Design of a Permanent-Magnet Synchronous …583871/FULLTEXT01.pdfSynchronous Machine with...

Degree project in

Design of a Permanent-MagnetSynchronous Machine with Non-

Overlapping Concentrated Windingsfor the Shell Eco Marathon Urban Prototype

DANIEL MARTÍNEZ

Stockholm, Sweden 2012

XR-EE-E2C 2012:020

Electrical EngineeringMaster of Science

Design of a Permanent-Magnet Synchronous Machinewith Non-Overlapping Concentrated Windings for the

Shell Eco Marathon Urban Prototype

DANIEL MART INEZ

Royal Institute of TechnologySchool of Electrical Engineering

Electrical Energy ConversionStockholm 2012.

XR-EE-E2C 2012:020

Design of a Permanent-Magnet Synchronous Machine with Non-OverlappingConcentrated Windings for the Shell Eco Marathon Urban PrototypeDANIEL MART INEZ

c© DANIEL MART INEZ, 2012.

School of Electrical EngineeringDepartment of Electrical Energy Conversion (E2C)Kungliga Tekniska HogskolanSE–100 44 StockholmSweden

Abstract

This thesis deals with the design of a permanent-magnet synchronous inner rotor motorfor an in-wheel application for the Shell Eco Marathon Urbanconcept vehicle.

First of all, concepts related to permanent magnet motors are studied. Likewise, dif-ferent features of permanent magnet motors are qualitatively evaluated in order to choosethe most suitable. A radial flux motor is selected based on itssolid, economic and accept-able characteristics.

Next, a detailed study of concentrated windings is carried out. Through this inves-tigation, undesirable configurations of pole and slot numbers due to unbalanced magneticpull or a low fundamental winding factor will be avoided and how to determine the differ-ent winding layouts for different pole and slots configuration will be explained. As well,based on this study, and the magnetic and electric behavior of the machine, an analyticalmodel is created. This model calculates the optimum size andcharacteristics of a machinein order to obtain lightweight design.

After that, the design of a program based on a finite element method that simulatesdifferent situations for the machine is accomplished, dealing with the difficulties that en-tails the concentrated windings.

Finally, through the use of this program, the machine calculated by the analyticalmodel is analyzed, specially regarding that it does not surpass some important margin inorder not to be demagnetized or not to surpass the maximum phase voltage supplied bythe batteries.

Keywords: Finite element analysis, in-wheel motor, concentrated windings,permanent-magnet synchronous machine, fundamental winding factor.

iii

Sammanfattning

Detta examensarbete fokuserar pa designen av en permanentmagnetiserad synkronmotorav ytterrotortyp. Motorn ar avsedd att anvandas som en hjulmotor i ett konceptfordon(Shell Eco Marathon Urban Concept). En motor av radialflodestyp studeras pa grund avkonceptets enkelhet och acceptabla prestanda. Sedan presenteras en noggrann studie kringkoncentrerade lindningar. Pol- och spartalskombinationer som ger upphov till obalanser-ade magnetiska krafter eller en lag fundamental lindningsfaktor harleds och en metodfor att bestamma lindningsutbredningen beskrivs. En analytisk modell tas sedan fram avmotorn vilken anvands for att minimera motorns aktiva vikt. Efter detta tas ett design-program baserat pa finita-elementmetoden fram i vilken denresulterande motordesignenharrorande fran den analytiska modelleringen utvarderas. Speciellt studeras risken for de-magnetisering av permanentmagneterna samt att fasspanningen inte overstiger den granssom stipuleras av givna batterispanningen.

Nyckelord: Finita elementmetoden, hjulmotorer, koncentrerade lindningar,permanentmagnetiserade synkronmotorer, fundamental lindningsfaktor.

v

Acknowledgements

First of all, I would like to sincerely thank Oskar Wallmark for all his help andguidance provided throughout these months of work. He has taught me a lot of things andtransmitted his knowledge and experience to me, making me a better engineer.

I am also thankful to Mats Leksell for his advices during the thesis and, aboveall, for his willingness to help me getting this Master Thesis in a moment that was quitedifficult and critical for me.

I am also very grateful to Rosa and Mercedes, for their help and advice along thesemonths, and for taking care of me in difficult times. I also appreciate them for creatingsuch a good work atmosphere in the office, which has contributed to perform this thesiseasier.

Within my group of friends, I would like to specially mentionGuillem and Elenafor all the help they have provided me with throughout the year, which has been priceless.I would also like to thank Manuel, Miguel, Alberto, Alejandro, Navarrete, Pere, Jesus,Lorena, Pastor, Sarah... their help and support throughoutthe year, especially in the mo-ments that have been critical to me, and for providing me withso many unforgettablemoments. Finally, the most important ones... I would like tothank my mother, my fatherand my sister for all the support, advice, help and patience Ihave received along this yearand throughout my life, which have made possible who I am right now. Thank you somuch!

Daniel Martınez JimenezStockholm, SwedenNovember 2012

vii

Contents

Abstract iii

Sammanfattning v

Acknowledgements vii

Contents ix

1 Introduction 11.1 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . .11.2 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Methodology and Material . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Shell Eco Marathon 2013 . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Low-Speed Direct-Drive Applications. In-Wheel PMSM 52.1 In-Wheel Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Low-Speed and High-Speed Machines . . . . . . . . . . . . . . . 62.2 PM Synchronous Machines . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Permanent Magnets . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Advantages and Drawbacks . . . . . . . . . . . . . . . . . . . . 102.2.3 Radial Flux PM Synchronous Machine (RFPM) . . . . . . . . . .122.2.4 Axial Flux PM Synchronous Machines, (AFPM) . . . . . . . . .14

2.3 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Shell Eco Marathon Circuit Analysis. Requirements 173.1 Track and Car Requirements . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Driving Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.2 Cycle Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Analysis of the Energy Consumption . . . . . . . . . . . . . . . . . .. . 213.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Non-Overlapping Concentrated Windings Analysis 25

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Contents

4.1 Concentrated or Distributed winding . . . . . . . . . . . . . . . .. . . . 254.2 Double or Single Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Concentrated Winding Layout . . . . . . . . . . . . . . . . . . . . . . .274.4 Fundamental Winding Factor . . . . . . . . . . . . . . . . . . . . . . . .31

4.4.1 Winding Factor Based on Cros’ Method. . . . . . . . . . . . . . 324.4.2 Winding Factor Based on Pole-slot Combinations. . . . .. . . . 33

4.5 Desirable Configurations of Poles and Slots . . . . . . . . . . .. . . . . 364.5.1 Unbalanced Magnetic Pull . . . . . . . . . . . . . . . . . . . . . 374.5.2 Cogging Torque and Torque Ripple . . . . . . . . . . . . . . . . 39

4.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Analytical Design 435.1 Geometric Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1.1 Magnet Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.1.2 Iron Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Electric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Loss Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3.1 Copper Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.3.2 Core Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3.3 Magnet Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.4 Analytical Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.4.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.4.2 Additional Equations . . . . . . . . . . . . . . . . . . . . . . . . 565.4.3 Design Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 575.4.4 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.4.5 Fixed Values and Constraints of Design . . . . . . . . . . . . .. 575.4.6 Matlab Program . . . . . . . . . . . . . . . . . . . . . . . . . . 575.4.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6 Finite Element Analysis 616.1 FEMM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.1.1 Input Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 616.1.2 Machine Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 636.1.3 Material Settings . . . . . . . . . . . . . . . . . . . . . . . . . . 646.1.4 Accuracy Parameters . . . . . . . . . . . . . . . . . . . . . . . . 656.1.5 Data Obtaining . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.1.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.1.7 Program Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2 Machine Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.2.1 Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.2.2 No Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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Contents

6.2.3 Rated Point Conditions . . . . . . . . . . . . . . . . . . . . . . . 716.2.4 Overload Condition . . . . . . . . . . . . . . . . . . . . . . . . . 736.2.5 Different Working Points . . . . . . . . . . . . . . . . . . . . . . 74

7 Conclusions and Further Work 797.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . .. 80

A Dynamic simulation 83

B Analytical Motor Data 85

C Fundamental Winding factor 87

D Iron Loss Constants 91

E List of Symbols, Subscripts and Abbreviations 93

References 99

xi

Contents

xii

Chapter 1

Introduction

1.1 Motivation and Objectives

It is widely recognized that permanent magnet synchronous machines (PMSMs) haveseveral advantages, such as a high efficiency, that make themsuitable for vehicle propul-sion. In fact, many automakers are implementing these systems in hybrid electric vehicles(HEVs) and electric vehicles (EVs). In-wheel (IW) motor technology is a novel solutionfor EVs. Through the use of this type of devices, gearboxes are removed from the drivetrain increasing the efficiency of the propelling system. The multi-pole PMSM designedin this thesis uses concentrated double-layer windings. This type of coil arrangements ex-hibits better properties compared to the traditional distributed windings, being their bestcharacteristic the reduction of copper inside the machine.This entails also a great reduc-tion of the losses of the machine as well as its cost. The aim ofthis thesis is to design anelectric propelling system to drive the ”Shell Eco MarathonUrban Concept”. Since themain objective of this contest is the reduction of energy consumption, the design will befocused on the machine and the architecture exhibited above. An analytical model able tocalculate the parameters of this motor is developed, and in order to test it, a tool based onFinite Element Method Magnetics (FEMMR©) is programmed to be able to verify differ-ent properties.

1.2 Outline of Thesis

The thesis is structured as follows:

• Chapter 2 reports a study of the possible solutions which canpropel the vehicle,showing their advantages and drawbacks. Permanent Magnets(PM) are studiedshowing their relevant properties.

1

Chapter 1. Introduction

• Chapter 3 explains how the requirements of the machine are obtained, based on thedriving cycle.

• Chapter 4 focuses on the design of a PMSM with non-concentrated windings. Meth-ods to calculate the fundamental winding factor and windinglayout are exposed.Undesirable effects on this windings are also studied.

• Chapter 5 shows the magnetic and the electric behavior that the electrical machineexperience, as well as the losses. Based on concentrated windings theory, an analyt-ical model is developed to calculate the parameters of the machine which is the aimof this thesis. The lightest design is selected and its election is motivated in order toachieve a lightweight design,

• Chapter 6 introduces a program based on the FEMMR© package that is developedto analyze the machine selected in the previous chapter in order not to surpasscertain constraints of design, such as the minimum flux density in the magnets orthe maximum phase-neutral voltage.

• Chapter 7 discusses the implementation of this machine in Shell Eco Marathonvehicle, and proposes further improvements for some elements of the program andthe machine designed.

1.3 Methodology and Material

The guideline followed during this project is quite similarto its structure. During the firststages, information of the different type of motors and IW concepts were gathered. Next,requirements were calculated based on the information gathered between the differentparts of the Shell Eco 2013 team through a simple simulation based on Newton dynam-ics. After that, an analytical model of a PMSM was carried out, choosing concentratedwindings as its coil arrangement. Fundamental winding factor based on different theoriesand winding layout were the main tasks of this part of the study. Next, the design of aprogram to analyze the machine designed by the analytical model was carried out. Thisprogram was implemented using the FEMMR© software. This task entailed several chal-lenges, such as the determination of the starting angle of the rotor, for the electrical period.Since FEMMR© calculates properties on static simulations, the program developed had toanalyze the different characteristics of the machine as well as in a whole electric period,in all operating points of the machine. In order to validate the results obtained by thisFinite Element tool, they were checked with EMETOR R© . To conclude, the machine wasanalyzed based on the analytical model data calculated and different information gatheredin the literature review.

2

1.4. Shell Eco Marathon 2013

1.4 Shell Eco Marathon 2013

Shell Eco Marathon is a student approached competition which challenge them to designand build a vehicle which aims to be as energy-efficient as possible. This contest takesplace in three annual events in different continents (America, Asia and Europe). In suchevents, young engineers have the opportunity of showing their technical skills to push theboundaries of fuel efficiency as well as other interesting issues as developing the competi-tion vehicle. For instance, some of the vehicles are able to drive a similar distance as fromParis to Moscow (2485 km) consuming only a single liter of fuel [24]. The origin of this

Fig. 1.1 Rotterdam Ahoy Urban Circuit [24].

competition was a friendly bet between employees of Shell Oil Company in 1939. Thewager consisted of who could travel furthest using the same quantity of fuel. Since then,it has been in constant evolution, expanding the contest to other propelling technologiessuch as bio-fuel, electric vehicles, hybrids, solar vehicles, etc [24].

The Shell Eco Marathon 2013 circuit will take place in Rotterdam, which is illus-trated in Fig. 1.1. The Ahoy urban track will challenge the different teams with five 90o

corners, so driving skills such as braking, accelerating and cornering will be critical fac-tors in order to succeed in the competition. Thus, it is essential that all members in thevehicle development will be involved when creating the vehicle [24].

The aim of ”Shell Eco Marathon 2013” is the design of an EV. This thesis is fo-cused on the propulsion system of the mentioned EV. In order to save as much energy aspossible, a novel approach is taken. The designed motor willbe located inside the wheel

3

Chapter 1. Introduction

(motor hub), enabling interesting advantages, such as the removal of the gearbox in thedrive train.

4

Chapter 2

Low-Speed Direct-Drive Applications.In-Wheel PMSM

The following chapter presents the most suitable alternatives to propel the Shell EcoMarathon vehicle. Several PMSM topologies are put forward,setting out their advan-tages and drawbacks. Finally, based on the literature exposed and concerning designingconstraints, a configuration is selected.

2.1 In-Wheel Motors

Nowadays, vehicles that use electric drive trains can locate the motor in different placesinside the car. A novel and original trend is to locate it inside the wheel, where the trans-mission path between the electric machine and the wheel can be reduced and even elimi-nated, avoiding a big source of losses. The fact that determines whether the transmissionis reduced or eliminated is the speed of the machine attachedto the rim [6]. Furthermore,the fact of bedding electric machines inside the hub gives the chance of obtaining bet-ter control over the driving wheels. Since IW vehicles usually have a motor hub in eachwheel, they can be controlled independently, performing the necessary amount of torquerequired at real-time. This is essential for vehicles that face traction problems, such asslippery surfaces or damps.

Nonetheless, setting an IW motor entails some problems. Theunsprung mass, whichrefers to all components not supported by the car’s suspension, is increased with thesetypes of motors, and this leads to problems with the suspension and steering, since it isnot shielded from bumps and potholes. Though there are some articles which deny thisunsprung mass problem and they conclude its analysis asserting that the loss of comfort,safety and drive steering is imperceptible for an average driver [2].

5

Chapter 2. Low-Speed Direct-Drive Applications. In-WheelPMSM

Rim

Rotor

Wheel Bearing

Suspension

Housing

Stator

PermanentMagnets

Fig. 2.1 Protean Hub motor exploded view [5].

2.1.1 Low-Speed and High-Speed Machines

Electrical machines are designed to perform a certain quantity of mechanical power. Nev-ertheless, the dimensions of the machine are not straight determined through this param-eter. Mechanical power is defined as a product of the torque performed by the motor atsome rotational speed. Hence, in order to perform the required power, electrical machinescan be designed to develop a high torque at low speed or low torque at high speed. The factthat mainly defines the size of these machines is the torque, as it is displayed in equation2.1

T = KD2Lact = KVrot (2.1)

whereK is a constant which depends on the parameters of the machine,D is the diameterof the rotor,L is the active length of the machine andVrot is the volume of the rotor [26][8] [11]. Therefore, electrical machine design can be approached in two different waystaking into account the torque performed, and consequentlythe working speed:

• High-speed machines.(Fig 2.2) Since their working speed is high, a fixed speedgear becomes necessary to achieve the needed velocity in thewheel. A planetarygear is usually used in this kind of IW motor and as all gearboxes, it is a source oflosses [6]. During the last few years, Michelin designed andpresented in 2008 its”active wheel” concept, as example of this type of technology [1].

• Low-speed machines.(Fig. 2.3) The torque that runs these machines is high, thusthey are bigger and heavier than high-speed machines. Within IW devices, the rotoris directly coupled to the wheel rim, therefore no gearbox isrequired [6]. As anexample of this approach of hub motor, Siemens designed the ”eCorner Module”[25].

6

2.2. PM Synchronous Machines

Brake disk

Electrical machine.30 kW

Brake Caliper

Electrical suspensionmotor

Suspension spring

In-wheelactive suspension

a) b)

Fig. 2.2 a) Michelin Active Wheel [1]; b) Cross-section of high-speed IW [6].

Another key point is that most of the commercial applications require a low rotating speedand a high torque in the load, i.e. wind mills, elevators, EVs. Thus, high speed machinesare usually coupled to a gearbox, wasting energy that could be useful. The use of directdrives, which are low-speed electric machines, entails several advantages mainly relatedto the removal of the gearbox [16] such as:

• Higher efficiency.This is the fact that justifies the implementation of this devices.The elimination of the gearbox take out the losses due to friction.

