar kunnen worden uitgebaat.

Met betrekking tot de beschikbare elektrischemachinetechnologie, wordt indit werk de permanentemagneetbekrachtigde machine met axiale flux voorgesteldvanwege haar positieve eigenschappen. Permanentemagneetbekrachtigde ma-chines met axiale flux hebben een zeer hoge flexibiliteit overeen zeer breedsnelheidsbereik. Dit maakt dat deze machine geschikt is voor zowel toepassingendie een hoog koppel bij lage snelheid als een laag koppel bij hoge snelheid vereisen.Bovendien zijn permanentemagneetbekrachtigde machines met axiale flux bijzon-der compact; de axiale lengte is veel korter in vergelijkingmet radiale machineswat de integreerbaarheid in de diverse toepassingen ten goede komt. Door

2Energy efficiency policy opportunities for electrical motor driven systems, International EnergyAgency, 2011 Working Paper in the Energy Efficiency Series, by Paul Waide and Conrad U. Brunner,OECD/IEA 2011

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xvi Samenvatting

de compacte en lichte bouwvorm hebben de permanentemagneetbekrachtigdemachines met axiale flux een relatief hoge vermogensdichtheid. Tenslotte, maarzeer belangrijk binnen dit werk, is dat permanentemagneetbekrachtigde machinesmet axiale flux zeer energie-efficient zijn.

Ondanks de diversiteit aan permanentemagneetbekrachtigde machinetypes metaxiale flux, richt dit werk zich tot het machinetype dat in hetvakjargon wordtaangeduid als de jukloze variant met gesegmenteerde statorof torusvariant metgesegmenteerde stator. De afwezigheid van een statorjuk versterkt de compactheidvan dit type en geeft aanleiding tot een nog hogere vermogensdichtheid. Tege-lijkertijd wordt er in de jukloze variant met gesegmenteerde stator een modulaireconstructiemogelijkheid geıntroduceerd: de verschillende statorelementen wordenindividueel vervaardigd, waarna ze van een wikkeling worden voorzien en uitein-delijk de stator kan worden opgebouwd uit de verschillende voorbereide elementen.De bewikkeling van de statorelementen kan hierbij buiten demachine uitgevoerdworden, wat het wikkelingsproces aanzienlijk vergemakkelijkt en bovendien re-sulteert in een hogere vullingsgraad van de statorgleuf metgeleidend materiaal.Hierbij wordt de productiekost gereduceerd en de energie-efficientie verbeterd.

Niettegenstaande de permanentemagneetbekrachtigde machine met axiale fluxin het algemeen, en de jukloze variant met gesegmenteerde stator in het bijzonder,een goede energie-efficientie hebben, is het de bedoeling in dit werk de energie-efficientie van dit machinetype verder te onderzoeken en, waar mogelijk, te ver-beteren. Hiertoe worden in dit werk hoogwaardige materialen ingevoerd en gecom-bineerd met nieuwe concepten en geavanceerde technologie¨en. Het evalueren vande impact van deze energieverbeterende maatregelen vereist de ontwikkeling vande geschikte wiskundige modellen. Omdat tijdens de voorbije decennia de klem-toon in het ontwerp van elektrische machines gericht was op radiale machines,wordt er uitgebreid aandacht besteed aan de wiskundige modellering van de axialevarianten. Dit omvat het invoeren van een meerlaags tweedimensionale model-leringstechniek waarbij de inherente driedimensionale geometrie van de axiale-flux machines wordt benaderd door tweedimensionale rekenvlakken die op ver-schillende diameters in de machine worden gedefinieerd. Hoewel in dit werkvoornamelijk berekeningen worden uitgevoerd met behulp van eindige elementen,wordt er toch aandacht besteed aan de opbouw van een analytisch model. Eenanalytisch model geeft beter inzicht in het achterliggendefysische mechanisme,waar een commercieel beschikbaar eindigeelementenpakketenkel een numeriekeoplossing aanreikt.

Omdat in het overgrote deel van de statorelementen de axialecomponent vande magnetische fluxdichtheid overheerst, wordt het niet-georienteerd gelamelleerdsiliciumstaal van middelgoede kwaliteit in de statorelementen vervangen door eengeorienteerde gelamelleerde variant van uitstekende kwaliteit. Om beide materi-alen correct te kunnen vergelijken, diende voor het georienteerde materiaal eenpassend wiskundig model opgesteld te worden dat rekening houdt met de niet-lineaire anisotrope eigenschappen. De gunstige lage verliezen en de hoge per-

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xvii

meabiliteit van het georienteerd materiaal in de (axiale)voorkeursrichting gevenaanleiding tot een sterke reductie van het energieverlies in de statorelementen eneen kleine toename in de koppelproductie. Ondanks de hogerekostprijs van hetgeorienteerd materiaal, kan de meerkost verantwoord worden door de merkbaarbetere energie-efficientie van de machine.

De permanente magneten worden vervaardigd uit een neodymiumlegering(NdFeB), een type zeldzaamaardemagnetischmateriaal, vanwege de bijzonderhoge haalbare energiedichtheid. Omdat NdFeB een goede elektrische geleideris en de permanente magneten aan het luchtspleetoppervlakte grenzen, zullenvariaties in de magnetische fluxdichtheid aanleiding geventot wervelstromenin de permanente magneten. Deze variaties van de magnetische fluxdichtheidin de luchtspleet zijn enerzijds toe te schrijven aan de statorgleufopeningen, enanderzijds aan het effect van aanwezige statorstromen. Wervelstromen in depermanente magneten moeten vermeden worden omdat zij de energie-efficientiedoen afnemen, maar nog belangrijker, kunnen resulteren in een onomkeerbaredemagnetisatie van de permanente magneten wanneer deze op te hoge temperatuurkomen. Om de wervelstromen in de permanente magneten te berekenen, wordtgebruik gemaakt van een tweedimensionaal model in het frequentiedomein datrekening houdt met de indringdiepte van de magnetische flux in de permanentemagneet. Om de wervelstromen in de permanente magneten te beperken, wor-den deze gesegmenteerd. De effectiviteit van deze segmentatie wordt hierbijkwantitatief berekend.

Optimalisatie van een beperkt aantal parameters die de geometrie van depermanentemagneetbekrachtigde machine met axiale flux bepalen, leidde toteen eerste machineontwerp. Toch zijn er nog geometrische parameters die eenbelangrijke invloed hebben op de energieverliezen in de machine. Zo is er debreedte van de statorgleufopeningen, die in sterke mate de energieverliezen inde statorelementen en de permanente magneten beınvloedt.Bovendien heeft eenvariatie van deze parameter tegengestelde invloeden op beide energieverliezen:een toename van de statorgleufopeningen geeft aanleiding tot een afname vanhet energieverlies in de statorelementen, maar verhoogt het energieverlies inde permanente magneten en omgekeerd. Ook belangrijk is de sterk merkbareinvloed van de statorgleufopeningen op de ongelijke verdeling van de magnetischeflux over de verschillende lamellen. Inzicht in de impact vande variatie van destatorgleufopeningen op beide energieverliezen wordt gegeven, en de resultatenvan een kwantitatieve studie worden besproken.

Omdat er tussen de fundamentele componenten van de elektromotorischekrachten van de verschillende windingen die tot een fase gecombineerd wor-den beduidende faseverschillen zijn, ligt de fundamentelewindingsfactor voorheel wat combinaties van statorgleufaantallen en pooltallen gevoelig lager daneen. Omdat een lage fundamentele windingsfactor resulteert in een verlagingvan de koppelproductie van de machine voor een gegeven stroomdichtheidin de wikkelingen, wordt een nieuw concept ingevoerd: de gecombineerde

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xviii Samenvatting

sterdriehoekschakeling. In deze gecombineerde sterdriehoekschakeling wordende verschillende windingen al naar gelang van hun fasehoek toegewezen totde in stergeschakelde reeks wikkelingen of de in driehoekgeschakelde reekswikkelingen. Door de natuurlijke fasehoek tussen de ster- en deltageschakeldewikkelingen, wordt een hogere fundamentele windingsfactor, en dus een hogerekoppelproductie, behaald. Ondanks de verhoging van de koppelproductie, geeftde sterke harmonische inhoud, waaronder homopolaire componenten, in deelektromotorische krachten van de in driehoekgeschakeldereeks wikkelingenaanleiding tot circulatiestromen. Een kwantitatieve studie die de verhoging van hetgeproduceerde koppel afweegt met de extra energieverliezen die voortkomen uitde circulatiestroom in de in driehoekgeschakelde reeks wikkelingen geeft aan datvoordelen van de gecombineerde sterdriehoekschakeling duidelijk de bovenhandhalen.

In de meeste huidige hoogwaardige elektrische aandrijvingen wordt gebruikgemaakt van een variabelefrequentievoeding. Een variabelefrequentievoedinglaat toe het koppel en de snelheid van de machine te regelen door de amplitudeen frequentie van de klemspanning aan te passen. Om een variabele span-ning te verkrijgen, maken de meeste variabelefrequentievoedingen gebruik vanpulsbreedtemodulatie. Als voorbeeld wordt in dit werk de permanentemag-neetbekrachtigde machine met axiale flux gevoed via een spanningsgestuurdeinverter die gebruik maakt van pulsbreedtemodulatie en waarin een veldorientatie-algoritme is geımplementeerd. Hoewel de frequentie van dedraaggolf voldoendehoog gekozen wordt, zijn de aan- en uit-tijden van de vermogenelektronischeschakelaars die zorgen voor de gepaste golfvorm van de spanning, ook zichtbaarin de golfvorm van de fasestromen. Hoewel de stroomrimpel eerder beperktblijft, geeft deze toch aanleiding tot een niet verwaarloosbare toename van dewervelstroomverliezen in de permanente magneten. Ook het tijdsverloop van demagnetische fluxdichtheid in de statorelementen wordt beınvloed, wat onmiddel-lijk resulteert in een toename van het energieverlies in de statorelementen. Variatievan de frequentie van de draaggolf geeft aan dat de bijkomende verliezen in demachine afnemen met toenemende frequentie van de draaggolf.

Tenslotte worden twee toepassingen besproken waarvoor de permanentemag-neetbekrachtigde machines met axiale flux uitermate geschikt zijn: kleine windtur-bines en elektrische voertuigen.

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List of Abbreviations

AF axial fluxAMM amorphous magnetic materialHFT Maxwell stress harmonic filterGO grain orientedIGBT insulated-gate bipolar transistorNdFeB Neodymium Iron BoronNN north northNO non orientedNS north southPM permanent magnetPMSM permanent magnet synchronous machinePWM pulse width modulationRD rolling directionRRF rotor reference frameSAT segmented armature torusSMC soft magnetic compositeSM2C soft magnetic mouldable compositeSRF stator reference frameTD transverse directionVSI voltage source inverterYASA yokeless and segmented armature2D two dimensional space3D three dimensional space

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xx List of Abbreviations

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List of Symbols

Torque, Speed, Power and Energy Efficiency

η energy efficiencyf frequency [Hz]HFT property calculated by Maxwell stress harmonic filter methodkc temperature coefficient electric conductivitykFe iron loss coefficient(s)kR ac/dc-resistance ratioΩ rotational speed (mechanical) [rad/s]ω rotational speed (electric) [rad/s]MAX property calculated by Maxwell stress tensorP active power [W]Pc copper losses in the conductors [W]Pcl classical loss component [W]Pm mechanical power [W]Pe electric power [W]Pexc excess loss component [W]PFe iron losses [W]Pfr friction losses [W]Phy hysteresis loss component [W]PPM eddy current losses in the permanent magnets [W]Q reactive power [VAr]Rac ac-resistance [Ω]Rdc dc-resistance [Ω]Te electromagnetic torque [Nm]θ angular rotor position [rad]

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xxii List of Symbols

Electromagnetic Modelling

A,Ar, Az magnetic vector potential [Vs/m]B,B magnetic flux density [T]BPM remanent magnetisation of the permanent magnet material [T]c property of conductor material / tooth coil windingδ skin depth [m]∇ del operator∆ Laplace operatorE,E electric field [V/m]e(x) unit vectorFe stator core material propertyH,H magnetic field [A/m]Hexc excess field [A/m]J, J current density [A/m2]k number of periods/layersK current sheetλ permeance functionn number of computation planesM ,M magnetisation [A/m]µ0 vacuum permeabilityµ(r) relative permeability or harmonic numberν harmonic number (Np pole representation)ph phase propertyPM,PM property of permanent magnet (material)ϕ azimuthal coordinateΨ gauge factorr referred in the rotor reference framer radial coordinatercp radius of computation plane [m]ρ electric resistivity [Ωm]S cross section area [m2]s referred in the stator reference frameσ electric conductivity [Sm−1]t time [s]tcp computation plane thickness [m]Wem magnetic energy [J]

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Flux Linkage, Voltage and Current

E(x) electromotive force corresponding to parameterx [V]I current [A]kC Carter’s factorNc number of windings placed on one core elementl number of phasesm number of tooth coilsψ flux linkage [Vs]ξ winding factorξp pitch factorξd distribution factor

Geometrical Parameters

αPM permanent magnet spanb0 width of the stator slot openings [m]bs slot width [m]Di inner diameter [m]Do outer diameter [m]g air gap thickness [m]hPM dimension of the permanent magnets in the magnetisation direction [m]lc electric conductor length [m]p number of pole pairsQs number of stator slotsτp pole pitch [m]τs slot pitch [m]

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xxiv List of Symbols

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Chapter 1

Introduction

1.1 Motivation

The current development in rotating electrical machine technology is dedicated tothe three E’s:ecodesign, energyandeconomy.

Within the focus onecodesign, an important role is drawn towards the devel-opment of new electrical motors and generators. Energy usage and the energy ef-ficiency of electrical motors in the industrial environmentare still important, how-ever, the development and integration of electrical motorsin electrical vehicles andthe renewable energy conversion had a significant increase in the past decade andis still a hot topic in ongoing research. New applications pose new challenges; inelectric vehicles there is a need for electrical motors witha high power densitybecause of the space and mass issues, while very high torquesare imposed to newdirect-drive wind energy systems.

Irrespective of the application, there is a need ofenergyefficient exploitation ofthe electrical machines. Recent legislation, the Directive 2009/28/EC of 23 April2009 and Regulation 640/2009/EC of 22 July 2009, requires electrical machinesto have a minimum energy efficiency as a function of the rated output power andnumber of poles.1

Whereasecodesignandenergyefficiency are likely to increase the price of theelectrical machines, a low life cycleeconomiccost is still indispensable.

Regarding these constraints, new adapted machine topologies (outer runnerradial flux, axial flux, transverse flux,...) have been introduced as better performingsurrogates for the classical dc and asynchronous machines.In most new machine

1In IEC 60034, an international standard of the International Electrotechnical Commission forrotating electrical machinery, the energy efficiency classes for single-speed, three-phase, cage-induction motors with 2, 4 or 6 poles are specified. It classifies three classes: IE1 (standard), IE2(high) and IE3 (premium). For each class the energy efficiency is defined for a rated output rangefrom 0.75 to 375 kW. In the European Community the IE2 class ismandatory for all new motorssince 16 June 2011. The IE3 class will be mandatory from 1 January 2015 (7.5-375 kW) and 1January 2017 (0.75-375 kW).

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2 Introduction

topologies, permanent magnets with a high energy density such as neodymiummagnets (NdFeB) are used to generate the magnetic field. For these machines, themagnetic field moves simultaneously with the rotor, which isindicated by the termsynchronous machine.

This research focusses on axial flux permanent magnet machines. This topol-ogy is particularly suitable for electric vehicles and small/medium power rangewind energy systems. The pancake shape,i.e. the high ratio of the diameter to ax-ial length, allows the integration of the machine in the wheels of electric vehiclesand in the nacelle of wind turbines. As the pole number is chosen sufficiently high,a high torque density combined with a good performance at lowspeeds is obtained.The efficient exploitation of the machine, even at very low speeds, permits directcoupling of the machine to the low speed application withoutusing a gearbox. Theresult can be a direct drive wind energy generator of which the maintenance andreliability can be improved significantly due to the absenceof a gearbox.

Among the axial flux PM machines, different topologies exist. In this re-search, the yokeless and segmented armature (YASA), sometimes also called thesegmented armature torus (SAT), is considered. The yokeless and segmented ar-mature topology combines the use of a double layer fractional slot winding, oftencalled tooth coil winding, with a modular construction of the stator. The fractionalslot winding is generally used in machines with a high power density, while themodular construction of the stator facilitates the construction of the machine.

Within this research on axial flux permanent magnet machines, the focus ison energy efficiency. Therefore, premium quality materialsare combined withadvanced technologies to decrease the power losses in the machine.

1.2 Objectives

To distinguish this research among other scientific contributions on axial flux PMmachines, the research topics were properly defined and the objectives were set:

• First of all, this research is limited to the electromagnetic design of axialflux permanent magnet machines. Thermal, mechanical and manufacturingissues are only mentioned briefly. The electromagnetic design includes theintroduction of the appropriate mathematical tools to model the axial fluxPM machine, which is quite challenging as these axial flux topologies havean inherent 3D geometry.

• Secondly, and most important within the scope of this research, is the intro-duction of different measures to decrease the power losses in the machine.Therefore, accurate mathematical models for the differentpower loss phe-nomena need to be introduced. Once these models are introduced, a com-parison is made between the original machine, and the machine that has thepower loss saving feature. This feature can be a premium grade material, a

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1.3 Outline 3

modification to the geometry of the machine, or the introduction of a newconcept.

• In all of the research topics, the aim was to introduce theories and developmathematical models that are valid for a wide range of axial flux PM ma-chines, rather than focussing on one specific application. Nevertheless, theintroduced ideas are illustrated for a predefined test case,having a specifiedparameter set. The comparison of the machine with and without the powerloss saving feature, is in the first instance a qualitative rather than a quantita-tive approach.

• The aim of this work is not to present the most energy efficient axial flux PMmachine at the end of this work for one particular application, nor to developa (commercial) prototype, but to develop mathematical tools and knowledgethat can be used in future electromagnetic machine designs.A preliminaryresearch prototype was developed during this research and anumber of ex-periments were carried out. Nevertheless these findings arenot included inthis thesis. For the experimental work, the interested reader is addressed tothe five relevant journal papers and three conference contributions listed atthe end of this chapter.

It should be noticed that many of the introduced models and features are notonly applicable for axial flux PM machines, but can also be used, directly or byminor adaptations, in other electrical machines such as radial flux PM machinesand axial flux switched reluctance machines.

1.3 Outline

The structure of this thesis comprises 3 major parts: introduction to the axial fluxPM machine, modelling the axial flux PM machine and the introduction of lossdecreasing measures.

As the axial flux PM machine topology is still quite unknown, an elaboratedoverview on axial flux PM machine technology is presented in Chapter 2. Here, thedifferent topologies of axial flux PM machines are introduced. Consequently, oneof the topologies is selected: the yokeless and segmented armature (YASA) topol-ogy. The benefits of this topology are highlighted, and a proposal for mechanicalconstruction and cooling is introduced, however, not described in detail.

The second major part comprises the mathematical modellingof the axial fluxPM machine and is discussed in Chapter 3. Here, the focus is onthe modellingof the inherent 3D geometry of the axial flux PM machine. A multislice 2D mod-elling technique is introduced as an efficient alternative for full 3D modelling. Thismultislice 2D technique is used for both analytical and finite element modelling.The basic models developed in this part will be used and extended in further partsof the work.

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4 Introduction

The third part is dedicated to the introduction of differentmeasures to decreasethe power losses. As this part distinguishes this work from other scientific contri-butions, it comprises several chapters. In each of the chapters, one specific powerloss mechanism is mathematically modelled, a loss decreasing measure is intro-duced and the effect on the power loss is compared with the original machine. Thisway allows to evaluate the performance of each loss decreasing measure individu-ally.

The first measure is introduced in Chapter 4. Here, a grain oriented materialis used in the stator cores instead of a non oriented material. To model the grain-oriented material, a nonlinear anisotropic material modeland core loss model basedon loss separation are developed. Later in this chapter, twomachines with the samegeometry of which one uses non oriented and the other grain oriented material inthe stator core elements are compared, showing a significantly better performanceof the case that uses the grain oriented material.

In Chapter 5, eddy currents in the permanent magnets are calculated. As theNdFeB-magnets have a good electric conductivity, eddy currents are induced inthe permanent magnets as the air gap magnetic flux density varies in time. Thebasic model of Chapter 3 is extended, in order to allow propermodelling of theinduced eddy currents. To decrease the losses in the permanent magnets, magnetsegmentation is introduced and evaluated.

Although the influence of a limited set of geometrical parameters on the en-ergy efficiency of the machine was examined in Chapter 3, someother geometricalparameters do have a major influence on the power losses as well. One of theseparameters, the width of the stator slot openings, is considered in Chapter 6. Thestator slot openings are found to have a contrary effect on the power losses in thestator cores and permanent magnets; wide slot openings lower the core losses butincrease the eddy current losses in the permanent magnets. Aquantitative analysisfor the predefined test case machine is performed.

The introduction of a new winding connection is the topic of Chapter 7. Thecommon wye connection of the different tooth coils is replaced in this chapter bya combined wye-delta connection in order to increase the fundamental windingfactor. This combined wye-delta connection results in an increase of the electro-magnetic torque output, while maintaining the same currentdensity in the windingsas for the common wye-connected tooth coils.

Chapter 8 focusses on the power losses introduced by pulse-width modulation(PWM). The appropriate mathematical models are introducedto model a voltagesource inverter powered machine. The influence of the high frequency pulse-widthmodulation on the stator core loss and the induced eddy currents in the permanentmagnets is studied.

Chapter 9 concludes this works and makes some proposals for future researchin the domain of axial flux (PM) machines with the focus on their applications.

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1.4 Scientific Publications 5

1.4 Scientific Publications

1.4.1 Articles in International SCI Journals

An overview of journal papers published in peer-reviewed international journals atthe time of publication of this thesis:

• H. Vansompel, P. Sergeant, and L. Dupre, ”Optimized design consideringthe mass influence of an axial flux permanent-magnet synchronous generatorwith concentrated pole windings,”Magnetics, IEEE Transactions on, vol. 46,no. 12, pp. 4101–4107, 2010

• H. Vansompel, P. Sergeant, L. Dupre, and A. Van den Bossche, ”Evalua-tion of a Simple Lamination Stacking Method for the Teeth of an Axial FluxPermanent-Magnet Synchronous Machine With Concentrated Stator Wind-ings,” Magnetics, IEEE Transactions on, vol. 48, no. 2, pp. 999–1002, 2012

• H. Vansompel, P. Sergeant, L. Dupre, and A. Van den Bossche, ”A Com-bined Wye-Delta Connection to Increase the Performance of Axial-Flux PMMachines With Concentrated Windings,”Energy Conversion, IEEE Trans-actions on, vol. 27, no. 2, pp. 403–410, 2012

• H. Vansompel, P. Sergeant, and L. Dupre, ”A Multilayer 2-D–2-D CoupledModel for Eddy Current Calculation in the Rotor of an Axial-Flux PM Ma-chine,” Energy Conversion, IEEE Transactions on, vol. 27, no. 3, pp. 784–791, 2012

• H. Vansompel, P. Sergeant, L. Dupre, and A. Van den Bossche, ”Axial FluxPM Machines with a Variable Airgap,”Industrial Electronics, IEEE Trans-actions on, in press

1.4.2 Publications in the Proceedings of International Conferences

An overview of most important conference papers:

• H. Vansompel, P. Sergeant, and L. Dupre, ”Efficiency optimization of anaxial flux permanent-magnet synchronous generator with concentrated polewindings,”Electromagnetic Field Computation (CEFC), 2010 14th BiennialIEEE Conference on, pp. 1–1, IEEE, 2010

• H. Vansompel, P. Sergeant, L. Dupre, and A. Van den Bossche, ”Improv-ing the torque output in radial-and axial-flux permanent-magnet synchronousmachines with concentrated windings by using a combined Wye-Delta con-nection,”Electric Machines & Drives Conference (IEMDC), 2011 IEEE In-ternational, pp. 936–941, IEEE, 2011

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6 Introduction

• H. Vansompel, P. Sergeant, and L. Dupre, ”Effect of segmentation on eddy-current loss in permanent-magnets of axial-flux PM machinesusing a multi-layer-2D–2D coupled model,”Electrical Machines (ICEM), 2010 XIX Inter-national Conference on, pp. 1–6, IEEE, 2012

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Chapter 2

Introduction to the Axial FluxPM Machine

2.1 Introduction

Many recent scientific publications suggest the use of an axial flux PM machine in(in-wheel) electric vehicle applications [1–4] and directdrive small/medium rangedwind energy systems [5–10], while other authors investigated the difference in per-formance between radial flux and axial flux PM machines based on different cri-teria [11–17], but generally axial flux machines are selected because of their mainadvantages [18]:

• Axial flux PM machines have an excellent flexibility at a variety of rota-tional speeds. Good performance at different rotational speeds is obtained bychanging the number of magnets on the rotor discs and varyingthe diameterof the machine; a large diameter with high number of permanent magnets isperfectly suitable for low-speed-high-torque applications [19] whereas high-speed-low-torque [20] applications require fewer poles and smaller diame-ters. This makes the axial flux PM machine perfectly suitablefor electricvehicle propulsion and direct-drive wind energy systems (low-speed-high-torque), and pump and fan purposes (high-speed-low-torque).

• Axial flux PM machines are very compact. The axial length of the machineis much smaller compared to radial machines, which is very often crucial inbuilt-in applications. The slim and light-weight structure results in a machinewith a relatively high power density. Applications in whicha high powerdensity is wanted are found in the electric vehicle applications, where massand space are important issues.

• Axial flux PM machines have a good energy efficiency. As the magneticfield is generated by the permanent magnets, no field excitation current isnecessary and the corresponding copper losses are absent.

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8 Introduction to the Axial Flux PM Machine

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Figure 2.1: Basic axial flux PM machine comprising one stator (1) and onerotor (2) on which the permanent magnets (3) are mounted.

In [21], the superiority of axial flux with respect to radial flux PM machines ontorque density and energy efficiency is claimed.

The interest in axial flux PM technology is not only found in scientific com-munities, but successful experimental prototypes have also resulted in the setup ofbusinesses providing axial flux PM technology commerciallysuch as Axco-motors(Finland) and YASA Motors Ltd (UK). These relatively small innovative venturesare characterised by their application specific custom designed machines.

2.1.1 Basic Axial Flux PM Machine

The most basic version of an axial flux machine is schematically shown in Fig.2.1. This machine comprises only one stator and one rotor on which the permanentmagnets are placed. In practice, the rotor with the permanent magnets is placedas closely as possible to the stator. The thin zone of air thatseparates the rotorfrom the stator is called the air gap. The air gap thickness ispreferably as small aspossible and is mostly limited by attainable mechanical tolerances.

As the permanent magnets are magnetised in the axial direction, i.e. the direc-tion perpendicular to the air gap plane, a magnetic flux is excited in the differentteeth of the stator. As adjacent permanent magnets are alternately magnetized, themagnetic fluxes in adjacent teeth are magnetized differently. Nevertheless, in theidealised axial flux machine, the net flux crossing the air gapis zero, which meansthat the flux in the stator and the rotor have to return internally. This return path iscarried by a stator yoke and the rotor itself. In the schematic representation of thebasic axial flux PM machine, the stator back iron and the different stator teeth are

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2.1 Introduction 9

a single entity.The operation principle in generation mode of an axial flux PMmachine is

similar to the one of radial flux PM machines; as an external force rotates the rotordisc with respect to the stator, a time varying magnetic flux is generated in thestator teeth. As a proper winding is fitted into the stator slots, this time variationin the magnetic flux induces a back electromotive force (voltage) in the differentstator windings. If this winding is connected to a load, power is transferred fromthe external force imposing the rotation to electric power dissipated in the load. Inmotor working, an external imposed current results in a magnetic field that interactswith the one produced by the permanent magnets and results ina rotation of therotor. In this case electric power is transferred into mechanical power, available atthe shaft of the machine.

As the magnetic flux in the rotor iron has mainly a constant value, i.e. a fluxproduced by the permanent magnets, the rotor can be made out of solid construc-tion steel as the risk for induced eddy currents is limited. In the stator, where atime-varying magnetic flux is present, the induction of eddycurrents is reduced byusing (isolated) laminated silicon steel or soft magnetic composite material (SMC)rather than solid construction steel. Care should be taken that the mechanical prop-erties of the materials and the construction of the stator and rotor are suitable towithstand the very high attractive forces between the stator and the rotor. Elasticdeformation of the stator and rotor parts should be considered when choosing theair gap thickness. The net force acting between stator and rotor is also crucial withrespect to the choice of the proper bearings. These bearingsshould be able to carryradial as well as axial forces.

2.1.2 Topologies

Next to the most basic axial flux PM machine with one stator andone rotor, manyother topologies [22] are found.

First of all, there is the double stator single rotor topology presented in Fig. 2.2.Here, the rotor is placed between the two stators. Eventually, the ferromagnetic partof the rotor can be omitted.

Another variant is the single stator double rotor (Fig. 2.3), in which the statoris placed between the two rotor discs. Two main classes in thestator constructioncan be distinguished: slotted windings and coreless stator[23–25]. In case ofslotted windings, the stator is made of high permeable material (laminated siliconsteel, soft magnetic composite, amorphous iron) and the winding is placed in thestator slots. The main disadvantage of using a slotted winding is the so calledcogging torque. This cogging torque is caused by the stator slot openings nearthe air gap. These slot openings result in a rotor position dependent torque asa result of the interaction (change in permeance) of the permanent magnets andthe stator slots. The absence or at least minimisation of these cogging torquesis crucial in distinct applications. For example, the cogging torque can prevent

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Figure 2.2: Schematic presentation of the double stator single rotor axialflux PM machine topology. Stator (1), rotor (2), permanentmagnet (3).

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Figure 2.3: Schematic presentation of the single stator double rotor axialflux PM machine topology. Stator (1), rotor (2), permanentmagnet (3).

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Figure 2.4: Schematic presentation of the NS axial flux PM machine topol-ogy. Stator (1), rotor (2), permanent magnet (3), winding (4).

generators in wind turbines to start up at low wind speeds andtherefore decreasethe annual energy yield of the wind turbine. In these applications, coreless (aircored) windings are introduced. In these axial flux PM machines the winding ismostly embedded in an epoxy resin having unit permeability.Despite the absenceof cogging, the lower mass of the machine and the absence of core losses, the highequivalent air gap thickness requires relatively high permanent magnet volumes toobtain sufficiently high magnetic fluxes in the windings. Thepermanent magnets(NdFeB), very often contain rare earth materials such as neodymium, are costlywith respect to silicon steel and copper. Therefore, the reduction of cogging torquein axial flux PM machines with slotted windings is a topic of much recent scientificresearch.

2.2 Yokeless And Segmented Armature (YASA) Topology

Within the scope on the axial flux PM machine topology with onestator and tworotors, special attention is drawn towards the slotted torus machines. In [26] thenorth south (NS) torus (Fig. 2.4) and north north (NN) torus (Fig. 2.5) were in-dicated to be the best performing axial flux PM machine topologies. In order toexplain the torus machines, the linear representation of anaxial flux PM machine(cfr. section 3.2) is introduced in Fig. 2.6. In the NN torus topology, the perma-nent magnets on both rotors are magnetised in the opposite direction, resulting ina magnetic flux in the stator yoke (axial symmetry). In the NS torus machine, no

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Figure 2.5: Schematic presentation of the NN axial flux PM machine topol-ogy. Stator (1), rotor (2), permanent magnet (3), winding (4).

significant magnetic fluxes are present in the stator yoke as the permanent magnetson both rotors are magnetised in equal direction (axial antisymmetry).

Despite the similarity of both machine topologies, the shorter stator yoke in theNS torus topology results in a increasing power density and lower stator core losscompared to the NN torus topology. In contrast, the winding arrangement in thestator slots is more complicated for the NS torus topology than for the NN torustopology, which generally results in a reduced filling factor of the conductors inthe stator slots. In the NN torus topology, the winding arrangement is particularlyeasy as the winding is wound toroidally around the stator yoke (Fig. 2.5). Goodfilling factors of the conductor in the stator slots are obtained, and moreover, theend winding length is significantly reduced. Generally, a round conductor sectionis used in the NS torus, whereas in the NN torus a rectangular conductor [1] can beused, which has a positive impact on the conductor filling factor.

In [1], a slightly higher power density and peak energy efficiency were foundby the NS torus topology with respect to the NN torus topology.

2.2.1 From Torus to YASA

The NS torus topology benefits from the short stator yoke, whereas the easy wind-ing arrangement is a big advantage of the NN torus machine topology. A combi-nation of the advantages of both machine topologies was introduced in [1], wherethe yokeless and segmented armature (Fig. 2.7) was derived.Starting from the

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Figure 2.6: Planar versions of the NS and NN torus machine topologies. The paths of the magnetic flux are indicated.

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Figure 2.7: The yokeless and segmented armature axial flux machine topol-ogy. Stator consisting of multiple stator core elements (1)around which a tooth coil winding (4) is wound, rotor (2), per-manent magnet (3).

original NS torus topology, the following manipulations are performed:

• The stator yoke is removed entirely. This can be done since the stator yoke inthe NS torus topology has no magnetic function; magnetic symmetry in themachine does not require a closing path for the magnetic flux.By removingthe stator yoke, the magnetic as well as the mechanical link between thedifferent stator teeth vanishes. This results in the existence of individuallysegmented armature elements.

• The number of slots per pole and per phase is reduced in orderthat the slotpitch approximates the pole pitch. This can be obtained by using a concen-trated (fractional pitch) winding rather than a distributed winding.

• By introducing a double layer concentrated fractional pitch winding, thewinding arrangement complexity is reduced significantly. Around each indi-vidually segmented armature element a winding is wound (Fig. 2.7), whichresults in an advanced modular construction. (cfr. section 2.2.3) As the eachwinding is wound around one stator core element this double layer concen-trated fractional pitch winding is very often called a toothcoil winding. 1

1Tooth coil windings can also be found in single layer concentrated fractional pitch windings.Here the stator is assembled by a sequence of cores with and without a tooth coil winding.

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The absence of a stator yoke has an advantageous influence on the power densityand the stator core loss, whereas the winding arrangement iseasy, enables shortend windings, and a good filling factor of the conductor in theslots is obtained byusing wires with a rectangular cross section. Hence, the yokeless and segmentedarmature torus machine topology has an overall better performance: a superiorpower density and an excellent energy efficiency.

Notwithstanding the superiority of the introduced yokeless and segmented ar-mature torus machine topology, some aspects in the machine design can be furtherimproved. The improvements to YASA-machine topology, introduced in this workare mainly addressed to a further improvement of the energy efficiency. Therefore,first, accurate modelling of the electromagnetic behaviourof the machine needs tobe introduced. Mechanical construction, thermal behaviour and production issuesare less elaborated in this work. Nevertheless, some concepts that may result in aglobal feasible machine design are introduced in the next sections of this chapter. Itshould be stressed once again that none of these features areoptimised, nor (exten-sively) tested experimentally, and are presented as concepts that may be consideredin future global designs of YASA-machines.

2.2.2 Structure Proposal

In previous figures 2.4-2.7, only the electromagnetic active components,i.e. ro-tor, stator, winding and permanent magnets are shown. Notwithstanding, a globalmachine design comprises more than just an electromagneticdesign. The differentparts need to be assembled solidly to withstand the high attractive forces betweenthe stator and the rotor, and integration of the machine in the application of in-terest. Furthermore, even if in the electromagnetic designof the machine specialattention is drawn towards energy efficiency, there is stilla need to remove the heatproduced by the different power losses out of the machine. Therefore, a coolingstrategy needs to be introduced.

A prototype machine concept is introduced in Fig. 2.8. Here,a cross section isshown to give an overview of the different parts in the machine. The machine hasthe YASA-construction and cooling is performed by internalair cooling.

Two major parts can be distinguished. First, there is the yokeless and seg-mented armature structure in the stator which is characterised by its modular con-cept. The latter is discussed in detail in section 2.2.3. Second, there is the rotordesign which adds the electromagnetic active partsi.e. rotor and permanent mag-nets to a forced cooling system. This cooling concept is elaborated in section 2.2.4.

The connection between the stator and rotors, allowing frictionless rotation, isperformed by ball bearings. To withstand the high axial forces between stator androtor, deep groove ball bearings or angular contact ball bearings are preferred. Theshaft connects both rotors and transmits the torque to the attached application. Ifnecessary, a significant part of the axial force between the stator and the rotors canbe carried by the shaft as well.

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Figure 2.8: Cross section giving an overview of the different parts in thesuggested yokeless and segmented axial flux PM machine topol-ogy; stator core element (1), winding (2), mechanical bracket(3), permanent magnet (4), rotor (5), shaft (6), ball bearing(7), inner stator structure (8), housing (9), rotor endshield (10),housing endshield (11), terminal box (12), locking nut (13).

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Figure 2.9: Preassembled stator core element provided with a double layerconcentrated fractional slot winding (tooth coil winding)and amechanical bracket. (Exploded view in 2.10)

To protect the crucial electromagnetic active machine parts from dust and mois-ture, the whole assembly is protected by a housing and accompanying endshieldsat both sides of the stator.

2.2.3 Modular Stator

In the yokeless and segmented armature torus machine, the stator is no longera massive entity, but consists of a number of individual coreelements. Theuse of a double layer concentrated fractional slot winding allows to producepre-manufactured stator elements. Such a stator element ispresented in Fig.2.9. To give a better overview on such a stator module, an exploded view ispresented in Fig. 2.10. In such a module, the segmented armature element orstator core element is provided with double layer concentrated fractional slotwinding. Because this winding is simply wound around such a core element, thenametooth coil windingis very often used. The benefits and layout of such adouble layer concentrated fractional slot winding are topic of Chapter 7, but themost important advantages with respect to the modular construction are mentionedbriefly. First there is the easy winding process that can be performed on theindividual core elements out of the machine, which is advantageous in obtaininghigh filling factors of the slots with conductors. Second, there is the short andsimple arrangement of the end windings, that allow to make a winding withconductors having a rectangular cross section. Again the rectangular cross sectionincreases the filling factor with respect to a round conductor section. Hence, thetooth coil winding is beneficial for both manufacturing of the core element and the

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Figure 2.10: Exploded view of a preassembled stator core element. Mechanical bracket (1), stator core element (2), toothcoil winding (3).

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energy efficiency of the machine.

Unlike the advantages of the modular stator on manufacturing and energy ef-ficiency, there are some important drawbacks as well. The main disadvantage isthe mechanical assembling of the different modular elements to a solid stator en-tity. Generally, the individual teeth are connected by means of this yoke, but asthis yoke is absent in the yokeless and segmented armature machine, alternativemeasures need to be introduced to fit the individual stator elements solidly. Animportant issue in this construction is the high axial forces acting between the coreelements and the rotor. Among the different constructed YASA-prototype ma-chines in recent literature, three variants are found. In [27], two endshields at bothsides of the stator keep the different elements together. These endshields are madeof high-strength synthetic material, as electrically conductive or magnetically per-meable materials would deteriorate the electromagnetic properties of the machine.A drawback of this method is the increase of the air gaps. A second solution isproposed in [3] where 2 holes are made into the core elements near the core tips(one at each side). The mechanical fixation is provided by bolts, connecting theindividual elements with an inner stator structure. Disadvantages are found in theinduced eddy currents in the electrically conductive material of the bolts and a ma-jor influence on the flux density pattern and possible local saturation in the coreelements near the stator tips. A third construction method is introduced in [6],where the shafts of the individual stator core elements are inserted into two slottedsupporting structures near the tooth tips. The axial movement is prevented by inter-nally clamping of the tooth tips between the two slotted supporting structures. Thishas the major disadvantage that the slot area near the tooth tips remains unwound,which increases the leakage flux. Moreover, there is the riskof induced eddy cur-rents in the electrically conductive slotted supporting structures as they are situatedvery close to the air gap.

