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

ABSTRACT 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

ACKNOWLEDGEMENTS This research work was carried out at the Laboratory of Electrical Engineering, Department of 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

ABBREVIATIONS AND VARIABLES

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 for defining bearing losses

k1 Factor for defining inductance

k2 Factor for defining inductance

k3 Factor for defining iron losses

k4 Factor for defining iron losses

k5 Factor for defining eddy current losses

L, l Physical length of the stator core, Inductance, Length

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

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

Qs Number of stator slots

q Slots per pole and per phase

Rph 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 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

δa Load angle

δeff Equivalent air-gap length

γk Phase shift

η Efficiency

Λso Permeance of upper layer

Λsu Permeance of lower layer

Λg Mutual permeance

Λgo Mutual permeance of upper layer

Λgu Mutual permeance of lower layer

λ Permeance factor

λe Reactance factor for the end windings

λw Reactance factor for the end windings

λ’n Permeance factor, describes all λ factors

λz Leakage inductance factor

PMδ,Φ Air gap flux created by permanent magnets

ρm Resistivity of the magnet

σδ Leakage factor

σ Conductivity

µ Permeability

µFe Permeability of iron

µr 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

ωs Angular frequency of stator field

ξν Winding factor, νth harmonic

ξ1 Winding factor, fundamental harmonic

ξd Distribution factor

ξp Pitch factor

ξsk Skewing factor

Ψ Flux linkage

Ψa Armature flux linkage

ΨPM Flux linkage due to permanent magnet

Ψs Stator flux linkage

Ψδ 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

EV Motor with embedded magnets in V-position

FEA Finite element analysis

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. 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 ....................................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

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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

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

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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 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.

15

-1.0

-0.5

0.0

0.5

1.0

0 1 2 3Air gap radius

Flux

den

sity

nor

mal

com

pone

nt (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 2τp (τp 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.

16

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.

17

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.

18

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

19

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

20

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

21

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

22

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

NS S

N

S

frame

permanentmagnet

11-slot woundarmature

stator frame3 phase 12-slot stator winding

4-pole permanentmagnet 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 (µr = 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

23

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

NS

NS

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

24

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

25

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).

26

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

27

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

[ ]m02.02

)217.1083.0( 1δb +

++=

pypDpl . (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

bb

y1

Dδ

hb

12

lb = 2hb + bb

12

x1

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

[ ]m01.0...005.0)(π1

s

1δb ++

+= x

QyDl . (1.2)

28

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).

29

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.

30

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 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.

nz

pmQq ==

2s , (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 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 n is an odd number the winding is a 1st-grade winding and if n is even then it is a

2nd-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.

31

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 2⋅Qs/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

nz

pmQq ==

2s

52

35212

=⋅⋅

89

3242162

=⋅⋅

227

311221

=⋅⋅

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 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 as q = 1/2, 1/4, 2/5 and 2/7.

32

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

-B2

4

6

7

8

9

10

11

12+A

-B

+A

+B

-C

+C

+C

αn = 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 1st-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

33

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 1st-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 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.

34

1

2

3

4

5

6

7

8

9

10

1112

13

15

17

1921

14

16

1820

-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

2021

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

+Bξd1 = 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, ξd1.

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.

35

12

2

13

3

14

4

15

5

16

617

7

8

9

1011

18

19

2021

1

+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 –1/2i. 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)

Rad

ial m

agne

tic st

ress

(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.

36

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 the

radial 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 ξξξξν ⋅⋅= , (2.2)

where ξp is the pitch factor, ξd is the distribution factor and ξsk is the skewing factor. The pitch

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

37

=

=

sp

sp

πsin2πsin

Qν

ττνξ . (2.3)

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

( )( )psk

psksk 2/π

2/πsinττν

ττνξ = , (2.4)

where τsk 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 3τs in a

q = 1 machine equals τp). 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 2nd-grade (if n is even, p* = n/2)

( )221+±= mg

npν . 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+±= mg

npν . 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 (ν = +1). Equation (2.6) is valid also for non-fractional one-layer

windings when the ± sign is removed. For q ∈ N (n = 1) the order numbers are ν = 1, -5, 7, …

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

38

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

by a two-layer winding (Tüxen, 1941).

Table 2.2. The harmonic waves developed by a two-layer winding (Tüxen, 1941) n ν/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) - 4

2 , 44 ,- 4

8 , 410 ,- 4

14 , 416 ,- 4

20 , 422 ,- 4

26 , …

5 - 51 (6g+1) - 5

1 , 55 ,- 5

7 , 511 ,- 5

13 , 517 ,- 5

19 , 523 ,- 5

25 , …

7 71 (6g+1) 7

1 ,- 75 , 7

7 ,- 711 , 7

13 ,- 717 , 7

19 ,- 723 , 7

25 , …

8 81 (6g+2) 8

2 ,- 84 , 8

8 ,- 810 , 8

14 ,- 816 , 8

20 ,- 822 , 8

26 , …

10 - 101 (6g+2) - 10

2 , 104 ,- 10

8 , 1010 ,- 10

14 , 1016 ,- 10

20 , 1022 ,- 10

26 , …

11 - 111 (6g+1) - 11

1 , 115 ,- 11

7 , 1111 ,- 11

13 , 1117 ,- 11

19 , 1123 ,- 11

25 , …

13 131 (6g+1) 13

1 ,- 135 , 13

7 ,- 1311 , 13

13 ,- 1317 , 13

19 ,- 1323 , 13

25 , …

14 141 (6g+2) 14

2 ,- 144 , 14

8 ,- 1410 , 14

14 ,- 1416 , 14

20 ,- 1422 , 14

26 , …

16 - 161 (6g+2) - 16

2 , 164 ,- 16

8 , 1610 ,- 16

14 , 1616 ,- 16

20 , 1622 ,- 16

26 , …

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 π radians from the next coil.

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

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

39

==2πsin

2πsin

2πsin

pd qmy

pqmp

q

mp νν

ν

ξξξν . (2.7)

In the equations y is 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

(Tüxen, 1941; Jokinen, 1973)

=2πsin

2πsin

2πsin

qmy

pnqmp

nq

mp νν

ν

ξν . (2.8)

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

2.52)

=

2cos

2πsin

2πsin

2πsin vην

ν

ννξν p

nmqpnq

mpp

, (2.9)

where ηv 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 factorξd = 1) and if the machine is not

skewed (ξsk = 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 (ξ1 = 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

40

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 ξp1 for concentrated windings (q ≤ 0.5)

Qs Poles

4

6 8

10

12

14

16

20

22

24

26

6 ξp1 q

0.866 0.5

** 0.866 0.25

0.5 0.2

** 0.5 0.143

0.866 0.125

0.866 0.1

0.5 0.091

** 0.5 0.077

9 ξp1 q 0.866

0.5 0.985* 0.375

0.985* 0.3

0.866 0.25

0.643 0.214

0.34 0.188

0.34 0.15

0.643 0.136

0.866 0.125

0.985 0.115

12 ξp1 q 0.866

0.5 0.996

0.4 ** 0.966

0.286 0.866 0.25

0.5 0.2

0.26 0.182

** 0.26 0.154

15 ξp1 q 0.866

0.5 ** 0.995*

0.357 0.995* 0.313

0.866 0.25

0.74 0.227

** 0.4 0.192

18 ξp1 q

0.866 0.5

0.94 0.429

0.985 0.375

0.985 0.3

0.94 0.273

0.866 0.25

0.77 0.231

21 ξp1 q 0.866

0.5 0.793 0.438

0.953* 0.35

0.997* 0.318

** 0.93 0.269

24 ξp1 q 0.866

0.5 0.95 0.4

0.991 0.364

** 0.991 0.308

* 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 ξ1 for concentrated two-layer windings (q ≤ 0.5)

Qs Poles

4 6

8

10

12

14

16

20

22

24

26

6 ξ1 q

0.866 0.5

** 0.866 0.25

0.5 0.2

** 0.5 0.143

0.866 0.125

0.866 0.1

0.5 0.091

** 0.5 0.077

9 ξ1 q

0.866 0.5

0.945* 0.375

0.945* 0.3

0.866 0.25

0.617 0.214

0.328 0.188

0.328 0.15

0.617 0.136

0.866 0.125

0.945 0.115

12 ξ1 q 0.866

0.5 0.933 0.4

** 0.933 0.286

0.866 0.25

0.5 0.2

0.25 0.182

** 0.25 0.154

15 ξ1 q 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 ξ1 q

0.866 0.5

0.902 0.429

0.945 0.375

0.945 0.3

0.902 0.273

0.866 0.25

0.74 0.231

21 ξ1 q 0.866

0.5 0.89 0.438

0.953* 0.35

0.953* 0.318

** 0.89 0.269

24 ξ1 q 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

41

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

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

or k ± 1/5 the winding factors can be solved as (Tüxen, 1941; Jokinen, 1973)

=2πsin

2πsin

2πsin

mqy

pnmqp

nq

mp νν

ν

ξν for odd ν/p (2.10)

and

±=2πsin

2πcos

2πcos

mqy

pnmqp

nq

mp νν

ν

ξν for even ν/p. (2.11)

When fractional ν/p are present their winding factors can be solved for k ± 1/4 by

−

=2πsin

2π

2πsin

2π

2πsin

mqy

ppnmqp

nq

pmp ννν

νν

ξν

m

. (2.12)

and for k ± 1/5 by

−

=2πsin

π2πsin

π2πsin

mqy

ppnmqp

nq

pmp ννν

νν

ξν

m

. (2.13)

Tüxen (1941) introduced winding factor equations also for two different q = k ± 2/5 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 qa qb qa qb qb for one

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

42

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

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

±

−

=2πsin

π22πsin

π2πsin

mqy

ppnmqp

nq

pmp ννν

νν

ξν . (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 ν/p as follows (Tüxen, 1941; Jokinen, 1973)

−

−=2πsin1πcos2

2πsin

2πsin

mqy

pnmqpnmqp

nq

mp ννν

ν

ξν (2.15)

and for fractional ν/p as follows

−

+

±

−

−=2πsin1π4πcos2

π22πsin

π2πsin

mqy

ppnmqppnmqp

nq

pmp ννννν

νν

ξν . (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 Tüxen (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 (Tüxen, 1941; Jokinen, 1973)

43

112 sslot +=+== g

pQmqg

pνν 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 (ν = 1). The first slot harmonic pair occurs as g = ±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 q equal 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 voltage vector graph are shown in Fig. 2.5. The result

obtained from the case q < 1 differs slightly from the result in which q ≥ 1. The harmonics of q

= 3 are all even numbers. The harmonics of (fractional slot q > 1) q = 3/2 includes all same

harmonics as q = 3 and also odd number of harmonics. There appears periodical series, but e.g.

for q = 1/2 all amplitudes of the winding factors are the same. According to Fig 2.5 there occurs

several high amplitude harmonics, as q < 1 and q is a fractional number for example q = 1/2 or

2/5. Appendix B presents the winding factors and the periodical behaviour of integer slot (q =

9) and fractional slot q > 1 (q = 9/2 and q = 9/4) windings.

44

0

0.5

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0.0

0.5

1.0

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

0.0

0.5

1.0

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

0.0

0.5

1.0

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

q = 3

q = 32

q = 12

odd

harmonic order number

q = 25

ξν

integer, odd ν/p = 1, 3, 5, 7, 11 ...

fractional ν/p = 1/2, 3/2, 5/2, ...

even

ξν

ξν

ξν

integer, even ν/p = 2, 4, 6, 8, 10 ...

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Fig. 2.5. The winding factors of a fractional slot q = 2/5, integer slot (q = 3), fractional slot q > 1 (q = 3/2)

and fractional slot q < 1 (q = 1/2) windings. In case of q = 3 there exist harmonic order numbers which are

all odd integer ν/p = 1, 5, 7, 11, 13 … shown as white bars. In case of q = 3/2 there exist also harmonic

order numbers which are even integer numbers ν/p = 2, 4, 8, 10, … shown as grey bars. In case of q = 2/5

there exist also fractional harmonic order numbers ν/p = 1/5, 7/5, 11/5, … shown as black bars. In this

example, there exist more harmonics with q < 1 than with q ≥ 1.

45

2.3.1. Winding factor according to the voltage vector graph

As the harmonic order numbers are known, it is possible to define the winding factors for the

harmonic waves. The voltage vector graph for the harmonic waves can be constructed in a

similar way as for the fundamental wave. The angle αn is multiplied by the harmonic order

number. The vectors of the voltage vector graph are numbered and the angle between the

vectors is now ναn. As an example, the winding factors are defined for a 12-a slot 10-pole

fractional slot concentrated winding using a voltage vector graph. When q = 2/5, the harmonic

waves created by the winding are 1/5, 5/5, 7/5, 11/5 … The voltage vector graph of the

fundamental (ν = 1) is shown in Fig. 2.6 a).

a) ν = 1 b) ν = 1/5

1

3

5

-A-A

-C

+B

-B2

4

6

7

8

9

10

11

12+A

-B

+A

+B

-C

+C

+C

α1

-A

-A+B

+B

-C

-C

+A

+A-B-B

+C

+C

1

11

9

-A-B

-C

+A

-A6

4

2

7

12

5

10

3

8+B

-C

+A

+C

-B

+C

+B

α1/5-C

-B+B

+A

-A

-C

+C

+B-A-B

+A

+C

Fig. 2.6. Voltage vector graph of a 12-slot-10-pole fractional slot concentrated (two-layer) winding a) for a

fundamental wave ν = 1 and b) for a harmonic wave ν = 1/5. The winding factors are defined as

geometrical sum/arithmetic sum of the vectors.

The angle α1 for the fundamental wave is 360ep/Qs = 150e. It is also shown how the geometrical

sum is used to calculate the winding factor ξ1 to be 0.933. The next harmonic order number

ν = 1/5 (which is a sub harmonic because ν < 1) will have an angle α1/5 = 1/5·150 e = 30e

between the vectors. Fig. 2.6 b) shows the arrows of ν = 1/5 harmonic. The vectors describing

phase A are drawn separately to define the winding factor for the harmonic ν = 1/5. Using the

geometrical sum as shown in Fig. 2.6 b) on the right side, the winding factor gets the value ξ1/5

= 0.067. Appendix C gives the harmonic order numbers of different fractional slot windings and

their winding factors. It may be noticed that most of the motors have few integer harmonics and

46

numerous non-integer harmonics, which are sub-harmonics if q is less than one. The winding

factors of the harmonics are quite small, except those created by the slot harmonic, since they

have the same winding factor as the fundamental harmonic. The q = 0.5 winding consists of

purely integer numbers of harmonics: 1, 2, 4, 5, 7, 8, 10… which all have a 0.866 winding

factor. The q = 0.25 winding consists of integer and non-integer numbers of harmonics: 0.5, 1,

2, 2.5, 3.5, 4, 5, 5.5, 6.5, 7, 8, … which have a winding factor of 0.866 or 0.5. Multiples of 3 are

not included in the harmonic order numbers of fractional slot windings.

2.4. Flux density and back EMF

By using the finite element analysis (FEA) it is possible to solve the electromagnetic state of the

machine. When the machine geometry is described into the FEA-software, the value of the flux

created by the permanent magnets PM,δΦ can be solved. This is an important value for the

calculating of the induced back EMF, EPM. The value of PM,δΦ depends greatly on the

equivalent air-gap length δeff, which can’t be obtained directly and accurately from the

analytical equations. Therefore, in this study the FEA is used to solve PM,δΦ , but it is also

calculated analytically in order to compare the results. The motor inductances are the most

critical parameters for calculating the pull-out torque achieved from the motor. The torque is

inversely proportional to the inductance. Therefore, it was the essential task to find an accurate

analytical method, which enables to correctly calculate the inductances for the fractional slot

machines. In the literature, methods for calculating the magnetizing and the leakage inductances

can be found for the integer slot machines, but there are no well-known methods described for

the fractional slot machines. Richter (1967) and Vogt (1996) presented some slightly different

methods to calculate the leakage inductance. The method introduced by Richter is not accurate

enough to be used for the permanent magnet structures treated in this study and it is assumed to

be more suitable for the integer slot machines. Both methods are described in this chapter.

As the winding arrangement is known and thereby the harmonic order numbers included in the

winding properties are known, the mmf created by the stator current can be calculated. The

mmf waveforms can be defined mathematically by using the equations given by Magnussen and

Sadarangani (2003a). The Fmν wave is separated into two waves rotating in opposite directions.

The waveform pattern should be drawn once over the whole symmetrical cycle of the machine

47

(therefore the minimum harmonic order is 2/p). Because the Fmν waveform is not the same for

each pole, a displacement – describing the phase shifts – factor ny is used as given by

31

6x

y −=q

nn , (2.18)

where nx is the physical displacement in the number of slots. The magnetomotive force Fmν for

a 3-phase machine with symmetrical phase windings is

( )∑ ∑∞

= =

−+ +=p

n

ν FFpNF

2

c

1cmmp

n1m π

4

νξ

ν (2.19)

where nc is the number of coils and Nn1 is the effective turns of a coil. The forward Fm+ and

backward Fm - rotating waves are defined as

( ) ( )

+−+−−=+

ycm 2π13

2πcos21)(cosˆ ntiF ννϑγνω (2.20)

( ) ( )

+++−+=−

ycm 2π13

2πcos21)(cosˆ ntiF ννϑγνω . (2.21)

The parameter γ is distance (along air-gap diameter) and ϑc the distance between the coil sides.

The waveforms of some fractional slot machines are shown in Fig 2.7. From the Fourier

spectrums of the waveforms – in Fig. 2.8 – the amplitudes of the harmonics can be observed.

The slot harmonics often appear in pairs and the amplitudes of the harmonics diminish almost

in proportion to 1/ν. In a waveform of an integer slot winding there exist harmonics with only

odd numbers, but in a waveform of fractional slot windings also even harmonics, fractional

harmonics and sub harmonics may appear. All these harmonics can be seen in the spectrum of

the Fm waveform. The waveform in this case is far from a sinusoidal as can be seen in Fig. 2.7.

48

-3

0

3

02π

All harmonicsFundamental

Fm

(p.

u.)

Air-gap peripherya)

-3

0

3

02π

All harmonicsFundamental

Fm

(p.

u.)

Air-gap peripheryb)

Fig. 2.7. The magnetomotive force wave of a) the fractional slot (q = 0.5) wound 4-pole 6-slot machine

wound in one-layer and b) 4-pole 6-slot machine, but wound in two-layers. Magnussen and Sadarangani

(2003a).

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30Harmonic order number

6-slot-4-pole one-layer

a)

Fm

(p.u

.)

