Analysis of interior permanent magnet motors with non-overlapping windings

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non-overlapping windings

Johannes Ja obus Germishuizen

Dissertation approved for the degree of Do tor of Philosophy

in Ele tri al Engineering at Stellenbos h University

Promoters:

Prof. M.J. Kamper

Ele tri aland Ele troni Engineering

University of Stellenbos h

South Afri a

Dr. A. Jö kel

Te hnology and Innovation

Loher GmbH

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By submitting this dissertation ele troni ally, I de lare that the entirety of the work

ontained thereinis my own, originalwork, that I amthe owner ofthe opyrightthereof

(unless to the extent expli itly otherwise stated) and that I have not previously in its

entirety or inpart submitted it for obtainingany quali ation.

Date:02.03.2009

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Analysis of interior permanent magnet motors with

non-overlapping windings

J.J.Germishuizen

Department of Ele tri al and Ele troni Engineering

University of Stellenbos h

Private Bag X1, Matieland,7602, South Afri a

PhD Dissertation

Mar h 2009

At present most of all existing variable speed drive systems are still based on indu tion

ma hines. In order to redu e energy loss, investment and maintenan e osts there is a

trendtorepla ethe indu tionmotorand gearboxwitha dire tdrivesystem. However, it

wasonly untilre ently that the progress in materials ien emade it possible to

e onom-i ally implementma hines with permanent magnet te hnology in a dire t drive system.

Interior permanent magnets motors with single layer non-overlapping windings have

ad-vantages,whi hmakeitattra tiveforthe useindire tdrives. Inthe presentdissertation

afast and a urate analysis methodology, suitablein an everyday designenvironment is

oered. The method makes use of both the analyti al and nite element method. Part

ofthe designpro edure requires asystemati algorithmtoallo atethe statorslots tothe

windingphase belts.

The a ura y of the proposed method was veried by means of a prototype tra tion

ma hine. Fromtheresultsitis on ludedthatthreetwo-dimensionalfun tions anbeused

to a urately al ulate the performan e of permanent magnet ma hines in an everyday

designenvironment. The analysis methodwhi h wasdeveloped isbased on real physi al

two-dimensional ma hine models. The unique ontributions of the present dissertation

are the performan e al ulation method and anexpression that denes a windingphase

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Analise en ontwerp van elektiese motors met permanente

binnemagnete en nie-oorvleuelende wikkelings

J.J.Germishuizen

Departement van Elektrieseen Elektroniese Ingenieurswese

Universiteit van Stellenbos h

Privaatsak X1, Matieland,7602, Suid-Afrika

PhD Proefskrif

Maart 2009

Die meeste veranderlike spoed aandryfstelsels is tans gebaseer opinduksiemasjiene. Om

energie verliese, beleggings en onderhoudskoste te verminder, is daar 'n tendens om die

induksiemasjien en ratkas te vervang met n direkte aandryfstelsel. Dit is eers onlangs

dat vooruitgangin die materiaalwetenskappeditmoontlikgemaak het ommasjienemet

permanent magneet tegnologie ekonomies in n direkte aandryfstelsel te implementeer.

Elektiese motors met permanente binnemagnete en nie-oorvleuelende wikkelings het

vo-ordele wat ditbaie aantreklik maak vir die gebruik in direkte aandryfstelsels. In hierdie

proefskrif word nvinnige en akkurate analise metode geskik vir n daaglikse

ontwerps-omgewing voorgestel. Die metode maak gebruik van beide die analitise en die eindige

element metode. n Deel van die prosedure benodig n sistematiese algoritme om die

statorgleuwe aan die wikkeling-fase-gordelstoe teken.

Die akkuraatheid vandie metode word geverieer deur middelvannprototipe

trak-sie motor. Uit die resultate word die gevolgtrekking gemaak dat drie twee-dimensionele

funksies gebruik kan word om in ndaaglikse ontwerpsomgewing die werkverrigting van

permanentmagneet elektriesemasjiene akkuraat te kan bereken. Die analisemetode wat

ontwikkel is, is gebasseer op die werklike siese twee-dimensionele masjien model. Die

unieke bydrae van hierdie proefskrif is die werkverrigtingsberekengsmetode en 'n

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I would like to thank my promoters, Prof. Maarten Kamper and Dr. Andreas Jö kel,

for their onstant support and guidan e throughout this proje t. I also a knowledge the

following persons and foundationfor their ontributions:

TheexternalexaminersProfessorsBerndPoni k(UniversityofHanover,Germany),

Konrad Rei hert (Swiss Federal Institute of Te hnology Züri h, Switzerland) and

Frikkie van der Merwe (Stellenbos h University, South Afri a) for their valuable

omments and advi e;

Prof. Heinri hvander Mes ht for ontinuous support with the dissertation layout;

My olleagues at Siemens AG (Vogelweiherstraÿe, Nürnberg), parti ularly Peter

E kert and ThomasS hmidt for their assistan e;

Prof.HansOttoSeins h(University ofHanover)and Prof.Zdeneke°ovský

(Te h-ni al University of Prague)who kindly provided me with detailson VilémKlíma;

IvanKlímaforproof-readingmyessayonhisfather,VilémKlíma,andforproviding

the photosof his father;

Willie Coetzeefor his un easing interest inmy work and

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De laration ii Abstra t iii Uittreksel iv A knowledgements v Dedi ations vi Contents vii

List of Figures xii

List of Tables xv

Nomen lature xvi

1 Introdu tion 1

1.1 Ba kground tothe study . . . 1

1.2 Problem statement . . . 3

1.3 Overlapping and non-overlapping windings . . . 3

1.4 Approa h to the problem . . . 5

1.5 S ienti ontributions of this dissertation . . . 6

1.6 Delimitationsof the study . . . 6

1.7 Layout of the dissertation . . . 7

1.8 Di ulties en ountered during the study . . . 7

1.9 Notes tothe reader . . . 8

2 Literature overview 9 2.1 A histori alview of windingdesign . . . 9

2.2 Tenden ies inma hine analysis . . . 11

2.2.1 Introdu tory remarks . . . 11

2.2.2 The nite elementmethodas adesign tool . . . 12

2.2.3 The inuen e of saturation ontransient solutions . . . 12

2.2.4 Present approa hes iniron loss al ulation . . . 13

2.3 A short ex ursion . . . 14

2.3.1 Coming a ross avery interesting sour e . . . 14

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3 Design and analysis of stator windings 17