To avoid the increase of the air gap width, the holes near the tooth tips and theincreasing leakage flux, the mechanical fixation is suggested in the middle part ofthe core element. There, a high strength epoxy mechanical bracket is clamped be-tween local widening (Fig. 2.10) in the shaft of the stator cores after the winding isput around the stator core element. The risk of induced eddy currents is eliminatedwhile the flux density pattern in the stator core is nearly unchanged. The furtherextension of the modular concept to the mechanical construction, yields some ad-vantages. Fig. 2.11 shows the easy construction of the stator assembly. Inversely,defective modules can be replaced easily, as each specific module can be shovedout of the other core elements. The space between the stator cores at the innerand outer radius avoids the induction of eddy currents in theadjacent electricallyconductive parts while these spaces are used for forced convection cooling.

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Figure 2.11:Construction concept of the stator, performed through compo-sition of the individual stator modules. Individual modules areshoved into another.

2.2.4 Air Cooling Possibilities

Despite the focus on energy efficiency in this work, present loss phenomena inelectrical machines will always result in local heat production in the machine. Thisheat should be abducted out of the machine. In current radialelectrical machines,the stator laminations are pressed into the external housing, and cooling of the sta-

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tor is mainly obtained by conductive heat transfer from the stator housing. In axialflux machines with a yoke, like the topology with one stator and one rotor in Fig.2.1, a significant part of the cooling can be obtained throughconduction. In theYASA-machine topology, conductive cooling through the yoke is not possible andalternative strategies are required. These are found in [27], where forced coolingwith a water jacket of the stator is implemented. Such a waterjacket cooling isnot simple to implement and requires additional equipment in order to operate.Therefore, forced air cooling is suggested in this YASA-machine. In good approx-imation the rotor disc(s) of an axial flux machine will act like a fan: ambient airwill be transferred from the inner to the outer diameter of the disc. This natural fanworking of the disc can be extended by adding blades to the backside of the rotordisc. A suggestion for the cooling mechanism is made in Fig. 2.12. The rotor discis a die cast workpiece, that combines three functions: electromagnetic, thermaland mechanical. First, the rotor is the yoke for the magneticflux on which thepermanent magnets are mounted. Second, holes below the inner diameter allow tosupply air from the inner side of the stator in axial direction and pump this air tothe outer radius of the disc. To limit the air flow from the exterior of the machine,an endshield is attached to the backside of the blades. Third, the introduced bladesact like spokes and provide mechanical stiffness combined with low weight.

Fig. 2.13 illustrates the intended flow of the air through themachine. Air istaken in axially at the inner radius of the disc, is pumped to the outer part of thedisc by fan working of the external blades and is deviated by the endshields of theexternal housing to the outer part of the stator. There, the air is blown over the endwindings of the tooth coils and is moving through the non-filled parts of the statorslots to the inner part of the stator where the air is recirculated through the holes inthe rotor.

As the air is recirculated within the machine, heat exchanges with the outerparts of the machine,i.e. external housing and endshields, are necessary to evac-uate the produced heat by the different loss phenomena in themachine. At theouter surface, the heat is assumed to be transferred to the surroundings by naturalconvection.

Crucial in PM machines is the temperature of the permanent magnets. Heat-ing of the permanent magnets may result in a reduced remanentflux density, andmoreover, the permanent magnets can be irreversibly demagnetised if the temper-ature becomes too high. In radial machines with surface mounted magnets, whereheat exchange between rotor and stator is present, the permanent magnet tempera-ture should be examined extensively in the design process ofthe machine. This isless crucial in axial flux PM machines with outer discs. Nevertheless special atten-tion to the calculation of eddy current losses in permanent magnets is the topic ofChapter 5.

Obviously this cooling strategy needs to be optimised. The air flow, unfilledslot space and axial length of the blades should be chosen as afunction of thewanted temperatures and air flow path in the machine. On one side, sufficient air

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Figure 2.12: Suggestion for the rotor construction (exploded view). Rotor endshield (1), die cast rotor with blades/spokes(2), permanent magnets (3).

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Figure 2.13: Illustration of the suggested cooling concept. Left side: machine without external housing endplates. Theblades (1) evacuate air from the inner part of the stator through (2) holes below the inner diameter of the rotor.Right side: cross section view on the stator. The generated air flow passes through the non-filled slot sections(3) from the stator outside diameter (4) to the inside diameter (5).

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Table 2.1: Parameter set of the example YASA axial flux PM machine. Val-ues are obtained through optimisation.

Parameter Symbol Value

Number of pole pairs p 8Number of stator slots Qs 15Rated speed Ω 2500 rpmRated torque < Tem > 18.5 NmRemanent flux density permanent magnetsBPM 1.26Outer diameter Do 148 mmInner diameter Di 100 mmAxial length core element htot 60 mmSlot width ls 11 mmAxial length slot hs 24 mmAxial thickness magnet hPM 5 mmAxial thickness rotor hr 8 mm

flow is necessary to provide good cooling, on the other side too extensive coolingresults in excessive ventilation loss in the machine. The thermal-fluid problem[18,28], governing the conductive, convective and radiative heat transfer combinedwith the Navier-Stokes equations is not considered in this work and is left for futureresearch.

2.3 Test Case Machine

Although the aim of this research to introduce mathematicalmodels describing theelectromagnetic power transfer in the machine and the different loss phenomena inthe machine, the introduced mathematical models and measures are evaluated ona test case machine having a specified parameter set. The parameter set of the testcase YASA axial flux PM machine2 are mentioned in Table 2.1. Most values inthis parameter set were obtained through optimisation using a genetic algorithm.This parameter set will be used in all future parts of this research, if not, it will bementioned explicitly.

2The example parameter set was adopted from the generator in the combined heat and powerapplication introduced in 2008 by prof. dr. ir. Alex Van den Bossche and prof. dr. ir. Peter Sergeantat the department Electrical Energy, Systems and Automation. Also the combination of 16 magnetsand 15 stator cores is adopted.

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2.4 Conclusion 25

2.4 Conclusion

In this chapter, an introduction to axial flux permanent magnet machines was pre-sented. The many different axial flux PM machine topologies were introducedbriefly, and finally, the yokeless and segmented armature topology (YASA) wasselected for further consideration. The light-weight and easy construction of theYASA-topology, enforcing the general properties of axial PM flux machines suchas high power density and good energy efficiency, were decisive in this selection.Moreover, the modular construction of the YASA-machine makes this topologyextremely interesting. The global construction of the YASA-machine, as well as apossible concept for forced air cooling, were introduced briefly. Finally, an exam-ple YASA-machine, having a specific parameter set, is introduced and will be usedin future parts of this work to explain and illustrate the introduced mathematicalmodels.

Bibliography

[1] T. Woolmer and M. McCulloch, “Analysis of the yokeless and segmentedarmature machine,” inElectric Machines & Drives Conference, 2007.IEMDC’07. IEEE International, vol. 1. IEEE, 2007, pp. 704–708.

[2] F. Caricchi, F. G. Capponi, F. Crescimbini, and L. Solero, “Experimen-tal study on reducing cogging torque and no-load power loss in axial-fluxpermanent-magnet machines with slotted winding,”Industry Applications,IEEE Transactions on, vol. 40, no. 4, pp. 1066–1075, 2004.

[3] W. Fei, P. Luk, and K. Jinupun, “A new axial flux permanent magnetsegmented-armature-torus machine for in-wheel direct drive applications,” inPower Electronics Specialists Conference, 2008. PESC 2008. IEEE. IEEE,2008, pp. 2197–2202.

[4] H. Kierstead, R. Wang, and M. Kamper, “Design optimization of a singlesided axial flux permanent magnet in-wheel motor with non-overlap concen-trated winding.”

[5] B. Chalmers and E. Spooner, “An axial-flux permanent-magnet generator fora gearless wind energy system,”Energy Conversion, IEEE Transactions on,vol. 14, no. 2, pp. 251–257, 1999.

[6] A. Di Gerlando, G. Foglia, M. F. Iacchetti, and R. Perini,“Axial flux pmmachines with concentrated armature windings: Design analysis and test val-idation of wind energy generators,”Industrial Electronics, IEEE Transactionson, vol. 58, no. 9, pp. 3795–3805, 2011.

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[7] Y. Chen, P. Pillay, and A. Khan, “Pm wind generator topologies,” IndustryApplications, IEEE Transactions on, vol. 41, no. 6, pp. 1619–1626, 2005.

[8] T. Chan and L. Lai, “An axial-flux permanent-magnet synchronous generatorfor a direct-coupled wind-turbine system,”Energy Conversion, IEEE Trans-actions on, vol. 22, no. 1, pp. 86–94, 2007.

[9] G. F. Price, T. D. Batzel, M. Comanescu, and B. A. Muller, “Design and test-ing of a permanent magnet axial flux wind power generator,” inProceedingsof the 2008 IAJC-IJME International Conference, 2008.

[10] A. Parviainen, J. Pyrhonen, and P. Kontkanen, “Axial flux permanent mag-net generator with concentrated winding for small wind power applications,”in Electric Machines and Drives, 2005 IEEE International Conference on.IEEE, 2005, pp. 1187–1191.

[11] A. Parviainen, M. Niemela, J. Pyrhonen, and J. Mantere, “Performance com-parison between low-speed axial-flux and radial-flux permanent-magnet ma-chines including mechanical constraints,” inElectric Machines and Drives,2005 IEEE International Conference on. IEEE, 2005, pp. 1695–1702.

[12] A. Cavagnino, M. Lazzari, F. Profumo, and A. Tenconi, “Acomparison be-tween the axial flux and the radial flux structures for pm synchronous motors,”in Industry Applications Conference, 2001. Thirty-Sixth IASAnnual Meeting.Conference Record of the 2001 IEEE, vol. 3. IEEE, 2001, pp. 1611–1618.

[13] S. Huang, J. Luo, F. Leonardi, and T. A. Lipo, “A comparison of power den-sity for axial flux machines based on general purpose sizing equations,”En-ergy Conversion, IEEE Transactions on, vol. 14, no. 2, pp. 185–192, 1999.

[14] R. Qu, M. Aydin, and T. A. Lipo, “Performance comparisonof dual-rotorradial-flux and axial-flux permanent-magnet bldc machines,” in Electric Ma-chines and Drives Conference, 2003. IEMDC’03. IEEE International, vol. 3.IEEE, 2003, pp. 1948–1954.

[15] A. Parviainenet al., “Design of axial-flux permanent-magnet low-speed ma-chines and performance comparison between radial-flux and axial-flux ma-chines,”PhD dissertation, Acta Universitatis Lappeenrantaensis, 2005.

[16] M. Aydin, S. Huang, and T. A. Lipo, “Torque quality and comparison of in-ternal and external rotor axial flux surface-magnet disc machines,”IndustrialElectronics, IEEE Transactions on, vol. 53, no. 3, pp. 822–830, 2006.

[17] A. Cavagnino, M. Lazzari, F. Profumo, and A. Tenconi, “Acomparison be-tween the axial flux and the radial flux structures for pm synchronous motors,”Industry Applications, IEEE Transactions on, vol. 38, no. 6, pp. 1517–1524,2002.

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[18] C. H. Lim, “Thermal modelling of the ventilation and cooling inside axialflux permanent magnet generators,” Ph.D. dissertation, Durham University,2010.

[19] F. Libert, “Design, optimization and comparison of permanent magnet mo-tors for a low-speed direct-driven mixer,”Licentiate Thesis, Royal Institute ofTechnology, TRITA-ETS-2004-12, ISSN-1650-674x, Stockholm, 2004.

[20] F. Sahin, A. Tuckey, and A. Vandenput, “Design, development and testing of ahigh-speed axial-flux permanent-magnet machine,” inIndustry ApplicationsConference, 2001. Thirty-Sixth IAS Annual Meeting. Conference Record ofthe 2001 IEEE, vol. 3. IEEE, 2001, pp. 1640–1647.

[21] K. Akatsuet al., “A comparison between axial and radial flux-motors pm mo-tors by optimum design method from the required output nt characteristics,”in Int. Conf. Electrical Machines, 2004, p. 161.

[22] M. Aydin, S. Huang, and T. Lipo, “Axial flux permanent magnet disc ma-chines: A review,” inSymposium on Power Electronics, Electrical Drives,Automation, and Motion (SPEEDAM), 2004, pp. 61–71.

[23] F. Caricchi, F. Crescimbini, O. Honorati, G. L. Bianco,and E. Santini, “Per-formance of coreless-winding axial-flux permanent-magnetgenerator withpower output at 400 hz, 3000 r/min,”Industry Applications, IEEE Transac-tions on, vol. 34, no. 6, pp. 1263–1269, 1998.

[24] S. M. Hosseini, M. Agha-Mirsalim, and M. Mirzaei, “Design, prototyping,and analysis of a low cost axial-flux coreless permanent-magnet generator,”Magnetics, IEEE Transactions on, vol. 44, no. 1, pp. 75–80, 2008.

[25] M. J. Kamper, R.-J. Wang, and F. G. Rossouw, “Analysis and performance ofaxial flux permanent-magnet machine with air-cored nonoverlapping concen-trated stator windings,”Industry Applications, IEEE Transactions on, vol. 44,no. 5, pp. 1495–1504, 2008.

[26] S. Huang, M. Aydin, and T. A. Lipo, “Torus concept machines: pre-prototyping design assessment for two major topologies,” in Industry Ap-plications Conference, 2001. Thirty-Sixth IAS Annual Meeting. ConferenceRecord of the 2001 IEEE, vol. 3. IEEE, 2001, pp. 1619–1625.

[27] T. Woolmer, “Electric machine-flux,” Feb. 12 2010, uS Patent App.13/148,863.

[28] F. Marignetti and V. D. Colli, “Thermal analysis of an axial flux permanent-magnet synchronous machine,”Magnetics, IEEE Transactions on, vol. 45,no. 7, pp. 2970–2975, 2009.

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Chapter 3

Modelling the Axial Flux PMMachine

3.1 Introduction

Crucial in the design of electrical machines is the development of proper and ac-curate mathematical tools. Although the design of an electrical machine is a mul-tiphysics problemi.e. involving electromagnetic, thermal and mechanical mod-elling, this work is limited to the development of mathematical models describingthe electromagnetic behaviour of the machine.

Numerous methods for the mathematical modelling of electrical machines havebeen developed over the years. The complexity of these mathematical modelsvaries from a rather simple analytical tool up to (full 3D) finite element analy-sis. Despite the development of decent finite element analysis tools, the interesttowards analytical tools is hardly decreased. Analytical models still benefit fromthe low computational time with respect to finite element modelling, and are veryoften accurate enough to be used in a preliminary machine design. Moreover, ana-lytical tools provide a good understanding in the ruling laws that define the electro-magnetic properties of the machine. Whereas in the past, thethin wire approxima-tion predicted the induced electromotive force assuming a sinusoidal magnetic fluxdensity in the air gap, the most recent analytical modellinguses magnetic vectorpotential equations to model the air gap magnetic flux density including the fullharmonic content of the air gap magnetic flux density. The least advanced versionsamong these analytical models assume a smooth stator surface with infinite per-meability. In more advanced versions the influence of the stator slot openings istaken into account by defining a vector potential in each of the slots and imposingthe proper boundary conditions. Last developments includethe modelling of sta-tor armature reaction by specifying a current density in theslot areas. Althoughthese techniques result in a highly accurate modelling of the air gap magnetic fluxdensity, the model complexity and calculation time increase significantly. These

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30 Modelling the Axial Flux PM Machine

modelling techniques were initially introduced for radialmachines, however, re-cently axial flux variants are found as well.

Whereas the modelling of electrical machines with a radial magnetic field canbe done accurately by using the planar representation, the inherent 3D geometryof the machines with an axial magnetic field increases the complexity. Therefore,many authors limit their models to an evaluation of the machine’s electromagneticproperties at the average radius. Nevertheless, the multislice 2D modelling whichis elected in this thesis, was introduced because of its excellent combination ofaccuracy and computation time.

Although these analytical tools provide an impeccable modelling of the elec-tromagnetic properties in the air gap, there are still some limitations. As the statorcore material is assumed to be infinitely permeable, the influence of nonlinear be-haviour of the stator core material is not taken into account. As in future partsof this research the nonlinear behaviour of the stator core material is crucial, theimplementation of nonlinear finite element computations isused frequently in thisresearch. As a lot of decent (2D) finite element computation packages are freelyor commercially available, the aim of this research was not to develop a finite el-ement solver. The mathematical models developed in this work, are introducedindependently of the used finite element analysis package.

Despite the inherent 3D geometry of the axial flux PM machine,the use of 3Dfinite element analysis is very limited in this research.

3.2 Multislice 2D Modelling

As the axial flux PM machine has a inherent 3D geometry, accurate simulationsneed 3D finite element computations. Despite the accuracy obtained through 3Dfinite element analysis, simulations take too much time to beused in design pro-cesses which may include optimisation algorithms, using many evaluations. Evenin case of magnetic symmetry, where only half of the axial fluxPM machinesneeds to be modelled,cfr. Fig. 3.1, simulation times are still too long. Therefore,sometimes 2D simulations are performed on the average diameter. These simula-tions may be fast, but generally yield a limited accuracy. Hence, the modelling ofaxial flux PM machines meets two contradictory constraints:speed and accuracy.A good trade-off between speed and accuracy was suggested in[1–4] where a socalled quasi-3D modelling was introduced. In this quasi-3Dmodelling, not only asingle evaluation at the average radius is performed, but simulations are performedat multiple radii. Fig. 3.1, (1) indicates such a computation plane at a random ra-dius. Analytical expressions can be directly used on these cylindrical computationplanes using a cylindrical coordinate system, while 2D finite element computationsrequire a cartesian coordinate system. Therefore the cylindrical computation planeindicated by (1) in Fig. 3.1 is unrolled to a planar 2D geometry, which can be usedstandardly in commercially available 2D finite element solvers.

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3.2M

ultislice2D

Modelling

31

1

2

Figure 3.1: Principle of the transformation of the 3D geometry of an axial flux PM machine to a 2D geometry, which canbe used in multislice 2D computation. At different radii (only one radius depicted), cylindrical surfaces (1) aredefined. The analytical modelling uses a cylindrical coordinate system for these surfaces (1), while the 2D finiteelement computations use an unrolled variant (2) which can be treated like a common planar 2D geometry incartesian coordinates as used in most finite element packages.

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In practice,n different radiircp,i, i = 1 . . . n are generally selected uniformlybetween the innerDi and outer diameterDo

rcp,i =Di

2+Do −Di

2n

(

i− 1

2

)

i = 1, . . . , n. (3.1)

Each computation plane has thus a thickness, which is the dimension perpendicularto the computation plane, of

tcp =Do −Di

2n. (3.2)

The number of radiin required in the multislice 2D modelling strongly dependson the variation of the geometrical parameters in the radialdirection. Dependingon the parameter of interest, a sensitivity analysis of the number of computationplanes with respect to the value of the parameter of interestcan be performed.

To obtain a global quantityΓ, which can be the flux linkage, induced electro-motive force, electromagnetic torque,etc., summation over the contributions bythe different computation planes is performed

Γ =

n∑

i=1

Γi (3.3)

Although the number of computation planes can be chosen high, they are stillsolved individually. This means that the radial componentsof magnetic fluxes arenot taken into account. The lack in modelling of these radialcomponents of themagnetic fluxes has a minimal effect when the variation of themagnetic propertiesover the radial direction is limited. Particularly in case of laminated silicon steelas stator core material, the decomposition into the different computation planesis permitted. The anisotropic behaviour of the laminated stator cores results in ahigh permeability in the axial and azimuthal direction, while the permeability inthe radial direction is poor. Therefore, magnetic fluxes will be parallel to the com-putation planes rather than perpendicular to these planes.Note that the presence ofmagnetic fluxes perpendicular to the lamination/computation plane, would induceeddy currents in the good electrically conductive laminated silicon steel sheets andthus increase the losses in the machine. Therefore, when designing an axial fluxPM machine, care should be taken that the magnetic flux density levels are as uni-form as possible over the radial direction.

3.3 Analytical multislice 2D Modelling

Despite the existence of finite element analysis, analytical modelling is still fre-quently used because of its fastness [5, 6]. In an early design stage, analytical ex-pressions are often accurate enough to get an overview of theinfluence of different

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3.3 Analytical multislice 2D Modelling 33

parameters on the machine’s performance. Current development and improvementof analytical modelling in PM machines, is concentrated on the accurate predictionof the air gap magnetic flux density. Therefore, the current research in analyticalmodelling for axial flux PM machines is dominated by magneticvector potentialequations [7–10]. The latest development is concentrated on air gap magnetic fluxdensity calculations in PM machines with a slotted stator. There, the magnetic vec-tor potential in each of the slots is linked with the magneticvector potential in theair gap and permanent magnet region. This procedure is done at no load [11–14],with no current density imposed in the slots, or more recently at load [15,16], witha current density imposed in the slots. Also recently, an analytical expression forthe magnetic field in the stator slots was used to calculate the strand level skin andproximity losses [17]. The benefit of this exact modelling ofthe slotting effect,is an accurate prediction of both the axial and azimuthal components of the mag-netic flux density, which is for example necessary for accurate prediction of thewaveforms of cogging torque and electromagnetic torque at load.

The aim of the analytical model presented in this work is to predict the domi-nant axial component of the magnetic flux density in the air gap correctly, to takeinto account multiple harmonic components rather than a model for the fundamen-tal frequency, and to keep the model simple to allow fast evaluation. Therefore,in the next section a current sheet is introduced to model thepermanent magnetsand armature reaction in the air gap. A permeance function isused to model theslotting effect of the stator. This modelling technique might not include the latestfeatures introduced in the previous mentioned papers, but is sufficiently accurateto understand the converging parameter dependencies in thefuture design of PMmachines.

Once the air gap magnetic field is modelled, it is used to derive quantities suchas flux linkage and induced electromotive force. The analytical model also includesa preliminary estimation of the losses in the machine.

3.3.1 Magnetic Vector Potential from a Current Sheet

The multislice 2D modelling technique reduces the complexity to simple 2D prob-lems. Fig. 3.2 illustrates the simplified 2D model that is used for the analyticalcalculations. In this 2D analytical model, current sheets are used to model both thepermanent magnet and armature current. The position of the current sheet dividesthe air gap into two regions, the lower region I and upper region II. The computa-tion of the magnetic vector potential from a generalised current sheet between thestator and rotor is the basic building block used in the analytical analysis. As an-alytical models are already subject of many publications, similar techniques havebeen used in [18–22] and more recently in [23–25].

The current sheet can be represented in Fourier series expansion in terms of

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ϕ

z

z1

z2 region II

region I

K(ϕ)

µ = ∞

µ = ∞

Figure 3.2: Planar representation of the calculation plane used in the ana-lytical modelling. The current sheet divides the air gap into twozones.

space harmonics along the circumferential coordinate, azimuthϕ

K(ϕ) =

∞∑

ν=1

Mν cos (νkϕ) +Nν sin (νkϕ) (3.4)

wherek represents the periodicity over the azimuthal direction; to express the per-manent magnets by a current sheet in the test case machine with 8 pole pairsp,k = p = 8. The values of the coefficientsMν andNν depend on the actual currentdistribution and are evaluated in section 3.3.2 for the permanent magnets and insection 3.3.3 for the armature currents.

The magnetic vector potential is a vector field defined by

B = ∇× A. (3.5)

In these cylindrical planes the magnetic flux densityB has only a circumferentialcomponent and a component along the z-axis (axial direction)

B = Bϕeϕ +Bzez, (3.6)

and, hence, the magnetic vector potential has only a component perpendicular tothe circumferential and z direction,i.e. the radial direction

A = Arer. (3.7)

Ampere’s circuital law with Maxwell’s correction

∇× H = J, (3.8)

together with Gauss’s law for magnetism

∇ · B = ∇ · (∇× A) = 0, (3.9)

and the constitutive relation (in free space) between the magnetic flux densityBand magnetic fieldH

B = µ0H. (3.10)

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results in∇× (∇× A) = µ0J. (3.11)

This equation can be transformed into

∇ (∇ · A)−∇2A = µ0J. (3.12)

To find a unique solution for the vector potential, the Coulomb gauge∇ · A = 0is chosen. Therefore, in the absence of an external current densityJ, the magneticvector potential equation becomes

∇2Ar = ∆Ar = 0 (3.13)

which is known as Laplace’s equation.Laplace’s equation in cylindrical coordinates is given by

1

r2∂2Ar∂ϕ2

+∂2Ar∂z2

= 0 (3.14)

and has a solution of the form

Ar(ϕ, z) =

+∞∑

ν=1

r

νkµ0

[

Cν cosh(

νkr z)

+Dν sinh(

νkr z)]

cos (νkϕ)

+[

Eν cosh(

νkr z)

+ Fν sinh(

νkr z)]

sin (νkϕ)

.

(3.15)The magnetic field is derived directly from the magnetic vector potential by

Hϕ =1

µ0

∂Ar∂z

; Hz = − 1

rµ0

∂Ar∂ϕ

. (3.16)

In both the lower region I and upper region II, a formulation for the magneticvector potential can be made. To determine the coefficientsCν , Dν , Eν andFνin the equation (3.15) of the vector potential in both the lower region I and upperregion II, four additional constraints are required. As theiron in the stator andthe rotor is assumed to have an infinite permeability, magnetic field lines are per-pendicular to these boundaries, and hence, the circumferential components of themagnetic field are zeroi.e.

HIϕ|z=0= 0; HII

ϕ |z=z2= 0. (3.17)

Next to these two boundary conditions, continuity of the z-component of the mag-netic flux density is required

BIz|z=z1= BII

z |z=z1 . (3.18)

The last constraint is obtained by applying Ampere’s circuital law on the circum-ferential component of the magnetic field

∆Hϕ|z=z1= HIϕ|z=z1−HII

ϕ |z=z1= K(ϕ) (3.19)

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whereK(ϕ) is defined by equation (3.4).The coefficients are

DIν = 0 (3.20)

F Iν = 0 (3.21)

CIν = Mν

cosh(

νkr (z2 − z1)

)

sinh(

νkr z2

) (3.22)

EIν = Nν

cosh(

νkr (z2 − z1)

)

sinh(

νkr z2

) (3.23)

for the lower region I, and

DIIν = −Mν cosh

(

νk

rz1

)

(3.24)

F IIν = −Nν cosh

(

νk

rz1

)

(3.25)

CIIν = Mν

cosh(

νkr z1

)

cosh(

νkr z2

)

sinh(

νkr z2

) (3.26)

EIIν = Nν

cosh(

νkr z1

)

cosh(

νkr z2

)

sinh(

νkr z2

) (3.27)

for the upper region II.The solution of Laplace’s equation gives the magnetic vector potential, and the

normal component of the magnetic field, in the lower and the upper region as:

Lower region I

AIr(ϕ, z) = −µ0

+∞∑

ν=1

r

νk

cosh(

νkr (z2 − z1)

)

sinh(

νkr z2

) cosh

(

νk

rz

)

· [Mν cos (νkϕ) +Nν sin (νkϕ)] (3.28)

HIz(ϕ, z) =

+∞∑

ν=1

cosh(

νkr (z2 − z1)

)

sinh(

νkr z2

) cosh

(

νk

rz

)

· [Nν cos (νkϕ)−Mν sin (νkϕ)] (3.29)

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Upper region II

AIIr (ϕ, z) = −µ0

+∞∑

ν=1

r

νk

cosh(

νkr z1

)

sinh(

νkr z2

) cosh

(

νk

r(z2 − z)

)

· [Mν cos (νkϕ) +Nν sin (νkϕ)] (3.30)

HIIz (ϕ, z) =

+∞∑

ν=1

cosh(

νkr z1

)

sinh(

νkr z2

) cosh

(

νk

r(z2 − z)

)

· [Nν cos (νkϕ)−Mν sin (νkϕ)] (3.31)

3.3.2 Permanent Magnet Magnetic Field

The magnetic field from the permanent magnets is calculated by representing thepermanent magnets by an equivalent current sheet. Therefore, each permanent

z

ϕ

2ǫ

µ = ∞

µ = ∞

I = BPMhPM

µ0µPM

z1 hPM

z2KPM

Figure 3.3: Representation of the permanent magnets by an equivalent cur-rent sheet, plotted for the computation plane at the inner diame-ter.

magnet is modelled by a current at its edges which is assumed to flow in a vanishingsmall area with thickness2ǫ. Subsequently the permanent magnets are divided intoa number of current sheets with widthdz1 placed at heightz1 with a linear current

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density:

KPM(ϕ, θm) =+∞∑

ν=1

JPM(ϕ, θ)dz1 (3.32)

=+∞∑

ν=1

JPMν dz1 sin (νp(ϕ+ θ)) (3.33)

=+∞∑

ν=1

KPMν sin (νp(ϕ+ θ)) (3.34)

=

+∞∑

ν=1

MPMν (θ) cos (νpϕ) +NPM

ν (θ) sin (νpϕ) (3.35)

whereJPM is the current density at each sheet.As the relative permeabilityµPM of the permanent magnets is greater than

unity, the equivalent magnet current has a magnitude of

BPM

µ0µPMhPM (3.36)

and a corresponding current density that varies with the radius given by

JPM(r) =BPM

µ0µPM

p

2ǫr. (3.37)

The individual current density harmonics are now obtained by Fourier series ex-pansion to calculate the equivalent current density distribution for each magnetcurrent sheet as:

JPM(ϕ, θ) =+∞∑

ν=1

8

πJPMǫ sin

(

νπαPM

2

)

sin (νp(ϕ+ θ)) (3.38)

=+∞∑

ν=1

4

π

BPM

µ0µPM

p

rsin(

νπαPM

2

)

sin (νp(ϕ+ θ)) (3.39)

The magnetic vector potential in the upper region II is then found by mergingequation (3.35) and equation (3.30), and integrating over the permanent magnetthicknesszPM to obtain:

APMr (ϕ, z) = −µ0

+∞∑

ν=1

(

r

νp

)2 4

π

BPM

µ0µPM

p

rsin(

νπαPM

2

)

· sinh(νpr hPM

)

sinh(νpr z2

) cosh(νp

r(z2 − z)

)

sin (νp(ϕ+ θ)) . (3.40)

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Subsequently, the axial component of the magnetic flux density in the upper regionII can be evaluated as

BPMz (ϕ, z) = µ0

+∞∑

ν=1

1

ν

4

π

Brem

µ0µPMsin(

νπαPM

2

)

· sinh(νpr hPM

)

sinh(νpr z2

) cosh(νp

r(z2 − z)

)

cos (νp(ϕ+ θ)) . (3.41)

The magnetic flux density in the upper region II is plotted in Fig. 3.4 for the pa-rameters corresponding to the inner diameterDi. In this eight pole magnetic fluxdensity, the harmonic numberν of the fundamental component is thus 8. It may al-ready be pointed out that the distribution of the magnetic flux density in the air gapis far from sinusoidal. Consequently, the clear presence ofadditional harmonicsother than the fundamental component, will have to be taken into account duringthe rest of the research.

BPM

[T]

Azimuthϕ []

0 45 90 135 180 225 270 315 360-1.2

-0.8

-0.4

0

0.4

0.8

1.2

Figure 3.4: Air gap magnetic flux density generated by the permanent mag-nets only. Plotted for the computation plane at the inner diame-ter.

3.3.3 Armature Magnetic Field and Current Distribution

Next to the magnetic flux density by the permanent magnets, the armature reactionwill also have a contribution in the air gap magnetic field. When the stator windingscarry a current, they will result in a magnetic flux in the stator core. Although most

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of the flux travels through the tooth tips to the adjacent stator cores, some parts ofthis magnetic flux go through the air gap up to the rotor. The ratio of both fluxesis strongly related to the geometrical parameters that influence the reluctance pathsfor both fluxes. Among these geometrical parameters, the airgap added with themagnet thickness and tooth tip opening widths are the most important.

As the magnetic flux is going through the permanent magnets, eddy currentswill be induced in the permanent magnets as they see a time variation in the mag-netic flux density resulting from the armature currents. Therefore, when calculat-ing eddy current losses in the permanent magnets, it is crucial to take the armaturereaction into account.

This paragraph focuses on the contribution of the armature currents to the airgap magnetic flux density. The current sheet is in this case directly linked to thecurrent density distribution in the stator slots. The arrangement of the differentphases into the stator slots is the topic of Chapter 7. In thischapter the concept ofa double layer concentrated fractional slot winding, sometimes referred to as toothcoil winding, is examined extensively. In this section it issufficient to understandFig. 3.5. Here, the current sheet for the present current density distribution in thestator slots of the axial flux PM machine is proposed. Each peak in this figure

KA

[A/m

]

Azimuthϕ []

0 45 90 135 180 225 270 315 360

×105

-3

-2

-1

0

1

2

3

Figure 3.5: Current sheet equivalent to the current density distribution inthe stator slots. Plotted for the computation plane at the innerdiameter.

represents the current density in the corresponding slot. As the current densities inthe stator slots are a function of time, the value of the peaksin Fig. 3.5 will be timedependent as well.

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For any winding configuration, a Fourier series expansion ofthis current sheetof the azimuthal coordinateϕ can be performed

KA(ϕ) =

∞∑

ν=1

MAν cos (νϕ) +NA

ν sin (νϕ) (3.42)

where the superscriptA indicates that it corresponds to the armature reaction.Equation (3.30) and (3.42) result directly into the vector potential

AAr (ϕ, z) = −µ0

+∞∑

ν=1

r

ν

cosh(

νr z1)

sinh(

νr z2) cosh

(ν

r(z2 − z)

)

·[

MAν cos (νϕ) +NA

ν sin (νϕ)]

(3.43)

of which the resultant magnetic field can be obtained throughequation (3.16)

HAz (ϕ, z) =

+∞∑

ν=1

cosh(

νr z1)

sinh(

νr z2) cosh

(ν

r(z2 − z)

)

·[

NAν cos (νϕ)−MA

ν sin (νϕ)]

. (3.44)

This magnetic field is plotted in Fig. 3.6 with the parametersof the inner diameterDi. As the armature reaction field is directly linked to the current density in the

HA

[A/m

]

Azimuthϕ []

0 45 90 135 180 225 270 315 360

×105

-1.5

-1

-0.5

0.5

1

1.5

Figure 3.6: Air gap magnetic field generated by the armature reaction only.Plotted for the computation plane at the inner diameter.

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stator slots, the armature reaction field is directly linkedto the load conditions ofthe machine, starting from no armature reaction at no load and maximum armaturereaction at full load. This willi.e. result in load dependent eddy current losses inthe permanent magnets.

3.3.4 Effect of Stator Slotting

In the previous described modelling, the air gap magnetic flux density was cal-culated for a smooth stator having an infinite permeability.In reality, the air gapmagnetic flux density is influenced locally around the statorslots. As mentioned inthe introduction, the effect of stator slotting can be introduced by defining a vectorpotential in each of the stator slots and by linking them to the solution in the airgap region [11–14]. As this set of coupled partial differential equations increasesthe complexity of the analytical model significantly, a closed analytical expres-sion cannot be derived anymore. Therefore, a permeance function using conformalmapping techniques is introduced in this section. In [26–28], conformal mappingtechniques were introduced for radial machines to model theinfluence on both theradial and tangential component of the magnetic flux densityby the slotting effectusing an complex relative permeance function. In this section, such a relative per-meance function is introduced to model the effect of stator slotting on the axialcomponent of the magnetic flux density in the air gap. Similartechniques wereintroduced in [29–32] for radial machines.

In the further analysis, an infinite permeability of the stator and the rotor partsis proposed and the axial length of the stator slots is assumed to be infinite. InFig. 3.7, a section of the air gap region in a PM machine is illustrated. Due tothe presence of the stator slot, the magnetic flux density decreases locally in theslot region, having a value less than in case no slotting would be present. The airgap magnetic field in case of absence of slots is indicated byBmax. Due to theslotting, a minimal value of the magnetic flux densityBmin is found in the centreof the slot. Due to the slotting, the average valueBave of the waveform ofB(ϕ)will have a value less thanBmax. The decrease ofBmax to Bave is equivalentto a fictitious increase of the air gap. For PM machines, the axial length of thepermanent magnets should be added to the air gapg, resulting in the effective airgapg′

g′ = g +hPMµPM

(3.45)

where a constant recoil permeability of the permanent magnet material is assumed.The factor expressing the fictitious increase of the air gap is given by

kC =τs

τs − γg′. (3.46)

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where

γ =4

π

b02g′

tan−1

(

b02g′

)

− ln

√

1 +

(

b02g′

)2

(3.47)

This factorkC is called the Carter coefficient, for whichkC > 1. In PM machines,the relatively high value of the effective air gap with respect to the rather smallslot openings, results in a Carter coefficient very close to 1and can generally beneglected.

To express the effect of stator slotting, an additional assumption needs to beintroduced. As Fig. 3.7 indicates, the effect of the stator slot has vanished beforethe effect of the neighboring slots is introduced. Therefore, the effect of each statorslot can be modelled individually. This individual modelling of the stator slotting isassumed in future calculations. Nevertheless, it is still acceptable for PM machines

b0

τtg′

Bmin

Bave

Bmax

Figure 3.7: Local influence of the stator slot openings on the magnetic fluxdensity.

in which the ratio of the air gap to the slot opening is high [29]. In other situations,a model using a series of slots needs to be introduced.

The relative permeance function is introduced by [29] as

λ(ϕ, z) =

Λ0

[

1− β(z)− β(z) cos(

π0.8ϕ0

ϕ)]

for 0 ≤ ϕ ≤ 0.8ϕ0

Λ0 for 0.8ϕ0 ≤ ϕ ≤ ϕt/2

(3.48)

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whereΛ0 = µ0/g′ and the functionβ(z) is defined at the air gap side of the stator

slots and depends on the axial position.β(z) can be determined by a Schwartz-Christoffel transformation, and is derived in the Appendixof this chapter as

β(z) =1

2

1− 1√

1 +(

b02g′

)2(1 + v2)

(3.49)

wherev is determined from

yπ

b0=

1

2ln

(√a2 + v2 + v√a2 + v2 − v

)

+2g′

b0arctan

(

2g′

b0

v√a2 + v2

)

(3.50)

and

a2 = 1 +

(

2g′

b0

)2

(3.51)

with y = g′ − z.The relative permeance function is then calculated from

λ(ϕ, z) =λ(ϕ, z)

Λ0=λ(ϕ, z)

µ0/g′. (3.52)

For further analysis, a Fourier series expansion over the azimuthal coordinateϕ of this function is done to be compatible with the earlier equations for the airgap magnetic flux density distribution due to the permanent magnets and armaturereaction. The relative permeance function is periodic withthe azimuthal widthcorresponding to the slot pitch. Therefore in case ofQs slots the equation for therelative permeance function becomes

λ(ϕ, z) = Λ0(z) +

∞∑

ν=1

Λν(z) cos (νQsϕ) (3.53)

where the coefficients are determined by Fourier series expansion of equation(3.48)

Λ0(z) = 1− 1.6β(z)b0τs

(3.54)

Λν(z) = −β(z) 4

νπ

0.5 +

(

ν b0τs

)2

0.78125 − 2(

ν b0τs

)2

sin

(

1.6νπb0τs

)

(3.55)

In Fig. 3.8, the relative permeance function is plotted for the computation planeat the inner diameter. Finally, the air gap magnetic flux density at no load can be

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λ

Azimuthϕ []

0 45 90 135 180 225 270 315 3600.5

0.6

0.7

0.8

0.9

1

Figure 3.8: Relative permeance function, plotted for the computation planeat the inner diameter.

obtained by multiplying the magnetic flux density by the permanent magnets (3.41)with the relative permeance function (3.53):

Bz(ϕ, z) = BPMz (ϕ, z) · λ(ϕ, z). (3.56)

The situation at no load is illustrated in Fig. 3.9. Here the local decreases ofthe magnetic flux density by the permanent magnets in the slotregions are clearlyvisible in comparison to Fig. 3.4. A similar approach is usedto model the machineat load where the sum of the magnetic flux density by the permanent magnets (3.41)and the armature reaction (3.44), is multiplied with the relative permeance function(3.53)

Bz(ϕ, z) =[

BPMz (ϕ, z) +BA

z (ϕ, z)]

· λ(ϕ, z). (3.57)

The previous discussion was only done for one random computation plane andthe corresponding figures were plotted for the computation plane at the inner di-ameter. The multislice 2D approach requires to evaluate theprevious equations atthen computation planes at the different radii. When the solutions of the differentcomputation planes are found, they can be superimposed to find the global solu-tion. In Fig. 3.10, the solution for the air gap magnetic field(3.56) as a function ofthe azimuthal coordinateϕ for 10 different radii in a cylindrical coordinate system.A more accurate representation is obtained in Fig. 3.11 by using a cartesian coor-dinate system. Here, the representation of the air gap magnetic flux density is morerealistic and similar to the real 3D air gap magnetic flux density distribution. Toget rid of the third dimension, a contour plot of the air gap magnetic flux density in

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λ·B

PM

[T]

Azimuthϕ []

0 45 90 135 180 225 270 315 360-1.2

-0.8

-0.4

0

0.4

0.8

1.2

Figure 3.9: Air gap magnetic flux density generated by the permanent mag-nets including the effect of stator slotting, plotted for the com-putation plane at the inner diameter.