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30Harmonic order number

6-slot-4-pole two-layer

Fm

(p.u

.)

b)

Fig. 2.8. The harmonics of the magnetomotive force of a 4-pole machine with 6 slots (q = 0.5): a) one-

layer winding and b) two-layer winding. There are more harmonic orders in the one-layer winding than in

the two-layer winding. Magnussen and Sadarangani (2003a).

49

The induced back EMF, EPM of the fractional slot permanent magnet motor is solved as

(Hendershot and Miller, 1994, p. 6-22)

2

ˆπ2 PM,δ1phs

PMΦNf

Eξ

= , (2.22)

where Nph is the amount of winding turns in series of stator phase, fs is the frequency of stator

field and PM,δΦ is the fundamental air-gap flux due to magnet. The PM,δΦ can be analytically

solved following e.g. the procedure by Heikkilä (2002). It can also be solved with the finite

element analysis, FEA. There are different methods to derive the equation of the magnetizing

inductance. It can be derived from the stator Fm and the stator flux linkages, according to Vogt

(1996), or from the air-gap reluctance and flux linkage, according to Grauers (1996).

2.5. Inductances

The direct-axis magnetizing inductance depends on the equivalent air-gap length δeff, which

depends on the mechanical air-gap length δ, the height of the magnets hm and the Carters

coefficient kC (and also the effect of the saturated iron, which is not included in Eq. 2.23).

Equivalent air-gap can be defined as

δeff = (δ + hm)kC. (2.23)

The magnetizing inductance for the whole machine is solved as (Gieras and Wing, 1997, p. 157)

( )2ph1ieff

p0md ππ

µ2 NLp

mL ξδ

τ= , (2.24)

where µ0 is permeability of air, τp is pole pitch and Li is the effective length of the stator core

(Li ≈ L + 2δ : L, physical length of the stator core). The accuracy of the analytical PM motor

torque calculation depends, in a considerable way, on the estimated inductances. The

magnetizing inductance can be calculated correctly only, if the equivalent air-gap length is

precisely known. The air-gap length, δeff is not easy to define, because it is not constant and

because there are many parameters which affect it, as e.g. the permanent magnets, the virtual

50

pole shoe above the magnets, the stray fluxes and the iron saturation in both the stator and rotor

iron (Heikkilä, 2002). The area for the air-gap for which Lmd is calculated is for an integer slot

machine the whole pole arc τp along which the flux density distribution may be assumed to be

even sinusoidal. Since the inductance is basically a stator-based quantity the rotor pole pitch or

pole pair number in Eq. (2.24) should not have any effect on the inductance of a fractional slot

machine. The coil inductance has three distinct components due to the three distinct areas - the

air-gap, the slots and the end windings - where magnetic fields are created by the coil current.

The ferromagnetic portions do not contribute to the inductance as long as their relative

permeability is high, according to Hanselman (2003). In concentrated wound fractional slot

machines the air-gap area which the flux travels through to produce flux linkage is the area

spanned by the coils (Qs/mτsLi), as illustrated in Fig. 2.9. By using these correlations the

magnetizing inductance for a three-phase machine can be solved as

( )2ph1i

effs

s0md

ππµ2 NL

mQ

mL ξδ

τ= . (2.25)

-A

-A

+A

+C

-C

-C

+C

-B

-B

+B

+B

+A

+C

-A

-C

-B+B+A

+A

-A

-C

+C

+B-B

τs

τs

τs

τs

Φ

Φ

Φ

Φ

Fig. 2.9. Flux paths of a 12-slot-10-pole machine for a A-phase. The air-gap area which the flux Φ travels

through to produce torque is the area spanned by the coils ((Qs/m)τsLi = 4τsLi).

2.5.1. Leakage inductance method 1

The leakage inductance Lσ can be calculated as the sum of its partial inductances. The leakage

inductances can be defined by the method given by Richter (1963, 1967). For the integer slot

51

windings, this method gives good results. Richter divides the leakage inductance into five

components:

• Lδ for the air-gap leakage inductance

• Ln for the slot leakage inductance • Lz for the tooth tip leakage inductance • Lw for the end winding leakage inductance

• Lχ for the skew leakage inductance.

The leakage inductance is the sum of all partial inductances and is defined below

χwznδσ LLLLLL ++++= . (2.26)

The structure of a two-layer winding is shown in Fig. 2.10 a). Fig. 2.10 b) shows a diagram to

define the leakage-factor σδ, which is needed to calculate the air-gap leakage inductance Lδ.

x4

slot pitch, τs

x1

y1

y12

y11

y3 y4

y5

y2

q = 2

q = 3

q = 4

q = 5

q = 6

8q =

0.03

0.026

0.012

0.008

0.004

0.0

σδ

y / τpa) b)0.7 0.8 0.9 1.0

Fig. 2.10. a) A two-layer winding with some important dimensions. In a two-layer winding there may be

slots that have coils of two different phases. b) A diagram to define the leakage-factor σδ for three phase

(q = integer) windings as a function of the coil pitch/pole pitch (Vogt, 1996, p. 267).

52

Air-gap leakage inductance

The stator mmf-harmonic content of a traditional (q = integer) machine is small compared to the

fractional slot machine (q < 1). Traditionally, an air-gap leakage-factor, σδ is calculated to

define the air-gap leakage inductance. According to Richter (1963), the air-gap leakage

inductance can be defined as

δ

2ph

iδ0

δ πµ σ

δ

=

pN

LDmL . (2.27)

Fig. 2.10 b) shows the values of σδ for a three-phase winding. The factor σδ can be calculated

from the winding harmonics content according to (Richter, 1954, p. 136)

∑+∞=

≠−∞=

=

ν

νν

ν

νξξσ

1

2

1δ . (2.28)

According to Richter Eq. (2.28) is valid only for the integer windings. In the case of a two-layer

fractional slot winding the air-gap leakage inductance cannot be calculated from the basic

equations, because it depends on the winding concerned. In the analytical computations the

harmonics and their winding factors were computed to estimate the amount of the leakage air-

gap inductance. As an example, a 12-slot-10-pole machine has the harmonic winding factors ξν5

= 0.933, ξν7 = 0.933, ξν11 = 0.067, ξν13 = 0.067 … and the factor σδ becomes

02.0..933.011

067.0933.07

933.0933.05

933.0 222

δ =+

⋅+

⋅+

⋅=σ . (2.29)

Slot leakage inductance

The fractional slot arrangement does not differ from that of the integer slot machine with

respect to the stator slot leakage. In the case of a two-layer winding, the inductance factor is

defined by integrating the magnetic field strength. The mutual inductance of both coils should

be taken into consideration as well as the fact that the coils may belong to different phases when

53

there is a phase shift between the currents flowing in them. The self-inductance of a two-layer

winding can be defined as (Richter, 1967, p. 269)

( )sosu21nn1 ΛΛNL += , (2.30)

where Nn1

is the number of effective turns of the coil, and Λsu

and Λso

are the permeances of the

upper and lower layers. They can be defined as

soi0sosui0su µandµ λλ LΛLΛ == , (2.31)

where Li is the effective length of the core, λsu

and λso

are permeance factors. The permeance

factors λsu

and λso

can be defined by using Richter’s (1967, p. 269) methods as follows:

4

43n

1

25

1

11su 3

4xy

xyy

xy

+++

+= λλ , (2.32)

4

43n

1

2

1

12so 3 x

yxy

xy

+++= λλ . (2.33)

A factor λn3 depends on the shape of the slot (round, square, …). For a sharp-angular slot the

factor may be defined (Richter, 1967) as

4

1

41

3n3 ln

xx

xxy−

=λ . (2.34)

Not only the self-inductance but also the mutual inductance must be taken into account. If there

are Nn1 winding turns in both parts of the coils the mutual inductance will be

( )gogu2n1n1 ΛΛNL += . (2.35)

Without the phase shift it can be stated that (the upper and lower coils belong to the same phase)

54

=

=

.µ

µ

goi0go

gui0gu

λΛ

λΛ

L

L (2.36)

Using the dimensions as shown in Fig. 2.10 a) the permeance factors can be defined as

4

43n

1

2

1

5ggogu 2 x

yxy

xy

+++=== λλλλ . (2.37)

The currents in the upper and lower coils do not always belong to the same phase shift.

According to Richter (1967), it is possible to calculate a factor

∑=

=n

kng

1kcos1 γ , (2.38)

which is multiplied with the permeance between the coils in the slot, to take into account the

difference of the phase shift of two coils in the same slot. The angle γk is the phase shift of the

coils. The summation includes all coils of one phase. So, the resultant inductance for one phase,

which has 2pq coils and a parallel branches of winding is (Richter, 1967, p. 271)

( )gsosu0i

21n

n 2µ2 λλλ gLa

NpqL ++

= . (2.39)

Using a symbol λ’n (to describe the effects of all λ factors)

42 gsosu'

nλλλ

λg++

= , (2.40)

and Nph to present the number of effective turns of one phase (in series)

apqNN 1n

ph2

= , (2.41)

the slot inductance of a two-layer winding can be written as

55

ns2phi0

sn µ4 λNL

QmL = . (2.42)

End winding inductance

The end winding inductance can be defined as (Richter, 1963)

s2phi0

sw µ4 λqNL

QmL = , (2.43)

where λs is defined as 2hbλe + bbλw. Factor hb is the height and bb is the width of the end

winding. Reactance factors for the end windings λe and λw depend on many parameters, such as

the winding structure, end winding layer orders, rotor type etc… There are several methods

available to estimate the values for these factors, as e.g. given by Richter (1963) and Jokinen

(1973). In this study, it was used the reactance factors λe = 0.518 [1/m] and λw = 0.138 [1/m],

which are defined for synchronous machines by Richter (1963). The width of the end winding

of the concentrated winding arrangement is the same as the slot pitch τs.

Tooth tip leakage inductance

In this case too, the traditional methods, defined for the integer slot machine, are directly

applicable. According to Richter (1963, p. 90), the leakage inductance factor can be defined by

δδλ/45

/5

4

4z x

x+

= . (2.44)

The tooth tip inductance of the phase coil is (Richter, 1963)

z2phi0

sz µ4 λNL

QmL = . (2.45)

56

2.5.2. Leakage inductance method 2

To estimate the amount of the leakage flux Vogt (1996) introduced a method, which is very

similar to the method described above. He noticed that in the case of a fractional slot machine

the estimation of the leakage flux amount was rather complicated. If there are two coils in the

same slot, there will be some interaction of the leakage components. Vogt made a study on the

slot opening effect appearing in fractional slot winding machines. He found out that, if the value

of the slot opening width per air-gap (x4/δ) is equal to 3, then the factor of the tooth tip leakage

λz is zero (Fig. 2.11). And if the value λz is greater than 3, then λz can be considered to be a

negative value.

The nature of the air-gap inductance is examined. If the value of q is large, it means that there

are several slots, which create the stepped magnetomotive force. The more slots there are (i.e.

the larger q is) the more sinusoidal the mmf will look and the smaller σ δ (Eq. 2.28) becomes –

as well as the air-gap inductance value. This stepped mmf is generated from different slots next

to each other. It is difficult to find in the literature any references to methods designed for the

calculating of this inductance in fractional slot applications. Vogt described many ways to

calculate the air-gap inductance, but only for integer slot windings. In the case of a concentrated

winding the situation is totally different, because there is no stepped mmf waveform at all. A

single coil is wound around one tooth and will produce half of the mmf of one main flux route.

This mmf is not strongly magnetically connected to the coils of the phases next to it. Therefore,

it is reasonable to assume that there is no air-gap inductance or it must be very low. At

saturation it might be wise to assume that some inductance will occur. (Vogt, 1996).

The higher the x4/δ value is the more the air-gap flux is drawn into the slot. Thereby, λz can

have negative values – to correct this effect. The factors, which are later needed to calculate the

slot inductance, can be solved by Vogt as

z4

4

k

3

1

2

1

1n 3

λλ ++++=xy

xy

xy

xy (2.46)

z4

4

k

3

1

2

1

12o 3

λλ ++++=xy

xy

xy

xy (2.47)

57

1

15z

4

4

k

3

1

2

1

11u

5.03 x

yyxy

xy

xy

xy +

+++++= λλ . (2.48)

Factor xk is (x1 – x4) and λz can be selected from the diagram in Fig 2.11. In two-layer windings

there may be slots where the coils are belonging either to the same phase, or to different phases.

This depends on the selected winding. The slot inductance of the upper slots (No) should be

calculated using the factor λo, for the calculation of the bottom slots (Nu) the factor λo should be

used and for the slots (Nn) with both coils belonging to the same phase the factor λn should be

used. The slot inductance is a combination of different slots λnz = Nu·λu + No·λo + Nn·λn. In

symmetrical windings having the same amount of upper and bottom slots the slot inductance

can be calculated as follows

1

5z

4

4

k

3

1

22

1

11ns 43 x

yxy

xy

xyk

xyk +

++++= λλ . (2.49)

Factors k1 and k2 are selected from the diagram shown in Fig. 2.11. A numerical example of

calculating inductances for a 24-slot-22-pole fractional slot wound motor is given in Appendix D.

1.0

0.8

0.6

0.4y /τp

0.6 0.8 1.0

k1

k2 k2

k1

1.0

0.8

0.6

0.4

0.2

0

-0.1

-0.2

1 2 3

2 6 10 14

λz

x4 / δ2/3

Fig. 2.11. Factors k1 and k2 for the calculation of the inductance, given by Vogt (1996), are shown on the

left. On the x-axis the ratio y /τp is the coil pitch per pole pitch. The dotted line is for machines with

doubled zone width and the dashed line for 2 phase machines. The diagram introduced by Vogt (1996, p.

254) for the selection of the factor λz is given as a function of slot opening width x4 per air-gap δ.

58

2.6. Torque calculation

When the inductances and induced back EMF, EPM are known the torque can be solved. The

torque developed by a synchronous motor is solved from (Gieras and Wing, 1997, p. 154)

( )

−+= a

dq

2

ad

PM2

s

2sin112

)sin( δδω LL

UL

UEmpT . (2.50)

In machines, where Ld = Lq the maximum torque is achieved with load angle, δa = 90°. In

PMSMs the maximum torque is often reached at load angles larger than 90°. When the supply

voltage U, induced back EMF, EPM and inductances are known the load angle can be solved

from the power equation, if Ld = Lq,

)sin()sin( asmd

PM

sa

d

PM

sδ

ωδ

ω σLLUEmp

LUEmpP

+== . (2.51)

2.7. Loss calculation

Resistive losses (copper losses) in windings may be defined by

2nphCu ImRP = . (2.52)

Iron losses in the stator and rotor are calculated as follows (Vogt, 1996)

2/32t10tFe,tFe,

2/32y10y Fe,yFe,Fe 50

ˆ50

ˆ

+

=

fbpkmfbpkmP , (2.53)

where f is frequency, kFe, y = 1.5, kFe, t = 1.2 and p10 = 2.7 (at 1 T 50 Hz, subscript y means yoke

and t means teeth). The bearing losses are defined as

( )

+=

606.0 s

prrbBrnLDkP τ . (2.54)

59

In the case of a surface cooled motor a factor krb of 8 … 10 Ws2/mm4 can be used for the

calculation of the bearing losses.

The stray losses according to Gieras and Wing (1997) can be calculated as

PP ⋅= 0075.0Str . (2.55)

The stray losses may also be calculated by using other methods. The stray losses are combined

of the pulsation losses PPu and the eddy current losses PEddy of the magnets. According to

Richter (1963), the pulsation losses for a machine of this frame size of 225 are about 10 W.

According to Nipp (1999), the eddy current losses for a surface magnet motor can be computed

as

m

32δ

2swmzmx

2Eddy

ˆπ2ρkbfnpnP = , (2.56)

where fsw is the switching frequency and ρm is the resistivity of the magnet. Value nmz is the

number of the magnets in z-axis direction (axial direction) of the machine. Value nmx is the

number of the magnets in x-axis direction along one pole arc. δb is the fundamental (peak)

component of the air-gap flux density wave. Factor k3 is solved from equation

.mm

4mm

3m

2m

2m

3mm

4m

m

mm

mm4mm

m3

6852541323

2ln)2(2

2916

++−−−

−

−+

−=

blblblblblb

blblblhk

(2.57)

Eq. (2.57) includes parameters, which are related to the magnet geometry: lm is the length of

magnet; bm is the width of magnet and hm the height of the magnet. A simplified factor k3 may

be used

12m

3mm

3lbhk ≈ . (2.58)

60

The eddy current losses may be calculated separately for the direct and quadrature-axis. The

eddy current losses are

2

mxm

32δ

2swmz

2d Eddy,

2cosˆπ4 ∑

=

nn

nkbfpnP αρ

(2.59)

2

mxm

32δ

2swmz

2q Eddy,

2sinˆπ4 ∑

=

nn

nkbfpnP αρ

(2.60)

q Eddy,d Eddy,Eddy PPP += . (2.61)

In Eq. (2.59) and (2.60) α is electrical angle and n is the number of magnets along one pole

pitch. There are also other additional losses e.g. windage losses and ventilation losses, which

may be taken into consideration in some cases. All loss components Ph can be summed up to

finally solve the efficiency (PPu ≈ 10 W).

BrStrCuFeh PPPPPP

PPP

++++=

+=η (2.62)

2.8. Finite element analysis

The finite element analysis program used in the computations is Cedrat’s Flux2D version 7.6. It

computes for plane sections (problems in the plane or problems with rotational symmetry) the

magnetic, electric or thermal states of devices. These states allow computation of several

quantities: field, potential, flux, energy, force etc. The quantities obtained would be difficult to

define by other methods (analytical computations, prototypes, tests, measurements). In the case

of a fractional slot machine the flux in the air-gap is difficult to estimate by any analytical

method, but may be easily solved with finite element analysis, FEA. The accuracy of the FEA

depends on the geometry, the quality of the FE-mesh and also on the time-step values. Time

stepping computations can be done with circuit couplings. In these computations ideal

sinusoidal current supplies and ideal voltage supplies have been used. In real measurements the

supplies usually have some harmonic components too, especially when a frequency converter is

used. When using a direct torque controlled -drive the current is close to sinusoidal.

