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Lappeenrannan teknillinen yliopisto

Lappeenranta University of Technology

Pia Salminen

FRACTIONAL SLOT PERMANENT MAGNET SYNCHRONOUS

MOTORS FOR LOW SPEED APPLICATIONS

Thesis for the degree of Doctor of Science(Technology) to be presented with duepermission for public examination andcriticism in the auditorium 1382 atLappeenranta University of Technology,Lappeenranta, Finland on the 20th of

December, 2004, at noon.

Acta Universitatis

Lappeenrantaensis

198

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ISBN 951-764-982-7ISBN 951-764-983-5 (PDF)

ISSN 1456-4491

Lappeenrannan teknillinen yliopistoDigipaino 2004

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ABSTRACT

Pia Salminen

FRACTIONAL SLOT PERMANENT MAGNET SYNCHRONOUS MOTORS FOR

LOW SPEED APPLICATIONS

Lappeenranta 2004

150 p.Acta Universitatis Lappeenrantaensis 198Diss. Lappeenranta University of TechnologyISBN 951-764-982-7, ISBN 951-764-983-5 (PDF), ISSN 1456-4491

This study compares different rotor structures of permanent magnet motors with fractional slotwindings. The surface mounted magnet and the embedded magnet rotor structures are studied.This thesis analyses the characteristics of a concentrated two-layer winding, each coil of which

is wound around one tooth and which has a number of slots per pole and per phase less than one(q< 1). Compared to the integer slot winding, the fractional winding (q< 1) has shorter endwindings and this, thereby, makes space as well as manufacturing cost saving possible.

Several possible ways of winding a fractional slot machine with slots per pole and per phaseless than one are examined. The winding factor and the winding harmonic components arecalculated. The benefits attainable from a machine with concentrated windings are considered.Rotor structures with surface magnets, radially embedded magnets and embedded magnets inV-position are discussed. The finite element method is used to solve the main values of the

motors. The waveform of the induced electro motive force, the no-load and rated load torqueripple as well as the dynamic behavior of the current driven and voltage driven motor aresolved. The results obtained from different finite element analyses are given. A simple analyticmethod to calculate fractional slot machines is introduced and the values are compared to thevalues obtained with the finite element analysis.

Several different fractional slot machines are first designed by using the simple analyticalmethod and then computed by using the finite element method. All the motors are of the same225-frame size, and have an approximately same amount of magnet material, a same ratedtorque demand and a 400 - 420 rpm speed. An analysis of the computation results gives newinformation on the character of fractional slot machines.

A fractional slot prototype machine with number 0.4 for the slots per pole and per phase, 45 kWoutput power and 420 rpm speed is constructed to verify the calculations. The measurement

and the finite element method results are found to be equal.

Key words: Permanent magnet synchronous motor, PMSM, machine design

UDC 621.313.323 : 621.313.8 : 621.3.042.3

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ACKNOWLEDGEMENTS

This research work was carried out at the Laboratory of Electrical Engineering, Department ofElectrical Engineering, Lappeenranta University of Technology.

I wish to express my deepest gratitude to Professor Juha Pyrhnen, head of the Department ofElectrical Engineering and the supervisor of this thesis, for his guidance and support.

The work is a research project of the Carelian Drives Motor Centre, CDMC. The project waspartly financed by ABB Oy. Special acknowledgements are due to M.Sc. Juhani Mantere, head

of the Electrical Machines Department of ABB Oy, for his guidance during this work and forthe co-operation facilities. I wish to express my gratitude to D.Sc. Markku Niemel, head of theCDMC, Lappeenranta.

I wish to express my special thanks to M.Sc. Asko Parviainen, D.Sc. Markku Niemel andProfessor Juha Pyrhnen for their support during the research work. They are the core of a largegroup of dear colleagues, which whom I had valuable and guiding discussions on the subject of

this thesis. I am also grateful to Mr. Harri Loisa for the manufacturing of the windings of theprototype machine.

I wish to express my gratitude to the pre-examinators of this thesis, D.Sc. Jarmo Perho, HUT,

and Professor Chandur Sadarangani, KTH, for their valuable comments and proposedcorrections. Their co-operation is highly appreciated.

My warm thanks are due to FM Julia Vauterin for the language review of this thesis.

I also wish to express my gratitude to my colleagues, friends and especially to my son Esa fortheir help and understanding during my work.

Financial support by the South-Karelian Department of Finnish Cultural Foundation, Jenny andAntti Wihuri Foundation, Foundation of Technology and Association of Electrical Engineers inFinland, Ulla Tuominen Foundation, Walter Ahlstrm Foundation is gratefully acknowledged.

Lappeenranta, December 2004

Pia Salminen

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k5 Factor for defining eddy current losses

L, l Physical length of the stator core, Inductance, LengthLd Direct axis inductance

Lq Quadrature axis inductance

Li Effective length of the core

Lmd Magnetizing inductance of the direct axis

Lmq Magnetizing inductance of the quadrature axis

Ln Slot leakage inductance

Ls Stator leakage inductance

Lz Tooth tip leakage inductance

L Leakage inductance, skewinglb Length of the end winding

lm Length of the permanent magnets, axial

m Number of phases, mass

mCu Mass of copper

mFe, y Mass of iron, yoke

mFe, t Mass of iron, teeth

N Natural number

Nn1

Effective turns of a coil

Nph Amount of winding turns in series of stator phase

n Denumerator of q(slots per poles and per phase), Speed

nc Physical displacement in the number of slots

nmx Number of magnets (tangential direction)

nmz Number of magnets (axial direction)

P Power

PBr Bearing losses

PCu Copper losses

PEddy Eddy current losses of the magnets

PFe Iron losses

Ph Total losses

Pin Input power

Pn Rated power

PPu Pulsation losses

PStr Stray losses

p Pole pair number

p10 Factor for defining iron loss

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Qs Number of stator slots

q Slots per pole and per phaseRph Phase resistance

s Slip

T Torque

t Time, Variable, defines the winding arrangement

Tp-p Peak-to-peak torque ripple % of average torque

U Voltage

x Width

x1 Slot width

x4 Slot opening widthy Coil pitch, height

y1 Slot height

y4 Slot opening height

z Numerator of q(slots per poles and per phase)

Greek letters

Electric angle, Magnet width (Magnet arc width / pole pitch, shown in Fig. 3.12)

Width of tooth, angle

Air-gap length, radial

a Load angle

eff Equivalent air-gap length

k Phase shift

Efficiency

so Permeance of upper layer

su Permeance of lower layer

g Mutual permeance

go Mutual permeance of upper layer

gu Mutual permeance of lower layer

Permeance factor

e Reactance factor for the end windings

w Reactance factor for the end windings

n Permeance factor, describes all factors

z Leakage inductance factor

PM, Air gap flux created by permanent magnets

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m Resistivity of the magnet

Leakage factor

Conductivity

Permeability

Fe Permeability of iron

r Relative permeability

0 Permeability of air (vacuum)

Harmonic

slot Slot harmonic

p Pole pitch

s Slot pitch

sk Skewing pitch

Electrical angular frequency

s Angular frequency of stator field

Winding factor, thharmonic

1 Winding factor, fundamental harmonic

d Distribution factor

p Pitch factor

sk Skewing factor Flux linkage

a Armature flux linkage

PM Flux linkage due to permanent magnet

s Stator flux linkage

Air-gap flux linkage

Acronyms

2D Two-dimensional

A Analytical calculationAC Alternating current

CD Compact disk

DC Direct current

DTC Direct torque control

DVD-ROM Digital videodisk read only memory

EMF Electro motive force

ER Motor with radially embedded magnets

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EV Motor with embedded magnets in V-position

FEA Finite element analysisHDD Hard disc drive

LCM Least common multiplier

mmf Magnetomotive force

Nd-Fe-B Neodymium Iron Boron -alloy

PM Permanent magnet

PMSM Permanent magnet synchronous motor

S Motor with surface mounted magnets

SM Synchronous motor

RMS Root mean square

Subscript

b End winding

d Direct

q Quadrature

r Rotor

s Stator

Leakage

1 Fundamental wave

Harmonic

n Rated

o Upper

u Lower

max Maximum

y Yoke

t Teeth

Superscripts

e Electric angle

Others

Upper case letters, in italic Root mean square value

Lower case letters, in italic Instantaneous value

p.u. Per unit value

_ Space vectors are underlined

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CONTENTS

ABSTRACT

ACKNOWLEDGEMENTS

ABBREVIATIONS AND VARIABLES

CONTENTS

1. INTRODUCTION....................................................................................................................13

1.1. Brushless motor types .......................................................... .........................................20

1.2. Location of the permanent magnets ................................................................ ..............22

1.3. Applications ................................................................ ..................................................24