λ·B

PM

[T]

Radiusr [m] Azimuthϕ []0 45 90 135 180 225 270 315 360

0.050.0563

0.06250.0688

0.075-1

-0.5

0

0.5

1

Figure 3.10:Polar representation of the magnetic flux density in the centerof the air gap caused by the T-shaped permanent magnet takingthe effect of stator slotting into account.

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λ·B

PM

[T]

y [m] x [m]-0.075-0.05

-0.0250

0.0250.05

0.075

-0.075-0.05

-0.0250

0.0250.05

0.075-1

-0.5

0

0.5

1

Figure 3.11: Cartesian representation of the magnetic flux density in thecenter of the air gap caused by the T-shaped permanent magnettaking the effect of stator slotting into account.

Fig. 3.11 is represented by Fig. 3.12. This figure clearly displays the geometry ofthe permanent magnet, and the local decrease in the air gap magnetic flux densityby the effect of stator slotting can also be distinguished.

In a last step, the rotational motion of the rotor with respect to the stator is takeninto account by changing the mechanical angleθ in equation (3.35), imposing theright armature reaction through equation (3.42) and starting the calculation of the1D air gap magnetic flux density calculations at the different radii all over again.This calculation procedure forms the basis of all future calculations in this chapteron analytical modelling.

3.3.5 Flux Linkage and Back Electromotive Force

An important parameter in the design of electrical machinesis the back electromo-tive force. In the past, the calculation of the back electromotive force was limitedto the fundamental component. As this machine has a non-sinusoidal air gap mag-netic flux density, this will result in a non-sinusoidal backelectromotive force aswell. Therefore this research focusses on the waveform of the back electromotive

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y[m

]

x [m]

-0.075 -0.05 -0.025 0 0.025 0.05 0.075-0.075

-0.05

-0.025

0

0.025

0.05

0.075

Figure 3.12:Contour plot of the magnetic flux density in the center of theair gap caused by the T-shaped permanent magnet taking theeffect of stator slotting into account. Color levels are equal tothose in Fig. 3.11.

force as a function of time, including the full harmonic content.Prior to the calculation of the back electromotive force, the flux linkage with

each of the coils needs to be determined. The use of flux linkage with a coil, or asfrequently found in recent research, tooth coil winding, ismore appropriate thanusing the thin wire approximation as presented in [25]. The flux linkage with themth tooth coil winding is expressed by

ψc,m(θ) = Nc

n∑

i=1

rcp,i

m 2πQs∫

(m−1) 2πQs

Bz,i(ϕ)tcpdϕ (3.58)

whereBz,i is calculated through equation (3.56) evaluated for the actual rotor po-sition θ andtcp is defined by equation (3.2).

Consequently, the induced back electromotive force in one of the tooth coil

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windingsEc,m can be calculated by

Ec,m(θ) =dψc,m(θ)

dt=

dψc,m(θ)

dθ

dθ

dt=

dψc,m(θ)

dθω (3.59)

whereω is the electric pulsationi.e. pΩ.In Fig. 3.22 the flux linkage and induced back electromotive force in one of

the tooth coil windings are plotted. Note that only one electric period is shown inthis figure: when the waveforms in Fig. 3.22 are repeatedp times, there is a rotordisplacement of a full revolution. Therefore,θ is introduced to indicate the electricangle, whileθ is used to indicate the mechanical displacement of the rotor. Theelectric angle is passedp times for each full revolution of the rotor, hence,θ = pθ.This also results in a shift in the Fourier series spectrum. Similar to the electricangle, a harmonic order number in the electric reference frame ν is introduced.Here,ϕ = pϕ. This notation will be maintained in the future parts of thistext.

The phase back electromotive forceEph is found by addition of the back elec-tromotive forces produced by each individual tooth coil winding assigned to thatphase. The assignment of the different tooth coil windings to one of the phasesdepends on the pole pair number and the number of tooth coils,and is left for dis-cussion in Chapter 7. If only the fundamental component is considered, the phaseback electromotive force can be found directly by using the fundamental windingfactor, which also depends on the pole pair number and the number of tooth coils.The calculation of the fundamental winding factor is also left for discussion inChapter 7.

3.3.6 Electromagnetic Torque and Electric Power Output

The electromagnetic torque is given by

Te(θ) =

n∑

i=1

rcp,i · tcp,i2π∫

0

Bz,i(ϕ)KAi (ϕ)dϕ (3.60)

whereBz,i(ϕ) is defined by (3.56) andKAi (ϕ) by (3.42). This equation for the

electromagnetic torque is similar to the equation of the electric power, which isdirectly related to the phase back electromotive force and phase currents

Pe =

Qs∑

k=1

Ec,k · Ic,k =3∑

l=1

Eph,l · Iph,l. (3.61)

Equation (3.60) is preferred in this work as it is directly linked to the air gapmagnetic flux density.

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3.3.7 Electric Resistance and Resistive Heating

In electrical machines, a significant part of the total losses can be addressed to thecopper losses, also known as Joule losses. In the design of electrical machines,care should be taken that the temperature in the windings remains at a safe level.This maximum winding temperature is determined by the temperature class of theinsulation material around the conductors.

To keep the temperature in the conductors at a safe level, thearrangement ofthe conductors in the stator slots needs to be done with care,taking into accountfrequency dependent effects such as skin effect and proximity effect, and appropri-ate cooling systems can be introduced to dispose excessive heat. Whereas energyefficiency is important in this research, the generation of copper losses should bereduced rather than using extensive cooling of the copper winding.

Generally, winding designs start with the calculation of the dc-resistanceRdc

of the winding, by using the law of Pouillet

Rdc =lc (1 + kc (Tc − 20))

σcSc. (3.62)

In this equation,lc is the length of the conductor,Sc the cross section of the conduc-tor, σc the electric conductivity of the conductor material at 20C, kc the temper-ature coefficient of the conductor resistivity andTc the operational temperature inthe conductor inC. As mostly copper is used as conductor material, the valuesofthe electric conductivity and temperature coefficient are 5.96×107 and 3.8×10−3

1/C respectively.Equation (3.62) shows a linear dependence of the dc-resistanceRdc with the

temperature in the conductor material. This temperature dependence is plotted inFig. 3.13 for a copper conductor up to 165 corresponding with the maximumtemperature in a material with insulation class H. As can by observed in Fig. 3.13,the dc-resistance increases rapidly with increasing temperature. Therefore in highenergy efficient electrical machines, the winding temperature should be maintainedas low as possible. As mentioned earlier, one solution is found in the windingdesign itself. According to equation (3.62), this can be done by reducing the lengthof the conductor. This is realised by choosing concentratedfractional slot windingsrather than conventional windings as their end windings aremuch smaller. Anotheroption is the increase of the slot width. Although a larger slot width results in lowercopper losses, generally higher core losses are found due tothe higher magneticflux density. Therefore, the slot width should be optimized taking both copper andcore losses into account.

Although the maximum values of the operational temperatureare limited bythe temperature class, the actual operational temperatureshould be calculated iter-atively by coupling the electromagnetic model with a thermal model.

The preceding discussion assumed only time invariant currents, however, in theactual machine, alternating currents are present. As a result, the slotted conductors

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Rdc(T

c)/R

dc(20C)

Tc[C]

H-165

F-145

B-120E-115

A-100

20 40 60 80 100 120 140 160 1801

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Figure 3.13: Increase of the dc-resistance as a function of the temperature.Temperature classes are indicated.

exhibit an alternating magnetic field which results in a non uniform current densityin the conductors. Therefore, the copper losses can increase significantly.

Further explanation will briefly explain the influence of these generated alter-nating fields, while a detailed discussion is left for the Appendix [33].

Assume the situation in Fig. 3.14i.e. a slot filled with only two conductorsof which the undermost is carrying a currentIs and the uppermost is carrying nocurrent. According to equation (3.143), the undermost conductor will be subjectedto a magnetic slot field given by

H = −Is1

bs

sinh (kx)

sinh (khc)(3.63)

and the current density distribution in the undermost conductor is given by

J = −Isk

bc

cosh (kx)

sinh (khc). (3.64)

The current carried by the undermost conductor is displacedto the upper part ofthe conductor due to its own generated magnetic slot field. The effect of currentdisplacement due to its own magnetic slot field is called skineffect.

Although there is no net current carried by the uppermost conductor, the cur-rent carried by the undermost conductor will result in a nonzero current density, a

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bc

bs

hs

hc

hc

⊙

⊙Is

I = 0

J [A/m2]

H [kA/m]

J [A/m2]

H [kA/m]

x[m

m]

H [kA/m] / J [A/m2]

x[m

m]

H [kA/m] / J [A/m2]

×105

×105

-5 -2.5 0 2.5 5 7.5 10 12.5 15

-5 -2.5 0 2.5 5 7.5 10 12.5 15

0

5

10

15

15

20

25

30

Figure 3.14:Magnetic field and electric current density in a stator slot withtwo conductors; undermost is carrying a current, uppermostiscarrying no total current.

circulating current, in the uppermost conductor

J = −Isk

bc

sinh(

k(

12h− x

))

cosh(

khc2

) (3.65)

and a corresponding magnetic slot field

H = −Is1

bc

cosh(

k(

12h− x

))

cosh(

khc2

) . (3.66)

This effect is called the proximity effect, as the current inan underlying conductorresults in a nonzero current density in an overlying conductor.

The situation in which both conductors carry the same current i.e. the sameamplitude and phase, is presented in Fig. 3.15. In this situation the current densityin the uppermost conductor is not only influenced by its own current, but also bythe current carried by the underlying conductor.

Although this discussion is only done for two conductors carrying equal cur-rents, this theory can be extended to more than two conductors and different cur-rents.

Due to the skin and proximity effect, the equivalent resistanceRac will increasecompared to the dc-resistanceRdc. In the Appendix, the following expression was

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bc

bs

hs

hc

hc

⊙

⊙Is

I = Is

J [A/m2]

H [kA/m]

J [A/m2]

H [kA/m]x[m

m]

H [kA/m] / J [A/m2]

x[m

m]

H [kA/m] / J [A/m2]

×105

×105

-5 -2.5 0 2.5 5 7.5 10 12.5 15

-5 -2.5 0 2.5 5 7.5 10 12.5 15

0

5

10

15

15

20

25

30

Figure 3.15: Magnetic field and electric current density in a stator slot withtwo conductors; undermost and uppermost are carrying thesame current.

derived to relate the equivalent resistance to the dc-resistance

Rac = Rdc

[

GR(χ) +Is I

∗s +

12 (I I

∗s + I∗ Is)

I I∗G

′

R(χ)

]

(3.67)

where

GR(χ) = χsinh (2χ) + sin (2χ)

cosh (2χ)− cos (2χ)(3.68)

G′

R(χ) = 2χsinh (χ)− sin (χ)

cosh (χ) + cos (χ)(3.69)

In these equation,χ is the reduced conductor height according to equation (3.152).In Fig. 3.16, the functionsGR andG

′

R are plotted as a function ofχ.In the slots wherem superimposed conductors carry the same currentIs the

following situation occurs:for the first (undermost) conductor applies

Rac

Rdc= GR(χ) (3.70)

for the second conductor applies

Rac

Rdc= GR(χ) + 2G

′

R(χ) (3.71)

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GR(χ

)

χ

GR(χ

)

χ

0 1 2 3 4 50 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 3.16:FunctionsGR andG′

R as a function ofχ.

for the third conductor applies

Rac

Rdc= GR(χ) +

(

22 + 2)

G′

R(χ) (3.72)

for thenth conductor applies

Rac

Rdc= GR(χ) +

[

(n− 1)2 + (n− 1)]

G′

R(χ) (3.73)

= GR(χ) + [n(n− 1)]G′

R(χ) (3.74)

Therefore, for the whole of them superposed conductors, the resistance ratio be-comes

kR =Rac,tot

mRdc(3.75)

= GR(χ) +1

m

[

1 + 22 + 32 + . . .+ (m− 1)2]

+ (3.76)

[1 + 2 + 3 + . . .+ (m− 1)]G′

R(χ) (3.77)

= GR(χ) +m2 − 1

3G

′

R(χ) (3.78)

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In Fig. 3.17, the resistance ratio is plotted for various conductor heights forthe frequency corresponding to the first harmonic (fundamental) component. Foreach conductor height, the number of superposed coils is indicated. For machines

kR

hc [mm]

m=12

m=13

m=16

m=19

m=24

m=32m=48

0.5 0.75 1 1.25 1.5 1.75 21

1.2

1.4

1.6

1.8

2

Figure 3.17: Resistance ratio for various conductor heights for the testcasemachine. Corresponding number of superposed coils is indi-cated.

with multiple poles at relatively high speed, Fig. 3.17 indicates that the increasein resistance due to skin and proximity effect is significant. The choice towardssmaller conductor heights is crucial to reduce the skin and proximity effect. Astoo many conductors in series will result in an excessive output voltage, multipleconductors with smaller heights are connected in parallel.When making parallelbranches, care should be taken that the induced voltage is equal in the differentparallel conductors. Unequal induced voltages in the different parallel conductorswould result in a circulating current, and thus, increase the copper losses. Thanksto the modular concept of the yokeless and segmented armature axial flux machinetopology, the winding can be put on the stator cores out of themachine. There,it is possible to wind in an automated procedure with multiple wires at the sametime. This winding procedure diminishes the risk of circulating currents in parallelbranches.

Although smaller conductor heights and parallel branches seem to overcomethe skin and proximity effect, a lower filling factor of the slots with conductive ma-terial will be achieved as the relative percentage of insulating material with respectto the conductor section increases.

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3.3.8 Stator Core Loss

Next to the copper losses, the stator core losses have a not negligible contributionto the total losses in electrical machines. Modelling of thelaminated silicon steelmaterial used in the stator core is examined extensively in Chapter 4. Therefore,this section is only giving a short introduction to the stator core loss calculation.

Despite many papers use the Steinmetz equations and variants on this empir-ical equation, here, the principle of loss separation is introduced. Here, the totalstator core loss is composed of a hysteresis loss, a classical loss and an excess losscomponent:

PFe = Phy + Pcl + Pexc. (3.79)

In Chapter 4 it will be evinced that no minor loops are presentin the magneticflux density loci, and hence, the hysteresis lossphy is determined by the peak valueof the magnetic flux densityBp

< phy >= kFe,1BkFe,2p f (3.80)

wheref is the frequency corresponding with the fundamental component.< phy >indicates the time average value of the hysteresis loss per kilogram over the funda-mental period.

The instantaneous classical power losses per kilogram depend on the timederivative of the magnetic flux densityB(t)

pcl(t) = kFe,3

(

dB

dt

)2

(3.81)

where the constantkFe,3 is determined by the conductivity and the thickness of thelaminated silicon steel material.

In [34], an expression of the instantaneous excess loss per kilogram was intro-duced

pexc(t) = kFe,4

(√

1 + kFe,5

∣

∣

∣

∣

dB

dt

∣

∣

∣

∣

− 1

)

∣

∣

∣

∣

dB

dt

∣

∣

∣

∣

. (3.82)

In this equation, the parameters rely on the statistics of the simultaneously activemagnetic objects,cfr. Chapter 4.

Whereas, in Chapter 4 for each computation plane the loss value is evaluated inmany points in the stator core geometry, here, a single pointevaluation of the lossper computation plane is done. After multiplication of the loss components with themassGi of the stator core assigned to that computation plane and summation over

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3.4 Finite Element Method 57

the computation planes, the components presented by equation (3.79) are found

Phy =n∑

i

< phy > Gi (3.83)

Pcl =

n∑

i

1

T

T∫

0

pcl(t)dt

Gi (3.84)

Pexc =

n∑

i

1

T

T∫

0

pexc(t)dt

Gi. (3.85)

These power losses are expressed in Watts.

3.4 Finite Element Method

3.4.1 3D Modelling

Despite the existence of 3D finite element solvers, the number of papers in which3D finite element models are used, is limited [35–41]. Moreover, most of thesepapers use the 3D finite element analysis to present an illustrative distribution of themagnetic flux density in the stator core made of soft magneticcomposite material,rather than performing accurate calculation of the magnetic torque in the air gap.Very long simulation times and limited accuracy discouragethe use of these 3Dfinite element analysis. Therefore, also in this work primacy is given to multislice2D modelling. Only in Chapter 5, full 3D finite element computation is used as averification model.

3.4.2 Multislice 2D Modelling

In the multislice 2D modelling using finite element analysis, the cylindrical sur-faces defined at different radii, are unrolled to planar surfaces as illustrated in Fig.3.1. In these computation planes, a 2D finite element solver is applied in a cartesiancoordinate system. This finite element solver calculates the magnetic vector poten-tial over the defined geometries. Similar to the analytical modelling, the magneticvector potential has only a component perpendicular to the computation plane,i.e.thez-direction

A = Azez. (3.86)

and the magnetic flux density defined by equation (3.5), can beexpressed by

B = Bxex +Byey. (3.87)

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8

1

2

3

3

4

5

6

7

x

y

Figure 3.18:2D finite element model. Dirichlet boundary condition (1),Neumann boundary condition (2), periodic boundary condition(3), air gap (4), rotor (5), stator elements (6), windings (7),permanent magnets (8).

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Thex- andy-direction are defined in Fig. 3.18. Equation (3.5) directlycalculatesthe magnetic flux density components from the solution of themagnetic vectorpotential

Bx =∂Az∂y

(3.88)

and

By = −∂Az∂x

. (3.89)

The implementation in finite element analysis, requires theimposition of thecorrect boundary conditions at the geometry edges and material properties to thedifferent subdomains. An illustrative overview of the finite element model is sug-gested in Fig. 3.18. To reduce the size of the model, axial magnetic symmetrywas introduced. Therefore, only half of the axial length of the machine needs to bemodelled. In order to take this axial symmetry into account,the Neumann bound-ary condition is applied along the symmetry edge indicated by (2). This Neumannboundary equation expresses that the component of the magnetic field tangential tothis boundary vanishes

n×H = 0. (3.90)

Furthermore it is assumed that the magnetic field lines do notexit the outer sideof the rotor disc. Therefore, the Dirichlet boundary condition is expressed at theedge indicated by (1). The Dirichlet boundary condition imposes that the normalcomponent to the boundary vanishes and is expressed through

Az = 0. (3.91)

As the original cylindrical surface is cut to obtain the planar representation,continuity in the solution for the magnetic vector potential along this edge, in-dicated by (3), should be included in the finite element model. This is done byintroducing a periodic boundary condition. IfAz1 andAz2 are the magnetic vectorpotentials at both sides of the model, then the periodic boundary condition requiresthat

Az1 = Az2. (3.92)

along this edge. The periodic boundary condition ensures continuity of the mag-netic field lines crossing the edge.

Next to the boundary conditions, different material properties are assigned tothe defined subdomains. In the air gap (4), the equation for the magnetic vectorpotential is defined by

∇×(

1

µ0∇× A

)

= 0. (3.93)

In the permanent magnets (8), the magnetic remanenceBPM as well as the rela-tive permeabilityµPM (estimated at 1.05 for NdFeB) are taken into account when

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defining the magnetic vector potential

∇×(

1

µ0µPM∇× A − BPM

)

= 0. (3.94)

As the winding sections (7) carry a current densityJ, the equation for the magneticvector potential in this region becomes

∇×(

1

µ0∇× A

)

= J. (3.95)

Finally there are the subdomains defining the stator cores and the rotor disc. Asboth parts are made of a steel grade with good relative permeability µr, the mag-netic vector potential in these subdomains is expressed by

∇×(

1

µ0µr∇× A

)

= 0. (3.96)

The relative permeability can be modelled by a constant value µr, when linearbehaviour of the material is sufficient in the modelling, or by a saturation dependentrelative permeabilityµr(B). The modelling of nonlinear behaviour is an importantadvantage of finite element computations over analytical modelling. Very oftenlocal saturation of the magnetic material results in additional harmonics in the airgap (saturation harmonics) or lowers the electromagnetic torque produced by themachine. Therefore, the use of a nonlinear material model inthe stator cores ispreferred. For the rotor disks a constant permeability is assumed. While this modelis still isotropic, all magnetic properties are the same in all directions which is agood assumption for non oriented steel grades. Anisotropicmaterial modelling isintroduced in Chapter 4 to model grain oriented steel grades.

All previously introduced equations for the magnetic vector potential were onlyvalid for static simulations. In this research, the multislice 2D simulations are onlyapplied for static simulations. Although only static simulations are used, motionof the rotor discs with respect to the stator can be included by redrawing the finiteelement model for the different positions of the rotor discs. In these static simu-lations, time dependent eddy currents in electric conductive materials such as theneodymium permanent magnets, cannot be calculated immediately. Nevertheless,time dependent properties can be calculateda posterioriusing the data retrievedby static simulations. By doing so, induced electromotive forces, eddy currents inthe laminations and the permanent magnets can be calculated. Within the scope ofthis procedure, it is assumed that the influence of the magnetic field produced bythe induced eddy currents is minor to the magnetic fields obtained through staticsimulations. This is the so called resistance limited approach.

In the following chapters, only finite element analysis is used for modellingwhereas the role of analytical modeling in these chapters isminor, only helping tounderstand the ruling phenomena. As in these chapters an elaborated discussion

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on the different loss phenomena in the machine is given, no further explanation isgiven in this section. Only the electromagnetic torque calculation is discussed inthe next subsection as it differs from the traditional approach.

3.4.3 Mathematical Modelling of Nonlinear Material

To express the nonlinearity and saturation feature of the magnetic material in thestator cores, the magnetic fieldH and the magnetic flux densityB are linkedthrough1

H(B) =1

µ0µr(B)B (3.97)

in which the relative magnetic permeabilityµr depends on the actual value of themagnetic flux density. For numerical stability reasons, themathematical expressionfor the magnetic permeabilityµr(B) as well as its first derivativedµr(B)

dB need to becontinuous functions.

In literature, many analytical approximations for the single value magnetisa-tion curve are found [42–45]. Here, a (non-full) power series equation is used.The single valued nonlinear constitutive relation of the soft magnetic material ismodelled by three material dependent parametersH0,B0 andν [46]:

H

H0=

(

B

B0

)

+

(

B

B0

)ν

. (3.98)

Consequently, the expression for the relative magnetic permeability of the softmagnetic materialµr is given by

µr(B) =B0

(

H0µ0

(

1 +(

BB0

)ν−1)) . (3.99)

The parametersH0 andB0 determine the initial value of the relative perme-ability of the materialµr,ini,

µr,ini =B0

H0µ0. (3.100)

WhenB = B0 the magnetic permeability of the soft magnetic value reaches halfof its maximum value.

The third parameterν defines the slope of the variation of the relative magneticpermeability as a function of the magnetic flux densityB. The higher the value ofν, the steeper the slope.

1This relation is preferred overB(H) = µ0µr(H)H as the magnetic flux density B can beobtained directly from the solution for the magnetic vectorpotentialB = ∇× A.

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3.4.4 Maxwell Stress Harmonic Filter Method

To calculate the electromagnetic torque using finite element analysis, expressionsbased on a closed surface integration in the air gap and expressions based on air gapvolume integration are used [47]. In case of the 2D computation planes, the closedsurface integration becomes a line integral in the air gap. The general expressionfor the Maxwell stress tensor is expressed by

TMax =

∮

S

1

µ0(B · n)B− 1

2µ0B

2n

dS. (3.101)

Only the component responsible for the electromagnetic torque is retained, theother component results in an axial attraction between stator and rotor and is notconsidered in this work:

FMax =1

µ0

2pτp∫

0

BxBydy (3.102)

whereBx andBy are expressed by (3.88) and (3.89) respectively. In Fig. 3.19, (1)indicates a path along which the line integral is calculated. Retrieving the value ofthe electromagnetic force includes integration of the magnetic flux density compo-nents over the closed integration line. Hereby, some remarks should be considered.As the solution for the flux density components was exact, anyintegration along

1

2

2

Figure 3.19:Definition of the lines used for the evaluation of the electro-magnetic torque; evaluation of the components of the magneticflux density in the Maxwell stress tensor method (1), evalu-ation of the magnetic vector potential in the Maxwell stressharmonic filter method (2).

a closed path in the air gap would result in the same value for the Maxwell stress

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tensor. Nevertheless, in most of the finite element solutions, these solutions areonly approximates. Therefore, each closed integration path in the air gap regionmay result in a different value for the Maxwell stress tensor.

If, for incidence, first order triangular elements were usedin the finite elementmodel, the magnetic vector potential varies linearly over each mesh element. Asthe components of the magnetic flux density are derivatives of the magnetic vectorpotentialA, cfr. equation (3.88) and (3.89), they will have a piecewise constantvalue over the mesh elements. In the air gap, where on a relatively small distance,boundaries between materials with very different permeability in the x-directionare present, this may result in a significant error in the magnetic flux density com-ponentBy.

A significant factor in the accuracy of the Maxwell stress tensor along the in-tegration path, is attributed to the quality of the mesh. Thecoarser the mesh inthe air gap region, the less accurate the evaluation of the Maxwell stress will be.Hence, accurate evaluation of the Maxwell stress tensor is obtained by increasingthe mesh elements in the air gap region. However, as this increases the simulationtime, a more practical solution should be introduced.

Solutions were introduced in [47, 48] for radial machines using a cylindricalcoordinate system. The idea presented in [47] is adopted in this paper to axial fluxmachines, in which a cartesian coordinate system is used in the different compu-tation planes. The calculation of the electromagnetic torque in [47] is called theMaxwell stress harmonic filter (HFT) method. Like in equation (3.102), the forceacting in the air gap is defined by

FHFT =1

µ0

2pτp∫

0

BxBydy. (3.103)

Different from the classical approach is thatBx andBy are not evaluated at the linein the air gap, but use is made of the formulation of the magnetic vector potential inthe air gap. The solution for the magnetic vector potential in a cartesian coordinatesystem is given by

A (x, y) = a0 +

∞∑

ν=1

[

aν exp

(

νπ

pτpx

)

+ bν exp

(

− νπ

pτpx

)]

·[

cν cos

(

νπ

pτpy

)

+ dν sin

(

νπ

pτpy

)]

. (3.104)

In this equation, four coefficients need to be identified. Therefore, the magneticvector potential is recorded along 2 lines in the air gap by making use of the solu-tion obtained through finite element analysis. These two lines are indicated in Fig.3.19 by (2). The magnetic vector potential recorded along the two lines in the air

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gap can be written as a Fourier series

A (x1, y) = a01 +

∞∑

ν=1

[

aν1 cos

(

νπ

pτpy

)

+ bν1 sin

(

νπ

pτpy

)]

(3.105)

and

A (x2, y) = a02 +

∞∑

ν=1

[

aν2 cos

(

νπ

pτpy

)

+ bν2 sin

(

νπ

pτpy

)]

. (3.106)

As these magnetic vector potential equations are a result ofFourier series expan-sion from finite element data, all coefficients are known.

The cross products of the coefficients in the magnetic vectorpotential equation(3.104) can be expressed by making use of the Fourier series expansion of therecorded magnetic vector potentials along the two predefined lines by

aνcν =1

ζν

[

aν1 exp

(

− νπ

pτpx2

)

− aν2 exp

(

− νπ

pτpx1

)]

(3.107)

bνcν = − 1

ζν

[

aν1 exp

(

νπ

pτpx2

)

− aν2 exp

(

νπ

pτpx1

)]

(3.108)

aνdν =1

ζν

[

bν1 exp

(

− νπ

pτpx2

)

− bν2 exp

(

− νπ

pτpx1

)]

(3.109)

bνdν = − 1

ζν

[

bν1 exp

(

νπ

pτpx2

)

− bν2 exp

(

νπ

pτpx1

)]

(3.110)

where

ζν = exp

(

νπ

pτp(x1 − x2)

)

− exp

(

− νπ

pτp(x1 − x2)

)

. (3.111)

Substitution of these factors in the general expression forthe magnetic vector po-tential in the air gap (3.104), and making use of equations (3.88) and (3.89) whichrelate the magnetic flux density to the magnetic vector potential, the expression forthe force (3.103) becomes

FHFT =2π2

µ0pτp

∞∑

ν=1

ν2 (bν2aν1 − aν2bν1) exp(

νπpτpg)

exp(

νπpτp

2g)

− 1. (3.112)

In this equation it is assumed thatx2 = x1+ g, g i.e. the two lines along which thevector potential is evaluated, are situated at both sides ofthe air gap. The Maxwellstress harmonic filter results in an expression for the forceacting in the air gap, bysummation over the different harmonic components. This expression also allows

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3.5 Comparison of Example Results 65

to distinguish the contributions of the different harmoniccomponents in the forcein the air gap. Due to the recording of the magnetic vector potential in the Maxwellstress harmonic filter, rather than making use of the recorded magnetic flux densitycomponents, a more accurate prediction of the electromagnetic force is possible.

With respect to the multislice 2D approximation, the resultant electromagnetictorque is found by multiplying the electromagnetic force ofeach computation planeFHFT,i by the thicknesstcp of the computation plane and the radiusrcp,i at whichthe computation plane is defined

Te =

n∑

i=1

rcp,i · (tcpFHFT,i) . (3.113)

3.5 Comparison of Example Results

As analytical models are built upon some idealized assumptions, the accuracy ofthese models should be checked. This section compares the analytical results withthose obtained through multislice 2D finite element analysis.

A first crucial element is the accurate prediction of the air gap magnetic fluxdensity. If the analytical model predicts the air gap magnetic flux density ade-quately, it can be directly used in Chapter 5 to calculate theeddy current lossesin the permanent magnets. In Fig. 3.20 and Fig. 3.21, the air gap magnetic fluxdensity are presented for respectively no load and load. Theanalytical expres-sion (3.56) of the air gap magnetic flux density is compared with those obtainedthrough finite element analysis. The use of a permeance function to model thestator slotting is acceptable as a good correspondence between the analytical andthe finite element method is found. Therefore, in chapter 5, no preference of thefinite element over the analytical modelling is given when the axial component ofthe magnetic field is evaluated in order to calculate the eddycurrents induced inthe permanent magnets.

Another important quantity is the calculation of the back electromotive force.The tooth coil flux linkage and corresponding tooth coil backelectromotive force,obtained through analytical and finite element modelling, is plotted in Fig. 3.22.As for the air gap magnetic flux density, a very good correspondence betweenboth modelling techniques is found. Considering the back electromotive force inFig.3.22, the analytical model gives an accurate prediction with respect to the har-monic content present in the waveform. Evaluation of the electric output power byequation (3.61), will thus result in a good estimation of theinstantaneous electricpower by the analytical model as well.

An important remark in this comparison is that linear material behaviour isconsidered in the finite element model. When taking into account saturation of themagnetisation in the stator core elements and rotor, the magnetic flux density in theair gap may decrease due to local saturation and additional harmonic contribution,

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PSfrag

fem

ana

BPM

z·λ

[T]

Azimuthϕ[]

90 105 120 135 150 165 180 195 210 225-1.5

-1

-0.5

0

0.5

1

1.5

Figure 3.20:Comparison of the air gap magnetic flux density obtainedthrough analytical and finite element analysis for no load. Plot-ted for the computation plane at the inner diameter.

fem

ana

BA z·λ

[T]

Azimuthϕ []

90 105 120 135 150 165 180 195 210 225-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Figure 3.21:Comparison of the air gap magnetic flux density obtainedthrough analytical and finite element analysis for load. Onlythe contribution by the armature reaction is shown. Plottedforthe computation plane at the inner diameter.

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3.6 Energy Efficiency 67

PSfrag

Ec [mWb], fem

ψc [mWb], fem

Ec [mWb], ana

ψc [mWb], anaψc

[mW

b]/E

c[V

]

θ [electric degrees]

0 45 90 135 180 225 270 315 360-60

-40

-20

0

20

40

60

Figure 3.22: Comparison of the flux linkage and back electromotive forceobtained through analytical and finite element analysis.

so called saturation harmonics, may be introduced. Nevertheless, the accuracy ofthe analytical model in the design parameters of interest ispositively evaluated.

3.6 Energy Efficiency

Central in the design of the axial flux PM machine is the energyefficiency. Thedefinition of an efficient energy conversion depends on the use of the machine asmotor or generator and is directly related to the losses present in the machine.

3.6.1 Motor Mode

If the machine is used as a motor, electric powerPe is transferred to the terminals ofthe machine. Very often a variable frequency drive (VFD) (cfr. Chapter 8) is usedto provide this proper electric power. The efficiency of the energy conversion is de-termined by the resultant mechanical output powerPm that can be produced for thatgiven electric power input. Due to losses present in the machine, the mechanicaloutput will be reduced with respect to the electric input. When discussing energyefficiency, very often the Sankey diagram is used to indicatethe present losses.The Sankey diagram for a motor is illustrated in Fig. 3.23. From the incomingelectric power, the copper losses and the stator core losseslimit the power that istransferred through the air gap. The mechanical power transferred to the shaftPm

decreases further due to the presence of eddy currents in thepermanent magnets

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Pe Pm

PFe Pc

PPM Pfr

Figure 3.23: Sankey diagram: motor mode.

and rotor. Moreover the rotation of the rotor results in friction lossesPfr. In thesefriction losses the ventilation losses, the losses in the bearings as well as the losseswith respect to reversible mechanical stresses are present. The aim of this researchis not to estimate these friction losses, however, when performing measurementson a test setup, the existence of these losses is very often not negligible.

The efficiency of the energy conversion from electric into mechanical powercan hence be expressed by

ηM =Pm

Pe=Pe − Pc − PFe − PPM

Pe. (3.114)

3.6.2 Generator Mode

In generator mode, the incoming mechanical powerPm provided at the shaft needsto be converted in outgoing electric power as well as possible. The Sankey diagramfor generator mode is illustrated in Fig. 3.24. Here, the input mechanical poweris decreased by the friction losses and the power losses due to eddy currents in thepermanent magnets and rotor. The remaining power is transferred through the airgap to the stator. In the stator, the copper losses and the stator core losses decreasethe power output. Only the remaining electric powerPe is provided to the terminalsof the machine. With respect to the integration of axial flux PM machines, the wildAC, i.e. AC power with wind speed dependent frequency and amplitude,is veryoften rectified by a diode rectifier [49].

The efficiency of the energy conversion from mechanical intoelectric power

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3.6 Energy Efficiency 69

can hence be expressed by

ηG =Pe

Pm=Pm − Pc − PFe − PPM

Pm. (3.115)

Pm Pe

PFePc

PPMPfr

Figure 3.24: Sankey diagram: generator mode.

3.6.3 Important Remarks on Energy Efficiency

As the different losses are depending on both speed and load,the efficiency of theenergy conversion needs to be evaluated for each working condition. The plot ofthe energy efficiency as a function of the speed and load results in an efficiencymap.

In this definition of energy conversion, the focus is only on the machine. Asmentioned, power electronics are connected to the terminals of the machine ex-changing energy with the power grid and a load can be connected to the mechan-ical side. So in an optimisation, not only the energy efficiency of the machineshould be considered, but also the total entity comprising the application, the elec-trical machine and the power electronics. This procedure isoften calleddrive trainoptimisation.

In the mathematical modelling, the analytical as well as finite element based,there are no losses included. Losses are calculateda posteriori, based on saved dataretrieved through the mathematical model. As the model is lossless, the differencebetween input power and output power vanishes.

Within the focus on mathematical (finite element) modelling, increasing theenergy efficiency is obtained by reducing the power losses inthe machine and by

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maximisation of the electromagnetic torque for a given current density. In thefollowing chapters, in which each an energy efficiency improving measure is in-troduced, the current waveform is maintained while decreasing one particular losscomponent, or the current waveform is maintained while increasing the electro-magnetic torque.

3.7 Parameter Optimisation

The benefits of a fast evaluating, but accurate, analytical model are found in prelim-inary designs which include optimisation. Here, analytical models provide rapidlyinsight on the influences of different parameters in the design. To illustrate thebenefits of an analytical model, while showing the influence of some importantgeometrical parameters, such a design process is topic of this section. This designprocedure has resulted in the parameter set in Table 2.1.

In the optimisation example only a limited set of parametersis considered: theouter diameterDo, the inner diameterDi, the axial length of a stator core elementht and one parameter that specifies the ratio copper to iron section: the slot widthbs. Generally the outer diameter is limited with respect to theintegrability in theapplication of interest. Therefore, the maximum value of the outer diameter was setto 148 mm. The axial length of the core elements and the slot width were chosen asdesign parameters as they are strongly related to the governing copper losses andcore losses in the stator element. The smaller the slot section, i.e. the axial lengthtimes the slot width, the lower the iron core losses but the higher copper losses andvice versa.

During the optimisation process, in which a genetic algorithm is used, the re-lation between the energy efficiency and the active mass of the machine is consid-ered. Active mass covers the parts of the machine which take part in the electro-magnetic energy conversioni.e. permanent magnets, rotor, stator cores and wind-ings. In the optimisation process, certain quantities suchas the output power andoutput voltage are fixed, while the number of windings and thephase current aredependent variables.

In order to force the active mass to its prescribed value, oneof the four designparameters is expressed as a function of the other three. Theoptimisation processin which the energy efficiency is considered, is repeated foreach value of the activemass.

The effect of the prescribed active mass on the energy efficiency, power lossesin the stator cores and conductors, as well as the values of the design parametersthat correspond to the design with the best energy efficiency, are presented in Fig.3.252.

2The optimisation of the design variables is performed for the original generator (2008 version)introduced for the combined heat an power application studied by prof. dr. ir. Alex Van den Bosscheand prof. dr. ir. Peter Sergeant at the at department Electrical Energy, Systems and Automation.These application was originally limited to 3.6 kW and had a rated speed of 2000 rpm. Later, the

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3.7 Parameter Optimisation 71

PFe

Pc

Slotwidth[mm]

Act

ive

mas

s[k

g]

Axiallengthcore[mm]

Act

ive

mas

s[k

g]

Innerdiameter[mm]

Act

ive

mas

s[k

g]

Outerdiameter[mm]

Act

ive

mas

s[k

g]

Powerlosses[W]

Act

ive

mas

s[k

g]

Efficiency[%]

Act

ive

mas

s[k

g]

03

69

1215

03

69

1215

03

69

1215

03

69

1215

03

69

1215

03

69

1215

6789101112

0255075100

125

150

60708090100

110

120

140

142

144

146

148

150

152

0

100

200

300

400

500

600

85

87.590

92.595

97.5

100

Fig

ure

3.25

:Th

eef

fect

of

the

pre

scri

bed

activ

em

ass

on

the

ener

gy

effic

ien

cy,

pow

erlo

sses

inth

est

ato

rco

res

and

cop

per

loss

esas

wel

las

the

valu

eso

fth

ed

esig

np

aram

eter

sth

atco

rre

spo

nd

toth

ed

esig

nw

ithth

eb

este

ner

gy

effic

ien

cy.

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i


With respect to the energy efficiency as a function of the active mass, it canbe observed that over a wide range of the active mass the energy efficiency isinfluenced in a limited way. This means that the active mass can be reduced signif-icantly without affecting the energy efficiency. This results in a machine which hasa comparable energy efficiency while the power density is higher. This observationis crucial for applications which require a great energy efficiency combined with ahigh power density, such as electric vehicles.

The considered power losses in the machine,i.e. the copper losses and thelosses in the stator core elements, show a huge decrease of the copper losses withincreasing mass, while the core losses become higher with increasing active mass.The core losses are directly related to the mass of the statorcore elements which islinked to the total active mass. For very low active mass, thestator slot openingsare very wide with respect to the core cross section, which may result in saturationof the magnetic material. As saturation is not taken into account in the analyticalmodel, the optimisation results may not be very reliable. Nevertheless, this verylow active mass also results in very high copper losses, which make these machinedesigns less attractive for applications in which an excellent energy efficiency isindispensable.