61

The iron losses of the stator and the rotor as well as eddy current losses of the magnets can be

computed with FEA. The equations used for the computation of the iron losses are explained

here. For more information about the other computation mechanisms of Cedrat’s Flux2D it is

referred to respective manuals. The iron losses can be calculated in a magnetic region during the

analysis. The losses, computed with Flux2D, include the hysteresis losses, the classical losses

(Joule losses) and the excess losses. In harmonic state (magneto dynamic applications) the iron

losses are defined as

67.8)ˆ()ˆ(6

πˆ 5.1e

22

22hFe ⋅++⋅= fbkfbdfbkP σ . (2.63)

In periodic state (time stepping magnetic applications over one complete period) the iron losses

are defined as

tttbk

ttbdk

TkfbkttP

T

TTd

d)(d

d)(d

121ˆd)(1

0

5.1

e

22

ff2

h0

Fe ∫∫

+

+= σ , (2.64)

where b is the maximum flux density at the element concerned, f the frequency, σ the

conductivity, d the lamination sheet thickness, kh the coefficient of hysteresis loss, ke the

coefficient of excess loss and kf is the filling factor. The factors depend on the steel material

used. In the computations (and later in prototype machine) laminated steel M600-50 is used.

The parameters used in the computations for M600-50 are

the conductivity, σ = 4⋅106 (1/ Ω m)

the lamination sheet thickness, d = 0.5 mm

the coefficient of hysteresis loss, kh = 152 (Ws/T2/m3)

the coefficient of excess loss, ke = 2.32 (W/Ts-1)1.5/m3

the filling factor, kf = 0.98.

The magnet material used in the computations is Neorem’s 495a. The B/H-curves of the magnet

material are shown in Appendix E.

62

3. COMPUTATIONAL RESULTS

In this chapter a performance analysis of several different fractional slot machines will be

given. The target is to find suitable constructions for a 45 kW, 400 rpm machine. These are

values that are typically applied in paper machines. Fractional slot motors have been studied

with a 2D finite element method, because the FEA (finite element analysis) proved to be an

accurate - though time consuming - method. Analyses were also carried out by using an analytic

method and the results will be compared to the results obtained with the FEA.

It is searched for concentrated wound machines that have the capacity of producing a high

torque and a good torque quality. According to the author’s knowledge, the following

statements must be valid when a 400 rpm, 45 kW, frame 225 machine is designed

1. The structure must have a large fundamental winding factor. (This, however, is not a

sufficient condition, since e.g. machines such as 21-slot-20-pole, 21-slot-22-pole or

24-slot-26-pole do not produce enough torque even though their fundamental winding

factors are equal to 0.951 or 0.949 (see chapter 2.2). These combinations also have

other unwanted properties, such as an unbalanced magnetic pull etc.)

2. The structure should produce the highest possible torque.

3. The structure should have a low cogging.

4. The structure should have a low torque ripple.

5. An unbalanced magnetic pull should be avoided. This excludes odd slot numbers.

6. The structure should use a low amount of PM material.

The computations were carried out as follows:

• The motor geometry is drawn for the FEA.

• The flux created by the magnets is computed; a static computation, no currents in the

stator.

• The number of coil turns is determined so that 180 (about 0.9EPM/ 3 ) volts no-load

induced phase voltage (EPM = 351 V) was achieved. This no-load voltage level was

found to be suitable to be used in DTC-controlled drives.

63

• The rated values are first estimated by analytical calculations with the method

described in Chapter 2.

• A time stepping computation is performed to solve the induced back EMF. In this

computation the rotor is rotating at a 400 rpm constant speed and there are no currents

in the stator. The cogging of the motor (as a function of the relative magnet width) is

also computed from this computation. The results are given in Chapters 3.3 and 3.4.

• The time stepping computations are carried out with circuit couplings.

o A circuit with sinusoidal current supply. The torque ripple peak-to-peak

values are solved out as a function of the relative magnet width.

o A set of voltage driven computations with different load angles are performed

for the motor. A purely sinusoidal voltage supply with 351 V terminal voltage

is used although, in real measurements, the voltage is supplied with a DTC

drive. From these computations the pull-out torque and the rated load angle

are solved.

Fig. 3.1 shows three of the studied concentrated wound machines. The figure illustrates the

three different structures: S = surface magnet motor, ER = radially embedded magnet motor

and EV = embedded magnet motor where the pole consists of two rectangular magnets in

V-position.

Fig. 3.1. Three different motor structures analysed in the study: a) 24-slot-22-pole, surface magnets (S), b)

12-slot-14-pole, radially embedded (ER) magnets and c) 12-slot-10-pole, embedded magnets in V-position

(EV). To model the whole electrical cycle of these machines, in fact, the whole machine geometry must be

described.

64

The investigated fractional slot machines are listed in Table 3.1, which also show the most

important FEA results: the pull-out torque (compared to rated torque), the obtained minimum

cogging torque and the minimum torque ripple peak-to-peak values (% of rated torque) for a

certain magnet width and the amount of the magnet material in the machine.

Table 3.1. Motor structures being analysed: A 45 kW, 400 rpm machine with sinusoidal voltage supply with 351 V terminal voltage. The pull-out torques are in p.u. values. Torque ripples (peak-to-peak values, % of rated torque) are computed with open slots.

Structure, Stator slots/ rotor poles S, EV, ER

q Winding factor

Pull-out torque (p.u.) / relative magnet width

Cogging (peak-to-peak%)/ relative magnet

width

Torque ripple (peak-to-peak%)/ relative magnet

width

PM-material amount

[kg] 12/8 S 0.5 0.866 1.7/0.85 4/0.78 13/0.77 10.6 12/10 S 0.4 0.933 1.7/0.81 1/0.73 2.5/0.87 10.3 12/10 ER 0.4 0.933 1.2 10.3 12/10 EV 0.4 0.933 1.1 12.5 12/14 S 0.286 0.933 1.2/0.84 0.2/0.93 1.5/0.76 10.5 12/16 S 0.25 0.933 1.0/0.7 3/0.73 3.5/0.75 10.5 18/12 S 0.5 0.866 2.1/0.7 10.3 18/14 S 0.429 0.902 1.8/0.72 1.2/0.81 10.3 21/22 S 0.318 0.951 1.1/0.85 10.3 24/16 S 0.5 0.866 2.0/0.78 3.8/0.77 10.3 24/20 S 0.4 0.933 1.8/0.83 1.7/0.89 10.3 24/20 ER 0.4 0.933 1.7 9.6 24/22 S 0.364 0.949 1.6/0.83 0.3/0.75 10.3 24/22 ER 0.364 0.949 1.5 10.3 24/26 S 0.308 0.949 1.0/0.84 0.3/0.81 10.3 24/28 S 0.286 0.933 1.3/0.81 0.8/0.75 10.3 36/24 S 0.5 0.866 1.7/0.71 3/0.73 2.0/0.78 10.3 36/30 S 0.4 0.866 1.5/0.77 1.5/0.7 10.3 36/42 S 0.286 0.933 1.0 0.05/0.72 0.6/0.9 10.4

Next, the process of computing the results will be explained. The content in this chapter will be

presented in the following order: First, the structures that produce the highest torque will be

determined. Secondly, the quality of the produced torque will be calculated. The effect of the

relative magnet width (pole arc per pole pitch, defined in Fig. 3.12) on the amplitude of the

cogging torque as well as the torque ripple values will be studied. From the results obtained the

effect of the slot opening width in the cases of a semi-closed slot opening and a totally open slot

will be analysed. Also the waveform of the induced back EMF will be calculated, because if the

curve is not sinusoidal, it may indicate a high torque ripple. In the third stage, the performance

of the surface permanent magnet motors will be compared to that of the embedded permanent

magnet motors. And the fourth stage will present an analysis of the losses.

65

3.1. Torque as a function of the load angle

A set of voltage driven computations were performed for the motor, so that the maximum

torque available could be solved. The torques obtained from the FE analysis were plotted as a

function of the load angle. The graph shows the available maximum torque and from the graph

it is also possible to determine the load angle at the rated point of the fractional slot motor. The

FEA with voltage driven model was carried out for several motors of the same frame size. In

order to obtain a fair comparison, some of the parameters reported in Table 3.2, were kept

constant. The air-gap diameter and the machine length were selected to be constant, so that the

area in which the torque is produced would be the same for the machines to be analysed. There

is one exception, which is the machine with 12 slots, since with this machine the air-gap

diameter has to be smaller to fit in the larger slots. In this case the stator inner diameter is 249

mm.

Table 3.2. Constant parameters for machines in voltage driven model computations

Constant parameter

Output power 45 kW

Speed 400 rpm

Rated torque 1074 Nm

Stator outer diameter 364 mm

Stator inner diameter 254 mm

Stator core length 270 mm

Magnet material mass 10.4 kg ±0.1 kg

The amount of the magnet material used was kept as equal as possible for each machine. The

amount of magnet material varies a little, from 10.3 up to 10.5 kg. While drawing the geometry

of the magnet a variation in the amount of magnet material may be possible. When comparing

the embedded magnet structures to the surface magnet structures magnet material amount of

9.5 kg has also been used. To be able to compare the effect of the pole numbers, a 12-slot-stator

was designed and computed using several rotors with different numbers of poles: 8, 10, 14 and

16, respectively. Also a 24-slot-stator was examined, using several rotors with different

66

numbers of poles, 16, 20, 22, 26 and 28, respectively. Furthermore a 36-slot-stator with 24, 30

and 42 poles was analysed.

In all, 16 different fractional slot surface magnet motors were computed with the FEA using a

voltage driven model and also with analytical methods. Some of the studied motors were also

computed using an embedded magnet rotor structure. Fig. 3.2 shows an example of the torque

curve plotted for a 24-slot-22-pole machine. The points are got from the voltage driven model.

The dashed line is computed from the Ld and Lq values obtained from FEA and the torque

equation (Eq. 2.50). The black line is the obtained analytical result, which was computed using

the inductances (Eq. 2.25 and Eq. 2.26), assumed that Lq equals to Ld and the torque equation

(Eq. 2.50). The inductances were computed with 4 different analytical methods. From these

methods it was selected the method the results of which were close to the FEA results. The

computation of the inductances is shown in Appendix D.

0

0.5

1

1.5

2

0 45 90 135 180Load angle (deg)

Torq

ue (p

.u.)

FEA

FEA points

Analytical

Fig. 3.2. Torque as a function of the load angle (electrical degrees) of 24-slot-22-pole surface mounted

machine. The points are got from the voltage driven FEA model. The dashed line is computed with the

torque equation (Eq. 2.50) and with the inductance (Ld and Lq) values obtained from FEA. The black line is

the obtained analytical result, calculated with the torque equation (Eq. 2.50) using the inductances

(Eq. 2.25 and Eq. 2.26) and assuming, that Lq equals to Ld. Appendix D shows the calculation of the

inductances.

67

The torque curves as a function of the load angle for the 24-slot surface magnet machines with

different pole numbers are shown in Fig. 3.3 and for the 12-slot surface magnet machines

in Fig. 3.4.

0

0.5

1

1.5

2

0 45 90 135 180

Load angle (deg)

Torq

ue (p

.u.)

16-pole

20-pole

22-pole

26-pole

28-pole

Fig. 3.3. Torque curves as a function of load angle for 24-slot surface magnet machines. Each machine has

10.3 kg magnet material. The curves are drawn according to the computation points from the FEA.

0

0.5

1

1.5

2

0 45 90 135 180

Load angle (deg)

Torq

ue (p

.u.)

8-pole

10-pole

14-pole

16-pole

Fig. 3.4. Torque curves as a function of the load angle for 12-slot surface magnet machines. The highest

curves are for the 12-slot-8-pole and the 12-slot-10-pole machines and the lowest for 12-slot-16-pole machine.

68

The values of pull-out torque and torque ripples ∆T (% of the rated torque, peak-to-peak value)

obtained from the FEA are shown in Table 3.3.

Table 3.3. The pull-out torque Tmax/Tn (p.u.) and torque ripple values ∆Tp-p (% of the rated torque, peak-to-peak value) for the surface mounted machines obtained from the voltage driven model. The machine parameters were 400 rpm, the magnet material about 10.3 kg and the voltage supply was sinusoidal with a 351 V terminal voltage.

Slots Poles Qs = 12 8 10 12 14 16 20 22 24 26 28 30 42 q 0.5 0.4 0.29 0.25 0.2 0.18 0.17 0.15 0.143 0.13 Tmax/Tn (p.u.) 1.66 1.66 1.17 1.0 ∆Tp-p (%) 16 2.5 7.5 13 Qs = 18 12 14 16 20 22 24 q 0.5 0.43 0.38 0.3 0.27 0.25 Tmax/Tn (p.u.) 2.1 1.79 ∆Tp-p (%) 16 6.5 Qs = 24 16 20 22 26 28 q 0.5 0.4 0.36 0.31 0.286 Tmax/Tn (p.u.) 2.0 1.79 1.56 1.0 1.3 ∆Tp-p (%) 8 2.5 6 >50 3 Qs = 36 24 26 28 30 42 q 0.5 0.46 0.429 0.4 0.286 Tmax/Tn 1.73 1.53 1.0 ∆Tp-p (%) 3.5 2 1

The highest pull-out 2.1 p.u. torque is achieved with the 18-slot-12-pole motor and the lowest

torque obtained is 1.0 p.u., as it is illustrated in Table 3.3. The high pull-out torque values are

achieved with q = 0.5 or close to that value. When q varies from 0.25 to 0.31 the pull-out torque

is less than 1.2 p.u.

The torque ripple values (with sinusoidal voltage supply) for a certain value of q are decreasing

as the pole and slot number are increasing, e.g. the torque ripple of a 12-slot-8-pole (q = 0.5)

motor is 16%, but the torque ripple of a 24-slot-16-pole (q = 0.5) motor is only 8% and 3.5%

with a 36-slot-24-pole (q = 0.5) motor.

Comparison can also be made (according to Table 3.3) of machines, which have the same

number of slots, e.g. machines with 24 slots and with 16, 20, 22, 26 or 28 poles. With a 24-slot

69

stator, which has the same frame size, air-gap diameter, output power and speed, the highest

pull-out torque 2.0 p.u. is achieved with the 24-slot-16-pole motor (q = 0.5). As the number of

poles increases, the maximum torque decreases. This may be due to the stray flux of the

magnets. Each magnet has stray flux components on both sides of the magnet. As the number of

slots is kept constant but the number of magnets is increased (while the amount of magnet

material is kept the same), the relative amount of the leakage flux increases. Each pole has some

leakage flux on the edges, so that, as the pole number increases, a larger leakage flux is

obtained.

3.2. Number of slots and poles

From the manufacturer’s standpoint it may be a benefit to have a motor with few poles and slots

so that the costs may be kept low. The fact is that, when the number of poles and slots is high,

there is more processing to do. A high pole number also needs a high supply frequency, which,

on the other hand, considerably increases the losses. Therefore, the torque production

capabilities were examined for the case when the number of slots and poles are multiplied by an

integer factor. This was made for a particular slot per pole and per phase number q for a surface

magnet motor equipped with semi-closed slots. A certain q value is examined with different

numbers of slot and poles. E.g. q = 0.4 is first observed with 12 slots and 10 poles and, then, the

number of slots and poles are doubled to 24 slots and 20 poles. The third computation is then

performed with 36 slots and 30 poles. Fig 3.5 illustrates the geometrical structures of q = 0.4

motors to be analysed.

72o 72o 72o

Fig. 3.5. The geometries (in principle) of q = 0.4 motors under study. One fifth of the machine is

illustrated for a) a 12-slot-10-pole, b) 24-slot-20-pole and c) 36-slot-30-pole motor. (In finite element

model the whole machine geometry was be described.)

70

The weight of the magnet material was kept around 10.3 kg. (For the case of 24-slot machines

also 9.5 kg magnet material would have been sufficient to produce a suitable back EMF.) The

examined slots per pole and per phase numbers were equal to 0.5, 0.4 and 0.286. All motors

studied have the same frame size, air-gap diameter, air-gap length and the same amount of

magnet material. The values computed for the machines used in this comparison are presented

in Table 3.4. As q is kept constant, but both the slot and pole numbers are increased, the

frequency increases causing also an increasing of the iron losses. This can be concluded

especially from the stator iron losses values. As the slot and pole numbers increase, the relative

value of the magnetizing inductance Lmd decreases and the leakage inductance Lsσ increases.

Table 3.4. The parameters and FEA results of 45 kW motors with q equal to 0.286, 0.4 and 0.5 with three different slot-pole combinations. Machines are with semi-closed slots and with surface magnets. The terminal voltage is 351 V and the frame size 225.

Slots 12 12 12 24 24 24 36 36 36

Poles 14 10 8 28 20 16 42 30 24

q 0.286 0.4 0.5 0.286 0.4 0.5 0.286 0.4 0.5

Winding factor, ξ1 0.933 0.933 0.866 0.933 0.933 0.866 0.933 0.933 0.866

Rated current (A) 91.5 88.4 88 86 82 83.5 92 90 95.2

Speed (rpm) 400 400 400 400 400 400 400 400 400

Frequency (Hz) 46.7 33.3 26.7 93.3 66.7 53.3 140 100 80

Power factor 0.93 0.91 0.96 0.975 0.98 0.97 0.97 0.98 0.90

Inductance, Ld (p.u.) 0.91 0.66 0.64 0.86 0.57 0.49 0.85 0.72 0.64

Lsσ / Ld 0.56 0.52 0.48 0.79 0.77 0.75 0.85 0.8 0.8

Back EMF (V) 180 179 180 192 188 185 184 189 168

Nph 104 104 120 104 104 112 96 96 108

Rph (Ω) 0.10 0.10 0.12 0.10 0.10 0.11 0.09 0.10 0.11

Tmax/Tn (p.u.) 1.17 1.66 1.66 1.3 1.79 2.0 1.02 1.53 1.73

mmagn (kg) 10.5 10.3 10.6 10.3 10.3 10.3 10.4 10.2 10.3

PFe, s (W) 274 258 262 507 374 350 984 696 511

PFe, r (W) 37 22 19 30 20 15 24 19 14

PCu (W) 2512 2344 2881 2219 2017 2364 2387 2430 2936

PStr (W) 225 225 225 225 225 225 225 225 225

Ph (W) 3048 2849 3387 2981 2636 2954 3620 3145 3686

Efficiency (%) 93.7 94.0 93.0 93.8 94.5 93.8 92.6 93.0 92.4

71

From the results given in Table 3.4 also different pole numbers for a certain stator can be

compared. In the case of the 12-slot stator, the pole numbers are changed from 8 poles to 14

poles. As the pole number increases, the obtained pull-out torque decreases, due to the

increased inductance. With a high inductance value it is not possible to achieve a high torque.

As the pole pair number p is high, also the frequency f is higher and this gives a higher

magnetizing inductance Lmd compared to a smaller pole number, as it is shown in Table 3.4.