1.4. End winding and stator resistance.................................................................................25

1.5. Scientific contribution of this work...............................................................................29

2. CALCULATION OF A FRACTIONAL SLOT PM-MOTOR................................................30

2.1. Two-layer fractional slot winding.................................................................................31

2.1.1. 1st-Grade fractional slot winding................................................................... ..32

2.1.2. 2nd-Grade fractional slot winding .................................................................. ..33

2.2. Winding arrangements................................................................ ..................................34

2.3. Winding factor ...................................................... ........................................................36

2.3.1. Winding factor according to the voltage vector graph ....................................452.4. Flux density and back EMF ....................................................... ...................................46

2.5. Inductances ...................................................................................................................49

2.5.1. Leakage inductance method 1 ........................................................... ..............50

2.5.2. Leakage inductance method 2 ........................................................... ..............56

2.6. Torque calculation.........................................................................................................58

2.7. Loss calculation.............................................................................................................58

2.8. Finite element analysis..................................................................................................60

3. COMPUTATIONAL RESULTS.............................................................. ...............................62

3.1. Torque as a function of the load angle .......................................................... ................65

3.2. Number of slots and poles............................................................................ .................69

3.3. Induced no-load back EMF...........................................................................................73

3.4. Cogging torque..............................................................................................................75

3.4.1. Semi-closed slot vs. open slot .........................................................................82

3.4.2. Conclusion.......................................................................................................86

3.5. Torque ripple of the current driven model ...................................................... ..............87

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3.5.1. Some examples................................................................................................89

3.5.2. The magnet width and the slot opening width.................................................92

3.5.3. Conclusion.......................................................................................................95

3.6. Surface magnet motor versus embedded magnet motor................................................97

3.6.1. 12-slot-10-pole motor......................................................................................97

3.6.2. 24-slot-22-pole motor and 24-slot-20-pole motor ...........................................101

3.6.3. Conclusion.......................................................................................................104

3.6.4. Slot opening.....................................................................................................106

3.6.5. Embedded V-magnet motors...........................................................................111

3.6.6. Conclusion.......................................................................................................112 3.7. The fractional slot winding compared to the integer slot winding................................113

3.8. Losses............................................................................................................................115

3.9. The analytical computations compared to the FE computations...................................117

3.10. Designing guidelines................ ................................................................ .....................119

4. 12-SLOT 10-POLE PROTOTYPE MOTOR...........................................................................121

4.1. Design of the prototype V-magnet motor .....................................................................121

4.2. No-load test........ ................................................................ ...........................................124

4.3. Generator test ................................................................ ................................................126

4.3.1. Temperature rise test ..................................................................... ..................127

4.3.2. Vibration measurement ...................................................................................129

4.4. Cogging torque measurement .......................................................................................129

4.5. Measured values compared to the computed values .....................................................130

4.6. Comments and suggestions...........................................................................................131

5. CONCLUSION........................................................ ........................................................... .....133

REFERENCES....................................................................................................................................136

APPENDIX A Winding arrangements.......................................................................................140

APPENDIX B Periodical behaviour of harmonics ....................................................................141APPENDIX C Winding factors .................................................................................................143

APPENDIX D Calculation example of inductances ..................................................................145

APPENDIX E B/H-curves for Neorem 495a.............................................................................147

APPENDIX F Torque ripples results from FEA .......................................................................148

APPENDIX G Prototype motor data..........................................................................................150

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1. INTRODUCTION

The appellation synchronous motor is derived from the fact that the rotor and the rotating field

of the stator rotate at the same speed. The rotor tends to align itself with the rotating field

produced by the stator. The stator has often a three-phase winding. The rotor magnetization is

caused by the permanent magnets in the rotor or by external magnetization such as e.g. a DC-

supply feeding the field winding. These motor types are called permanent magnet synchronous

motors (PMSMs) and separately excited synchronous motors (SM), correspondingly.

Depending on the rotor construction the motors are often called either salient-pole or non-

salient-pole motors. The performance of the synchronous motor is very much dependent on the

different inductances of the motor. Different equivalent air-gaps in the direct and quadrature-

axis cause different inductances in the directions of the d- and q-axis. The direct-axis

synchronous inductance Ld consists of the magnetizing inductance Lmd and the leakage

inductanceLs. Correspondingly, the quadrature-axis synchronous inductance Lqis the sum of

the quadrature-axis magnetizing inductance Lmqand the leakage inductanceLs. The values of

these two synchronous inductances mainly determine the character of a synchronous motor.

The flux created by the stator currents depending on the construction of the permanent magnet

motor is typically only 0.1 0.6 of the amount of the flux created by the permanent magnets.

Thus, the armature flux (or armature reaction) is typically small. This is the reason why, for the

permanent magnet synchronous motor, the torque can be adjusted flexibly by changing the

stator current. Also for this reason, the permanent magnet motor has an obvious advantage over

the induction motor. The small armature reaction involves also the following difficulty; the field

weakening is often difficult in PMSMs. Moderate field weakening properties are achieved in

motors with embedded magnets and with a large number of poles. In these cases, the

synchronous inductance easily reaches a p.u. value of about 0.7. This means that the rated

current in the negative d-axis direction gives a 0.3 p.u. flux value.

The history of permanent magnet motors has been dependent on the development of the magnet

materials. Permanent magnets have been first used in DC motors and later in synchronous AC

motors. After the rare earth magnets were developed for production in the 1970s, it was

possible to manufacture also large PM synchronous motors. The industrial interest to

manufacture permanent magnet motors arose in the 1980s as the new magnet material

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Neodymium-Iron-Boron, Nd-Fe-B was developed. As the magnet materials have been further

developed and their market prices decreased, the use of permanent magnet machines has been

growing. The first machine applications of the PM motor were small-sized, cylindrical rotor

synchronous motors. In the 1990s, the permanent magnet remanence flux density Br= 1.2 T

was considered to be a high value. In practice, also magnets with low Brvalues have been used

to save costs. Nowadays, the best Nd-Fe-B grades can reach Br of 1.5 T. This, again, will

certainly give new design aspects. Considering the properties of steel, the demagnetization

curve of the present-day permanent-magnet materials and the maximum energy product as well

as the best utilisation of the permanent-magnet material, it may be stated that the motor designer

might be satisfied, when it is available for various use a permanent magnet material which has aremanence flux density of nearly 2 T. This value should guarantee an air-gap density of about

1 T, full use of the steel mass and good use of the permanent magnet material in case of a

surface magnet motor. The permanent magnet materials have nowadays almost all desired

properties and create a strong flux. Of course, the motor designer will ask for still a larger

remanence and temperature independency as well as for even better demagnetization properties,

but the present-day materials are, nevertheless, quite well suited for permanent magnet motor

applications.

This thesis introduces a performance comparison of different permanent magnet motor

structures equipped with fractional slot windings in which the number of slots per pole and per

phase is lower than unity, q < 1. For a motor with q(the number of slots per pole and per phase)

less than unity, the flux density distribution in the air-gap over one pole pitch can consist of just

one tooth and one slot, as for example the 24-slot-22 pole motor, Fig. 1.1.

The main flux can flow through one tooth from the rotor to the stator and return via two other

teeth. The resulting air-gap flux density distribution is not sinusoidal, as it is illustrated in

Fig. 1.1 b). As a consequence, for the cogging torque or the dynamic torque ripple, problemsmay be expected to appear. In a well-designed fractional slot motor the voltages and the

currents may be purely sinusoidal.

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-1.0

-0.5

0.0

0.5

1.0

0 1 2 3

Air gap radius

Fluxdensitynormalcomponent(T)

2 4 6

32.62

Air-gap periphery

Fig. 1.1 a) Flux lines of a fractional slot motor with 24 slots and 22 poles, q= 0.364. One electrical cycle

of 2is equal to 2p(pis the pole pitch). b) The corresponding normal (radial) component of the air-gap

flux density along the air-gap periphery.

The magnetomotive force (mmf) waves of three different 22-pole motors are illustrated in Fig.

1.2. On the top, a q= 2 motor with 132 slots is illustrated, in the middle a q= 1 motor with 66

slots and at the bottom a fractional slot q= 24/(322) = 4/11 motor with 24 slots.

q= 1

q= 2

q= 0.364

Fig. 1.2. The magnetomotive force waves of 22-pole motors with q= 2, q= 1 and q= 4/11 at an instance

when the stator phase currents i1= 1 and i2= i3= .

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The figure reveals clearly the pulse-vice nature of the mmf of the fractional slot winding and

that the harmonic content of the mmf is large. There exist also low order sub-harmonics in the

mmf, which is not the case for the integer slot windings.