• Reduce maintenance.Gearboxes are one of the main causes of failures of me-chanic systems. Moreover, they do not need to be lubricated as regularly as theones with gears.

• Higher reliability.

• Reduction of noise.Gearboxes are noisy since the teeth of the gears collide at highspeed. Again, this problem is reduced with IW motors.

• Simpler design.The number of pieces on the power train is reduced.

2.2 PM Synchronous Machines

2.2.1 Permanent Magnets

There are two types of magnetic materials, soft materials and hard materials. On the onehand, soft materials, also called ferromagnetic materials, can be easily magnetized and

7


PMSM

Active suspension

Brake

Active steering

Fig. 2.3 Siemens eCorner [1] and low-speed hub motor scheme [6].

demagnetized and they are used to facilitate the magnetizing guidance. Laminated steelis an example. On the other hand, hard materials are magnetized and demagnetized withdifficulty. They are characterized by a wide hysteresis loop[7] [19]. Regarding the micro-scopic structure of hard materials, they are characterizedby the anisotropy of the materialand the shape of the crystals they are composed of, which impedes the orientation of itsWeiss domains, once they are oriented. This is the reason, from a microscopic point ofview, for a high coercivityHci and a high magnetic hardness [11]. Hence, they can bemagnetized and produce magnetic field in the air gap without the use of excitation wind-ing. Permanent magnets (PMs) are magnetized in quadrant I (or III) of the B-H hysteresisloop (Fig. 2.4) and they operate in quadrant II (or IV). Data sheets that manufacturingcompanies normally provide only depict the II quadrant. Within Fig. 2.4, the main param-eters which characterize a PM are described [7] [11].

• Saturation magnetic flux density,Bsat and Hsat. All the magnetic moments ofthe PM domains are aligned in the direction of the external magnetic field applied.

• Remanent magnetic flux density or remanence,Br. It is the magnetic flux densitywhen zero magnetic field intensity is applied,H = 0. The higher remanence, thehigher magnetic flux density in the air gap can support the magnet.

• Coercive field strength or coercivity,Hc. This is the value of the magnetic fieldapplied to bring the magnetic flux density to zero. The higherthis value, the thinnermagnets can be implemented to withstand the same demagnetizing field.

• Intrinsic demagnetization curve,Bi(Hi). Bi = B − µoHi

• Intrinsic coercivity, iHc. It is the value needed to permanently demagnetize thematerial.

8


H

B

Operation(IV)

Operation(II)

Magnetizing(I)

Magnetizing(III)

Br

Normal

curve

B –Hm m

Intrinsic demagnetizationB -Hi i curve

HcHci

Bknee

Hknee

recoil line

Fig. 2.4 B-H Hysteresis loop and characteristic parameters for a permanent magnet [19].

• Knee magnetic flux density,Bknee. Once this threshold is exceeded, the materialenter in a region where it is demagnetized.

• Recoil magnetic permeability,µrec B = Br + µrecµoH

• Maximum magnetic energy per unit of volume,(BH)max. This value shows arelative measure of the strength of the PM.

• Temperature coefficients ofBr and HcJ . Magnets are temperature sensitive, andtheir properties vary fromTamb = 20C to the operating temperature.

• Resistivity.PM are exposed to magnetic field variations in electric machines, there-fore, they are a source of eddy currents. It is important thatthis value is high.

• Chemical characteristics.There are some hard materials which cannot be usedin corrosion atmosphere. NeFeB magnets have low corrosion resistance, thus theyreact quicker in high temperature or humidity conditions. Furthermore, adhesiveswith acid content must be avoided, since they lead to fast decomposition of the PM.SmCo Magnets are more tolerant with corrosion issues [7].

PMs are designed to operate in the recoil line (Fig. 2.4), thelinear part of the demag-netization curve betweenBr andHc. Hence, the operating point is located along thementioned line. The linearity of this recoil line finishes when the magnetic field den-sity reachesBknee. If B is higher than this value, magnetizing and demagnetizing cyclesare reversible. IfB is further reduced, the magnet is partially demagnetized and it falls

9


down in a new recoil line and thus, in an irreversible cycle. It is important to highlightthatBknee depends on the temperature: the higher the temperature, thehigherBknee andtherefore, the easier demagnetization as is illustrated inFig. 2.5 [19]. In general, regard-

NeF

eB

SmCo

Ferrit

e

AlN

iCo

B

H

0.2

0.4

0.6

0.8

1

1.2

1.4

0.2

0.4

0.6

0.8

1

1.2

0.2

0.4

0.6

0.8

1

1.2

-200-400-600-800-1000-200-400-600-800

B

Hfield strength(kA/m)

flux d

ensi

ty(T

)ΔHc

ΔBrem

ΔT20ºC

70ºC120ºC

flux d

ensi

ty(T

)

field strength(kA/m)

Fig. 2.5 a) Effect of the temperature on PM demagnetization curve. b)Demagnetization curves ofcommon permanent magnets at 20oC. Adaptation from [19].

ing the properties shown above, depending the application which the PMSM is built for,a good PM has to have a good trade-off wide hysteresis loop, highBr, high (BH)maxand elevate working temperature since the operating point has a deep impact on the PMcharacteristics. In Table 2.1 general properties of the most common PMs are displayed.

Table 2.1: General properties of permanent magnets [19]Br Hc TCurie Tmax density (BH)max ∆Br / ∆T[T] [kA/m] [ C] [ C] [kg/m3] [kJ/m3] [%/C]

Ferrite 0.38 250 450 300 4800 30 -0.20AlNiCo 1.20 50 860 540 7300 45 -0.02SmCo 0.85 570 775 250 8300 140 -0.04NdFeB 1.15 880 310 180 7450 260 -0.12

2.2.2 Advantages and Drawbacks

The principal difference of PMSM relies on the rotor, since the stator used is the same asthe normal AC machines stator. PMs are used as excitation system in the rotor substitutingthe electromagnetic excitation. This can lead to an improvement of several characteristicson synchronous motors [20] [19].

• Higher torque density and/or power density.The space occupied by the PMs inthe rotor is smaller than the electromagnetic excitation. Hence, the machine can bemore compact performing the same power and torque.

10


• Higher efficiency. Copper losses caused by the electromagnetic excitation arere-moved. Nevertheless, eddy currents are induced in the PM (itis a conductive mate-rial) due to the variation of the magnetic field on them. However, these losses canbe minimized by cutting axially the magnets [11].

• Simplification of construction and maintenance.The absence of conductors inthe rotor makes the construction easier. Moreover, the rotor is more solid and themaintenance is easier as well.

• Better dynamic performancedue to higher magnetic flux density in the air gap.

Nevertheless, the use of this materials as excitation system involves also some disadvan-tages [20]:

• The high price of PM. The price of NeFeB has increased during the last years,reaching a peak of 800% of its value in 2007. At the present, ithas been slightlyreduced to values between 300-200% compare to the one in 2007as it is depictedin Fig. 2.6.

Fig. 2.6 Neodymium price evolution [18].

• Magnets are temperature sensitive.As depicted in Fig. 2.5 the properties of thePMs are deteriorated when the temperature rises. This can vary the properties of themachine during the operating function. It is important to avoid magnets trespassingcertain temperature since they can be permanently demagnetized, as it has beenmentioned above.

• A sensor position or rotor angle estimation method is required.

11


• Excitation cannot be controlled directly.

Depending where the PM is placed in the rotor, it leads to a variation on the magnetizationdirection (radially or axially). Therefore, there are several topologies of PMSM. The mostsuitable for the PMSM which is designed in this thesis are theRFPM and the AFPM.

2.2.3 Radial Flux PM Synchronous Machine (RFPM)

RFPMs are the most common PMSM due to their rotor simplicity and their similarityto the synchronous AC motors. The stator of this machines is the same one as in thesynchronous AC motors. As has been mentioned above the difference lies in the rotor.Since the electromagnetic excitation is removed, the direct control of the excitation froma DC disappears. Instead, the flux is controlled with the stator currents [16]. Fig. 2.7shows the path followed by the flux through the airgap. It crosses radially the airgap. Thecurrents of the stator flow in the same direction of the rotating axis of the machine.

Iron

PM

shaft

rotation

current direction

flux direction

Fig. 2.7 Flux, current direction and rotation in a Radial flux PMSM [16].

Rotor Configurations

Notwithstanding, despite the stator is the same as in the synchronous AC machines, therotor configurations can assume different topologies depending where the magnets arelocated. They can be divided in three main groups [19]:

• Surface-mounted PM machine (SPM).(Fig. 2.8a) The main advantage of thesemachines is the simplicity of construction and therefore, aconsequently reductionof cost compared to other magnets topologies. Nevertheless, this configuration ex-poses the PM to demagnetization fields. BesidesLd ≃ Lq, since the relative perme-ability of the magnet is similar to air (µr = 1− 1.05), the rotor is isotropic. Hence,this configuration is not able to perform reluctance torque [16].

12


a) c)b)

Fig. 2.8 Different RFPM synchronous motors. a) SPM motor; b) Inset PMmotor; c) IPM motor..

• Inset-PM machine. (Fig. 2.8b) The rotor is similar to the rotor of SPM motors.The difference is that there is iron between two adjacent PMs. Since the relativepermeability of the iron is higher compared with the magnet permeability, the rotoris anisotropic. Thus,Ld 6= Lq and the torque is composed by two sources: the torquecomponent that the SPM motors exhibit (PM torque) and the reluctance torque.

• Interior-PM machine (IPM). (Fig. 2.8c) The rotor is completely different to theothers shown above. PMs are located inside the rotor, and they can be adaptedto different shapes depending on the direction of magnetization inside the rotor,which can be tangential or radial. All IPM motors are characterized by their abilityto perform reluctance torque due to the different permeancepaths that they exhibit.Moreover, they can achieve a high open-circuit air gap flux density, specially theones magnetized tangentially, and they are protected against demagnetization andmechanical stress. However, they are not as simple as the other machines, and thusleads to more expensive solutions.

Despite IPM and Inset-PM motors seem to exhibit better properties for the purpose of thisthesis, SPM synchronous topology is the final choice due to its simplicity, its solid and itscost.

Inner vs Outer rotor

The rotor of RFPM machines can be placed as the outside or the interior part of the PMSMas depicted in Fig. 2.9. Both of them have different properties, which are important to takeinto account [16].

• Suitability for mechanical drives. Outer rotor design is usually a better solutionfor applications such as IW motors and wind turbines. The rotor can be directlyattached to the wind turbine or the rim. This makes the systemreally compact.

13


• Thermal design. In some applications it is a critical aspect. As it has been men-tioned above, magnets are temperature sensitive and winding isolation cannot sur-pass certain temperature (Isolation H can handle a maximum temperature of180C).Otherwise, both of them can be damaged. The differential factor between them liesin where the main source of losses is located, the stator windings. Inner rotor config-urations are easier to refrigerate, as the cooling surface which is directly in contactwith the copper is larger and directly attached to the aluminium housing. Outer rotorcommercial designs are usually refrigerated by inner coolant.

• Magnetic design.Outer rotor PMSM usually has largerD for the same externaldiameter, as it is illustrated in Fig. 2.9. Since the torque is directly proportional tothe square value ofD and the activeLact (equation (2.1)) the axial length of outerrotor designs is smaller. Hence, a reduction of active weight is achieved. Outer rotorPMSM motors are usually 15% lighter than inner rotor designs[16].

airg

apd

iam

eter

Fig. 2.9 Comparison ofDgap in Inner and Outer design. Adapted from [16].

• Winding wound. It is slightly easier to wind outer rotor than inner rotor designssince the teeth point outwards.

• PM detachment. PMs withstand centrifugal force due to the rotation of the ma-chine. This force is proportional to the distance they are located from the rotationaxis and to the square of the speed. Despite the air gap diameter being usually higherin outer rotor design, magnets push against the rotor yoke. On the other hand, innerrotor magnets are only glued to the rotor surface, and there is not iron part opposingforce. That is for this reason that they are usually bandagedor use some protectionto avoid the detachment.

2.2.4 Axial Flux PM Synchronous Machines, (AFPM)

Unlike RFPM machines, the flux inside this machine flows axially through the air gapwhile the currents flow in the radial direction [16]. This type of motor is commonly usedin solar car competitions, due to its high efficiency and low weight, as the Aurora Car. Itperforms efficiencies close to 98% and its torque density is the highest among all types ofarchitecture [21]. AFPMs have several advantages over the counterparts RFPMs [10] [14]:

14

2.3. Summary and Conclusions

Windings

Stator Iron

Rotor Iron

PM

Shaft

Direction of magnetization

Current direction

Rotation

Fig. 2.10 Flux, current direction and rotation in a Axial flux PMSM. Adapted from [16].

• Higher power density. RFPMs do not exploit the rotor yoke, which is hardly usedas magnetic circuit. AFPMs take advantage of the machine’s iron in a better waythan RFPMs, thus it is possible to design more compact machines.

• Planar and adjustable air gap. AFPMs have the capability of adjust the air gap.

• Lower vibrations and noise levels.

• Better heat removal. Since the inner diameter of AFPMs is greater than the RFPMs,their capability of ventilation and cooling is expected to be better.

• Larger diameter to length ratio relation. AFPMs usually have larger diameters thanRFPM but an smaller axial length.

• Higher number of poles. Since the outer diameter of the core is larger, a highernumber of poles can be placed, making this choice really interesting for low-speedapplications.

Its main drawback lies in the manufacturing procedure. The manufacturing process ofthe stator is difficult, due to the need of avoiding eddy currents, and thus, it is extremelyexpensive.

2.3 Summary and Conclusions

Throughout this chapter, the different possibilities for an IW motor are raised, both motorarchitecture and different types of PM motors. Regarding the motor architecture, in orderto remove losses and increase the efficiency of the system, the most suitable option is an

15


IW low-speed machine directly attached to the rim. This removes any gearbox from themotor to the rim, removing unnecessary losses.

Related to the type of machine selected, it is important to take into account that themachine is air-cooled. It is an essential fact to be aware of during the design procedure,since the magnets and the winding insulations cannot reach certain threshold temperatureof (150C), as well as the difficulty of attaching the rotor to the carbon fiber rim which iscurrently constructed for the ”Shell Eco 2012”. Thus, RFPM with an outer rotor design isdiscarded since the risk of thermal heating is higher than inthe two proposed alternatives.Between the remaining designs, AFPMs provides more advantages than the RFPM ofinner rotor design. AFPMs are lighter since their power density ratio and their coolingsystem is better. Vehicles involved in solar car contests, as the ”Aurora”, usually usethem. However, since the manufacturing cost of the stator isextremely high, it is alsodiscarded. Therefore, the machine selected is an RFPM of inner design. This is the mostconservative and solid solution for accomplishing the goalof a light-weight and efficientdesign. Despite RFPM is not the best solution in terms of light-weight and efficiency, ithas a good trade-off power and torque density as well as a goodefficiency ratio as well asit is the harder machine.

16

Chapter 3

Shell Eco Marathon Circuit Analysis.Requirements

In this chapter, the main requirements of the machine are calculated. 1) Peak torqueTpeak, 2) Steady state torqueTsteady and 3) Maximum rotational speedωmax. Based on anideal simulation of the vehicle and its maximum requirements, these are estimated. Thisestimation of the parameters is done based on a trade-off reduction of energy per cycleand feasibility of the machine

3.1 Track and Car Requirements

Before calculating the motor requirements, which have to bedefined as torque and speedvalues, the contest requirements and the circuit characteristics are depicted to further meetthe machine requirements. In Table 3.1 the principal characteristics of the ”Ahoy UrbanCircuit 2013” are summarized. Two possible vehicle concepts are available in this com-

Table 3.1: ”Ahoy Urban Circuit 2013” characteristics [24].CircuitDistance 1630 [m]Slope 0 [%]90 corner 5

petition and both have to fulfill different requirements. This thesis focuses on the ”UrbanConcept”. Regarding ”Shell Eco Marathon” rules, the machine that will be designed hasto fulfill several conditions, taking into account some relevant constructive aspects of therace, listed below:

• 10 laps of 1630 meters at 25 km/h. That means 234 seconds per lap.

• Reduction of energy consumption as much as possible.

17

Chapter 3. Shell Eco Marathon Circuit Analysis. Requirements

Table 3.2: Different vehicle concepts [24].Prototype Concept Urban concept

Wheels 3-4 4Min Driver Weight 50 [kg] 70 [kg]Max Vehicle Weight 140 [kg] 205 [kg]

• Be able to drive through corners. It would be desired not to use the breaks duringcornering.

• The ratio between peak torque and steady state torque cannotbe extremely high.The machine is designed taking the steady state torque as a reference. If the peaktorque is so high it is possible that the machine cannot perform this due to the ironsaturation.

In order to further determine the car requirements, it is assumed that some characteristicsof this vehicle are the same as the last ”Shell Eco Marathon Urban Concept 2012”. Like-wise, some parameters are expected and estimated by the Shell Eco Marathon Team, suchas the mass of the vehicle. These parameters are detailed in Table 3.3.