With respect to the design parameters, energy efficient machines with a lowactive mass have a short ring shaped geometry: large inner diameter combinedwith a large outer diameter and short axial length; energy efficient machines witha high active mass have a long cylindrical shaped geometry: small inner diametercombined with large outer diameter and high axial length. Note that in generalthe optimisation of the energy efficiency results in an outerdiameter that sticksto the prescribed upper boundary limit. Indeed, at the outerdiameter the torqueproduction is more effective than at the inner diameter.

As a conclusion of this parameter optimisation consideringthe mass influenceof the machine, the parameter set as presented in Table 2.1 was selected.

3.8 Conclusion

The mathematical modelling of the axial flux PM machine was elaborated in thischapter. A multislice 2D modelling technique, in which only2D mathematicalcalculations are involved, is introduced to model the inherent 3D structure of theaxial flux machine topology. Although the majority of the calculations in this workare performed with finite element analysis, an analytical model is introduced as itgives a good insight in the ruling physics, whereas finite element analysis ratherbehaves like a black box.

In the analytical modelling, an equivalent current sheet isused to model thepermanent magnets and armature reaction. The influence of the stator slotting ef-

rotational speed was increased to 2500 rpm and output power up to 5 kW while maintaining theoriginal set of geometry parameters.

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3.8 Conclusion 73

fect is modelled by means of a relative permeance function. Acomparison ofthe simulation results obtained through analytical and finite element computationwas presented to evaluate the accuracy of the analytical model. Due to its lowcomplexity, the analytical based modelling evaluates fastand is perfectly suitablefor preliminary designs that may include optimisation processes. Such a prelimi-nary design study, considering the mass influence on the energy efficiency of themachine, is illustrated at the end of this chapter. Althoughits particular use in pre-liminary studies, more advanced studies in the next chapters will use the nonlinearfinite element analysis based modelling.

Appendix

Expression forβ(z) using the Schwarz-Christoffel Transformation

The determination ofβ(z) derived by [29], is adopted in this Appendix. How-ever, as this theory was intended for radial-flux machines, minor adaptations areintroduced to meet the needs for axial-flux variants.

Based on the idealized single-slot model of Fig. 3.26, the interior airgap regionin the complex planez is first transformed to the upper half of the complex planew, and subsequently to the complex planet, by the conformal transformation.

(w < −1) (w > 1)

(−1 < w < 0) (0 < w < 1)

ϕ

ϕmo

ψ

v

u

u1

u2 u3 u4 u5

-a -1 a1

x

yz1

z2z3 z4z5

bo

g′

(a) (b) (c)

00 0

Figure 3.26: Conformal transformation (a)z-plane, (b)w-plane, (c)t-plane.

The first transformation is given by

z =b0π

arcsin(w

a

)

+g′

b0ln

[√a2 − w2 + 2g′

b0w

√a2 − w2 − 2g′

b0w

]

(3.116)

which after some deduction becomes

z =b0π

j

2ln

(

1− j w√a2−w2

1 + j w√a2−w2

)

+ j2g′

b0arctan

( −j2g′wb0√a2 − w2

)

(3.117)

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wherej is the complex operator, and

a =

√

1 +

(

2g′

b0

)2

(3.118)

The second transformation is

t =ϕPM,0

πln

(

1 + w

1− w

)

(3.119)

whereϕPM,0 = (BPM/µ0µPM)hPM is the magnetic potential between the statorand rotor iron surfaces. The flux density in thez-plane is then given by

|Bz| = µ0|Hz|= µ0

∣

∣

∣

∣

dt

dz

∣

∣

∣

∣

= µ0

∣

∣

∣

∣

dt

dw

∣

∣

∣

∣

∣

∣

∣

∣

dw

dz

∣

∣

∣

∣

(3.120)

= µ0ϕmo

g′1

√

1 +(

b02g′

)2−(

b02g′w

)2(3.121)

By lettingw = ±1, the maximum flux densityBz max atx = ±∞ is obtained as:

Bz max = µ0ϕPM,0

g′. (3.122)

The variation of flux density against axial position exhibits a minimum at the airgap side of the stator slots, the value depending on the axialdistance, and is ob-tained by lettingw = jv, i.e.,

Bz min = µ0ϕPM,0

g′1

√

1 +(

b02g′

)2+(

b02g′ v

)2(3.123)

= Bz max1

√

1 +(

b02g′

)2(1 + v2)

(3.124)

and

β(z) =Bz max −Bz min

2Bz max(3.125)

=1

2

1− 1√

1 +(

b02g′

)2(1 + v2)

(3.126)

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3.8 Conclusion 75

v is determined by solving (3.117) withz = jy andw = jv

jy =b0π

j

2ln

(

1− j jv√a2+v2

1 + j jv√a2+v2

)

+ j2g′

b0arctan

( −j2g′(jv)b0√a2 + v2

)

. (3.127)

Therefore

yπ

b0=

1

2ln

(√a2 + v2 + v√a2 + v2 − v

)

+2g′

b0arctan

(

2g′

b0

v√a2 + v2

)

(3.128)

wherey = g′ − z.At the rotor surfaceBz min is determined by settingv = 0, to give

Bz min = Bz max1

√

1 +(

b02g′

)2. (3.129)

Hence

β(z)|z=g′=1

2

1− 1√

1 +(

b02g′

)2

. (3.130)

Skin and Proximity Effect in the Slotted Conductors

In this section, the effect of an alternating magnetic slot field on the current den-sity distribution in slotted conductors is discussed. In this discussion, the problemdefinition is simplified by making some assumptions:

• the permeability of the core material is infinite;

• the core is perfectly laminated such that no eddy currents in the core materialexist;

• the temperature of all conductors in one slot is equal.

The variables used in this discussion are defined in Fig. 3.27.The Maxwell equations and the constitutive relations result in

∂Hz

∂x= −bc

bsJy (3.131)

and∂Jy∂x

= −µoσc∂Hz

∂t. (3.132)

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bc

bs

hs

hc

hc

⊙

⊙Is

I x

Figure 3.27: Indication of the parameters defining the stator slot geometry.

As the magnetic slot field varies sinusoidally in time with frequencyω, the timeharmonic equations for the magnetic field and current density become

Hz = H(x) exp (jωt) (3.133)

Jy = J(x) exp (jωt) (3.134)

Therefore the time harmonic equivalents of relation (3.131) and (3.132) become

dH

dx= −bc

bsJ (3.135)

anddJ

dx= −jωµoσcH (3.136)

and result ind2H

dx2= k2H (3.137)

The general solution for this equation is

H = C1 cosh (kx) + C2 sinh (−kx) (3.138)

where

k =1 + j

δ′(3.139)

with δ′

the skin depth of the conductors material

δ′

=

√

2

ωµoσ′

c

. (3.140)

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3.8 Conclusion 77

In this equationσ′

c is used instead ofσc, as it takes the paretical filling of the slotwidth into account according to Fig. 3.27

σ′

c = σcbcbs

(3.141)

The constantsC1 andC2 are defined by the boundary conditions, which are

bsH|x=0= −Is; bsH|x=hc= − (Is + I) . (3.142)

Some calculations result in an expression for the magnetic slot field and currentdensity in the range0 ≤ x ≤ hc:

H = − 1

bs

(

Isinh (kx)

sinh (khc)+ Is

cosh(

k(

hc2 − x

))

cosh(

k hc2)

)

(3.143)

J = − bsbc

dH

dx(3.144)

= − k

bs

(

Icosh (kx)

sinh (khc)− Is

sinh(

k(

hc2 − x

))

cosh(

k hc2)

)

. (3.145)

The time average of the complex power per unit area that enters a specifiedvolume is given by the Poynting vector

S =1

2ρcJ H

∗ (3.146)

Subsequently a volume with lengthlc in the slot and between the horizontal planesx = 0 andx = hc is considered, and the Poynting vector Sis integrated over thisvolume

Svol = −bslc2σc

[J(hc) H∗(hc)− J(0) H∗(0)] (3.147)

which becomes after some calculations

Svol = − lc2σcbch

I I∗khc coth(khc)

+

[

Is I∗s +

1

2(I I∗s + I∗ Is)

]

2khc tanh

(

1

2khc

)

. (3.148)

In this equation

Rdc =lc

σcbchc(3.149)

represents the resistance of the conductor in case the current density distributionwas uniformi.e. the dc-resistance.

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As khc = (1 + j)(hc/δ′

) is complex, the following notations are introduced

khc coth(khc) = GR(χ) + jGL(χ) (3.150)

2khc tanh

(

1

2khc

)

= G′

R(χ) + jG′

L(χ) (3.151)

where

GR(χ) = χsinh (2χ) + sin (2χ)

cosh (2χ)− cos (2χ)(3.152)

GL(χ) = χsinh (2χ)− sin (2χ)

cosh (2χ)− cos (2χ)(3.153)

G′

R(χ) = 2χsinh (χ)− sin (χ)

cosh (χ) + cos (χ)(3.154)

G′

L(χ) = 2χsinh (χ) + sin (χ)

cosh (χ) + cos (χ)(3.155)

In these equations,χ = hc/δ′

is called the reduced conductor height.As S = P + jQ, expressions for the activeP and reactive powerQ can be

obtained

P =1

2Rdc

I I∗GR(χ) +

[

Is I∗s +

1

2(I I∗s + I∗ Is)

]

G′

R(χ)

(3.156)

Q =1

2Rdc

I I∗GL(χ) +

[

Is I∗s +

1

2(I I∗s + I∗ Is)

]

G′

L(χ)

. (3.157)

Bibliography

[1] S. Gair, A. Canova, J. Eastham, and T. Betzer, “A new 2d femanalysis ofa disc machine with offset rotor,” inPower Electronics, Drives and EnergySystems for Industrial Growth, 1996., Proceedings of the 1996 InternationalConference on, vol. 1. IEEE, 1996, pp. 617–621.

[2] G. Cvetkovski, L. Petkovska, M. Cundev, and S. Gair, “Quasi 3d fem in func-tion of an optimisation analysis of a pm disk motor,” inInternational confer-ence on electrical machines, 2000, pp. 1871–1875.

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[4] A. Parviainen, M. Niemela, and J. Pyrhonen, “Modeling ofaxial fluxpermanent-magnet machines,”Industry Applications, IEEE Transactions on,vol. 40, no. 5, pp. 1333–1340, 2004.

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[6] A. Proca, A. Keyhani, A. El-Antably, W. Lu, and M. Dai, “Analytical modelfor permanent magnet motors with surface mounted magnets,”Energy Con-version, IEEE Transactions on, vol. 18, no. 3, pp. 386–391, 2003.

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[9] G. Price, T. Batzel, M. Comanescu, and B. Muller, “Designand testing of apermanent magnet axial flux wind power generator,” inProceedings of the2008 IAJC-IJME International Conference, 2008.

[10] T. Chan, L. Lai, and S. Xie, “Field computation for an axial flux permanent-magnet synchronous generator,”Energy Conversion, IEEE Transactions on,vol. 24, no. 1, pp. 1–11, 2009.

[11] H. Tiegna, A. Bellara, Y. Amara, and G. Barakat, “Analytical modeling of theopen-circuit magnetic field in axial flux permanent-magnet machines withsemi-closed slots,”Magnetics, IEEE Transactions on, vol. 48, no. 3, pp.1212–1226, 2012.

[12] A. Bellara, Y. Amara, G. Barakat, and B. Dakyo, “Two-dimensional exact an-alytical solution of armature reaction field in slotted surface mounted pm ra-dial flux synchronous machines,”Magnetics, IEEE Transactions on, vol. 45,no. 10, pp. 4534–4538, 2009.

[13] T. Lubin, S. Mezani, and A. Rezzoug, “Exact analytical method for magneticfield computation in the air gap of cylindrical electrical machines consideringslotting effects,”Magnetics, IEEE Transactions on, vol. 46, no. 4, pp. 1092–1099, 2010.

[14] Z. Liu and J. Li, “Accurate prediction of magnetic field and magnetic forces inpermanent magnet motors using an analytical solution,”Energy Conversion,IEEE Transactions on, vol. 23, no. 3, pp. 717–726, 2008.

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accounting for slotting effect and load,”Magnetics, IEEE Transactions on,vol. 48, no. 1, pp. 107–117, 2012.

[16] T. Lubin, S. Mezani, and A. Rezzoug, “2-d exact analytical model forsurface-mounted permanent-magnet motors with semi-closed slots,”Magnet-ics, IEEE Transactions on, vol. 47, no. 2, pp. 479–492, 2011.

[17] P. Reddy, Z. Zhu, S. Han, and T. Jahns, “Strand-level proximity losses in pmmachines designed for high-speed operation,” inElectrical Machines, 2008.ICEM 2008. 18th International Conference on. IEEE, 2008, pp. 1–6.

[18] F. Caricchi, F. Crescimbini, E. Fedeli, and G. Noioa, “Design and construc-tion of a wheel-directly-coupled axial-flux pm motor prototype for evs,” inIndustry Applications Society Annual Meeting, 1994., Conference Record ofthe 1994 IEEE. IEEE, 1994, pp. 254–261.

[19] S. Huang, M. Aydin, and T. Lipo, “A direct approach to electrical machineperformance evaluation: torque density assessment and sizing optimization,”in Proceedings of International Conference on Electrical Machines, ICEM02, 2002.

[20] F. Marignetti and M. Scarano, “Mathematical modellingof an axial-flux pmmotor wheel,” inInternational conference on electrical machines, 2000, pp.1275–1279.

[21] A. Smith, H. Willsamson, N. Benhama, L. Counter, and J. Papadopoulos,“Magnetic drive couplings,” inElectrical Machines and Drives, 1999. NinthInternational Conference on (Conf. Publ. No. 468). IET, 1999, pp. 232–236.

[22] A. Wallace, A. von Jouanne, S. Williamson, and A. Smith,“Performance pre-diction and test of adjustable, permanent-magnet, load transmission systems,”in Industry Applications Conference, 2001. Thirty-Sixth IASAnnual Meeting.Conference Record of the 2001 IEEE, vol. 3. IEEE, 2001, pp. 1648–1655.

[23] J. Bumby, R. Martin, M. Mueller, E. Spooner, N. Brown, and B. Chalmers,“Electromagnetic design of axial-flux permanent magnet machines,” inElec-tric Power Applications, IEE Proceedings-, vol. 151, no. 2. IET, 2004, pp.151–160.

[24] R. Di Stefano and F. Marignetti, “Electromagnetic analysis of axial-flux per-manent magnet synchronous machines with fractional windings with exper-imental validation,”Industrial Electronics, IEEE Transactions on, vol. 59,no. 6, pp. 2573–2582, 2012.

[25] F. Marignetti, G. Tomassi, and J. Bumby, “Electromagnetic modelling of per-manent magnet axial flux motors and generators,”COMPEL: Int J for Com-putation and Maths. in Electrical and Electronic Eng., vol. 25, no. 2, pp.510–522, 2006.

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[26] D. Zarko, D. Ban, and T. Lipo, “Analytical calculation of magnetic field distri-bution in the slotted air gap of a surface permanent-magnet motor using com-plex relative air-gap permeance,”Magnetics, IEEE Transactions on, vol. 42,no. 7, pp. 1828–1837, 2006.

[27] M. Hafner, D. Franck, and K. Hameyer, “Static electromagnetic field compu-tation by conformal mapping in permanent magnet synchronous machines,”Magnetics, IEEE Transactions on, vol. 46, no. 8, pp. 3105–3108, 2010.

[28] D. Zarko, D. Ban, and T. Lipo, “Analytical solution for cogging torque in sur-face permanent-magnet motors using conformal mapping,”Magnetics, IEEETransactions on, vol. 44, no. 1, pp. 52–65, 2008.

[29] Z. Zhu and D. Howe, “Instantaneous magnetic field distribution in brushlesspermanent magnet dc motors. iii. effect of stator slotting,” Magnetics, IEEETransactions on, vol. 29, no. 1, pp. 143–151, 1993.

[30] D. Lin, S. Ho, and W. Fu, “Analytical prediction of cogging torque insurface-mounted permanent-magnet motors,”Magnetics, IEEE Transactionson, vol. 45, no. 9, pp. 3296–3302, 2009.

[31] X. Wang, Q. Li, S. Wang, and Q. Li, “Analytical calculation of air-gap mag-netic field distribution and instantaneous characteristics of brushless dc mo-tors,” Energy Conversion, IEEE Transactions on, vol. 18, no. 3, pp. 424–432,2003.

[32] M. Chung and D. Gweon, “Modeling of the armature slotting effect in themagnetic field distribution of a linear permanent magnet motor,” ElectricalEngineering (Archiv fur Elektrotechnik), vol. 84, no. 2, pp. 101–108, 2002.

[33] J. Melkebeek,Bouw en berekening van elektrische machines, 2008.

[34] E. Barbisio, F. Fiorillo, and C. Ragusa, “Predicting loss in magnetic steelsunder arbitrary induction waveform and with minor hysteresis loops,”Mag-netics, IEEE Transactions on, vol. 40, no. 4, pp. 1810–1819, 2004.

[35] S. Das, D. Arnold, I. Zana, J. Park, M. Allen, and J. Lang,“Microfabri-cated high-speed axial-flux multiwatt permanent-magnet generators - part i:Modeling,” Microelectromechanical Systems, Journal of, vol. 15, no. 5, pp.1330–1350, 2006.

[36] A. Letelier, D. Gonzalez, J. Tapia, R. Wallace, and M. Valenzuela, “Coggingtorque reduction in an axial flux pm machine via stator slot displacement andskewing,” Industry Applications, IEEE Transactions on, vol. 43, no. 3, pp.685–693, 2007.

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[38] Z. Wang, R. Masaki, S. Morinaga, Y. Enomoto, H. Itabashi, M. Ito, andS. Tanigawa, “Development of an axial gap motor with amorphous metalcores,”Industry Applications, IEEE Transactions on, vol. 47, no. 3, pp. 1293–1299, 2011.

[39] W. Fei, P. Luk, and K. Jinupun, “A new axial flux permanentmagnetsegmented-armature-torus machine for in-wheel direct drive applications,” inPower Electronics Specialists Conference, 2008. PESC 2008. IEEE. IEEE,2008, pp. 2197–2202.

[40] G. Liew, N. Ertugrul, W. Soong, and D. Gehlert, “Analysis and performanceevaluation of an axial-field brushless pm machine utilisingsoft magneticcomposites,” inElectric Machines & Drives Conference, 2007. IEMDC’07.IEEE International, vol. 1. IEEE, 2007, pp. 153–158.


[42] A. Adly and S. Abd-El-Hafiz, “Utilizing particle swarm optimization in thefield computation of non-linear magnetic media,”Applied ComputationalElectromagnetics Society Journal, vol. 18, no. 3, pp. 202–209, 2003.

[43] I. A. A. Trejo, Numerical modeling and evaluation of the small magnetometerin low-mass experiment (SMILE). School of Electrical Engineering, RoyalInstitute of Technology, 2007.

[44] D. A. Tziouvaras, P. McLaren, G. Alexander, D. Dawson, J. Esztergalyos,C. Fromen, M. Glinkowski, I. Hasenwinkle, M. Kezunovic, L. Kojovic et al.,“Mathematical models for current, voltage, and coupling capacitor voltagetransformers,”Power Delivery, IEEE Transactions on, vol. 15, no. 1, pp. 62–72, 2000.

[45] A. Pulnikov, V. Permiakov, R. Petrov, J. Gyselinck, G. Langelaan, H. Wis-selink, L. Dupre, Y. Houbaert, and J. Melkebeek, “Investigation of residualstresses by means of local magnetic measurement,”Journal of magnetism andmagnetic materials, vol. 272, pp. 2303–2304, 2004.

[46] A. M. A. Abdallh, “An inverse problem based methodologywith uncertaintyanalysis for the identification of magnetic material characteristics of electro-magnetic devices,”PhD dissertation, Ghent University, 2012.

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[47] M. Popescu, “Prediction of the electromagnetic torquein synchronous ma-chines through maxwell stress harmonic filter (hft) method,” Electrical Engi-neering (Archiv fur Elektrotechnik), vol. 89, no. 2, pp. 117–125, 2006.

[48] R. Mertens, U. Pahner, K. Hameyer, R. Belmans, and R. De Weerdt, “Forcecalculation based on a local solution of laplace’s equation,” Magnetics, IEEETransactions on, vol. 33, no. 2, pp. 1216–1218, 1997.

[49] K.-N. Areerak, S. Bozhko, G. Asher, and D. Thomas, “Dq-transformationapproach for modelling and stability analysis of ac-dc power system withcontrolled pwm rectifier and constant power loads,” inPower Electronics andMotion Control Conference, 2008. EPE-PEMC 2008. 13th. IEEE, 2008, pp.2049–2054.

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Chapter 4

Comparison of Non Oriented andGrain Oriented Material

4.1 Introduction

For decades, the stator cores of electrical machines have been made of laminatedsilicon steel. These machines were generally radial flux machines of which thestator consisted of a stack of identical laminated geometries. With the recent emer-gence of new machine topologies, of which axial flux is an important example,the feasibility of laminated silicon steel as stator core material is under discus-sion in many recent papers. In these papers two important alternative materials areconsidered: soft magnetic composites (SMC) and amorphous magnetic materials(AMM).

Soft magnetic composite materials are composed of grains made of magneticmaterial, which are insulated electrically by means of a resin. The mixture ofgrains and resin is compacted at high pressure. During compacting, random con-tacts between the grains may be introduced which result in a low, but not negligible,electric conductivity of the soft magnetic composite material [1–3]. Whereas softmagnetic composite material is processed afterwards into the required geometry,soft magnetic mouldable composites (SM2C) allow direct manufacturing of thegeometries, which makes this production technique advantageous towards batchmanufacturing [4, 5]. Moreover, as the grains are randomly distributed, propertiessuch as electric conductivity, magnetic permeability and thermal conductivity ofsoft magnetic composite materials are isotropic [6, 7],i.e. independent of the di-rection. Especially the isotropic magnetic properties areadvantageous: whereasin stator core made of laminated silicon steel only flux pathsparallel to the lam-ination plane are prevailing, 3D fluxes are possible in soft magnetic compositestator cores [8, 9]. Nevertheless, soft magnetic compositematerials do have dis-advantages as well. Table 4.1 shows some data of a commercially available softmagnetic composite material Somaloy produced by Hoganas. The permeability

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i

86C

ompa

rison

ofN

onO

rient

edan

dG

rain

Orie

nted

Mat

eria

l

Table 4.1: Physical properties of iron based Somaloy soft magnetic composite material [10].

Somaloy Material Resistivity TRS B@10000 A/m µmax Core Losses @ 1 T [W/kg][µΩ·m] [MPa] [T] 5x5mm 15x15mm

100 Hz 400 Hz 1000 Hz 1000 Hz

Baseline1P Somaloy 130i 8000 35 1.40 290 12 54 145 1471P Somaloy 700 400 40 1.56 540 10 44 131 1581P Somaloy 700 HR 1000 35 1.53 440 10 46 134 145High strength, high permeability3P Somaloy 700 200 125 1.61 750 10 46 137 1893P Somaloy 700 HR 600 120 1.57 630 11 48 137 1573P Somaloy 1000 70 140 1.63 850 10 46 144 287Lowest losses5P Somaloy 700 HR 700 60 1.57 600 6 32 104 115

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4.1 Introduction 87

Table 4.2: Physical properties of iron based Metglas amorphous magneticmaterial [17].

Quantity 2605CO 2605SA1

Saturation magnetic flux 1.8 1.59 annealeddensity [T] 1.57 castSpecific core losses less than 0.28 about 0.125at 50 Hz and 1 T [W/kg]Specific 7560 7200 annealeddensity [kg/m3] 7190 castElectric conductivity [S/m] 0.813× 106 0.769× 106

Hardness in 810 900Vicker’s scaleElasticity modulus [GN/m2] 100. . .110 100. . .110Stacking factor less than 0.75 less than 0.79Crystallization 430 507temperature [C]Curie temperature [C] 415 382Maximum service 125 150temperature [C]

of even the high permeability materials is relatively low compared to laminatedsilicon steel. Also the magnetic flux density at saturation is not as high as for lami-nated silicon steel, which limits the machine’s power density. But more important,as this research focusses on energy efficiency, are the relatively high losses at typi-cal frequencies (< 1 kHz) for electrical machines [11]. As soft magnetic compositematerial is an emerging technology, these properties may improve in the next yearsto come [12].

On the other hand there is amorphous magnetic material whichis, despiteits advantages, still very limited used in rotating electrical machine technology;only [13–16] made use of amorphous magnetic material in rotating electrical ma-chines. Amorphous magnetic material is metallic glass which is produced byquenching molten steel at about one megakelvin per second. This is done bypouring molten alloy steel onto a rotating cooled wheel. As the temperature de-crease over time is that high, rapid solidification preventscrystallisation and re-sults in a non-crystalline frozen liquid. In Table 4.2, the properties of Metglasamorphous magnetic materials are presented. The most important advantage ofamorphous magnetic materials are the very high permeability and the extreme lowcore losses [18]. Therefore, they are very promising for usein energy efficientelectrical machines. Nevertheless, amorphous magnetic materials do have a ma-jor disadvantage: machinability. Due to the very high Vickers hardness number,standard cutting techniques like guillotine cutting and punching are not applica-

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88 Comparison of Non Oriented and Grain Oriented Material

ble [17]. Lasercutting causes local heating which results in detrimental crystallisa-tion in the amorphous magnetic material. The only suitable cutting techniques arewaterjetcutting and chemical treatment. Although the material is machined withoutcracking, melting or crystallization, this technique is particularly costly and limitsmass production. The use of amorphous magnetic materials for its specific excel-lent magnetic properties is thus limited due its very bad machinability. Anotherdisadvantage is the relatively small stacking factor due tothe limited thickness (afew tens of microns) of the amorphous magnetic material.

The applicability due to the specific properties of both materials is reflectedby the number of presented prototype axial flux PM machines: only [13,19] is us-ing amorphous magnetic material while [20–24] are using soft magnetic compositematerials. Despite the existence of both materials, the majority of the stator coresin presented prototype axial flux machines [25–32] is still made of laminated sili-con steel. The limitation of 2D flux paths is amply compensated by the relativelylow material cost, the relatively low losses and the good andcheap machinabilitythrough punching. Therefore, laminated silicon steel is considered in this research.Recently, combinations of amorphous and soft magnetic composite material, andlaminated silicon steel and soft magnetic composite material are found to improvethe overall performance of the machine [14,33].

Among the laminated silicon steel materials, two major types exist: non ori-ented (NO) and grain oriented (GO) materials. The grain oriented material hasadvantageous properties such as the low core losses and highsaturation flux den-sity in one particular direction,i.e. the rolling direction (RD). In applications inwhich the magnetic flux is unidirectional, such as power transformers, core ele-ments are made of grain oriented material. Due to the low corelosses, large powertransformers are exploited at a particular high energy efficiency. So far, classi-cal rotating electrical machines do not use grain oriented material as the flux indifferent parts of the stator is not unidirectional.

In this chapter, an analysis of the magnetic flux density in the stator core of ayokeless and segmented armature (YASA) axial flux PM machinewill indicate thatthe magnetic flux density is unidirectional in most parts of the stator core. There-fore, the proposition is made to use grain oriented materialin the stator core of theyokeless and segmented armature axial flux PM machine. In order to take advan-tage of the grain oriented material, the rolling direction is chosen parallel to thedirection of the magnetic fluxi.e. the axial direction. To evaluate the performanceof the grain oriented material, an anhysteretic anisotropic material model based onthe magnetic energy and a core loss model using loss separation are introduced. Toillustrate the superiority of grain oriented material, a comparison is made with anon oriented M400-50A material.

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4.2 Magnetic Flux Density in the Stator Teeth 89

4.2 Magnetic Flux Density in the Stator Teeth

The magnetic flux density in the stator teeth is determined bythe permanent mag-nets and the current density in the stator windings. As both sources are a functionof the rotor position, the flux density in any point of the stator core can be ex-pressed as a function of the rotor position. In Fig. 4.1, magnetic flux density lociin different points in the stator are presented; each point of such a locus representsa different rotor position.

1

23

4

5

6

78

8

7

6

5

4

3

2

1 By

[T]

Bx [T]

By

[T]

Bx [T]

-1.5 -1 -0.5 0 0.5 1 1.5-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5

-1

-0.5

0

0.5

1

1.5

Figure 4.1: Magnetic flux density loci in the(Bx, By)-plane at different po-sitions in a stator core for the NO stator core material for noload.

With respect to these loci, some important remarks can be made. The mostimportant observation is that the magnetic flux density in most parts the stator corehas mainly a component in the direction of the long axis of thestator corei.e.the axial direction. The component in the direction of the short axis is limited.Although this observation is distinct in the center part of the stator cores and lessprevailing in the tips, the volume of the center part outnumbers the volume of thecore tips. Moreover, for nearly each locus a main direction of the magnetic fluxcan be indicated. In Fig. 4.2, the relative presence of the different main directionsof the magnetic flux loci are presented. The presence of magnetic fluxes nearlyparallel to the axial direction is significant. Making use ofgrain oriented materialoriented parallel to the axial direction, thus may be interesting as in more than75 % of the stator core material the magnetic flux is quasi axial. Only in the stator

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Rel

ativ

epr

esen

ce[%

]

Dominant direction of the magnetic flux density []

0≤15 15≤30 30≤45 45≤60 60≤75 75≤900

10

20

30

40

50

60

70

80

Figure 4.2: Relative presence of the dominant directions of the magneticflux density loci in the stator core material where 0 is therolling direction.

tips the use of grain oriented material is not beneficial as the flux in this part has adirection significantly different form the axial direction.

Next to the observation that the magnetic flux is mainly axialin most parts ofthe stator core, some other important conclusions can be made by considering theflux density loci. In addition to the mainly axial direction of the magnetic flux in thestator core, it can be observed that in most other points a distinct main direction canbe observed: even in case elliptical loci appear, a significantly different transverseand conjugate diameter is found. These outcome will justifythe implementationof an unidirectional loss model and neglect the influence of elliptical and rotationaleffects on the losses in the stator core.

Another important finding is that minor loops in the magneticflux density lociare absent for no load as well as load. Consequently, to calculate the core losses,the considered loss model should deal with arbitrary magnetic flux patterns, butnot with minor loops. Preisach modelling taking into account the effect of minorloops on the hysteresis losses, as done in [34], is not considered. Instead, theloss calculation based on the principle of loss separation is introduced. Hereby,hysteresis loss is only determined by the peak value of the magnetic flux density,and the main direction of the magnetic flux density in the stator core material whenusing grain oriented material.

The next two chapters focus on the development of an anisotropic materialmodel based on the magnetic energy and a core loss model basedon the principle

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4.3 Anisotropic Material Model Based on the Magnetic Energy 91

of loss separation. Both models have parameters that need tobe fitted throughmeasurements. For the grain oriented material, several Epstein strips were cut froma laminated silicon steel sheet in seven different directions. The rolling direction ischosen as a reference and is assigned as the angle 0. The other directions enclosean angle of 15, 30, 45, 60, 75 and 90 with respect to the rolling direction.The 90 is often referred to as the transverse direction (TD). Epstein measurementswere preferred over single sheet measurements,cfr. [35], because the dimensionsof the Epstein strips are in fairly good approximation congruent to the shape of thestator core laminations in Fig. 4.1(a).

For all grain oriented strips, several quasi-static hysteresis loops up to7200 A/m were measured on the Epstein frame, which results inB(H) charac-teristics along the seven considered directions as presented in Fig. 4.3. Thesehysteresis loops are used in the development of the anisotropic material modelbased on the magnetic energy in the next section. To estimatethe parameters ofthe core loss model, several dynamic loops were measured up to 200 Hz.

For the non oriented material, that is used as a reference material, quasi staticand dynamic hysteresis loops were measured on a ring core, for magnetic fields upto 4000 A/m and 200 Hz.

4.3 Anisotropic Material Model Based on the MagneticEnergy

To achieve a reliable magnetic flux density computation in the stator core elements,an accurate material model should be used in the finite element analysis. Theanisotropic non linear behaviour of the grain oriented laminated silicon steel ismodelled using the magnetic energy. This model results in anaccurate representa-tion of the magnetisation processes which take place in the grain oriented material.

4.3.1 Anisotropic Material Model

To implement the non linear behaviour of the grain oriented material in the finiteelement analysis, the magnetisation vectorM is used instead of specifying a nonlinear relative permeability which is used to model the non oriented material. Themagnetic vector potential formulation becomes

∇×(

1

µ0∇× A − M

)

= J (4.1)

whereB = ∇× A (4.2)

As mentioned already in this chapter, in these planar 2D finite element computa-tions both the magnetic vector potentialA and the external current densityJ only

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90

(TD

)

75

60

45

30

15

0

(RD

)

B [T]

H[A

/m]

0200

400600

8001000

12001400

16001800

20000

0.2

0.4

0.6

0.8 1

1.2

1.4

1.6

1.8 2

Figure

4.3:Sin

gle

valuedB

H-ch

aracteristicsin

the

sevenco

nsid

eredd

irection

sm

easured

on

the

Ep

steinfram

e.

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4.3 Anisotropic Material Model Based on the Magnetic Energy 93

have a component perpendicular to the calculation plane:

A = Aez (4.3)

andJ = Jez. (4.4)

respectively.The magnetisation vectorM in equation (4.1) is a function of the magnetic flux

density vectorB

M = M (B) =1

µoB − H (B) , (4.5a)

and can be split up into a component along the x- and y- direction:

Mx =1

µ0Bx −Hx (B) (4.5b)

My =1

µ0By −Hy (B) . (4.5c)

In equation (4.5b) and (4.5c),Hx respectivelyHy are a function of the magneticflux density vectorB i.e. they rely on both the amplitude and direction of the mag-netic flux density. In non linear isotropic materials the magnetic field would onlydepend on the amplitude of the magnetic flux density. This nonlinear anisotropicdependence of the magnetic field vectorH on the magnetic flux density vectorB ismodelled using the magnetic energy.

4.3.2 Magnetic Energy

In [36] a similar model was used to model a non linear anisotropic material be-havior. There, the magnetic flux density vectorB was expressed as a function ofthe magnetic field vectorH using the magnetic coenergy. Here, the magnetic fieldvectorH is expressed as a function of the magnetic flux density vectorB, and thus,requires the use of the magnetic energy.

The magnetic energy is defined by

Wem (B) =

B∫

0

H (B) · dB. (4.6)

The numerical construction of the magnetic energy functionWem (B) is done usingthe single valuedBH-characteristics in the seven considered directions presentedin Fig. 4.3. The single valued data for each direction are retrieved by using thepeak values of the measured hysteresis loops. Here, the values ofWem (B) arereconstructed from the experimental data in the following discrete points of the

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B-plane: |B| ejϑ with ϑ corresponding to one of the seven considered directions.As the energy is only calculated in the seven considered directions, interpolationis used to get an energy map in an equidistant rectangular grid of points in the(Bx, By)-plane. Such an energy map is presented in Fig. 4.4. Each input of a

log(W

em)

[J/m

3]

By[T ] Bx[T ]0

0.51

1.52

0

0.5

1

1.5

2-5

0

5

10

15

Figure 4.4: Logarithm of the magnetic energyWem as a function ofBx andBy.

magnetic flux density vectorB results in a unique value of the magnetic energyWem.

Once the magnetic energy map is constructed, the magnetic field vectorH cor-responding with a magnetic flux density vectorB can be calculated by taking thegradient from the energy functionWem (B)

H (B) = ∇BWem (B) (4.7a)

where∇B denotes the gradient operator inB-space,

∇B = ex∂

∂Bx+ ey

∂

∂By. (4.7b)

Equation (4.7a) can be rewritten as

H (B) = ex∂Wem (B)∂Bx

+ ey∂Wem (B)∂By

(4.7c)

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4.4 Core Loss Model (Loss Separation) 95

and hence the components of the magnetic field in thex- andy-direction can beexpressed as

Hx (B) =∂Wem (B)∂Bx

(4.7d)

and

Hy (B) =∂Wem (B)∂By

(4.7e)

respectively.

4.3.3 Epstein Measurement Restriction

In Epstein measurements, the magnetic field vectorH is always imposed in thedirection along the Epstein strip, but the measured magnetic flux density vectorB‖represents only the component of the real magnetic flux density vectorB parallel tothe direction of the magnetic field vectorH. The component of the magnetic fluxdensity vector perpendicular to the magnetic field vectorH, i.e. B⊥, is thus nottaken into account. Consequently this results in an inaccurate prediction ofB⊥.

Only in the specific case in which the magnetic flux density vector B is parallelwith the magnetic field vectorH, Epstein measurements introduce no error. This isthe case for measurements in the 0 direction,i.e., the rolling directions.

Measurements on a measurement setup for rotational fields, which allows mea-surements of the real vectors of magnetic fieldH and magnetic flux densityB fordifferent directions, might be more appropriate.

4.4 Core Loss Model (Loss Separation)

An accurate calculation of the losses in the stator core is crucial when modellinghigh energy efficient machines. The evaluation of the core loss is donea posterioriusing a statistical loss model [37, 38]. Here, the input of this model are the fluxdensity locicfr. Fig. 4.1 in different parts of the stator core. As a multislice2D technique is used, the losses are determined individually for each slice andafterwards summed over the different layers.

Analysis of the magnetic flux density loci in the stator core material, illustratedin Fig. 4.1, shows that no minor loops are present in the magnetic flux density loci.Consequently, the proposed core loss model must deal with arbitrary magnetic fluxdensity waveforms, but not with minor loops. Moreover, in most parts of the coregeometry unidirectional field patterns are found. Only in a limited part of thecore geometry nonunidirectional field patterns exist, which justify neglecting theinfluence of elliptical or rotational effects on the electromagnetic losses in the corematerial.

According to [37, 38], the core losses are split into the three governing losscomponents: a quasi-static loss component, a classical dynamic loss component

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and an excess dynamic loss component. In the following threesubsections, eachloss component is discussed in detail.

4.4.1 Quasi-Static Loss Component

As illustrated in Fig. 4.1, no minor loops are present in the magnetic flux densityloci. The absence of minor loops results in a hysteresis lossthat is independent ofthe magnetic flux density waveform. In this case, the hysteresis loss is determinedby the peak value of the magnetic flux densityBp.

For fast evaluation of the loss model, a 2D lookup table was composed thatcontains the hysteresis loss data for several values of the peak vectorBp located inthe(Bx, By)-plane.

4.4.2 Classical Dynamic Loss Component

The classical energy loss per cycle depends on the time derivative of the magneticflux densityB(t):

Wcl =1

12σd2

T∫

0

(

dB

dt

)2

dt = PclT. (4.8)

In this equationd represent the lamination thickness andσ the electric conductivityof the stator core material.

The conductivity was measured based on the 4-point method:σ =(l/S)(I/V ), where l and S are the length and cross section of the strip,Iis the imposed dc current andV is the measured voltage.

For the GO material,σ is 2.0 MS/m, and equal in the RD and TD direction.For the NO material,σ is 3.2 MS/m.

4.4.3 Excess Dynamic Loss Component

An expression for the instantaneous excess loss was taken from [37]:

pexc(t) =noVo2

(√

1 +4σGS

n2oVo

∣

∣

∣

∣

dB

dt

∣

∣

∣

∣

− 1

)

∣

∣

∣

∣

dB

dt

∣

∣

∣

∣

. (4.9)

Whereno and Vo are functions of the peak value of the magnetic flux densityBp and are fitted from measured magnetisation loops at multiplefrequencies andamplitudes.no is the number of simultaneously active magnetic objects fora fre-quencyf → 0 andVo defines the statistics of the magnetic objects. The dimen-sionless coefficientG and the lamination cross sectionS are known constants.

The fitting of no and Vo is done separately for multiple peak values of themagnetic flux densityBp and for each considered orientation of the magnetic field.

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4.5 Simulations and Comparison 97

From [38, pp. 424–425], a linear dependence betweenn and the excess fieldHexc

is suggested as a first approximation [39]:

n (Hexc) = no +Hexc

Vo. (4.10)

Whereno andVo can be fitted starting from the loss of a measured loopPmeas

that is performed for multiple peak values of the magnetic flux densityBp andfor multiple frequencies. Given the measured lossPmeas, the hysteresis loss andclassical loss calculated as explained above, the excess field Hexc can be retrievedfrom:

Hexc =Pmeas − Ph − Pcl

4Bpf(4.11)

where in this equationPh = fWh is the hysteresis loss in W/m3 and the classicalloss is obtained by multiplying equation (4.8) byf 1.