The only feasible motor type that, in practice, may run equipped with a fractional slot winding

is the synchronous motor the rotor conductivity of which should be as low as possible. Even the

permanent magnet material should be as poorly conducting as possible. The rotor magnetic flux

carrying parts must also be made of laminated steel in order to avoid excessive rotor iron losses

due to the fluctuating flux in the rotor. The machine type produces anyway losses in the rotor

and is, therefore, inherently best suited for low speed applications. The popularity of low speed

applications is increasing as the use of direct drive systems in industry and domestic

applications as well as in wind power production, commerce and leisure is growing.

In low speed applications it is often a good selection to set a high pole number. It has the

advantage that the iron weight per rated torque is low due to the rather low flux per pole. A high

pole number with conventional winding (q 1) structures involves also a high slot number,

which increases the costs and, in the worst case, leads to a low filling factor since the amount of

insulation material compared to the slot area is high. The fractional slot winding (q < 1)

solution, instead, does not require many slots although the pole number is high, as a result of

which both the iron and the copper mass can be reduced. Compared to the conventional

windings (q1) with the same slot number it can be shown that the length of the end winding

is less than one third in concentrated fractional wound motors. This offers a remarkable

potential to reduce the machine copper losses. If the copper weight can be reduced, also the

material costs, correspondingly, will decrease, because the raw material cost of copper is about

6 times the cost of iron. Some fractional slot motors offer relatively low fundamental winding

factors and create harmonics and sub-harmonics causing extra heating, additional losses and

vibration. It has been studied the use of these machine types merely in applications with small

power and, in some cases, with 1 or 2 phase systems, so their use at high power ratings has thus

far been not very common. Because the problem of selecting the geometry and winding

arrangements of the fractional slot motor remains still partly unsolved, it is important to further

study in detail the fractional slot motors. Therefore, the importance of manufacturing a

prototype machine of considerable power should be stressed.

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It was the authors objective to design a low speed motor for the specific application, in which a

high torque and 45 kW output power could be achieved from a restricted motor volume. In

order to be able to fulfil these conditions the multi-pole machine with fractional slot windings

should be studied carefully. One of the designs studied was verified with the prototype machine.

The given performance comparison is based on several 2D-finite element computations made

on the 45 kW, 400 rpm, 420 rpm and 600 rpm machines. The torque, torque ripple and back

EMF waveforms are analysed. The machine design relies on an efficient forced air-cooling

which brings an over 5 A/mm2stator current density at rated load. A two-layer concentrated

winding type, in which each stator tooth forms practically an independent pole, was selected for

manufacturing. The most significant advantage of this winding type is that it minimizes thelength of the end windings. Almost all copper is contributing to the torque production of the

machine. The fundamental winding factors for some concentrated windings (where two

different coils are placed in the same slot) for different rotor pole (2p) and stator slot (Qs)

combinations are given. It may be noticed that only a few combinations of Qsand 2pproduce a

high fundamental winding factor. Analytical calculations and the finite element analysis (FEA)

were carried out for several types of the fractional slot motor.

Hendershot and Miller (1994) studied the variations of possible pole and slot numbers for

brushless motors in terms of how the cogging may be resisted. It was noticed that the minimum

cogging torque was not dependent on whether the machine is of the fractional-slot or integer-

slot type. If qis an integer every leading or lagging edge of poles lines up simultaneously with

the stator slots causing cogging, but in fractional slot combinations fewer pole-edges line up

with the slots. A fractional slot winding minimizes the need for skewing of either the poles or

the lamination core to reduce the cogging. This actually precludes one of the best-known

brushless motors, the 12-slot-4-pole motor, as well as all the derivates from the 3 slots per pole

series. Hendershot and Miller also paid attention to the winding pitch character. Since the coils

can be wound only over an integer number of slots, dividing the number of slots by the numberof poles and rounding off to the next lower or higher whole number determine the winding

pitch. Obviously, the end turns are most short when the pitch is one or two slot-pitches. Any

number above two requires a considerable overlapping of the end turns. This may make some

slot/pole combinations more difficult, but one-slot- and two-slot-pitch windings can be

fabricated economically while using needle winders. The actual pole arc can make this situation

either worse or better. It is obvious that the end turns are most short when the pitch is one or

two-slots and that is why some two-layer constructions may be useful.

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Spooner and Williamson (1996) have studied multi-pole machines, since direct-coupled

generators were needed in wind turbines. In an application like that, the machine must fit within

the confined space of a nacelle; also a high efficiency and a power factor over a wide range of

operating power are demanded. The authors compared different structures taking into

consideration the easiness of construction as well as the manufacturing costs. They first built

prototype machines of a smaller size with 16 poles and 26 poles (rotor diameters 100 mm and

150 mm) and then designed a 400 kW machine with 166 poles (rotor diameter 2100 mm). The

efficiency of this machine was reported to be 90.8 (at rated power).

Lampola and Perho (1996) made a study of PM generators in wind turbine applications using

fractional slot windings. They used a 500 kW, 40 rpm generator with frequency converter. The

efficiency of the generator at rated load was 95.4%. Lampolas (2000) study focuses on the

electromagnetic design of the generator and the optimisation of the radial flux permanent

magnet synchronous generators with surface mounted magnets. He analysed machines with

different powers: 500 kW, 10 kW and 5.5 kW. The rated speeds of the machines were quite low

varying from 40 rpm to 175 rpm. The finite element method was used in computations and

genetic algorithms were used to optimise the costs, the pull-out torque and the efficiency

separately. According to the optimisations, the conventional machine has a higher efficiency

and smaller costs of active materials compared to the unconventional ones. The unconventional

fractional slot generator has a simple construction, it is easy to manufacture and it has a small

pole pitch, a small diameter, a smaller demagnetization risk and a low torque ripple. Therefore,

it is competitive for some PM generators. According to Lampola (2000), the choice between

these two types of machines depends on the mechanical, electrical, economic and

manufacturing requirements.

Cros and Viarouge (1999, 2002) studied different fractional slot PM motors with concentrated

windings. The details of the motors designed are not given in their papers. Therefore, acomparison between the fractional slotted designs introduced by the authors is difficult. From

the given torque curves, it can be estimated, that with q= 0.5 the torque ripple is about 15%

peak-to-peak and with q= 1 about 20% (30 slots 10 poles). It was noticed that machines with q

equal to 0.5 have a relatively low performance with sinusoidal currents. Such machines are

recommended for low power applications since the winding factor of these machines is only

0.866 and the torque ripple is high. According to Cros and Viarouge, machines with qbetween

1/2 and 1/3 generally produce a high performance. The machine with 10 poles and 12 slots is of

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particular interest, because it can support a one-layer concentrated winding and the torque ripple

of the machine is low. Moreover, these structures also give a no-load cogging of low amplitude

although the frequency is relatively high.

Cros et al. (2004) also studied brushless DC motors with concentrated windings and segmented

stator. According to his studies, by using concentrated windings it is possible to save 17%

copper material, 24% iron material and to reduce the total copper losses up to 17% compared to

the integer slot wound machine.

Kasinathan (2003) made a study of fractional slot machines, which have a slotted stator inside

and in the outer side a rotor constructed of permanent magnets. The thesis primarily analyses

the practical limits for the force density in low-speed permanent magnet machines. These limits

are imposed by the magnetic saturation and heat transfer. The author studied the force densities

of fractional slot motors with qequal to 0.375, 0.5 and 0.75 as well as an integer slot motor with

qequal to 1. An experimental in-wheel motor for a wheelchair application was built and tested

and it was shown that the design specifications were met. The motor has 42 slots and 28 poles

(q = 0.5) with one slot pitch skew. At a 150 rev/min rated speed the output power was

approximately 600 W and the torque 42 Nm. The results were promising and showed a

remarkable increase in performance compared to the existing conventional geared drive used in

wheelchair applications. Unfortunately, the author was not granted permission to include the

details of field-testing of the experimental motors or prototypes in his thesis.