Table 3.3: Parameters of the Shell Eco Marathon 2013.Vehicle Characteristics

Mass 110 [kg]Drag coefficient,cd 0.3Rolling resistance coefficient,µrr 0.014Frontal Area 0.8 [m2]Number of driven wheels 1Radius of the wheel 275 [mm]

3.1.1 Driving Cycle

First of all, the speed cycle has to be determined. The principal aim of this cycle is toperform a good trade-off between two of the issues describedabove: the reduction ofenergy consumption and the torque ratio. Moreover, since the maximum speed is notexpected to be higher than 30 km/h, the car should not need to use the brake in any ofthe corners of circuit. Thus, the shape of the cycle proposedis illustrated in Fig. 3.1.It has to be pointed out that this simulation cycle is an idealcycle which does not takeinto account some important matters as the efficiency map of the motor, points which theconverters works at highest efficiency, and the importance of the braking energy recovery.This simple cycle is the most suitable for the competition since there is only one period

18

3.1. Track and Car Requirements

V(t)

tΔt

T=234 sΔt

vav=25km/h

Fig. 3.1 Driving cycle.

of acceleration and a deceleration one. In terms of energy consumption, this is a goodchoice because a lot of energy is used during acceleration and not all is recovered duringthe deceleration period, since the energy from the wheels tothe battery have to passthrough some elements which generate losses, as the motor orthe converters. Regardingthe peak torque issue, it can be reduced by tempering the speed profile at the final part ofthe acceleration slope. This can be decreased if the final part of the acceleration slope istempered as in Fig. 3.2.

V(t)

t

Fig. 3.2 Tempered driving cycle.

3.1.2 Cycle Optimization

Once the main shape of the driving cycle is selected, it has tobe optimized to accomplishwith the requirements exposed above. In order to optimize this cycle and calculate the mo-tor requirements, an ideal simulation of the vehicle is carried out using SIMULINK R© . Theenergy consumption during the complete cycle, the energy consumption of each contribu-tion of power (dynamics, drag resistance and rolling resistance) and the torque profile areestimated. Furthermore, the maximum speed (or the steady-state speed in this case) thatthe vehicle reaches is calculated. The equations which thissimulation model is based onare attached in Appendix A. The main difference between the diverse driving cycles is the

19


acceleration and deceleration time,∆t (both periods are considered equal, because it is anideal simulation). Since the average speed is fixed if∆t varies, consumption,Tpeak, Tsteady,energy consumption distribution and other magnitudes change. Thus the study is carriedout throughout∆t = 1, 2...234/2. Facing the energy consumed and theTpeak in every cy-

0 50 100 150 2000

20

40

60

v(k

m/h

)

time (s)

Fig. 3.3 Different driving cycles. Average speed of both cycle is thesame.

cle, one can realize that there is a wide area where the energyconsumption is minimum,below 120 Wh as it is illustrated in Fig. 3.4. Regarding Fig. 3.2 and Fig. 3.3 this minimumregion is extended form 20 Nm to 70 Nm. This interval ofTpeak corresponds to aTsteadyof 6-7 Nm, as is illustrated in Fig. 3.5. Since the machine cannot be overloaded more than3-4 times its nominal torque value, and theTsteady is around 7 Nm, the feasible area isthe one belonging to 20-30 Nm ofTpeak. Moreover, the maximum speed that is reachedin this area is lower than the one reached in speed cycles withsmaller peak torque, as isFig. 3.5 shows. That means that the losses related to frequency issues would be reduced(Iron and magnet losses).

0 10 20 30 40 50 60 70110

120

130

140

150

E(W

h)

Tpeak

d)

Fig. 3.4 Relation between energy consumption and peak torque.

20

3.2. Analysis of the Energy Consumption

3.2 Analysis of the Energy Consumption

The energy supplied by a battery is not only used to move the car, but also to overcomethe drag and the rolling resistance. In the whole process of the energy transformation,there are some losses in the electrical system (iron, copperand semiconductors losses)as well as mechanical losses (bearings and gears). This section shows how the energyis ideally used (without losses), and its distribution in the different cycles. RegardingApp. A, the power performed by the machine is divided in threecomponents, the drag,the rolling resistance and the dynamic power, and consequently, the torque. Looking atFig. 3.6, the energy used to overcome the rolling friction isconstant since it only dependson the gravity and the mass of the car. However, the energy consumption due to the dragincreases almost the double since the drag power depends on speed. App. A illustratesthe relation between the energy consumption and the interval of acceleration for a sameaverage speed. Likewise, in Fig. 3.6, one can realize that the energy consumption of theacceleration period increases when the interval of acceleration is higher. The vehicle willnot be able to recover the whole amount of dynamic energy since there will be losses inthe motor, the power electronics, despite the fact that in this ideal simulation the dynamicenergy consumption is zero as illustrates Fig. 3.6. Furthermore, Fig. 3.4 illustrates theTpeak tendency to decrease quite quickly, maintaining the energyconsumption in similarvalues, then, it reaches a point whereTpeak is stabilized and the energy consumption rises.As a result of this, some points with the sameTpeak consume different energy. This canbe explained by looking at Fig. 3.7. For small acceleration periods, the dynamic compo-nent of the torque leads the torque composition used to move the car. However, as longas this acceleration interval increases, dynamic featuresdecrease since the accelerationneeded is smaller and the drag component continuously increase. For big time intervalsof acceleration, where the maximum speed of the cycle is almost double than the smallerperiods, that the drag component is the one that leads the torque distribution, and thereis an acceleration interval which is a minimum (Fig. 3.7). Hence, there are points withdifferent energy consumption and the sameTpeak. The dynamic component of the torquedecreases and the drag component increases quicker, as wellas the energy consumed bythe drag resistance increases raises as well.

Mass and Area Sensitivity

An important issue to analyze is how penalizes, in terms of energy consumption, theincrease of weight or area of the vehicle, or on the other hand, what benefits their re-duction can provide. The energy consumption of every sourceof energy per lap thatthe motor has to face is calculated in Appendix A. Mass and area are linearly depen-dant to energy consumption. Therefore, the sensitivity of increasing the mass,∆mv, isαmass= µrrgvavTnlap = 0.6695 Wh/kg and the sensitivity of increasing the frontal area∆Af is αarea = 54.422 Wh/m2. Both sensitivities have been calculated using∆t = 15s.

21


Thus, if this value changes, so does the sensitivity.

3.3 Conclusion

The selection of the driving cycle for the ”Shell Eco Marathon 2013” has been proved tobe an important part of the design of the motor. Depending on the cycle selected, differentrequirements for the machine are needed and varied energy consumptions are achieved.The cycle selected is the one with an acceleration period of∆t = 15. The correspondingdesign parameters are illustrated in Table 3.4

Table 3.4: Requirements for the PMSM.Requirements

Tpeak 22 [Nm]Tr 7 [Nm]ωr 275 [rpm]

22

3.3. Conclusion

0 20 40 60 80 100 1200

5

10

15

0 20 40 60 80 100 1200

200

400

600

0 20 40 60 80 100 1200

25

50

75

0 20 40 60 80 100 120100

125

150

a)Tsteady(N

m)

b)

ωmax(r

pm

)Tpeak(N

m)

c)

E(W

h)

∆t(s)

d)

Fig. 3.5 a) Variation ofTsteady with the interval of acceleration; b) Variation ofwmax with the in-terval of acceleration; c) Variation ofTpeak with the interval of acceleration; d) Variationof the energy consumption with the interval of acceleration.

23


0 20 40 60 80 100 120

0

20

40

60

80

En

erg

y(W

h)

∆t (s)

Fig. 3.6 Different energy components. a) Drag: GREEN; b) Rolling: RED; c) Dynamic: BLUE;d) Dynamic peak: MAGENTA.

0 20 40 60 80 100 1200

4

8

12

16

20

T(N

m)

∆t(s)

Fig. 3.7 Evolution of the different sources of resistance torque, atoverload conditions. BLUE:Peak torque; GREEN: Drag resistance torque; RED: Rolling resistance torque; MA-GENTA: Acceleration torque.

24

Chapter 4

Non-Overlapping ConcentratedWindings Analysis

Along this chapter concentrated windings are described. Different methods to calculatetheir winding layout are explained as well as the fundamental winding factor. Furthersome undesirable effects due to the PM and the winding layoutare depicted and it isexplained how to avoid them. This chapter mainly shows how tochoose a good configu-ration of poles and slots.

4.1 Concentrated or Distributed winding

The type of stator winding used in an electrical machine is determined by the numberof slotsQs, the number of polesp, and the number of phasesm [12]. These parametersof the machine determineq, the number of slots per pole and phase (4.1). Depending onwhether the value ofq is an integer or a decimal, the winding type is different.

q =Qs

mp(4.1)

• Distributed winding. (q = integer). It is the conventional winding. The higher thevalue ofq, the more sinusoidal MMF-wave is produced by the windings [23]. Thefundamental winding factor,kw1, in this type of winding is 1 whenq = 1.

• Concentrated windings.(q 6= integer). The machines that use this type of windingcan have each coil wound around one tooth. Depending the typeof winding, theycan be single-layered or double-layered as Fig. 4.1.

Concentrated winding presents better properties for low-drive applications.

• Short end-windings. Since copper losses are much higher compared with ironlosses for frequencies lower than 100 Hz (low-speed applications) [12], it is im-portant to reduce them as much as possible in order to increase the efficiency of

25

Chapter 4. Non-Overlapping Concentrated Windings Analysis

A A

A

A

B

B

B

C

C

AA

A

A

AA

C

B

C

C

C

B

B

A

A

A

Fig. 4.1 End winding distribution of different winding topologies.a) Distributed winding withQs = 36, p = 12 andq = 1; b) one-layer concentrated winding withQs = 12, p = 8andq = 0.5; c) double-layer concentrated winding withQs = 12, p = 8 andq = 0.5 Slotdivided vertically. While one-layer concentrated windingwound alternate teeth, double-layer concentrated winding wound all the teeth in the machine.

the motor. Concentrated windings contain smaller end-windings compared to dis-tributed windings, since they are wound around one tooth. Thus, there is less copperin the machine and the losses decrease [16] [12], as they are directly related to thevolume of copper, as it is exhibited in equation (5.41).

• Low cogging torque.This oscillatory torque which can be seen in absence of cur-rents, is owing to the tendency of the magnets to line up with the stator trying tomaximize the permeance of the magnetic circuit from the standpoint of the perma-nent magnets [16] [12]. According to [20], it is possible to realize that the rippleis quite small in concentrated windings compared to distributed windings, whichusually need skewing on the rotor.

• Good fault-tolerant capability. Distributed windings are overlapped, hence thereis a risk of failure if the insulation fails. On the other hand, since the coils in con-centrated windings are wound around one tooth, the contact between conductors ofdifferent phases is drastically reduced, even eliminated in single-layer winding.

• High constant power speed can be achieved[20].

• Insulation and manufacturing systems are easier in concentrated windings [16][12].

• Reduction of mutual coupling among the phases[9].

26

4.2. Double or Single Layer

4.2 Double or Single Layer

In Fig. 4.2 cross-sectional area of of the stator is shown. Itcan be appreciated the dif-ferent wound between single-layer and double-layer windings concentrated windings.Depending on the final application of the machine, it is preferred one of these type of

a) b)

Fig. 4.2 Teeth in both type of windings. a) single-layer. b) double layer [16].

winding or the other. They exhibit different characteristics which can adapt better for aparticular application. Characteristics shown in Table 4.1 are the most important onesconcerning this thesis. As it has been mentioned above, light-weight and efficiency are

Table 4.1: Comparison of properties between single-layer and double-layer windings [16]

.

Single-layer Double-layerFundamental winding factor higher lowerEnd-windings longer shorterEddy currents higher lowerOverload Torque capability higher lowerHarmonic content of EMF higher lowerTorque ripple higher lower

critical in the design of this motor. In order to accomplish both objectives, double-layerwinding seems to be more appropriated. Since the end-windings are shorter and the eddycurrents computed in the double-layer winding are lower, the losses that this machine per-forms during its operation are significantly lower than its counterpart single-layer wind-ing. Moreover, the value of the torque ripple is lower in double-layer windings and thereare more possible combinations of slots and poles [16]. However, it is important to remarkthat the overload torque capability is lower in double-layered windings. This point is im-portant in this design because, as it will be further described in Chapter 5, the machine isdesigned for the steady-state, and it has to be able to reach apeak torque between three orfour times the steady-state torque, as mentioned in Chapter3.

4.3 Concentrated Winding Layout

Unlike distributed windings, concentrated windings perform a winding factor below theunit. This means that due to the different distribution of slotsQs and polesp, the electro-

27


motive force (EMF) induced in each phase is not the addition of the absolut value of theEMF induced in each conductor, since they are phase shifted.According to the Faraday’sInduction law [11]

e =dψ

dt= nskw1

dφ

dt(4.2)

wheree is the electromotive force induced. This entails an effect on the torque perfor-mance, since the fundamental winding factor is directly related to the torque. Therefore,the stator needs to be fed with higher currents in order to deliver the same torque com-pared with those with larger winding factor. The fundamental winding factorkw1(Qs, p),depends on the number of poles and the number of slots, as wellas the winding layout.But there are other properties of the machine that also depend on this configuration, suchas the torque ripple and the unbalanced magnetic pull. Thus,the selection of the numberof poles and slots the PMSM is designed for, is a critical issue to perform a good trade-offof all properties described above [16] [30].

First of all, two methods are proposed to place the conductors of each windingphase. Both methods dispose the conductors following some key rules taking into accounteach configuration ofQs andp:

• Obtain the maximum amplitude of the main EMF harmonic waveform. This meansthe highest winding factor possible.

• The waveform has to be equal in each phase.

• The phases of the winding have to be electrically displaced by 2π/m radians,among them.

Winding Layout Using Cros’ Method

This method is based on the decomposition ofq in its most simplified fraction. It isbased on the method used in large synchronous machines with afractional value ofq.The method is described in the following steps and it is illustrated in Fig. 4.3 [16] [13].The parameters used in this example areQs = 18 andp = 14, as well as a double-layerwinding.

1. Simplification ofq. The number of slots per pole per phase,q, is canceled down toits lowest terms.q = b

c; d = c− b.

2. Partial one-zeros sequence. In this step, a sequence ofb ones (”1”) andc− b zeros(”0”) is distributed as regularly as possible.

3. Complete ”ones-zeros sequence”. The sequence obtained in the last part, is repeated3p/d = Qs/b times, and is compared to the layout of the distributed winding,q = 1.

28

4.3. Concentrated Winding Layout

4. Comparison to distributed winding layout. As it is illustrated in Fig. 4.3 conductorsfrom the distributed winding that corresponds to ”1” in the ”one-zero sequence” arekept and they form one side of the double-layer layout. The other side is obtainedby writing the returning coil in the other side of the tooth.

5. S vector. A vector~S that describes the layout of one phase of the machine is created.The final layout obtained is numbered by slots, from 1 toQs. For each slot whichcontains a conductor of the A phase, the number of the slot with the sign of the phaseis written in ~S. Thus,~S has2Qs/3 elements. This last step is essential to calculatethe fundamental winding factorkw1, as it can be appreciated in equation (4.8).

B'B A'C'CA

a) q=3/7=b/d3 x "1"

4 x "0"

b) 1010100

c) 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0...

A C' B A' C B' A C' B A' C B' A C' B A' C B' A C' B... '

ACB

B'B A'C'CA ACBA' B' C' C A B B' C'd)

e) S = [-1 1 -2 -5 6 9 -10 -10 11 14 -15 -18 ]

1 2 3 4 5 6 7 8 9slotnumber

A'

Fig. 4.3 Illustration of winding layout Cros’ method determinationfor Qs = 18 andp = 14

This method is not suitable to be used in single-layer winding, since there are some layoutswhich are difficult to be found [16]

Winding layout from the Star of Slot

This method, as the one explained above, is based on the design of large synchronous ma-chines with a high number of poles. Its principal aim is to maximize the main harmonicsof the EMF induced in the windings [4]. It illustrates the phasor representation of the mainEMF harmonic induced in each coil in each slot. Some parameters have to be calculatedbefore implementing this method [22].

• Machine periodicity.

t = GCDQs, p/2 (4.3)

29


• Number of spokes.

Nspk =Qs

t(4.4)

• Angle between two adjacent slots.

α =2π

Qs(4.5)

αe =p/2

Qs

α (4.6)

• Angle between two spokes.

αph =2π

Qs/t=

αep/2

t (4.7)

Moreover, it has to be taken into account that the winding is feasible only when there arethe same number of spokes per phase. Procedure [22]:

• Phase Region. Phasors’ space is divided into2m equal sectors, each sector cover-ing π/m radians. Opposite sectors belong to the same phase and each sector andits opposite have different polarity (Fig. 4.4). To determine the other phases, bothsectors have to be rotated by an angle of2kπ/m radians,k = 1, 2, ..(m− 1)

A-

B+ C-

B-C+

A+

Fig. 4.4 Phase Region.