With this calculated excess fieldHexc, the number of simultaneously activemagnetic objects is given by

n (Hexc) =4σGSBpf

Hexc. (4.12)

In Fig. 4.5 the simultaneously active magnetic objectsn as a function of the excessfieldHexc are depicted for the GO material in the RD for different peak values ofthe magnetic flux density.

By using the equations (4.11) and (4.12), the curves (4.10) can be fitted asshown in Fig. 4.6 for GO material. The parameterS in equation (4.9) is chosenequal to 1/G and as mentioned before a linear dependency between simultaneouslyactive magnetic objectsn and the excess fieldHexc is assumed.

For the NO material, similar curves forno andVo were determined.

4.4.4 Unidirectional Model

The proposed loss model is unidirectional, while Fig. 4.1 indicates that the mag-netic field and flux density have an elliptical locus in some parts of the geometry.However, in most parts of the geometry the elliptical loci clearly have a signif-icantly different transverse and conjugate diameter. Therefore, the direction towhich the unidirectional loss model is applied, is chosen parallel to the transversediameter of the elliptical locus.

4.5 Simulations and Comparison

In this section, a comparison is made between a grain oriented material and a nonoriented material. The grain oriented material is a relatively good quality material:

1For a pure sinusoidal magnetic flux density, the classical energy loss can be calculated asPcl =σπ2d2B2

pf2/6

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PSfrag

1.8 T

1.6 T

1.2 T

0.6 T

0.1 T

n(H

exc)

Hexc

0 10 20 30 40 50 60

×108

0

0.5

1

1.5

2

2.5

3

Figure 4.5: Simultaneously active magnetic objectsn as a function of theexcess fieldHexc for the GO material in the RD for differentpeak values of the magnetic flux density.

0.3 mm thickness, 1 W/kg in the rolling direction at 50 Hz and 1.5 T, while thenon oriented material is a M400-50A grade of 0.5 mm thicknessand 4 W/kg at50 Hz and 1.5 T. By comparing these two markedly different materials, a trade-offis made between the better performance and the extra cost of the higher qualitymaterial.

To allow proper comparison between both materials, the samegeometrical pa-rameter values of the axial flux PM machine are used. The only difference is thatequation (3.79) is used to model the stator core material in case of the non orientedmaterial and equation (4.1) to model the grain oriented material.

Within the scope of this research, the influence of the material on the stator corelosses is considered as most important. In Fig. 4.7, the corelosses for both the nonoriented and grain oriented material are presented as a function of the rotationalspeed of the axial flux PM machine. These simulations were done at no loadi.e.the magnetic fluxes in the stator cores are caused by the permanent magnets only.

Within the considered speed range, the losses in the grain oriented material areonly one third of those in the non oriented material. This is in quite good agreementwith the prescribed losses per kilogram, knowing that prescribed losses in the grainoriented material are referred to the rolling direction. This significant reduction ofthe core loss is considered as crucial in this research towards energy efficient axialflux PM machines; the significantly lower losses do compensate the extra cost ofthe grain oriented material.

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906030150

906030150

Vo

[A/m

]

Bp [T]

no

Bp [T]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

×10−6

×107

0

1

2

3

4

5

0

2.5

5

7.5

10

12.5

15

Figure 4.6: Parametersno andVo as a function of the peak flux densityBp

for different directions in the GO material.

Next to the lower losses, the grain oriented material has some additional ad-vantages as well. In the rolling direction, which is chosen parallel with the axialdirection of the stator cores, the flux density at which the material saturates is muchhigher than for the non oriented material. In Fig. 4.8, the magnetic fieldH andthe magnetic flux densityB are plotted in the center part of the stator core as afunction of the rotor position. Again, only the permanent magnets are consideredas a source of magnetic fluxes in the stator core.

Not very surprising, in the non oriented material much higher magnetic fieldsare found than in the grain oriented material for a similar flux density, which in-dicate saturation of the stator core in case of the non oriented material while thisis hardly present in the grain oriented material. This localsaturation of the stator

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PFe [W], GO

Pexc [W], GO

Pcl [W], GO

Ph [W], GO

PFe [W], NO

Pexc [W], NO

Pcl [W], NO

Ph [W], NO

P[W

]

Ω [rpm]

P[W

]

Ω [rpm]

0 250 500 750 1000 12501500 1750 2000 2250 2500

0 250 500 750 1000 1250 1500 1750 2000 22502500

0

10

20

30

40

50

0

30

60

90

120

150

Figure 4.7: Quasi-staticPh, classical dynamicPcl, excess dynamicPexc

power loss components and total power lossPFe in the statorcore at no load as a function of the rotational speedΩ for theNO and GO material.

core material also results in a slight decrease of the peak value of the magnetic fluxdensity. In this center part, where the flux is parallel to therolling direction of thelaminated silicon steel, the grain oriented material clearly shows an advantage overthe non oriented material. This may also influence the optimised geometry of thestator cores; the usage of grain oriented material allows tomake the center part ofthe core section narrower for the same flux. Narrowing the center section allows tomake the stator slots wider, which can be used for a higher copper section of thewinding. Choosing the right core center section to slot width is a trade-off betweencore losses and copper losses: a narrower center section of the core increases the

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PSfrag

B [mT], GO

H [A/m], GO

B [mT], NO

H [A/m], NO

H[A

/m]/

B[m

T]


0 45 90 135 180 225 270 315 360-3000

-2000

-1000

0

1000

2000

3000

Figure 4.8: Magnetic fieldH and magnetic flux densityB in the center ofa stator core in function of the electric angleθ at no load for theNO and GO material.

flux density and accordingly the stator core losses, but lowers the copper losses andvice versa.

Although the flux is in the rolling direction in the major partof the stator core,fluxes with directions different from the rolling directionoccur in the stator tips.Nevertheless these parts represent a minority of the statorcore volume, they mighthave an influence on the electromagnetic torque. As where theproperties of grainoriented material are advantageous in the rolling direction, they are much worse inthe other directions. In Fig. 4.9, a view on the finite elementmodel is plotted forthe axial flux PM machine at load. In the presented rotor position, the magnets arein quadrature with the stator tips and the stator current reaches its maximum value.This results in a flux direction that is very different from the rolling direction. Theplot of the magnetic fieldH, clearly indicates the high magnetic field in one sideof the stator tip. On this side of the stator tip, the flux produced by the permanentmagnets is increased with the armature reaction flux due to the stator currents. Inthe non oriented material, the magnetic field is less significant than in case of thegrain oriented material. There, the less good magnetic properties in a direction dif-ferent from the rolling direction of the grain oriented material may results in localsaturation of the core material near the stator tips. This local saturation can result ina lower average electromagnetic torque output and an increase of the torque ripple.

In Fig. 4.10, the flux linkage and electromotive force at no load are presented.The higher induced electromotive force that could be expected based on the flux

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×104

0

0.5

1

1.5

2

2.5

3

(a) NO material

×104

0

0.5

1

1.5

2

2.5

3

(b) GO material

Figure 4.9: Magnetic field in the first layer at theθ=0 position for a 7 A si-nusoidal current injected in phase with the electromotive force.The magnetic field is expressed in A/m.

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PSfrag

Ec [V], GO

ψc [mWb], GO

Ec [V], NO

ψc [mWb], NOψc

[mW

b]/E

c[V

]


0 45 90 135 180 225 270 315 360-60

-40

-20

0

20

40

60

Figure 4.10: Coil flux linkage ψc and coil electromotive forceEc at2500 rpm for the NO and GO material.

< Te > [Nm], GO

Te [Nm], GO

< Te > [Nm], NO

Te [Nm], NO

Te

[Nm

]/<Te>

[Nm

]


0 45 90 135 180 225 270 315 36018.3

18.4

18.5

18.6

18.7

18.8

18.9

19

Figure 4.11: Electromagnetic torqueTe as a function of the electric angleθgenerated by a 7 A sinusoidal current injected in phase with theelectromotive force. The average value of the electromagnetictorque is indicated by< Te >.

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density waveform in the center of the stator core in Fig. 4.8,is somewhere van-ished. This may be a result of the lower performance of the grain oriented materialnear the stator core tips. Notwithstanding, Fig. 4.11 indicates a clear increase ofthe average electromagnetic torque in case of the grain oriented material. The us-age of grain oriented material in axial flux PM machines thus not only results in adecrease of the stator core losses but also contributes to a higher electromagnetictorque. It should be added that the local saturation near thetips of the stator resultin a slightly higher torque ripple.

With the focus on energy efficiency, the use of grain orientedmaterial insteadof non oriented material is convincing, even when considering the extra cost. Thelosses in the stator can be reduced significantly, while the same current density inthe stator windings results in a higher electromagnetic torque.

4.6 Conclusion

Despite the emerging soft magnetic composite and amorphousmaterial technology,the applicability of laminated silicon steel in axial flux PMmachines is advanta-geous: laminated silicon steel is cheap, has relatively lowlosses in the frequencyrange of the considered application, has a high saturation flux density, has a highpermeability and is easy machinable. Whereas classical rotating electrical ma-chines are using non oriented laminated silicon steel, grain oriented material canbe used in axial flux PM machines as the flux is in the axial direction in the majorpart of the stator core.

In order to perform accurate simulations, an anisotropic material model basedon the magnetic energy and a core loss model based on loss separation were intro-duced. Both models are built using data obtained through Epstein measurementson material strips in seven directions along the grain oriented material.

A comparison of a non oriented and grain oriented material showed the overallbetter performance of the grain oriented material. This is mainly due to the superiormagnetic properties of the grain oriented material in the rolling direction. Despiteits higher cost, the grain oriented material has significantlower core losses than thenon oriented material. Moreover, when considering the samegeometry the grainoriented material has a slightly higher electromagnetic torque for the same currentdensity in the stator, in comparison with non oriented material. Both lower corelosses and higher electromagnetic torque results in an improved energy efficiency.

Bibliography

[1] F. Marignetti, V. Colli, and S. Carbone, “Comparison of axial flux pm syn-chronous machines with different rotor back cores,”Magnetics, IEEE Trans-actions on, vol. 46, no. 2, pp. 598–601, 2010.

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4.6 Bibliography 105

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[4] J. Cros, P. Viarouge, Y. Chalifour, and J. Figueroa, “A new structure of uni-versal motor using soft magnetic composites,”Industry Applications, IEEETransactions on, vol. 40, no. 2, pp. 550–557, 2004.

[5] J. Wang, T. Ibrahim, and D. Howe, “Prediction and measurement of iron lossin a short-stroke, single-phase, tubular permanent magnetmachine,”Magnet-ics, IEEE Transactions on, vol. 46, no. 6, pp. 1315–1318, 2010.

[6] O. de la Barriere, C. Appino, F. Fiorillo, C. Ragusa, L. Rocchino,H. Ben Ahmed, M. Lecrivain, M. Gabsi, F. Mazaleyrat, and M. LoBue,“Characterization and prediction of magnetic losses in soft magnetic com-posites under distorted induction waveform,” 2012.

[7] C. Cyr, P. Viarouge, S. Clenet, and J. Cros, “Methodologyto study the influ-ence of the microscopic structure of soft magnetic composites on their globalmagnetization curve,”Magnetics, IEEE Transactions on, vol. 45, no. 3, pp.1178–1181, 2009.

[8] W. Fei and P. Luk, “Cogging torque reduction techniques for axial-fluxsurface-mounted permanent-magnet segmented-armature-torus machines,” inIndustrial Electronics, 2008. ISIE 2008. IEEE International Symposium on.IEEE, 2008, pp. 485–490.

[9] A. Chebak, P. Viarouge, and J. Cros, “Analytical computation of the full loadmagnetic losses in the soft magnetic composite stator of high-speed slotlesspermanent magnet machines,”Magnetics, IEEE Transactions on, vol. 45,no. 3, pp. 952–955, 2009.

[10] O. Andersson, “Hoganas ab, s-263 83 hoganas, sweden paul hofecker northamerican hoganas, 111 hoganas way hollsopple, pa 15935-6416, usa.”

[11] J. Wang and D. Howe, “Influence of soft magnetic materials on the designand performance of tubular permanent magnet machines,”Magnetics, IEEETransactions on, vol. 41, no. 10, pp. 4057–4059, 2005.

[12] G. Cvetkovski and L. Petkovska, “Performance improvement of pm syn-chronous motor by using soft magnetic composite material,”Magnetics, IEEETransactions on, vol. 44, no. 11, pp. 3812–3815, 2008.

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[13] Z. Wang, R. Masaki, S. Morinaga, Y. Enomoto, H. Itabashi, M. Ito, andS. Tanigawa, “Development of an axial gap motor with amorphous metalcores,”Industry Applications, IEEE Transactions on, vol. 47, no. 3, pp. 1293–1299, 2011.

[14] N. Dehlinger and M. Dubois, “Clawpole transverse flux machines with amor-phous stator cores,” inElectrical Machines, 2008. ICEM 2008. 18th Interna-tional Conference on. IEEE, 2008, pp. 1–6.

[15] L. Wei, Q. Xiaohui, L. Junfang, and A. Fei, “The high frequency character-istic of amorphous iron induction motor,” inComputer and CommunicationTechnologies in Agriculture Engineering (CCTAE), 2010 International Con-ference On, vol. 2. IEEE, 2010, pp. 439–442.

[16] L. Wang, J. Li, S. Li, G. Zhang, and S. Huang, “Development of the newenergy-efficient amorphous iron based electrical motor,” in Computer Dis-tributed Control and Intelligent Environmental Monitoring (CDCIEM), 2011International Conference on. IEEE, 2011, pp. 2059–2061.

[17] J. Gieras, R. Wang, and M. Kamper,Axial flux permanent magnet brushlessmachines. Springer, 2008.

[18] R. Lenke, S. Rohde, F. Mura, and R. De Doncker, “Characterization ofamorphous iron distribution transformer core for use in high-power medium-frequency applications,” inEnergy Conversion Congress and Exposition,2009. ECCE 2009. IEEE. IEEE, 2009, pp. 1060–1066.

[19] R. Caamano, “Electric motor and generator having amorphous core piecesbeing individually accommodated in a dielectric housing,”Nov. 16 1999, uSPatent 5,986,378.


[21] G. Liew, N. Ertugrul, W. Soong, and D. Gehlert, “Analysis and performanceevaluation of an axial-field brushless pm machine utilisingsoft magneticcomposites,” inElectric Machines & Drives Conference, 2007. IEMDC’07.IEEE International, vol. 1. IEEE, 2007, pp. 153–158.

[22] F. Marignetti and V. Colli, “Thermal analysis of an axial flux permanent-magnet synchronous machine,”Magnetics, IEEE Transactions on, vol. 45,no. 7, pp. 2970–2975, 2009.

[23] W. Fei, P. Luk, and K. Jinupun, “A new axial flux permanentmagnetsegmented-armature-torus machine for in-wheel direct drive applications,” in

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Power Electronics Specialists Conference, 2008. PESC 2008. IEEE. IEEE,2008, pp. 2197–2202.

[24] G. Cvetkovski, L. Petkovska, M. Cundev, and S. Gair, “Improved design of anovel pm disk motor by using soft magnetic composite material,” Magnetics,IEEE Transactions on, vol. 38, no. 5, pp. 3165–3167, 2002.

[25] M. Aydin, S. Huang, and T. Lipo, “Design, analysis, and control of a hybridfield-controlled axial-flux permanent-magnet motor,”Industrial Electronics,IEEE Transactions on, vol. 57, no. 1, pp. 78–87, 2010.


[27] A. Letelier, D. Gonzalez, J. Tapia, R. Wallace, and M. Valenzuela, “Coggingtorque reduction in an axial flux pm machine via stator slot displacement andskewing,” Industry Applications, IEEE Transactions on, vol. 43, no. 3, pp.685–693, 2007.

[28] A. Di Gerlando, G. Foglia, M. Iacchetti, and R. Perini, “Design criteria ofaxial flux pm machines for direct drive wind energy generation,” in ElectricalMachines (ICEM), 2010 XIX International Conference on. IEEE, 2010, pp.1–6.

[29] Y. Yang, Y. Luh, and C. Cheung, “Design and control of axial-flux brushlessdc wheel motors for electric vehicles-part i: multiobjective optimal designand analysis,”Magnetics, IEEE Transactions on, vol. 40, no. 4, pp. 1873–1882, 2004.

[30] B. Chalmers and E. Spooner, “An axial-flux permanent-magnet generator fora gearless wind energy system,”Energy Conversion, IEEE Transactions on,vol. 14, no. 2, pp. 251–257, 1999.

[31] D. Gonzalez-Lopez, J. Tapia, R. Wallace, and A. Valenzuela, “Design andtest of an axial flux permanent-magnet machine with field control capability,”Magnetics, IEEE Transactions on, vol. 44, no. 9, pp. 2168–2173, 2008.

[32] W. Fei and P. Luk, “An improved model for the back-emf andcogging torquecharacteristics of a novel axial flux permanent magnet synchronous machinewith a segmental laminated stator,”Magnetics, IEEE Transactions on, vol. 45,no. 10, pp. 4609–4612, 2009.

[33] P. Lemieux, O. Delma, M. Dubois, and R. Guthrie, “Soft magnetic compositewith lamellar particles-application to the clawpole transverse-flux machine

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with hybrid stator,” inElectrical Machines, 2008. ICEM 2008. 18th Interna-tional Conference on. IEEE, 2008, pp. 1–6.

[34] J. Gyselinck, L. Dupre, L. Vandevelde, and J. Melkebeek, “Calculation of no-load induction motor core losses using the rate-dependent preisach model,”Magnetics, IEEE Transactions on, vol. 34, no. 6, pp. 3876–3881, 1998.

[35] F. Fiorillo, L. Dupre, C. Appino, and A. Rietto, “Comprehensive model ofmagnetization curve, hysteresis loops, and losses in any direction in grain-oriented fe-si,”Magnetics, IEEE Transactions on, vol. 38, no. 3, pp. 1467–1476, 2002.

[36] T. Pera, F. Ossart, and T. Waeckerle, “Field computation in non linearanisotropic sheets using the coenergy model,”Magnetics, IEEE Transactionson, vol. 29, no. 6, pp. 2425–2427, 1993.

[37] E. Barbisio, F. Fiorillo, and C. Ragusa, “Predicting loss in magnetic steelsunder arbitrary induction waveform and with minor hysteresis loops,”Mag-netics, IEEE Transactions on, vol. 40, no. 4, pp. 1810–1819, 2004.

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[39] L. Dupre, F. Fiorillo, J. Melkebeek, A. Rietto, and C. Appino, “Loss versuscutting angle in grain-oriented fe–si laminations,”Journal of magnetism andmagnetic materials, vol. 215, pp. 112–114, 2000.

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Chapter 5

Eddy Current Loss in thePermanent Magnets

5.1 Introduction

In permanent magnet machines with a fractional slot winding, the eddy currentlosses in the permanent magnets can have a peculiar importance [1]. Due to therelatively high electric conductivity of rare-earth magnets, the high interaction ofthe permanent magnets with stator slots and by stator magnetomotive force space(and time) harmonics, resultant eddy current losses can be significant. Therefore,these eddy current losses are no longer negligible in comparison with the iron corelosses and copper losses, and thus, will deteriorate the energy efficiency. More-over, they can result in excessive heating of the permanent magnets and, thereby,increase the risk of partial irreversible demagnetization[2]. Especially machineswith a high electric loading, a high rotational speed or a high pole number aresusceptible [3]. Therefore, models solving coupled electromagnetic, thermal andfluid dynamical problems are introduced in [4, 5] to estimatethe temperatures indifferent parts of the machine.

Among the rare-earth magnets, two types exist: neodymium and samarium-cobalt. The most relevant properties of both materials are listed in Table 5.1.Samarium-cobalt magnets have a weaker magnetic field strength with respect toneodymium magnets. Moreover the economic cost of samarium-cobalt is higher.Nevertheless in applications where high operation temperatures in the permanentmagnets are present, preference is given to samarium-cobalt. Another advantage ofsamarium-cobalt is its high resistance to oxidation. Neodymium magnets alwaysrequire a protective coating against oxidation which can bea gold, nickel, zinc ortin plating or epoxy resin coating. Because a high power density is important inthe design of the axial flux PM machine, neodymium magnets areconsidered inthe further parts of this thesis.

When designing axial flux PM machines with a fractional slot winding, these

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110

Edd

yC

urre

ntL

oss

inth

eP

erm

anen

tMag

nets

Table 5.1: Comparison of the characteristics of neodymium magnets andsamarium-cobalt magnets [6].

Quantity Neodymium magnets SmCo magnets

Composition Nd, Fe, B, etc. Sm, Co, Fe, Cu, etc.Production Sintering SinteringEnergy product 199-310 kJ/m3 255 kJ/m3

Remanence 1.03-1.3 T 0.82-1.16 TIntrinsic coercive force,HcJ 875 kA/m to 1.99 MA/m 493 kA/m to 1.59 MA/mRelative permeability 1.05 1.05Reversible temperature coefficient of remanence -0.11 to -0.13%/K -0.03 to -0.04%/KReversible temperature coefficient of coerciveHcJ -0.55 to -0.65%/K -0.15 to -0.30%/KCurie temperature 320C 800CDensity 7300-7500 kg/m3 8200-8400 kg/m3

Coefficient of thermal expansion in magnetizing direction 5.2×10−6/K 5.2×10−6/KCoefficient of thermal expansion normal to magnetizing direction -0.8×10−6/K 11×10−6/KBending strength 250 N/mm2 150 N/mm2

Compression strength 1100 N/mm2 800 N/mm2

Tensile strength 75 N/mm2 35 N/mm2

Vickers hardness 550-650 500-550Resistivity 110-170×10−6 Ω cm 86×10−6 Ω cmConductivity 590000-900000 S/m 1160000 S/m

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5.1 Introduction 111

eddy current losses in the permanent magnets should be takeninto account. Inorder to calculate these eddy current losses, many computational methods havebeen introduced going from estimations based on analyticalexpressions [7–17] toaccurate simulations of eddy currents with numerical techniques, mainly finite el-ement techniques [18–31]. Among the analytical expressions, [10] only takes theeffect of stator slotting into account. In [11], this model is extended for the statorarmature reaction field. Extensive research towards the effect of armature reac-tion in fractional slot windings on eddy currents in the permanent magnets is donein [12–14]. Very often these models assume the permanent magnets to have infi-nite dimensions in the axial direction. The limited dimensions of the permanentmagnets are discussed in [15] by defining the eddy current resistance path. Moreadvanced analytical models [16, 17] also include the effectof circumferential andaxial segmentation of the permanent magnets, as a method to reduce the eddy cur-rents. Comparable topics are studied using finite element analysis: in [18–23] ananalysis of the magnet eddy current loss in fractional slot windings is performed,the effect of circumferential and axial segmentation on theeddy currents in thepermanent magnets are topic of [24–26] and some models [27, 28] even introducecopper cladding of the permanent magnets as a supplementarymeasure to reducethe magnet eddy currents. Different from the analytical models, the nonlinear be-haviour of the stator core material is modeled in [29–31] to inspect the influenceof additional magneto motive force harmonics due to local saturation of the statorcore material.

All previously introduced models, except [15], deal with radial machines. Inmost of these models, only 2D models of a cross section are considered. In particu-lar, in the models [29,31] the geometry is extended into the axial direction. For theaxial flux topology of [15], the proposed analytical model was verified with a 3Dtime stepped finite element analysis that took about one week. In this finite elementmodel, the stator was modelled using a scalar vector potential while the permanentmagnets were modelled using edge elements. In order to simulate the effect ofarmature reaction, a vector potential formulation should be used, taking even morememory and calculation cost. Therefore, new approaches towards eddy currentcalculation in the permanent magnets of axial flux permanentmagnet machines arenecessary to reduce the memory and calculation constraints.

In this chapter, the eddy currents in the permanent magnets of an axial fluxpermanent magnet machine are calculated by using only 2D finite element com-putations. This calculation method uses the multislice 2D technique to calculatethe air gap magnetic flux density. Consequently, this air gapmagnetic flux densityis imposed to a 2D finite element model of the permanent magnetto calculate theeddy currents. In this model, the eddy currents in the permanent magnets are as-sumed to be resistance limited [32]. These 2D finite element computations havethe advantage of lower computational time compared to a 3D finite element anal-ysis, while the nonlinear behaviour of the stator core material in the multislice 2Dtechnique can still be applied. Finally, this model is used to evaluate the perma-

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112 Eddy Current Loss in the Permanent Magnets

nent magnet eddy current losses for different segmentationgrades of the permanentmagnets.

5.2 Analysis of the Air Gap Magnetic Flux Density

As in the electrically conductive permanent magnets eddy currents are induced dueto the interaction with the stator slots and by stator armature reaction, this sectionfocusses on the origin of these eddy current losses,i.e., the variation of the mag-netic flux density in the permanent magnets. As the rotor and permanent magnetsare moving with respect to the stator, distinction between the two coordinate sys-tems is made. First, there is the coordinate system that is fixed to the stator, whichis indicated by the cartesian coordinateys. Second, there is the coordinate sys-tem fixed to the rotor, indicated by the cartesian coordinate1 yr. Both coordinatestems will by indicated by the stator reference frame (SRF) and the rotor referenceframe (RRF) respectively. The transformation between bothcoordinate systems isexpressed by

ys = yr +pτpπ

Ωt. (5.1)

This transformation indicates the rotation of the rotor with respect to the stator atthe synchronous rotational speedΩ. This transformation will have a major impacton the variation of the magnetic flux density in the permanentmagnets; harmon-ics components produced by the stator, each having a specificamplitude and syn-chronous rotational speed, will due to the relative motion of the rotor with respectto the stator, result in a different harmonic content in the rotor reference frame.The next two paragraphs study the influence of two causes of the variation of theair gap magnetic flux density: stator slotting effect and armature reaction. A thirdcause, time harmonic variation of the air gap magnetic flux density due to pulsewidth modulation (PWM) or injection of high frequency voltage pulses of sen-sorless control [33], also contributes to the variation of the air gap magnetic fluxdensity. The influence of pulse width modulation on the eddy current losses in thepermanent magnets is left for discussion in Chapter 8.

5.2.1 Influence of Stator Slotting

A first variation of the magnetic flux density in the permanentmagnets is caused bythe presence of the stator slot openings near the air gap. Extensive analysis of theeffect of stator slotting openings on the air gap magnetic flux density was alreadypresented in Chapter 3. The analysis was performed using analytical modelling inthe reference frame fixed to the stator.

1In a cylindrical coordinate system, the cartesian coordinate y is equivalent to the azimuthalcoordinateϕ.

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5.2 Analysis of the Air Gap Magnetic Flux Density 113

In this stator reference frame analysis, the stator slotting effect was modelled bya permeance function (3.53) represented in Fig. 3.8. To takeinto account the effectof the stator slotting, here, a non constant air gap reluctance function is defined

1

g(ys)=

1

g′

1 +∞∑

µ=1

ǫµ cos

(

µ2π

τsys

)

=1

g′

1 +

∞∑

µ=1

ǫµ cos

(

µQsπ

pτpys

)

. (5.2)

As for the permeance function, this function is periodic with the slot pitchτs.With respect to the rotor,i.e. in the rotor reference frame, the stator slots pass

by during movement of the rotor. As a permanent magnet passessuch a stator slotopening, this results in a local decrease of the magnetic fluxdensity. The effect ofstator slotting on the air gap magnetic flux density in the rotor reference frame isillustrated in Fig. 5.1.

A prediction of the present harmonic components in the variation of the air gapmagnetic field near the permanent magnets, can be obtained through analytical ex-pressions. In the rotor reference frame, the magnetomotiveforce of the permanentmagnets can be expressed by Fourier series expansion

fPM =∞∑

ν=1

FPMν sin

(

νπ

pτpyr

)

. (5.3)

A random permanent magnet magnetomotive force harmonicν

fPMν = FPMν sin

(

νπ

pτpyr

)

(5.4)

subjected to the stator slotting effect, results in an air gap magnetic flux density

bPMν (yr, t) =µ0g(yr)

fPMν

= Bν sin

(

νπ

pτpyr

)

+Bν sin

(

νπ

pτpyr

)

·∞∑

µ=1

ǫµ cos

(

µQsΩt+ µQsπ

pτpyr

)

. (5.5)

The derivation of equation (5.5), required the transformation of equation (5.2) tothe rotor reference frame through equation (5.1).

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114

Edd

yC

urre

ntL

oss

inth

eP

erm

anen

tMag

nets

Bg,l

oad

[T]

θ [rotor position in degrees]

Bg,n

olo

ad[T

]


0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360

0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360

0.6

0.8

1

1.2

1.4

0.6

0.8

1

1.2

1.4

Figure 5.1: Air gap magnetic flux density taking into account stator slotting effect only (upper), and both the stator slottingeffect and armature reaction (lower) as a function of the rotor position. Data are expressed in the rotor referenceframe,i.e., position fixed with respect to the rotor; in this case the center of the permanent magnet.

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Considering the analytical expression for the air gap magnetic flux density inthe rotor reference frame (5.5), two terms can be distinguished. The first termexpresses the air gap magnetic flux density by the permanent magnets themselves,and hence, is constant with respect to the rotor position. The second term expressesthe influence of the air gap reluctance function on the flux density by the perma-nent magnets and therefore contains the reluctance harmonics. These reluctanceharmonics comprise only harmonic numbers that fulfill

νδ = µQs µ = 1, 2, . . . . (5.6)

This means that only harmonics with multiples of the slot number result in a varia-tion of the magnetic flux density near the permanent magnets.The speed at whichthese harmonics move with respect to the rotor isΩ, i.e. the synchronous rotationalspeed. A Fourier spectrum of the variation in the air gap magnetic field near thepermanent magnets is presented in Fig. 5.2. As predicted by the analytical ex-

Bg,ν

ν

0 50 100 150 200 250 300 350 400 450 50010−3

10−2

10−1

Figure 5.2: Amplitudes of air gap magnetic flux density considering the sta-tor slotting effect only as a function of the harmonic orderν atthe center of the permanent magnet. Note thatQs=15.

pression only harmonics with multiples of the stator slot number are present. Thisobservation will have major implications on the further calculation of the time har-monic simulations of the eddy currents in the permanent magnets; simulations areonly performed for the discrete numbers of present harmonics.

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5.2.2 Influence of Armature Reaction

At no load, the variation of the air gap magnetic flux density near the permanentmagnets depends on the stator slotting effect only. The armature current at load,however, results in an additional variation of the air gap magnetic flux density nearthe permanent magnets.

As for the stator slotting effect, Chapter 3 introduced the armature reactionin the stator reference frame. For a sinusoidal current, a magnetomotive forceharmonic component can be expressed by

fAν = Fν sin

(

pΩt− νπ

pτpys

)

. (5.7)

A random magnetomotive force harmonic of the armature reaction will directlyinfluence the air gap magnetic flux density near the permanentmagnets, but alsothe interaction of a random magnetomotive force harmonic with the slotting effectcan cause additional variations. In Fig. 5.1, the air gap magnetic flux density nearthe permanent magnets is illustrated taking into account both the slotting effect andarmature reaction.

An analytical expression for the air gap magnetic flux density near the perma-nent magnets comprising the armature reaction is given by

bAν (yr, t) = Bν sin

(

(p− ν)Ωt− νπ

pτpyr

)

+Bν

∞∑

µ=1

1

2ǫµ

[

sin

(

(p− ν + µQs)Ωt− (ν + µQs)π

pτpyr

)

+ sin

(

(p− ν − µQs)Ωt− (ν − µQs)π

pτpyr

)]

. (5.8)

In this expression for the air gap magnetic flux density near the permanent magnetsdue to the armature reaction, two main contributions can be distinguished; themagnetomotive force harmonic itself and a set of reluctanceharmonics due to theinteraction of the armature reaction with the stator slots.The first term is equalto equation (5.7), but expressed in the rotor reference frame. The second term,the reluctance harmonics, represents two series of travelling waves with harmonicorders

νδ = ν ± µQs µ = 1, 2, . . . . (5.9)

An interesting situation occurs whenν = p; the first term does not result in avariation of the air gap magnetic flux density near the permanent magnets as thearmature air gap flux density is moving synchronous with the permanent magnets.Notice that even in caseν = p, the reluctance harmonics generally result in avariation of the air gap magnetic flux density near the permanent magnets.

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5.3 Multislice 2D - 2D Coupled Modelling 117

Bg,ν

ν

0 50 100 150 200 250 300 350 400 450 50010−3

10−2

10−1

Figure 5.3: Amplitudes of air gap magnetic flux density considering boththe stator slotting effect and armature reaction as a function ofthe harmonic orderν at the center of the permanent magnet.

In Fig. 5.3 the Fourier spectrum is shown in case stator slotting as well asarmature reaction is consideredi.e. a loaded machine. As can be observed in Fig.5.3, in fractional slot windingscfr. Chapter 7, the harmonics can have an orderlower than the number of pole pairsp as well. They are called subharmonics. Asthey have a high wavelength, these harmonics have flux paths that go through amajor part of the rotor.

5.3 Multislice 2D - 2D Coupled Modelling

In Chapter 3 the multislice 2D modelling technique was introduced for both ana-lytical and finite element analysis. The idea to quantify a global value as the sumof contributions of the multiple slices was used there to predict flux linkage, elec-tromotive force and electromagnetic torque. In radial machines eddy currents inpermanent magnets are obtained from single slice simulations. There, very oftenthe magnets are assumed to be infinitely long in axial direction. This assumptionwould lead to inaccurate results in axial flux machines as theslice thickness is gen-erally very thin. Even in the limit of one slice, the slice thickness is not sufficientlylarge to neglect the end effects near the upper and inner radius. Although 3D timestepped finite element analysis is commercially available,the correctness of thesolution strongly depends on the mesh refinement and used time step and generallythe end user is facing memory constraints and very long simulation times. There-

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fore, a technique that makes again use of the multislice 2D simulations is intro-duced. Once the multislice 2D simulations are finished, the 1D air gap flux densitydata of the different slices are collected. Using the 1D air gap flux density data ofthe different slices, the 2D air gap flux density in a cylindric cross section segmentis reconstructed. Finally, this reconstructed air gap flux density is imposed to a 2Dfinite element model of the permanent magnet in the cylindriccross section. Totake the depth of penetration of the magnetic flux density into account, time har-monic 2D finite element analysis is performed. In this technique, eddy currents areassumed to be resistance limited. As this technique calculates the eddy currentsas a global quantity by using only multislice 2D simulationsfor the cylindric sur-faces to obtain the air gap flux density data and 2D simulations in a cylindric crosssection to calculate the eddy currents in the permanent magnets, this technique iscalledmultislice 2D - 2D coupled modelor multilayer 2D - 2D coupled model. Asit approximates 3D modelling, these techniques are often referred to asQuasi 3Dand generally combine a good accuracy with low memory and time constraints.

5.3.1 Multislice 2D Simulations

The multislice 2D simulations are used to obtain the air gap magnetic flux densitynear the surface of the permanent magnet. In Chapter 3, this multislice 2D tech-nique was examined extensively for both analytical and finite element modelling.There, the air gap magnetic field obtained through analytical and finite elementmodelling showed comparable results. Nevertheless, when the effect of saturationin the magnetic core on the air gap magnetic flux density needsto be considered,nonlinear finite element analysis as introduced in Chapter 3is necessary.

The air gap magnetic flux density data are obtained for different positions ofthe rotor, using analytical expressions or static finite element computations. Onlythe axial component, this is the component perpendicular tothe permanent magnetsurface, is considered to be responsible for the induced eddy current.

Here, static finite element computations are used to obtain the air gap magneticflux density data. Retrieving the air gap magnetic flux density data can be donesimultaneously with the calculation of flux linkage, electromotive force and elec-tromagnetic torque. Calculation of the eddy currents in thepermanent magnets isdone in postprocessing.

5.3.2 Air Gap Magnetic Flux Density Reconstruction

In order to impose the harmonic air gap magnetic flux density data to the 2D finiteelement model used for the eddy current calculation in the permanent magnets,some handling is required. A selection over one pole pitch ofthe 1D air gap mag-netic flux density from the multislice 2D simulations is collected, taking into ac-count the movement of the permanent magnet in a rotating machine. As the air gapmagnetic flux density is only known at the different slices, the data are extended

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with zero values beyond the inner and outer radius. A comparison between thereconstructed air gap magnetic flux density obtained through multislice 2D dataand an air gap magnetic flux density generated with a commercial 3D finite ele-ment package is made in Fig. 5.4 and Fig. 5.5. Here, the air gapmagnetic flux

Bg

[T]

y [m] x [m]-0.02-0.01

00.01

0.02

0.04

0.05

0.06

0.07

0.08-1.5

-1

-0.5

0

0.5

1

1.5

Figure 5.4: Reconstructed 2D air gap magnetic flux density obtained bycombining multislice 2D results. The T-shape of the permanentmagnet is clearly visible. The presence of a stator slot decreasesthe magnetic flux density in the center of the permanent magnet,while the armature reaction influences the magnetic flux densityover the azimuthal direction: decrease at the left side of the per-manent magnet, increase at the right side.

density is illustrated at load when a stator slot is in front of the permanent magnet.In the unfiltered air gap magnetic flux density pattern generated with commercial3D finite element package there is clearly some noise due the rather coarse meshand linear elements, while this is absent in the reconstructed pattern due to the finemeshing and the second order elements in the multislice 2D finite element simu-lations. More important, the multislice 2D finite element simulations are able toreconstruct the air gap magnetic flux density adequately.

The air gap magnetic flux density patterns as presented in Fig. 5.4 and Fig.5.5 are a function of the rotor position. However, a time harmonic finite element

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Bg

[T]

y [m] x [m]-0.02-0.01

00.01

0.02

0.04

0.05

0.06

0.07

0.08-1.5

-1

-0.5

0

0.5

1

1.5

Figure 5.5: Air gap magnetic flux density obtained with a commercial 3Dfinite element package to verify the reconstructed 2D air gapmagnetic flux density obtained by combining multislice 2D re-sults. The relatively coarse mesh and use of linear elementsaddsnumerical noise to the solution. Nevertheless, good correspon-dence with the 2D air gap magnetic flux density in Fig. 5.4 isfound.

analysis is used to calculate the eddy currents in the permanent magnets. Thus, therotor position dependent air gap magnetic flux density should be transformed intothe frequency domain by use of fourier series. For each pointin the air gap, theharmonic with orderν of the magnetic flux densityBν is a scalar complex value.These harmonic magnetic flux density patterns will be imposed to the permanentmagnet surfaces in the 2D time harmonic eddy current simulations.

5.3.3 2D Magnet Eddy Current Calculation

For the calculation of the eddy currents in the permanent magnets the 2D modelillustrated in Fig. 5.6 is used. In this model, appropriate material properties aregiven to the permanent magnet area, and the area enclosing the permanent magnetsis modelled as air. The Dirichlet boundary condition is applied at the outer edges ofthe geometry. The problem to be solved is a time harmonic in plane electric current

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1

1

1

1

2

Figure 5.6: Time harmonic 2D finite element model used to calculate theeddy currents in the permanent magnets and their correspondinglosses. (1) Diriclet boundary condition and (2) impositionof theair gap magnetic flux density over the permanent magnet area.

calculation of which the source is a magnetic flux density specified perpendicularto that plane. This problem is very similar to the previouslymentioned simulationsin which an in plane magnetic flux density is calculated, given an external currentdensity.

Starting from time harmonic Faraday’s law

∇× E = −jνΩBν , (5.10)

the modified electric Gauss’ law

∇ · E = 0, (5.11)

and the constitutive relationE = ρPMJ, (5.12)

the equation to be solved for the time harmonic in plane electric current calculationcaused by an external magnetic flux density perpendicular tothat plane becomes

∇× (ρPM∇× F) = −jνΩBν ν = 1, 2, . . . (5.13)

whereJ = ∇ × F. In these equations,E is the in plane electric field,J is the inplane electric current density,ν the harmonic order,Ω the mechanical rotationalspeed in rad/s,Bν the harmonic component with orderν of the magnetic flux den-sity perpendicular to the plane, andρPM the electric resistivity of the permanentmagnets. In analogy to the magnetic vector potentialA, F is called the electricvector potential. In planar 2D calculationsF only has a component perpendicu-lar to the calculation plane. Fig. 5.7 illustrates the eddy current density in the

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×105

0

0.5

1

1.5

2

Figure 5.7: Electric current density in the permanent magnets due to the15th harmonic of the air gap magnetic flux density, illustratedfor a permanent magnet consisting of two segments. The elec-tric current density is expressed in A/m2.

permanent magnets at no load forν=15 and a permanent magnet consisting of 2segmentations.