Magnussen and Sadarangani (2003a) and Magnussen et al. (2003b, 2004) introduced a study of

machines, where a slotted armature is the rotating part and the permanent magnets are

assembled in a non-rotating outer part of the machine. A fractional 15-slot-14-pole prototype

motor was designed for a hybrid vehicle application. The rated torque of the motor was 85 Nm

and the estimated torque peak-to-peak ripple 3.5% of the rated torque. Magnussen et al. (2003a)

compared conventional integer slot windings with fractional slot windings. Three winding

structures were studied. The first structure is a theoretical reference machine, where the

fundamental winding factor is unity and which has a distributed winding with q= 1 (integer slot

winding). The second and the third machine are equipped with concentrated one-layer and two-

layer windings. The winding factor of the reference winding is 1= 1, but the fractional slot

wound motors have a fundamental winding factor 1 = 0.866. As the winding factor of the

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fractional slot winding is lower than that of the integer slot winding, also the torque developed

is lower, unless there will be more winding turns or a higher current density in the fractional

slot wound machine. The machine with a winding factor 1= 0.866 has a 15.5% higher current

density and 33.3% higher copper losses compared to the reference machine for the same torque,

assuming that the machines have equal slot filling factors and a comparable magnetic design

and also that the end windings are disregarded. As the machines were compared concerning

their slot filling factors, other parameters were calculated for each motor. In the fractional slot

machine the length of the end winding is smaller and the filling factors can be higher than those

of integer slot windings. Therefore, the relative winding losses (DC losses) of both fractional

machines were smaller than in the integer slot machine. It was also stated that these copperlosses diminish as the pole pair number is increased.

1.1. Brushless motor types

A brushless motor is a motor without brushes, mechanical commutator or slip rings, which are

required in a conventional DC motor or synchronous AC motor for connection to the rotor

windings. According to Hendershot and Miller (1994), there are several motors, which satisfy

this definition, as e.g. the

AC induction motor,

Stepping motor,

Brushless DC motor and

Brushless AC motor.

The most common of these is the AC induction motor, in which the current in the rotor

windings is produced by electromagnetic induction. The AC induction motor employs a rotating

magnetic field that rotates at a synchronous speed set by the supply frequency. The larger the

number of slots per pole and per phase qis, the more the properties of the induction motor will

improve. The larger the q value is, the lower super-harmonic magnetomotive force content,

created by the winding, will be and the torque production will be smooth. However, the rotor

rotates at a slightly slower speed because the process of electromagnetic induction requires

relative motion slip between the rotor conductors and the rotating field. Because the rotor

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speed is no longer exactly proportional to the supply frequency the motor is called an

asynchronous machine. The induced rotor current increases the copper losses, which, again,

heat the rotor and decrease the efficiency proportionally to the slips. The variation of the rotor

resistance with the temperature causes the effective torque to vary, which actually makes the

motor control difficult, as it is e.g. in high-precision motion control applications at least in the

absence of a position encoder. Hendershot and Miller (1994) state, that the brushless permanent

magnet motor overcomes the above-described restricting characteristics of the AC induction

motor.

The stepping motor is also a commonly used brushless motor type. In most structures, the rotor

has permanent magnets and laminated soft iron poles, while all windings are in the stator. The

torque is developed by the tendency of the rotor and stator teeth to pull the poles into alignment

according to the sequential energization of the phases. One of the advantages of the stepping

motor control is that an accurate position control may be achieved without a shaft position

feedback. Stepping motors are designed with small step angles, a fine tooth geometry and small

air-gap to achieve stable operation and enough torque. The disadvantages of the stepping motor

are its cost and acoustic noise levels.

The operation of the brushless DC motor is based on the rotating permanent magnet passing a

set of conductors. Thereby, it may be comparable with the inverted DC commutator motor, in

which the magnets rotate while the conductors remain stationary. In both of the motor types, the

current in the conductors must reverse polarity every time a magnet pole passes by, to ensure a

unidirectional torque. The commutator and the brushes are used to perform reverse polarity in

the case of the DC commutator motor. The polarity reversal of the brushless DC motor is

performed by power transistors, which must be switched on and off in synchronism with the

rotor position. The performance equations and speed as well as the torque characteristics are

almost identical for both motor types. When the phase currents in the brushless DC motor areswitching polarity as the magnet poles pass by, the motor is said to operate with square wave

excitation and the back EMF is usually arranged to be trapezoidal. In another operation mode,

the phase currents are sinusoidal and the back EMF should be, in the ideal case, sinusoidal. The

motor and its controller appear physically similar as in previous case, but there is an important

difference. The motor with sine waves operates with a rotating field, which is similar to the

rotating magnetic field in the induction motor or the AC synchronous motor. This brushless

motor type is a pure synchronous AC motor that has its fixed excitation from the permanent

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magnets. This motor is more like a wound rotor synchronous machine than a DC commutator

motor, and is, thereby, often called brushless AC motor. Different names may be used in the

literature on the subject or by the manufactures in different countries for the motors described

above. Two cross-sections used in different motor types are shown in Fig. 1.3.

N

N

S S

N

S

frame

permanent

magnet

11-slot wound

armature

stator frame

3 phase 12-slotstator winding

4-pole permanent

magnet rotora) b)

Fig. 1.3. a) Motor cross-section of a DC commutator motor and exterior rotor brushless DC motor. b)

Cross section for an interior rotor brushless DC motor and brushless AC motor. (Hendershot and Miller,

1994).

The motor cross-section used for a DC commutator motor is shown in Fig. 1.3 a), but it can also

be used for an exterior rotor brushless DC motor. Fig. 1.3 b) shows a cross section of an interior

rotor brushless DC motor and the same cross section can also be used for a brushless AC motor.

The study in his thesis is mainly focused on a brushless AC motor, which is a synchronous

motor equipped with an interior rotor with permanent magnets.

1.2. Location of the permanent magnets

Nowadays, the most commonly used construction for the PM motors is the rotor construction

type which has the permanent magnets located on the rotor surface. Herein, this motor type will

be called surface magnet motor for simplicity reasons. In a surface magnet motor the magnets

are usually magnetized radially. Due to the use of low permeability (r= 1 1.2) Nd-Fe-B

rare-earth magnets the synchronous inductances in the d- and q-axis may be considered to be

equal which can be helpful while designing the surface magnet motor. The construction of the

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motor is quite cheap and simple, because the magnets can be attached to the rotor surface. The

embedded magnet motor has permanent magnets embedded in the deep slots. There are several

possible ways to build a surface or an embedded magnet motor as shown in Fig. 1.4.

N

S

d

q

S

N

SNNS

S

d

q

N

N

S

d

q

S

N

SNNS

N

S

d

q

S

N

SNS

S

S

N N

N

S

N

S

N

N

S

S

d

q

N

S

N

SS

d

q

N

N S

N

S

S

N

d

q N

S

N S

a) b) c)

d) e) f) g)

Fig. 1.4. Location of the permanent magnets: a) Surface mounted magnets, b) inset rotor with surface

magnets, c) surface magnets with pole shoes, d) embedded tangential magnets, e) embedded radialmagnets, f) embedded inclined V-magnets with 1/cosine shaped air-gap and g) permanent magnet assisted

synchronous reluctance motor with axially laminated construction. (Heikkil, 2002)

In the case of an embedded magnet motor, the stator synchronous inductance in the q-axis is

greater than the synchronous inductance in the d-axis. If the motor has a ferromagnetic shaft a

large portion of the permanent magnet produced flux goes through the shaft. In this study the

embedded-magnet motor is equipped with a non-ferromagnetic shaft in order to increase the

linkage flux crossing the air-gap. Another method to increase the linkage flux crossing the air-

gap is to fit a non-ferromagnetic sleeve between the ferromagnetic shaft and the rotor core

(Gieras and Wing, 1997).

Compared to the embedded magnets, one important advantage of the surface mounted magnets

is the smaller amount of magnet material needed in the design (in integer-slot machines). If the

same power is wanted from the same machine size, the surface mounted magnet machine needs

less magnet material than the corresponding machine with embedded magnets. This is due to

following two facts: in the embedded-magnets-case there is always a considerable amount of

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leakage flux in the end regions of the permanent magnets and the armature reaction is also

worse than in the surface magnet case. Zhu et al. (2002) reported that the embedded magnet

structure facilitates extended flux-weakening operation when compared to a surface magnet

motor with the same stator design (both machines are equipped with an integer slot winding).

He also stated that the iron losses of the embedded magnet machine were higher than that of the

machine with surface magnet rotor. However, there are several other advantages that make the

use of embedded magnets favourable. Because of the high air-gap flux density, the machine

may produce more torque per rotor volume compared to the rotor, which has surface mounted

magnets. This, however, requires usually a larger amount of PM-material. The risk of

permanent magnet material demagnetization remains smaller. The magnets can be rectangularand there are less fixing and bonding problems with the magnets: The magnets are easy to

mount into the holes of the rotor and the risk of damaging the magnets is small. (Heikkil,

2002). Because of the high air-gap flux density an embedded magnet low speed machine may

produce a higher efficiency than the surface magnet machine.

1.3. Applications

When many poles are used it is possible to increase the air-gap diameter since less space is

needed for the stator yoke. The capacity of producing the motor torque grows up rapidly with

the increased air-gap diameter. Additionally, the copper losses of the stator diminish by

decreasing the end winding length and the winding resistance. Therefore, the torque per volume

ratio of these motors can be especially high. This may be described with the rotor surface

average tangential stress, which in these cases easily reaches values between 30 50 kN/m2.