3.1 Denition of the working harmoni . . . 17

3.2 Classi ation of symmetri alwindings . . . 17

3.2.1 Slots and oils perpoleand phase . . . 18

3.2.2 Average oilpit h . . . 18

3.2.3 Classi ation s heme . . . 19

3.3 Chara teristi s of symmetri alwindings . . . 20

3.3.1 Basi winding . . . 20

3.3.2 Winding symmetry . . . 22

3.3.3 Redu ed number of polepairs . . . 22

3.4 Rotating mmf . . . 23

3.4.1 The mmf of a single turn oil . . . 23

3.4.2 The mmf of three single turn oils . . . 24

3.4.3 Denition of the mmf envelope fun tions . . . 27

3.4.4 Phase beltdenition . . . 27

3.4.5 Higher order harmoni s. . . 27

3.5 Matrix representation of a winding . . . 28

3.5.1 Slot ve tor . . . 28

3.5.2 Phase belt onstraint . . . 29

3.5.3 Algorithm ow hart . . . 30

3.5.4 Matrix element assignment. . . 30

3.6 Properties of the windingmatrix . . . 32

3.6.1 Winding fa tor . . . 32

3.6.2 Current sheet anti-node axis . . . 33

3.6.3 Magneti axis . . . 33

3.6.4 Slot mmf. . . 34

3.7 Examples . . . 34

3.7.1 Winding fa tor table . . . 34

3.7.2 Slot mmf and urrent sheet . . . 36

3.7.2.1 Single layernon-overlapping . . . 36

3.7.2.2 Double layernon-overlapping . . . 37

3.7.2.3 Single layeroverlapping . . . 37

3.7.2.4 Double layeroverlapping . . . 41

3.8 Determination of the winding axes . . . 41

3.9 Summary . . . 41

4 Nonlinear magneti ir uit analysis 44 4.1 Introdu tory remarks . . . 44

4.2 Material properties . . . 45

4.2.1 Laminated steel . . . 45

4.2.1.1 Epstein framemeasurements . . . 45

4.2.1.2 Extrapolation of the B(H) Curve . . . 46

4.2.1.3 Three-term ore lossmodel . . . 48

4.2.1.4 Two-term ore loss model . . . 50

4.2.1.5 A ounting for the sta king fa tor in 2D FEA . . . 50

4.2.2 Permanentmagnets . . . 51

4.3 Nonlinear eld solutions . . . 52

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4.3.1.1 Ampère'slaw . . . 53

4.3.1.2 Faraday's law . . . 53

4.3.2 Boundary setupand sour es . . . 53

4.3.2.1 Diri hlet boundary onditions . . . 53

4.3.2.2 Periodi boundary onditions . . . 54

4.3.2.3 Current sour es . . . 54

4.3.2.4 Voltage sour es . . . 55

4.3.3 Flux linkage al ulation . . . 56

4.3.4 Indu tan e al ulation . . . 56

4.3.4.1 Denitions for energy al ulation . . . 57

4.3.4.2 Indu tan e matrix . . . 57

4.4 Determination of the winding axis: se ond method . . . 59

4.5 Harmoni analysis. . . 60

4.5.1 Denition of the dis rete Fourier transform . . . 60

4.5.2 Time harmoni s . . . 61

4.5.3 Spatial harmoni s . . . 61

4.6 Example of a spatialharmoni analysis . . . 63

4.6.1 Air gap ux density . . . 63

4.6.1.1 Two-dimensionalairgap uxdensity fun tions . . . 63

4.6.1.2 Harmoni omponents . . . 65

4.6.2 Flux perpole . . . 68

4.7 Summary . . . 68

5 Tra tion ma hine ase study 69 5.1 Introdu tory remarks . . . 69

5.2 Torque versus speed hara teristi . . . 70

5.3 Con eptional design. . . 71

5.3.1 Winding type . . . 71

5.3.1.1 Form-wound versus random-wound oils . . . 71

5.3.1.2 Single versus double layernon-overlapping windings . . . . 73

5.3.2 Sizing equations . . . 73

5.3.3 Numberof stator slots . . . 74

5.3.4 Ele tromagneti design . . . 74

5.3.5 Winding properties . . . 75

5.3.5.1 Ee tivenumberof turns per slot . . . 75

5.3.5.2 Series numberof turns . . . 75

5.3.5.3 End windingleakage indu tan e. . . 76

5.3.5.4 Stator d. . windingresistan e . . . 76

5.3.5.5 Winding layout . . . 77

5.3.6 Geometri al dimensioning . . . 77

5.3.6.1 No-load ux linkage versus tooth width . . . 78

5.3.6.2 Torque pulsation . . . 78

5.3.6.3 Rotor design . . . 79

5.3.6.4 Staggered rotor design . . . 81

5.3.6.5 Stator parameters . . . 82

5.4 Loss al ulation . . . 82

5.4.1 Lo aleddy urrent loss . . . 82

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5.4.1.2 Single layerwindings . . . 84

5.4.1.3 Round wire . . . 84

5.4.2 Cir ulating urrent loss. . . 86

5.4.3 Stator windinglosses . . . 86

5.4.4 Iron loss . . . 86

5.4.5 Eddy urrents inthe permanentmagnets . . . 87

5.5 Two-dimensional ma hine hara teristi fun tions . . . 88

5.5.1 Park transformation . . . 89

5.5.2 General motorvoltage and torque equations . . . 91

5.5.3 Equivalent ir uit . . . 92

5.5.4 Solution domain. . . 93

5.5.5 Torque versus frequen y hara teristi . . . 93

5.5.6 S aling of the solutiondomain . . . 96

5.6 Realisationof a prototype . . . 97

5.6.1 Manufa turedmotor . . . 97

5.6.2 Stator d. . resistan e measurement . . . 98

5.6.3 No-load measurements . . . 98

5.6.3.1 Open terminaltest . . . 98

5.6.3.2 Magnetising urrent . . . 100

5.6.4 Short ir uit test . . . 101

5.6.5 Removed rotor test . . . 101

5.6.6 Torque speed hara teristi . . . 103

5.6.7 Temperature rise test . . . 103

5.6.7.1 Measured results . . . 106

5.6.7.2 Adiabati rotor loss . . . 106

5.6.7.3 TransientFEA . . . 108

5.6.7.4 Loss al ulations . . . 108

5.7 Summary . . . 110

6 Con lusions and re ommendations 111 6.1 Introdu tory remarks . . . 111

6.2 Answering the resear h sub-questions . . . 111

6.2.1 What isthe mathemati alexpression to allo ate the stator slots? . 111 6.2.2 How an a windinglayout be represented ina ompa t form? . . . 112

6.3 Answering the main resear h question . . . 112

6.4 Re ommendations for further resear h . . . 113

Appendi es 114 A Klíma's losed expression 115 A.1 Introdu tory remarks . . . 115

A.2 Distribution fa tor . . . 115

B Slot ombinations 116 C Revision of the Air Gap Element 119 C.1 Introdu tory remarks . . . 119

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C.2.1 AGE stiness matrix . . . 121

C.2.2 Time-savings heme. . . 121

C.2.2.1 Simplied Fourier oe ients . . . 122

C.2.2.2 Stepped AGE . . . 123

C.2.3 Periodi ity onditions . . . 123

C.2.4 Ele tromagneti torque . . . 124

C.2.5 Air gap ux density . . . 125

C.3 Cambridgesoftware . . . 125

C.3.1 History of the Cambridgesoftware . . . 126

C.3.2 Alternative formof the time-savings heme . . . 126

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1.1 Double layeroverlapping and single layernon-overlapping windings . . . 4

1.2 Double and single layernon-overlapping windings . . . 5

2.1 Leakage ompensationinthe stator teeth . . . 11

2.2 VilémKlíma, 10.04.1906-06.10.1985 . . . 16

3.1 Classi ationof symmetri al

m

-phase windings . . . 21

3.2 Spatialmmf distribution fora

N

t

-turn oil . . . 23

3.3 The fundamentalspatial mmf distributionfor a three-phasema hine . . . 26

3.4 Rotationof the resultantmmf . . . 26

3.5 Star of slots . . . 29

3.6 Flow hart to allo atethe statorslots . . . 31

3.7 Non-overlapping single layer winding . . . 38

(a) Winding layout . . . 38

(b) Slot mmf and windingfa tors . . . 38

3.8 Non-overlapping double layerwinding. . . 39

(a) Winding layout . . . 39

3.9 Single layer overlapping winding . . . 40

(a) Single layeroverlapping winding layout . . . 40

3.10 Double layeroverlapping winding . . . 42

(a) Double layer overlapping windinglayout . . . 42

3.11 Determination of the windingaxes . . . 43

4.1 Measurement setup to measure

J(H)

for laminated steel . . . 46

4.2 The measured data for M470P65A laminationsteel . . . 47

(a) Frequen y dependen y of

J(H)