Calculation of the eddy current losses in the permanent magnets is based on theassumption that the permanent magnets are surface mounted and the eddy currentsare resistance limited [32],i.e., the relatively high resistivity and low permeabilityof the permanent magnets will limit the amplitude of the induced eddy currents andthe reaction field produced by the eddy currents is negligible. The skin depth of theeddy current distribution is much greater than the axial thickness of the permanentmagnets. The rare-earth permanent magnets are sintered neodymium iron boron(NdFeB) magnets, of which the relative permeability is typically µPM=1.03-1.05[34] and the electric resistivityρPM is in the range of 100-200µΩ·cm [35]. Thedepth of penetration of the harmonic with orderν in the permanent magnet materialis defined by

δν =

√

2ρPMνΩµ0µPM

. (5.14)

In Fig. 5.8, the depth of penetration is plotted as a functionof the harmonic orderat the upper rotational speed of 2500 rpm. Only for very high harmonic orders, theskin depth reaches the axial thickness of the permanent magnets which is 5 mm.

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Therefore, the eddy currents can be assumed to be resistancelimited.δ

[mm

]

ν

0 50 100 150 200 250 300 350 400 450 5000

10

20

30

40

50

60

Figure 5.8: Depth of penetration in the permanent magnet material as afunction of the harmonic orderν. Even for the higher harmon-ics, the depth of penetration exceeds the axial length of theper-manent magnets, which justifies the eddy currents to be calcu-lated as resistance limited.

The harmonic with orderν of the air gap magnetic flux density causes at thepermanent magnet surface an eddy current densityJν,0. This eddy current den-sity at the permanent magnet surface is obtained through finite element analysisas depicted in Fig. 5.7. As the magnetic flux density penetrates into the perma-nent magnet, the eddy current density in the permanent magnet will decrease. Thedependency of the eddy current density with the depth of penetration is expressedby

Jν(z) = Jν,0 exp

(

− z

δν

)

(5.15)

wherez is the axial direction,i.e. the direction in which the magnetic flux densitypenetrates the permanent magnet.

As the eddy current loss density is proportional to the current density squared,the eddy current loss corresponding to each harmonic eddy current density can be

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calculated by integration over the permanent magnet volume

PPM,ν =ρPM2

∫

SPM

hPM∫

0

J2νdzdSPM

=ρPM2

∫

SPM

hPM∫

0

J2ν,0 exp

(

−2z

δν

)

dzdSPM. (5.16)

The harmonic eddy current losses are represented in Fig. 5.9at rated load andspeed. Losses are mainly assigned to the multiples of the slot number,i.e. ν=15,30, 45,. . . and the lower harmonic numbers.

PPM,ν

[W]

ν

0 50 100 150 200 250 300 350 400 450 50010−4

10−3

10−2

10−1

100

101

Figure 5.9: Eddy current loss in a single permanent magnet as a functionof the harmonic orderν during load. Losses are mainly causedby eddy currents induced by air gap magnetic flux density har-monics for the harmonic orders that are a multiple of the slotnumber.

Subsequently, the total eddy current loss is found by summation over the har-monic loss components

PPM =

∞∑

ν=1

PPM,ν . (5.17)

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5.3.4 Model Verification

To verify the accuracy of the multislice 2D - 2D coupled modelling, the resultsare compared with those obtained through 3D finite element analysis. Althoughcommercial software is claiming to solve these problems outof the box, a plausiblesolution was never found, and therefore, an alternative solution was introduced.Instead of making use of moving mesh techniques on a time stepped 3D finiteelement model of the full geometry, the time stepping is performed on a geometrythat only models the permanent magnet and a segment of the rotor (Fig. 5.11).The boundary conditions include the imposition of the vector potential. Thesevector potential data are obtained through static 3D finite element analysis on thefull geometry of the axial flux PM machine. This method creates a good trade-offbetween calculation time and accuracy.

The model used for the static 3D simulations is presented in Fig. 5.10. Again,due to magnetic symmetry, only half of the axial flux PM machine needs to bemodelled. In the static simulation the vector potential equation

1

1

2

3

3

3

Figure 5.10: 3D Finite element model used for the static simulations. Be-cause of the axial symmetry, only half of the machines is mod-elled. Dirichlet boundary condition (1), Neumann boundarycondition (2), and boundaries at which the vector potentialdatais stored (3).

∇×(

1

µ0µr∇× A

)

= J (5.18)

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where

B = ∇× A (5.19)

is solved. For simulations modelling load conditions, the external current densityJis imposed in the stator coils. A linear behaviour of the stator core material is usedin order to reduce the simulation time. When the solution forthe vector potentialis found, the vector potential at the boundaries indicated in Fig. 5.10 is stored.

As the static simulations are performed for each rotor position separately, thegauge functionΨ may be different for each solution and has an impact on the vectorpotential by

A = A +∇Ψ. (5.20)

As the vector potential data are imposed at the boundaries ofthe time steppedfinite element model of the permanent magnet and a segment of the rotor, the gaugefunctionΨ should by chosen appropriately in such a way that the vector potentialA is unique. Therefore, the Helmholz’s theorem should be fulfilled, i.e., both∇·Aand∇× A are defined. This is done by setting the Coulomb gauge∇ · A = 0.

Due to the rather coarse mesh and the use of linear elements, the components ofthe vector potential contain some numerical noise. Therefore, a smoothing spline isfit into the different vector potential data. Storage of the spline coefficients allowsfast evaluation of the vector potential data at the boundaries on which the vectorpotential data are imposed in the model presented in Fig. 5.11. This model is used

1

1

1

2

22

Figure 5.11:3D Finite element model of a permanent magnet and a seg-ment of the rotor used for the time simulations. (1) Dirichletboundary condition and (2) imposition of the magnetic vectorpotential.

to calculate the eddy currents in the permanent magnets.In this time stepped finite element analysis, the solution for the vector potential

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A, is found by solving the following equation

1

ρPM

∂A∂t

+∇×(

1

µ0µPM∇× A

)

= J. (5.21)

After a transition effect, the steady state time waveforms of the eddy current lossesin the permanent magnets are obtained.

As the multislice 2D - 2D coupled modelling was introduced asa responseto the accuracy, memory and time constraints related to timestepped 3D finiteelement analysis, a benchmark is made. In this benchmark, both models are usinga linear material characteristic, a permanent magnet without segmentation, and theeddy currents are calculated at rated speed and load. The 120 rotation of therotor, i.e. the fundamental period, is performed using 480 steps. The multislice 2Dmodel has about 30.000 quadratic elements per slice and usesthe COMSOL 3.5aUMFPACK-solver, while the full 3D model uses over 300.000 linear elements anduses the COMSOL 3.5a SPOOLES-solver.

Despite the difference in modelling, comparable results for the eddy currentlosses in the permanent magnets are found; the total magnet eddy current loss,i.e.for all 32 magnets, is rated at 15.41 W for the multislice 2D - 2D coupled modellingand 15.12 W for the 3D modelling. The advantage of the multislice 2D - 2D cou-pled modelling is significant when considering the simulation times. In Table 5.2,the simulation time of each individual process is mentioned2. Summation over thedifferent processes results in 176.289 s, equivalent to more than 2 days, for the 3Dmodelling and only 21.451 s, equivalent to less than 6 hours for the multislice 2D- 2D coupled modelling. Making use of the multislice 2D - 2D coupled modelling,

Table 5.2: Calculation time required for each process in multislice 2D- 2Dcoupled modelling and the 3D modelling. Numbers: 480 rotorpositions, 6 computation planes in the multislice-2D model, 3components for the vector potential, and 500 harmonic compo-nents.

Process multislice 2D - 2D 3D

Static simulation 480×6×5s 480×325sData handling 51s 3×1224sTime harmonic/transient 500×14s 16617s

thus results in a reduction of the simulation time by a factor8 while the accuracyof the simulation results is maintained. It should be mentioned that the used 3Dmodel is already an optimised version in memory and time constraints comparedto time stepped 3D finite element analysis on a full geometrycfr. [15] where a timestepped 3D finite element analysis on a full geometry took about one week.

2Processor: Intel(R) Core(TM)2 Quad CPU Q9650 @3.00 GHz. Installed memory (RAM)8.00 GB. System type: 64-bit operation system

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5.4 Magnet Segmentation

Sintered neodymium iron boron (NdFeB) permanent magnets have a relatively lowelectric resistivity in the range of 100-200µΩ·cm [35]. Moreover, the thermal con-ductivity of NdFeB is relatively poor, only about 9 W/mK [36]. This may causeproblems in surface mounted structures, where variations in the magnetic flux den-sity induce eddy currents in the permanent magnets. As the heat transfer in theseNdFeB permanent magnets is relatively low, heating of the permanent magnetsoccurs. Initially a temperature increase in the permanent magnets lowers the rema-nent flux density. However as the temperature in the permanent magnets reachesa critical temperature, irreversible demagnetisation of the permanent magnets mayoccur. These eddy currents in the permanent magnets should thus be limited asthey reduce the energy efficiency of the machine anyhow.

In contrast to sintered permanent magnets, plastic bonded permanent magnettechnology is emerging, offering a practically lossless alternative, high mass pro-duction in any shape using polymer molding technology is possible, and the plasticresin protects the corrosive NeFeB towards harmful environments [37–41]. Never-theless, current available technology in plastic bonded permanent magnets gener-ally produces a remanent flux density that is significant lower than 1 T. In [36], aplastic bonded NdFeB permanent magnet was used with a remanent flux density of0.65 T and a coercive force of 430 kA/m. As a reference, the considered sinteredNdFeB permanent magnets have a remanent flux density of 1.26 Tand a coerciveforce of 955 kA/m (20C). Because of the high energy product, sintered NdFeBis here preferred above plastic bonded NdFeB. As a consequence, rather expensivesegmentation [42] of the permanent magnets is introduced tolimit the eddy currentlosses in the permanent magnets.

In radial machines, circumferential and axial segmentation is examined exten-sively in [16, 17] using analytical modelling and in [24–26]using finite elementanalysis. Some models [27, 28] even introduce copper cladding of the permanentmagnets as a supplementary measure to reduce the eddy currents in the permanentmagnets. Segmentation of permanent magnets is mostly done by combining multi-ple small permanent magnets which are electrically insulated externally. However,recently a realisation of an internal insulation techniqueis presented in [43].

Here, the segmentation of permanent magnets in axial flux PM machines isinvestigated. In axial flux PM machines, the bulky permanentmagnet is segmentedin both radial and azimuthal direction. To investigate the influence of the numberof segments, 4 segmentation grades presented in Fig. 5.12 are considered.

Using the multislice 2D - 2D coupled model has the advantage that the staticmultislice 2D simulation, that consumes most of the overallprocess time, onlyneeds to be performed once for each load condition. Once the 2D air gap magneticflux density is reconstructed, it can be imposed to the 2D timeharmonic modelsmodelling the different segmentation grades.

For each segmentation grade, an evaluation of the eddy current losses is made

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5.4 Magnet Segmentation 129

(a) (b)

(c) (d)

Figure 5.12: Considered magnet segmentation grades: (a) bulky: no seg-mentation, (b) 2 segments, (c) 4 segments, and (d) 14 seg-ments.

in the two extreme working conditions,i.e. no load and load. Both simulations areperformed at the rated speed. These two working conditions allow to make a cleardistinction between eddy current losses caused by the slotting effect and losses dueto armature reaction.

In Table 5.3, an overview of the eddy current losses at no loadand load forthe different segmentation grades is presented. For the bulky permanent magnet,

Table 5.3: Effect of segmentation on the eddy-current losses.

magnet eddy current loss# segments no-load load

1 8.1435 15.41492 6.6104 11.75614 5.8773 10.145014 2.8881 4.0904

relatively high eddy current losses are found, which are notnegligible with respectto the losses in the iron core and the resistive losses in the copper winding. Dueto the semi open slots, the eddy current losses are already quite high at the rated

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speed of 2500 rpm. The use of open slots, which are used in manypapers as anadvantage of fractional slot windings for the winding process, would even resultin higher eddy current losses. On the other hand, the armature reaction of thefractional slot winding has a significant contribution to the eddy current losses inthe permanent magnets. Fractional slot windings thus result in significant eddycurrent losses. Nevertheless, similar as in radial machines, segmentation of thepermanent magnets results in a significant reduction of the eddy current losses inthe permanent magnets. Segmentation of the permanent magnets can reduce theeddy current losses in the permanent magnets to a level whichno longer resultsin harmful heating of the permanent magnets. Moreover, within the scope of thisresearch, permanent magnet segmentation also results in animprovement of theenergy efficiency of the machine.

5.5 Rotor Back Iron Loss

As the rotor yoke is situated right under the permanent magnets, it is still exposedto parts of the varying magnetic flux density. The part of the varying magnetic fluxdensity that reaches the rotor yoke strongly depends on the machine geometry andwinding arrangement. The effect of stator slotting on the eddy current losses in therotor core will be much stronger for machines with open slotsthan for machineswith semi closed slots. Therefore, in machines with semi closed slots, the eddycurrent losses in a solid rotor yoke remain limited while lamination of the rotoryoke is mostly necessary for machines with open slots to limit the eddy currents.

To estimate the eddy current losses in the rotor yoke, a similar multislice 2D -2D coupled model as for the eddy current losses in the permanent magnets couldbe used. However as the multislice 2D - 2D model takes only theaxial componentof the varying magnetic flux density near the surface of the rotor back iron intoaccount, its applicability is limited as the varying magnetic flux density will have acomplex (3D) path into the rotor back iron [44]. Moreover, whereas in the perma-nent magnets the depth of penetration depth was higher than the axial length of thepermanent magnet, the depth of penetration in the iron yoke is much lower thanthe axial length of the rotor yoke. The depth of penetration in the solid steel of therotor yoke is much smaller due to its high relative permeability (estimated at 2000)and the low electric resistivity (estimated at 50µΩ·cm). The eddy currents are nolonger resistance limited, and the skin limited [10] behaviour of the eddy currentsshould be considered: skin limited eddy currents do not entirely penetrate throughthe rotor yoke, and the reaction field of the eddy currents should be considered.This shielding effect of the rotor yoke to the varying air gapmagnetic flux den-sity is taken into account when performing time-stepped finite element simulationsusing a full 3D model.

Here, the influence of the reaction field is neglected and the multislice 2D -2D coupled modelling technique is applied. As the air gap fluxdensity data werealready recorded to calculate the eddy currents in the permanent magnets, only the

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5.6 Conclusion 131

time harmonic 2D finite element eddy current loss calculations need to be done.This 2D model of the rotor yoke includes the complete rotor surface to take intoaccount the subharmonic induced eddy currents since the subharmonics of the mag-netic flux density travel a significantly larger path throughthe stator compared tothe fluxes corresponding the fundamental component.

Simulations applying the multislice 2D - 2D coupled modelling technique es-timated the rotor losses less than 2 W. Therefore, they are neglected in future partsof this thesis.

5.6 Conclusion

Fractional slot windings are generally known for their highmagnetomotive har-monic content. As these harmonics induce eddy currents in the relatively wellconductive NeFeB permanent magnets, attention should be paid towards the tem-perature of the permanent magnets. An increasing temperature of the permanentmagnets has a reducing effect on the remanent magnetisation, and the permanentmagnets may be even permanently demagnetised if the temperature becomes toohigh.

Thus, when designing axial PM machines with fractional slotwindings, theseeddy current losses in the permanent magnets should be considered. As commer-cially available 3D finite element packages take too much time, and thus makethem less useful in preliminary machine designs or optimisation, a less time con-suming method to calculate the eddy current losses in the permanent magnets wasproposed. There, multislice 2D simulations are used to reconstruct the 2D air gapmagnetic flux density, which is consequently used to calculate the eddy current inthe permanent magnets using time harmonic 2D finite element modelling. There-fore, this technique is calledmultislice 2D - 2D coupled modelor multilayer 2D -2D coupled model. As it approximates 3D modelling, these techniques are oftenreferred to asQuasi 3D. This modelling technique combines a good accuracy withlow memory and time constraints.

As this research focusses on energy efficiency, the eddy current losses in thepermanent magnets are decreased by introducing segmentation of the permanentmagnets. Therefore, 4 segmentation grades including azimuthal and radial seg-mentation are examined. As in radial machines, permanent magnet segmentationsignificantly reduces the eddy current losses and thus improves the energy effi-ciency of the machine.

Bibliography

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Chapter 6

Influence of the Stator SlotOpenings

6.1 Introduction

Research towards the influence of geometrical parameters was only mentionedbriefly in Chapter 3. There, an optimisation process regarding a limited set ofparameters was performed. In this optimisation, it was already noticed that someparameters have a contrary effect on the different losses inthe machine. For ex-ample, a high axial length of the stator cores combined with large slot openings isbeneficial with respect to the copper losses, however, it results in higher stator corelosses.

A comparable situation is caused by the width of the stator slot openings (Fig.6.5). As will be illustrated in this chapter, this parameterhas a major influence onthe stator core losses and eddy current losses in the permanent magnets. It will beillustrated that the influence of the stator slot openings width on both loss mech-anisms is contrary; widening of the stator slot openings will result in lower statorcore losses, but will increase the eddy current losses in thepermanent magnets andvice versa.

To investigate the influence of the stator slot openings width on the losses, themathematical models that were introduced in previous chapters are applied. Themultislice 2D model using finite element analysis, introduced in Chapter 3, is usedto calculate the stator core losses. The eddy current lossesin the permanent mag-nets are evaluated using the multislice 2D - 2D model, introduced in Chapter 5. Inthese simulations the stator slot openings width is varied from nearly closed slots(1 mm) to nearly open slots (11 mm) using a domain scan. As willbe illustrated, avariation of the stator slot openings width has also a minor impact on the electro-magnetic torque. Therefore, a final comparison is made in which the total losses inthe machine are placed against the output power of the machine.

Although the losses in the stator core elements strongly depend on the grade of

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138 Influence of the Stator Slot Openings

the considered laminated silicon steel, and the eddy current losses in the permanentmagnets on the segmentation grade, only one material and segmentation grade isused in the evaluation of the losses. Consequently, the aim of this chapter is topoint out the important effect of the stator slot openings width on the losses in thedesign stage of the machine, focusing on the acting loss phenomena rather thanaiming an extensive quantitative analysis.

Within this research topic, the authors of [1] very recentlyinvestigated theinfluence of stator slotting on the performance of radial fluxpermanent magnetmachines with concentrated windings, and in [2, 3] one introduced soft magneticwedges in the stator slot openings to vary the no load performance.

6.2 Stator Core Losses

The influence of the stator slot openings width has a direct impact on the geometryof the different stator core laminations, and therefore, also on the magnetic fluxdensity pattern in the stator core elements. As the magneticflux density can besubdivided into a component by the permanent magnets and by the armature reac-tion, both contributions are first being examined separately before considering thetotal flux. To illustrate the findings, two values are chosen in all further examples:rather closed slots having a slot opening of only 3 mm and rather open slots havinga 9 mm slot opening.

6.2.1 Permanent Magnets

In Fig. 6.1 the magnetic flux density pattern in the stator core element is illus-trated in case only permanent magnets were present and aligned with the statorcore element in case of 3 mm slot opening width. In case of alignment of the per-manent magnets with the stator core, the overall maximal values of the magneticflux density are found in the stator core material. Due to the shape of the perma-nent magnets1, the magnetic flux density in the laminations near the inner radius isfound to be higher than the one at the outer radius. This is caused by the variabletooth pitch as a function of the diameter combined with a constant slot width. Fluxleveling over the laminations by radial magnetic flux components, is limited dueto the very poor permeability of the stator cores in the direction perpendicular tothe lamination planes. Consequently, radial flux components remain limited. How-ever, when some lamination regions become saturated, magnetic fluxes will moveto the adjacent less saturated lamination regions. As the corresponding magneticfluxes are in the radial direction, this may result in significant eddy current lossesin the plane of the laminations. Although the presence of these radial fluxes isnot examined in this work, a conservative attitude is maintained towards unequal

1The permanent magnet span is generally chosen to have a constant value, for example a span of0.8τp

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6.2 Stator Core Losses 139

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Figure 6.1: Magnetic flux density in the stator core when the permanentmagnet is aligned with the stator core for the 3 mm stator slotopenings width. Magnetic flux density values in T.

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magnetic fluxes over the different lamination planes in general and saturation oflamination regions in particular.

The influence of increasing the slot opening width from 3 to 9 mm is illustratedin Fig. 6.2. At the inner diameter lamination regions, the magnetic flux density hasdecreased with an average value of 0.3 T as the smaller tooth tips catch less mag-netic flux. In the laminations at the outer diameter region, this effect is less visibledue to the relatively larger tooth tip width, having only an average decrease withless than 0.1 T. As a result, the magnetic flux density is better leveled over the dif-ferent lamination layers and possible saturation of the inner diameter laminationsis avoided. However, as the total magnetic flux in the stator core element reduces,this will have an influence on the flux linkage with the tooth coils.

6.2.2 Armature Reaction

Whereas the previous magnetic flux density patterns were generated by the perma-nent magnets only, here only the armature reaction is considered. Therefore, theremanent magnetic flux density of the permanent magnet regions in the multislice2D finite element model is set to zero.

The magnetic flux density pattern is plotted in Fig. 6.3 for the 3 mm slotwidth opening and the nominal 7 A current. Apparently the magnetic flux is hardlycrossing the air gap, but is transferred through the tooth tips to the adjacent teethwhere the flux is in the opposite direction. This is a general observation in surfacemounted PM machines where the air gap is generally significantly bigger than theslot opening width. As this flux is not crossing the air gap, itcan be considered asleakage flux. Due to the relatively low reluctance for these leakage flux through thetooth tips, the level of the magnetic flux density in the stator core is not negligible.Moreover, as for the magnetic flux by the permanent magnets, higher magnetic fluxdensities are found at the inner diameter regions compared to the outer diameterregions. For machines with a high electric loading, saturation of the inner diameterlamination might take place while the upper laminations arestill unsaturated.

Increasing the slot openings width to 9 mm results in the magnetic flux den-sity pattern in Fig. 6.4. Due to the increase of the slot openings, the reluctancepath for the leakage flux through the tooth tips is seriously increased. Therefore,the magnetic flux density levels in the stator core elements are strongly reduced.The reduction of the magnetic flux density is higher at the inner diameter regionscompared to the outer diameter regions. For the same currentdensity, the risk ofsaturation of the stator cores is hence less in case of wide slot openings.

6.2.3 Combination

The previous flux density patterns, both taken at the maximumvalue of the mag-netic flux, showed lower magnetic flux density levels for the machine with thewider slot openings. As the peak values of the magnetic flux density are strongly

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Figure 6.2: Magnetic flux density in the stator core when the permanentmagnet is aligned with the stator core for the 9 mm stator slotopenings width. Magnetic flux density values in T.

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0.2

0.3

0.4

0.5

0.6

0.7

0.8


0.2

0.3

0.4

0.5

0.6

0.7

0.8


Figure 6.3: Magnetic flux density in the stator core taking only armaturereaction into account for the 3 mm stator slot openings width.Magnetic flux density values in T.

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0.2

0.3

0.4

0.5

0.6

0.7

0.8


0.2

0.3

0.4

0.5

0.6

0.7

0.8


Figure 6.4: Magnetic flux density in the stator core taking only armaturereaction into account for the 9 mm stator slot openings width.Magnetic flux density values in T.

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related to the stator core losses, lower stator core losses are expected for the ma-chine with the wide slot openings. In contrary, the flux linkage with the tooth coilswill be influenced as well, which will result in a variation ofthe electromagnetictorque.

6.3 Analysis of the Air Gap Magnetic Flux Density

Next to the effect of the stator slot openings on the magneticflux density pattern inthe stator cores, the stator slot openings have also a major influence on the air gapmagnetic field. The analytical expression for the permeancefunction, introducedin Chapter 3, will be used to explain the influence of an increase in the stator slotopenings width on the air gap magnetic flux density.

As will be illustrated in the next paragraphs, an increase ofthe stator slot open-ings results in a bigger variation of the air gap magnetic fluxdensity. This increasein the variation of the air gap magnetic field induces higher eddy currents in thepermanent magnets, and hence, causes additional losses. Moreover, the increasingslot openings result in a higher reluctance difference overthe circumference of themachine which will result in a higher cogging.

6.3.1 Permeance Function

The effect of the stator slot openings on the air gap magneticflux density can beexplained by considering the permeance function which was introduced in Chapter3. This permeance function modelled the local decrease of the permeance near thestator slot openings. In Fig. 6.5, the permeance function isplotted for the 3 mmand 9 mm slot openings.

An increase of the stator slot openings not only includes a wider azimuthalregion in which the permeance function is affected, but has also a major influenceon the local permeance drop. Wider slot openings will hence result in an increaseof the variation in the air gap magnetic flux density near the slot openings, andtherefore, induce higher eddy currents in the permanent magnets.

Moreover, the wider slot openings result in a increase of Carter’s factor. Thedecrease of Carter’s factor can be directly linked to the decreasing flux linkage ofthe tooth coils. Based on this analytical expression, a reduction of the electromag-netic torque as a function of the slot opening would be expected.

6.3.2 Eddy Current Losses in the Permanent Magnets

In Fig. 6.6 the air gap magnetic field in the rotor reference frame is plotted for the9 mm slot opening for both no load and rated load. Comparison of this figure withFig. 5.1, which differs only on the 3 mm slot opening, revealssome interestinginformation.

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b0

λ,bs0=

9mm

Azimuthϕ []

λ,bs0=

3mm

Azimuthϕ []

0 8 16 24 32 40 480 8 16 24 32 40 480.4

0.5

0.6

0.7

0.8

0.9

1

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6.5: Influence of the stator slot openings on the permeance function.The permeance function is evaluated at the inner computationplane and in the center of the air gap; (left) for 3 mm, and (right)for 9 mm slot opening width.

Although these air gap magnetic fields are simulated using finite element anal-ysis, the predictions based on the simple analytical expression for the permeancefunction are confirmed. The analytical permeance function predicted the effect ofstator slotting on the air gap magnetic field over a wider azimuthal region and haslower values. Indeed, the local drops in the air gap magneticflux density near thestator slots are wider and deeper, from 0.8 T for the 3 mm slot openings to 0.6 T forthe 9 mm slot openings. Due to the wide and deep drops in the airgap magneticflux density, the average value of the air gap magnetic field will be lower. Thisreduction of the average value of the magnetic flux density isin agreement withthe reduction of the Carter’s factor.

Next to the observations that can be explained through the permeance function,another observation can be done. In Fig. 5.1, correspondingto the 3 mm slot open-ing, there is a significant difference of the no load and ratedload air gap magneticfield due to the armature reaction. In Fig. 6.6, the difference is less observable.Consequently, this observation is expected to have its consequences with respectto the eddy current losses in the permanent magnets. For the 3mm slot openings,the difference between no load eddy currents losses in the permanent magnets and

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146

Influ

ence

ofth

eS

tato

rS

lotO

peni

ngs

Bg,l

oad

[T]


Bg,n

olo

ad[T

]


0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360

0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360

0.4

0.6

0.8

1

1.2

1.4

0.4

0.6

0.8

1

1.2

1.4

Figure 6.6: Air gap magnetic flux density taking into account stator slotting effect only (upper), and both the stator slottingeffect and armature reaction (lower) as a function of the rotor position. Data is expressed in the rotor referenceframe,i.e., position fixed with respect to the rotor; in this case the center of the permanent magnet.

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6.4 Evaluation and Comparison 147

eddy current losses at load were relatively high,cfr. Table 5.3. With increasing slotopenings, the relative difference between the eddy currentlosses in the permanentmagnets at no load and load is expected to decrease.

6.3.3 Cogging

A general disadvantage of PM machines with a slotted stator is the inherent pres-ence of cogging. The influence of the stator slotting on the cogging in axial fluxPM machines was studied extensively in the past, and as an answer many coggingtorque reducing measures were presented in [4–13].

Therefore, no elaboration on cogging torque is done in this work. Neverthelessthe 15-slot-16-magnet combination, avoiding circumferential symmetry, and the T-shaped permanent magnets are chosen in order to reduce cogging in the suggestedtest case machine.

Moreover, very recent papers [14–16] illustrated the massive impact of eccen-tricity faults and manufacturing dissymmetry effects on the cogging torque of themachine.

6.4 Evaluation and Comparison

Whereas the previous sections were introduced to acquire some understanding onthe mechanisms causing the different losses qualitatively, a quantitative analysis isperformed in this section.

For the material in the stator core elements, a laminated steel with M600-50Agrade is chosen and the permanent magnets are assumed to consist of four elec-trically isolated segments. The stator slot openings are varied between 1 mm and11 mm.

To evaluate the electromagnetic torque, stator core lossesand eddy currentlosses in the permanent magnets, the multislice 2D model is used and coupledwith the 2D model for the eddy currents in the permanent magnets. Simulationswere performed for both no load and the rated load current of the machine. Allrelevant data are presented in Table 6.1.

6.4.1 Stator Core Losses

Analysis of Table 6.1 indicates that the stator core losses for both no load and loadworking conditions decrease as the stator slot openings become wider. It shouldbe noticed that the effect at no load is less significant as forworking at rated load.Indeed, the analysis of the magnetic flux density patterns showed minor changefor the flux density caused by the permanent magnets comparedto those by thearmature reaction. The effect of the increased reluctance path in the tooth tipsby increasing stator slot openings has a dominant position in the decrease of themagnetic flux density in the stator core and the corresponding stator core losses.

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148

Influ

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ofth

eS

tato

rS

lotO

peni

ngs

Table 6.1: Influence of the stator slot openings width on the machine’s performance.

Stator slot Average Average Load lossesa No load lossesb

openings torque power Core PM Totalc Core PM Totalwidth [mm] [Nm] [kW] [W] [W] [W] [W] [W] [W]

1 18.2170 4.7692 148.0341 5.4458 214.2799 124.4200 0.1691 124.58913 18.5022 4.8439 135.1230 10.1450 206.0680 124.5130 5.8773130.39035 18.3766 4.8110 127.2899 21.9092 209.9991 120.3200 18.4188 138.73887 18.0888 4.7356 118.7487 33.4548 213.0035 113.4800 30.3333 143.81339 17.6532 4.6216 109.2007 40.5223 210.5230 104.8290 37.5455 142.374511 17.0738 4.4699 98.9526 42.5613 202.3139 95.0712 39.6920134.7632

aEvaluated at the rated speed of 2500 rpm, and the rated current of 7 A.bEvaluated at the rated speed of 2500 rpm.cCopper losses in the windings are estimated at 60.8 W.

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6.4 Evaluation and Comparison 149

The difference between the core losses with closed slots andopen slots is massive;a decrease of more than 30% is achieved. At the same time the flux density levels inthe stator core elements are reduced, eventually resultingin mitigation of saturationof the lamination profiles at the inner diameter region.

6.4.2 Eddy Current Losses in the Permanent Magnets

Despite the losses in the stator cores decrease, Table 6.1 indicates that the lossesin the permanent magnets due to eddy currents increase seriously. Increasing eddycurrents losses in the permanent magnets as a function of increasing stator slotopenings are found at no load as well as at rated load. For the nearly closed slots,the no load eddy current losses almost vanish as the small slot openings hardlyinfluence the air gap magnetic field. For these small stator slot openings, the ef-fect of the armature reaction is dominating the losses. Thisis in agreement withthe relative big differences in the air gap magnetic field at no load and load thatwere presented in Fig. 5.1, having the 3 mm stator slot openings. As the statorslot openings become wider, the effect of stator slotting dominates the armaturereaction resulting in a lower increase of the losses at load compared to those at noload.

6.4.3 Electromagnetic Torque and Power

In Fig. 6.1 and Fig. 6.2 it was illustrated that the stator slot openings influencethe magnetic flux density in the stator core elements. Comparison showed that thetotal flux in the stator core elements decreases with increasing stator slot openings.Moreover it can be expected that a variation of the stator slot openings has an im-pact on the harmonic content of the coil flux linkage and coil electromotive force.Indeed, Fig. 6.7 presents the coil flux linkage and coil electromotive force for thestator slot openings width set to 3 mm and 9 mm respectively. The figure illustratesa none negligible reduction of the peak value of the coil flux linkage for larger sta-tor slot openings, while comparison of the coil electromotive force clearly indicatesthe change in the harmonic content. The electromotive forcecorresponding to the9 mm stator slot openings has less harmonic content and is a better approximationto the pure fundamental sinusoidal component.

Consequently, the stator slot openings have also an effect on the average elec-tromagnetic torque and torque ripple. In Fig. 6.8 the electromagnetic torque isplotted for the 3 mm and 9 mm stator slot openings. Here, the electromagnetictorque is evaluated by the Maxwell stress harmonic filter method (3.103).

Although the electromagnetic torque has decreased for the 9mm with respectto the 3 mm width of the stator slot openings, it is wrong to conclude that themaximum electromagnetic torque is obtained for fully closed slots. Table 6.1 indi-cates that the maximum of the electromagnetic torque for the7 A load current isreached for the 3 mm stator slot opening. A similar conclusion in [1] confirms this

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PSfrag

Ec [V], 9 mm

ψc [mWb], 9 mm

Ec [V], 3 mm

ψc [mWb], 3 mmψc

[mW

b]/E

c[V

]


0 45 90 135 180 225 270 315 360-60

-40

-20

0

20

40

60

Figure 6.7: Comparison of the coil flux linkage and coil electromotive forcefor the stator slot openings width set to 3 mm and 9 mm.

observation.

< Te > [Nm], 9 mm

Te [Nm], 9 mm

< Te > [Nm], 3 mm

Te [Nm], 3 mm

Te

[Nm

]/<Te>

[Nm

]


0 45 90 135 180 225 270 315 36017.6

17.8

18

18.2

18.4

18.6

18.8

Figure 6.8: Comparison of the electromagnetic torque for the stator slotopenings width set to 3 mm and 9 mm.

Although large slot openings are likely to decrease the electromagnetic torque,

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6.5 Conclusion 151

Fig. 6.8 indicates a reduction of the torque ripple for the wide stator slot openings.It should be mentioned that the cogging torque in the prototype machine has almostvanished due to the choice of the 15-slot-16-magnet configuration and the T-shapedpermanent magnets.

6.4.4 Overview

The selection of the appropriate stator slot opening is a trade-off between averagepower output, stator core losses and eddy current losses in the permanent magnets.Within the focus on this work, it is obvious to choose for the configuration withthe best energy efficiency. The highest energy efficiency is obtained for the 3 mmstator slot openings. Nevertheless other requirement in the design such as thermalbehaviour of the machine can be decisive in the selection of the stator slot openings.If the stator cannot be cooled efficiently it might be a good choice to move thelosses to the rotor where the power losses are removed by convection.

On the other hand it should be mentioned that these comparisons used only onegrade for the stator core material and one segmentation grade for the permanentmagnets. Moreover the electric loading of the machine will have a major influenceon the selection of the stator slot openings. The selection of the appropriate statorslot openings by quantitative analysis is left for the machine designer and will notbe elaborated in this work.

6.5 Conclusion

This chapter investigated the influence of the stator slot openings on the powerlosses in the stator cores and in the permanent magnets. An increase of the statorslot openings results in a reduction of the leakage flux, which results in an overalldecrease of the magnetic flux density in the stator core elements. Therefore, thecore losses decrease with increasing stator slot openings.On the other hand, largerstator slot openings have a bigger influence on the magnetic flux density in the airgap near the stator slot openings. Larger stator slot openings increase the variationof the magnetic flux density and, hence, increase the eddy current loss in the per-manent magnets. As a conclusion, the variation of the statorslot openings has acontrary effect on both power losses in the axial flux PM machine.

To illustrate the effect of a variation of the stator slot openings on both lossesquantitatively, a domain scan on the geometry of the test case machine is per-formed. Here, the stator slot openings are varied from nearly closed slots to nearlyopen slots and the power losses are calculated for each case.It was observed thatalthough the variation of both losses is significant, the sumof both losses is lessvarying. On the other hand, for very wide stator slot openings, a slight decrease ofthe electromagnetic torque output is found. For the test case machine, the optimalvalues for the sum of the losses and the torque output was found for semi closedstator slot openings.

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Bibliography

[1] H. Vu Xuan, D. Lahaye, H. Polinder, and J. Ferreira, “Influence of stator slot-ting on the performance of permanent-magnet machines with concentratedwindings,” Magnetics, IEEE Transactions on, vol. 49, no. 2, pp. 929–938,2013.

[2] G. De Donato, F. G. Capponi, and F. Caricchi, “No-load performance of ax-ial flux permanent magnet machines mounting magnetic wedges,” IndustrialElectronics, IEEE Transactions on, vol. 59, no. 10, pp. 3768–3779, 2012.

[3] G. De Donato, F. Giulii Capponi, and F. Caricchi, “On the use of magneticwedges in axial flux permanent magnet machines,”Industrial Electronics,IEEE Transactions on, vol. early acces, 2013.

[4] J. Chen, Z. Zhu, S. Iwasaki, and R. P. Deodhar, “Influence of slot openingon optimal stator and rotor pole combination and electromagnetic perfor-mance of switched-flux pm brushless ac machines,”Industry Applications,IEEE Transactions on, vol. 47, no. 4, pp. 1681–1691, 2011.

[5] T. Lubin, S. Mezani, and A. Rezzoug, “2-d exact analytical model forsurface-mounted permanent-magnet motors with semi-closed slots,”Magnet-ics, IEEE Transactions on, vol. 47, no. 2, pp. 479–492, 2011.

[6] W. Fei and P. Luk, “An improved model for the back-emf and cogging torquecharacteristics of a novel axial flux permanent magnet synchronous machinewith a segmental laminated stator,”Magnetics, IEEE Transactions on, vol. 45,no. 10, pp. 4609–4612, 2009.

[7] A. B. Letelier, D. A. Gonzalez, J. A. Tapia, R. Wallace, and M. A. Valen-zuela, “Cogging torque reduction in an axial flux pm machine via stator slotdisplacement and skewing,”Industry Applications, IEEE Transactions on,vol. 43, no. 3, pp. 685–693, 2007.

[8] G. Barakat, T. El-Meslouhi, and B. Dakyo, “Analysis of the cogging torquebehavior of a two-phase axial flux permanent magnet synchronous machine,”Magnetics, IEEE Transactions on, vol. 37, no. 4, pp. 2803–2805, 2001.

[9] D. A. Gonzalez, J. A. Tapia, and A. L. Bettancourt, “Design consideration toreduce cogging torque in axial flux permanent-magnet machines,”Magnetics,IEEE Transactions on, vol. 43, no. 8, pp. 3435–3440, 2007.

[10] F. Caricchi, F. G. Capponi, F. Crescimbini, and L. Solero, “Experimen-tal study on reducing cogging torque and no-load power loss in axial-fluxpermanent-magnet machines with slotted winding,”Industry Applications,IEEE Transactions on, vol. 40, no. 4, pp. 1066–1075, 2004.

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[11] M. Aydin, Z. Zhu, T. Lipo, and D. Howe, “Minimization of cogging torque inaxial-flux permanent-magnet machines: Design concepts,”Magnetics, IEEETransactions on, vol. 43, no. 9, pp. 3614–3622, 2007.

[12] T. S. El-Hasan and P. C. Luk, “Magnet topology optimization to reduce har-monics in high-speed axial flux generators,”Magnetics, IEEE Transactionson, vol. 39, no. 5, pp. 3340–3342, 2003.

[13] J. H. Choi, J. H. Kim, D. H. Kim, and Y. S. Baek, “Design andparametricanalysis of axial flux pm motors with minimized cogging torque,” Magnetics,IEEE Transactions on, vol. 45, no. 6, pp. 2855–2858, 2009.

[14] S. M. Mirimani, A. Vahedi, F. Marignetti, and E. De Santis, “Static eccentric-ity fault detection in single stator-single rotor axial fluxpermanent magnetmachines,”Industry Applications, IEEE Transactions on, vol. 48, no. 6, pp.1838–1845, 2012.