What kind of the winding structure should be, this depends a lot on the application conditions

for the motor to be used in: how much space is available, which is the speed desired and how

many poles will be used. With an integer slot winding it is possible to adjust the winding turn

amount only by chording the coils. Usually, integer slot windings are used with q= 2 6. Theselection of qis done according to the mechanic limitations the numbers of poles and slots

suitable for the motor size. More possibilities to select qcan be found if fractional slot windings

are used. In cases where there is already a slotted rotor or stator of suitable size available, it may

be easier to adjust the pole number by using fractional slot windings than produce new steel

laminations. According to several scientific publications fractional slot wound machines are

often used in vehicles, such as for example the hybrid electric vehicle application by

Magnussen et al. (2003b), the fractional slot wound PM-machine for train application by Koch

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and Binder (2002). Koch and Binder (2002) discovered the fractional slot wound motor to be a

suitable motor for their application requirements: it has a direct gearless drive, low speed, high

torque and low mass per torque. There are some applications with only one or two phases.

According to Cho et al. (1999), a brushless DC motor with permanent magnets has been used as

a spindle motor in diskette driving systems such as CD/DVD-ROM, HDD etc. and as a direct

drive motor in e.g. washing machines. Direct drive permanent magnet generators used in wind

turbines, as e.g. the surface magnet machine by Lampola (2000) and embedded magnet machine

by Spooner and Williamson (1996) are examples of applications where fractional slot windings

are used. Today, fractional slot machines have been used also in converter fed high torque, low

speed machines for elevators, machining and ski lift drives with torque ratings up to 200 kNm,Reichert (2004).

1.4. End winding and stator resistance

Some possible machine structure sizes are illustrated in Fig 1.5. The machine with the air-gap

diameterDequal to the length of the core L, is illustrated in a) with a conventional winding

and b) with a concentrated fractional slot winding. The end winding of the conventional lap

winding a) is as long as the length of the core L. With fractional slot windings, shown in Fig.

1.5 b), the end winding length is about 1/5 of the length of the machine. In longer machines the

relative end winding length may be much smaller than in short machines and, therefore, the end

winding length may be a less important parameter in such cases. Fig. 1.5 c) shows a long

machine, which has a higher pole number than the machine in Fig. 1.5 b).

+ A

- A

- A

+ A

+ A

- A- A

+ A

L

a) b) c)

Fig. 1.5. The machine structures a) conventional winding, where p= 2, q= 1, b) concentrated fractional

slot winding, where p= 4, q= 0.5 (short machine) and c) a winding, where q= 1 and the pole number is

high (long machine where the relative end winding length is short despite of the traditional winding).

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According to Bianchi et al. (2003), when the number of poles is high the concentrated winding

is convenient only when the stator length is smaller than the air-gap diameter. Bianchi et al.

(2003) calculated the D/L values for a fractional slot machine to estimate in which

circumstances the use of concentrated windings may be beneficial. He compared a full-pitch

winding to a concentrated fractional slot winding taking into consideration the capacity of

torque production and the amount of copper losses. Research has been done also on special

machine types that are equipped with concentrated windings and have an irregular distribution

of the slots with two widths, e.g. by Cros and Viarouge (2002), and Koch and Binder (2002).

Cros and Viarouge (2002) discovered that this motor type has a higher performance than the

motor type with regular distribution of the slots. The copper volume and copper losses in theend windings are reduced. The end winding arrangements and the copper losses of a fractional

slot machine were studied and the results were compared to an integer slot machine. First, the

45 kW fractional wound (q= 0.4) prototype motor with 12 slots and 10 poles was compared to

a motor with q= 1. A fractional slot motor with q= 0.4 can have at least three different winding

constructions:

a) one-layer winding

b) two-layer winding, where the slots are divided horizontally

c) two-layer winding, where the slots are divided vertically.

The end windings of one phase of a 10-pole-machine with different winding constructions are

shown in Fig. 1.6. It is easy to see that the length of the end windings of motor a) are about

three times as long as in motor b) or c).

+A

- A- A

+A+A

-A

a) b) c)

+A

-A

Fig. 1.6. End windings of one phase of a 10-pole-machine: a) a traditional one-layer winding with Qs= 30

and q= 1, b) a one-layer fractional windingQs= 12 and q= 0.4 and c) a two-layer fractional slot winding

with Qs= 12 and q= 0.4, where the slot is divided vertically.

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The end windings of a traditionally wound machine need more space (which, again, requires

more copper volume and mass), because different phase coils cross each other. In the

concentrated fractional slot wound machine the space needed for the conductors to travel from

one slot to the next one is as small as possible, as the example illustrates in Fig. 1.6 b) where the

coil is wound around one tooth. However, the two-layer winding type produces the smallest end

windings as it is shown in Fig. 1.6 c). The average length of the end winding, lbof a cylindrical

machine can be calculated, according to Gieras and Wing (1997, p. 409), with

[ ]m02.02

)217.1083.0( 1b ++

+=

p

ypDpl . (1.1)

VariableDis the air-gap diameter,pis pole pair number andy1is the height of the stator slot.

It may be possible to measure the lengths of one particular motor. This is one method but also

the proper way to do in the case of a concentrated winding, because some equations do not

function well if qis less than one. If a coil is wound around one tooth the average end winding

length is simply the length between two slots (measured from middle) and the width of the slot

as illustrated in Fig. 1.7.

x1

2.5 - 5 mm

bb

y1

D

hb

1

2

lb = 2hb + bb

1

2x1

Fig. 1.7. Definition of the length of the end winding lb.Variable x1is the width of the stator slot,y1is theheight of the stator slot,Dis the air-gap diameter, hbis the height and bbthe width of end winding.

The equation below can be used for the concentrated winding

[ ]m01.0...005.0)( 1s

1b ++

+= x

Q

yDl . (1.2)

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wherex1is the width of the stator slot and y1is the height of the stator slot. As the conductors

come out from the slot they cannot twist directly to the next slot but there should be a small, e.g.

5 mm, gap between the core end and the innermost winding turns. The end winding

constructions of the four different 10-pole-machines are compared in Table 1.1. The four

different 10-pole-machines are:

a) concentrated two-layer winding with vertically divided slots (12 slots, 10 poles),

b) concentrated two-layer winding with horizontally divided slots (12 slots, 10 poles),

c) one-layer winding (12 slots, 10 poles, q= 0.4),

d) one-layer winding (30 slots, 10 poles, q= 1).

Table 1.1. 10-pole-machines 45 kW, machine core length 270 mm, stator outer diameter 364mm, air-gap diameter 249 mm (Nph= 132)

q(slots per pole and per phase)

a

0.4

b

0.4

c

0.4

d

1

End winding length (mm)

(with a 5mm minimum distance from the core)

118 130 130 330

End winding copper Mass (kg) 8.5 12.3 12.3 34.7

Copper mass in slots (kg) 28.5 28.5 28.5 28.5

Copper in the whole motor (kg) 37 41 41 63

End winding copper mass / Copper mass in slots 0.30 0.43 0.43 1.22

End winding mass per total copper mass (%) 23 30 30 55

The least amount of copper was needed for the end windings of the motor a) with a

concentrated wound fractional slot winding. The mass of copper in the end windings was only

8.5 kg in comparison to the non-fractional winding d) in which the mass was over 30 kg. The

end windings of the concentrated wound fractional slot machine are 2030% of the totalcopper weight of the machine in comparison to the end windings weight of the traditional

machine (q= integer) which are typically over 50%. The copper losses were calculated at a 90

A current with Wye connection. It was noticed that the copper losses of the stator diminish with

the decreasing of the end winding and the copper resistance. The copper losses of a 10-pole-

machine with q= 1 would be two times as high as those of the q= 0.4 machine (If the current

density of the machines is about the same, then the copper losses are directly comparable to the

copper weight).

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1.5. Scientific contribution of this work

The popularity of industrial permanent magnet motors is growing. They have been increasingly

used especially in low speed direct drive applications, where the fractional slot winding

structure proved to be an attractive solution. There is, however, not available much knowledge

on the fractional winding arrangements concerning PM motors, if q< 1. Traditionally, in the

literature on fractional slot machines the issue has usually been treated in the form where qis

larger than unity. E.g. q = 1.5 and q = 2.5 are popular traditional fractional slot winding

arrangements. In those modern applications where multi-pole machines are needed, the

fractional slot winding arrangement with q < 1 is an attractive alternative for traditional

solutions some of these applications have been studied in recent papers. The literature on the

subject poorly offers criteria for the selection of motor design variables. Here, a study is made

on fractional slot wound permanent magnet motors, because this type of motor can be used in

various applications. The main objective of this work is to compare different pole and slot

combinations applied to a machine, which has a fixed air-gap diameter and a 45 kW output

power. The performance analysis is done for machines having concentrated winding, where coil

is around tooth and qis equal or less than 0.5.