. . . 47

(b) Spe i loss . . . 47

4.3 The extrapolationof the

B(H)

for M470P65A laminationsteel . . . 48

4.4 Determination of oe ients

k

h

and

k

c

. . . 50

4.5 Permanent magnetmaterial properties . . . 51

4.6 Finiteelementmodel showing the dierent materialsand boundary onditions 54 4.7 Contours

C

1

and

C

2

to al ulate the uxlinkage and indu tan e . . . 56

4.8 Denitionsfor energy al ulation . . . 57

4.9 Saved permeability for a given operating point . . . 59

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4.11 Even and odd boundary onditions . . . 62

4.12

B

d

and

B

q

as two-dimensional fun tions . . . 64

(a)

B

d

asa fun tionof

(b) Air gap ux density in ludingthe stator slots . . . 66

4.14 Air gap harmoni analysis with

I

1 6= 0

. . . 67

(a) Spatial harmoni analysis with

i

d

> 0

. . . 67

(b) Spatial harmoni analysis with

i

q

> 0

. . . 67

5.1 Tra tion ma hine torque speed hara teristi . . . 71

5.2 Typi alsingle and double layerwinding layouts . . . 72

5.3 Geometri al ma hine parameters . . . 75

5.4 End winding parameters . . . 76

5.5 Winding layout . . . 77

5.6 No-load eld solutionusing FEMP . . . 79

5.7 Tooth widthas design parameter . . . 80

(a)

ψ

1

as a fun tionof tooth width . . . 80

(b) Coggingtorque . . . 80

5.8 Air gap uxdensity harmoni s due to the magnets . . . 81

5.9 Staggered rotor . . . 82

5.10 Condu tors lo atedin a stator slotof anele tri alma hine . . . 85

(a) Single and doublelayer overlapping windings . . . 85

(b) Single layer on entrated oil . . . 85

5.11 Eddy urrentsin the permanent magnets . . . 88

5.12 Redu tion fa tor for the permanent magnet's ondu tivity . . . 89

5.13 Design loop . . . 90

(a) Existing pro ess. . . 90

(b) Proposed pro ess . . . 90

5.14 Cross se tionof ma hine and ve tor diagramfor

qd

-variables . . . 91

5.15 Steady-state

d

- and

q

-axes equivalent ir uits . . . 92

5.16 Torque as a fun tionof

i

d

and

i

q

. . . 94

5.17 Two-dimensional uxlinkage fun tions . . . 95

(a)

ψ

d

asa fun tion of

i

d

and

i

q

. . . 95

(b)

ψ

q

as afun tion of

i

d

and

i

q

. . . 95

5.18 Findinga validoperating point . . . 96

5.19 Prototypetra tion ma hine . . . 97

5.20 Measured and al ulatedno-load voltageat

62.5 Hz

. . . 99

5.21 Measured and al ulatedno-load losses . . . 100

5.22 No-load magnetising urrent at

40 Hz

. . . 101

5.23 Removed rotor test . . . 104

(a) Measured inputpower with

100 A

. . . 104

(b) FEA resultsfor the removed rotor test . . . 104

5.24 Measured stator urrent for torque hara teristi . . . 105

5.25 Measured and al ulatedterminal voltage . . . 105

5.26 Measured and al ulatedtorque . . . 106

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5.28 Cal ulated results fromTab. 5.13 (

U

p

= 337.8 V

p = 15

. . . 118

C.1 Finiteelementmesh with Air Gap Element. . . 120

C.2 Ve torpotential solutionwith positiveperiodi boundary onditions . . . 124

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1.1 Dieren e between single and doublelayer windings . . . 4

1.2 Typi alma hine quantities and their German translation . . . 8

2.1 Klíma'sentry in the listof le turers inthe ghetto Theresienstadt . . . 14

3.1 Properties of singleand doublelayer windings . . . 18

3.2 Constraintsfor windingsymmetry. . . 22

3.3

ξ

p

× 10

−3

for single layernon-overlapping windings . . . 35

3.4

ξ

p

× 10

−3

for doublelayer non-overlapping windings . . . 35

3.5 Dierent polepair ombinationswith

Q

s

= 30

. . . 36

4.1 Multiple regression results forM470P65A laminationsteel . . . 49

4.2 Spe i lossfor M470P65A at

1 T

. . . 50

4.3 Properties of VAC677AP . . . 51

5.1 Drivespe i ation (maximum values) . . . 70

5.2 Maindieren es between form- and random-wound oils . . . 72

5.3 Summaryof single and double layerwinding oils . . . 73

5.4 Slot and pole pair ombinations for ase study design . . . 74

5.5 Winding fa tors when hangingthe slot pit h (

xτ

s

) . . . 78

5.6 Parameters of the prototype stator . . . 83

5.7 Ironloss al ulation at

20

(

B

r

= 1.08 T

) . . . 87

5.8 Resistan e measurement of the windings . . . 98

5.9 No-load measured data with open terminals . . . 99

5.10 Short ir uittest measured results . . . 102

5.11 Removed rotor loss al ulations . . . 103

5.12 Temperaturerise test at

70 Hz

. . . 107

5.13 Cal ulated FEAresults at

70 Hz

. . . 108

5.14 Tempeaturerise test loss al ulations . . . 109

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Variables

a

number of parallelbran hes, (5.3.2)

a

w

number of parallelwires, (5.3.2)

k

b

Boules's redu tion fa tor, Fig. 5.12

k

r

average loss oe ient for lo aleddy urrents,(5.4.1) - (5.4.4)

k

rc

N

t

number of turns per oil, se tion 3.4.1

p

number of polepairs, se tion 3.2.1

p

b

number of polepairs of the basi winding, (3.3.2)and (3.3.3)

q

number of slots per poleand phase, se tion 3.2.1

q

c

number of oils perpoleand phase, se tion3.2.1

Q

b

number of slots of the basi winding,(3.3.2)

Q

s

number of statorslots, se tion 3.2.1

t

greatest ommondivisor, (3.3.1) and (3.3.4)

y

d

oilpit h,(3.2.3)

y

p

average oilpit h,(3.2.4)

z

average numberof turns perslot, (5.3.2)

ν

harmoni oder, (3.4.14)

ξ

windingfa tor, (3.6.3)

µ

r

relativepermeability, (4.2.12)

Variables with units

A

Cu

opper ross se tion area,(5.3.7) . . . [

m

H

magneti eld intensity, Fig. 4.5 . . . [

A·m

−1

℄

H

c

magneti eld oer ivity intensity, Fig. 4.5 . . . [

A·m

peak phase urrent,Fig. 5.14 . . . [

A

℄

I

1 rms

phase urrent,(5.5.7) . . . [

A

℄

J

spe i iron loss, (4.2.6) . . . [

W·kg

−1

℄

P

F e

N·m

W

energy, (4.3.17) . . . [

J

℄

(18)

α

anglebetween

I

ˆ

1

and the

d

-axis,Fig. 5.14 . . . [

rad

℄

β

anglebetween

U

ˆ

1

and the

q

-axis, Fig.5.14. . . [

rad

λ

s

µ

0

permeability of freespa e

µ

0 = 4π10 × 10

−7

. . . [

H·m

−1

℄

τ

s

slotpit h,Fig. 3.6 . . . [

m

℄

τ

p

polepit h . . . [

m

℄

φ

anglebetween

U

ˆ

and

I

ˆ

A

ve tor potentialat AGE nodes, (C.2.22)

i

d

2D

d

-axis urrent matrix, (5.5.12)

i

q

2D

q

-axis urrent matrix, (5.5.12)

L

indu tan ematrix, se tion 4.3.4.2

M

windingmatrix, (3.5.1)

S

stiness matrix, (C.2.1)

T

used in(C.2.22) fortorque al ulation

T

2D torque matrix, (5.5.12)

v

slotve tor,(3.5.4)

X

regression matrix, (4.2.8)

Symbols

C

omplex numbers

N

positive integer numbers,

n ∈ Z

+

Z

integer numbers

. . . , −2, −1, 0, 1, 2, . . .