[15] A. Di Gerlando, G. M. Foglia, M. F. Iacchetti, and R. Perini, “Evaluationof manufacturing dissymmetry effects in axial flux permanent-magnet ma-chines: Analysis method based on field functions,”Magnetics, IEEE Trans-actions on, vol. 48, no. 6, pp. 1995–2008, 2012.

[16] S. M. Mirimani, A. Vahedi, and F. Marignetti, “Effect ofinclined static ec-centricity fault in single stator-single rotor axial flux permanent magnet ma-chines,”Magnetics, IEEE Transactions on, vol. 48, no. 1, pp. 143–149, 2012.

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Chapter 7

Combined Wye-Delta Connection

7.1 Introduction

For construction reasons, yokeless and segmented armature(YASA) machines arecomposed of a limited number of stator core elements. As thisresults in a ratherlow slot number, fractional slot windings become interesting. Moreover, for theYASA machines a double layer concentrated fractional slot winding is particularlyinteresting [1]:

• The end windings are much shorter than those of a conventional winding.Less end winding length results in less copper losses for thesame electro-magnetic torque, and thus a better energy efficiency. Additionally shorterend windings result in less copper weight and correspondingly less materialcost. The lower copper weight increases the power density while the materialcost of copper, which is about 6 times the cost of iron, reduces the overallcost of the machine.

• Each stator core element can be wound individually before placing it into thestator. This individual winding process allows to attain a good filling factorof the copper, which results in lower copper losses for the same electromag-netic torque, and correspondingly a better energy efficiency. Moreover, thewinding process is simple and thus it reduces the manufacturing cost.

• Some configurations enable a low torque ripple.

However, fractional slot windings have also some drawbacks. Some fractionalslot windings offer relatively low fundamental winding factors, create additionalharmonics and sub-harmonics which result in higher eddy current losses in therotor, and result in mechanical vibrations.

The use of these fractional slot windings was initially limited to applications ofsub fractional power (lower than 50 W) such as motors for electric fans or computerperipherals [2, 3]. Extensive research to larger three phase machines is only done

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156 Combined Wye-Delta Connection

recently. Different papers studied the winding factors of different pole and slotnumbers [4], the use of irregularly distributed teeth [4, 5], winding factors for onelayer and double layer windings [6], influence of different pole and slot numbers ontorque ripple [7] and cogging [8], multi-phase concentrated windings [9], specificpurposes such as low speed applications and direct drive wind energy conversion[10].

This chapter focusses on the fundamental winding factor. A machine with a lowfundamental winding factor needs to compensate its low electromagnetic torqueoutput with a high current or with more winding turns, which both are inverselyproportional to the fundamental winding factor. Thus a low fundamental windingfactor deteriorates the electromagnetic torque output of the machine.

To illustrate the issue concerning the fundamental windingfactor, the 15-slot-16-pole fractional slot two layer winding is considered. InFig. 7.1, the voltagephasor graph for the fundamental component and winding arrangement are pre-sented. The phasors are numbered from 1 toQs so that the phasor 2 is placed to360p/Qs electric degrees, now 192 electric degrees, from the phasor1 and so on.The coils are ordered into positive and negative values -C, A, -B, C, -A and B. Thisresults in an unequal number of positive and negative coil sides for each layer ofthe two layer winding. By shifting the winding arrangement of the first layer overone slot, the winding arrangement of the second layer is obtained.

The fundamental winding factor can be expressed as [5]

ξ1 = ξp,1ξd,1 (7.1)

whereξp is the pitch factor andξd is the distribution factor. The pitch factor resultsfrom the phase shift between the first and the second layer of the two layer winding.Therefore, as illustrated in Fig. 7.2, the sum of the phasorsfor the fundamentalcomponents of both coil sides will result in a phasor of whichthe amplitude islower than the sum of the amplitudes of the phasors of both coil sides. The ratio inwhich the resulting amplitude differs from the sum of the amplitudes is expressedby the pitch factor.

Analytically, the pitch factor is defined for a concentratedtwo layer windingas [5]

ξp,ν = sin

(

νπτs2τp

)

= sin

(

νπ

Qs

)

(7.2)

For the 15-slot-16-pole fractional slot two layer winding,the fundamental (ν=8)pitch factorξp,1 is 0.9945.

The resulting phasor from the voltage phasor graph in Fig. 7.3 represents thefundamental componentEc,1 of the coil electromotive forceEc i.e. the fundamen-tal component of the voltage induced in the coil at no load fortwo consecutivecoils. Notice that harmonic components, different from thefundamental, requireother voltage phasor graphs.

As indicated by the voltage phasor graph in Fig. 7.1, also phase shifts betweenthe coil sides phasors, and thus the coil electromotive forces, exist. In Fig. 7.3, the

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-A

-A-A

+A

+A

+C

+C

-C

-C

-C

+B

+B

-B

-B

-B

1

11

2

3

4

5

6

7

8 9

10

12

13

14

15

-A

-A

-A

-A -A

+A +A+A

+A

+A

+C

+C+C

+C

+C

-C

-C

-C-C

-C+B

+B

+B

+B

+B

-B

-B

-B

-B

-B

1

11

2

3

4

5

6

789

10

12

13

14

15

Figure 7.1: Voltage phasor graph and winding arrangement for the 15-slot-16-pole fractional slot two layer winding.

coil electromotive forces and their fundamental components of two consecutivecoils are presented, in which the phase shift of the fundamental component can beobserved. As each phase is composed out of 5 coils in series, the voltage phasorgraph of the fundamental component of one phase will look like Fig. 7.4. Similar

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+A

-A 12

12

Ec,1

E−A,1

E+A,1

Figure 7.2: Illustration of the pitch factorξp,1: the sum of the phasors forthe fundamental components of both coil sides will result inaphasor of which the amplitude is lower than the sum of the am-plitudes of the phasors of both coil sides.

Ec,1

[V], coil 8

Ec [V], coil 8

Ec,1

[V], coil 7

Ec [V], coil 7

Ec

[V]/E

c,1

[V]

θ [electric degrees]0 45 90 135 180 225 270 315 360

-80

-60

-40

-20

0

20

40

60

80

Figure 7.3: Coil electromotive forcesEc and their fundamental componentsEc,1 of two neighboring coils. A phase shift of 12 exists be-tween the fundamental components of coil electromotive forces.Apart form the fundamental component, the coil electromotiveforces contain some additional components with significantam-plitudes as well.

as for the pitch factor, the amplitude of the fundamental component of the phaseelectromotive forceE 1 will differ from the sum of the amplitudes of the funda-mental components of the coil electromotive forceEc,1. The distribution factorξdis defined by the ratio between these two.

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E1

Ec,1

Ec,1

Ec,1

Ec,1

Ec,1

12

12

Figure 7.4: Voltage phasor graph for the fundamental component of onephase for the 15-slot-16-pole fractional slot two layer winding.The fundamental components of coil electromotive forces areindicated byEc,1, the resulting phase electromotive force byE1.

Analytically, the distribution factor is defined for a concentrated two layerwinding as [11]

ξd,ν =sin(

νpπ2m

)

nq sin(

νp

π2mnq

) (7.3)

In this equation,q, representing the number of slots per pole and per phase, is givenby

q =Qs

2pm=z

n(7.4)

wherem is the number of phases,z is the numerator ofq andn is the denominatorof q reduced to the lowest terms.

For the 15-slot-16-pole fractional slot two layer winding,the fundamental(ν=8) distribution factorξd,1 is 0.9567. Resulting in a fundament winding factorξ1 equal to 0.9514.

As mentioned before, high winding factors are important forthe performanceof the machine. An increase in the winding factor would allowthe machine to pro-duce a higher torque for the same current, and thus increasesthe energy efficiency.Therefore, in the next section, a technique is introduced toobtain a higher funda-mental winding factor. Although this technique will be illustrated extensively for a15-slot-16-pole fractional slot winding, its validity is extended afterwards to mostslot and pole combinations.

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7.2 Combined Wye-Delta Connection

As described in the previous section, the fundamental winding factor ξν is theproduct of the pitch factorξp and the distribution factorξd. The pitch factor isdirectly linked to the double layer winding,i.e. a coil that is wound around a statorcore element. As a consequence, the pitch factor cannot be modified. In contrast,the way in which the different coils are connected can still be modified. In previousparagraph, a common non grounded wye connected was used, which results in aseries connection of 5 consecutive coils, leading to a distribution factor of only0.9567 because of the phase shift of 12 electric degrees between each phasor.

A connection technique of the coils which compensates for the phase shifts be-tween the different fundamental components of the coil electromotive forcesEc,1should solve this issue. Therefore, a multi-phase system was introduced in [12].In this multi-phase system, the nine coil electromotive forces are connected to aconvertor consisting of 9 legs. Although this technique allows to obtain a fun-damental distribution factor equal to one, a custom made complex convertor isrequired. Three-phase machines that already use phasor shifting techniques arepower transformers; by changing the internal connection, different phase shifts be-tween the primary and secondary grid are realised. Comparable techniques wereused in [13,14] to build multiple-step wye-delta winding connections to get an ad-justable flux. The same technique was used in [15] to improve the fundamentalwinding factor and reduce the harmonic content in inductionmotors.

In this section, the idea of a combined wye-delta winding connection is in-troduced for a fractional pitch winding. The principle of the combined wye-deltaconnected is illustrated in Fig. 7.5. Instead of making a single wye or single deltaconnection, a combination of both is made in the same machine. The combineduse of wye and delta connections in the same machine allows tosubdivide the fun-damental coil electromotive forcesEc,1 into two groups: a first group of coils ofwhich the phase of the fundamental component fits best to the wye connection anda second group of coils of which the phase of the fundamental component fits bestto the delta connection. To visualise the internal relations between the differentfundamental coil electromotive phasors in the combined wye-delta connection, athree-phase equivalent of Fig. 7.5 is given in Fig. 7.6.

In the example of a 15-slot-16-pole fractional slot two layer winding, threecoils are assigned to the wye connection and two to the delta connection. Bythis combined connection technique, the fundamental distribution factor ξd,1 isincreased to 0.9891. This is an increase of 3.4% compared to the fundamental dis-tribution factor of the single wye connection in previous paragraph. As the torquechanges proportional with the winding factor, the combinedwye-delta connectionproduces 3.4% more torque than the single wye connection forthe same phasecurrent.

The idea in previous paragraph is only valid if the coils in the delta connectionhave

√3 times the number of windings of the wye connected coils. As the slot

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7.2 Combined Wye-Delta Connection 161

EY,1

E∆,1

Ec,1

Ec,1

Ec,1

Ec,1

Ec,1

12

12

6

30

Figure 7.5: Voltage phasor graph for the fundamental component of onephase for the 15-slot-16-pole fractional slot two layer windingusing the combined wye-delta connection. The coil electromo-tive forces are indicated byEc,1, the resulting phase electromo-tive forces of the wye and delta connected coils byEY,1 andE∆,1 respectively.

width remains the same for both wye and delta connected coils, the wire sectionof the delta connected coils should be reduced by a factor

√3. Although the wire

section of the delta connected coils is reduced by a factor√3 while the length is

increased by a factor√3, the copper losses remain unchanged as the current in

the delta connected is reduced by a factor√3 compared to the current in the wye

connected coils.Some criticism may arise that the implementation of the combined wye-delta

connection introduces some additional difficulties. Threeconstraints,i.e. properoutput voltage, good filling factor of the copper winding andthe

√3 relation be-

tween number of turns and wire diameter, should be fulfilled at the same time.These issues are refuted partially by the many wire diameters that are commer-cially available. Moreover the easy winding process out of the machines makes itpossible to wind with multiple wires at the same time enabling parallel branches.Last but not least, non filled space in the stator slots is usedfor stator cooling.Therefore, a slight unbalance between the winding section of the wye and deltacoils may result in a better cooling of the surrounding coilswhich partially com-pensates the diameter unbalance.

Generally, a combined wye-delta connection of the different coils will result ina higher winding factor. In Table 7.1, the fundamental winding factors are givenfor some suitable fractional slot two layer winding. It should be mentioned that in

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Ea,Y,1

Ea,∆,1

Eb,Y,1

Eb,∆,1

Ec,Y,1

Ec,∆,1

Ec,1

Ec,1

Ec,1

Ec,1

Ec,1

30

30

30

Figure 7.6: Three-phase variant of Fig. 7.5 displays the internal relationsbetween the different fundamental coil electromotive phasors inthe combined wye-delta connection.

casesq = 2/5 andq = 2/7, the distribution factor becomes equal to one, and thefundamental winding factor increases with more than 3.5%. As theseq = 2/5 andq = 2/7 are found a lot in current presented prototypes and in many commercialavailable machines with fractional slot two layer windings, the usability of thiscombined wye-delta is significant.

In contrast to [13–15], the coil electromotive force of the fractional pitch wind-ing has a high harmonic content. Fig. 7.7 shows the Fourier spectrum of the elec-tromotive force induced in a coil. The presence of the 3th and 9th harmonic com-ponent will require some extra attention, as the triple harmonics present in the coilelectromotive forces will result in a circulating current and corresponding copperlosses in the delta connected coils. Moreover the flux pattern in the delta connectedstator core elements will change due to the absence of tripleharmonic fluxes, andthus will influence the core losses. Also the permanent magnets will be exposedto a changed harmonic content of the air gap magnetic flux density, and thus themagnet eddy current losses will change as well. Furthermore, the combined wy-delta connection will have an influence on the ripple in the electromagnetic torqueoutput. Therefore, during the evaluation of the combined wye-delta connection itwill be necessary to take all these phenomena into account.

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7.2C

ombined

Wye-D

eltaC

onnection163

Table 7.1: Fundamental winding factors (ξ1 ≥ 0.8660) for fractional slot two layer windings.

Qs Number of poles4 6 8 10 12 14 16 18 20 22 24

6 ξ1 0.8660 0.8660ξ∗1

- -q 1/2 1/2

9 ξ1 0.8660 0.9542 0.9452 0.8660ξ∗1

- 0.9748 0.9748 -q 1/2 3/8 3/10 1/2

12 ξ1 0.8660 0.9330 0.9330 0.8660ξ∗1

- 0.9659 0.9659 -q 1/2 2/5 2/7 1/2

15 ξ1 0.8660 0.9514 0.9514 0.8660ξ∗1

- 0.9836 0.9836 -q 1/2 5/14 5/16 1/2

18 ξ1 0.8660 0.9019 0.9452 0.9452 0.9019 0.8660ξ∗1

- 0.9302 0.9748 0.9748 0.9302 -q 1/2 3/7 3/8 3/10 3/11 1/2

21 ξ1 0.8660ξ∗1

-q 1/2

24 ξ1 0.8660 0.9330ξ∗1

- 0.9659q 1/2 2/5

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PSfrag

Volta

ge[V

]

ν

0 5 10 15 20 25 30 35 40 45 500

10

20

30

40

50

60

70

Figure 7.7: The spectrum of the electromotive force induced in a coil.

7.3 Simulations and Comparison with Wye-Connection

The combined wye-delta connection proposed in previous chapter, predicts a sig-nificant increase in the fundamental winding factor. Nevertheless, the concept ofthe combined wye-delta connection was introduced theoretically based on the pha-sor diagram for the fundamental component. This and next section investigatesome practical issues using finite element analysis. In thissection, the effects ofharmonic components, different from the fundamental, on electromagnetic torqueoutput and ripple, are investigated. Two machines which only differ in the con-nection of the stator coil windings are compared; as a reference the common wyeconnection is used. In the combined wye-delta connection the number of turns percoil in the two delta connected coils is multiplied by a factor

√3 in order to main-

tain the same output voltage, and the section of the wires is reduced by a factor√3,

as this leads to equal copper losses.In these simulations the effect of the circulating current in the delta connected

coils is taken into account. In the common wye connection andthe combinedwye-delta connection, no zero sequence stator currents arepossible in the wyeconnected coils. Therefore, the stator currents are directly imposed

ia(t) = ℜ

I√2ejωt

ib(t) = ℜ

I√2ej(ωt−

2π3 )

ic(t) = ℜ

I√2ej(ωt+

2π3 )

.

(7.5)

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7.3 Simulations and Comparison with Wye-Connection 165

Different from the wye connected coils, there is a zero sequence circulating currentic in the delta connected coils. The stator currents are given by

ia(t) = ic(t) + ℜ

I√2ejωt

ib(t) = ic(t) + ℜ

I√2ej(ωt−

2π3 )

ic(t) = ic(t) + ℜ

I√2ej(ωt+

2π3 )

(7.6)

where the circulating currentic has to be determined iteratively for each rotor po-sition and for the given load conditions and speed, in order that the sum of thevoltages of the delta connected coils assigned to the three phases vanishes.

In Fig. 7.8, the induced electromotive forces in both the wyeand delta con-nected coils are illustrated at rated speed. In the wye connected coils, the highamplitude of the third harmonic component is visible, whilein the delta connec-tion the triple harmonic components have vanished. Here only the presence ofsome harmonic components (mainly 5th, 7th, 9th, 11th and 13th) with relativelysmall amplitude are present.

E∆,1

E∆

EY,1

EY

EY

[V]/E

∆[V

]


0 45 90 135 180 225 270 315 360-250

-200

-150

-100

-50

0

50

100

150

200

250

Figure 7.8: Induced electromotive forces in the wyeEY and deltaE∆ con-nected coils. Their fundamental components are indicated byEY,1 andE∆,1 respectively.

To calculate the electromagnetic torque, the same load current is imposed toboth machines. In this way, apart form the copper losses due to the circulatingcurrent in the delta connected windings, the copper losses of both machines areequal which allows proper comparison. The electromagnetictorque outputs ofboth winding arrangements is depicted in Fig. 7.9; the common wye connection

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produces an average electromagnetic torque of 18.5 Nm, while the combined wye-delta connection 19.4 Nm. This increase of 3.47%, is in very good comparisonwith the 3.4% expected by theoretical analysis based on the phasor diagram for thefundamental component.

< Te > [Nm], Y∆

Te [Nm], Y∆

< Te > [Nm], Y

Te [Nm], Y

Te

[Nm

]/<Te>

[Nm

]


0 45 90 135 180 225 270 315 36018.3

18.5

18.7

18.9

19.1

19.3

19.5

Figure 7.9: Comparison between the electromagnetic torque outputs of thecommon wye and combined wye-delta connection.

Despite the higher average electromagnetic torque output for the combinedwye-delta connection, a higher torque ripple is found. Thistorque ripple is mainlycaused by the torque produced by the circulating current in the delta connectedcoils. The periodicity of this torque ripple is six times thefundamental period, andthus two times this of the circulating current.

Despite the higher torque ripple, [16] introduced an expression to calculatethe optimal current waveform resulting in a constant torquewhile minimising thecopper losses.

7.4 Influence on the Machine’s Losses

In previous chapter it was illustrated that the electromagnetic torque output, pro-duced by a sinusoidal current with fundamental frequency, is increased by using acombined wye-delta connection. At the same time, it was observed that the torqueripple increased, due to the presence of the circulating current in the delta con-nected coils. Primary, there are the copper losses in the delta connected coils dueto the circulation current, that cause additional losses. But secondary the circulat-ing currents will influence the magnetic flux density patternin the stator core and

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7.4 Influence on the Machine’s Losses 167

the air gap magnetic flux density. Therefore, the combined wye-delta connectionwill have an influence on the core losses and the eddy current losses induced inthe permanent magnets and rotor. The next three sections will focus on each of theloss phenomena,i.e. copper losses, stator core losses and eddy current losses inthepermanent magnets. Finally, the energy efficiency of a machine with the combinedwye-delta connection is compared with these of a common wye connection.

7.4.1 Circulating Current in the Delta-Connected Coils

The circulating current in the delta connected windings hasto be determined it-eratively for each rotor position and for the given load conditions and speed, inorder that the sum of the voltages of the delta connected coils assigned to the threephases vanishes. In Fig. 7.10, the circulating current is given for no-load at ratedspeed working condition.

I c[A

]


0 45 90 135 180 225 270 315 360-1.5

-1

-0.5

0

0.5

1

1.5

Figure 7.10: Waveform of the circulating current in the delta connectedcoils of the combined wye-delta connection. The circulatingcurrent was evaluated at rated speed and no load working con-dition.

The dominant 3th harmonic component in the coil electromotive forcecfr. Fig.7.7, is mainly responsible for the circulating current in the delta connected wind-ings. The circulating current strongly depends on the rotational speed of the ma-chine. In Fig. 7.11, the amplitude of the circulating current is plotted as a functionof the frequency. A first order behaviour is clearly observed: at low frequenciesthe resistance of the delta current path determines the amplitude, while at highfrequencies the current is determined by the inductance.

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This observation allows a simplification of the iteration process necessary toobtain the circulating current in equation (7.6). At rated speed, the first zero se-quence harmonic component has already three times the frequency of the funda-mental component and allows the use of the high frequency approximation. There,the constraint that the sum of the voltages of the delta connected coils assigned tothe three phases vanishes, reduces to the constraint that the sum of the flux linkagesof the delta connected coils assigned to the three phases vanishes. This is a less de-manding constraint, as for the flux linkages no time steppingfinite element analysisis required, but static simulations for each rotor positionare required. Individualstatic simulations for the different rotor positions are necessary since the value ofthe inductance depends on the saturation level of the statorcore. Practically thiscan be implemented by using a nonlinear search function thatminimises the zerosequence flux linkage in the delta connected coils by changing the value for thecirculating current. This procedure results in an overestimation of the circulatingcurrent in the lower speed ranges, and thus, overestimates the copper losses in thedelta connected windings due to the circulation current.

jωL

R

R + jωL

I c[A

]

Frequency (Hz)

0 100 200 300 400 500 600 700 800 90010000

0.25

0.5

0.75

1

1.25

1.5

Figure 7.11:Amplitude of the circulating current in the delta connectedcoils of the combined wye-delta connection. The first orderbehaviour shows resistive behaviour at low speeds, while athigh speeds the inductance of the delta connected coils domi-nates.

As the electric resistance of a delta connected coil is 0.2Ω, the worst case valueof the copper losses due to the circulating current is less than 2 W (1.8750 W) atno-load. This value is negligible to the power that is gainedby using the combinedwye-delta connection.

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7.4 Influence on the Machine’s Losses 169

As there is a circulating current in the delta connected coils, a correspondingelectromagnetic torque is generated. At load, this resultsin an additional torqueripplecfr. Fig. 7.9, but also at no load this torque ripple exists. Fig. 7.12 illustratesthe torque ripples at no load and rated speed for a common wye and combinedwye-delta connection. In case of the common wye connection there is no torqueripple, only some cogging torque is possible, while the torque ripple in the wye-delta connection is quite high. It should be mentioned that the time average valueof this electromagnetic torque is zero. Although the amplitude of this torque ripple

Te [Nm], Y∆

Te [Nm], Y

Te

[Nm

]


0 45 90 135 180 225 270 315 360-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Figure 7.12: Torque ripple at no load and rated speed for the common wyeand the combined wye-delta connection. Torque ripple is ab-sent in the wye connected machine, while the circulating cur-rent in the delta connected windings of the combined wye-delta connection results in torque ripple.

is limited at low speeds, it is, like cogging torque, unwanted in certain applicationssuch as small wind turbines where it limits the startup at lowwind speeds.

7.4.2 Stator Core Losses

The existence of a circulating current in the delta connected coils in the combinedwye-delta connection will influence the pattern of the magnetic flux density in thestator core elements. Also in the core elements with the wye connected coils thepattern of the magnetic flux density will change slightly dueto the small phase shiftin which the current is injected. In the common wye connectedmachine the currentis in phase with the electromotive force of the middle one of the five successive

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coils, while in the combined wye-delta connection the current is in phase with theelectromotive force of the middle one of the three successive wye connected coils.

In the common wye connected machine, the core losses at load in five succes-sive coils assigned to one phase are respectively, 7.9327, 8.2365, 9.0936, 9.8987and 9.8670 W. Summation of the three phases results in 135 W core losses. Inthe combined wye-delta connection, the core losses in the three successive wyeconnected coils are respectively 8.5003, 9.1007 and 9.4559W and 8.5218 and8.7889 W in the delta connected coils. This results in a totalstator core lossof 133 W. Therefore, the combined wye-delta connection has slightly lower corelosses compared to the common wye-connection.

7.4.3 Magnet Eddy Current Losses

Like for the core losses, the combined wye-delta connectioninfluences the mag-netic flux density pattern in the stator cores, so the combined wye-delta connectionwill have an effect on the air gap magnetic field. The air gap magnetic flux densityfor a machine with a combined wye-delta connection is plotted in Fig. 7.13, the airgap magnetic flux density for a machine with a wye connection was plotted earlierin Fig. 5.1. For the no load working condition of the axial fluxPM machine, theair gap magnetic flux density is the same when the permanent magnet passes thefirst three, wye connected coils. However, when the permanent magnet passes thedelta connected coils, a change in the air gap magnetic flux density is found. Whenpassing these delta connected coils, the permanent magnet is subjected to an airgap magnetic flux density that is influenced by the circulating current.

Also at load, comparison of Fig. 7.13 with Fig. 5.1, indicates a change inthe air gap magnetic flux density. Here, not only the influenceof the circulatingcurrent on the air gap magnetic flux density can be detected inthe region of thedelta connected coils, but also a change in the air gap magnetic flux density in theregion of the wye connected coils. This change can be addressed to the phase shiftof the current with respect to the coil electromotive forces, as discussed in previoussection.

As the air gap magnetic flux density is changing due to the combined wye-delta connection, an impact on the eddy current losses in thepermanent magnetsis expected. In Table 7.2, the eddy current losses in the permanent magnets at noload and load working conditions are listed for the common wye and combinedwye delta connection.

Due to the increased harmonic content of the air gap magneticflux density,higher magnet eddy current losses are found as well. Howeverthe eddy currentlosses in the combined wye-delta connection increase with about 6% compared tothe common wye connection, the absolute values of the lossesremain less than1 W.

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7.4Influence

onthe

Machine’s

Losses

171

Bg,l

oad

[T]


Bg,n

olo

ad[T

]


0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360

0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360

0.6

0.8

1

1.2

1.4

0.6

0.8

1

1.2

1.4

Figure 7.13: Air gap magnetic flux density taking into account stator slotting effect only (upper), and both the stator slottingeffect and armature reaction (lower) as a function of the rotor position. Data are expressed in the rotor referenceframe,i.e., position fixed with respect to the rotor; in this case the center of the permanent magnet.

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Table 7.2: Effect of segmentation on the eddy-current losses.

# segments magnet eddy current lossno-load load

Y Y∆ Y Y∆

1 8.1435 8.5693 15.4149 16.32832 6.6104 6.8082 11.7561 12.20794 5.8773 5.9172 10.1450 10.195814 2.8881 2.8586 4.0904 4.1221

7.4.4 Higher Energy Efficiency

Overlooking the pro’s and con’s with respect to the combinedwye-delta connec-tion, it is clear that, with the focus on energy efficiency, the combined wye-deltaconnection is superior to the common wye connected variant.For the suggestedmachine, the increase in output power with more than 235 W outnumbers the smallincrease in copper losses and magnet eddy current losses. The effect of the circu-lating currents on the overall losses remains limited.

7.5 Conclusion

The use of fractional slot windings in yokeless and segmented armature (YASA)axial flux machines has some important benefits: short end windings result in lowcopper losses and a winding process out of the machine enables easy windingplacement. Nevertheless, fractional slot windings have generally relatively lowfundamental winding factors. To obtain higher fundamentalwinding factors, acombined wye-delta connection is introduced. This combined wye-delta connec-tion combines wye and delta connected coils in the same machine. This regroupingof individual coils, as illustrated in Fig. 7.6, results in ahigher fundamental wind-ing factor.

Due to the increase in fundamental winding factor, a significant increase in thetorque output is found for most fractional slot windings. Despite the increase inelectromagnetic torque, there are some issues as well: in these fractional slot ma-chines harmonics other than the fundamental one will resultin a circulating currentin the delta connected coils. This circulating current results in supplementary cop-per losses in the delta connected coils, has an influence on the core losses, slightlyincreases the eddy current losses in the permanent magnets and increases the ma-chine’s torque ripple.

With the focus on energy efficiency the increase in power output, for the sameload current, outnumbers the slight increase in additionallosses. As a result thecombined wye-delta connection will be more energy efficientthan the same ma-chine with a common wye connection.

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Bibliography

[1] F. Libert, “Design, optimization and comparison of permanent magnet mo-tors for a low-speed direct-driven mixer,”Technical Licenciate report, RoyalInstitute of Technology, Stockholm, 2004.

[2] T. Kenjo and S. Nagamori, “Permanent-magnet and brushless dc motors,”1985.

[3] B. Huang and A. Hartman, “High speed ten pole/twelve slotdc brushlessmotor with minimized net radial force and low cogging torque,” Oct. 7 1997,uS Patent 5.675.196.

[4] J. Cros and P. Viarouge, “Synthesis of high performance pm motors withconcentrated windings,”Energy Conversion, IEEE Transactions on, vol. 17,no. 2, pp. 248–253, 2002.

[5] T. Koch and A. Binder, “Permanent magnet machines with fractional slotwinding for electric traction,” inProc. of 15th Int. Conf. on Electrical Ma-chines (ICEM), vol. 25, no. 28.08, 2002.

[6] F. Magnussen and C. Sadarangani, “Winding factors and joule losses of per-manent magnet machines with concentrated windings,” inElectric Machinesand Drives Conference, 2003. IEMDC’03. IEEE International, vol. 1. IEEE,2003, pp. 333–339.

[7] F. Magnussen, P. Thelin, and C. Sadarangani, “Performance evaluation ofpermanent magnet synchronous machines with concentrated and distributedwindings including the effect of field-weakening,” inPower Electronics, Ma-chines and Drives, 2004.(PEMD 2004). Second InternationalConference on(Conf. Publ. No. 498), vol. 2. IET, 2004, pp. 679–685.

[8] H. JR Jr and T. Miller, “Design of brushless permanent-magnet motors,”1996.

[9] J. Cros and P. Viarouge, “New structures of polyphase claw-pole machines,”Industry Applications, IEEE Transactions on, vol. 40, no. 1, pp. 113–120,2004.

[10] P. Salminenet al., “Fractional slot permanent magnet synchronous motorsfor low speed applications,”PhD dissertation, Acta Universitatis Lappeen-rantaensis, 2004.

[11] K. Vogt, Berechnung elektrischer maschinen. VCH, 1996.

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[12] S. Brisset, D. Vizireanu, and P. Brochet, “Design and optimization of anine-phase axial-flux pm synchronous generator with concentrated windingfor direct-drive wind turbine,”Industry Applications, IEEE Transactions on,vol. 44, no. 3, pp. 707–715, 2008.

[13] M. Cistelecan, F. Ferreira, and M. Popescu, “Adjustable flux three-phase acmachines with combined multiple-step star–delta winding connections,”En-ergy Conversion, IEEE Transactions on, vol. 25, no. 2, pp. 348–355, 2010.

[14] R. Gjota, “Winding arrangement of a stator and/or rotorof a three-phase gen-erator or electromotor with improved performances,” Dec. 11987, uS Patent4.710.661.

[15] J. Chen and C. Chen, “Investigation of a new ac electrical machine winding,”in Electric Power Applications, IEE Proceedings-, vol. 145, no. 2, 1998, pp.125–132.

[16] A. P. Wu and P. L. Chapman, “Simple expressions for optimal current wave-forms for permanent-magnet synchronous machine drives,”Energy Conver-sion, IEEE Transactions on, vol. 20, no. 1, pp. 151–157, 2005.

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Chapter 8

Losses in VSI-PWM fed machines

8.1 Introduction

Unlike line start permanent magnet synchronous machines, most permanent mag-net machines are not directly connected to the grid. Therefore, permanent mag-net machines with limited power range are very often fed by a current controlledvoltage source inverter (VSI). Generally, the variable voltage source is obtained bypulse width modulation (PWM). By combining the permanent magnet synchronousmachine with a VSI-PWM inverter, a highly dynamic controlled electric drive atvariable speed is obtained.

Making use of PWM, a fundamental period of the output voltageis composedof a number of positive blocks with appropriate width duringthe first half periodand a number of negative blocks with appropriate width during the second half pe-riod. The width and position of the positive and negative pulses are appropriatelychosen to achieve the desired amplitude of the output voltage while moving thehigher harmonic components sufficiently high into the frequency domain. Never-theless, some harmonic components present in the output voltage of the inverterstill affect the performance of the machine. Although the high harmonic compo-nents in the phase voltages of the machine will by filtered by the inductances in themachines, a not negligible ripple in the phase current will still be present.

Consequently, these current ripples will have their influence on the losses inthe machine. Therefore, this chapter focusses on the lossesintroduced in the ma-chine by VSI-PWM feeding. The current ripple will have a direct influence on thecore losses [1–3] and the induced eddy currents in the permanent magnets [4]. Tomodel these losses, the mathematical models introduced in previous chapters areimplemented. However, as the ripple in the current waveformcauses internal loopsin the magnetic field, a Preisach model for the correct evaluation of the hysteresis(losses) is introduced.

Crucial in this evaluation of the losses in VSI-PWM fed machines, is to obtainthe correct current waveform. As all previously introducedmathematical models

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176 Losses in VSI-PWM fed machines

had the current as input, a voltage driven model for the axialflux PM machine isintroduced. This model predicts, given the voltage at the terminals of the machine,the corresponding phase currents. In this modelling, no direct finite element eval-uations are performed to speed up the simulations. Nevertheless, the exact currentand position dependent behaviour of the machine’s flux linkages as well as elec-tromagnetic torque are modelled by lookup tables. This makes the machine modelperfectly suitable for simulations that include time stepping.

As this work is limited to the losses in the machine, the aim ofthis chapteris not to implement a VSI-PWM control strategy nor to model the losses of theconverter. Therefore, the voltage waveform used in the timestepping is generatedby applying pulse-width modulation directly to the voltagewaveform which resultsin the desired current waveform.

8.1.1 Field-Oriented Control

As mentioned in the introduction of this chapter, generallyPM synchronous ma-chines are not able to start up when they are directly connected to the grid. Onthe other hand, nowadays less and less electrical machines are directly connectedto the grid as it allows no (synchronous) or very limited (asynchronous) speedcontrol. The introduction of the inverter had a major impacton this: feeding ofboth asynchronous and synchronous machines with a variablefrequency results ina high performance electric drive, which can be efficiently operated over a widespeed range. It should be added that with invertor fed machines, the implementa-tion of a control strategy is essential.

Here, field-oriented control of the axial flux PM machine is considered. Withthe field-oriented control strategy, analog behaviour as for a torque controlled (cur-rent controlled) dc machine is obtained. In order to obtain this behaviour with PMmachines, the current should be controlled independently and the angle betweenthe fundamental component (ν = 1) of the magnetic flux density by the perma-nent magnets and the current sheet by the armature reaction (currents) should beconstant over time. Moreover, if this angle equals zero, themaximum values ofboth the magnetic flux density by the permanent magnets and the current sheet arereached simultaneously, which results in the maximal producible electromagnetictorque.

This control strategy requires full control of the armaturecurrent, which is doneby feeding the machine with a current source inverter (CSI) or with a voltage sourceinverter (VSI-PWM) with a current controller having a sufficiently high bandwidth.In this chapter a voltage source inverter (VSI-PWM) with a current controller aspresented is considered. The control scheme for such a field-oriented control of apermanent magnet synchronous machine is illustrated in Fig. 8.1. In this controlscheme two control loops are present; the control loop for the amplitude of thecurrent and the loop that controls the phase of the current. Different from classicalsynchronous machines with wound rotor is the lack in controlling the rotor flux.

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iaibic

θ

θ∗

i∗a i∗

bi∗c

1

2

3

4

5

6

T ∗

s1s2s3s4s5s6

PMSM

CurrentCurrentControllerController

Vector ControlAlgorithm

PWM-VSI

Figure 8.1: Control scheme of a field-oriented control of a permanent mag-net synchronous machine using a VSI-PWM.

Knowledge of the actual position of the rotor is necessary toimpose the currentswith the desired phase. Knowledge of the actual rotor position is obtained by anencoder. Alternatively, because of the vulnerability of these components, more andmore sensorless control algorithms are found.

In precedent parts, a pure sinusoidal current in phase with the fundamentalcomponent of the back electromotive force was always assumed. Choosing thepure sinusoidal currentI 1 in phase with the fundamental component of the backelectromotive forceE1 results in the maximal corresponding power output. In thiscase, the power factor is not equal to one and the convertor will have to deliverreactive power in order to compensate the inductive behaviour of the PM machine.On the machine side, the fundamental components of the current and the electro-

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motive force in phase results in the best achievable energy efficiency as the outputtorque is maximised for the same copper losses. Taking into account both the lossesin the machine and convertor, the optimum of energy efficiency might be obtainedfor a different power factor. Notwithstanding, the fundamental components of thecurrent and the electromotive force will be maintained in phase in this chapter.

8.1.2 Pulse Width Modulation

To control both the frequency and the amplitude of the outputvoltage, the VSIstudied in this work, is using pulse width modulation. PWM allows to controlthe voltage without significant increase of the harmonic components with lowerorders. These harmonic components with lower orders shouldbe avoided as theyare detrimental for the connected load.

The modulation technique is illustrated in Fig. 8.2. The simplest way to gen-erate a PWM signal is the intersective method. This method only requires a carrier(modulation waveform) and a comparator. When the value of the reference wave-form is higher than the modulation waveform, the PWM waveform is in the highstate, otherwise it is in the low state.

The aim of this chapter is to investigate the losses due to PWM. Therefore, thevoltage waveform that results in a pure sinusoidal current in phase with the funda-mental component of the back electromotive is chosen as the reference waveform.This voltage waveform can be obtained directly through finite element analysis;the desired current waveform is imposed and the resulting voltage waveform iscalculateda posteriori.

Subsequently an arbitrary carrier waveform, in this case a triangular waveform,with a frequencyfc is introduced. The amplitude of the carrier waveform is chosenslightly higher than the peak value of the reference waveform. The frequency ofthe carrier has to be chosen sufficiently high in order not to affect the load. As willbe illustrated in a future section of the work, the switchingfrequency has a majorimpact on the ripple in the phase current and consequently onthe losses in themachine. In Fig. 8.2 the frequency of the carrier waveform was set to 10 kHz,cfr.the fundamental component of the voltage reference waveform has a frequency of333 Hz. Whereas the reference voltage waveform hardly includes higher harmoniccontent, the Fourier spectrum of the PWM waveform presentedin Fig. 8.3 clearlyindicates the presence of higher harmonic components.

As could be expected intuitively, the harmonic components near the frequencyof the carrier waveform and its multiples strongly dominatethe high harmoniccontent. The Fourier spectrum includes the possible harmonic numbers

ν = kfcf

± l (8.1)

corresponding with the frequencies

fs = kfc ± lf (8.2)

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8.1Introduction

179

Vc

Vref

VPWM

Vref

Volta

ge[V

]


Volta

ge[V

]


0 30 60 90 120 150 180 210 240 270 300 330 360

0 30 60 90 120 150 180 210 240 270 300 330 360

-600

-400

-200

0

200

400

600

-600

-400

-200

0

200

400

600

Figure 8.2: Voltage reference waveform and carrier (modulation waveform) (upper),Voltage reference waveform and gener-ated PWM signal by the intersective method by using a comparator (lower). One fundamental (electrical) periodof the voltage reference waveform is illustrated.

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Vν

[V]

ν

0 15 30 45 60 75 90 105 120 135 150101

102

103

Figure 8.3: Amplitudes of air gap magnetic flux density considering the sta-tor slotting effect only as a function of the harmonic orderν atthe center of the permanent magnet forfc

f= 30.

with k ≥ 0, l ≥ 0 andk + l odd.Indeed, in the Fourier spectrum of the PWM waveform presented in Fig. 8.3,

next to the fundamental component of the reference voltage waveformν = 1 , theharmonic components near the frequency of the carrier waveform ν = fc/f = 30and its multiples have high amplitudes:30±2 (fc±2f), 60±1 (2fc±1f), 60±3(2fc ± 3f), . . . .

Although the PM machines is fed with PWM voltage waveforms that include ahigh harmonic content, the low pass filter action of the inductances of the machinelimits the ripple in the current waveform. The simulation ofthe current waveformin the VSI-PWM fed machine is discussed in the next paragraphs.

The losses in the VSI-PWM are not investigated.