The values of the pull-out torque are illustrated in Fig. 3.6. It can be seen that the amount of

pull-out torque is slightly different as the number of slot and poles varies for a certain q value.

1

1.5

2

1 2 3

Pull-

out t

orqu

e (p

.u.)

q = 0.5 q = 0.4

q = 0.286

poles 8 16 24 10 20 30 14 28 42slots 12 24 36 12 24 36 12 24 36

Fig. 3.6. The calculated pull-out torques in p.u. values got from the voltage driven model for fractional slot

machines with q equal to 0.5, 0.4 or 0.286.

The highest values are achieved with the 24-slot machines. For this machine size the 24-slot

stator would be a good alternative. (As shown in Table 3.3 also an 18-slot-12-pole structure

would be a suitable alternative, because it has a 2.1 p.u. pull-out torque) With a 24-slot stator

and by decreasing the pole number from 20 to 16, it is not possible to achieve the same back

EMF with the same number of the stator winding turns. The results from the FEA are shown in

Fig. 3.6. The pull-out torque, achieved with 8 poles, is as high as the pull-out torque achieved

with 10 poles. An 8-pole motor with 12 slots is difficult to fit into this frame size, since the

winding factor is low and the numbers of coil turns needed are high. Therefore, the cross-

sectional area of a slot becomes large if the current density in the slot is to be kept constant.

This could be a possible structure, but requires a reduced air-gap diameter, and this, on the other

hand, decreases the maximum torque.

72

The values of the cogging torque ripples obtained at no-load (no current, speed 400 rpm) and

torque ripples at rated load (rated current, speed 400 rpm) were also analysed for these

machines. The amplitude of the cogging torque decreases as the number of slots and poles

increases as shown in Fig. 3.7 a). Also, the torque ripple decreases, as the numbers of slots and

poles increase. This is described in Fig. 3.7 b), where the results from the voltage driven model

are given.

0

2

4

6

8

1 2 3

Cog

ging

torq

ue (%

) of r

ated

torq

ue q = 0.5

q = 0.4

q = 0.286

poles 8 16 24 10 20 30 14 28 42slots 12 24 36 12 24 36 12 24 36

0

5

10

15

20

1 2 3

Torq

ue ri

pple

(%) o

f rat

ed to

rque q = 0.5

q = 0.4

q = 0.286

poles 8 16 24 10 20 30 14 28 42slots 12 24 36 12 24 36 12 24 36

a) b)

Fig. 3.7. a) The peak-to-peak values of the cogging torque in (%) of the rated torque and b) torque ripple

in (%) of the rated torque for fractional slot machines with q equal to 0.5, 0.4 and 0.286. The torque ripple

values are the results obtained from the voltage driven model.

It must be noted that, if the number of poles is increased, the height of the stator yoke as well as

the iron area in the rotor (below magnets) could be decreased – in this study these values were

kept constant. As an example, the 36-slot-30-pole motor was also computed with an air-gap

diameter, which was 8% larger. The stator yoke iron thickness as well as the slot depth was

decreased, which can cause a high current density and to a weak mechanical structure. After

increasing the air-gap diameter by 8%, the FEA with the voltage driven model showed that the

pull-out torque increased about 20% from 1.53 p.u. to 1.85 p.u.

Next, the machines with a same pole number but with a different number of slots in the stator

will be compared. It should be remembered that the torque ripple is dependent on the machine

geometry, which can affect the torque ripple amplitude and the cogging torque amplitude. This

is more closely examined in chapters 3.3 and 3.4. Some of the slot-pole combinations are

73

differing up to 20 percent in the torque ripples values, depending on the relative magnet width

and the relative slot opening width. Hendershot and Miller (1994) showed that every time the

number of poles is doubled the required thickness of the rotor yoke or back-iron inside the

magnets is reduced by one half, as is the thickness of the stator yoke. As the number of poles is

increased, the number of effective winding turns decreases in inverse proportion, so that the

synchronous reactance decreases in motors with high pole number.

A 14-pole motor with 12 slots generates a pull-out torque of 1.17 p.u., while a 14-pole (with the

same rotor geometry) with 18 slots generates a pull-out of 1.79 p.u, according to the results

obtained from the FEA. A 16-pole motor with 12-slots generates a pull-out torque of 1.0 p.u.,

while with 24-slots it generates a pull-out torque of 2.0 p.u. The same behaviour may be

observed with the 22-pole motor; with 21 slots the pull-out torque is smaller - 1.1 p.u. – than

with the higher slot number 24, where the pull-out torque is 1.56 p.u. The torque ripple

decreases as the number of slots is increased if the pole number is kept constant.

Conclusion: Increasing the pole number and keeping the slot number constant reduces the

developed pull-out torque in most of the analysed cases, as the magnet material and the machine

size (and air-gap diameter) were kept practically constant. Further increasing of the slot number

and keeping the pole number constant increases the developed pull-out torque. The highest

obtained pull-out torque for the examined machine size was 2.1 p.u. for a 18-slot-12-pole motor

with q equal to 0.5.

3.3. Induced no-load back EMF

The waveform of the induced back EMF can indicate the quality of the torque produced. If

either the voltage or current is non-sinusoidal, some torque ripple may be expected. It could

also be noticed that the smaller q value yields a more sinusoidal back EMF waveform. When

studying 24-slot machines, it is obvious that with 28, 26 and 22 poles the back EMF is very

sinusoidal, as it might be seen in Fig. 3.8. The worst waveform was offered by a 24-slot-16-pole

machine, with q = 0.5. There is a pattern that continues from one pole pair to the other: one

magnet is completely under a tooth and the next magnet is completely under a slot – which is

reflected in the induced waveform. Another slot-pole combination (q = 0.5) for the machine

was implemented: a 42-slot-28-pole machine. Although this machine has high pole and slot

74

numbers, the back EMF waveform is far away from a sinusoidal curve, which probably causes

that the machine has a high torque ripple. Machines with q equal to 0.5 produce a distorted back

EMF, as it can be seen from the EMF waveforms in Fig. 3.9.

-300

-150

0

150

300

0 ω mek

Bac

k EM

F (V

)

24 slots 28 poles (0.286)

24 slots 26 poles (0.308)

24 slots 22 poles (0.364)

24 slots 20 poles (0.4)

24 slots 16 poles (0.5)

q = 0.5q = 0.4

Fig. 3.8. Induced back EMFs of fractional slot machines with a 24-slot-stator. The results are given for a

surface magnet machine with semi-closed slots.

-300

-200

-100

0

100

200

300

0 ω mek

Bac

k EM

F (V

)

42 slots 28 poles (0.5)

24 slots 16 poles (0.5)

12 slots 8 poles (0.5)

Fig. 3.9. Induced back EMFs of fractional slot machines with the number of slots per pole and per phase

equal to 0.5. The results are given for a surface magnet machine with semi-closed slots.

75

According to the FEA results shown in Figs 3.8 and 3.9, some of the back EMFs of the

analysed fractional slot machines are not sinusoidal, but flattened at the top. If there is a

sinusoidal current and a sinusoidal voltage the power produced should be constant and the

torque ripple should be small. But, e.g. in the case of the machines with q equal to 0.5, torque

oscillations are expected because of the non-sinusoidal voltage waveform, as shown in Fig 3.9.

3.4. Cogging torque

Cogging is an oscillatory torque, which is caused by the tendency of the rotor to line up with the

stator in a particular direction where the permeance of the magnetic circuit from the standpoint

of the permanent magnets is maximized. Cogging occurs even when there is no current in the

stator. The manual rotation of a disconnected machine gives an indication about of the cogging

torque. When the motor is running there are also other additional oscillatory torque components

resulting from the interaction of the magnets with the space-harmonics created by the winding

layout and with the magnetomotive forces created by the current harmonics. These additional

oscillatory torque components are generally referred to as torque ripple, while the term cogging

is often used for the no-current situation.

The torque ripple can be reduced by several methods (Hendershot and Miller (1994), Li and

Slemon (1988)): by using an increased air-gap length, or a fractional slots/pole, or larger

numbers of slots/pole, or thick tooth tips to prevent saturation, by keeping the slot opening to a

minimum, by using magnetic slot wedges, by skewing the stator core or the permanent magnets,

by forming or chamfering the magnet poles, by forming or chamfering the stator tip or punch

holes in tooth tips, by varying the magnetization of the magnet poles, by using bifurcated teeth,

by using a low magnetic flux density and compensating the cogging by modulating the drive

current waveform. With a large number of slots/pole slightly skewing is usually sufficient to

eliminate most of the cogging.

When the number of slots/pole is close to 1, the slot geometry becomes more important. Then,

the width of the slot opening can be adjusted to minimize the cogging effect. It could be

expected that a fractional slot machine with many slots and poles would have a very small

cogging torque. Jahns and Soong (1996) introduced technique guidelines for the motor design

as well as for the control to minimize the torque ripple. Zhu et al. (2003) have studied the

76

cogging torque of some interior-magnet (q < 1) brushless machines. According to Cros and

Viarouge (2002), the cogging torque is dependent on the geometry when q (slots per pole and

per phase) is close to 0.3 (and Qs ≈ 2p). The authors stated that the performance of q ≈ 0.3

motors is relatively low if a supply with sinusoidal currents is used. In the case of a rectangular

current supply and a smooth rotor with surface magnets, the no-load EMF generated in the

windings does not produce a flat portion with a sufficient width.

According to Cros and Viarouge, a machine with a number of slots per pole and per phase

between 1/2 and 1/3 generally produces a high performance and can have relatively high

fundamental winding factors. A machine with these structures can also produce a low no-load

cogging torque. The cogging frequency in these constructions is high. The number of cogging

periods per rotor revolution is dependent on the least common multiplier LCM of Qs and 2p.

The LCM values for some fractional slot machines are given in Table 3.5. These machines have

q = 0.5 or q < 0.5 because they are concentrated wound machines. When both the number of

slots and the number of poles are doubled, also the LCM increases. This means that the torque

ripple is lower for the machines with multiple poles and slots compared to the simpler

structures. As an example (shown bolded in table), if Qs is 9 and 2p is 6 the LCM number is 18,

but if the numbers are doubled - Qs is 18 and 2p is 12 - the LMC number becomes 36.

Table 3.5. The least common multiplier LCM (of Qs and 2p) values for concentrated wound fractional slot machines

Qs\2p 2 4 6 8 10 12 14 16 20 22 26 28 3 6 12 6 24 30 12 42 48 60 66 78 84 6 12 18 24 30 36 42 48 60 72 78 84 9 18 72 90 36 126 144 180 198 234 252

12 48 60 72 84 48 60 132 156 168 15 30 60 210 240 60 330 390 420 18 36 126 144 180 198 234 252 21 42 336 420 462 546 84 24 96 120 264 312 168

The waveform of the cogging torque as a function of electric angle of a 24-slot-20-pole motor is

shown in Fig. 3.10 at the rated speed of 400 rpm. Fig. 3.10 describes the cogging torque from

which it can be counted that there are 12 periods in the waveform during a cycle over one pole

77

pair, giving a total number of 120 periods over one mechanical cycle. This number 120 is

exactly the number of the LCM for the 24-slot-20-pole motor. For the 12-slot-8-pole machine

there are 12 waves over one pole pair, which gives 48 waves during one mechanical cycle.

-0.04

-0.02

0.00

0.02

0.04

0 2π

Cog

ging

torq

ue (p

.u.)

12 slots 8 poles

24 slots 20 poles

Electric angle

Fig. 3.10. Cogging torques of 12-slot-8-pole and 24-slot-20-pole motor as a function of electric angle.

The smallest cogging torques are achieved for the machines with several slots and poles, as it is

shown in Fig. 3.11. Fig. 3.11 gives the peak-to-peak cogging torque values for surface magnet

machines with semi-closed slot openings and with a relative magnet width of about 0.85 p.u. If

the basic structure is multiplied the amplitude of the cogging torque will decrease, but its

frequency increases. As an example, a 12-slot-8-pole motor (q = 0.5) has a 6% cogging torque,

but a 24-slot-16-pole (also q = 0.5 machine) has 5% and a 42-slot-28-pole has only 0.1%. The

cogging torque (% of the rated torque) values obtained from the FEA and the LCM ratios for

the surface magnet motors are shown in Fig. 3.11. In this case the LCM ratio corresponds to a

scaled value 240/LCM. Such a scaling is used in order to reasonably fit the values into the same

plot. According to the results obtained from the FEA, the cogging torque is inversely

proportional to the LCM numbers. The LCM is a measure of the frequency of the cogging

torque. The LCM value for the motor is easy and fast to calculate, and is, therefore, a useful

parameter for the designer of fractional slot windings. Cogging was also computed for open slot

machines, because it is obvious that the slot opening width has effect on the cogging torque

values.

78

0

2

4

6

12-8 12-16 12-10 12-14 24-16 24-20 18-14 24-28 42-28 24-22 24-26 21-22slots-polesq

Cog

ging

torq

ue (%

)LCM ratio

Computed cogging torque (%)

0.5 0.25 0.5 0.4 0.286 0.4 0.429 0.286 0.5 0.364 0.308 0.318

LCM

ratio

Fig. 3.11. Cogging torque (peak-to-peak values, % of rated torque) for several structures with semi-closed

slots and surface magnet rotors according to the FE analysis. Semi-closed slot openings and a relative

magnet width of about 0.85 p.u. were used. The LCM ratios (scaled from the LCM numbers shown in

Table 3.5, LCM ratio = 240/LCM) on y-axis are proportional to the cogging results.

The effect of the relative magnet width and the slot opening width on the performance of a

fractional slot wound PM motor was studied. In the literature several investigations for

conventional wound (q = integer) machines are reported, e.g. Li and Slemon (1988), and

Ishikawa and Slemon (1993) discussed the selection of a suitable relative magnet width. The

definition of the relative magnet width (pole arc length/ pole pitch) is used to describe the

magnet width, as it is shown in Fig. 3.12 a). Fig. 3.12 b) illustrates the definition of the relative

slot opening width (x4/τs) used in this study.

pole pitch

MAGNET

pole arc

x4

slot pitch, τsa) b)

bm

hm

Fig. 3.12. The definition of the relative permanent magnet width (pole arc / pole pitch) and the definition

of the relative slot opening width (x4/τs). The pole arc is measured along the curved side of magnet.

79

For traditionally wound (q = integer) machines the suitable permanent magnet width may be

calculated as given by Ishikawa and Slemon (1993)

p

sw )(

ττ

α kk += , (3.1)

where τs is the slot pitch, τp the pole pitch, k is an integer and kw is the constant number 0.17 or

0.14 suggested by Li and Slemon (1988). As an example, for the integer slot machine, where

q = 1, it is often wise to select the relative magnet width to be 0.67 in order to diminish the

cogging torque (as the 3rd harmonic is cancelled).

The theoretical behaviour of the cogging for an integer slot machine is given in Fig. 3.13. The

suitable relative magnet width is not an exact number; it is rather a suitable range. Some α

values computed with method by Ishikawa and Slemon (1993) are gathered in Table 3.6, which

gives results for kw equal to 0 and equal to 0.17. Even thought Ishikawa’s equation is designed

for integer slot machines, it is here examined, if it is also suitable for fractional slot machines.

The cogging torque appears even with no currents in the stator winding, and therefore the

equation may be valid also for a fractional slot winding.

k = 1αp = 0.195

k = 2αp = 0.362

k = 3αp = 0.528

k = 4αp = 0.695

k = 5αp = 0.862

0.2 0.4 0.6 0.8

T

α

Fig. 3.13. Cogging torque as a function of the relative permanent magnet width for an integer wound

machine, q = 2 with kw = 0.17.

80

Table 3.6. Relative magnet widths α (by Ishikawa and Slemon, 1993) for kw equal to 0 and 0.17.

Qs - 2p 12 - 8 12 - 10 12 - 16 24 - 20 24 - 22 12 - 8 12 - 10 12 - 16 24 - 22

q 0.5 0.4 0.25 0.4 0.364 0.5 0.4 0.25 0.364

k\kw 0 0 0 0 0 0.17 0.17 0.17 0.17

0 0.67 0.83 1 0.83 0.92 0.78 0.98 … …

1 0.33 0.67 0.67 0.67 0.83 0.39 0.78 0.78 0.98

2 0.5 0.33 0.5 0.75 0.59 0.56 0.88

3 0.33 0.33 0.67 0.56 0.78

… … …

For the 12-slot-8-pole machine there exists only two minimum points, being 0.33 and 0.67

(if kw is 0), were the cogging might achieve the minimum. Also the 12-slot-16-pole machine has

only few possible minima. In contrast, a 24-slot-22-pole machine, for example, has several

possible and also useful minimum values for the relative magnet width. It can also be seen from

Table 3.6 that machines with the same q have the same values of α, e.g. for q is 0.4, the 12-slot-

10-pole (shadowed column) and 24-slot-20-pole machine have the same values. This indicates

that the minimum torque values are achieved with the same relative magnet width, although the

slot and pole numbers were doubled. Which relative magnet width should be chosen for an

individual motor depends on the application in question. Reliable knowledge of the cogging

may be achieved by the FEA.

Ackerman et al. (1992) explained the criteria of selecting a suitable relative magnet width to

reduce cogging in the case of brushless DC motors. The magnet widths are selected using

similar methods, as introduced by Li and Slemon (1988), Ishikawa and Slemon (1993), but

Ackerman et al. (1992) also gives some guidance for the selecting of a suitable tooth width. He

developed his method especially for machines where Qs ≈ 2p. The behaviour of the cogging

torque in the case of a brushless DC motor can be expected to be similar to brushless PM motor,

because the cogging is analysed at no-load situation, where the currents are not effecting. A

suitable relative magnet width for a vanishing cogging torque is proposed by Ackerman et al.

(1992)

81

Kp

QN −=2

sα ( 10 << α ), (3.2)

where N = 1, 2, …, 2p –1 and K = 0, 1, 2, … , Qs – 1. A suitable width of a tooth is according to

Ackerman et al. (1992)

NQp

K −=s

2β ( 10 << β ), (3.3)

where N = 0, 1, 2, …, 2p –1 and K = 1, 2, … , Qs – 1. The values α and β are given in Table 3.7

for some 12-slot and 24-slot fractional slot machines. When comparing the values to those

given in Table 3.6 (Ishikawa and Slemon, 1993), it is noticed that the given magnet widths are

similar.

Table 3.7. A suitable magnet width α and a suitable tooth width β according to Ackerman’s equations, Eqs. 3.2 and 3.3.