Z

+

integer numbers

1, 2, 3, . . .

Z

∗

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Introdu tion

Analysisofele tri alma hinesisaverywell-knownsubje tinele tri alengineeringandat

arstglan eonemightthinkthatthiseld annotbeembarkedonanyfurther. However,

this is not the ase, sin e advan es in materials su h as permanent magnets oer new

possibilities whi h might not have been e onomi al a few years ago. In addition to the

development of the materials used in ele tri al ma hines, the mathemati al and physi s

software toolsfor solving omplex problems andthe visualisationof the results have also

improved. Furthermore,the te hnologiesdriving themi ropro essorsand storagedevi es

usedinpersonal omputers have ontributed tosolve di ultmathemati alfun tions by

meansofnumeri almethods. Allthesefa torshaveinuen edthewayele tri alma hines

aredesignedandwillbedesignedinthefuture. Theresear hpresented inthisdissertation

is a typi al example of how advan es in other elds an lead to new ideas in ele tri al

engineering,and espe iallyele tri alma hines.

1.1 Ba kground to the study

At present most of all existing variable speed drive systems are still based on indu tion

ma hines. A typi alexample are tra tion drivesfor rail vehi les. Today, the state of the

artdrivesystem of metrotrainswith underoortra tionequipment onsistsof a3-phase

IGBT onverterfeedingtwoorfourindu tiontra tionma hinesinparallel. Forthetorque

transmissiontothe wheel-seta gearbox isneeded, be ausethe indu tionma hine annot

develop su ient torque in the available spa e. For the transport operator the gearbox

means investment, maintenan e, undesired noise and losses. Furthermore, Jö kel et al.

(2006a) mentioned that the very high torque developed during a terminal short ir uit,

whi harisesfromaninverterfailure,leads toover-sizedme hani al omponents. Inorder

to redu e energy loss, investment and maintenan e osts there is a trend to repla e the

indu tion motor and gearbox with a dire t drive system as pointed out by Jö kel et al.

(2006 ). This not only improvese ien y and performan e, but alsothe life y le osts,

asshown by Germishuizen et al. (2006).

The ma hine that is used in a dire t drive system requires a high torque to weight

ratio, high e ien y and a wide speed range for onstant power. The permanent

mag-net syn hronous ma hine fulllsthese requirements. Gieras and Gieras (2000) ompared

dierent ma hine types and showed that the permanent magnet syn hronous ma hine

has the highest torque density and highest e ien y. The ma hine an alsobe used for

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ma hinethepermanentmagnetsyn hronous ma hine anbedesignedwithagreater

vari-etyof windinglayouts. Theseadvantagesmakeitthereforefeasible todesigndire tdrive

tra tion ma hines using permanent magnet te hnology. Usually su h designs lead to a

ma hine with a high polenumberand often with afra tional slotwinding.

Ko hand Binder (2001)investigatedthe performan eof permanentmagnetma hines

for high speed trains. The suggested ma hine has a double layer winding omprising

form-wound oils. This winding type has a high opper ll fa tor and although it was

foundthat thiswindingtypehas agoodperforman eitisstillquiteexpensivedue tothe

highnumberofform-wound oils. Analternativewindingtypeisanon-overlappingsingle

layer on entratedwinding. Here onlyevery se ondstatortoothhas a oilwound around

it. Advantages are the shorter end-winding overhang, the simplied winding insulation

and a redu ed number of stator oils as pointed out by Cros and Viarouge (2002) as

well as Huth and Qian (2004). The shorter overhang means that the a tive length an

be in reased. As a result the amount of opper in the end-windings is redu ed. These

advantages improve the e ien y and lead to redu ed manufa turing osts. Drawba ks

are the lower power fa tor and possible rotor heating due to the harmoni ontent of

the air gap magnetomotive for e (mmf). In general the higher harmoni s in the air gap

(andasa resultthe in reased leakage indu tan e) ause higher ore lossesas reportedby

Magnussen and Sadarangani(2003).

Thedesignof thewindingused inanele tri alma hine isoftenunderestimated. This

doesnotne essarilyneedtobeseenasnegative,sin etypi allyamanufa turerofma hines

willuse approved designs. Su h apoli yhas a lotto ommend it. Resorting toapproved

designs means saving development osts and even a redu ed design phase. With new

materials available, older ideas whi h seem to be forgotten, an be investigated. Only

then the real omplexity of designing windings omesto the fore.

The idea to repla e the indu tion motor and gearbox with a dire t drive is ertainly

not new. However, it was only until re ently that the progress in material s ien e made

it possible to e onomi ally implement su h a on ept. On 12 August 2008 the Muni h

publi transport ompany MVG started the operation of an underground train that is

equipped with dire t drive te hnology as reported by Tibudd (2008). It is the rst of

its kind and is a pra ti al example of an innovative strategy and approa h to integrate

tra tion, bogie and braking te hnology. This new motor bogie te hnology weighs

30 %

less than onventional state of the art motor bogies. The expe ted redu tion in energy

lossis estimated to be

20 %

less than the present system in operation and willbe tested over a periodof twoyears.

Due to the in reasing demand to redu e manufa turing ost and weight, ma hine

designstendtobe omemore ompa t. Thisleadstoanin reaseinthetorquepervolume

ratio. As aresult,newmaterialswith highmagneti energydensitysu has

Neodymium-Iron-Boron(NdFeB)allowavery ompa tdesigninwhi h aseparti ularlythepermanent

magnetsyn hronous ma hinebenets. Inaddition,theemphasisona urateperforman e

estimations be omes essential and requires in many ases the use of numeri al methods

su h as the nite element method. Compa t designs alsomean that the laminated parts

are highly saturated and as a onsequen e the approximation of the material behaviour

be omes an essential design omponent. And be ause of the la k of a ura y analyti al

solutionsmay lead to inadequate designs. Moreover, redu ing osts may be a hieved by

the use of simple omponents whi h allow easy manufa turing. Therefore, a ma hine

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1.2 Problem statement

Designingthe windingis froman ele tromagneti pointof viewthe mostimportantpart

inma hine design. In this step the oilsides of the phasesneed to be assignedto an

ap-propriatestatorslotwhi hallowsarotatingmagneti eldwhenphasedispla ed urrents

areappliedtothe winding. On e the assignment isdone,adja ent oilsides belongingto

the same phase are alled a phase belt. This is the denition used in modern textbooks

su h asthatwrittenby Fitzgerald et al.(1992)and wasalready inuseinearlypapers by

Kauders (1932) 1

.

When designing interior permanent magnet ma hines with non-overlapping

on en-tratedwindingsitisne essary tohaveaninexpensivemethodwhi ha urately al ulates

the ma hine's performan e over a wide speed range. In addition, it is desirable to use

theniteelementmethod(FEM)sin eittakesintoa ountthea tualma hinegeometry

aswell asthe non-linear materialproperties. Additionally, itshould be fast and suitable

inan everyday design environment. Therefore the main resear h question of the present

dissertationis:

How an an analysis method be des ribed to a urately al ulate

the performan e of interior permanent magnet motors with

non-overlapping windings in an everyday design environment?