• Star of slots. There areNspk spokes that have to be equally divided over the phasorspace. The angle between spokes isαph. One of the spokes have to be placed at 0o,coinciding with the common ”x axis”. After that, the phasorshave to be numbered.The first one has to be positioned in 0o. The angle between two adjacent slots isαe,so starting from phasor ”1”, the following have to be shiftedαe non-clockwiseQs

times.

30

4.4. Fundamental Winding Factor

1

8

6

4

2 9

7

5

3

1 4 7 10

2

5

8

11

36

8

12

10

11

12

13

1415

16

17

18

b)a)

αse

α αph se=

αph

Fig. 4.5 Star of Slots. a)Qs = 12, p = 8. With this parameters the results are:t = 4, αse =αph = π/3. Thus, there are 4 spokes with 4 phasors each; b)Qs = 18, p = 14. With thisparameters the results are:t = 1, αse = 7αph = 7π/9. Thus, there are 18 spokes with 1phasors each

• Overlaying both draws (Fig. 4.6), the phase and the sign of all coils are determined.The slots that are inside the different sectors of the phase regions, belong only to thisphase and sign. With this method, only one coil side is determined, the return sideof each coil is in the adjacent slot, with the opposite polarity. This method is widelyexplained in [4], and it is also extended to a single-layer winding. It is important tonotice that there are some configurations where some spokes are in the boundariesof the phase region, thus the phase and sign of these spokes are not determined. Asolution to this problem is to rotate the phase region in non-clockwise forαph/2.This happens whenαph = 1

kπ6; k = 1, 2..., N (for a three phase winding).

4.4 Fundamental Winding Factor

This parameter is one of the most important factors of this type of windings, since itdetermines how efficiently the conductors have been arranged throughout the stator. Itexplains how much fundamental flux of each harmonic the winding extracts from the fluxwave [30]. Several methods have been found to calculate the fundamental winding factorand their harmonics. One is based on the Cros’ method and it uses the vector~S calculatedin the last step of this method [16] [13]. Other analytical methods, which are based onthe proportion of slots and poles, are described in [30] [22], and they explain the differentfactors that are involved in this fundamental winding factor.

31


1

8

6

4

2 9

7

5

3

1 4 7 10

2

5

8

11

3

6

8

12

10

11

12

13

1415

16

17

18

b)a)

A-

B+ C-

B-C+

A+A-

B+ C-

B-C+

A+

Fig. 4.6 a) Layout of one coil-side:ABC|ABC|ABC|ABC; b) Layout of one coil-side:ACC−B−A−ACBB−A−C−CBAA−C−B−B

4.4.1 Winding Factor Based on Cros’ Method.

This method uses the EMF phasors to calculate the winding factor. Looking up to theCros’ method, in the last step a vector~S was calculated. This vector, which contains thenumber of slot that belongs to one phase of the machine, is used to obtain the electromo-tive force (EMF) phasor of conductor i from the phase A,~Ei, for the main harmonic orderof p/2 or the fundamental.

~Ei = sign( ~S(i))ejπp|~S(i)/Qs| (4.8)

Then, the fundamental winding factorkw1 is calculated as

kw1 =

∣∣∣∣∣∣

2Qs/3∑

i=1

~Ei

∣∣∣∣∣∣

nlQs/3(4.9)

wherei is an element of~S andnl the number of layers, since the analysis is carried out ina double-layer winding (nl = 2). Continuing with the example started above, the sum ofthe EMF phasors is calculated in equation 4.10 and illustrated in Fig. 4.8.

12∑

i=1

~Ei = ejπp

Qs

(

2− ejπpQs − e4jπ

pQs + e5jπ

pQs − 2e9jπ

pQs + e10jπ

pQs + e13jπ

pQs − e14jπ

pQs − e17jπ

pQs

)

(4.10)

The result for the of the fundamental winding factor iskw1 = 0.902

32


A-

B+C-

B-C+

A+A-

B+ C-

B-C+

A+ 1

2

3

4

5

6

7

8

9

10

11

12

a) b)

Fig. 4.7 a) Overlapping of spokes and boundaries of the phase region.b) Phase region shiftedπ/12. Qs = 12,p = 10. With these parameters the results are:t = 1, αse = 5αph =5π/6. Thus, there are 12 spokes with 1 phasors each. The layout of one coil-side is thefollowing one:ACC−B−BAA−C−CBB−A−

151411

10109

652

1 118

number ofslots

Fig. 4.8 Sum of EMF phasors for one phase of a 14 poles 18 slots SPM synchronous machine

4.4.2 Winding Factor Based on Pole-slot Combinations.

As is explained in [22] [30], the winding factor can be calculated as the product of thepitch factorkp, the distribution factorkd, and the skew factorks.

kw = kpkdks (4.11)

This method is detailed in [30] where it can be implemented toPMSM withm-phases andsingle-layer and double-layer winding independently. Factorskp andkd are expressed indifferent formulas between [30] and [22], but they are related by trigonometric relations;thus they are the same equation. The calculation of the different parts is detailed

• Distribution factor.Distribution factor is given by the ratio of the MMF performedin the concentrated windings compared to the distributed windings. Coils in thestator are displaced from each other by a certain electricalangle, each coil produce a

33


sinusoidal MMF with a shift angleγ. For distributed windings, this angle isγ = 2π

and for concentrated windingsγ 6= 2kπ (k = 1, 2..). Therefore, the sum of theMMF of the distributed windings is the multiplication of thevalue of MMF of onecoil and, on the other hand, it is a sum of vectorsNv, since they are displaced.kd isderived after some calculations [30].

kd =sin[(Nvγ)/2]

Nv sin(γ/2)(4.12)

Nv andγ have to be linked with the values of the machine. In order to validate 4.11for all the harmonic ordern, it is not as simple as multiplyingγ per the harmonicmultiple numbern′, wheren′ = n

GCD(Qs,p/2). Therefore, after some calculations

cited in [30], the distribution factor for eachnth harmonic ordern is derived from

kdn =sin(nπ/2m)

z sin(nπ/2mz)(4.13)

wherem is the number of phases andz:

z =Qs

GCD(Qs, mp)(4.14)

kdn is the distribution factor for thenth harmonic.

• Pitch factor or coil-span factor.The pitch factor is defined as the ratio between theflux linked by a short pitch coil compared to a full-pitch coil, as illustrated inFig. 4.9 [30]

kpn =Bnshort

Bnfull= sin(α/2) (4.15)

whereBnshort is the magnetic flux density produced by a short pitch coil andBnfull

slot pitch: α12

Fundamental flux distribution, B1

The 3-rd order harmonic, B3

Fig. 4.9 Flux density distribution to short-pitch coils

34


is the magnetic flux density produced by a full pitch coil. In order to calculate thepitch factor, the slot pitch angle,α12 is defined as

α12 =2π

Qs

p

2=

π

mq(4.16)

The coil-span angleε is illustrated in Fig. 4.10

ε = π − α12 (4.17)

To conclude, the pitch factor can be calculated for anynth harmonic order from [22]

α12α12

π-ε π+ε

π

ε/2

π

ε/2 ε/2 ε/2

a) b)

Fig. 4.10 a) Coil span is shorter than pole pitch ofπ; b) Coil span is larger than pole pitch ofπ

kp = cos(nε

2) (4.18)

• Skew factor.A PMSM usually suffers torque ripple, and it cannot be removedthrough the winding distribution, due to several issues further explained in sec-tion 4.5.2. In order to remove this ripple, the magnets or thestator windings aresometimes displaced a certain angle. The fundamental skew factorks is defined asthe ratio between the MMF produced by the machine skewed and the MMF pro-duced by the machine unskewed [30].

ks =Bnskew

Bnnonskew=

sin(σ/2)

σ/2(4.19)

whereBnskew is the magnetic flux density produced by the skewed machine,Bnnonskew

is the magnetic flux density produced by the non skewed machine andσ is the skewangle.

35


Based on both ways of calculating the fundamental winding factor, results are deployedin App. C. If these results are faced against the number of slots per pole per phaseq(Fig. 4.11), it can be realized that there are a lot of the combinations that are not suitablefor the design of concentrated winding because of their low fundamental winding factor.To achieve a fundamental winding factor higher thankw1 ≥ 0.866, q has to belong to arange of0.25 ≤ q ≤ 0.5. Thus, all the combination which are not within this range arenot considered further [22].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

kw1

q

Fig. 4.11 Fundamental winding factor as function of the number of slots per pole and phase,kw1(q)

Winding Space Harmonics

Depending on the pole-slot configuration, as well as performing different fundamentalwinding factors, different winding factors for all harmonic spectrum are achieved. Ac-cording to the definition of the EMF, this can be used , somehow, as a filter to avoiddifferent undesirable frequencies in the back EMF. Fig. 4.12 illustrates the winding fac-tors for all the frequencies in a machine with 18 slots and 14 poles [22].

4.5 Desirable Configurations of Poles and Slots

The selection of the number of poles and the number of slots isa critical issue when usingconcentrated windings. As mentioned above, not only the fundamental winding factor isinvolved in this selection, but there are some properties that are seriously dependent onthis configuration such as the torque ripple and the unbalanced magnetic pull. Being care-ful about this selection can avoid undesired forces which can create noise and vibrationsas well as the reduction of some sources of the torque ripple (cogging torque).

36

4.5. Desirable Configurations of Poles and Slots

0 10 20 30 40 500

0.5

1

kwn

nth

Fig. 4.12 Winding factor spectrum for a double-layer concentrated winding withQs = 18 andp = 14

4.5.1 Unbalanced Magnetic Pull

This is an important aspect in machines with concentrated windings, since they are a frac-tional winding type. Whenq is not an integer there are rotating attractive forces in RFPMmachines. As it has been mentioned above, noise and vibration depend on the selectionof the number of poles and slots. These are mostly due to the unbalanced magnetic forcesin the airgap. These forces are computed using the Maxwell stress tensor, using the fluxdensity distribution in a line along the airgap [16] [22] [9].

~F =

∫

S

1

2µo[B2

n(θ, t)−B2t (θ, t)]ds~n +

∫

S

1

µoBn(θ, t)Bt(θ, t)ds~t (4.20)

where equation (4.20) is the integral through the airgap,θ is the mechanical angle of thestator,t is the time. This force in the airgap has two components:

• Radial component.It is the one which mainly causes the noise and the vibrations.

~Fn =

∫

S

1

2µo[B2

n(θ, t)− B2t (θ, t)]ds~n (4.21)

• Tangential component.It is the component involved in the torque.

~Ft =

∫

S

1

µoBn(θ, t)Bt(θ, t)ds~t (4.22)

The importance of the pole and slot combination resides in the flux densityB along theairgap. As it has been mentioned above, the radial force depends on both componentsof the flux density in the airgap. Since the flux density waveform varies from one poleslot combination to another, radial magnetic forces can be not regularly distributed alongthe airgap, resulting in unbalanced pulling forces that generates noise and vibration inthe machine, since the rotor eccentricity is increased [16][9]. Moreover, is important

37


to take into account that these forces do not excite any resonant frequency of the ma-chine [22]. This unbalanced force is calledunbalanced magnetic pull. Machines whichexperiment this unbalanced magnetic pull are characterized by their lack of symmetry intheir winding layout, without taking into account the conductors orientation. Accordingto equation 4.21, magnetic forces are directly related to the square of the flux density,therefore the conductor orientation has no influence on these forces [16]. The periodicityof the winding layout and consequently, the structural periodicity on the airgap is givenby the greatest common divisor between the number of poles and the number of slots,X ′ = GCD(p,Qs). When the value ofX ′ is 1, there is a high resulting unbalanced force,since there is no symmetry on the winding layout. The combinations withQ = p ± 1

and some with oddQs are the ones which accomplish this value. Thus, they have to beavoided in order not to perform unbalanced magnetic pull in the machine, as illustratedin Fig. 4.13 [16]. The higherX ′ the machine configuration has, the better the balanceconcerning radial forces. SinceX ′ describes the symmetry of the radial forces, it can

Fig. 4.13 Magnetic forces in the stator of a PMSM with different pole-slot configuration. a) and b)68 poles and 69 slots; c)60 poles and 72 slots; d)10 poles and 12 slots. Figure extractedfrom [16]

be noticed that there is a periodic force waveform. The fundamental wave length of theattractive force can be computed as [22]:

τF,1 =2π

|Qs − p| (4.23)

38

4.5. Desirable Configurations of Poles and Slots

4.5.2 Cogging Torque and Torque Ripple

The torque performed by the machines is not perfectly constant, it oscillates around amean value, varying its value with the rotor position. It is caused by different sources [16]:

• The cogging torque.

• Variation of the permanence due to the magnetic saturation.

• Time harmonics. The inverter produces harmonics in the current stator windings.Therefore, a pulsating torque is caused when this interactswhit the rotor field.

• Space harmonics. The field produced by the PM contains some harmonics thatinteract with the magnetic field produced by the windings.

• Imperfections in the machine, such as not regular magnetization of the magnetsand rotor eccentricity.

The reduction of this torque ripple is important, since it isa source of noise and vibration,specially in low-speed drives. This vibrations can cause problems in the control system,since the estimation of the position and the speed of the rotor can be distorted [20]. In

Fig. 4.14 Definition of Cogging and Ripple torque [16].

order to reduce this ripple, several methods can be implemented. These methods try toreduce the source of torque ripple shown above [20] [19].

• Reduction of cogging torque. Several strategies can be adopted to reduce it.

39


– Correct selection of the number of poles and slots [16] [20] [19] [9].

– Skew the rotor magnets or the stator windings.

– PM Pole arc with different width.

– Notches in the stator teeth.

– Shifting of the PMs.

• Reduction of the space harmonics, changing the shape of the magnets, to performa more sinusoidal flux in the airgap.

• Feed the windings with pulsating currents which create a pulsating torque oppositeto the ripple [20].

Since the machine designed is based on concentrated windings, its ripple should be small,as [20] [9] point out. This thesis is only focused on the reduction of the cogging torque bythe selection of the correct number of poles and slots.

Cogging Torque

Cogging is the ripple torque which occurs in absence of currents in PMSMs, when therotor tries to align with the stator in order to maximize the permanence of the magneticcircuit regarding the PMs [9] [19].

Tcog =∂Wm

∂θm(4.24)

whereWm is the magnetic energy in the airgap. This variation is not homogenous along

PM

PM middlepoint

θm

Wm

τedge

θm

θm

Fig. 4.15 Elementary torqueτedge illustration due to the variation of the magnetic energyWm

along the relative position of the rotorθm

the airgap,Tcog has to be considered as the interaction of each edge of the rotor with the

40

4.6. Summary and Conclusion

stator slot openings. Since each PM edge is located in a different position regarding theslot opening, they have to be considered independently and then, sum their contribution.Fig. 4.15 shows the variation of theWm when the PM edge reaches the slot opening. Thismagnetic energy is formed by the contribution of the PM and the air, and as Fig. 4.15shows they depend on the angular position of the rotorθm. The variation ofWm is lowwhen the PM is in the middle of the slot or far away from it. Thus, normally there is a peakof variation in the edge of the PMτedge, although it is not a rule of thumb. So adding alltheτedgethat occurs along the airgap of this interaction between PM and teeth the coggingtorque is obtained [19]. Therefore, all the applications mentioned above that reduce thecogging torque are mainly focused in the counteraction of the different variations of theWm or avoiding the superimposing of them. One of the sorts of decreasing this source oftorque ripple is selecting an appropriate combination of the number of poles and slots,np,which is given by

np =p

GCD(Qs, p)=LCM(Qs, p)

Qs(4.25)

whereLCM(Qs, p) is the number ofTcog periods per revolution. The lowernp, the moreτedge will be superimposed (negatives and positives), yielding ahigh Tcog. On the otherhand, ifnp is high the elementary torquesτedgeare distributed and not superimposed alongthe slot pitch, performing a lowTcog. Therefore, combinations of lowGCD(Qs, p) or highLCM(Qs, p) are sought in order to perform lowTcog [9] [19] [12]. Another rules to selectthe correct number of poles and slots.

• The number of poles have to be an even number.

• The number of pole pairs cannot be multiple of the phase number, since this leadto unbalanced windings.

• The number of poles cannot be equal to the number of slots.

4.6 Summary and Conclusion

Properties of concentrated winding show that they are more suitable than distributed wind-ing in order to be built in the machine. However, the design ofthe winding layout is notas intuitive as the one for distributed windings. When a design process of a PMSM basedon concentrated windings is carried out, the selection of the number of poles,p and thenumber of slotsQs is critical to succeed. The selection of this configuration can lead tomachines with good performance characteristics as a high fundamental winding factorkw1and low cogging torque, as well as avoid some undesired effects as the unbalanced mag-netic pull. In Table 4.16 the parameters that have to be takeninto account in the selectionprocess of the winding layout are summarized.