Qs - 2p Qs - 2p Qs - 2p Qs - 2p Qs - 2p Qs - 2p Qs - 2p

12 - 8 12 - 10 12 - 14 12 - 16 24 - 26 24 - 22 24 - 20

q = 0.5 q = 0.4 q = 0.286 q = 0.25 q = 0.308 q = 0.364 q = 0.4

α β α β α β α β α β α β α β

0.67 1 0.83 1 0.83 0.857 1 1 0.917 0.923 0.92 0.919 0.83 1

0.33 0.5 0.67 0.8 0.67 0.714 0.67 0.75 0.83 0.846 0.83 0.818 0.67 0.8

0.5 0.6 0.5 0.571 0.33 0.5 0.75 0.769 0.75 0.727 0.5 0.6

0.33 0.4 0.33 0.429 0.25 0.67 0.692 0.67 0.636 0.33 0.4

0.2 0.286 0.583 0.615 0.58 0.545 0.2

0.5 0.538 0.5 0.455

It was examined, whether the method introduced by Li and Slemon (1988), and Ishikawa and

Slemon (1993) or the theory suggested by Ackerman et al. (1992) are appropriate for applying

to fractional-slot PM machines. In order to achieve a high flux density in the air-gap and,

thereby, a high torque, the optimal magnet width should be selected to be as wide as possible.

The magnet width used for the magnets of a fractional wound surface PM motor is often the

width of the tooth. A wider magnet may cause leakage flux, which is the case e.g. if the slot arc

is much narrower than the pole arc.

82

Fig. 3.14 shows the cogging torque values, obtained from the FEA, for 12-slot-motors with

semi-closed slots, relative slot opening of 0.08. The minimum cogging torque value for a 12-

slot-8-pole motor is found with a 0.78 relative magnet width. The expected value in Tables 3.6

is 0.78 if kw is 0.17. Therefore, it can be stated that the method introduced by Ishikawa and

Slemon (1993) can be used for this motor. The 12-slot-10-pole motor instead has three minima

0.59, 0.76 and 0.92. The minima in the Tables 3.6 and 3.7 are 0.5, 0.67 and 0.83, which means

that kw is 0.09. Logically, for the 12-slot-14-pole motor the factor kw would be 0.03 and for the

12-slot-16-pole motor -0.074. The highest cogging torque level of the studied 12-slot motors

with semi-closed slot opening was found for the 12-slot-16-pole motor type and the lowest for

the 12-slot-14-pole motor type.

0

4

8

12

16

0.5 0.6 0.7 0.8 0.9 1.0

Relative magnet width

Cog

ging

torq

ue %

of r

ated

12-08

12-16

12-10

12-14

Fig. 3.14. Cogging torque peak-to-peak values (% of the rated torque) for 12-slot-8-pole, 12-slot-10-pole,

12-slot-14-pole and 12-slot-16-pole machines with semi-closed stator slot openings.

3.4.1. Semi-closed slot vs. open slot

The effect of the width of the slot opening is studied for several motor types. Fig. 3.15 gives the

cogging torque values with open slots for the same 12-slot motors (with semi-closed slot

openings) given in Fig. 3.14. The cogging torques of 12-slot-8-pole and 12-slot-16-pole

machines are lower with open slots than they were with semi-closed slot opening, when relative

magnet width is 0.75.

83

0

4

8

12

16

20

0.5 0.6 0.7 0.8 0.9 1.0Relative magnet width

Cog

ging

torq

ue %

of r

ated

12-8

12-16

12-10

12-14

Fig. 3.15. Cogging torque (peak-to-peak values % of rated torque) for 12-slot-8-pole, 12-slot-10-pole,

12-slot-14-pole and 12-slot-16-pole machines with open slots.

The cogging torque values for a 12-slot-10-pole motor are given in Fig. 3.16 a) and for a 12-

slot-14-pole motor in Fig. 3.16 b). The curves indicate that there are minimum values the

existence of which depends on the slot open width. The semi-closed slot opening width is 0.08

of the slot pitch and the open slot width is 0.63 of the slot pitch. Changing the slot opening

width changes the place of the minima.

0

2

4

6

0.5 0.6 0.7 0.8 0.9 1.0Relative magnet width

Cog

ging

torq

ue %

of r

ated

Open slot

Semi-closed

a)

0

2

4

6

0.5 0.6 0.7 0.8 0.9 1.0Relative magnet width

Cog

ging

torq

ue %

of r

ated

Open slot

Semi-closed

b)

Fig. 3.16. The cogging torques (peak-to-peak ripples % of rated torque) for a semi-closed-slot and for an

open slot machine: a) 12-slot-10-pole machine (q = 0.4) and b) 12-slot-14-pole machine (q = 0.286).

84

12-slot-16-pole machine

As an example, the cogging torque of a 12-slot-16-pole machine (q = 0.25) was studied closer.

The results of the FEA for the 12-slot-16-pole motor as a function of the magnet width are

shown in Fig. 3.17. The semi-closed solutions have 0.08 and 0.25 relative slot openings widths.

The open slot width is 0.5. The cogging of the semi-closed slot machine with the 0.08 relative

slot opening width has the minimum values, as the relative magnet width is 0.6 or 0.92. With

the 0.25 relative slots opening the minimum is at 0.68. With the open slot structure the cogging

torque ripple minimum is at 0.73. The average level of the cogging torque for both the 12-slot-

16-pole machines is much higher than for the 12-slot-10-pole and 12-slot-14-pole machine

studied earlier. The methods for integer slot machines (given by e.g. Ishikawa and Slemon,

1993 and Ackerman et. al, 1992) with 12 slots and 16 poles give a minimum cogging value at

0.67 or 1. According to FE analyses presented here, in Fig 3.17, the minimum point of cogging

in the case of the 12-slot-16-pole motor varies from 0.6 to 0.73 depending on the slot opening

width. Ackerman’s method gives a value that corresponds well to this FE analysis, because with

a 0.75 tooth width the cogging minimum is estimated to be at 0.67 and in the FE computations

it was 0.68.

0

5

10

15

20

25

0.5 0.6 0.7 0.8 0.9 1.0

Relative magnet width

Cog

ging

torq

ue %

of r

ated

Open slot 0.5

Semi-closed 0.25

Semi-closed 0.08

Fig. 3.17. The cogging torque peak-to-peak ripples (% of rated torque) for semi-closed-slots and for an

open-slot 12-slot-16-pole machine (q = 0.25). The semi-closed relative slot opening widths are 0.08 and

0.25 of the slot pitch. The totally open slot p.u. width is 0.5.

85

36-slot-24-pole machine

The cogging torque of q = 0.5 (36 slots and 24 poles) was studied with different slot openings

widths. The flux lines of different analysed structures are shown in Fig. 3.18. It is shown that

with the open slots (in Fig. 3.18 b)) the machine gives considerably high cogging values, e.g.

the value 20%, in some areas of the examined relative magnet width. Close to the value 0.73

there is a minimum for the cogging torque of the open slot structure, which can be seen from

the torque ripple values in Fig. 3.19. The cogging torque of the semi-closed structure was

between 2 to 11 % of the average torque (depending on the relative magnet width).

Fig. 3.18. Semi-closed-slot and open-slot structures for a 36-slot-24-pole machine (q = 0.5). The structure

a) with semi-closed slots gives a 4% cogging torque peak-to-peak of the rated torque, but the structure b)

with open slots gives a 25% cogging torque.

0

5

10

15

20

25

0.5 0.6 0.7 0.8 0.9 1.0

Relative magnet width

Cog

ging

torq

ue %

of r

ated

Open slot

Semi-closed

Fig. 3.19. The cogging torque (% of the rated torque, peak-to-peak values) of a 36-slot-24-pole machine

(q = 0.5). The semi-closed relative slot opening widths are 0.09 and open slots 0.42.

a) b)

86

36-slot-42-pole machine

The cogging torque of a fractional slot machine with high number of poles may be lower than

0.1%. The cogging torques level of q = 0.286 (36 slots and 42 poles) was low compared to other

studied machines as shown in Fig. 3.20. All cogging values regardless of the relative magnet

width are less than 1%. The cogging torque is about 0.05% at the minima.

0

0.2

0.4

0.6

0.8

1

0.60 0.70 0.80 0.90 1.00

Relative magnet width

Cog

ging

torq

ue %

of r

ated

Fig. 3.20. In the case of a 36-slot-42-pole (q = 0.285) machine with semi-closed slots, all cogging torque

values are less than 1%. The semi-closed relative slot opening widths are 0.09.

3.4.2. Conclusion

A study was carried out to investigate, if the method suggested by Li and Slemon (1988) and

Ishikawa and Slemon (1993) or the theory introduced by Ackerman et al. (1992) can be used for

fractional-slot PM machines. The cogging torques appears to behave as expected producing a

curve with minima. It depends of the slot opening width where the minima appear. For each

machine a kw factor can be calculated to estimate the minimum points. The cogging torque

values of the analysed fractional slot motor types that are studied can be less than 1% of the

rated torque, especially in the case of multi-pole machines. The effect of the slot opening is

therefore studied closer with machines under load, as the torque ripple is usually higher than at

no-load.

87

3.5. Torque ripple of the current driven model

The effect of the magnet width and the slot opening width on the torque ripple of the fractional-

slot PM motor is studied. The torque ripple peak-to-peak value is computed using the FEA with

a current driven model at 1000 Nm load and current density J ≥ 5 A/mm2. The parameters that

remain the same are the magnet mass (about 10.4 kg), the stator inner and stator outer diameter,

the slot surface area and current density. The 12-slot motors have a 249 mm stator inner

diameter. In the following section, some examples of the results obtained from the FEA will be

given. Furthermore, a comparison between the torque ripples of several machines with current

driven model is given. Some waveforms of the torque are shown in Fig 3.21. It should now be

remembered that all the results are valid for stators and rotors without any skew. The torque

ripple can be reduced to some extent when skewing is applied.

0.9

0.95

1

1.05

1.1

0.000 0.005 0.010

Torq

ue (p

.u.)

12-slot-14-pole

12-slot-10-pole

24-slot-26-pole

24-slot-22-pole

Time (s)

Fig. 3.21. Torque as a function of time for different fractional slot – surface magnet - machines. The

results are given for the surface magnet structures 12-14 (slots-poles), 12-10, 24-26 and 24-22.

The torque ripple peak-to-peak values for a set of surface magnet and embedded magnet motors

are given in Table 3.8. The relative magnet width of the surface magnet motors is fixed to 0.85

to ensure a high torque.

88

Table 3.8. The results of the current driven model for the relative magnet width of 0.85. ∆Tp-p is the peak-to-peak ripple in % of the average torque, at 1000 Nm load. S = surface magnet motor, ER = radially embedded magnet motor and EV = embedded magnet motor where the pole consists of two magnets in V-position. (Salminen et al., 2004)

Slots Poles q Magnet ∆Tp-p (%)

12 8 0.5 S 16

12 10 0.4 S 2.8

12 10 0.4 EV 2.4

12 14 0.286 S 4

12 14 0.286 ER 1.9

24 20 0.4 S 2.5

24 22 0.364 S 4.5

24 22 0.364 ER 3.8

24 26 0.308 S 4.5

24 26 0.308 ER 3.2

24 28 0.286 S 6.1

42 28 0.5 S 19

42 28 0.5 ER 29

Considering the surface magnet machines, the lowest torque ripple values 2.5% and 2.8% were

obtained for the 24-slot-20-pole and 12-slot-10-pole machines. As to the embedded magnet

machines, the lowest torque ripple obtained was 1.9% for the 12-slot-14-pole machine. The

highest values were obtained for the machines with q equal to 0.5, e.g. 19% for a 42-slot-28-

pole machine and 16% for a 24-slot-16-pole machine. In most of the cases, the machines with

embedded magnets generate lower ripples compared to the corresponding machines with

surface magnets. The only exception among the analysed q values was a fractional slot 42-slot-

28-pole machine with q equal to 0.5. The ripple of this motor was the worst, a 29% ripple for

the embedded solution and a 19% ripple for the surface magnet solution. (Salminen et al.,

2004).

89

3.5.1. Some examples

Torque ripple of a 12-slot-16-pole machine

The effect of the magnet width of a 12-slot-16-pole machine (q = 0.25) was first studied with a

0.08 and 0.43 relative slot opening width. The torque ripple as a function of relative magnet

width is shown in Fig. 3.22 a). The harmonic components of the torque ripple with semi-closed-

slot structure and open slots are shown in Fig. 3.22 b). Two magnet widths, 0.7 and 0.75, are

examined. The results given in Fig. 3.22 show that one value is differing from the others; this is

the value of the machine with open slots and with a 0.75 magnet width, which is one minimum

ripple point for this machine. The 4th harmonic of this machine disappeared, but the 6th

harmonic and the 12th are slightly higher than others.

0

5

10

15

20

0.5 0.6 0.7 0.8 0.9 1.0

Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

semi-closed 0.08

open slot 0.43

a)

0

2

4

6

1 2 3 4 5 6 7 8 9 10 11 12Harmonic order number

Torq

ue ri

pple

% o

f rat

ed to

rque

semi-closed 0.7

semi-closed 0.75

open slot 0.7

open slot 0.75

b)

Fig. 3.22. Torque ripple (% of the average torque, peak-to-peak) of a 12-slot-16-pole machine as a) a

function of relative magnet width. The 0.08 semi-closed slot opening width and 0.43 open slot were

analysed. B) Harmonics (from fast Fourier transform) are presented for the 0.7 and 0.75 magnet widths.

Fig. 3.23 describes a) a 12-slot-16-pole machine with semi-closed slots and with a magnet

width of 0.79 and b) a 12-slot-16-pole machine with open slots and with a 0.75 magnet width.

The current and rotor angle are the same for both of the machines. The flux paths are periodical

and four symmetrical areas appear in this machine type. One symmetrical area consists of

4 poles and 3 slots along the periphery of the machine. Fig. 3.23 shows some interesting areas,

where the flux lines of the a) semi-closed slot and b) open slot machine differ from each other.

90

The areas are marked with the circulated letters A, B and C. For the semi-closed slot shown in

Fig. 3.23 a), there occur stray flux lines in the area A, which do not appear for the open slot

machine shown in Fig 3.23 b). The area B (in Fig. 3.23 a)) shows that the tooth tip leaves a

wider path for the flux to flow from the stator into the rotor magnet compared to the area in

Fig. 3.23 b) where the flux path is narrower since the tooth tip is narrower. On the other hand,

the flux lines in Fig 3.23 a) in the area C, illustrate the disadvantage of wide tooth tips. There is

an easy path for the flux to flow from one magnet to another, which creates a zigzag stray flux.

In Fig 3.23 b) there are less stray flux lines (in the same area C). For the machine type with

semi-closed slots, the stray flux in the vicinity of the air-gap is always larger compared to the

machine with open slots. This explains the high torque ripple values of the 12-slot-16-pole

machine type with semi-closed slot structures.

a)

b)

Fig. 3.23. Flux lines of a 12-slot-16-pole machine with a) semi-closed slots and with a 0.79 magnet width

and b) with open slots and with a 0.75 relative magnet width.

Torque ripple of 24-slot-26-pole machine

A 24-slot-26-pole machine (q = 0.364) was examined, as for this machine type Qs ≈ 2p. For the

torque ripple, it might be a benefit that the pole and slot numbers are almost the same. The slot

A

B

C

A

B

C

91

openings were similar to those of the semi-closed type with 0.09 slot opening and 0.4 of the

totally open type. The torque ripples (% of the average torque) are given in Fig. 3.24 a) and the

harmonics of the torque in Fig. 3.24 b) with different widths of the magnet. In Fig. 3.24 it can

be seen that for semi-closed there is a minimum at 0.9 relative magnet width and for the open

slot there is a minimum at a 0.82. The average torque ripple level of the open slot motor type is

lower than the corresponding level of a semi-closed slot opening. The highest harmonic

component in both cases is the 6th harmonic, as it is shown in Fig. 3.24 b).

0

2

4

6

8

10

0.5 0.6 0.7 0.8 0.9 1.0Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

Semi-closed

Open slot

0

0.5

1

1.5

2

2.5

0.7 0.8 0.9 0.7 0.8 0.9Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

6 harmonic

12 harmonic

Semi-closed slot

Open slot

th

th

a) b)

Fig. 3.24. a) The torque ripples (% of the average torque peak-to-peak) and b) the 6th and 12th harmonic

components of a 24-slot-26-pole machine with different relative widths of the magnet. The 0.09 semi-

closed slot opening width and 0.4 open slot were analysed.

A study of motors with q equal to 0.5

In the literature several representations are given of machines with q equal to 0.5 used in

different applications, as, for example, Koch and Binder (2002), Kasinathan (2003). It is

observed that this particular motor structure gives a good torque to volume ratio, but the torque

ripple can be high. Therefore, also the dynamic behaviour, for instance the torque ripple values,

of this motor type was examined here. The motors to be analysed are the 12-slot-8-pole, 24-

slot-16-pole, 36-slot-24-pole and 42-slot-28-pole machines. All these motors have the same

frame size, 400 rpm speed and 45 kW output power demand. For the 42-slot-28-pole machine

with q = 0.5 calculation was performed in order to define the effect of the magnet width on the

92

torque ripple. The current density was 5.2 A/mm2 with 93 Hz supply frequency. The torque as a

function of time is given in Fig. 3.25.

0

0.2

0.4

0.6

0.8

1

1.2

0 2π

Torq

ue (p

.u)

magnet width 0.81

magnet width 0.68

electric angle

Fig. 3.25. The torque as a function of electric angle for the 42-slot-28-pole machine with 0.81 and 0.68

relative magnet widths.

The amplitude of the torque pulsations is quite high; when the relative magnet width is 0.81 the

peak-to-peak torque ripple is 27% and with a 0.68 relative magnet width the peak-to-peak

torque ripple is 19%. The minimum torque peak-to-peak ripple with open slots (Table 3.1)

obtained for a 12-slot-8-pole machine was 13%, for a 24-slot-16-pole 3.8% and for the 36-slot-

24-pole 2% (10% with semi-closed slot openings). A 13% ripple can be considered to be high.

The torque ripples as a function of relative magnet width obtained from the current driven

model for the 24-slot-16-pole and 36-slot-24-pole machine are shown in Appendix F.

According to FEA the torque ripple minima of q = 0.5 semi-closed machines is close to relative

magnet width of 0.68 and with open slots close to relative magnet width of 0.77. These magnet

widths are similar to magnet widths 0.67 and 0.78 presented in Table 3.6.