In order to answer the main resear h question, it is ne essary to have a systemati

algorithm to allo ate the slots. Then, after doing this, the on entrated oils an be

inserted in the appropriate slots. The immediate and essential sub-question that rst

needsto be answered is:

What is the mathemati al expression to allo ate the stator slots

belonging to a phase belt?

Anothersub-questionarisesfromtherequirementtoallo atetheslotsthatbelongtoa

phasebelt. Espe iallywhen usingFEM toanalyse ma hines,the enteringof the winding

arrangementis a omplex pro ess. Theproblem isexa erbated if dierent windingtypes

aretobeanalysed. Furthermore,inthe on eptualdesignphasea ompa trepresentation

of the winding ould simplify the hoi e of the initialgeometri al parameters. Therefore

the se ondsub-question is:

How an a winding layout be represented in a ompa t form?

1.3 Overlapping and non-overlapping windings

Fig.1.1(a)shows adoublelayeroverlapping winding. Inthe drawingsomeofthe oilsare

removed whi hmakes iteasier to identify the two layers. In(b) anon-overlapping single

layerwindingis shown. Onlyea h se ond statortooth has a oilwound aroundit.

Non-overlapping windings are a sub-set of fra tional slot windings and need some

extra dis ussion. It is often ategorised as on entrated. This is not ne essarily wrong

but is in ongruent with the formaldenition. The opposite ofa on entrated winding is

a distributed winding. In the latter ase a the oils of a given phase are distributed in

1

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PSfragrepla ements

doublelayer oil singlelayer oil

(a)Overlappingwinding (b)Non-overlappingwinding Statortooth

Figure 1.1: Double layer overlapping and single layernon-overlapping windings

severalslots. Whenreferringtooverlappingandnon-overlappingwindingsa on entrated

winding ould be dened as follows:

1. Formallya on entratedwindingisonewherethenumberofslotsperpoleandphase

equals one. In this ase the oil pit h equals the pole pit h and it is ategorised as

overlapping. This meansthatea h oilsideof thewindingispla edinasingleslot.

If the oilspans a polepit h, itis alled afull-pit h on entrated winding.

2. Non-overlapping ould also be lassied as on entrated, but then it is not done

in terms of the formal denition. Con entrated in this ase means that a oil is

on entrated around astator tooth as shown in Fig. 1.1(b).

Asimpliedillustrationofsingleanddoublelayernon-overlappingwindings 2

isshown

inFig. 1.2 and the dieren ebetween them is given inTab. 1.1. For the purposes of the

present dissertationa double layer windingis dened as follows:

Denition 1.3.1 Adoublelayer windinghastwo oilsidesperstatorslot. Doublelayer

in the ase of overlapping windingsmeans that the oil sides in a slot are pla ed radially

in two layers. In the ase of non-overlapping windings a double layer winding has two

oil sides side byside.

Table 1.1: Dieren e between single and double layerwindings

Single layer Double layer

Inthe ase ofthesinglelayerwindingea h

stator slot has only one oil side assigned

toitas shown in Fig. 1.2(a).

A double layer winding has two oil sides

assigned to a stator slot as shown in

Fig. 1.2(b).

In the literature dierent denitions of terms are in use for non-overlapping

on en-tratedwindings. The most ommon of themare:

2

InGermanthesewindingtypesare ommonlyreferredtoasZahnspulenwhi hmeanstooth oils.

Althoughtooth oilsseemtobeaverydes riptivenameforthesewindings,itisoften alled on entrated

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0000000000000000000000000000

1111111111111111111111111111

0000000000000000000000000000000000

1111111111111111111111111111111111

0

1

0

1

PSfragrepla ements

(a)Singlelayer (b)Doublelayer

xτ

s

2τ

s

τ

s

Figure1.2: Double and single layer non-overlapping windings

Magnussen and Sadarangani (2003): on entrated windings;

Magnussen et al. (2004): on entrated fra tionalpit hwindings;

Salminenet al.(2004a): fra tional slotwound; and

Bian hiand Bolognani (2004): fra tional slot.

Ifthe oils aretobeequallyinshape,whi h simpliesmanufa turing,itisre ognised

thatasinglelayer ouldeasilyhaveavariableslotpit h 3

. Furthermore,the oils ouldbe

implementedas either form-woundor round-wound. Returning to the variable oilpit h

ofthe singlelayer, itis importanttomentionthat this isanextra degreeof freedomthat

oerstobeanattra tivedesignparameter. Theairgapuxthatlinksthe oil ouldthus

bein reased and torque ripple an be improved.

It is therefore helpful to lassify a on entrated winding either as an overlapping

or non-overlapping on entrated winding. Cros and Viarouge (2002) ertainly aroused

interestin non-overlapping on entrated windings with theirpaperentitled Synthesis of

HighPerforman ePMMotorsWithCon entratedWindings,sin ethisisapaperwhi his

veryoftenusedasareferen eonthesewindingtypes. This ouldbeapossibleexplanation

forthe use of the term on entrated windings rather than tooth oilwindings.

The expansion of the lassi al winding types used in ma hines by non-overlapping

windingsoers newpossibilitiesinespe iallytra tionma hine design. Themainresear h

question in se tion 1.2 suggests a design algorithm that takes into a ount the

non-overlapping type. Obviouslya method that isvalidforall types is required. In addition,

itshould be easyto integrateintowhi heverpro ess isin use.

1.4 Approa h to the problem

As a rst step towards the analysis of interior permanent magnet syn hronous ma hines

withnon-overlapping on entratedwindings,anindepthoverviewofthesystemati sof

m

-phasewindingsisrequired. ThepaperofKauders(1932)andthebookofNürnberg(1952)

are two very good examples 4

. A histori aloverview of winding design and tenden ies in

3

The on eptofavariable oilpit hisexplainedin hapter3. Alsoreferto Fig.3.5for regularand

irregulardistributed slots.

4

Thisliteratureis unfortunatelywrittenin German,whi hmeansthat itis noteasily a essiblefor

(24)

ma hine analyses are given in hapter 2.

Addressing the resear h questions in se tion 1.2, a good starting point would be to

derive the air gap mmf using the lassi al single oil approa h and then expand it for

three phases. It would be ideal to nd a systemati algorithm for assigning the oils to

the stator slots. The derivation of the air gap mmf and winding theory is presented in

hapter3.

The new perspe tive on the winding representation and its properties oered in the

present dissertation are then applied to a tra tion ma hine ase study. Here only the

degreeoffreedom thatsinglelayernon-overlappingwindings provide,isfo usedon. This

is explained in hapter 5. Some important remarks on non-linear ma hine analysis to

support the designresults are explainedin hapter4.

A prototypeand itsmeasured results toverify the design pro ess are given in se tion

5.6. Finally, the on lusions and re ommendations for further studies are presented in

hapter6.

1.5 S ienti ontributions of this dissertation

Thedesignof permanentmagnetmotorsoftenresultsinthe use ofafra tionalslot

wind-ing. Dierentmethodsare availabletodesignthesewindings(whi hare oftengraphi ally

presented),ofwhi hnonetakesintoa ountsinglelayernon-overlappingwindings(whi h

ould have a variableslot pit h). Moreover, due to the dierent slot ongurations,

ma-terialnon-linearitiesand saturationee ts, itis impossibleto nd analyti almethods to

fullltheserequirements. Inaneverydaydesignenvironmentitisne essary tohave

a u-rateand reliabletools. The s ienti ontributionofthis dissertation an be summarised

asfollows:

A general method todesign single layernon-overlapping windings with a xed and

variable slotpit h is oered. The winding design is presented in its most ompa t

form and has all the information on the physi al layout as well as the winding

harmoni s. Itwillbeshownthatthedevelopedmethodappliesto

m

-phasewindings in generalas well.