41


High winding factor

Symetries in winding layout

Low cogging torque

Maximize torqueper Ampere

Avoid unbalancedmagnetic pull

Minimize coggingtorque

Fig. 4.16 Selection process of concentrated windings dependingQs andp, adapted from [16].

42

Chapter 5

Analytical Design

This chapter explains the magnetic and electric propertiesthat determine the dimensionsand the choice of some important parameters of the machine. An analytical model thatselects the dimensions of the machine is built, based on the theory framework exhibitedalong the previous, and the current chapter.

5.1 Geometric Design

Before developing the analytical model which selects the appropriate dimensions for thePMSM, in order to model the machine it is important to describe the relations among thedifferent parameters of the machine, such as the number of slotsQs, the number of polesp, dimensions of the different ferromagnetic portions inside the machine, as well as thevolume of the magnets and the size of the conductors. Fig. 5.1displays the more relevantconstructive parameters of a surface-mounted RFPM machine.

5.1.1 Magnet Sizing

An important parameter involved in the design of a PMSM is thefundamental flux densityexpected in the air gap,Bδ. This parameter determines the dimension of the PM in aPMSM, both thicknesshm, and coverage angle,2α. Assuming that the shape of the fluxdensity above the magnetsBm is rectangular, as illustrated in Fig. 5.2, and the absence offlux leakage,Bδ is given by the following expression [23]

Bδ =4

πBm sinα (5.1)

whereα is the half angle of the magnet of the magnet, defined in electrical degrees. Thispole angle2α is usually designed to120 since the highest flux density in the airgapper kilogram of magnet is obtained. I.e if the pole angle is increased to180 the magnetvolume increases by 50% and the airgap flux density only is increased in 14% [23]. The

43

Chapter 5. Analytical Design

Dext

h ss

hm

h ry

τs

2α

δ

h sy

D i

wtooth

b ss1

bss2

D

Fig. 5.1 Cross-section of a SMPM synchronous machine of 4 poles.

dependence betweenhm onBδ is studied through the analysis of the magnetic flux densityabove the magnetsBm, which can be expressed as

Bm = Br,m1

1 + µrδehm

(5.2)

whereBr,m is the remanence flux of the magnets at operating temperaturewhich typicallyis 15 − 20 below the winding temperature [23],µr is the relative permeability of themagnet andδe is the equivalent airgap length (including the effect of thestator slotting).Magnets are temperature sensitive, so that the remanence flux Br,m variation is reflectedin

Br,m = Br,0[1− Tk(Tr − T0)] (5.3)

whereBr,0 is the flux remanence at ambient temperature,Tk is the temperature coefficientfor the flux remanence density [1/C], Tr is the temperature of the magnets at the operating

44

5.1. Geometric Design

Bδ

Bm

2α θ

B( )θ

Fig. 5.2 Fundamental flux density above the magnets in an electric period.

point andT0 is the ambient temperature. Indeed, manufacturers of PMs usually give thedata of the magnet according to these parameters exhibited above, taking as referenceT0 = 20 C, as it is illustrated in [28]. Therefore, considering the equations (5.1) and (5.2)the thickness of the magnet which is necessary to produce certain Bδ in the airgap is

hm =µrδe

Br,m4 sinα

Bδπ− 1

(5.4)

With regard to the airgap length, it can be chosen more freelycompared to the asyn-chronous machines since the power factorcos φ of the PMSM is not penalized. The phys-ical airgapδ which is suitable for PMSM machines belong to the range of 1–3mm. Inorder to choose within this range, some properties which depends on the airgap length areexhibited [23]:

• Magnet height.The larger the airgap length, the more magnet material is necessaryin order to create the required flux density in the airgap. Thus, the magnet heighthas to be higher and the magnet volume increases, which lead to an increase of themachine cost.

• Magnetic inductance.When the airgap is larger, the magnetic inductance of themachine decreases. Field-weakening is related to the magnetic inductance and adrop of its value shortens the field-weakening operation range.

• Eddy current losses.The larger airgap, the more sinusoidal airgap flux density andthus, less harmonics in this wave. As it is explained later, eddy currents depend onthe variation of the magnetic field. Therefore, the less content of harmonics, thelower losses are performed.

Considering the information related to the airgap properties, the most suitable election forthe machine of this thesis is an small airgap, since the quantity of PM required is lower,and consequently the weight of the machine.

45


5.1.2 Iron Sizing

This section describes the guidance of the magnetic flux through the ferromagnetic mate-rial in the motor. It is assumed the absence flux leakage amongthe different iron parts andthere is not saturation in the ferromagnetic material. Concerning the design of electricalmachines, the magnetic flux density in the different parts ofthe machine is usually a con-straint of the design. In a PMSM, the values of the flux densityin the different parts of themachine,Bry, Bsy andBtooth varies between 1.2–1.5 T [11]. As depicted in Fig. 5.3, allthe flux produced by the magnetsφm passes through the stator yoke and the rotor yoke

φm = BmwmLact = Bm2α

p(D − 2δ)Lact (5.5)

whereBm is the magnetic flux density on the magnets,wm the width of the magnets,2αthe magnet coverage andLact the active length of the machine. The flux which passesthrough the stator yoke and the rotor yoke is given by

φsy = BsyhsykjLact (5.6)

φry = BryhrykjLact (5.7)

wherekj is the stacking factor of the iron lamination andhsy andhry are the height of

Φm

Φry

Φsy

Φtooth

Φry

Φsy

Fig. 5.3 Flux guidance through a ferromagnetic portion of the machine.

the stator and the rotor. As Fig. 5.3 illustrates, the flux created passes through the teeth,φtooth, and it splits in two equal fluxes in the stator yoke,φsy. These two equal parts, inabsence of leakage return to the magnet through the rotor yoke,φry

φry = φsy = φm/2 (5.8)

46

5.1. Geometric Design

According to this relation and (5.6) and (5.7),hry andhsy can be determined

hry =αBm(D − 2δ)

pkjBry≈ αBmD

pBry(5.9)

hsy =αBm(D − 2δ)

pkjBsy

≈ αBmD

pBsy

(5.10)

It can be appreciated in equations (5.10) and (5.9) thathsy andhry are quite similar.They are usually equal since the magnetic flux which they are designed for is the same,in order to exploit all the ferromagnetic material. To calculate the width of the tooth thealternative direction of the magnets is ignored. The total flux that passes from the magnetsto the stator through the airgap is

φtotal = BgAg = pφm (5.11)

whereBg, is the flux density in the airgap andAg, is the area of the airgap. The wholeflux created by the magnets is divided among the total number of teeth in the stator,Qs,independently whether the flux is incoming or outcoming.

φtotal = Qsφtooth = pφm (5.12)

φtooth = BtoothwtoothkjLst (5.13)

Linking (5.12) and (5.13)wtooth can be obtained as

wtooth =Bm2α(D − 2δ)

QsBtoothkj≈ Bm2αD

QsBtooth(5.14)

Equations (5.14), (5.10) and (5.9) play a fundamental role in determining the cross-sectionarea available for the windings. These parametershry, hsy andwtooth show a linear depen-dence on the diameter of the gapD and the magnetization of the magnets. However, it isundoubtedly that there is a relationship between tooth width, stator yoke and rotor yoke ifsome assumptions are taken into account. Taking care of exploiting as much as possiblethe ferromagnetic material, the design for all the parts of the machine has to be done forthe same flux density, henceBtooth = Bry = Bsy and absence of flux leakage is assumed.Eq 5.16 shows this relationship.

φtotal = pφm = p2φsy = Qsφtooth (5.15)

φtoothφry

=2p

Qs=wtoothhry

⇒ wtooth = 2p

Qshry (5.16)

47


The main difference of both is thathry andhsy exhibit an inversely dependence on thenumber of poles whereaswtooth on the number of slotsQs. Thus, the number of polesand slots for each diameter of machine is restricted, sincehry, hsy andwtooth can be smallenough to be involved in mechanical troubles.

Once the cross-sections which guide the flux are calculated,the other dimensionsof the machine can be fixed or derived (Di, Dext andhss). They have to be controlled inthe analytical process in order to not exceeding the design constraints.

Dext = D + 2hss + 2hsy (5.17)

Di = D − 2(δ + hm + hry) (5.18)

Once the dimensions of the teeth are calculated, the width ofthe slots can be determinedtaking into account that the width of the tooth is constant.

τs =πD

Qs

(5.19)

Whereτs is the slot pitch. The width of the slot in the closes part of the toothbss1 is givenby

bss1 = τs − wtooth (5.20)

The width of the slot which is closest to the stator yoke is given by

bss2 = πD + 2hss

Qs

− wtooth (5.21)

To conclude the definition of the dimensions of the machine, the area of the slots is ex-pressed as

Asl =bss1 + bss2

2hss (5.22)

The relation of how much bare copper is inside the slot is given by the fill in factor,ff (Fig. 5.4). This factor takes into account the air between conductors as well as theisolating layer which the conductors are sealed with. The maximum ideal fill in factorffreaches 0.79 [8].

Acu = ffAslot (5.23)

whereAcu is the bare copper area in the slot. Despite this ideal maximum, concentrated-winding machines reachff values between 0.5 and 0.6 (round wire). This fill in factorcan vary depending the manufacturing technique used to implement the wire in the stator.The conductor area is expressed as

Acond =Acuns

(5.24)

48

5.2. Electric Model

a) b)

Fig. 5.4 a) Slot filled with conductors; b) Maximum ideal fill factor with round wire.

wherens is the number of coil turns. The height of the conductorhss can be chosenfollowing different guidelines. Some ideas point out to reach the maximization of thetorque per ampere [26], whereas other try to reduce the copper resistance as much aspossible, sinceRAC depends on the height of the slot [11]. An important constraint oncethe cross-sectional area of the machine is fixed is the product nsI. This describes theMMF produced by each coil withns turns carryingI Amps [8].

nsI = JAcu (5.25)

J is the current density. Since this parameter is fixed, neither ns nor I can be changedindependently to maximize the torque, or reduce the copper losses as it can be appreciatedfurther, in equation 5.41. This constant is defined once the cross-sectional area of themachine is defined andJ is established as a design constraint. The current density isusually an important design constraint since it determinesthe cooling system which themachine is designed for. This constant varies depending thetype of machine built [11].

Table 5.1: Permitted rms values for current densities for PMSM machines depending thetype of cooling system [11]

Irms[A]Air cooling 2-3.5Force cooling 7-10

5.2 Electric Model

Here are deployed the equations which model the electric behavior of the machine. Ma-chine behavior is normally explained through the use of adq frame reference. If harmonicsare neglected, steady stated- andq- voltage can be presented as

ud = Rsid − ωψq = Rsid − ωLqiq (5.26)

uq = Rsiq + ωψd = Rsiq + ω(ψm + Ldid) (5.27)

49


The peak value of the fundamental phase voltageE (back EMF) is obtained from thederivation of the magnetic flux linkage at not no-load

E = max

[dψmdt

]

= N

[dφmdt

]

(5.28)

whereN set the number of turns per phase andφm is the fundamental magnetic flux.These magnitudes can be calculated as

N =p

2qnskw1 (5.29)

φm = φm sin(ωelt) = BδLπ(D − δ)

p

2

πsin(ωelt) (5.30)

whereωel is the electrical angular speed of the machine. Gathering equations 5.28, 5.29and 5.30 in equation 5.31 the result is

E = qnskw1ωelBδL(D − δ) (5.31)

As it has been shown in Chapter 2, SPM machinesd- andq- inductance is quite similarsince the permeability of the magnetsµr is close to one. Thus,Ld ≃ Lq = Ls. Thesynchronous direct inductance is made up by the magnetizinginductanceLm and theleakage inductanceLσ. The magnetizing inductance is calculated in [23] as

Lm =3

π(qnskw1)

2 µo

δe +hmµr

L(D − δ) (5.32)

The stator leakage inductanceLσ can be computed from the following equation which isdetailed in [17]

Lσ = pqn2sµoLλ1 (5.33)

where the ratioλ1 is the specific permanence coefficient of the slot opening. Due to sim-plicity and the following 2D analysis end winding inductance is neglected. The externalvoltage that the motor can perform is limited by the voltage range that the motor can per-form. Since the DC supply is linked through a PWM converter, the maximum voltage inthe converter terminals is

Umax =UDC,min√

3=

√

u2d + u2q =

√

(E +RsImax)2 + (ωelLsImax)

2 (5.34)

whereid has been neglected, since field weakening is out of scope in this thesis. It isimportant to take into account that the calculation of the synchronous inductance is anapproximation since its value is reduced when the machine issaturated [17].

50

5.3. Loss Model

5.3 Loss Model

One of the most important parameters of the electrical machines is how much power theywaste to produce power on the output shaft. In low-speed applications copper losses aremainly the main source of wasting energy. Furthermore, magnets in SMPM synchronousmachines are exposed to harmonics, and they produce quantifiable losses. This section tryto show some models from this three different sources and suggest some ways of reducingthem. Stray and bearing losses are not modeled in this section.

Ploss = Pcu + Piron + Pmag (5.35)

5.3.1 Copper Losses

Model a copper resistance of an electrical machine is important not only because of thelosses but also because its electric properties. The resistance is a propriety of all materialswhich show the opposition of the material to let the current flow through it

R =ρl

A(5.36)

whereρ is the resistivity [Ωm], l the length of the section andA the section of the con-ductor. The resistivity depends on the temperature which the material stands. Dependingthe type of material, this dependency can be linear or exponential. For materials as thecopper [8]

ρ(T ) = ρo(To)[1 + αt(T − To)] (5.37)

whereρo is the resistivity at ambient temperatureTo. In the design process it is reallyimportant to be aware about the warming-up temperature of the machine , since it deter-mines the value of the resistance, the copper losses and thusthe cooling system that themachine needs. For copper wire,ρo,20 = 1.68 ·10−8 Ωmαt = 3.7 ·10−3C−1 Taking intoaccount equation (5.36), and Fig. 5.5 the slot resistance for a concentrated winding canbe calculated as

Rslot =ρntoothLcoil

Acoil=ρntoothLcoil

Acu

2/ns

= ρn2s

Lcoil2

1

Acu(5.38)

wherentooth is the number of turns per tooth,ns is the number of turns per slot,Lcoil isthe length of the coil, and the phase resistance is

Rs = RslotQs

3= ρn2

s

Lcoil2

1

Acu

Qs

3= ρn2

spqLcoil2

1

Acu(5.39)

Eq. (5.39) shows that the resistance is dependance with the numbers of the turns squared.Thus, the copper losses are defined as

Pcu = 3I2Rs = ρQsLcoil2

1

Aslot(nsI)

2 (5.40)

51


whereI is the rms value of the current in the machine. Introducing the constant (5.25)in equation (5.40) confirms that the copper losses only depend on the volume of copperused for the windings, the number of slots and the squared value of the current density

Pcu = 3I2Rs = ρQsVcuJ2 (5.41)

whereVcu is the volume of the copper wires. Equations (5.40)– (5.41) shows that copperlosses for a cross sectional area fixed cannot be improved varying the number of turns, andit only depends on the squared value of the constantnsI. The length of coil is calculatedas follows [3]

1 2 Qs

ns Acu

τs wtooth

Lact

τs

Average coil

a) b)

Fig. 5.5 a) Cross-section of a stator with concentrated winding; b) Cross-section of a tooth withconcentrated windings. The striped area is the tooth.

Lcoil = 2Lact + 2Lend,av (5.42)

whereLact is the active length of the machine andLend,av is the average length of theend-winding. Glancing Fig. 5.5Lend,av is given by

Lend,av =1

2

[

wtooth + wtooth + πτs − wtooth

2

]

=1

2

[

wtooth(2−π

2) + π

τs2

]

(5.43)

Introducingwtooth from equation (5.14) in (5.43) it can be concluded thatLend,av =

ktoothDQs

. Thus, the value ofLcoil is given by

Lcoil = 2Lact + ktoothD

Qs(5.44)

and copper losses

Pcu = 3I2Rs = ρ(QsLact + ktoothD)1

Acu(nsI)

2 (5.45)

52

5.3. Loss Model

It is important to emphasize that copper losses do not dependon the speed of the machine,if the increase of the machine temperature is neglected during field weakening operation[17]. In order to decrease this source of losses it is important to fill the slot as muchas possible (highestff ), increasing the contact between wires. This makes better the heattransfer between wires, therefore the temperature of the windings get lower, the resistancedecreases and also the copper losses [17].