3.5.2. The magnet width and the slot opening width

In order to define the effect of the slot opening width on the torque ripple, a series of FEA

computations were carried out in which the magnet width was varied. The semi-closed 0.09 slot

opening and an open slot structure were analysed. In some of the cases, also other relative slot

opening widths were analysed. The torque ripples (% of the average torque peak-to-peak value)

were recorded from the current driven computations at current density of 5.2 A/mm2. The

93

results for machines with 12-slots and semi-closed slot opening are shown in Fig. 3.26 a) and

for the open slot machines the results are shown in Fig. 3.26 b). It is shown that the 12-slot-8-

pole and 12-slot-16-pole machines produce higher torque ripples than the others with semi-

closed slot structure. On the other hand, the 12-slot-16-pole machine with open slot structure

has a local minimum at the value of 0.75, which is a magnet width that is found to be suitable to

use in some applications. The semi-closed 12-slot-10-pole structure has minima all over the

analysed relative magnet width range. For the open slot structure there are two minima at 0.73

and 0.87. For 12-slot machines, the lowest torque ripples are obtained for the 12-slot-14-pole

motor with open slots.

0

5

10

15

20

25

0.5 0.6 0.7 0.8 0.9 1.0

Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

12-slot-8-pole

12-slot-16-pole

12-slot-10-pole

12-slot-14-pole

0

5

10

15

20

25

0.5 0.6 0.7 0.8 0.9 1.0Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

12-slot-8-pole

12-slot-16-pole

12-slot-10-pole

12-slot-14-pole

a) b)

Fig. 3.26. Torque ripples (% of the rated torque, peak-to-peak values) as a function of the relative magnet

width for the 12-slot stator with a) semi-closed slot and b) open slot.

The torque ripple values were analysed next for a series of 24-slot machines. The curves of the

torque ripples for the 24-slot machines are given in Fig. 3.27 a) with semi-closed slots and b)

with open slots. With semi-closed slots the 20-pole and 28-pole machine have curves with local

minima, but the curves of the 22-pole and 26-pole machines are almost straight lines crossing

all over the analysed magnet width range. Comparing Fig. 3.27 a) with b) it can be seen that the

open slot structures produce lower torque ripples than the semi-closed structures, with

exception of the 24-slot-20-pole motor. The machines with open closed slots have a low torque

ripple as the relative magnet width varies from 0.7 to 0.8. With open slots and a 0.8 relative

magnet width the 24-slot-26-pole machine achieves a torque ripple that is as low as 0.3%.

94

Low torque ripple values are also achieved with the 24-slot-22-pole machine with open slots;

at relative magnet width of 0.75 the torque ripple is 0.25%.

For the open slot structure examined with 12-slot and 24-slot stators, the minimum torque ripple

is achieved at rated load as the relative magnet width varies from 0.7 to 0.8. For the 12-slot-14-

pole, 12-slot-16-pole and 24-slot-22-pole motor, the minimum for open slots in the stator is

achieved at 0.75.

0

2

4

6

8

0.5 0.6 0.7 0.8 0.9 1.0Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

24-slot-20-pole24-slot-22-pole24-slot-26-pole24-slot-28-pole

0

2

4

6

8

0.5 0.6 0.7 0.8 0.9 1.0Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

24-slot-20-pole

24-slot-22-pole

24-slot-26-pole

24-slot-28-pole

a) b)

Fig. 3.27. Torque ripples (% of the rated torque, peak-to-peak values) as a function of the relative magnet

width for the 24-slot stator a) semi-closed slot and b) open slot.

It is given an example that compares the torque ripples at a certain magnet width. Fig. 3.28

shows the torque ripple values for the studied surface magnet machines with relative magnet

widths 0.82 and 0.85. From the values in Fig. 3.28, it is possible to compare the different

combinations of the slots and poles, which have the same relative magnet width.

95

0

5

10

15

20

25

0.5 0.4 0.286 0.250 0.429 0.5 0.4 0.364 0.308 0.286 0.5 0.4 0.286

Torq

ue ri

pple

% o

f rat

ed to

rque Relative magnet width 0.82

Relative magnet width 0.85

q

Q s-2p 12-8 12-10 12-14 12-16 18-14 24-16 24-20 24-22 24-26 24-28 36-24 36-30 36-42

Fig. 3.28. Peak-to-peak values of the torque ripple obtained from the FE computations for the surface

magnet motors with the 0.82 and 0.85 relative magnet widths and with semi-closed slots.

In Fig. 3.28 it can be seen that the highest ripples occur for q = 0.5 and q = 0.25. The different

behaviour of the 0.25 and 0.5 fractional slot machines may be due to the harmonic order

numbers and the winding factors. Appendix C shows the harmonic order numbers of different

fractional slot machines and the winding factors related to them. It is shown that most of the

motors have only few integer harmonics and many non-integer harmonics. The winding factors

are quite small, except if there is a slot harmonic, which has the same winding factor as the 1st

order harmonic. The q = 0.5 winding includes purely integer numbers of harmonics: 1, 2, 5, 7,

8, 10… with the 0.866 winding factor. The q = 0.25 winding includes integer and non-integer

numbers of harmonics: 0.5, 1, 2, 2.5, 3, 3.5, 4, 5.5, 6.5, 7, 8, … with the 0.866 winding factor.

3.5.3. Conclusion

A comparing study is done for several fractional slot machine constructions of which the

relative magnet widths and the relative slot opening widths were varied. The lowest ripple

values (% of the rated torque, peak-to-peak values) obtained for different machines are repeated

in Table 3.9.

96

Table. 3.9. The lowest torque ripple values (% of the rated torque, peak-to-peak) obtained for different surface magnet machines at 1000 Nm load. Ripple values ≤ 1% are bolded.

Open slot Semi-closed slot

Qs 2p q Torque Ripple (%)

Relative magnet width

Torque Ripple (%)

Relative magnet width

Relative slot opening, x4

12 8 0.5 13 0.77 11 0.71 0.08

12 10 0.4 2.5 0.55, 0.72, 0.87 2.5 0.82 0.08

12 14 0.286 1.5 0.76 3.5 0.91 0.08

12 16 0.25 3.5 0.75 4 0.6, 0.67 0.08

18 14 0.429 1.2 0.81 5 0.56, 0.66 0.07

24 16 0.5 3.82 0.77 3.2 0.7 0.09

24 20 0.4 1.7 0.71, 0.89 1.72 0.66, 0.84 0.09

24 22 0.364 0.25 0.753 1.9 0.56 0.09

24 26 0.308 0.3 0.81 4.2 0.9 0.09

24 28 0.286 0.8 0.71, 0.75 1.6 0.63, 0.82 0.09

36 24 0.5 2 0.78 10 0.7 0.09

36 30 0.4 1.5 0.7 1 0.7 0.09

36 42 0.286 0.6 0.9 1 0.69 0.09

The values in Table 3.9 are obtained from the torque ripple curves shown in this chapter and in

Appendix F. For the machines with semi-closed slots there seems to be several relative magnet

widths that, when used, make it possible to achieve a minimum for the torque ripple. It may be

stated that the value 0.75 for the relative magnet width is to be recommended if the stator

includes open slots and a fractional slotted structure. With open slots and Qs ≈ 2p (q close to

0.33) the torque ripples are small compared to other q values. In most of the analysed cases, the

machines have a torque ripple average level for open slots lower than with semi-closed slots.

This does not mean that the open slot construction were always the better alternative. When

using open slots - having the same geometry as the semi-closed slots – the induced voltage is

lower than when semi-closed slots are used, if in both of the cases the number of coil turns is

the same. For some fractional slot machines it is not possible to use the same air-gap diameter,

and, in such a case, the torque developed may be lower than what is required. According to the

results given in Table 3.9 for a particular slot number, the peak-to-peak value of the torque

ripple grows as the number of poles increases. This means that Table 3.9 can be used to

estimate the ripple of the motor and to decide whether the motor should be skewed.

97

According to the computations referred to here, as concerns the fractional slot surface mounted

permanent magnet machine, the open slot structure generates lower torque ripples than the

semi-closed slot structure. For most of the analysed fractional slot machines, the lowest torque

ripple was achieved when the width of the magnet was selected to be slightly wider than the

width of the tooth.

3.6. Surface magnet motor versus embedded magnet motor

A surface magnet motor (S) is compared with a radially embedded magnet motor (ER) and with

an embedded magnet motor (EV), which has the magnets in V-position. The surface magnet

motor is the most commonly used PM rotor type. For some applications it might be the

beneficial alternative to manufacture a motor with embedded structure, e.g. in such cases when

the magnets need to be safely embedded inside the rotor. In some applications a low torque

ripple is required and the torque ripple of a machine with the magnets inside the rotor is usually

lower than the torque ripple of a machine with surface magnets. The performance of fractional-

slot PM machines were studied with two different rotors: the rotor with surface mounted

permanent magnets and the rotor with embedded magnets. The FEA was performed at no-load

situation for a surface magnet motor and for an embedded V magnet structure in order to solve

the flux created by the magnets. For surface mounted magnet rotor it is quite simple to solve the

flux produced by the magnets also analytically, but for the rotor with embedded magnets this is

more complicated. Magneto static and dynamic computations were then carried out in order to

define the torque production capability of differently designed motors. First, a 12-slot-10-pole

machine is designed with a surface magnet rotor and with several different embedded magnet

rotors. Later, the surface and embedded rotor structures of 24-slot-22-pole, 24-slot-20-pole and

12-slot-8-pole machines will be compared.

3.6.1. 12-slot-10-pole motor

Several configurations are possible to design a rotor with embedded magnets. The rotor

structures can have a smooth rotor surface or formed surface with cosine shape. There can be

rectangular magnets or two magnets in V-position. Fig. 3.29 illustrates one surface magnet rotor

and three embedded rotors: a) a surface magnet rotor and, b) smooth rotor surface with radially

98

embedded magnets c) a smooth rotor surface with magnets in V-position and d) a cosine formed

rotor surface with magnets in V-position.

a)

b)

c)

d)

Fig. 3.29. The geometry and the flux lines, obtained with Flux2D from the current driven model at rated

load 1074 Nm, of 12-slot-10-pole motors a) a surface magnet rotor (S) and, b) smooth rotor surface with

radially embedded rectangular magnets (ER) and a non-magnetic rotor core c) a smooth rotor surface with

magnets in V-shape (EV) and d) a 1/cosine formed air-gap with magnets embedded in V-position (EV).

The flux lines of the machines at rated load are given in Fig 3.29. Some differences in the flux

paths can be seen when comparing the surface structures, Fig. 3.29 a), with the embedded

structures, Fig 3.29 b). With the surface magnet structure the stator flux flows through a tooth

and tooth tip to the air-gap and from the air-gap the flux will try to find its way to the rotor. It

can flow from the air-gap to a magnet or to air. If the tooth tip is located above the air-gap in a

position where there is only air below the tooth tip, there is no ‘easy’ path available for the flux

to flow. With embedded magnets the flux can travel through rotor iron as it travels through the

air-gap from the stator tooth tips to the rotor side. Since the flux may flow circumferentially in

the rotor pole, armature reaction may appear.

99

Different types of rotor structures were designed for the 12-slot-10-pole machine. At first, the

FEA was performed at no-load situation for a surface magnet motor and for an embedded V

magnet structure, in which the pole consists of two embedded rectangular magnets. For both the

machines 10.5 kg magnet material is needed. The 0.92 T flux density in air-gap generated by

the permanent magnets of the surface magnet machine is higher than the 0.86 T flux density

generated by the embedded magnet machine. A steady state calculation was performed with the

FEA. Constant phase currents were applied to the stator slots and the rotor was rotated step-by-

step over one pole pitch. Fig. 3.30 shows the torque curves as a function of the rotor angle for

several 12-slot-10-pole motors. The motors described in this figure are: a smooth rotor surface

with radially embedded magnets (10.5 kg), a cosine formed rotor surface with magnets in V-

position (12.5 kg), a surface magnet rotor (10.7 kg), a cosine formed rotor surface with magnets

in V-position (10.8 kg) and a smooth rotor surface with magnets in V-position (10.8 kg).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 45 90 135 180

Rotor angle (deg)

Torq

ue (p

.u.)

embedded rectangular magnet 10.5 kg

Surface magnet 10.7 kg

V-magnet 12.49 kg formed pole

V-magnet 10.8 kg formed pole

V-magnet 10.8 kg smooth rotor surface

V magnets

surface magnets

rectangular magnets

Fig. 3.30. Torque (p.u.) as a function of the rotor angle for several 12-slot-10-pole motors as constant

phase currents are applied in the stator slots (steady state calculation from FEA). The torques of the motors

in this figure are ranging from the highest to the lowest value: a smooth rotor surface with rectangular

magnets (10.5 kg), a cosine formed rotor surface with magnets in V-position (12.5 kg), a surface magnet

rotor (10.7 kg), a cosine formed rotor surface with magnets in V-position (10.8 kg) and a smooth rotor

surface with magnets in V-position (10.8 kg). Current density is 5.3 A/mm2.

100

Fig. 3.30 shows that with 10.5 kg to 10.8 kg magnet material, the highest torque value is

produced by the motor with rectangular embedded magnets. This construction is, however,

difficult to manufacture because it needs a non-ferromagnetic inner rotor core. The lowest value

is achieved by the motor with embedded V-magnets and with smooth rotor surface. The

maximum torque is achieved at load angle higher than 90 degrees for the rotor structure with

smooth rotor surface and embedded V-magnets; this can be explained through the reluctance

difference between the d- and q-axis. Forming the rotor pole can diminish this armature

reaction. As formed rotor pole shoe with V-magnets and 10.8 kg magnets were used, the

obtained curvature became more symmetrical. Because the 10.8 kg V-magnet motor was not

capable of producing the rated (1074 Nm) torque the amount of magnet material had to be

increased. The amount of magnet material for the V-magnet motor was increased from 10.8 kg

to 12.5 kg, as a result of which the flux density created by the magnets increased up to 0.935 T

and the maximum torque increased.

The FEA was performed with the voltage driven model for a 12-slot-10-pole motor with surface

magnet and with embedded magnet structures. The machine parameters and the results obtained

from the FEA are given in Table 3.10 and in Fig. 3.31.

0

0.5

1

1.5

2

0 45 90 135 180Load angle (deg)

Torq

ue (p

.u.)

Surface

Embedded R

Embedded V

Fig. 3.31. Torque curves as a function of the load angle of 12-slot-10-pole machines obtained from the

voltage driven model. The torque curves described in the figure are those of a surface magnet motor with a

mass of 10.3 kg magnet material, of a motor with radially embedded rectangular magnets with a mass of

10 kg magnet material and of a motor with embedded V-magnets with a mass of 12.5 kg magnet material.

101

The surface magnet structure gave the highest pull-out torque compared to embedded

structures. The results show that, in the case where embedded magnets are used, the rotor iron

losses are larger than in the case where surface magnets are used. The torque ripple is the lowest

in the case of the embedded magnet motor with cosine formed rotor surface and magnets in V-

position. (The embedded V-magnet structure, introduced in Table 3.10, has a 74 degrees angle

between the magnets of one pole. Other possible angles were studied; for example, a narrower

angle of 60° would give a 1.2 p.u. pull-out torque, according to the FEA.)

Table 3.10. 12-slot-10-pole machine parameters, Power 45 kW, speed 420 rpm

Magnets Surface Embedded V Embedded rectangular

Slots-poles 12 - 10 12 - 10 12 - 10

Slot opening Semi-closed Semi-closed Semi-closed

Magnet mass (kg) 10.3 12.5 10.5

Load angle at nominal point (deg) 35 72 75

Rated torque (Nm) 1023 1023 1023

Tmax/Tn (p.u.) 1.66 1.056 1.18

Nurber of turns, Nph 104 104 80

Rated current (A) 88 97 103

Power factor, cos(ϕ) 0.92 0.83 0.83

Inductance, Ld (p.u.) 0.72 1.06 1.28

Inductance, Lq (p.u.) 0.67 1.05 1.1

Torque ripple (%) of rated torque 5.2 2.4 3.8

Iron losses in stator (W) 260 260 250

Iron losses in rotor (W) 22 98 51

Copper losses (W) 2230 2820 2737

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

The performance of a 24-slot-22-pole fractional-slot PM machine is described for the motor

structure with surface mounted permanent magnets and with radially embedded magnets. The

fundamental value (obtained from the Fourier spectrum) of the flux density normal component

in air-gap were 1.01 T and 1.17 T for the surface magnet motor and for the embedded magnet

motor, respectively. The R.M.S. values were 0.738 T and 0.92 T. With the same amount of

102

magnet material – 10.3 kg – the embedded magnet solution gives clearly higher flux density

values than the surface magnet solution at no-load. The normal component of the flux density

was solved along the whole air-gap for a loaded machine. The result for the surface magnet

motor is shown in Fig. 3.32. It can be seen that the curve of the flux density wave in the air-gap

has a different character above each of the magnets. (Salminen et al., 2003)

Fig. 3.32. The flux density normal component along the air-gap diameter, for a surface magnet motor

q = 0.364 at rated load.

The results of the FEA computations for the best surface and for the best radially embedded

magnet motor in terms of torque production capability are presented in Table 3.11.

Table 3.11. Motor parameters of 24-slot-22-pole machines (Salminen et al., 2003)

Surface magnet Radially Embedded magnet

Slots-poles 24 - 22 24 - 22 Stator radius inner (mm) 127 127 Winding factor 0.96 0.96 Rated current (A) 86.4 86.1 Main voltage (V) 351 351 Winding turns per phase 104 88 Air-gap length (mm) 1.25 1.25 Phase resistance (Ω) 0.1 0.07 Back EMF (V) 192.4 188 Air-gap flux density (T), due to permanent magnets 1.01 1.17

Frequency (Hz) 73.33 73.33 Output power (kW) 45 45 Efficiency (%) 93 94 Power factor 0.93 0.91 Magnet mass (kg) 10.3 10.3 Slot area (mm2) 805 805 Load angle (deg) 42 48 Rated torque (Nm) 1074 1074

103

The curves of the torques versus load angle of these motors are shown in Fig. 3.33. At rated

1074 Nm load the tangential stresses of the analysed motors are 39 kN/m2. According to the

computations performed with the voltage driven model the air-gap torque at a load angle of 42°

is for the surface magnet motor 1090 Nm and for the embedded magnet motor 1000 Nm.

0.0

0.5

1.0

1.5

2.0

0 45 90 135 180

Load angle (deg)

Torq

ue (p

.u)

24-22 Embedded24-22 Embedded FEA24-22 Surface24-22 Surface FEA

a)

0.0

0.5

1.0

1.5

2.0

0 45 90 135 180

Load angle (deg)

Torq

ue (p

.u)

24-20 Embedded24-20 Embedded FEA24-20 Surface24-20 Surface FEA

b)

Fig. 3.33. The load angles of a) a 24-slot-22-pole surface and radially embedded magnet motor are similar

at 1 p.u. rated load – close to 38 degrees – but the pull-out torque is clearly higher when surface magnets

are used: 1.56 p.u. instead of 1.45 p.u. In b) a 24-slot-20-pole surface magnet motor with a 0.09 slot

opening width is compared to a radially embedded magnet motor with totally open slots.