8.2 Simulation of the Current Waveform

In this work, the static multislice-2D modeling is at the basis of all simulations.This model can be considered as a system having a specified setof inputs andoutputs. The inputs are

• the phase currents in the windings;

• the angular position of the rotor;

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8.2 Simulation of the Current Waveform 181

and the outputs are

• the flux linkages with the windings;

• the magnetic flux density pattern in the stator core elements;

• the magnetic vector potential evaluated at two lines in theair gap;

• the axial component of the magnetic flux density at the surface of the perma-nent magnets and rotor.

In a pre-processing the relation between the phase currentsand the angular po-sition of the rotor is specified. In this work, field-orientedcontrol of the machine isassumed: the sinusoidal phase currents are imposed in quadrature to the axis of thepermanent magnets. This results in a phase current in phase with the fundamentalcomponent of the back electromotive force.

In a post-processing

• the flux linkages are used to calculate the phase voltages;

• the magnetic flux density pattern in the stator core elements is used to calcu-late the stator core losses;

• the magnetic vector potential is evaluated at two lines in the air gap to cal-culate the electromagnetic torque through the Maxwell stress harmonic filtermethod;

• the axial component of the magnetic flux density at the surface of the per-manent magnets and rotor is used to calculate the eddy current losses in thepermanent magnets and rotor.

The static multislice-2D modeling is hence only able to simulate the voltageoutput, giving a specified current input. Generating the reference waveform usedin the PWM intersective methodcfr. Fig. 8.2, can be obtained by simulations usingthe static multislice-2D modeling. Subsequently this reference waveform can beused to generate the PWM waveform.

8.2.1 Machine Model

To calculate the current waveform for a given voltage waveform, one possibilityis to couple a finite element time stepping model with an electric network. Asthis finite element time stepping model is very time consuming, in this researchpreference is given to a state space model of the axial flux PM machine.

In this section, a state space model with linear coefficientsis presented andused to simulate the current waveform. Nevertheless, in theappendix the nonlinearvariant is discussed.

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For a PM machine, the basic equation which expresses the relation betweenvoltageV, back electromotive forceE, and phase currentI is given by

V = E + RI +d

dt(LI ) (8.3)

The voltageV, back electromotive forceE, and phase currentI are column vectorsrepresented in theabc-system:

V =

VaVbVc

, E =

Ea

Eb

Ec

, I =

IaIbIc

. (8.4)

The resistance matrixR is given by

R =

Rs

Rs

Rs,

(8.5)

and the inductance matrix by

L =

Ls Ms Ms

Ms Ls Ms

Ms Ms Ls.

(8.6)

As the surface1 permanent magnets have a relative permeability very close to unityand saturation of the magnetic circuit is neglected in the linear model, the induc-tance matrix is not dependent of the angular rotor position.In a balanced threephase system, the diagonal elements represent the inductance coefficients of thetooth coilsLs, the other elements represent the mutual inductance coefficientsMs.The inductance matrix is hence symmetric.

The inductance matrix taking into account the nonlinear behaviour of the statorcore material is presented in the appendix. Although the nonlinear model describesthe PM machine more accurately, the usage of the linear modelis maintained in thissection.

The electromotive force is calculated by taking the time derivative of the per-manent magnet flux linkageψPM i.e. the flux linkage caused by the permanentmagnets only, so without armature reaction

E =dψPM(θ)

dt=

dψPM(θ)

dθ

dθ

dt=

dψPM(θ)

dθω (8.7)

where

ψPM(θ) =

ψPMa (θ)ψPMb (θ)ψPMc (θ)

(8.8)

1In PM machines with embedded magnets the inductance matrix will be a function of the rotorposition.

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8.2 Simulation of the Current Waveform 183

The values ofdψPM(θ)dθ are calculated using the static finite element model and are

stored in a lookup table. The back electromotive force hencetakes into account thepresence of additional harmonic components. This permanent magnet phase fluxlinkageψPM and corresponding phase back electromotive forceE were alreadypresented in Fig. 3.22.

The tooth coils inductance coefficient and mutual inductance coefficient arecalculated by making use of the definition of the chord inductance

Ls =ψi(Ii)− ψPM

i

Ii, i = a, b, c (8.9)

Ms =ψi(Ij)− ψPM

i

Ij, i = a, b, c, j = a, b, c, j 6= i (8.10)

In the calculation of the elements of the inductance matrix in case of nonlinearmaterial behaviour, the tangent induction coefficient definition is used.

With respect to the calculation of the inductance coefficients by making useof the multislice-2D modelling technique, it should be noticed that the inductancecoefficients are underestimated. The multislice-2D modeling technique considersonly paths of the magnetic flux in the computation plane, the flux path in the endwindings at the inner and outer diameter are not taken into account. Accurateprediction of the inductance coefficients hence requires full 3D modelling of theaxial flux PM machine.

The basic equation for the PM machine (8.3), is subsequentlytransformed intoa state space model

dIdt

= −L−1RI + L−1 (V − E) . (8.11)

In this state space model, the output and state is the phase current vectorI , whilethe model has(V − E) as an input.

The state space model with linear coefficients presented in this section has theadvantage of very short evaluation times. The simulation time is mainly determinedby the search and interpolation operation in the lookup table for the flux linkage ofthe permanent magnet flux.

8.2.2 Simulated Current Waveform

Calculation Steps

The calculation of the current waveform in case of a VSI-PWM fed machine in-cludes three steps:

• simulation of the output voltage for a given current input without PWM(field-oriented control) by using the static multislice-2Dmodelling;

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• generation of the PWM-signal. Here the output voltage of the staticmultislice-2D modelling is used as the reference signal in the intersectivemethod;

• using this PWM-signal the state space model calculates thecorrespondingcurrent waveform.

Step two is preferred rather than implementing a full VSI-PWM including thetuning of the parameters of the current controller.

The simulations for the current waveform are performed at rated load andspeed; the waveform for the current is set to a 7 A having only afundamentalfrequency of 333 Hz (field-oriented control).

Phase Current Waveforms for different Carrier Frequencies

In the PMW intersective method, a carrier frequency of 10kHzis applied. Thevalues for the permanent magnet phase flux linkageψPM(θ) stored in the lookuptable are presented in Fig. 3.22. The value for the phase resistanceRs is estimated

Pha

secu

rren

ts[A

]


0 45 90 135 180 225 270 315 360-15

-10

-5

0

5

10

15

Figure 8.4: Phase current waveforms for the VSI-PWM fed machine; car-rier frequency of 10 kHz.

to be 0.2Ω, and the values for the inductance and mutual inductance coefficientscalculated by equation (8.9) and (8.10) are 7.4 mH and 323µH respectively.

Given the PWM-signal as an input, the state space model (8.11) calculatesthe corresponding current waveform. For the 10 kHz frequency of the carrier, thephase current (in phase a) is presented in Fig. 8.4. The ripple in the current due

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8.3 Core Losses 185

Pha

secu

rren

ts[A

]


0 45 90 135 180 225 270 315 360-15

-10

-5

0

5

10

15


to the PWM-signal (voltage) is clearly distinguishable. Asthe current ripple is afunction of the carrier frequency, the current waveforms for a 5 kHz and 20 kHzcarrier frequency are illustrated in Fig. 8.5 and Fig. 8.6 respectively. An increasingcarrier frequency has an indisputable impact on the mitigation of the current ripple.Focusing on the machine only, the carrier frequency should by taken as high aspossible. Notwithstanding, a high frequency of the carrierresults in high switchinglosses in the IGBT’s of the VSI. The minimisation of the losses in the combinedsystem, machine and power electronics, will not be examinedin this research.

8.3 Core Losses

To calculate the corresponding stator core losses, the current waveforms obtainedthrough the state space model are used in the multislice 2D finite element simula-tions and an evaluation of the core losses is performed as illustrated in Chapter 4.The calculations are performed for a M600-50A material.

8.3.1 Internal Loops

Chapter 4 used the loss separation method, which subdividedthe stator core lossesinto a hysteresis loss component, a classical loss component and an excess losscomponent. The calculation of the classical loss componentand excess loss com-ponent could be performed on any waveform of the magnetic fluxdensity, how-

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secu

rren

ts[A

]


0 45 90 135 180 225 270 315 360-15

-10

-5

0

5

10

15


ever, for the hysteresis loss component it was assumed that no local minima ormaxima were present in the waveform of the magnetic flux density. In Chapter4 this assumption was valid as there were no local minima or maxima observedin the waveform of the magnetic flux density. Nevertheless, in the VSI-PWM fedmachine local minima or maxima are observed in the waveform of the magneticflux density. In Fig. 8.7 the waveform of the magnetic flux density in the centerpart of the stator core is illustrated.

The calculation of the hysteresis loss component based on the peak value ofthe magnetic flux density as presented in Chapter 4, is only a first approximationas it does not take into account the local minima or maxima in the waveform ofthe magnetic flux density. To check the error that is made by neglecting the localminima or maxima in the waveform of the magnetic flux density,the hysteresis ismodelled by the Preisach model presented in the appendix [5,6], and the hystere-sis loss component is calculateda posteriori. Here, it is illustrated that the localminima and maxima in the waveform of the magnetic flux densityresult in minorhysteresis loops. The presence of these minor loops introduces supplementary hys-teresis losses in the stator core. For the 10 kHz carrier waveform, the hysteresislosses were estimated at 109 % of these in the machine withoutPWM. Neglectingthe minor hysteresis loops, hence, results in an underestimation of the hysteresisloss component with (only) 9 %, which is about 5 W. As these 5 W are very lim-ited compared to the total core loss of about 165 W, the complexity of the Preisachmodelling surpasses the improvement of the accuracy. Therefore the previously

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8.3 Core Losses 187

Mag

ticflu

xde

nsity

[T]


0 45 90 135 180 225 270 315 360-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 8.7: Waveform for the magnetic flux density in the center part of thecore. Values correspond to the inner diameter laminations.

introduced model for the hysteresis component based on the peak value of themagnetic flux density is maintained in further calculations.

8.3.2 Simulation Results

The results of the simulations of the core losses for different frequencies of the car-rier are presented in Table 8.1. The values between bracketsrepresent the relativeincreases with respect to the machine where PWM was not considered. The effect

Table 8.1: Effect of carrier frequency on the stator core losses including lossseparation.

stator core loss (M600-50A)# segments 5 kHz 10 kHz 20 kHz

Phy 53.1870 (+2%) 53.1870 (+2%) 52.6370 (+1%)Pcl 99.3600 (+42%) 94.6460 (+36%) 84.6680 (+21%)Pexc 18.1990 (+38%) 17.3900 (+13%) 15.7180 (+7%)PFe 170.7460 (+26%) 165.2230 (+22%) 153.0230 (+13%)

of the PWM is most visible in the increase of the classical losses. The increaseof the excess loss component and the hysteresis component, even if the Preisachmodel is used, is less pronounced.

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On the other hand, a positive influence of a high carrier frequency on the corelosses in the machine can be concluded.

8.4 Eddy Current Loss in the Permanent Magnets

Consequently, the eddy current losses in the permanent magnets are calculated us-ing the multislice 2D - 2D computation technique that was introduced in Chapter 5.Although the skin depth was taken into account during the calculation of the eddycurrents in the permanent magnets, the validity of this model is only limited as it isassumed that the eddy currents induced in the permanent magnets do not have aninfluence on the resultant magnetic flux density. This assumption will by violatedas for high frequencies the limited skin depth makes the surface of the permanentmagnets to act like a shield for the penetrating magnetic fluxdensity. Therefore,the multislice 2D - 2D model will be less accurate for VSI-PWMfed machines.Nevertheless, taking into account the influence of the magnetic flux density gener-ated by the eddy currents in the permanent magnets, requiresthe use of transienttime simulations on the full 3D finite element model [7,8]. The feasibility of thesesimulations was rejected in Chapter 5, based on simulation time and memory con-straints. The simulation time aspect is even enforced as themaximum time step isstrongly decreased in case of the small periods of the carrier.

In Table 8.2, the eddy current losses in the permanent magnets are summarizedfor the three carrier frequencies and for different segmentation gradescfr. Fig.5.12.

Table 8.2: Effect of carrier frequency on the eddy-current losses includingsegmentation of the permanent magnets.

magnet eddy current loss# segments 5 kHz 10 kHz 20 kHz

1 25.9966 (+69%) 20.5264 (+33%) 17.7079 (+15%)2 16.6699 (+42%) 14.1229 (+20%) 12.8784 (+10%)4 12.8923 (+27%) 11.4678 (+13%) 10.8314 (+7%)14 4.7153 (+15%) 4.3841 (+7%) 4.2385 (+4%)

As for the loss in the stator cores, an increasing carrier frequency results ina decrease of the eddy current loss in the permanent magnets.Therefore, it canbe concluded that an increasing carrier frequency results in lower power lossesin the machine. Despite the lower power loss in the machine for higher carrierfrequencies, the switching losses in the IGBT’s of the VSI-PWM increase withincreasing carrier frequencies. Therefore, the power losses in the entire electricpart of the drive traini.e. power electronic devices plus electrical machine, shouldbe considered when selecting the most appropriate frequency for the carrier.

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8.5 Conclusion 189

On the other hand, the affectivity of the segmentation of thepermanent magnetsis clearly visible in case of a VSI-PWM fed machine. In Fig. 8.8 and Fig. 8.9, theeddy current losses corresponding to each harmonic component are illustrated forthe bulky (no segmentation) permanent magnet and the permanent magnet with 14electrically isolated segments.

PPM

[W]

ν

0 50 100 150 200 250 300 350 400 450 50010−4

10−3

10−2

10−1

100

101

Figure 8.8: Eddy current losses in a single permanent magnet with no seg-mentation as a function of the harmonic orderν during load.

Whereas it was observed in Chapter 5 that the segmentation ofthe permanentmagnets results in a decrease of the eddy current losses in general, comparisonof Fig. 8.8 and Fig. 8.9 clearly indicates that the decrease of the harmonic losscomponents is most significant for the higher numbers,i.e. the higher frequenciesdue to the PWM. A high segmentation grade is hence a very effective measure toreduce the additional losses due to PWM.

8.5 Conclusion

As in most current high performance electric drive trains, PM machines are fed byvariable frequency drives, this chapter investigated the power losses in case of anVSI-PWM fed axial flux PM machine. Although the harmonic content present inthe PWM-voltage signal is significantly reduced by the low pass filter propertiesof the reactances of the machine, there is still a visible ripple present in the phasecurrents. As these current ripples affect the magnetic flux density pattern in thestator cores and the air gap magnetic flux density, they will result in increasingstator core losses and eddy current losses in the permanent magnets.

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PPM

[W]

ν

0 50 100 150 200 250 300 350 400 450 50010−4

10−3

10−2

10−1

100

101

Figure 8.9: Eddy current losses in a single permanent magnet consistingof14 electrically isolated segments as a function of the harmonicorderν during load.

To model a VSI-PWM fed axial flux PM machine, a state space model forthe machine was introduced. This state space model was used to calculate thephase currents corresponding to the imposed PWM signal as a voltage input. Con-sequently, the obtained phase currents were imposed to the multislice 2D (-2D)model to calculate the losses in the stator cores and the eddycurrent losses in thepermanent magnets. The simulations were performed for three different frequen-cies of the PWM-carrier.

The simulations showed a non negligible influence of the PWM on both the sta-tor core losses and the eddy current losses in the permanent magnets. The variationof the carrier frequency suggested to increase the frequency as this has a reducingimpact on the losses in the machine. In order to limit the eddycurrent losses inthe permanent magnets, the segmentation of the permanent magnets has a majorinfluence.

Appendix

Nonlinear Inductance Matrix

Although a linear inductance matrix was used in the state space model of the axialflux PM machine, a nonlinear variant needs to be considered totake the effect ofsaturation into account [9,10]. In the linear variant it is assumed that the elements

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8.5 Conclusion 191

of the inductance matrix do not depend on the rotor position,as the relative perme-ability of the permanent magnet material is very close to unity. Notwithstandingthe position of the permanent magnets is the main source of magnetic flux in thestator core elements. Therefore, the position of the permanent magnets with re-spect to the core elements will have a major influence on the flux density pattern inthe stator core elements, which will result in a position dependency of the elementsin the inductance matrix.

Next to the magnetic flux by the permanent magnets, the magnetisation stateof the stator core material is also affected by the armature reaction currents. Thiseffect was extensively examined in Chapter 6.

Therefore, the elements in the inductance matrix are a function of the rotor po-sition and the actual values of the phase currents. A new notation of the inductancematrix is introduced

L =

Laa Lab Lac

Lba Lbb Lbc

Lba Lbb Lbc

. (8.12)

In order to model the nonlinear behaviour of the stator core material on the el-ements of the inductance matrix more accurately, the tangent rather than the chordinductance. Calculation of the tangent inductance is performed by applying a cur-rent perturbationδI in the working point.

Laa =ψa

(

θ, Ia +δI2 , Ib, Ic

)

− ψa

(

θ, Ia − δI2 , Ib, Ic

)

δI(8.13)

Lab =ψa

(

θ, Ia, Ib +δI2 , Ic

)

− ψa

(

θ, Ia, Ib − δI2 , Ic

)

δI(8.14)

Lac =ψa

(

θ, Ia, Ib, Ic +δI2

)

− ψa

(

θ, Ia, Ib, Ic − δI2

)

δI(8.15)

Lba =ψb

(


)

− ψb

(


)

δI(8.16)

Lbb =ψb

(


)

− ψb

(


)

δI(8.17)

Lbc =ψb

(


)

− ψb

(


)

δI(8.18)

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Lca =ψc

(


)

− ψc

(


)

δI(8.19)

Lcb =ψc

(


)

− ψc

(


)

δI(8.20)

Lcc =ψc

(


)

− ψc

(


)

δI(8.21)

The applicability of the nonlinear inductance matrix remains limited as for eachindividual state in the state space model six static nonlinear finite element evalua-tions are necessary. Although parallelization of the finiteelement computation ispossible, simulation times are still very high [11].

Preisach Modelling

The Preisach model discussed in [5,6] was used to examine theinfluence of minorloops on the hysteresis loss in the stator core elements.

As the permanent magnets cause a magnetic flux in the laminations of the statorcores, the average value of the magnetic flux densities obtained through the staticfinite element computations for different rotor positions are taken as an input forthe hysteresis model. To model the variation of the magneticfield and magneticflux density over the cross section of the lamination, a 1D diffusion equation isused.

The output of the Preisach model at the center and the edge of alamination isillustrated in Fig. 8.10. Obviously the presence of minor loops is less visible at thecenter section compared to the edges. The minor loops resultform the PWM havea high frequency and are hence displaced to the edges of the laminations.

Bibliography

[1] W.-C. Tsai, “A study on core losses of non-oriented electrical steel lamina-tions under sinusoidal, non-sinusoidal and pwm voltage supplies,” in TEN-CON 2007-2007 IEEE Region 10 Conference. IEEE, 2007, pp. 1–5.

[2] S. Khomfoi, V. Kinnares, and P. Viriya, “Investigation into core losses due toharmonic voltages in pwm fed induction motors,” inPower Electronics andDrive Systems, 1999. PEDS’99. Proceedings of the IEEE 1999 InternationalConference on, vol. 1. IEEE, 1999, pp. 104–109.

[3] L. T. Mthombeni, P. Pillay, and N. A. Singampalli, “Lamination core lossmeasurements in machines operating with pwm or nonsinusoidal excitation,”in Electric Machines and Drives Conference, 2003. IEMDC’03. IEEE Inter-national, vol. 2. IEEE, 2003, pp. 742–746.

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8.5B

ibliography193

Mag

netic

flux

dens

ity[T

]Magnetic field [A/m]

Mag

netic

flux

dens

ity[T

]

Magnetic field [A/m]

-800 -600 -400 -200 0 200 400 600 800-800 -600 -400 -200 0 200 400 600 800-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 8.10: Hysteresis loops obtained through Preisach modelling at the center of a lamination (left) and the edge of alamination (right) Values are taken for a lamination at the inner diameter region of the machine.

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[4] M. Valtonen, A. Parviainen, and J. Pyrhonen, “Inverterswitching frequencyeffects on the rotor losses of an axial-flux solid-rotor coreinduction motor,” inPower Engineering, Energy and Electrical Drives, 2007. POWERENG 2007.International Conference on. IEEE, 2007, pp. 476–480.

[5] D. Philips, L. Dupre, J. Cnops, and J. Melkebeek, “The application of thepreisach model in magnetodynamics: theoretical and practical aspects,”Jour-nal of magnetism and magnetic materials, vol. 133, no. 1, pp. 540–543, 1994.

[6] D. A. Philips, L. Dupre, and J. Melkebeek, “Magneto-dynamic field compu-tation using a rate-dependent preisach model,”Magnetics, IEEE Transactionson, vol. 30, no. 6, pp. 4377–4379, 1994.

[7] K. Yamazaki and A. Abe, “Loss investigation of interior permanent-magnetmotors considering carrier harmonics and magnet eddy currents,” IndustryApplications, IEEE Transactions on, vol. 45, no. 2, pp. 659–665, 2009.

[8] K. Yamazaki and Y. Fukushima, “Effect of eddy-current loss reduction bymagnet segmentation in synchronous motors with concentrated windings,”Industry Applications, IEEE Transactions on, vol. 47, no. 2, pp. 779–788,2011.

[9] F. De Belie, “Vectorregeling van synchrone machines metpermanente-magneetbekrachtiging zonder mechanische positiesensor,” 2010.

[10] F. De Belie, J. Melkebeek, K. Geldhof, L. Vandevelde, and R. Boel, “A gen-eral description of high-frequency position estimators for interior permanent-magnet synchronous motors,” inConference Proceedings of the 16th Inter-national Conference on Electrical Machines, 2004.

[11] P. Sergeant, F. De Belie, and J. Melkebeek, “Effect of rotor geometry andmagnetic saturation in sensorless control of pm synchronous machines,”Magnetics, IEEE Transactions on, vol. 45, no. 3, pp. 1756–1759, 2009.

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Chapter 9

Concluding Remarks

9.1 Conclusion

The design of an energy efficient axial flux PM machine was considered in thiswork. After an introduction to the axial flux PM technology and the differenttopologies, the yokeless and segmented armature (YASA) topology was selectedbecause of its distinct advantages: an excellent power density and a high efficiency.Moreover, the YASA-topology introduces a modular stator construction which isadvantageous for the manufacturing of the stator cores and windings.

The second part of this work was dedicated to the mathematical modelling ofthe axial flux PM machine. In this mathematical modelling, both analytical andfinite element analysis were discussed. Different from radial machines, axial fluxmachines have an inherent 3D geometry, and therefore, require appropriate mod-elling techniques. To take into account the 3D geometry, a multislice 2D modellingtechnique was introduced. This modelling technique definedmultiple computationplanes at different diameters on which general 2D mathematical models can beapplied. As in this modelling technique only a limited number of 2D computa-tions are involved, it is much faster and uses only a fractionof the memory thatis required for full 3D (finite element) modelling. Once the multislice 2D calcula-tions are finished, the global quantities are found by summation over the differentcontributions obtained by the different 2D models. This multislice 2D modellingtechnique was considered as the basic model, and is constantly used and extendedin the following chapters.

After an introduction to the axial flux PM technology and the multislice 2Dmodelling technique, the following chapters discussed theenergy efficiency of theaxial flux PM machine.

The first topic on energy efficiency describes the comparisonof non orientedand grain oriented silicon steel sheets as stator core material. The advantageousproperties of the grain oriented material in the rolling direction are introduced inthe axial flux PM machine by choosing the rolling direction ofthe grain oriented

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196 Concluding Remarks

material parallel to the axial direction of the stator cores. To model the anisotropicproperties of the grain oriented material, additional mathematical techniques wereintroduced to calculate the magnetisation state and lossesin the material. Con-sequently, the material model is used in the previously introduced multislice 2Dmodel to calculate the no load back electromotive force, theoutput torque and thelosses in the stator cores. During the comparison of the non oriented and the grainoriented material, the superior properties of the grain oriented material are foundin a slightly higher output torque and a significant reduction of the core losses.The use of grain oriented material in axial flux PM machine hence results in asignificant increase of the energy efficiency of the machine.

A second topic focuses on the losses in the permanent magnetsdue to eddycurrents. As the NdFeB permanent magnet material has a good electric conductiv-ity, eddy currents are induced in the permanent magnets whenthey are subjected toa magnetic flux density that varies in time. The origin of thistime variation in themagnetic flux density has two causes: the stator slotting effect and the armature re-action. To calculate the eddy currents in the permanent magnets, the multislice 2Dmodel is used to calculate the magnetic flux density in a planeparallel to the air gapthat moves synchronous with the permanent magnets. Subsequently, a Fourier se-ries expansion is used to split the air gap magnetic field in its different components.For the different frequency components, the eddy currents in the permanent mag-nets are calculated. In the evaluation of the losses, the skin depth of the magneticflux density in the permanent magnets is taken into account. Finally, segmentationof the permanent magnets was introduced to reduce the eddy current losses.

Some geometrical parameters have a major impact on the losses in the ma-chine. Therefore, an optimisation of a limited set of geometry parameters wasperformed making use of the analytical model. Nevertheless, the influence of oneof these parameters,i.e. the stator slot opening width, on the different losses inthe machine was studied extensively. The stator slot openings are found to havea major impact on the stator core losses as the stator slot openings affect the pathof the magnetic flux in the stator cores. Also the losses in thepermanent magnetsare affected significantly. As the slot openings become wider, the variation of theair gap magnetic flux density increases which results in larger induced eddy cur-rents in the permanent magnets. The effect of an increasing stator openings widthis found to be contrary with respect to both losses: the losses in the stator coresdecrease while the losses in the permanent magnets increase. Moreover, a minorimpact on the electromagnetic torque output of the machine is found.

A fourth improvement of the energy efficiency is introduced by a combinedwye-delta connection. Whereas traditionally a wye (or delta) connection of thedifferent machine windings is chosen, a combination of bothconnection types isintroduced. The fundamental components of the electromotive forces of the differ-ent stator windings have small phase shifts which result in afundamental windingfactor that differs from unity. This effect is extremely visible in fractional slotwindings where poor fundamental winding factors are found for some combina-

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9.2 Energy Efficiency Map 197

tions of pole and stator slot numbers. By introducing the combined wye-deltaconnection, the phase shifts of the different coils are partially compensated by thephase shifts between the wye and delta connected coils. Therefore, the combinedwye-delta connection increases the fundamental winding factor of the machine.For the same phase current, the combined wye-delta connection is found to havea higher electromagnetic torque. On the other hand, the highharmonic contentwhich is typically present in the back electromotive forcesof fractional slot wind-ings, results in a circulating current in the delta connected coils. Evaluation of theadditional losses caused by this circulation current, showed only a negligible in-crease of the losses. Consequence, a combined wye-delta connection results in asignificant increase of the energy efficiency of the YASA-machine.

In current highly dynamic drive trains, the electrical machines are fed by powerelectronic devices. In this work the machine is assumed to befed by a voltagesource inverter (VSI), that makes use of puls-width modulation (PWM) to vary thevoltage output. The high frequency content of the PWM voltage signal, resultsin a ripple in the phase current which affects the different losses in the machine.The previously introduced models for the stator core lossesand eddy current lossesin the permanent magnets are used to evaluate the additionallosses due to PWM.Therefore, the phase currents corresponding to a VSI-PWM fed machine were cal-culated using a state space model. Consequently, these obtained phase currentswere imposed to the multislice 2D model. This was done for different frequenciesof the PWM. It was found that the PWM introduces a none negligible increase ofthe losses. Nevertheless, an increase of the PWM-frequencyhas a reducing impacton the losses in the machine.

9.2 Energy Efficiency Map

In this thesis, mathematical models were introduced to model the axial flux PMmachine as well as to evaluate the different losses. Also some measures wereintroduced to reduce the losses in the machine.

The power losses present in the axial flux PM machine were onlyevaluated attwo single working points of the machine; at no load and ratedspeed, and ratedload and rated speed. As in most high performance drives the torque output aswell as the rotational speed is varied over a wide range. Not only the energy ef-ficiency at these single points is sufficient. Therefore an energy efficiency mapis introduced, which specifies the energy efficiency as a function of the rotationalspeed and torque output of the machine. These energy efficiency maps can be builtby measurements on a prototype machine, or can be calculated. In the calculatedenergy efficiency map, the different electromagnetic powerloss components,i.e.copper losses, stator core loss and eddy current loss in the permanent magnets, arecalculated individually (Chapter 3, 4, 5). Such a calculated energy efficiency mapis presented in Fig. 9.1 for the test case machine.

This energy efficiency map illustrates that the energy efficiency of the axial

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Figure 9.1: Calculated energy efficiency map of the test case machine.

flux PM machine is not only excellent at the rated load and speed, but applies to awide range of load and speed. Consequently, it can be concluded that the axial fluxPM topology allows an efficient energy conversion under varying load conditions.

At the end of this work, the objectives of this work are checked again. First,this thesis was dedicated to the electromagnetic design of the axial flux PM ma-chine. Crucial in the electromagnetic design was the introduction of the multislice2D modelling, which reduces the demanding 3D finite element computation to sim-ple 2D (finite element) calculations. Second, the focus within the electromagneticdesign of the axial flux PM machine was on the energy efficiency. Mathematicalmodels were introduced to model and to estimate the different losses and, moreimportant, multiple measures were introduced to increase the efficiency of the en-ergy conversion: optimisation of geometrical parameters to improve the energyefficiency, a high grade oriented laminated steel in the stator cores to reduce thecore loss, segmentation of the permanent magnets to limit the eddy current loss,and a combined wye-delta connection to increase the torque output.

The introduced mathematical models are valid over a wide range of axial fluxPM machines, which was the third objective of this thesis. A test case machinewas introduced to illustrate the mathematical models and toestimate the impact ofthe energy efficiency increasing measures. The models are applicable to all YASAmachines, most of the models also to all axial flux PM machinesand some modelscan be used in radial flux PM machines as well.

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9.3 Drive Train Optimisation 199

9.3 Drive Train Optimisation

No direct application was suggested in this thesis. To illustrate the necessity ofgeneral applicable mathematical tools, two classes of applications are considered:small wind turbines and electric vehicles. In these applications the axial flux PMmachine becomes a part of the entire drive train.

9.3.1 Small Wind Turbines

Small wind turbines typically have a tower height less than 15 m, turbine bladeswith a diameter less than 10 m and have an output power which isgenerally lessthan 10 kW. [1] The drive train of a small wind turbine is illustrated in Fig. 9.2.The wind power is captured by the blades of the wind turbine, which are connected

AC

AC DC

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PMSM

Figure 9.2: Drive train of a small wind turbine.

to the shaft of the generator. This connection can be made through a gear box toconvert the low speed rotation into a sufficiently high speedsuitable for a standardgenerator, or, a custom designed generator can be connecteddirectly (direct drive)[2, 3] to the shaft of the wind turbine. This custom designed generator has tobe adopted to the low-speed-high-torque output of the turbine. Because of theexcellent flexibility, axial flux PM machines are perfectly suitable as direct drivegenerators [4–7].

The relation between the wind speed and the power transferred to the turbine isgiven by the power coefficientCp(λ) of the turbine. As the speed of the generatordepends on the incoming wind power, the generator will produce a voltage withvarying amplitude and frequency. This wild AC is generally rectified by a dioderectifier [8–10], and consequently converted to ac power that fits the standards ofthe grid.

In an optimisation, the drive train should be considered globally rather thanfocussing on its components individually. Optimisation ofa wind turbine meansthat, given the annual wind energy, the electric energy delivered to the grid shouldbe maximised. The input of this system is the wind power to theturbine [11].

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As the wind speed varies over time, the annual wind energy is approximated bythe introduction of a wind speed distribution (Fig. 9.3) that indicates the relativepresence of each wind speed in the annual wind energy profile.Consequently,

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Figure 9.3: Distribution of wind speed.

the power coefficientCp(λ) indicates the power that is captured by the bladesand represents the input of the axial flux PM generator. As thespeed as well asthe torque (power) of the axial flux PM generator vary, it is crucial to optimisethe generator for the overall best performance. Therefore,the generator needs tohave its best performance for low wind speeds as they occur most of the time,however, the generator needs to withstand the large power athigh wind speedsas well, electromechanically but also mechanically and thermally [12]. A goodknowledge of the electromagnetic behaviour of the axial fluxPM machine is crucialin this design of the generator. The energy efficiency map as afunction of the loadand speed will be used for the maximisation of the annual energy production. Anestimation of the power losses in the different parts of the generator as a function ofthe load and speed, will be the inputs of a thermal model. For example, during highwind speeds and/or wind gust, the losses in the machine will reach their maximalvalue. Depending on their duration and the thermal inertia of the machine, thedesign needs to be adapted in order to keep the temperature inthe different machineparts below a safe limit.

On the other hand, the (diode) rectifier will influence the current waveform inthe machine, while the maximum power point tracking specifies the load of thegenerator. Both will have an influence on the losses in the machine.

Therefore, during the maximisation of the annual energy output, an optimisa-

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9.3 Drive Train Optimisation 201

tion of axial flux PM machine cannot be done separately from the wind turbine andpower electronics; the complete drive train needs to be considered.

9.3.2 Electric Vehicles

Another application for which axial flux PM machines are perfectly suitable areelectric vehicles [13–15]. Regardless the power ratio of the application, going fromabout 200 W for electric bikes to tens to hundreds of kW for full electric passengercars, the drive train (Fig. 9.4) consists of a battery pack that is connected to aconverter that feeds the electrical motor. Depending on theapplication, one central

AC

DCPMSMBAT

Figure 9.4: Drive train of an electric vehicle.

motor can be connected, eventually through a gearbox, to thewheels, while otherapplications use an in wheel mounted motor. Particularly for the built-in wheeldrives, the axial flux PM machines is advantageous because ofits slim shape andnoticeable compactness [14,16,17].

As for the small wind turbine, optimisation of the entire system should be con-sidered rather than focussing on each component individually. In the optimisationof the drive train, the aim is to run through a specified cycle with a minimumamount of energy. Currently these drive cycles, NEDC (Europe) (Fig. 9.5), EPAFederal Test (US), ... , are already defined for passenger cars, to compare the fuelconsumption and emission values of a car. These driving cycles are similar to thewind energy probability in the optimisation of the annual energy output of the smallwind turbine.

Different than for the small wind turbine is that the power flow can occur inboth directions. During the driving in general and acceleration in particular, electricpower is transferred from the battery pack to the wheels. During this conditionthe axial flux PM machine is used as a motor. When deceleratingthe car, thestored inertia can be transferred back to the battery pack. The axial flux machine ishereby used as a generator. In the optimisation, both working conditions should beconsidered. Again, there is a need for a good mathematical model predicting theefficacy and losses for each working point of the machine.

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Figure 9.5: New European Driving Cycle.

9.4 Recommendations for Future Research

This work focused on the electromagnetic design of the axialflux PM machineonly. The entire design of the machine requires a thermal andmechanical study aswell.

The mechanical issues are mainly caused by the modular structure of the sta-tor. The different stator core elements should be assembledto a solid stator first,and furthermore this assembly has to be integrated in a housing or application.Some possibilities were discussed, however, no extensive mechanical analysis isperformed yet.

The thermal aspect of the machine is also an important issue.If the powerlosses are not sufficiently conveyed to the surroundings of the machine, local tem-perature increases affect the electromagnetic performance as well as the mechani-cal properties of the machine. Thermal design has to take into account the differenttransport mechanisms of heat: conduction, convection and radiation. Sometimes,in a so calledmultiphysicsapproach, finite element packages solve the electromag-netic and thermal problem simultaneously. Nevertheless, thermal studies of themachine are generally carried out separately given the different power losses in themachine as an input.

Next to the electromagnetic losses, there are mechanical losses in the bearingsdue to friction and ventilation losses in the machine due to fan working of the rotordiscs. In the design towards a maximised energy efficiency these losses should beestimated using mathematical modelling or empirical formulas.

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Also in the electromagnetic design, there are still many possibilities for im-provement. The use of amorphous and soft magnetic compositematerials in thestator core elements can be considered. This will involve the proper modelling ofthe different materials. Also the combination of differentmagnetic materials in asingle stator core element is an option that is not investigated yet.

Bibliography

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[2] T. Chan and L. Lai, “An axial-flux permanent-magnet synchronous generatorfor a direct-coupled wind-turbine system,”Energy Conversion, IEEE Trans-actions on, vol. 22, no. 1, pp. 86–94, 2007.

[3] A. Di Gerlando, G. Foglia, M. Iacchetti, and R. Perini, “Design criteria ofaxial flux pm machines for direct drive wind energy generation,” in ElectricalMachines (ICEM), 2010 XIX International Conference on. IEEE, 2010, pp.1–6.

[4] J. Azzouzi, G. Barakat, and B. Dakyo, “Quasi-3-d analytical modeling ofthe magnetic field of an axial flux permanent-magnet synchronous machine,”Energy Conversion, IEEE Transactions on, vol. 20, no. 4, pp. 746–752, 2005.

[5] G. F. Price, T. D. Batzel, M. Comanescu, and B. A. Muller, “Design and test-ing of a permanent magnet axial flux wind power generator,” inProceedingsof the 2008 IAJC-IJME International Conference, 2008.

[6] A. Parviainen, J. Pyrhonen, and P. Kontkanen, “Axial fluxpermanent mag-net generator with concentrated winding for small wind power applications,”in Electric Machines and Drives, 2005 IEEE International Conference on.IEEE, 2005, pp. 1187–1191.

[7] M. Andriollo, M. De Bortoli, G. Martinelli, A. Morini, and A. Tortella, “Per-manent magnet axial flux disc generator for small wind turbines,” inElectri-cal Machines, 2008. ICEM 2008. 18th International Conference on. IEEE,2008, pp. 1–6.

[8] S. Brisset, D. Vizireanu, and P. Brochet, “Design and optimization of anine-phase axial-flux pm synchronous generator with concentrated windingfor direct-drive wind turbine,”Industry Applications, IEEE Transactions on,vol. 44, no. 3, pp. 707–715, 2008.

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[9] A. Di Gerlando, G. Foglia, M. F. Iacchetti, and R. Perini,“Analysis and testof diode rectifier solutions in grid-connected wind energy conversion systemsemploying modular permanent-magnet synchronous generators,” IndustrialElectronics, IEEE Transactions on, vol. 59, no. 5, pp. 2135–2146, 2012.

[10] M. Andriollo, G. Martinelli, A. Morini, and A. Tortella, “Performance as-sessment of a wind pm generator-rectifier system by an integrated fem-circuitmodel,” inElectric Machines & Drives Conference, 2007. IEMDC’07. IEEEInternational, vol. 1. IEEE, 2007, pp. 358–363.

[11] S.-Y. Jung, H. Jung, S.-C. Hahn, H.-K. Jung, and C.-G. Lee, “Optimal designof direct-driven pm wind generator for maximum annual energy production,”Magnetics, IEEE Transactions on, vol. 44, no. 6, pp. 1062–1065, 2008.

[12] F. Marignetti and V. D. Colli, “Thermal analysis of an axial flux permanent-magnet synchronous machine,”Magnetics, IEEE Transactions on, vol. 45,no. 7, pp. 2970–2975, 2009.


[14] Y.-P. Yang, Y.-P. Luh, and C.-H. Cheung, “Design and control of axial-fluxbrushless dc wheel motors for electric vehicles-part i: multiobjective optimaldesign and analysis,”Magnetics, IEEE Transactions on, vol. 40, no. 4, pp.1873–1882, 2004.

[15] S. Javadi and M. Mirsalim, “Design and analysis of 42-v coreless axial-fluxpermanent-magnet generators for automotive applications,” Magnetics, IEEETransactions on, vol. 46, no. 4, pp. 1015–1023, 2010.

[16] C. Versele, Z. De Greve, F. Vallee, R. Hanuise, O. Deblecker, M. Delhaye,and J. Lobry, “Analytical design of an axial flux permanent magnet in-wheelsynchronous motor for electric vehicle,” inPower Electronics and Applica-tions, 2009. EPE’09. 13th European Conference on. IEEE, 2009, pp. 1–9.

[17] Y.-P. Yang and D. S. Chuang, “Optimal design and controlof a wheel motorfor electric passenger cars,”Magnetics, IEEE Transactions on, vol. 43, no. 1,pp. 51–61, 2007.

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