5.3.2 Core Losses

The presence of an alternate magnetic field in the iron causeslosses from different origin.Hysteresis losses (Physt ∼ f ) are caused by the continue variation of the operating pointin the iron along the hysteresis loop. Eddy currents losses (Peddy ∼ f 2) are caused bythe EMF induced in the iron by the alternating field due to Faraday’s law. Modelling thislosses is not as straightforward as the copper losses [8]. Losses data given by manufac-turers is often only available at 50–60 Hz only for sinusoidal current and there are partsof the machine which transport flux with a high level of harmonics. But these estimationsare good enough to determine strategies on the machine in order to reduce these losses. Ifit is assumed that the magnetic field density in the machine issinusoidal, iron losses canbe computed as

piron = physt + peddy = khystBβstf + keddyB

2f 2 (5.46)

wherephyst andpeddy are the hysteresis and the eddy current loss density,khyst andkeddyare hysteresis and eddy current constants,βst is the Steintmetz constant andf is the elec-trical frequency of the machine. The value of these constants depends on the ferromag-netic material used as well as the thickness of the lamination. Typical values for ironlaminations varies fromkhyst = 40 − 55, keddy = 0.04 − 0.07 andβst = 1.8 − 2 [17].These parameters are usually calculated by fitting them to the data given from the manu-facturers for sinusoidal conditions [8]. Consideringβst = 2, khyst andkeddy show a lineardependence in (5.58) and these coefficients can be determined using linear least squares.Assuming that there are not minor hysteresis loops, the hysteresis loss density in the wholeferromagnetic material parts of the machine can be calculated as

physt = khystBβstf (5.47)

However, eddy currents are caused by other phenomena where harmonics play a funda-mental role. According to Faraday’s law the current inducedin a conductor is due to thevariation of the flux densitydB

dtand the losses are directly related to the main square value

of the rms ofdBdt

, similar to the copper losses. The expression shown above (5.58) has tobe changed, since this is only valid for sinusoidal flux wavesand should be valid for allthe parts of the machine

B(t) = B sin(2πft) (5.48)

53


This is the sinusoidal flux density. Its derivative expression is

dB(t)

dt= 2πfB cos(2πft) (5.49)

and the square value of the rms value is

⟨dB(t)

dt

⟩2

= 2π2f 2B2 (5.50)

Using (5.50) in the eddy current loss part in (5.58) leads to amore general expression ofeddy currents

peddy =keddy2π2

⟨dB(t)

dt

⟩2

(5.51)

Since eddy currents losses increase with the variations of the magnetic field, the teeth arethe ferromagnetic part of the machine that much suffer this phenomena. The flux den-sity in the teeth rises quickly when the edge of the magnet is approaching and decreasesquickly when the lagging edge of the magnet passes long the teeth, leading the flux den-sity from maximum to zero. This extreme variation leads to calculate teeth eddy currentlosses density in the following way [23]

peddy,teeth =4D

πpwtoothkeddyf

2B2ts (5.52)

Some strategies can be adopted on the design process in orderto decrease this source oflosses as much as possible [17] [8]

• Reducing lamination thickness. Eddy current losses are directly proportional to thesquare of the thickness.

• Reduce the level of flux density in the ferromagnetic parts ofthe machine.

• Reduce the number of magnets poles. Have a great number of poles in the machineentails lots of advantage i.e. reduction of the machine size, but it entails disadvan-tages too, as increasing the electric frequency and consequently, the losses. Thus, agood trade-off of these proprieties have to be achieved withthe number of poles.

• Increase the resistivity of the laminations in order to reduce eddy currents.

• Annealing laminations after they have been cut or stamped, since it eliminates theinfluence of mechanical stress on iron losses.

54

5.4. Analytical Program

5.3.3 Magnet Losses

Rare earth permanent magnets are conductive materials, 5–10 times more conductive thanelectric steel (110− 170 · 10−8[Ωm]) and they are exposed to an alternate magnetic field,that is the reason why eddy current are induced in them. Indeed, the location of the per-manent magnets in SMPM synchronous machines contributes larger in these losses sincethey are exposed to slotting effect and great amount of harmonics. Moreover, harmoniccontent in concentrated windings is higher than distributed windings. Nonetheless, it isdifficult to give an accurate analytical model of this losses. However, some techniquescan be applied to reduce the losses in the magnet [20] [11]

• Magnets can be segmented axially or circumferentially in order to limit eddy currenteffects.

• The use of plastic bonded magnets in stead of sintered magnets can eliminate eddycurrent. However, the magnetic proprieties of them are worse and the amount ofmagnetic material needed is longer.

5.4 Analytical Program

Once the requirements are calculated and the theory of concentrated windings, magneticand electric behavior has been deployed, the constructive parameters of the machine canbe calculated. The design procedure is carried out by an own numerical tool in MAT-LAB R©, mainly based on the theory mentioned above. The machine is designed in order toreachTsteady. Thus, it has to be checked that they are able to reach theTpeak through theuse of FEMMR© .

5.4.1 Objective

With regard to the objective of this thesis, different selection criteria can be set up to de-sign the motor. This design can be focused on one of the principal aims of this machine,lightweight or efficiency. Motor efficiency seems to be more important than lightweightconsidering the consumption of energy point of view. Nevertheless, since the motor islocated inside the wheel (unsprung mass problem) and the setof losses is not accurateenough (PM losses), this program is concentrated on achieving the lightest design possi-ble. Furthermore, a light design entails the advantage of less material, and as a result acheaper design.

55


5.4.2 Additional Equations

Torque

The obtention of the torque equation derived from [23] is explained in

T = πD2

4kw1BδSLact sin β (5.53)

whereS is the current linear density of the motor andsin β is the saliency of the machine.In the design process, the torque is fixed to its rated value. The other parameters shownin equation 5.53) reveals somehow the relation between the constructive parameters fromthe machine.S is directly related to the slot area. For a fixed external diameter, the areaof the slot influences the rotor diameter. The torque is not designed for field weakeningoperation,id = 0, and this means thatsin β = 1. The active length of the machineLacthas a great influence on the machine weight, hence, it should be limited. If the constantnsI is considered asSτs = nsI = AcuJ [23], the torque can be expressed as

T =D

4kw1BδAcuJQsLact (5.54)

where the torque performed by the machine is derived from their constructive parameters.

Slot Height

As has been mentioned in Section 5.1.2, some parameters haveto be established to calcu-late the whole geometry of the machine. In [26], it is explained that the torque performedby a PMSM per ampere can be maximized based on the slot ratio,d = Dgap

Dgap+hss. The

shear stress in the airgap is considered as [26] [11]

σ = BδS (5.55)

This shear stress can be considered as the responsible of developing the torque on therotor

T = σAgD

2=π

2D2Lactσ = 2Vrotσ (5.56)

whereVrot is the volume of the rotor andAg is the area of the airgap. As it has beenmentioned in Chapter 2, the torque is proportional to the volume of rotor, where the shearstress is the product of magnetic and electric loading. Thisvalue should not overpass48000 Pa [11]. Hence, it is easy to realize that torque performance is obtained trading-off the volume of the rotor and the available area for the stator conductors. In fact, for afixed external diameterDext fixed, the larger the diameter of the rotorD, the lower spacefor the conductors will be. Therefore, it has to exist an optimum value of the rotor outerdiameter that maximize the torque performance. As mentioned above, stator teeth have

56


parallel sides to exploit all the ferromagnetic material. The electric current linear densityS is roughly proportional to the area of the slot, thusA ∼ (1 − d2). So the expression ofthe torque is given by

T ∼ σVr ∼ (1− d2)d2 (5.57)

This equation 5.57 has maximum value ind = 1√2. Thus, the value of this proportion of

the slot in order to maximize the torque should be 0.71, but regarding machines designedfor a great number of poles it is convenient to increase this number between 0.8-0.85 soas to reduce the quantity of magnetic material and consequently the iron losses [26].

5.4.3 Design Variables

These are the input variables that calculates all the geometric parameters of the machine ineach loop, using some fixed values and keeping the results within some constraint values.They are shown in Table 5.2.

Table 5.2: Design variables.Variable Symbol RangeNumber of poles p 4≤ p ≤ 60,p is evenSlots per pole and phaseq 0.25≤ q ≤ 0.5Diameter of the Rotor D 100≤ D ≤ 300 mm

5.4.4 Constants

These values are considered constant in the analytical program. They are properties fromthe materials used in the machine.

5.4.5 Fixed Values and Constraints of Design

There are some values fixed for the design calculations. The parameters calculated haveto be inside some margins of design.

5.4.6 Matlab Program

In Fig. 5.6 it is depicted how the analytical program based onthe information shown aboveworks. After calculations shown in the big square of Fig. 5.6, the matrix is tidy regardingthe weight of the designs. After that, machines which are till 0.3 kg heavier are analyzed.These machines have to be checked that they reach the peak torqueTpeak under overloadcapacity. As well, since the models for inductance of the machine are not very accurate,

57


Table 5.3: Constants.Constant Symbol Value

Magnet

Remanent flux density Br 1.2 TRelative permeability µr 1.05Temperature sensitivity Tk 0.001 T/KMagnet density ρm 7700 kg/m3

Magnet temperature Tmag 120 C

Iron

Steel density ρiron 7700 kg/m3

Eddy loss constant keddy 4.26·10−5

Hysteresis loss constantkhyst 0.0269Steinmetz constant β 2

Winding

Copper density ρcu 8930 kg/m3

Copper resistivity rcu 2.392·10−8 Ω·mCopper fill factor ff 0.6Copper Temperature Tcu 150

Table 5.4: Fixed values and constraints.Peak fund. flux density in stator yokeBsy 1.6 TPeak fund. flux density in teeth Bst 1.6 TPeak fund. flux density in rotor yoke Bry 1.6 TPeak fund. flux density in airgap Bδ 0.9 TMax. peak fund. current density J 3·106 A/m2

Air gap length δ 1 mmElectrical magnet angle α 120

Rotor back thickness hry hry ≥ 5 mmActive length L L ≤ 20 mmMagnet Temperature Tmag 120 C

the number of turnsns it has to be also checked if the phase voltage of the machine doesnot reach the minimum voltage range of the battery times 0.9,in order to allow voltagefor the control. Otherwise, they have to be modified to fulfillthis requirement.

Uph ≤UDC,min0.9√

3(5.58)

5.4.7 Results

From the program shown above, different motor models are obtained. The lightest modelis analyzed further through an own-developed program over FEMM R© package. In App. Bare detailed the measures and the parameters of the lightestdesign.

58


INPUT

Tsteady

Setting design variables variation

Concentrated windings

Fixed values

Design Constraints

CALCULATION MACHINE

GEOMETRIC PARAMETERS

WEIGHT & LOSSES

fulfill constraints? OUT

Keep in result Matrix

Selection

Criteria

variable data each loop

Short Matrix

OK

LightestResults-0.3 kg

maximize

torque

Cogging Torque

HIGH

CHECKOverload Capacity

FEMM

Fig. 5.6 Flow program design

59


60

Chapter 6

Finite Element Analysis

Throughout this chapter, the machine calculated in the analytical model is evaluated.This evaluation is carried out through the use of an own-developed program based on theFinite Element Method Magnetics package (FEMM R©). In addition, the tooth shape of themachine is selected through the analysis of the cogging torque. To conclude, the designedmachine is analyzed an evaluated at different load requirements.

6.1 FEMM Model

The Finite Element Method environment used in this thesis isFEMM R©. It is a finiteelement software developed by David Mecker which can process magnetics, electrostaticand heat transfer problems. However, unlike other finite element magnetic softwares asFLUX2D R© and JMAGR©, FEMM R© does not provide any model of the wide varietyof electrical machines. For this reason, a SPM synchronous machine has to be modeledin FEMM R© environment. MATLAB R© language is used to design this model followingthe command guidelines Octave FEMMR© [15]. The purpose of the designed programis to obtain different results such as the back EMF, the phasevoltage of the machine,the torque, the inductances, etc. for a given input, which gather, on the one hand theconstructive parameters of the machine and on the other handthe power input representedin adq frame reference. In the following steps it is described how the program is designedand how it works.

6.1.1 Input Parameters

Constructive and Geometric Parameters

First of all, the machine parameters calculated in the analytical model have to be ac-quired by the program. This is computed as an input vector which contains the geometricparameters, the number of poles, the number of slots and the number of turns. It is im-

61

Chapter 6. Finite Element Analysis

portant to point out that the number of turnsns calculated in the analytical model arenumber of turns per slot. When the machine is drawn in FEMMR©, the next step is label-ing each block with its material properties. Since the machine designed uses double layerconcentrated-windings, each slot is divided in two blocks.Therefore, it is suggestible toconsider programming the number of turns per tooth to make easier the definition of thematerials.

Power Input Parameters

Once the machine is completely modeled (shape and materialsdetermined), dynamics hasto be simulated. Hence, the machine input has to be related tothe torque and the speedperformed by the machine. First, the mechanical speed,ωmech is a direct input. On theother hand, the torque performed leads to a current input,q- current

T =3

4pψmiq (6.1)

In order to extractiq from this equation, firstψm is calculated as the flux linkage funda-mental harmonic at no load, and used as input parameter. For this reason, adq currentsframe reference is implemented as input. Nevertheless, since the machine is designedwith concentrated windings, some assumptions to implementa dq model are not accom-plished [16]

• Windings should be sine distributed.

• Magnetic circuit should not be saturated.

Despite this fact, in [16], states that thedq model can be applied to concentrated windings.In order to transform thesed andq input currents to phase currents, the Park transforma-tion has to be applied. This is necessary because current blocks in FEMMR© have to belabeled with phase currents. The Park transformation can change three-phase componentsfa, fb, andfc, which can be either currents, voltage or fluxes into 2 variables,fd andfq,which are in other frame reference, and viceversa [29]. Thisis described in 6.2

fafbfc

︸︷︷︸

fph

=

cos(θ) − sin(θ) 1

cos(θ − 2π3) − sin(θ − 2π

3) 1

cos(θ + 2π3) − sin(θ + 2π

3) 1

︸︷︷︸

Tdq,ph

fdfqfo

︸︷︷︸

fdq

(6.2)

whereθ is the angle between thed axis of the rotor which is the direction of the north poleof the rotor, and the starting position of the electric period in electric degrees. The startingposition is defined where is found the maximum magnetic flux linkage of the phase A,as illustrated in 6.2. In a distributed winding, this starting position is straightforward tobe calculated, as described in Fig. 6.1a. However, as illustrated in Fig. 6.1b and Fig. 6.2,

62

6.1. FEMM Model

θd

A A-

Initial positionof the electricperiod

θ

d

θi

a) b)

Fig. 6.1 a) Distributed winding rotor angle; b) Concentrated winding rotor angle and initial elec-tric angle period,θi

.

it is difficult to calculate this starting position for concentrated windings,θi. One wayto determineθi is calculatingψa for the whole electric period and establish where isthe maximum. This is because concentrated winding layouts follow different patterns, asillustrated in 6.2 Through the use of this own-developed program, different configurations

θm

Initial positionof the rotor

B'A'BAAC A'B C' C C'B'

ψA

θelΔθi,el

No loadanalysis

Fig. 6.2 Initial position of the rotor in a SPM synchronous machine and the initial phase differencebetween thed− axis of the rotor and the flux linkage of A phase.

of slot and poles have been tested in order to obtain the anglewhere the flux linkage islarger.

6.1.2 Machine Geometry

A sector that covers one slot pitch of the machine,τs has to be plotted in FEMMR© usingthe input vector above. After that, this sector is copiedQs times for the stator andp timesfor the rotor. It is important to delimit properly the different areas, in this case speciallythe conductors, since each slot is going to allocate two different types of wire phase.

63


6.1.3 Material Settings

For each region created in the step above, material characteristics have to be set. Threemain issues have to be taken into account for a PMSM of concentrated winding. First ofall the direction of the magnets has to be established, taking into account that it is a partof the machine that rotates. Moreover, the iron is not linear, so the hysteresis curve of thesteel has to be set as well. Finally, the windings have to be set according to the obtentionof the winding layout shown in Chapter 3 through the Cros’ method. Here the materialsused during FEMMR© processing are detailed.

Magnets

The selection of the magnet is conditioned by the temperature risk. In order to deal withthis problem, VACODYM 890 TP has been the choice of this thesis since this type of Ne-FeB magnets can withstand higher temperatures. In Fig. 6.3 the demagnetization curves

Fig. 6.3 Demagnetization curves B-H for VACODYM 890 TP at different temperatures [28].

for different temperatures are deployed. As well, it will beanalyzed if the magnet over-passes this threshold. It is important that the magnets do not reach this point of demag-netization since they can be irreversible demagnetized andconsequently the motor wouldbehave different from how it was designed.

Electrical Steel

The ferromagnetic material chosen for guiding the magneticflux density is laminatedsteel. It is important to choose the laminations as thin as possible, in order to decrease the

64

6.1. FEMM Model

iron losses. The selected iron is the M235-35A (.35 mm) provided by Surahammars BrukAB [27]. BH curve is provided in Fig. 6.4 and loss density curves at different frequencies

0 2000 4000 6000 8000 10000 120000

0.5

1

1.5

2B

(T)

H(A/m)

Fig. 6.4 BH Hysteresis curve at 50 Hz for M235-35 steel [27].

are depicted in Fig. 6.5.