The scientific contribution of this work can be summarized to be the following:

A comprehensive study of the winding design of concentrated wound fractional slot

machines. Winding arrangements and winding factors are given for concentrated wound

fractional slot machines.

A performance comparison of concentrated wound fractional slot machines in a same

machine size. Different slot-pole (Qs - 2p) combinations for concentrated wound

fractional (q 0.5) slot machines are analysed to find out, which slot-pole combinations

have a high pull-out torque. The cogging torque and torque ripple are also analysed.

A comparison of different rotor structure performances.

A 45 kW prototype motor was manufactured to verify the computations.

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2. CALCULATION OF A FRACTIONAL SLOT PM-MOTOR

In this chapter, different methods to calculate fractional slot wound machines are studied. The

winding is called fractional slot winding if qis not an integer number. In this study, both the

one-layer and two-layer windings are discussed. In a two-layer winding a slot can be divided

into two different parts in which the coils may belong to different phases. It is also possible to

wind the fractional slot wound motor in such a way that the slots include only coils of one phase

or the slots are divided to embed two coil sides belonging to two different phases. The

fundamental winding factor 1 of a fractional slot wound machine is often lower than the

winding factors of an integer slot wound machine. The value 0.95 is considered to be a high

value for a winding factor of the fractional slot machine. Vogt (1996) introduced methods to

design fractional slot windings. He divided these windings in to two groups: the 1st-grade and

2nd

-grade winding. Some definitions are needed to describe whether the winding is a 1st-grade

or a 2nd-grade winding. These definitions may be defined through closer examination of the

term q (slots per pole and per phase),as it is shown below.

n

z

pm

Qq ==

2

s , (2.1)

where mis the number of phases, zis the numerator of qand nis the denominator of qreduced

to the lowest terms. The winding definitions introduced by Vogt (1996) concerning the

fractional slot windings are given in Table 2.1. The 1st-grade winding is always built up based

on one straight method (see Table 2.1), but for the 2nd

-grade windings there are different

definitions depending on whether the winding is a one-layer or a two-layer winding. If the

denominator nis an odd number the winding is a 1 st-grade winding and if nis even then it is a

2nd

-grade winding.

A variablet is needed to calculate other values as e.g.Q* andp*. Q* is the number of slots in a

symmetrical base winding.p* is the number of poles in a symmetrical base-winding. t* is the

number of base windings in a stator winding. Base windings have the same induced voltage,

phase shift angle and they may be paralleled, if required.

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Table 2.1. Winding definitions (Vogt, 1996)

1st-Grade 2nd-Grade 2nd-Grade

Denominator, n Odd Even Even

t p/n 2p/n 2p/n

Layer One or two One Two

Q* Qs/t 2Qs/t Qs/t

p* n n n/2

t* 1 2 1

As the winding definitions are known, a voltage vector graph for the machine may be drawn.

The winding factor can be solved using this graph. This is described in Chapter 3.2.1. The

winding definitions for some of the analysed machines are given in Table 2.2.

Table 2.2. Numerical examples of winding definitions

1st-Grade 2nd-Grade 2nd-Grade

Qs 12 162 21

p 5 24 11

n

z

pm

Q

q == 2s

5

2

352

12

= 89

3242

162

= 227

3112

21

=

Denominator, n Odd Even Even

t 5/5 = 1 224/8 = 6 211/22=1

Layer One or two One Two

Q* 12 54 21

p* 5 8 11

t* 1 2 1

2.1. Two-layer fractional slot winding

Two-layer windings are divided in two groups: The 1st-grade and the 2nd-grade windings. In this

chapter, to the procedure of designing a two-layer winding will be discussed. The winding

arrangements of a 12-slot-10-pole and 21-slot-22-pole machine are described. In Appendix A

more winding arrangements are given, such asq= 1/2,1/4, 2/5 and 2/7.

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2.1.1. 1st-Grade fractional slot winding

Fig. 2.1 shows step-by-step how to select a suitable two-layer winding for a fractional slot

wound motor. At first, a voltage vector graph is drawn with Q*phasors.

1

3

5

-A

-A

-C

+B

-B

2

4

6

7

8

9

10

11

12

+A

-B

+A

+B

-C

+C

+C

n = 150e

a)

12

3

4

5

67

8

9

10

11

12

-A

-A

+A

+C

-C

-C

+C

-B

-B

+B

+B

+A

b)

12

3

4

5

67

8

9

10

11

12

-A

-A

+A

+C

-C

-C

+C

-B

-B

+B

+B

+A

+C

-A

-C

-B +B

+A

+A

-A

-C

+C

+B-B

c)

Fig. 2.1. a) A voltage vector graph of a 12-slot-10-pole fractional slot two-layer winding of the 1 st-grade.

b) The coil sides of the lower layer are placed first. c) Also the coil sides of the upper layer in the slots.

As an example, a voltage vector graph consisting of 12 phasors is drawn for a 12-slot-10-pole

machine. The phasors are numbered from 1to Q*so that the phasor number 2 is placed to

360p/Q*electric degrees, now 150electric degrees, from the phasor 1 and so on. The coil sides

are ordered into positive and negative values A, +B, -C, +A, -B and +C. Depending on the slot

number, there can be a different number of coils next to each other. With 12 slots there are 4

slots per phase: 2 positive ones and 2 negative ones. The voltage vector graph in Fig. 2.1 a)

shows, how the different coil sides of different phases are placed in the slots. The vectors

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belonging to the same phase must be adjacent (see vectors A A +B +B C C). Based on the

slot numbering illustrated in Fig. 2.1 a), the phase coils are placed into the lower winding layer,

which is located on the bottom of the slot, as is shown in Fig. 2.1 b). After having placed all 12

coils, the illustration of the lower winding layer is ready. The upper winding layer is

constructed from the lower winding layer by rotating the lower winding layer and by changing

the sign of each coil. (Because it is a tooth wound coil, the other coil side must be in the

adjacent slot). For example, from slot number 1 the -A coil side is connected to the +A coil side

located in the upper layer of slot 2. The required rotation angle is equal to a slot angle. Now, the

12-slot-10-pole winding is ready and is shown in Fig 2.1 c).

2.1.2. 2nd

-Grade fractional slot winding

For a two-layer winding of the 2nd-grade, there can occur a situation in which the width of the

zone is not a constant. A one-layer may include a different number of positive and negative

phase coils, e.g. for a 21-slot stator there may be 7 slots per phase in a lower layer and 7 in an

upper layer. It can be selected so that you have 4 positive and 3 negative phase coils for a layer.

Otherwise, the winding is built as it was explained before for the 1 st-grade winding. Next, the

winding arrangement is build for a 2nd

-grade winding, in this example, of a 21-slot-22-pole

motor, in which the q= 7/22 = 0.318 (n= 22).

First, a voltage vector graph is drawn with Q* = 21 phasors as it is shown in Fig 2.2. The

phasors are numbered from 1to Q* so that the phasor 2 is placed to 360p/Qselectric degrees,

now 188.6electric degrees, from the phasor 1 and so on. The coils are ordered into positive and

negative values c, a, -b, c, -a and b. Now, there are 7 coil sides in the one-layer forming the

bars of one phase, therefore, there will be an unequal number of positive and negative coil sides

in both layers (4 and 3, 3 and 4). The coil arrangements are shown in Fig. 2.2. The fundamental

winding factor can be solved to be 0.953 and the distribution factor to be 0.956.

It must, however, be remembered that this winding is not, despite of its high winding factor, to

be recommended for proper use. The winding produces a large unbalanced magnetic pull since

all the coil sides of one phase are located on one side of the stator. This will be discussed briefly

in the next chapter.

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1

2

3

4

5

6

7

8

9

10

1112

13

15

17

1921

14

16

18

20

-A-A

-A

-A

+A

+A+A

+C

+C

+C

-B

-B

-B-B

-C

-C

-C

-C

+B

+B

+B

12

2

13

3

14

4

15

5

16

617

7

8

9

1011

18

19

20

211

-A

-A

-A

+A

+A

+A

-A

+A

+A

+A

+A

-A

-A

-A

+C

+C

+C

+C+C

-C-C

-C-C

+C

-C

-C

-C

+C

-B

-B

-B

-B

+B

+B

+B

+B

+B

+B

-B

-B

-B

+B

d1= 0.956

Fig. 2.2. Placing the coils for a 21-slot-22-pole fractional slot two-layer winding of the 2nd-grade. The

drawing on the right hand side illustrates how to solve the distribution factor, d1.