The torque ripple (% of the rated torque) peak-to-peak value for the surface magnet motor with

a 0.85 relative magnet width and for the radially embedded magnet motor is 5.7% and 4.5%,

respectively. In this case, the radially embedded magnet motor gives less torque than the surface

magnet motor at the same load angle. A series of computations with voltage control were

carried out for the surface magnet motor with different load angles. Based on the results, it

could be stated that the maximum torque available from this machine is 1675 Nm. The radially

embedded magnet solution with voltage control gave a little less torque so that the maximum

torque was 1545 Nm, which is 8% less than the maximum torque of the surface magnet motor.

Both the machines exceed the requirement made and the overloading capacity is fulfilled. The

number of turns and the phase resistance of the embedded magnet structure are smaller than

those of the surface magnet structure. Thereby, the copper losses of the embedded motor are

smaller.

104

Different 24-slot-machines with 20 poles were also compared. The surface mounted magnet

motor has a small slot opening of 0.09, but the radially embedded magnet motor is here equipped

with totally open slots. Fig. 3.33 b) shows that, now, the radially embedded magnet structure

gives a higher pull-out torque than the surface magnet structure. The current at the rated torque

is practically the same for both of the machines. The torque ripple (and the cogging torque

ripple) with the radially embedded rotor structure is lower than with the surface rotor structure.

3.6.3. Conclusion

The pull-out torques got from the FEA of several fractional slot machines are presented in

Table 3.12. According to the computations of 24-slot-22-pole and 24-slot-20-pole machines,

similar pull-out torques can be achieved with the surface magnet structure as well as with the

radially embedded magnet structures. These radially embedded structures have less copper

losses and lower torque ripples than the corresponding surface mounted structures. As the slot

and pole number was smaller in the case of 12-slot-10-pole motor, the developed pull-out

torque of the radially embedded magnet structure was 30% smaller than with the surface

magnet structure of the same frame size. The 12-slot-10-pole radially embedded magnet motor

with open slots needed more stator coil turns than the surface structure to induce the same

amount of back EMF.

Table 3.12. The pull-out torques obtained from the FEA of several 45 kW, 400 rpm fractional slot machines. S = surface magnet motor and ER = radially embedded magnet motor.

Slots-poles 24-22 24-22 24-20 24-20 12-10 12-10 12-8 12-8 Rotor S ER S ER S ER S ER Pull-out torque ( )

1.56 1.45 1.79 1.73 1.66 1.18 1.66 1.02 Slot opening Semi Semi Semi Open Semi Semi Semi Semi

The question is, why do these 12-slot radially embedded magnet motors have a low pull-out

torque. The solution may be found by comparing the motors at rated load. A surface magnet and

a radially embedded magnet 12-slot-8-pole motor at rated load are shown in, respectively, Fig.

3.34 a) and b). In the surface magnet motor the current is 83.5 A, and in the radially embedded

magnet motor it is 92.9 A. In the case of the radially embedded magnet motor, the flux is higher

and it travels longer paths in the rotor than it does in the case of the surface magnet motor,

according to Fig. 3.34.

105

1.46 T

0.7 T

1.46 T

1.46 T

0.55 T

1.3 T

1.7 T

1.59 T1.02 T

1.6 T

1.59 T

0.4 T

1.68 T

2.0 T

a) b) Fig. 3.34. The flux line plots and flux density magnitudes of a 12-slot-8-pole a) surface magnet motor and

b) radially embedded magnet motor at rated load. Both machines are presented at same load angle.

In the 12-slot-8-pole radially embedded magnet motor, there is also a large portion of the flux

travelling in the rotor without passing through the permanent magnet. This may be regarded as

the quadrature-axis armature reaction that deteriorates the motor performance. Since, on the

contrary, the 24-20 radially embedded machine has a high performance, there must be found the

optimal pole dimension geometry for the ER machines. A space vector diagram of the surface

magnet motor is shown in Fig. 3.35 a) and of the radially embedded magnet machine in Fig.

3.35 b), both at rated load. The armature reaction Ψa is high with the radially embedded magnets

as a result of which the load angle is bigger. Therefore, the developed torque diminished in the

case of the radially embedded magnet motor.

us

is

Ψs

ΨPM

us

isΨs

ΨPMδa

a) surface magnet motor b) radially embedded magnet motor

δa

Ψa

Ψa

Fig. 3.35. A space vector diagram of a 12-slot-8-pole a) surface magnet motor and b) radially embedded

magnet motor at rated load. The load angle, between the stator flux linkage vector Ψs and the flux linkage

vector due to permanent magnets ΨPM, is 28° in case of a) surface magnet motor and 70° in case of the

radially embedded motor. The us is stator voltage vector and is stator current vector.

106

For the 24-slot-22-pole and 24-slot-20-pole machines, the achieved pull-out torques were high

1.56 p.u. and 1.79 p.u. (Table 3.12), which is a result independent of the position of the

magnets. Thereby, a closer examination of the 24-slot-20-pole machine will be done. When the

motor is equipped with 20 magnets, the distance between the magnets is obviously smaller than

in the previous motor, which had only 8 magnets. Thereby, the armature reaction will not be as

large. Fig. 3.36 shows the 24-slot-20-pole embedded magnet motor at rated load situation.

There is only a high reluctance route for the quadrature armature reaction and thus this machine

will give a high torque of 1.73 p.u. (the 24-slot-20-pole surface magnet solution gives 1.79

p.u.).

a)

us

is

Ψs

ΨPM

radially embedded magnet motor

δa

Ψa

b)

Fig. 3.36. a) The flux lines of the 24-slot-20-pole radially embedded magnet motor at rated load and b) the

corresponding space vector diagram. The load angle, between the stator flux linkage vector Ψs and the flux

linkage vector due to permanent magnets ΨPM, is about 36°.

3.6.4. Slot opening

The effect of the slot opening width was studied more closely, because it appeared from earlier

investigations, which were performed at no-load and with the current driven model, that the slot

opening does have some effect on the torque ripple values. For this reason, it was studied if the

slot opening width has some effect on the torque production, too. For the manufacturing of the

winding, the slot opening width is an important parameter. In the case of a concentrated

winding the coil is wound around tooth. This allows automatically winding of the machine.

Needle winders can be used to wind lap windings. The coils of the phases can be separately

wound and then inserted (by hand or automatically) in the stator lamination core. This option is

107

possible when there is a totally open slot structure. It can be useful, in some cases, to

manufacture machines with totally open slots in order to keep stator structure simpler, which

then, obviously, also reduces the production costs. Special manufacturing methods are

discussed by, among others authors, Jack et al. (2000). He studied the possibility to use

powdered iron cores and pre-pressed windings. He manufactured and tested a servo motor

design, which obtained a high winding filling factor and gave a high torque.

In the case of a conventional lap winding, it normally occurs that the reluctance of the magnetic

circuit is reduced if the slot opening is narrow unless the teeth or stator yoke iron are saturated.

The tooth tips area gives a suitable space for the flux to flow. If the slot opening is wide, the

equivalent air-gap length δeff and the air-gap reluctance are increased. This reduces the amount

of the flux producing the back EMF. In the case of a fractional winding, the situation is a little

more complicated. Because in some combinations there is nearly just one pole per slot, this

causes the flux paths to be dependent on the geometry of the slot. It was noticed that the current

driven computations do not - for all of the fractional slot machines – give a reliable result

because the fact is that currents are not purely sinusoidal. Thereby, a FEA with the voltage

driven model was carried out. All studied motors have a terminal voltage of 351 V. As an

example, a 12-slot-10-pole machine is studied with semi-closed slots and with open slots. To

design a 12-slot-10-pole with open slots similar to the 12-slot-10-pole semi-closed structure was

not easy and therefore several different open slot designs were done. In this chapter, the surface

magnet rotor is first investigated and later also embedded V-magnet rotor will be examined.

The geometry of the surface magnet motor with semi-closed slots is shown in Fig. 3.37. In the

example where the surface magnet 12-slot-10-pole motor is calculated, the parameters, which

are kept constant, are

• Stator outer diameter 364 mm

• Core length 270 mm

• Rotor geometry

• Speed 400 rpm

• Power 45 kW

• Frequency 35 Hz

• Magnet material mass 10.5 kg.

39 mm25.8 mm

18 mm

α = 0.83relative magnet width,

Fig. 3.37. Geometry of 12-slot-10-pole motor

108

The results are shown in Table 3.13 and the geometries of the motors marked with the

corresponding letters a, b, c, d, e and f are illustrated in Fig 3.38.

Table 3.13. Results obtained from the FEA for a 45 kW, 400 rpm 12-slot-10-pole surface magnet motor, voltage driven model.

Slot open width Semi-closed Open slot

Picture in Fig. 3.38. a b c d e f

Air-gap diameter (mm) 249 249 249 231 231 220

Slot area (mm2) 1900 1900 1750 1550 1900 2300

Winding turns, Nph 104 104 104 104 104 120

Rated current (A) 88 101 104 95 104 89

Power factor 0.93 0.86 0.81 0.92 0.84 0.93

Back EMF (V) 179 <160 154 <160 <160 173

Copper losses (W) 2344 3700 4700 3100 3900 3327

Efficiency (%) 94.0 90.5 90.5 - - 92.3

Pull-out torque (p.u.) 1.66 1.91 2.0 - 1.91 1.57

Current at 90 deg (A) 194 257 265 - 257 195

The semi-closed slot geometry a) shown in Fig. 3.38 a) is modified to open slots structure b)

shown in Fig. 3.38 b). A comparison is made between the geometries a) and b). Both machines

have the same slot area, which is 1900 mm2, and the coil turn count in series per phase Nph is

104. The motor with the semi-closed slots gives the rated torque at 88 A (η is 0.94) and the

maximum torque is 1.66 p.u. The motor with totally open slot gives the rated torque at 101

amperes (η is 0.905) and the maximum torque available is 1.91 p.u. The desired 45 kW power

is achieved, but the current density with the open-slot-version is 6.5 A/mm2, while it was only

5.4 A/mm2 with the semi-closed slots. The induced phase back EMF of the structure with semi-

closed slots was 179 volts, which is about 90% of the supply phase voltage. With totally open

slots only 154 volts was induced in the stator windings while the winding arrangements were

the same as before (about 75% of the supply phase voltage). To achieve a high efficiency and a

low current density, it was then studied whether the machine performance with open slots can

be improved by modifying the slot shape. Parameters that were varied to design an open slot

machine are shown in Fig. 3.38. First, the structures with the same winding and the same slot

109

area were studied in order to obtain fairly comparable data. But, it was soon discovered that

more coil turns are needed, so that the required back EMF 180 V can be achieved. Therefore,

the area of the slot was increased in order to fit in more coil turns and to keep the value of the

current density as the same as the previously i.e. 5.4 A/mm2. The computation results of the

motor f) with 220 mm air-gap diameter and 2300 mm2 slot area and of the original motor a)

with 249 mm stator radius and 1900 mm2 slot area are shown in Table 3.13.

= 220 mmDδ

x4 = 0.09

= 249 mm

d) e) f)

δ

x4 = 0.63 x4 = 0.63

a) b) c)

= 249 mm = 249 mm

= 231 mm = 231 mm

x4 = 0.6 x4 = 0.63

DδDδ

Dδ Dδ Dδ

x4 = 0.63

Fig. 3.38. The slot geometry of different 12-slot-10-pole-machines: a) semi-closed slot, 0.09 relative slot

opening b) open slot with 1900 mm2 slot area, c) open slot with 1750 mm2 slot area, d) open slot with 1550

mm2 slot area, e) open slot with 1900 mm2 slot area and f) open slot with 2300 mm2 slot area.

It can be seen that the motors a) and f) can induce almost the required back EMF and have

almost the same current density. The pull-out torque of the totally open slot motor f) is about

10% less than that of the semi-closed slot motor a). This is due to the fact that the air-gap

diameter was diminished by 10% from 249 mm to 220 mm. The developed torque of the motor

f) at a 45 degrees load angle is 20% less than of the motor a). Another difference in the

machines is the efficiency - for machine a) 94.0% and for machine f) 92.3% - a difference that

may be explained as follows: In the machine with tooth tips there is wider area for the flux to

flow into the stator teeth. This also means that the flux density in the narrow part of a particular

tooth can be higher than without the tooth tips. For the studied motor a) as the machine was at

110

rated load the highest flux density magnitude in the tooth tips was 1.89 T. In the yoke area the

maximum was 1.55 T. In a motor with 12 slots and 10 poles these maximum values are

obtained just in 2 or 3 teeth and in 2 parts of the yoke, the other teeth have a value that is even

less than 0.5 T. When the machine is rotating the areas of low/high flux values move with the

speed of the machine generating iron losses if the frequency is high.

Considering the 12-slot-10-pole surface magnet machine and according to the results given in

Fig. 3.39, the open slot structure gives less torque and a higher rated current than the semi-

closed slot structure. The semi-closed structure at rated torque has a 5.4 A/mm2 current density

and the motor with totally open slots has a 5.6 A/mm2 current density. It is also to be considered

that the increased current means that the copper losses are bigger and the efficiency is going to

be smaller if the slot opening width is increased.

0

0.5

1

1.5

2

0 45 90 135 180Load angle (deg)

Torq

ue (p

.u.)

Semi-closedSemi-closed, FEAOpen slotsOpen slots, FEA

Fig. 3.39. The torque as a function of the load angle for a 12-slot-10-pole motor with surface magnets. The

motor with semi-closed slots has a 5.4 A/mm2 current density and the motor with open slots has a 5.6

A/mm2 current density. The points represent the results obtained from the FEA and the lines are drawn

using the torque equation. Both machines have the same 225-frame size, the same terminal voltage of 351

V and the same amount of magnet material 10.3 kg.

111

3.6.5. Embedded V-magnet motors

A study is now made on a 12-slot-10-pole embedded machine with magnets in V-position with

semi-closed slots (dimensions of semi-closed structure are shown in Appendix G) and with

open slots. The results obtained are compared to the results of the corresponding surface magnet

motor. The overall geometries of the motors for which this comparison is made remain the

same; the only geometrical change is the width of the slot. Here, the rotor has cosine formed

pole shoes and the amount of the permanent magnet material is 12.5 kg. The rated current of

both machines is practically the same and also the amount of conductors – 104 – is the same.

The obtained torques of the motors with semi-closed slots and open slots are shown

in Fig. 3.40. The pull-out torque of the embedded V-magnet motor with open slots is about 20%

higher than the pull-out torque of the motor with semi-closed slots.

0

0.5

1

1.5

2

0 45 90 135 180

Load angle (deg)

Torq

ue (p

.u.)

Open slot

Open slot, FEA

Semi-closed

Semi-closed, FEA

Fig. 3.40. The torque as a function of the load angle for the embedded-V-magnet 12-slot-10-pole motor

with semi-closed slots and open slots. The points represent the results obtained from the FEA and the lines

are drawn using the torque equation. Both machines have the same 225-frame size, the same terminal

voltage of 351 V and the same amount of magnet material 12.5 kg.

112

3.6.6. Conclusion

The results of the surface magnet 12-slot-10-pole motor and the embedded magnet 12-slot-10-

pole motor computations with semi-closed slots and with totally open slots are given in Table

3.14. The surface magnet rotor gives the highest pull-out torque. However, the efficiency of the

motor with open slots is low. On the other hand, this motor type is more practical to

manufacture. The efficiency of the motor with semi-closed slots was 94% and its current

density 5.4 A/mm2, but when the slot opening structure was changed to be an open slot one the

efficiency dropped to 86.2% and the current density rose up to 8.3 A/mm2. Therefore, also other

parameters than just the slot opening width were modified. The number of coil turns was

increased from 104 to 120; the slot area was increased from 1900 mm2 to 2300 mm2 and the

124.5 mm stator inner radius was decreased to 110 mm. Finally, the efficiency of the surface

magnet motor with totally open slots got a value of 92.3%.

Table 3.14. 12-slot-10-pole motor with surface and embedded magnets (the same rotor for both the embedded magnet motors). FEA results are given with semi-closed slots and with totally open slots.

Machine, Qs - 2p 12 - 10 12 - 10 12 - 10 12 - 10

Magnets Slot opening Slot opening width (p.u.)

Surface Semi-closed

0.09

Surface Open slot

0.63

Embedded V Semi-closed

0.09

Embedded V Open slot

0.63 Magnet mass (kg) 10.3 10.3 12.5 12.5 Speed (rpm) 400 400 420 420 Load angle at nominal point (deg) 35 41 72 58 Rated torque (Nm) 1074 1074 1023 1023 Rated current (A) 88.4 89 97 92.2 Nph 104 120 104 104 Rph (Ω) 0.1 0.14 0.1 0.1 Pull-out torque (Nm) 1780 1690 1080 1280 Tmax/Tn (p.u.) 1.66 1.57 1.06 1.26 PFe stator/rotor (W) from FEA 258/22 129/18 259/98 207/116 PFe (W) from FEA 280 148 357 323 PFe (W) analytical computation 305 209 328 328 PCu (W) 2344 3327 2820 2550 PStr (W) 225 225 225 225

Efficiency (%), (PFe anal., PCu, PStr) 94.0 92.3 93.0 93.5

113

Of all the structures studied here, it is the open slot surface motor that has the smallest stator

iron losses PFe but that has also the worst efficiency due to the high current which causes high

copper losses PCu. The stator iron losses of the embedded magnet machines are slightly smaller

when the stator has open slots than when it has semi-closed slots.

The torque ripple of the surface mounted permanent magnet motor with semi-closed slots was

2.5% and with totally open slots it was 3% calculated with the voltage driven model. The torque

ripples of the embedded magnet rotor structures were 6%, regardless of the slot opening type.

The embedded magnet structure produces a low pull-out torque if the slot opening is small;

when the slot is totally open the pull-out torque is high. The obtained difference in the pull-out

torques is related to the difference in the inductance values. With a large inductance it is not

possible to achieve a high torque, according to the power vs. load angle equation (Eq. 2.51).

The synchronous inductance, Ld of the surface magnet motor is 30% smaller that of the

embedded V-magnet structure. The open slot structure combined with the embedded V-magnet

rotor gives a good torque to volume ratio and also a small torque ripple. The benefit of this

motor type is unquestionably the ease with which the stator and rotor can be manufactured. The

stator coils can be manufactured separately and plugged around the teeth, thereby the winding is

simpler to construct than in a solution with tooth tips.