0 0.5 1 1.50

10

20

30

40

50

frequencyincrease

Loss

den

sity

(W/k

g)

B(T)

Fig. 6.5 Iron loss density of the M235-35A magnetic steel at different frequencies 50, 100, 200,400, 1000 and 2500 Hz [27].

6.1.4 Accuracy Parameters

Mesh Size

Mesh size is a very sensitive parameter. On the one hand the size of the mesh defines howaccurate the calculation will be. If the mesh size is big enough it can contribute to poorcalculations. On the other hand, this high accuracy with extremely small mesh size canlead to long calculation times. Thus, it is important to achieve a good trade-off between theaccuracy of the mesh and the time consumption. Before defining the mesh in the differentblocks of the machine it is important to take into account that there are some parts ofthe machine which contribution to the final computation result is larger. Therefore, theairgap and the magnets have to use a smaller mesh size in orderto achieve good results.

65


This program uses a general mesh of 3 mm for all the elements inthe machine excludingthe airgap and the magnets, for which a mesh size of 0.3 mm is used, as it illustrated inFig. 6.6

Fig. 6.6 Mesh sizing.

Iterations

This is the number of steps in which the rotor completes2π electrical degrees. In eachiteration data is obtained for further processing. In the manner of the mesh size, it has tobe a good trade-off between the accuracy of the final wave obtained and the number ofiterations. For this machine, each electric period entails360 iterations. When currents arevaried, 20 iterations per period is good enough.

6.1.5 Data Obtaining

FEMM R© is able to get different data in each iteration. Flux linkage, current and voltagedrop of each phase can be obtained from the circuit. From the remaining parts of themachine (stator back, rotor back, magnets, airgap and stator teeth) magnetic data fromrelevant points is acquired. The torque can be computed in different ways, either usingthe Maxwell stress tensor applied on the rotor or using a lineintegral of the force insidethe airgap [15]. As well, it can be obtained based on the electric power consumed by eachphase but it entails error since the back EMF is not directly taken from the FEMMR©.

6.1.6 Analysis

Data obtained in each iteration is kept in different vectorsfor its later analysis. It is impor-tant to realize that the results obtained are not continuous, but discrete. Thus the accuracyof the post processing results depends on the number of iterations. In order to derive the

66

6.1. FEMM Model

flux linkage, the discrete derivative is used, as described in equations 6.3 and 6.4

dψadt

=dψadθ

dθ

dt= ω

dψadθ

(6.3)

where the discrete derivative from the flux linkage is

(dψadθ

)

i

=ψa,i − ψa,i−1

∆θ(6.4)

6.1.7 Program Flow

Fig. 6.7 shows the different features that the program calculates, keeps as data and plotsin MATLAB R© .

Linear or No Linear?

POSITION- Angle [0.360]- q current- d current

PERIOD

- ωmech

- q current- d current- Iterations

NOMINAL POINT- Max current-- Iterations/period- Iterations/Imax

ωmech

-Flux density curvesof certain regions( )B ,B

δ magnet

-Flux density analysisscaled.-Airgap strengthanalysis

- ψ ( )d θel

- ψ (θq el)- T( )θel

- Power ( )θel

- Iron losses- Copper losses- B ( ), B ( ),sy el ry elθ θB ( )tooth elθ

- B ( ), B ( ),δ

θ θel m el

- ψ θ θ θA A A( ), I ( ) ,U ( )el el el

- d /dt( )ψ θel

- ψ ( ): Ld d di- ψq q q(i ): L- T(i )q

- Peak2Peak(i )q

- T (i )ripple q

- Bsy, toothB

INPUT

Fig. 6.7 Interface menu and different features offered by the program.

67


6.2 Machine Analysis

6.2.1 Target

The importance of this analysis lies in the fact of not surpassing certain values in deter-minate parts of the machine. First of all, two design conditions have to be differentiated.At rated point conditions, magnetic flux parameters set in the analytical model have to becheck. On the other hand, overload capacity also has to be tested, which entails severaltasks

• Phase voltage must not exceed0.9Udc,min√3

, since this is the lowest value that the bat-tery can operate, giving also some margin for the control system.

• Magnetic flux on the magnets must not surpassBknee defined for150C, in orderto avoid the demagnetization effect.

• Saturation on the core entails bigger current to achieve thedesignedTpeak.

Moreover, different important properties of the machine are analyzed through this chapter,in order to show all the properties performed by the designedmachine.

6.2.2 No Load

Tooth Shoe

Although the analytical model calculates most of the parameters of the machine, the toothshape of the machine is not considered. As it has been mentioned during chapter three,cogging torque depends on the variation of the energy in the airgap. This variation washigher when the reluctance changed, in other words, at the end of the tooth tip. These vari-ations of reluctance are produced in all the teeth, and they can be cancelled by the torqueproduced in other teeth, or increased by the same reason. This is the primary purposefor tooth shoes [8]. Thus, if the proportion between the tooth opening and the slot width,kopen is varied from 0 to 1, there must exist a configuration where the cogging torque issmaller. Different slots openings have been tested in the machine designed, concludingthat the most suitablekopen is 0.75, as shows Fig. 6.8. So the final shape of the machineanalyzed is displayed in Fig. 6.9

Cogging

Some authors consider the cogging torque as the variation that occurs in the magneticenergy in the airgap, as mentioned in Section 4.5.2. Concurrently, other authors considerthe cogging torque as a result of the variation of the reluctance along the gap [8]

Tcog = −1

2φ2dR

dθ(6.5)

68

6.2. Machine Analysis

0 0.2 0.4 0.6 0.8

0.08

0.16

0.24

0.32

∆T

(Nm

)

Kopen

Fig. 6.8 Influence of the open width of the teeth with the cogging torque.

A+A-

A+A-

C+

C-B+

B-

B-

B+

A-

A+

C-

C+

C+

C-B+

B-A+ A+

A- A-C-

C+

B+

C-

C-

B+B+

B-

B-

B-C+

C+

A+

A-

Fig. 6.9 SPM Synchronous Machine designed. Cross-section.

Considering both statements, the variation undergone is related to the number of slotsthat contains the machine. Their slot openings is the sourceof this undesirable torque.Therefore, the frequency of this torque can be considered as

fcog =p

2Qsf (6.6)

wheref is the frequency performed by the machine. Fig. 6.10 shows the cogging torqueperformed by the machine in an electric period. It can be appreciated through its FastFourier Transformation (FFT) analysis, the main harmonic is the18th, as the number ofslots. This was the cogging frequency expected.

Airgap Flux

Slotting effect creates a great harmonic content in the sinusoidal flux density wave. Itis revealed through the analysis of the magnetic field density in the airgap, exhibited inFig. 6.11. Fig. 6.11 shows different peaks and wave-shapes far from a sine-wave, which

69


0 60 120 180 240 300−0.03−0.02−0.01

00.01

0 5 10 15 20 25 30 35 400

0.005

0.010.015

θe (rad)

T(N

m)

T(N

m)

nth harmonic

Fig. 6.10 No load torque.

indicates a great harmonic content. These harmonics are faced by the shoe of the teethand by the magnets. This great harmonic content as well as thelack of sine-distributedwindings, make concentrated winding machines to have a highquantity of losses in toothtips and magnets.

Back EMF

As it is illustrated in Fig. 6.12, it is a sine-wave harmonicsfree. According to equa-tion 5.28, EMF is related to the winding factor of all the harmonics and the flux producedby the magnets in the airgap. Regarding Fig. 4.12, which shows the whole spectrum ofwinding factors, shows that the5th harmonic is 0, and their neighbors are relatively low.In Fig. 6.11, it can be realized that the5th harmonic is the bigger one. Both sources ofare cancelled. In Table 6.1 some important parameters shownin the no load analysis onthe FEMMR© are exhibited. All the values are lower than it was expected in the analyticalmodel. This can be due to the change of tooth tip. Increase thetooth tip involves that theflux leakage between two neighboring tips also rises. This contributes to the drop of thesevalues. Likewise, slotting effect can also participate on the decline of the magnetic fluxabove the magnets,Bm.

Table 6.1: No-load characteristics valuesAnalytical values FEMM values

Eo (V) 11.77 10.97 (7.29%)Bδ (T) 0.9 0.82 (9.75%)Bm (T) 0.8162 0.7302 (11.77%)ψm (Vs) 0.0526

70


0 60 120 180 240 300

−0.5

0

0.5

0 5 10 15 20 25 30 35 400

0.20.40.60.8

1

θe

B(T

)B

(T)

nth harmonic

Fig. 6.11 Magnetic flux density in the airgap.

0 60 120 180 240 300 360

−10

0

10

E0

(V)

θe (rad)

Fig. 6.12 Back EMF.

6.2.3 Rated Point Conditions

Regarding Fig. 6.13, it can be noticed that the magnetic flux density in the machine issymmetrical,GCD(Qs, p) = 2. This can exhibit that the machine does not perform anunbalanced torque as explained in Section 4.5.1. Forces performed inside the machine arecompensated for opposite sizes. Moreover, concerning the magnetic flux density valuesperformed by the machine in an electric period, Table 6.2, their values are lower than itwas expected in the analytical design. This probably is attributable to the lack of linearityin the iron, since values of magnetic flux density around 1.5 Tindicates a little saturation.The torque performed in Fig. 6.14 is close to 7 Nm, considering that it exists a little sat-uration level on the machine. The ripple performed by the machine at this designed pointis quite low, 4.28%, which is acceptable. As is depicted in Fig. 6.15, the harmonic fluxcontent in the magnet is considerable. The main harmonic of the magnet does not generateany eddy currents, due to it magnet itself, but the harmonic does. Through the FFT anal-

71


Fig. 6.13 Cross-section of the machine at nominal conditions, at maximum flux linkage in phaseA.

Table 6.2: Magnetic flux density at rated pointAnalytical values FEMM values

Btooth(T) 1.6 1.505Bsy (T) 1.6 1.521Bry (T) 1.6 1.506

ysis of this figure, it can be noticed the great amount of harmonics. Since flux harmonicsare a critical aspect for the eddy currents generation, magnets should be segmented inorder to reduce this source of losses as much as possible. This was expected, since SPMsynchronous machines with concentrated windings perform agreat quantity of harmon-ics in the airgap flux, as well as the magnets are not shielded from the slotting effect. InTable 6.3 the values performed by the machine at rated point are described. Despite thecore losses are quite short since the electric frequency of the machine is low, the copperlosses are big enough to make the efficiency be under 90% of efficiency. Considering thatthe value of the output power of this machine is close to 200 W,mainly because it is alow-speed drive, it is easy that a low quantity of copper losses can lead to low efficiencies.

Table 6.3: FEMMR© values at rated point

T (Nm) I(A) Ua(V) J (A/mm2) dψdt

(V) Pcu(W) Piron(W) η(%) felec(Hz)7 13.58 12.9 2.74 11.24 34.32 1.53 82.21 32.1

72


0 60 120 180 240 300 360

6.86.9

77.1

0 5 10 15 20 25 30 35 400

0.050.1

0.150.2

T(N

m)

θe (rad)

Tripple

(Nm

)

nth harmonic

Fig. 6.14 Torque at load condition.

6.2.4 Overload Condition

As it is depicted in Fig. 6.16, the machine is widely saturated. Iron losses will arise aswell as higher current will be need to achieve the peak torqueestimated. This will leadto a low efficiency. As well, since the losses are critically increased at this point, theinside temperature of the machine will rise and some parameters calculated may vary,as the resistance of the copper. This temperature will affect to the magnets properties,and it is essential to evaluate where the value ofBknee is, in order to determine whetherthe magnets can be demagnetized or not. In Table 6.4, maximumvalues achieved bythe magnetic flux density waveform are illustrated. Fig. 6.17shows the different sources

Table 6.4: Magnetic flux density at rated pointFEMM values

Btooth(T) 1.72Bsy (T) 1.705Bry (T) 1.71

of voltage in the machine. It can be appreciated that the phase difference between thedifferent sources of voltage is the one expected, compared to Fig. 6.18. As mentionedabove, magnets cannot surpass certain magnetic flux limit,Bknee, in order to not beingdemagnetized. Through the study of Fig. 6.19, it can be appreciated that the lowest valueof magnetic flux density in the magnet is 0.4 T. Thus, regarding Fig. 6.3, magnets canreach 240oC, which is far from the estimated designed temperature. In this figure, it canbe appreciated that not all the points in the magnet experience the same flux density. In

73


0 60 120 180 240 300 3600.6

0.7

0.8

0.9

0 5 10 15 20 25 30 35 400

0.020.040.060.080.1

θe

B(T

)B

(T)

nth harmonic

Fig. 6.15 Magnetic flux density in the middle point of the magnet at rated conditions and its FFTanalysis

Table 6.5 are summarized the most important points of the machine at overload capacity.The current has been increased more than 3 times, since the machine is saturated. Aswell, the value of the phase voltage of the machine is lower than 18.7 V, which is themaximum voltage that the machine can reach. Copper losses aniron losses have beenwidely affected, since this is not the point they are designed for.

Table 6.5: FEMMR© values at overload point

T (Nm) I(A) Ua(V) dψdt

(V) Pcu(W) Piron(W) η(%) felec(Hz)22 43.45 18.51 13.92 368.89 1.75 41.497 32.1

6.2.5 Different Working Points

Throughout the following figures, different operating points of the machine are described,showing their characteristics. First of all, in Fig. 6.20 itcan be appreciated how the ma-chine torque does not evolute linearly with the current, dueto the saturation, as it wasexpected. The moreiq current on the machine, the more saturated it becomes. This can beappreciated in Fig. 6.21. This figure describes the evolution of the magnetic flux densityin the stator back (green line) and in the tooth (blue line). At no load operating point, theflux density in the stator back is higher than the one in the tooth. While iq increases, sodoes both magnet densities, but since the saturation on the tooth is lower at the beginning,the rise of this magnetic flux is higher. When the q current is close to 22 A, both magneticflux densities are equalized. Wheniq is close to the overload point (30 A), both flux den-sities grow equally. Furthermore, looking at the evolutionof ψq, it is also noticeable thesaturation effect. Fig. 6.22 shows the drop of the inductance wheniq reaches 25 A, which

74


Fig. 6.16 Cross-section of the machine at overload conditions, at maximum flux linkage in phaseA.

is almost double the designed current. Fig. 6.23 shows how the ripple grows according tothe definition ofTripple shown in Fig. 4.14.

75


0 60 120 180 240 300−20

−10

0

10

20

0 5 10 15 20 25 30 35 40 450

5

10

15

20

Ua,dψa

dt,R

sI a

(V)

θe

Ua(V

)

nth harmonic

Fig. 6.17 Voltage sources of the machine at overload conditions. RED:RsIa; GREEN: dψa

dt ;BLUE: Ua.

q

d

-wL iq q

R is q

Eo

Ua

d /dtψa

R is q

Fig. 6.18 Phasor diagram of the voltage in the machine in a dq frame reference, overiq > 0 andid = 0.

76


0 60 120 180 240 300 360

0.50.60.70.80.9

0 10 20 30 40 500

0.1

0.2

θe

B(T

)B

(T)

nth harmonic

Fig. 6.19 Magnetic flux density in the weakest point of the magnet, in three electric periods.

0 5 10 15 20 25 30 35 40

5

10

15

20

25

T(N

m)

ıq(A)

Fig. 6.20 Torque evolution with q-current and its linear behavior.

0 5 10 15 20 25 30 35 40

1.4

1.5

1.6

1.7

1.8

B(T

)

ıq(A)

Fig. 6.21 Evolution of the peak value of the magnetic flux density in thetooth (blue line) and inthe stator back (green line).

77


0 5 10 15 20 25 30 35 40

0.02

0.04

0 5 10 15 20 25 30 35 40

0.9

0.95

1

ψq(V

s)Lq(m

H)

ıq(A)

ıq(A)

Fig. 6.22 Evolution of the value of the q-flux linkageψq and the q-inductance of the machineLq.

0 5 10 15 20 25 30 35 400

2

4

6

Tripple

(%)

iq

Fig. 6.23 Evolution of torque ripple.

