Fractional slot permanent magnet synchronous motors for low speed applications

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 due permission for public examination and criticism in the auditorium 1382 at Lappeenranta University of Technology, Lappeenranta, Finland on the 20^th of December, 2004, at noon.

Acta Universitatis Lappeenrantaensis 198

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

ISSN 1456-4491

Lappeenrannan teknillinen yliopisto Digipaino 2004

Pia Salminen

FRACTIONAL SLOT PERMANENT MAGNET SYNCHRONOUS MOTORS FOR LOW SPEED APPLICATIONS

Lappeenranta 2004 150 p.

Acta Universitatis Lappeenrantaensis 198 Diss. Lappeenranta University of Technology

ISBN 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 slot windings. 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 end windings 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 phase less than one are examined. The winding factor and the winding harmonic components are calculated. The benefits attainable from a machine with concentrated windings are considered.

Rotor structures with surface magnets, radially embedded magnets and embedded magnets in V-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 torque ripple as well as the dynamic behavior of the current driven and voltage driven motor are solved. The results obtained from different finite element analyses are given. A simple analytic method to calculate fractional slot machines is introduced and the values are compared to the values obtained with the finite element analysis.

Several different fractional slot machines are first designed by using the simple analytical method and then computed by using the finite element method. All the motors are of the same 225-frame size, and have an approximately same amount of magnet material, a same rated torque demand and a 400 - 420 rpm speed. An analysis of the computation results gives new information 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 kW output 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

Electrical Engineering, Lappeenranta University of Technology.

I wish to express my deepest gratitude to Professor Juha Pyrhönen, head of the Department of Electrical 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 was partly 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 for the co-operation facilities. I wish to express my gratitude to D.Sc. Markku Niemelä, head of the CDMC, Lappeenranta.

I wish to express my special thanks to M.Sc. Asko Parviainen, D.Sc. Markku Niemelä and Professor Juha Pyrhönen for their support during the research work. They are the core of a large group 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 the prototype 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 proposed corrections. 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 for their help and understanding during my work.

Financial support by the South-Karelian Department of Finnish Cultural Foundation, Jenny and Antti Wihuri Foundation, Foundation of Technology and Association of Electrical Engineers in Finland, Ulla Tuominen Foundation, Walter Ahlström Foundation is gratefully acknowledged.

Lappeenranta, December 2004 Pia Salminen

Symbols

a Number of branches of winding B Flux density

Br Remanence flux density b Width bb Width of end winding bm Width of magnet cosϕ Power factor D Diameter Dδ Air gap diameter

d Lamination sheet thickness

EPM Induced back electro magnetic force (EMF) Fm Magnetomotive force

f Frequency

fs Frequency of stator field fsw Switching frequency g Factor, Index number

H, h Magnetic field strength, height hb Height of the end winding, radial hm Height of permanent magnet I, i Current

In Rated current k Index kC Carter’s coefficient ke Coefficient of excess loss kf Filling factor

kFe, t Factor for defining iron losses in teeth

kFe, y Factor for defining iron losses in yoke

kh Coefficient of hysteresis loss krb Factor fordefining bearing losses k1 Factor for defining inductance k2 Factor for defining inductance k3 Factor for defining iron losses k4 Factor for defining iron losses

Ld 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

Lsσ Stator leakage inductance Lz Tooth tip leakage inductance Lχ Leakage inductance, skewing lb 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 N_n1 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

Rph Phase resistance s Slip

T Torque

t Time, Variable, defines the winding arrangement

∆T_p-p Peak-to-peak torque ripple % of average torque

U Voltage

x Width

x1 Slot width

x4 Slot opening width y Coil pitch, height

y1 Slot height

y4 Slot opening height

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

Greek letters

α Electric angle, Magnet width (Magnet arc width / pole pitch, shown in Fig. 3.12) β Width of tooth, angle

δ Air-gap length, radial δ_a Load angle

δ_eff Equivalent air-gap length γ_k Phase shift

η Efficiency Λso Permeance of upper layer Λsu Permeance of lower layer Λg Mutual permeance

Λgo Mutual permeance of upper layer Λ_gu Mutual permeance of lower layer λ Permeance factor

λ_e Reactance factor for the end windings λ_w Reactance factor for the end windings λ’n Permeance factor, describes all λ factors λz Leakage inductance factor

PM

Φδ, Air gap flux created by permanent magnets

σ Conductivity

µ Permeability µ_Fe Permeability of iron

µ_r Relative permeability µ₀ Permeability of air (vacuum) ν Harmonic νslot Slot harmonic τp Pole pitch

τs Slot pitch τsk Skewing pitch

ω Electrical angular frequency ω_s Angular frequency of stator field ξ_ν Winding factor, ν^th harmonic

ξ₁ Winding factor, fundamental harmonic ξd Distribution factor

ξp Pitch factor ξsk Skewing factor

Ψ Flux linkage

Ψ_a Armature flux linkage Ψ_PM Flux linkage due to permanent magnet Ψ_s Stator flux linkage

Ψ_δ Air-gap flux linkage Acronyms

2D Two-dimensional A Analytical calculation AC 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

HDD 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 σ Leakage

1 Fundamental wave

ν 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

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

2.1.2. 2^nd-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 ...45

2.4. 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

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 ...141

APPENDIX 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

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 inductance Lsσ. Correspondingly, the quadrature-axis synchronous inductance Lq is the sum of the quadrature-axis magnetizing inductance Lmq and the leakage inductance Lsσ. 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 1970’s, it was possible to manufacture also large PM synchronous motors. The industrial interest to manufacture permanent magnet motors arose in the 1980’s as the new magnet material

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 1990’s, the permanent magnet remanence flux density Br = 1.2 T was considered to be a high value. In practice, also magnets with low Br values 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 a remanence 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, problems may be expected to appear. In a well-designed fractional slot motor the voltages and the currents may be purely sinusoidal.