2.2. Winding arrangements

Fig. 2.3 shows the winding arrangements of 21-slot-20-pole (q = 7/20) and 24-slot-20-pole

(q= 8/20 = 2/5) machines. Let us compare the winding arrangements. In the 21-slot-20-pole

machine all the coils of phase A are next to each other. The 7 coils of each phase are

concentrated to one area of the machine causing asymmetrical distribution of the coils. The

coils of a 24-slot-20-pole machine are symmetrically divided around the machine. An

asymmetrical placement of coils must be disadvantageous, because in a load situation there may

occur unwanted forces.

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12

2

13

3

14

4

15

5

16

617

7

8

9

1011

18

19

20

211

+B

+B

+B

-B

-B

-B

-A

+A

+A

+A

+A-A

-A

-A

-C

+C

-C

+C

+A +C-C

+C

-C

+B

+C

+A

+A

-A

-A -B

-B

-B

-B

+B

+B

+B

-C

-C+C

+C

-C

-A

12

2

13

3

14

4

15

5

16

6

17

7

8

9

10

11

18

19

20

21

1

22

2324

+C

+B

-B

-A

+A

+B

-B

-C

+C

-A

+A

-B+B

+C

-C+A

+A

-A -A

+B+B

-B

-B

+C

+C+A

+A -A

-A

-A -A

+A

+A

-C

-C

-C

-C

-B

-B

+B+B

-B

-B

+B+B

+C+C

+C

+C

-C

-C

a) b)

Fig. 2.3. The winding arrangements of a) 21-slot-20-pole (q = 7/20) and b) 24-slot-20-pole (q= 8/20).

Both machines have two-layer windings.

At one instant of time, as the machine is loaded, the situation occurs, where in phase A the peak

current is+i, and in phase B and C the peak current is only 1/2 i. In a situation like that the

unwanted effect called unbalanced magnetic pull may occur. Fig. 2.4 illustrates the radial

magnetic stress along the air-gap diameter (from finite element analysis, FEA) for a 21-slot-20-

pole machine. It is obvious that the radial forces on the air-gap periphery do not cancel each

other and an unwanted magnetic pull bending in the rotor and the stator is affected.

0

100

200

300

400

500

600

0 90 180 270 360

Mechanical angle (deg)

Radialmagneticstress(Nm/m

)2

Fig. 2.4. The radial magnetic stress along the air-gap diameter (obtained from the FEA) for a 21-slot-20-

pole machine.

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Experimental results of the unbalanced magnetic pull effect in a fractional slot machine are

described by Magnussen et al. (2004). He designed and tested a 15-slot-14-pole machine and

noticed that the asymmetrical placement of the coils causes unwanted forces. According to

Magnussen, the number of poles should not be selected to be almost equal to the number of

slots in the case of a concentrated three phase winding with an odd number of slots e.g. 9-slots-

8-pole, 15-slot-14-pole and 21-slots-20-pole. Jang and Yoon (1996) discovered that also the

9-slot-8-pole and 9-slot-10-pole brushless dc-motor generates the same unwanted forces. Also

Libert and Soulard (2004) studied radial forces and magnetic noise of concentrated wound

machines having 60, 62 and 64 poles. Asano et al. (2002) presented some results of vibrations

measurements of concentrated wound machines and he introduced methods to decrease theradial stress. Because of the unbalanced pull effect, the motor designer should carefully

consider whether to select an odd number of slots when fractional slot two-layer windings are

used.

2.3. Winding factor

In this chapter it is solved winding factors for the fractional slot windings, especially for

concentrated (two-layer) windings, where q< 1. The winding factors of an electrical machine

are proportional to the generated electromagnetic torques. So, the fundamental winding factor

of the machine must be high and its sub- and super-harmonic winding factors as low as

possible. A machine with a low fundamental winding factor needs to compensate its low torque

with a high current or with more winding turns, which both are inversely proportional to the

winding factor. The winding factor can be defined through a voltage vector graph or it can be

solved from the analytical equations. (When the winding factors of a particular machine are to

be solved by using the equations, it should be remembered that this must be done accurately,

because there are different equations to be applied for the different winding types.)

Analytically, the winding factor can be solved from (Koch and Binder, 2002)

skdp = , (2.2)

where pis the pitch factor, dis the distribution factor and skis the skewing factor. The pitch

factor pis defined for concentrated two-layer winding as (Koch and Binder, 2002)

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=

=

sp

sp

sin2

sin Q

. (2.3)

The skewing factor can be solved from the equation (Vogt, 1996, p. 401)

( )pskpsk

sk2/

2/sin

= , (2.4)

where sk is the skewing pitch. Skewing is used to minimize the cogging torque. As to the

concentrated fractional slot machine, there are cases, where the amplitudes of the cogging

torque are low, as it will be shown later. This is due to the fact that in a fractional slot machine

the different stator slot pitch multiples do not coincide with the rotor pole pitch (as if 3sin a

q= 1 machine equals p). The effect of skewing the fractional slot machine is studied e.g. by

Zhu and Howe (2000). A new universal method was introduced to solve the harmonic content

of an AC machine and may be successfully applied to fractional slot machines, (Huang et al.,

2004). However, in this thesis the matter is researched by using a conventional method. At first,

it is estimated which harmonics arise from these fractional slot windings. According to Jokinen

(1973), the harmonics are for the 2

nd

-grade (if nis even,p* = n/2)

( )221 += mgnp

. g= 0, 1, 2, 3, (2.5)

The harmonics created by fractional two-layer windings of the1st-grade two-layer winding (if n

is odd,p* = n)are

( )121

+= mgnp

. g= 0, 1, 2, 3, (2.6)

The sign in Eq. (2.5) and (2.6) is chosen to be + or to make the equations yield the positive

sign for the fundamental ( = +1). Equation (2.6) is valid also for non-fractional one-layer

windings when the sign is removed. For qN(n= 1) the order numbers are = 1, -5, 7,

The fractional slot winding q N generates also sub-harmonics ( < 1) and integer order

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harmonics including both even and odd numbers. Table 2.2 lists the harmonic waves developed

by a two-layer winding (Txen, 1941).

Table 2.2. The harmonic waves developed by a two-layer winding (Txen, 1941)

n /p Harmonics

1 6g+1 1, -5, 7, -11, 13, -17, 19, -23, 25, -29, 31,

2 3g+1 1, -2, 4, -5, 7, -8, 10, -11, 13, -14, 16, -17,

4 -41 (6g+2) -

42 ,

44 ,-

48 ,

410 ,-

414 ,

416 ,-

420 ,

422 ,-

426 ,

5 - 51 (6g+1) -

51 ,

55 ,-

57 ,

511 ,-

513 ,

517 ,-

519 ,

523 ,-

525 ,

7 71

(6g+1) 71

,- 75

, 77

,- 711

, 713

,- 717

, 719

,- 723

, 725

,

8 81 (6g+2)

82 ,-

84 ,

88 ,-

810 ,

814 ,-

816 ,

820 ,-

822 ,

826 ,

10 - 101 (6g+2) -

102 ,

104 ,-

108 ,

1010 ,-

1014 ,

1016 ,-

1020 ,

1022 ,-

1026 ,

11 -111 (6g+1) -

111 ,

115 ,-

117 ,

1111 ,-

1113 ,

1117 ,-

1119 ,

1123 ,-

1125 ,

13 131 (6g+1)

131 ,-

135 ,

137 ,-

1311 ,

1313 ,-

1317 ,

1319 ,-

1323 ,

1325 ,

14141 (6g+2)

142 ,-

144 ,

148 ,-

1410 ,

1414 ,-

1416 ,

1420 ,-

1422 ,

1426 ,

16 - 161 (6g+2) -

162 ,

164 ,-

168 ,

1610 ,-

1614 ,

1616 ,-

1620 ,

1622 ,-

1626 ,

The harmonics generate unwanted forces and additional losses in the machine (Vogt, 1996). In

a three-phase winding not all integer harmonics are present. From the air-gap spatial harmonic

spectrum all the harmonics which are multiplies of three are missing since their sinusoidal

waves locally cancel each other in symmetrical operation of a non-salient three-phase machine.

In the mmf waveform there appear also harmonic waves with even order numbers. These even

harmonics can cancel each other as the phase coils are constructed from the individual coils.

This happens especially in most of the two-layer windings, because the bunch coil of one pole

is shifted by an angle of radians from the next coil.