Developingananalyti almodel forthe magneti ir uitof interiorpermanent

mag-netmotorsisnearlyimpossible. Abettermethodistomakeuseofboththe

analyti- alandniteelementmethod. Thema hineis hara terisedbythree(niteelement

analysis generated) two-dimensional fun tions and frequen y dependent losses are

al ulated analyti ally.

The methodology fromthe study is appliedto a

150 kW

prototypemotor designed for a tra tion appli ation. The manufa tured ma hine has

30

stator slots and

20

poles. Pra ti al measurements were done to verify the omputational results

pre-sented.

1.6 Delimitations of the study

Theaimof thisstudy isthe analysisofinteriorpermanentmagnetsyn hronous ma hines

with single layer non-overlapping windings. A omparison with other ma hine types is

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namelythe variable oilpit h. Sin e thisis nottypi alforother 5

windingtypes, this will

betheonlydesign aspe t that willbefo used on. Adetailedma hineoptimisationisnot

provided.

1.7 Layout of the dissertation

The resear hresults are presented in six haptersas follows:

Chapter 2: A short literature overview whi h gives a brief histori al view of winding

design is presented. This is followed by general tenden iesin ma hine analyses.

Chapter 3: Analgorithmisderivedtopresentawindinginamatrixform. This ompa t

formallowsthe al ulationofthewindingfa torsforallharmoni s. Thebasisofthe

algorithmisthe phase beltsequen e and the phasebelt onstraint whi h isderived

from the airgap mmfenvelope fun tions.

Chapter 4: This hapterdes ribesthe nonlinearmagneti ir uitanalysisoftheinterior

permanent magnet ma hine. The hosen method is that of the nite elements.

The hapter starts o with an introdu tion to the nonlinear materials used in the

design, followed by a detailed al ulation of the relevant ma hine quantities whi h

are ne essary for aperforman e al ulation. The hapter on ludes with adetailed

explanation ofhow toperform aharmoni analysis.

Chapter 5: FEAand analyti almethodsare ombined tointrodu e adesign pro edure

suitable in anan everyday design environment. The hapter on ludes with a ase

study ofa prototype tra tionmotorwith non-overlapping singlelayer on entrated

windingsandinteriorpermanentmagnets. Pra ti almeasurementsontheprototype

are ompared to al ulated values and dis ussed.

Chapter 6: This hapter entails on lusion and re ommendations forfurther study.

Where appli ableallthe explanations are a ompaniedby anexample based onthe

pro-totype ma hine whi h has

30

stator slots and

20

poles. A detailed des ription for this ombinationis given in hapter5.

1.8 Di ulties en ountered during the study

Themajordi ultyen ounteredduringthestudywasthelimitationsof ommer ialnite

elementsoftware to generate and post-pro ess nite element analysis data in ane ient

way. Even the integration thereof in ustomised software tools was nearly not possible.

This di ulty was over ome by using non- ommer ial nite element software, FEMP

(Finite Element Method Program). However, it was rst ne essary (through intensive

Fortran programming) to adapt the software to in lude permanent magnets and allow

positive boundary onditions.

5

Whenreferringtootherwinding types,itmeansdoublelayernon-overlappingaswellas singleand

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Initiallythewindingfa torandresultsfromthedis reteFouriertransformweretreated

asabsolutevalues. Keeping these values as omplexnumbers greatly simpliesthe

anal-ysis, sin e (through a proper hoi e of referen e axis) information on the winding axes

and the

dq

-variablesare given.

1.9 Notes to the reader

The work presented in this thesis was done in Nürnberg, Germany. Consequently many

of the literature used were in German. To name only a few, books by Nürnberg (1952),

Klamt (1962) and Jordan et al. (1975) are still ommonly used in design o es. It is

noti eable that even thoughthere exists a list of symbols for various quantities, English

and German have in some ases dierent symbols for the same quantity. This is mainly

duetothe fa t thatlanguages develop individualand of ourse olloquiallanguagerules.

Even adire t translation does not ne essarilygive the typi alword used. An example is

theGermanwordFelderregerkurve. Adire ttranslationwouldbeeldex itation urve,

whi h of ourse is not wrong. However, the eld ex itation urve in ele tri al terms

is usually known as the magnetomotive for e (mmf). Tab. 1.2 gives some ommon

ele tri alma hine quantities in English and German,and the dierent symbols inuse.

Table1.2: Typi alma hinequantities,thereGermantranslationand ounterpartsymbols

Quantity English German Unit

magnetomotivefor e (Dur hutung)

F

Θ

A

uxlinkage (Flussverkettung)

λ

ψ

V·s

voltage (Spannung)

V

U

V

spe i resistan e (spezis her Widerstand)

ρ

̺

W

·m

spe i weight(spezis her Masse)

ρ

γ

kg·m

−3

ondu tivity (Leitfähigkeit)

σ

κ

S·m

−1

ross se tionarea (Quers hnitt)

A

Q

m

2

turn number(Windungszahl) N W

-It is ommendablethat the Germanterminology onele tri alma hines allows a very

pre isedes riptionofalmostallaspe tsrelatedtothesubje t. Inmany asesitisdi ult

to nd an equivalent te hni al term in English, be ause it simply does not exist. The

problem of dierent terminologies even exists within a language: dierent s hools use

dierentterms whi hmakesthe study of ma hinerelated books not easy. Alsohistori al

hanges needto be kept inmind.

The term ma hine as used inthe present dissertation means it ould be a ma hine

that is operated either as a motor or as a generator. In the title the term motor is

used, sin e only the measured results of motor operation are presented. However, in the

theoreti alse tions, the word ma hine is preferred.

The do ument is optimised for a two-sided printout. A ip-book shows that the

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Literature overview

In this hapter a brief overview of the history of stator windings (layout and design) is

given. Sin e this topi evolved over many years anin depth history is beyond the s ope

of the present dissertation. However, a fewimportantworks need tobementioned.

2.1 A histori al view of winding design

In the rst of his two remarkable papers Kauders (1932) explains the systemati s of

stator windings and the al ulation of the winding fa tors. The aim in this work was

to determine the parameters that hara terise the air gap mmf of the winding. Also in

this paper the indu ed voltage in the oil sides is already mentioned and represented

as a ve tor. The resultant ve tor diagram was alled the star of oil groups (German:

Spulengruppenstern). The adja ent ve tors onsu h a diagramthat belong tothe same

phaseis alled aphase belt (German: Zone).

Twoyears later,in these ond paper by Kauders(1934), thealgebrai methods

devel-oped in the rst paperwere visualised by means of Tingley's diagram. The latter ould

bereferred toas alinear representationof the now alled star ofslots (German:

Nuten-stern). The star of slots is onstru ted using the ele tri al angle between two adja ent

slots. Computerte hnologyasweknowittodaywasnotavailableatthe timeandtheuse

of graph paper ertainly was ommon. Furthermore, su h graphi al methods denitely

ontributed tothe subje tof stator windings.