-1.0 -0.5 0.0 0.5 1.0

0 1 2 3

Air gap radius

Flux density normal component (T)

2π 4π 6π

32.6°^2π

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 2π is equal to 2τ_p (τ_p is 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/(3⋅22) = 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 = −½.

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 (q ≥ 1) 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.

It was the author’s 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/mm² stator 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 the length 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 Qs and 2p produce 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 q is 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 number of 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.

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%. Lampola’s (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, a comparison 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 q between 1/2 and 1/3 generally produce a high performance. The machine with 10 poles and 12 slots is of

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 q equal to 0.375, 0.5 and 0.75 as well as an integer slot motor with q equal 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 ξ1 = 1, but the fractional slot wound motors have a fundamental winding factor ξ₁ = 0.866. As the winding factor of the

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 ξ₁ = 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 copper losses 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 q is, 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

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 slip s. 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 are switching 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

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-slot

stator winding

4-pole permanent magnet rotor a) 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 (µ_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

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

S N N

S

S d q

N N

S d q

S N

S N N

S

N S d q

S N

S N S

S S

N N

N S

N S N

N S

S

d q

N S N

S S

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 radial magnets, f) embedded inclined V-magnets with 1/cosine shaped air-gap and g) permanent magnet assisted synchronous reluctance motor with axially laminated construction. (Heikkilä, 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

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 rectangular and 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. (Heikkilä, 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. The selection of q is done according to the mechanic limitations – the numbers of poles and slots suitable for the motor size. More possibilities to select q can 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

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 diameter Dδ equal 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).

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 Dδ/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 the end 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 winding Qs = 12 and q = 0.4 and c) a two-layer fractional slot winding with Q_s = 12 and q = 0.4, where the slot is divided vertically.

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

[ ]

) 217 . 1 083 . 0

( ^{δ 1}

b = + + +

p y p pD

l . (1.1)

Variable Dδ is the air-gap diameter, p is pole pair number and y1 is 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 q is 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

b_b y₁

Dδ

h_b

1

2 l_b = 2h_b + b_b

1 2x₁

Fig. 1.7. Definition of the length of the end winding lb. Variable x1 is the width of the stator slot, y1 is the height of the stator slot, Dδ is the air-gap diameter, hb is the height and bb the width of end winding.

The equation below can be used for the concentrated winding

[ ]

005 . ) 0

( π

s 1 1

b = δ+ +x +

Q y

l D . (1.2)

where x1 is the width of the stator slot and y1 is 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 364 mm, 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 5 mm 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 20…30% of the total copper 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).

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 q is 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 q is 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.

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 q is 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 ξ₁ 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 1^st-grade and 2^nd-grade winding. Some definitions are needed to describe whether the winding is a 1^st-grade or a 2^nd-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 q= Q =

2

s , (2.1)

where m is the number of phases, z is the numerator of q and n is the denominator of q reduced to the lowest terms. The winding definitions introduced by Vogt (1996) concerning the fractional slot windings are given in Table 2.1. The 1^st-grade winding is always built up based on one straight method (see Table 2.1), but for the 2^nd-grade windings there are different definitions depending on whether the winding is a one-layer or a two-layer winding. If the denominator n is an odd number the winding is a 1^st-grade winding and if n is even then it is a 2^nd-grade winding.

A variable t is needed to calculate other values as e.g. Q* and p*. 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.

Table 2.1. Winding definitions (Vogt, 1996)

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

Denominator, n Odd Even Even

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

Layer One or two One Two

Q* Qs/t 2⋅Q_s/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

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

Qs 12 162 21

p 5 24 11

n z pm q= Q =

2

s

5 2 3 5 2

12 =

⋅

⋅ 8

9 3 24 2

162 =

⋅

⋅ 22

7 3 11 2

21 =

⋅

⋅

Denominator, n Odd Even Even

t 5/5 = 1 2⋅24/8 = 6 2⋅11/22=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 1^st-grade and the 2^nd-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 as q = 1/2, 1/4, 2/5 and 2/7.

2.1.1. 1^st-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

α_n = 150^e

a)

1 2

3

4

5 7 6

8 9 10 11

12 -A

-A

+A

+C

-C

-C +C

-B

-B

+B +B

+A

b)

1 2

3

4

5 7 6

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 1…to Q* so that the phasor number 2 is placed to 360⋅p/Q* electric degrees, now 150 electric 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

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. 2^nd-Grade fractional slot winding

For a two-layer winding of the 2^nd-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 2^nd-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 1…to Q* so that the phasor 2 is placed to 360p/Qs electric degrees, now 188.6 electric 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.

1

2

3

4

5

6

7

8

9

10 12 11

13

15 17 21 19

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 17 6

7

8 9 11 10 18

19 20

21 1

-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 ξ_d1= 0.956

Fig. 2.2. Placing the coils for a 21-slot-22-pole fractional slot two-layer winding of the 2^nd-grade. The drawing on the right hand side illustrates how to solve the distribution factor, ξ_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.