For a symmetrical integer slot winding (n = 1) the winding factor can be solved from the

equation (Txen, 1941; Jokinen, 1973, Eq. (19))

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39

==2

sin

2

sin

2

sin

pdqm

y

p

qmpq

mp

. (2.7)

In the equationsyis the coil pitch, which is one for concentrated two-layer windings. For a 1st-

grade two-layer winding (two coils in the same slot) the winding factor can be solved as follows

(Txen, 1941; Jokinen, 1973)

=2

sin

2

sin

2

sin

qm

y

p

nqmpnq

mp

. (2.8)

For a 2nd-grade two-layer winding the winding factor can be solved as follows (Vogt, 1996, Eq.

2.52)

= 2

cos

2

sin

2

sin

2sin v

p

nmqpnq

mpp

, (2.9)

where v is an angle from voltage vector graph. Eq. (2.9) is valid only for the equal zone

widths. If the zones of the phase are unequal, the winding factor can be found with the voltage

vector graph. The pitch factors (calculated with Eq. (2.3)) for some concentrated windings of

different pole and slot combinations are given in Table 2.3 and the fundamental winding factors

for some two-layer windings are given in Table 2.4. According to Koch and Binder (2002), the

pitch factor can be used as a fundamental winding factor for a concentrated one-layer winding,

if the teeth widths are equal (thereby the distribution factord = 1) and if the machine is not

skewed (sk= 1). The highest value for a certain pole number is bolded in the Table 2.4. When

equipped with an 18-pole rotor only the 27-slot-18-pole machine (1= 0.866) allows

concentrated windings. There are also many other slot-pole combinations with several slots and

poles; Table 2.4 can be continued as it is done by Libert and Soulard (2004). Some windings

with unbalanced windings are marked with*in Table 2.3 and in Table 2.4, because there is a

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risk of unbalanced pull effect. Combinations, where the denominator n(q=z/n) is a multiple of

the number of phases m, are not recommended and therefore not presented (marked with **in

Table 2.3 and in Table 2.4). Libert and Soulard (2004).

Table 2.3. Pitch factors p1for concentrated windings (q 0.5)

QsPoles

4 6 8 10 12 14 16 20 22 24 26

6p1

q0.866

0.5** 0.866

0.25

0.50.2

**0.5

0.143

0.8660.125

0.8660.1

0.50.091

**

0.50.077

9p1q

0.8660.5

0.985*0.375

0.985*0.3

0.8660.25

0.6430.214

0.340.188

0.340.15

0.6430.136

0.8660.125

0.9850.115

12 p1q

0.8660.5

0.9960.4

** 0.9660.286

0.8660.25

0.50.2

0.260.182

** 0.260.154

15p1q

0.8660.5

**0.995*0.357

0.995*0.313

0.8660.25

0.740.227

**0.4

0.192

18p1

q0.866

0.5

0.940.429

0.9850.375

0.9850.3

0.940.273

0.8660.25

0.770.231

21p1

q0.866

0.5

0.7930.438

0.953*0.35

0.997*0.318

**

0.930.269

24p1

q0.866

0.5

0.950.4

0.9910.364

**

0.9910.308

*not recommended because of the unbalanced magnetic pull

**not recommended because the denominator n(q=z/n) is a multiple of the number of phases m.

Table 2.4. Fundamental winding factors 1for concentrated two-layer windings (q 0.5)

QsPoles

4 6 8 10 12 14 16 20 22 24 26

61q

0.866

0.5**

0.8660.25

0.50.2

**0.5

0.1430.8660.125

0.8660.1

0.50.091

**0.5

0.077

91q

0.866

0.5

0.945*

0.375

0.945*0.3

0.866

0.250.6170.214

0.3280.188

0.3280.15

0.6170.136

0.8660.125

0.9450.115

121q

0.8660.5

0.9330.4

**0.9330.286

0.8660.25

0.50.2

0.250.182

**0.250.154

151q

0.866

0.5**

0.951*

0.357

0.951*

0.313

0.866

0.25

0.711

0.227**

0.39

0.192

18 1q 0.8660.5 0.9020.429 0.9450.375 0.9450.3 0.9020.273 0.8660.25 0.740.231

211q

0.866

0.5

0.89

0.438

0.953*

0.35

0.953*

0.318**

0.89

0.269

241q

0.866

0.5

0.933

0.4

0.949

0.364**

0.949

0.308*not recommended because of the unbalanced magnetic pull

**not recommended because denominator n(q=z/n) is a multiple of the number of phases m.

q> 0.5

q> 0.5

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Txen (1941) and Jokinen (1973) discussed some special cases where the fractional slot

machine has q= k 1/2, k1/4 or k1/5. In the equations kis an integer. For q= k 1/2, k1/4

or k1/5 the winding factors can be solved as (Txen, 1941; Jokinen, 1973)

=2

sin

2

sin

2

sin

mq

y

p

nmqpnq

mp

for odd /p (2.10)

and

=2

sin

2

cos

2

cos

mq

y

p

nmqpnq

mp

for even /p. (2.11)

When fractional /pare present their winding factors can be solved for k1/4 by

=2

sin

2

2

sin

2

2

sin

mq

y

p

pnmqpnq

pmp

m

. (2.12)

and for k1/5 by

=

2

sin

2

sin

2

sin

mq

y

ppnmqp

nq

pmp

m

. (2.13)

Txen (1941) introduced winding factor equations also for two different q= k2/5 windings. It

is possible to arrange these windings in two ways, depending on the phase spreads qa= q 3/5

and qa= q 2/5. The first winding type has a sequence of phase spreads qaqbqaqbqbfor one

phase and in the second winding type qaqaqbqbqb. Eq. 2.12 (for /p= odd) is valid for the first

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type, but with a negative sign. For n = 5 there are no even harmonics. For fractional /p the

winding factors can be solved by (Txen, 1941; Jokinen, 1973)

=2

sin

22

sin

2

sin

mq

y

p

pnmqpnq

pmp

. (2.14)

The second winding type q = k 2/5 with a sequence of phase spreads qa qa qb qb qb has

winding factors for odd /pas follows (Txen, 1941; Jokinen, 1973)

=2

sin1

cos2

2

sin

2

sin

mq

y

pnmqp

nmqpnq

mp

(2.15)

and for fractional /pas follows

+

=2

sin14

cos2

22

sin

2

sin

mq

y

ppnmqp

pnmqpnq

pmp

. (2.16)

The sign of the harmonic must be used in the equations. The sign depends on the selected

origin place. The start point origin lies in the middle of the coil group. (The start point is

used for building the Fourier series of the mmf. There may be different widths of coil groups in

two-layer windings: the start point can be selected to be in the middle of the shorter or longer

coil group.) Factor y is also an important parameter in these equations, because it takes into

account the width between two slots in the same group, and it is not a constant parameter it

depends always on the winding arrangement selections. Also Txen (1941) presented winding

arrangement solutions and winding factor equations for the 3-phase two-layer fractional slot

windings as well as for the one-layer windings with integer or fractional coil arrangements. For

both the fractional slot windings and integer slot windings there occur also slot harmonics.The

slot harmonics are defined according to (Txen, 1941; Jokinen, 1973)

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112s

slot +=+== gpQ

mqgp

g= 1, 2, 3, 4, (2.17)

Slot harmonics occur in pairs. The winding factor of a slot harmonic is the same as for the

fundamental harmonic (= 1). The first slot harmonic pair occurs asg= 1 and the second pair

as g= 2. In a harmonic pair, one harmonic rotates in the same direction as the fundamental

wave does and the other one rotates in the opposite direction. The winding factors can be

organized in tables or series according to their order numbers. This means that there can be

found some periodical behaviour for the winding factors of the fractional slot windings. This

will be shown next with the help of some examples.

The harmonic waves created by the winding with q= k 2/5 (2nd-grade) were studied, because

one of the motors used for the comparisons in this thesis (the prototype motor) has qequal to

2/5, with 12 slots and 10 poles. Differently to the previous studies, now the fractional slot

numbered waves (1/5, 7/5, 11/5, ) do not achieve exactly the same amplitudes as the integer

slot waves (1, 5, 7, ). The winding factors of the waves created by the fundamental wave

(e.g. 1, 5, 7) and the slot harmonic waves always remain the same amplitude. The amplitudes

of the harmonics between them can have different amplitudes in different wave groups. The

winding factors and wave groups of the 1st-grade windings are always periodical, but in some

special cases of the 2nd-grade windings (e.g. q= 2/5) they are not. This study concentrates on

windings in which q is less than unity. As an example, the mmf harmonics created by these

windings are studied using a comparison of the fractional slot q= 2/5 winding with integer slot

(q= 3), fractional slot q>1 (q= 3/2) and fractional slot q< 1 (q= 1/2) windings. The results for

the winding factors solved from the vo