Vilém Klíma (Wilhelm Kauders) died on 6 O tober 1985 and in an obituary by

Frohneand Seins h (1985) it is mentioned that Klíma's equation for the distribution

fa tor 6

of fra tionalslotwindings isnot found in textbooks. Another remarkin the

obit-uary is that in some referen es it is stated that it is not possible to nd a losed-form

expression for the winding fa tor of fra tional slot windings. In the rest of this

litera-ture overview, none of the authors (ex ept Kremser) refers to or makes use of Klíma's

losed-formexpression 7

.

MorethanftyyearslaterthanKauders(1934),Kremser(1989)introdu eda

ompre-hensivestudy offra tionalslotwindingsand the urrentsinthe parallelpaths ofrotating

ma hines. The reasonforthis bigtime gap doesnot meanthat nothinghad happened in

themeantimeinwindingdesign;itshouldbekept inmindthatthis overview onlypoints

out some important aspe ts. In the rst part of Kremser's do toral thesis a detailed

6

Alsoknownasthebreadthfa tor.

7

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algebrai des ription of single and double layer fra tional slot windings is presented. In

ordertoallo atethe oils tothe statorslots,modulararithmeti ,whi huses theso- alled

ommutatorpit h,isused. This anbeseenasasteptowardusing omputerprogramsto

automati allyperformingwindingdesigns. Inthestar ofslotsthe ve torsstartwrapping,

andon eput onagraph,twove torshavinganglesof

45

°and

405

°respe tivelylieexa tly onea h other. The starof slots isthusthe same asthe remainderafterdividing anangle

by

360

° whi h an easily be implementedby means of the modulo fun tion.

Wa h (1997) explains another algebrai method to design stator windings whi h has

mu hin ommonwith thatpresented by Kremser(1989) 8

. Themethodspresented inthe

paper is hara terised by the matrix representation of a winding. The winding matrix

simplifythe winding fa tor al ulation and avoids omplex equations. Basi ally the

ve -tors of the star of slots is used in matrix form. What is alled the windingmatrix ould

beseen as the matrix representation of Tingley's diagram. A drawba k of the method is

that ea h matrix entry for double layer windings has two values. In omputer

program-ming this should be solved by either a multi-dimensionalmatrix or two two-dimensional

matri es. In another paperbyWa h(1998),the fo us isontheoptimisationof fra tional

slot windings. In spite of the fa t that this paper is a very good sour e on the topi of

windingdesign,none of the literature inthe rest of this se tionrefers toit.

Cros and Viarouge(2002)showed thatthe useofstator oilswound aroundthestator

teeth has some attra tive advantages for manufa turing. The oils donot overlap whi h

means that the manufa turing is simplied and the winding overhang is less than that

of overlapping windings. Ele tri ally the shorter windingoverhang means the use of less

opper. Furthermore, the non-overlapping windings are derived from single or double

layer overlapping windings. It is also pointed out that in the ase of single layer

non-overlapping windings the oil pit h an be in reased and used as a design parameter 9

.

The latter, however, is not taken into a ount in the winding fa tor al ulation. This

paper is referen ed quite often, proving the interest among ma hine designers in

non-overlapping windings.

Salminen(2004)didanextensivestudyonfra tionalslotwindingsforlowspeed

appli- ations. Themainobje tiveinthedo toralthesisisto omparedierentslot ombinations

ofama hinewith axed airgap diameter inthe

45 kW

range. Althoughboth singleand doublenon-overlapping windings are taken into a ount,the fo us is onthe double layer

type. Sin e round wire oils are used inthe investigated appli ation it is appropriate to

use a double layer rather than single layer. If form-wound oils were used the hoi e of

double layer windings would have aused some di ulties when inserting the oils into

the stator slots. Also, the stator had semi- losed slots. The star of slots isreferred to as

the voltage ve tor graph in Salminen'sthesis. A very important omment by the author

isthat areshouldbetakenwhenthewindingfa torsfordierentwindingtypesaretobe

al ulated. Dependingon the windingtype the rightset of equationsshould be hosen.

Skaar et al.(2006)proposedamethodto al ulatethewindingfa torsfor on entrated

oils without knowledge of the winding layout. This is of ourse su ient to qui kly

ompare dierent slot and pole ombinations. Using this method it is shown that the

best numberof slots per poleand phaseshould be inthe range ¼

≤ q ≤

½.

The tutorial presented by Bian hiet al.(2007) is anin-depth study of fra tional slot

8

It should howeverbe noti ed that the authors Wa h and Kremserare from Poland andGermany

respe tively.

9

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windingsingeneral. In theirtutorialnotesadistin tionismadebetween overlappingand

non-overlapping windings. Here too the non-overlapping type is derived from a double

layeroverlapping winding.

2.2 Tenden ies in ma hine analysis

When analysing permanent magnet ele tri al ma hines it is not ne essary to start from

s rat h. Sin e the indu tion ma hine has been known for more than

100

years the devel-oped methods serve as a good basis. A well-known book on indu tion ma hines is Die

Asyn hronmas hine ( ommonly known as the indu tion ma hine) by Nürnberg (1952).

The book gives a very detailed des ription of modelingte hniques whi h an be used in

the design of indu tion ma hines (whi h is of ourse not limited to indu tion ma hines

only).

2.2.1 Introdu tory remarks

It is important to mention that Nürnberg (1952) uses equivalent

B(H)

urves for the al ulation of the mmf's in the teeth and yoke. This is not wrong and indeed helps to

understandthephysi albehaviourofthema hine. However,the urvesevolvedovermany

years and are based onmeasured results and are only validfor steady-state operation.

Fig.2.1illustratestheprin ipleofleakage ompensation(German: Zahnentlastung).

Due to saturation, ux lines leave the teeth tips and are parallel to the slot wall. This

means that the ux density in the stator teeth de reases. Sin e the single layer

non-overlapping winding ould have avariable oilpit h the statorhas two tooth types. The

validity of the ompensation urves given by Nürnberg (1952) and applied to ma hines

with dierent stator slots was not investigated and ould be a very interesting topi for

furtherstudy. Theapproa hinthepresentdissertationistousetheniteelementmethod,

whi h al ulates the exa t ux densitiesby means ofa physi al model.

00000

11111

PSfragrepla ements

B

1 B

2 B

3 B

4 B

5

Fluxlines Yoke

H/A·m

−

1 B/T

B

1 B

5 H

1 H

5

Slotwall

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2.2.2 The nite element method as a design tool

Frommostre ent onferen e ontributionsthereisanoti eabletenden y tousenumeri al

methodsliketheniteelementmethodintheinitialdesignphase. Germishuizen and Kamper

(2007) proposed anew al ulation methodfor al ulatingthe ma hine performan e from

ux linkage and torque fun tions. These fun tions are obtained from a nite element

analysis (FEA).

Centner (2008)introdu ed a software toolfor the design of high-speed indu tion

ma- hines. Forsu hma hinestheloss al ulationandtheinuen eofthe materialproperties

are very important. Therefore, to a ount for these aspe ts the nite element method

is ombined with analyti al expressions. Hafner et al. (2008) suggested a similar design

strategyfor the same reasons.

Rei hert (2004) explains a simplied approa h to determine ma hine hara teristi s

by means of the nite element method. In doing so the non-linear materialbehaviour is

a ounted for and is still fast and a urate. Garbe et al. (2008)also ombines FEM and

analyti alexpressions to al ulate motor parameters.

2.2.3 The inuen e of saturation on transient solutions

Seifert and Strangmüller(1989)showedthattheuseofindu tionmotorparameters

al u-latedinthe lassi alwayleadstoina uratepredi tionsofma hinebehaviourunderfault

onditions su h as terminalshort ir uits. Experien ed based fa tors were introdu ed to

a ount for the la k of a ura y. Again, this is an example of a situation where results

must be known in order toobtain useful saturation fa tors.