78

Chapter 7

Conclusions and Further Work

This chapter summarizes the conclusions obtained in this thesis, as well as criticize it.Recommendations for further work are deployed here

7.1 Conclusions

Throughout this thesis, an electric motor prototype for theShell Eco Marathon has beendesigned. After creating an analytical model which calculates the parameters of the ma-chine, it is analyzed in FEMMR©. First of all, the topology of the machine was selected.In order to improve the drive train as much as possible, gearboxes should be removed.This leads to an low-drive direct IW motor design. Several topologies of PM machine arequalitative studied, radial (SPM, Inset, IPM) and axial, showing their advantages and theirdrawbacks concerning the design. Finally, based on this qualitative study and some designconstraints, such as the air-cooling system or economic reasons (axial design), a radialSPM synchronous machine of inner rotor was chosen for the design. After that, the re-quirements of the machine are calculated. According to the requirements of the Shell EcoMarathon 2013 urban concept prototype and based on dynamic Newton’s law, the max-imum speed of the machine and the rated torque and the overload torque are calculated.Regarding the winding arrangements, concentrated windings have been demonstrated thebest solution in efficiency terms, since they have the smallest end-windings. However, theselection of the number of the poles and slots, since they canperform unbalance forcesor required with a high level of torque ripple. In order to avoid this configurations, lotsof configurations cannot be implemented. Following, an analytical model is developed tocalculate the dimensions of the machine and some parametersof the machine, based onits magnetic and electric behavior, and some design constraints. The machine resulting isanalyzed through an own developed program created in FEMMR©. This program for con-centrated windings, entailed several difficulties as the design of the winding layout andto determine the initial position of the rotor to apply the Park transformation. Finally, themachine calculated in the analytical program, is implemented in the FEMMR© program,

79

Chapter 7. Conclusions and Further Work

and adjusted some parameters such as the width of the tooth and the number of turns tofulfill the design constraints. The efficiency obtained at the rated point is not as high asexpected, mainly because the required power of the vehicle is small. Furthermore, thepermanent magnets have been analyzed at overload conditions, showing that the temper-ature that the magnets have to reach in order to be demagnetized has to be higher than200oC. The magnets should not achieve this temperature because according to [23], thecopper wires operate on higher temperatures, and the highest limit temperature for the Hinsulation is 180C. Moreover, the machine performs the requirements calculated withoutsurpassing the maximum phase voltage provided by the minimum voltage of the battery.The overload torque is reached over saturation conditions in the iron. To conclude, there isnot an analytical model implemented on the iron losses on themachine (almost neglectedsince it is a low-speed machine) and on the magnets. Permanent magnets exhibits a greatquantity of harmonics, which can lead to a big source of losses. Thus, they should be seg-mented axially. Regarding the results, iron losses are negligible if they are compared tocopper losses, because the IW is a low-speed application andthe electric frequency per-formed is quite low (32.1 Hz). The efficiency is lower than it was expected. The reasonof this is that the design of the motor is focused on a lightweight design, instead of a highefficiency design. Thus, it would be recommendable to decrease the current density andfind a better trade-off between a lightweight and a efficiencydesign.

7.2 Recommendations for Future Work

• The analytical model used could be improved. Teeth shoes have not been taken intoaccount, and they can vary the weight of the machine and the amount of PM needed,due to their relation to the karter factor. The inductance model can also be improvedalso. Besides, an analytical model for the torque ripple could be implemented.

• FEMM R© does not provide a result for iron losses in absence of frequency since thedesign of the FEMMR© has been done setting the frequency parameter to 0 due tospecifications of the program for PM. Thus, the iron model should be tested in otherprogram as FLUX R©, or JMAGR©.

• A thermal model should be implemented to check the most vulnerable parts of themachine. This could be accomplished through the creation ofa lumped circuit, andtest it with MOTOR-CAD R© , or on the other hand, in order to improve the programdesigned in this thesis, create an thermal program based on FEMM R©.

• PM losses model could be implemented in FEMMR© and tested it in a reliablesoftware.

• In the program developed, end-winding has been introduced in the drop voltageof the windings. However, leakage winding inductance has not been taken into ac-

80

7.2. Recommendations for Future Work

count. An analytical model could be designed to be implemented in FEMMR© andchecked the end winding effect on FEMMR© with a 3D FEM software.

• Other alternatives discussed on Chapter 2 could be analyzed, since some of themmight seem better to achieve the targets of ”Shell Eco Marathon 2013”. SPM innerdesign improves the properties of this model, but its study should be coupled witha careful study of the thermal transient of the machine, since it is air cooled.

81

Chapter 7. Conclusions and Further Work

82

Appendix A

Dynamic simulation

The simulation created in Simulink which is attached further in this appendix, is an idealsimulation only based on Newton’s second law which does not take into account importantparameters as the efficiency map of the motor and the converters. The power that the motorhas to perform is described in (A.1).

Pmot = Pdyn + Pdrag+ Prr (A.1)

whereFmot is the driving force performed by the motor,Fdyn is the dynamic force,Fdrag isthe wind resistance andFrr is the rolling resistance. The dynamic force which is the oneinvolved in the acceleration of the vehicle is expressed as

Pdyn = mvdv

dtv (A.2)

wheremv is the mass of the vehicle, included the driver, anddvdt

is the acceleration of thevehicle. The rolling resistance force can be expressed as

Prr = µrrmvgv (A.3)

whereµrr is the rolling resistance coefficient andg is the gravity acceleration. The windor drag resistance is given by

Pdrag =1

2cdAfρairv

3 (A.4)

wherecd is the drag coefficient,Af is the vehicle’s front area,ρair is the density of theair andv is the speed of the vehicle. Since there is only one wheel hub and there is nogearbox, the torque required by the motor is.

Tm =R

v[mv

dv

dtv + µrrmvgv +

1

2cdAfρairv

3]. (A.5)

whereR is the radius of the wheel.

83

Appendix A. Dynamic simulation

Drag Consumption

One of the purposes of the cycle, is finishing the cycle with a fixed average speedvav =

27km/h. Depending on the interval of acceleration∆t illustrated in Fig. 3.1 the maximumspeed varies its value

vmax = vavT

T −∆t(A.6)

and as well the acceleration varies with different intervals

a =vmax

∆t(A.7)

To calculate the energy dissipated by the wind resistance

Edrag =

T∫

0

Pdrag =

T∫

0

Cv(t)3dt (A.8)

whereC = 12cdAfρair. Integrating this expression throughout the speed cycle, the result is

the following

Edrag = Cv3max(T − 3

2∆t) = C

(

vavT

T −∆t

)3(

T − 3

2∆t

)

= C2

T − 32∆t

(T −∆t)3(A.9)

Rolling Resistance Consumption

This section gives an expression of the energy consumed by the rolling resistance.

Err =

T∫

0

Prr =

T∫

0

µrrmvgv(t)dt (A.10)

Err = µrrmvgvavT (A.11)

Acceleration Consumption

This section computes the energy used during the acceleration of the vehicle.

Edyn =

T∫

0

Pdyn =

∆T∫

0

mvav(t)dt (A.12)

Edyn =1

2mvvav

(T

T −∆t

)2

(A.13)

84

Appendix B

Analytical Motor Data

Motor 1Motor CharacteristicsRated torque Nm 7Overload torque Nm 22Rated speed rpm 275Max. fund. phase rated current (rms) A 13.48Max. fund. phase overload current (rms) A 43.45Max. fund. current density (rms) A/mm2 3/

√2

Min. dc-link voltage V 32.4Geometrical spec.Number of slots - 18Number of poles - 14Physical air gap length mm 1Shaft diameter mm 124.878Outer Stator diameter mm 192.826Core diameter mm 135.829Total slot width mm 15.17Total slot depth mm 18Tooth width mm 11.01Thickness of stator back mm 5.48Thickness of rotor back mm 5.48Active length mm 20.3Magnet thickness mm 3.84Active weight kg 2.89Electrical data

85

Appendix B. Analytical Motor Data

Phase resistance mΩ 130.2Permanent magnet flux linkage Vs 0.0526d−axis air gap inductance mH .95q−axis air gap inductance mH .95d−axis current at rated operation A 0q−axis current at rated operation A 13.48Loss dataCopper losses at rated operation W 34.32Iron losses at rated operation W 1.53Efficiency at rated operation % 82.21Winding spec.Number of poles - 14Number of slots - 18Number of turns per slot - 58Fundamental winding factor - 0.902Copper fill factor - 0.577Flux density dataAir gap peak fund. density - 0.81Peak fund. flux density in stator back T 1.505Peak fund. flux density in stator teeth T 1.521Peak fund. flux density in rotor back T 1.506Magnet dataRemanent flux density at operating temp. T 1.18Relative permeability - 1.05Magnet density kg/m3 7700Steel dataEddy current loss density constant Ws2/kgT2rad2 4.26·10−5

Steinmetz constant - 2Hysteresis loss density constant Ws/kgTβrad 0.0269Steel density kg/m 7700Other spec.Copper resistivity at operating temp. nΩm 23.9Copper density kg/m3 8390

86

Appendix C

Fundamental Winding factor

In this appendix characteristics of different pole-slots combinations are deployed. Fig. C.3describes the values of the fundamental winding factorkw1 for all the pole-slots combi-nation, Fig. C.2 shows the configurations which are not desirable due to the unbalancemagnetic pull and Fig. C.1 illustrates the variation of theLCM(Qs, p), which is re-lated with the cogging torque. The contents of the followingtables have been retrievedfrom [16] [13].

Fig. C.1 Lowest common multiple of different pole-slots combinations,LCM(Qs, p). The higherthis value, the lower is the cogging torque.

87

Appendix C. Fundamental Winding factor

Fig. C.2 Great common divisor of different pole-slots combinations, GCD(Qs, p). Configura-tions thatGCD(Qs, p) = 1, perform unbalance magnetic pull.

88


Fig. C.3 Fundamental winding factor of different pole-slots combinations for concentrated wind-ings. Different colors illustrates some winding layout which is repeated.kw1(q)

89


90

Appendix D

Iron Loss Constants

As mentioned in section 5.3.2, eddy current constant,ke, and hysteresis constant,kh,depend on the material used. In this appendix both constantsare calculated by the leastsquare method, considering the Steintmetz constant,βst = 2. Table D.1 shows the lossdensity data given by Surahammars Bruk AB. Changing equation 5.58 to the conditions

Table D.1: Iron loss density data from SURAR© M235-35A [27]B(T) W/kg W/kg W/kg W/kg W/kg W/kg

(50Hz) (100Hz) (200Hz) (400Hz) (1000Hz) (2000Hz)

0,1 0,02 0,04 0,08 0,19 0,93 3,890,2 0,06 0,14 0,32 0,87 3,55 14,30,3 0,11 0,3 0,73 1,88 7,45 29,60,4 0,2 0,49 1,21 3,17 12,3 50,20,5 0,29 0,71 1,78 4,73 18,5 76,70,6 0,38 0,97 2,44 6,56 25,8 1100,7 0,5 1,25 3,19 8,67 34,6 1530,8 0,62 1,57 4,03 11 45 2050,9 0,77 1,92 4,97 13,8 57,2 2701 0,92 2,31 6,01 16,9 71,5 3491,1 1,1 2,75 7,19 20,3 88,31,2 1,31 3,26 8,54 24,31,3 1,56 3,88 10,1 28,91,4 1,92 4,67 12,2 34,81,5 2,25 5,54 14,4 41,21,6 2,531,7 2,751,8 2,94

exhibited above

piron = khystB2f + keddyB

2f 2 (D.1)

91

Appendix D. Iron Loss Constants

This equation can be adapted to calculateke andkh. Consideringpf the iron loss densityfor each frequency

pf =

0.12

...1.82

[f f 2

]

︸︷︷︸

[Af ]

[khystkeddy

]

(D.2)

wheref is the frequency, and

0.12

...1.82

is the flux density for every loss density in the table.

Since[Af ] is not an square matrix, in order to obtainke andkh, there must to be multipliedby its transpose matrix to be able to calculate its inverse

[Af ]T pf = [Af ]

T [Af ]

[khystkeddy

]

(D.3)

[khystkeddy

]

=[

[Af ]T [Af ]

]−1

[Af ]T pf (D.4)

92

Appendix E

List of Symbols, Subscripts andAbbreviations

Symbolsa accelerationbss1 slot top widthbss2 slot base widthcd drag coefficientcf Carter factorcp specific heat at constant pressured ratio between rotor diameter and slot height diametere induced electromotive forcefelec electric frequencyfcog cogging frequencyff copper fill factorg gravity accelerationhm magnet heighthry rotor yoke heighthss stator slot heighthsy height of the stator yokei peak fundamental phase current, integerkw1 fundamental winding factorkj stacking factorkopen slot openingmv vehicle weightns number of turns per slotnl number of winding layersntooth number of coils per toothp number of poles

93

Appendix E. List of Symbols, Subscripts and Abbreviations

q number of slots per pole and phaser resistivityt time, gearbox ratio, machine periodicityu peak fundamental voltagev velocitywtooth tooth widthw magnet segment widthAf vehicle front areaAg airgap areaAcond stator slot areaAcu bare copper area in stator slotsB magnetic flux densityBm magnetic flux density above the magnetsBr remanent magnetic flux density, radial flux densityBr,m remanent magnetic flux density at rated temperatureBr,0 remanent magnetic flux density at ambient temperatureBt tangential flux densityBknee demagnetization magnetic flux density in the magnetsBry magnetic flux density in the rotor yokeBsy magnetic flux density in the stator yokeBtooth magnetic flux density in the stator teethBδ peak fundamental air gap flux densityD rotor diameterDext external diameterDi shaft diameterE peak fundamental phase voltageFd driving forceFrr rolling resistance forceFw wind resistance forceH magnetic field intensityHc coercive field strengthiHc intrinsic coercivityJ peak fundamental current densityLact motor active lengthLcoil coil lengthLend end winding lengthLm magnetic inductanceLs synchronous inductanceLσ leakage inductanceN number of turns per phase

94


Nspk number of spokesP power lossQs number of slotsR wheel radiusRs stator resistanceRslot stator slot resistanceS peak fundamental current loading~S Cros’ method layout vectorT temperatureTe electro-mechanical torqueT0 ambient temperatureTpeak overload torqueTcog cogging torqueTr rated temperatureTripple ripple torqueTsteady steady-state torqueVrot volume of the rotorWm magnetic energy in the airgapα electric angle covered by one magnet, coefficient, angle between two adjacent slotsαe electric angle between two adjacent slotsαelec electrical angle covered by one magnetαph angle between two spokesβ Steinmetz constantδ airgap lengthδe effective airgap length∆t acceleration intervalθ rotor angle in adq referenceθm mechanical angleθel electrical rotor angle (position)γ shift electric angle of stator coilsλ1 specific coefficient of the slot openingµ magnetic permeabilityµ0 magnetic permeability of vacuumµr relative magnetic permeabilityµrr rolling resistance coefficientρ electrical resistivity, densityρcu copper densityρm magnet densityρiron steel densityσ shear stress

95


τs slot pitchτedge torque contribution of each slot in cogging torqueφ magnetic fluxφm magnetic flux in the magnetsφsy magnetic flux in the stator yokeφry magnetic flux in the rotor yokeφtooth magnetic flux in the stator teethψ flux linkageψm permanent magnet flux linkageω electrical rotor speed, angular frequency of the current

Subscriptsa phase a componentd direct-axis component0 ambient temperatureq quadrature-axis componentr rated valueav averagecu coppereddy eddy currentsmag magnethyst hysteresismax maximummin minimum

Abbreviationsdc direct currentrms root mean squareCW concentrated windingsEMF electromotive forceEV electric vehicleFEM finite element methodFFT fast Fourier transformationGCD greatest common divisorHEV hybrid vehicleIPM interior mounted permanent magnetIW in-wheel motorLCM lowest common multipleMMF magnetomotive forcePM permanent magnet

96


PMSM permanent magnet synchronous machineSPM surface mounted permanent magnet

97


98

References

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[4] N. Bianchi and M. D. Pre, “Use of the star of slots in desingning fractional-slotsingle-layer synchronous motors,”IEE Proc.-Electr. Power Appl., Vol. 153, No. 3,May 2006.

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[6] C. C. Chan and K. T. Chau,Mordern electric vehicle technology. New York, USA:Oxford Science Publications, 2001.

[7] J. F. Gieras and M. Wing,Permanent magnet motor technology. New York, USA:Marcel Dekker, Inc., 2002.

[8] D. D. Hanselman,Brushless Permanent Magnet Motor Design. Orono, USA: TheWriters’ Collective., 2003.

[9] P. F. J. A. Guemes, A. M. Iraolagoitia and M. P. Donsion,“Comparative study ofpmsm with integer-slot and fractional-slot windings,”ICEM, XIX International Con-ference on Electrical Machines.

[10] R.-J. W. Jacek F. Gieras and M. J.Kamper,Axial Flux Permanent Magnet BrushlessMachines. Springer Science, 2008.

[11] T. J. Juha Pyrhonen and V. Hrabovcova,Design of Rotating Electrical Machines.Lappeenranta University of Technology, USA: John Wiley andSons, 2008.

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[12] H. Jussila, P. Salminen, M. Niemela, and J. Pyrhonen, “Guidelines for design-ing concentrated winding fractional slot permanent magnetmachines,” POW-ERENG2007 International Conference on Power Engineering -Energy and Elec-trical Drives Proceedings, pp. 191–194, Apr. 2007.

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