Besides the saturation of the materials used the modelingthereof is alsoessential for

ma hine behaviour. The material behaviour up to saturation is explained in se tion 4.2

and,whereappli able, referen ed tothe literature. Poni k(1993) showed thatnegle ting

theee tofsaturationintransientsolutions anleadtoina urateresults. Itissuggested

thatnot onlytheindu tan esinuen ed by saturation,butthe hange in urrentaswell,

should be taken intoa ount. Avoiding saturation ould result in undersizing of riti al

me hani al omponents. Depending on the ma hine appli ation, undersized me hani al

parts an be ome asafety problem, forexample in tra tiondrives for railvehi les.

Retièreand Ivanès(1998)oeredstudymethodsofthree-phaseshort ir uitsof

indu -tionma hines. Thereversepeaktorqueduringashort ir uit an auseseriousme hani al

stress. The authors show that for indu tion ma hines the saturation and the skin ee t

in the rotor bars should be a ounted for in the al ulation of the reverse peak torque

and that it is ne essary to take into a ount the saturation ee ts. Brauer et al. (2004)

ompared the dynami short ir uits of indu tion ma hines (IM) to that of permanent

magnetsyn hronous ma hines (PMSM). An important out ome of their study is that in

the ase of the two-phase short ir uit a PMSM starts to os illate and ontinues to do

so until it is isolated from the inverter. It is alsoemphasised that these fault onditions

should be al ulated using FEM. Ei hler et al. (2008) alsodes ribes that linear ma hine

models annotbe used to simulatewind generators in the

5 MW

range.

Thereisala k ofexperien e inthedevelopment ofpermanentmagnetma hines

om-paredtoindu tionma hines. Thismeansthattheuseofforexample ompensation urves

toa ountforsaturationshouldbeavoided,sin eitmightleadtoina uratedesign

al u-lations. Furthermore,itshouldbenoti edthatsu h ompensation urveswere developed

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any-moreandthe useof stateofthe artte hnologyisadvisablefora urate predi tions. This

iseven more importantas prototyping (and measurementsto nd ompensationfa tors)

isto be kept to a minimum for e onomi reasons.

2.2.4 Present approa hes in iron loss al ulation

The huge number of s ienti ontributions found on iron loss al ulations emphasises

the omplexity of iron losses. Two aspe ts that denitely ontribute to this fa t is that

the material data annot be des ribed analyti ally and is usually presented in graphi al

form. Se ondly, the inuen e of the manufa turingpro ess isdi ult toa ount for.

It is important tomention that a goodapproximation of the spe i loss is found in

Nürnberg(1952). Here the spe i lossof one kilogramofdeburred laminatedsteel in

W

isgiven by

p

F e

=

k

h

f

50 + k

c

f

50

2 !

B

1 T

2

(2.2.1)

The oe ients

k

h

and

k

c

a ount for the hysteresis and lassi al eddy urrent losses respe tively. Sin e the frequen y in the stator is onstant for a given operating point

it is onvenient to separate the frequen y and ux density omponents of the loss as in

(2.2.1). In Nürnberg's book it isalsomentioned that the iron loss al ulated inthis way

isless than the measured loss using the Epstein 10

frame. To a ount for this ina ura y

it is suggested to multiply the al ulated iron loss in the yoke and teeth by

1.5

and

3

respe tively. Thisholdsforstatorslotswhi hissemi- losed. Foropenstatorslotstheloss

inthe teeth tends to in rease even further. Due to this un ertainty it is understandable

that Nürnberg (1952) advised the following 11

: Wegen der erhebli hen Unsi herheit bei

derEisenverlustbere hnung gehemanaberni htindieFeinheiten. This shows thelimits

of empiri almodels beforete hniques likethe nite elementmethodwere introdu ed.

Bertotti (1988) introdu ed what is alled the ex ess loss and thereby expanded the

analyti alexpression for spe i loss by a

k

e

(f B)

1.5

term. Thus, (2.2.1) hanges to

p

F e

= k

h

f + k

c

f

2 B

2 + k

e

(f B)

1.5

(2.2.2)

Linet al. (2003) used the expression (2.2.2) in a transient nite element analysis. The

advantageindoingsoisthat thea tualuxdensitiesaredire tlyobtained fromthenite

element solution. Germishuizen and Stanton (2008) showed that the loss al ulation in

(2.2.2) is only a theoreti al loss estimation based on extrapolated data obtained from

resultsmeasured inthe Epsteinframe. Instead of usingdierentfa tors forthe yokeand

teethtoa ountforthemanufa turinginuen e,asinglemanufa turingfa torshouldbe

applied to the overall result. For ma hines in the

200 kW

range a manufa turing fa tor of two is found to be adequate. Another out ome of this paper is that inadequate loss

omputations ould easilylead to misinterpretation of results.

10

InNürnberg(1952)theEpsteinframeiswronglyspeltEppstein(whi hisasmalltowninGermany

notfarfromFrankfurtamMain).

11

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2.3 A short ex ursion into the life of one of the

authors onsulted

AtthispointIwouldliketomakeashortex ursionandreportaveryinterestingin iden e

forthe following reasons:

Nearlynoneofthereferen es onsultedtooknoteofVilémKlíma's losedexpression

for the al ulationof the distribution fa tor.

Vilém Klíma'slife storyis aninspiration for young resear hers.

Published resear habout theintelle tuallifeoftheJewisheliteintheghetto

There-sienstadt led me to this dis overy.

If one takes a loser look at the referen es listed in the last paper by Klíma (1979),

there is an entry entitled Systematik der Drehstromwi klungen and the author is given

as V. Klíma (Kauders). To supply a name between bra kets is not typi al, and this

o urren emademe suspi ious, sin ethe onlypaperI ouldndwith thistitle iswritten

by Wilhelm Kauders. As I ould not explain the enigma, I started to ask questions

and never expe ted to nd answers about tragi events that o urred duringthe Se ond

World War. Sin e I grew up in South Afri a whi h, ompared to other ountries, was

littleee ted by the war,this dis overy made the experien e extraordinary.

2.3.1 Coming a ross a very interesting sour e

Nowadays it is typi al touse the well-known Google sear h engine. Tryinga few

ombi-nations of the names Kauders and Klíma, I ame a ross a list of le turers in the ghetto

of Terezin(Theresienstadt). The entry details forKauders are given inTab. 2.1.

Table 2.1: Klíma'sentry in the listof le turers inthe ghetto Theresienstadt

Name and Title Kauders (Klima) (Vilem)Dr.

Birth Date 10.04.1906

Deportedto Terezin 04.12.41

Deportedfrom Prague

Deportedfrom Terezin 01.11.44

Survived in Grossrosen

This led me to the book University Over The Abyss: The story behind 520 le turers

and 2,430 le tures in KZ Theresienstadt 1942-1944 by Makarova et al. (2004). A very

interesting detail from the book is that Dr. Golds hmied and Dr. Kauders were se retly

taken to Germany to improve the performan e of German radar 12

. A witness, Gerda

Haas, remembered the following:

12

A ordingto IvanKlíma (Klíma'sson),VilémKlímawassenttogetherwith twoother spe ialists:

Mr. Golds hmiedandMr. Kohntothe on entration ampin Grossrosen. Mr.Golds hmiedandKlíma

stayedonlyforafewweeks inGrossrosen. Asaresultoftheeva uationofthe ampbothofthemwere