3.7. The fractional slot winding compared to the integer slot winding

For this study, a prototype motor was manufactured and tested at laboratory: a fractional slot

machine with q = 0.4, a shaft power of 45 kW, speed 420 rpm and frame size of 225. In an

earlier investigation another machine with the same shaft power 45 kW was manufactured to

the same frame size, an integer slot machine q = 2 (speed 600 rpm). It is compared the

parameters of these embedded V magnet machines with q = 0.4 and q = 2, which are

manufactured to different applications, because they have the same frame size. The parameters

of these machines are shown in Table 3.15. Table 3.15 proposes the values for two surface

magnet motor designs, also for the frame size 225.

114

Table 3.15. Parameters of 45 kW motors: Embedded V-magnet prototype motors, which are manufactures at LUT and two proposed surface magnet motor designs (not prototypes).

Winding type Prototype/Motor design Magnets

Fractional Prototype

Embedded V

Integer Prototype

Embedded V

Fractional Motor design

Surface

Integer Motor design

Surface

Slots-poles q

12-10 0.4

48-8 2

12-10 0.4

60-10 2

Speed (rpm) 420 600 400 420

Rated torque (Nm) 1023 715 1074 1023

Tmax/Tn (p.u.) 1.1 1.9 1.66 1.44

Air-gap diameter (mm) 249 250 249 250

Core length (mm) 270 270 270 270

Copper in end windings (kg) 8 14 8 22

Copper in the machine (kg) 31 32 31 44

Efficiency (%) 91.5 93 92.2 -

Induced back EMF (V) 351 351 351 -

Current (A) 97 78 88.5 -

Power factor, cosϕ 0.832 0.973 0.91 -

Torque ripple, ∆Tp-p (%) 3 - 3 22

The prototype motor with q = 2 has a higher pull-out torque and better efficiency than the q = 0.4

machines, because the q = 2 machine has a 600 rpm speed instead of 420 rpm. The copper

needed for the integer slot machines is higher than for the fractional slot machine. When

comparing the designed surface magnet motors with a 400 and 420 rpm speed, it can be seen

that the fractional slot machine obtained a higher pull-out torque than the integer slot machine.

In the current driven model of the integer slot machine (q = 2) the torque ripple is 22% while it

is for a fractional slot machine, e.g. for a q = 0.4 machine only 2.5%. In the voltage driven

model the integer slot machine has a 9% torque ripple while the value for the fractional slot

machine is 2.5%. The cogging torque of the integer slot machine is 3%, which is much higher

than the values of fractional slot machine, which is less than 1%. According to the FEA, it can

be expected that the torque ripple values are smaller for machines of this size and with

fractional slot windings. The pull-out torque achieved for the machine with fractional slot q

equal to 0.4 is a little higher than the corresponding integer slot q = 2 machine. Calculations

115

were also made on the amount of copper that is needed in the slots and in the end windings of

the fractional slot 45 kW machine as well as of the integer wound 45 kW machine. The copper

amount needed for the q = 2 machine and for the q = 0.4 surface magnet machine is 44 kg and

embedded magnet machine 31 kg, respectively. The copper amount that can be saved by using

fractional slot windings is about one fourth of the amount needed for the integer windings.

Bianchi et al. (2004) compared a 9-slot-8-pole fractional slot machine to a 24-slot-8-pole

integer slot machine. The torque ripple of the integer slot (q = 1) machine with surface magnets

was 24.2% and the torque ripple of the fractional slot (q = 0.375) machine only 2.8%. The

corresponding values for the machines with radially embedded magnets were 42.6% for the

integer slot machine and 3.6% for the fractional slot machine.

3.8. Losses

The loss components were calculated using the FEA and analytical methods. Table 3.16 shows

the losses of some surface magnet motors (S) and a radially embedded magnet motor (ER). The

iron losses of the stator and rotor can be obtained from the FEA, but they can also be calculated

analytically from the flux density values. The iron loss values obtained for several machines

types indicated that the iron losses of the embedded magnet motors are about 20 to 30% higher

than the iron losses of the corresponding surface magnet motor at rated load. Zhu et al. (2002),

in his study on integer slot 18-slot-6-pole machines, stated that with embedded magnet rotors

have higher iron losses than with surface magnet rotors. This is due to the high harmonic

content of the armature reaction field.

From the analysis shown in this thesis, it could also be noticed that for some motor types the

iron losses were lower - even 50% lower - with open slots than with semi-closed slot openings.

The copper losses vary from 1550 to 2880 W depending on the winding turns needed. The

radially embedded 24-slot-22-pole machine has less copper losses than all the other studied

machines, because it was capable of producing enough back EMF with only 88 winding turns.

116

Table 3.16. Losses of surface magnet motors (S) and a radially embedded magnet motor (ER). All machines have the same magnet mass of 10.3 kg.

Magnets S S ER S S S S S Slots-poles 24-28 24-22 24-22 24-20 24-16 12-14 12-10 12-8

q 0.286 0.364 0.364 0.4 0.5 0.286 0.4 0.5

Rated current (A) 86 86.4 86 82 83.5 91.5 88.4 88

Frequency (Hz) 93.33 73.33 73.33 66.67 53.33 46.67 33.33 26.67 Nph 104 104 88 104 112 104 104 120 Tmax/Tn (p.u.) 1.3 1.56 1.45 1.79 2.0 1.2 1.66 1.66 PFe, stator (W) 507 500 650 374 350 274 258 262 PFe, rotor (W) 30 23.5 75 20 15 36.5 22 19 PCu (W) 2219 2239 1553 2017 2364 2512 2344 2881 PEddy (W) 200 175 155 105 60 420 405 235

η, efficiency (%) * 93.8 93.8 94.7 94.5 93.8 93.7 94.0 93.0

* Efficiency is computed with a constant PStr = 225 W (for each machine), for simplification.

Considering the eddy current loss computation results obtained with the FEA, it must be noted

that embedded magnet motors have usually lower eddy current losses than surface magnet

motors. For the 12-slot-10-pole machine, the 12-slot-8-pole machine and the 24-slot-22-pole

machine, each with radially embedded magnet structure; the eddy current losses were 390 W, 70

W and 155 W, respectively. It was also observed that machines with open slot structures have

higher eddy current losses than those with semi-closed structures. Fig. 3.41 shows a plot of the

flux densities of a) a 24-slot-16-pole and b) a 24-slot-20-pole surface magnet machine. It can be

seen that the flux densities vary a lot in the magnet areas, especially in case of the 24-slot-20-

pole machine.

a)

0 - 0.0020.002 - 0.0030.003 - 0.0550.055 - 0.550.55 - 0.730.73 - 0.910.91 - 1.091.09 - 1.281.28 - 1.461.46 - 1.641.64 - 1.831.83 - 2.02.0 - 2.22.2 - 2.372.37 - 2.562.56 - 2.742.74 - 2.92

Color shadeFlux density (T)

b)

Fig. 3.41. Flux density plot from FEA for a) a 24-slot-16-pole and b) 24-slot-20-pole machines.

117

Some of the studied structures have high frequencies. This causes that the flux densities vary

rapidly in the magnet, which may cause computational problems in FEA. Therefore, it is to be

recommended to carefully interpret the FEA results. Analytical methods of calculating the eddy

current losses in permanent magnets are introduced e.g. by Nipp (1999) and Atallah et al.

(2000). Fig. 3.42 shows the efficiencies of several studied surface magnet machines. All the

machines in Fig. 3.42 have the same frame size, the same main voltage and the same amount of

magnet material. The figure gives also the values of the pull-out torques to illustrate the

machines capability of producing torque. According to Fig. 3.42 the amount of obtained pull-

out torque, in most of the analysed machines, increases as q increases from 0.25 to 0.5. It can

also be noted that q ≈ 0.3 offers a low performance (low efficiency and low pull-out torque),

when comparing to machines, which have a 225 frame size, a 45 kW shaft power and a 420 rpm

speed.

0.90

0.95

1.00

0.25 0.286 0.286 0.286 0.318 0.364 0.4 0.4 0.4 0.429 0.5 0.5 0.5

Efficiency

Pull-out torque

slots 12 12 24 36 21 24 12 24 36 18 12 24 36 q

2.0

1.0

Pull-out torque (p.u)Ef

ficie

ncy

Fig. 3.42. The efficiencies and the pull-out torques of the studied fractional slot machines with surface

magnets. Machines have semi-closed slots, 225-frame size, 351 V terminal voltage and the amount of

magnet material is 10.3 kg.

3.9. The analytical computations compared to the FE computations

The analytical computations are compared to computations carried out with the FEA in order to

see if the values correspond to each other. The values are presented in Table 3.17. A comparison

is made of the 45 kW motors that are discussed in earlier chapters and which have a 1074 Nm

rated torque and a 400 rpm rated speed. In the analytical computations of the maximum torque

118

92.5% efficiency was used. It can be seen from Table 3.17 that the analytical computation

results are close to the FEA results. In the analytical computation it is important to accurately

estimate the value of the inductances in order to solve the maximum available torque.

Table 3.17. Analytical computations (A) compared to the FE analysis (FEA) for 45 kW surface magnet motors.

A FEA A FEA A FEA A FEA

Magnet Surface Surface Surface Surface Surface Surface Surface Surface

Slots 24 24 24 24 24 24 24 24

Poles 28 28 22 22 20 20 16 16

q 0.286 0.286 0.364 0.364 0.4 0.4 0.5 0.5

Winding factor 0.933 0.933 0.949 0.949 0.933 0.933 0.866 0.866

Rated current (A) 89 84.5 85.4 86.4 82.1 80 88 83.5

Frequency (Hz) 93.33 93.33 73.33 73.33 66.67 66.67 53.33 53.33

Power factor 0.92 0.98 0.93 0.98 0.98 0.98 0.96 0.98

Synchronous inductance, Ld (p.u.) 0.92 0.9 0.71 0.70 0.59 0.57 0.49 0.5

Back EMF (V) 191 192 190 183 184 188 183 184

Magnet mass (kg) 9.8 9.8 10.3 10.3 9.6 9.6 10.3 10.3

Tmax/Tn (p.u.) 1.23 1.2 1.59 1.56 1.65 1.69 1.9 2.0

PFe (W) 710 742 570 524 400 394 541 365

PCu (W) 2376 2323 2188 2239 1982 1920 2602 2364

Both the calculation (analytical and FEA) methods give results for the synchronous

inductances, Ld that are about the same. The magnetizing inductance and the leakage inductance

were computed separately following the equations (method 2) given in chapter 2. The

inductances were computed with 4 different analytical methods. From these methods it was

selected the method the results of which were close to the FEA results. As an example, the

procedure to solve the inductances for a 24-slot-22-pole machine is shown in Appendix D. It

was noticed that, when the inductance was set close to the value 1 p.u., the analytical

calculation result differs from the FEA. This is also due to the fact that there is saturation in the

motor, which the FEA can take this into account but not the analytical method. The analytical

method could be improved by using reluctance circuits. (If saturation occurs, the equivalent air-

gap length is increased.)

119

3.10. Designing guidelines

All the motors in Table 3.18 are designed to frame size of 225. They all have about the same

amount of magnet material and about the same air-gap diameter. When comparing the

performances of these motors, some differences can be seen. The highest torque obtained is 2.1

p.u. and the lowest 1.0 p.u. The results in Table 3.18 show that for a certain slot number the

highest torque achieved is usually with a machine having a small pole number.

Table 3.18. The pull-out torque Tmax (p.u.) and torque ripple values ∆Tp-p (% of the rated torque, peak-to-peak values) for the surface mounted machines obtained from the voltage driven model. The LCM and fundamental winding factors ξ1 for concentrated two-layer windings are presented.

Poles Slots 8 10 12 14 16 20 22 24 26 28 30 42

q 0.5 0.4 - 0.29 0.25 Tmax (p.u.) 1.66 1.66 1.17 1.04

12 ∆Tp-p (%) 15.9 2.5 7.5 12.9 LCM 48 60 84 48 ξ1 0.866 0.933 0.933 0.866

q 0.5 0.43 Tmax (p.u.) 2.1 1.79

18 ∆Tp-p (%) 16 6.6 LCM 36 126 ξ1 0.866 0.902 q 0.318 Tmax (p.u.) 1.1

21 ∆Tp-p (%) >50 * LCM 462 ξ1 0.951 q 0.5 0.4 0.36 0.31 0.286 Tmax (p.u.) 2.0 1.79 1.56 1.0 1.3

24 ∆Tp-p (%) 8 2.5 5.7 >50 3 LCM 96 120 264 312 168 ξ1 0.866 0.933 0.949 0.949 q 0.5 0.4 0.286 Tmax (p.u.) 1.73 1.53 1.02

36 ∆Tp-p (%) 3.5 1.8 1 LCM 180 144 252 ξ1 0.866 0.933 0.933

* Not recommended because of the unbalanced magnetic pull effect.

120

Although a high fundamental winding factor is used, this does not guarantee that the machine

will have the capacity of producing a high torque. As an example, the 18-slot-12-pole machine

has a 0.866 winding factor and, yet, it can produce a torque as high as 2.1 p.u. High pull-out

torques given by machines of this size category were obtained with the 18-slot-12-pole and 24-

slot-16-pole machines for both of which q is equal to 0.5. The lowest torque ripple obtained

with the voltage driven model was 1% and the highest torque ripple was over 50% of the rated

torque (peak-to-peak values). A high LCM number indicates that the value of the torque ripple

is small, except in some special cases where Qs ≈ 2p, because there may appear unbalanced pull

effect. With low LCM or with q equals 0.5 or 0.25 it can be expected that the torque ripple is

high.

When designing a machine with Qs ≈ 2p, the risk of unwanted forces must be taken into

account. The motor structure with an odd number of slots is a special case, especially when the

number of slots and poles is almost equal. The winding arrangement of a 21-slots and 20-poles

machine may consist of several coils from same phase that are next to each other. It is possible

to amend the unbalanced magnetic pull in the machine. Magnussen et al. (2004) described this

unwanted effect for the case of a 9-slot-8-pole and 15-slot-14-pole machines. The pull effect is

caused by the radial forces that are much higher on one side of the machine than on the other

side.

If an embedded motor is to be designed, it is recommended not to select the smallest number for

the slots and poles in order to avoid the risk of causing a high armature reaction effect, which

would reduce the machine capacity of producing the high pull-out torque. The embedded

magnet machine with open slot structure is to be preferred. With the embedded magnet machine

a higher torque was achieved for most of the analysed structures even though, in this case, the

rotor losses were higher than in the case of the embedded structure with semi-closed slot

openings.

121

4. 12-SLOT 10-POLE PROTOTYPE MOTOR

Fractional slot permanent magnet machines can be manufactured with the short end windings in

the axial direction of the machine. The axially longer stator core can thus be mounted into the

same frame size. In low speed applications this may be advantageous because of the increased

air-gap area. Because of the short end windings the amount of active copper is increased

compared to conventional windings, thus a corresponding reduction of the copper losses may be

expected. It must be noted that the line frequency of (practical) fractional slot machines is in

some applications higher than that of conventional wound one- or two-pole pair machines.

Therefore, in high-speed applications the use of fractional slot machines may be unpractical

because of the high iron losses. Fractional slot windings have another advantage over

conventional windings, which is the possibility to use concentrated wound coils – one coil

around each tooth. Such a winding is easy to make and the manufacturing can be automated,

which reduces the manufacturing costs.

A prototype machine with 12-slot-10-pole structure was constructed in order to obtain practical

experiences concerning the manufacturing of these machine types. Furthermore, the

measurement results provide the essential information needed to verify both the analytical

calculations and the computations made with the 2D FEA. The designing process of the 12-slot-

10-pole prototype machine was discussed in Chapter 3.6.1, where a comparison was done of the

surface magnet and embedded magnet structures. The prototype motor was also discussed in

Chapter 3.7, where the prototype motor q = 0.4 was compared to the q = 2 motor of the same

frame size and with a 45 kW shaft power.

4.1. Design of the prototype V-magnet motor

According to the voltage driven model discussed in Chapter 3, a high pull-out torque and low

torque ripple can be achieved with a 24-slot-20-pole surface permanent magnet motor with q =

0.4 for the slots per poles per phase. To save manufacturing time and also costs, it was selected

a proto motor, where the torque ripple would be low and no skewing would be needed. That

was the reason to avoid q = 0.5 motor to be a proto motor although a q = 0.5 usually gives high

pull-out torque. Therefore, it was decided manufacture a fractional slot machine with q = 0.4 to

be operated as prototype machine. The decision to manufacture a 12-slot-10-pole machine type

122

of the q = 0.4 machine was partly based on the manufacturing costs. Stators with a small slot

number are a more attractive alternative for the manufacturer since a smaller amount of slots

must be punched. This facilitates a faster manufacturing process of the lamination core. There is

the same concern with the rotor lamination. Using a small slot number reduces the number of

required coils and offering further savings in manufacturing time. For these reasons, a 12-slot-

10-pole machine was selected to function as the prototype machine, even though the 24-slot-20-

pole or 24-slot-16-pole machines of the same frame size may have offered a better capacity for

producing the torque.

It was selected for the 12-slot-10-pole machine a rotor construction with embedded V-magnets.

It may be mentioned several advantages that favour the use of embedded magnets: 1) the risk of

permanent magnet material demagnetization becomes smaller. 2) The magnets can be

rectangular and there are less fixing and bonding problems with the magnets: The magnets are

easy to mount into the slots of the rotor and the danger of damaging the magnets is smaller.

FEA computations were carried out to find the best geometrical solution for an embedded V-

magnet motor.

The 12-slot-10-pole machine was designed with the object to achieve a high rated torque from a

small volume for a relative low-speed application. According to the calculation result, this goal

was reached. The surface magnet structure, shown on the left in Fig. 4.1, produces a 1.66 p.u.

pull-out torque in the voltage driven FEA computations, but the V-magnet structure, on right in

Fig. 4.1 does not seem to achieve as high a torque. This is because of the relative high

inductance, which results in low power. The torque ripple of the embedded rotor structure was

only half of the corresponding value of the surface magnet structure. The problem with the

surface magnet structure is that the permanent magnets are facing a high reluctance tooth tip

structures in the air-gap region – the tips are saturated. If there were steel lamination on the

permanent magnet surface, this would improve the utilization of the permanent magnets but,

unfortunately, the stator inductance would grow immediately and the potential benefit would be

ineffectual.

123

Fig. 4.1. A surface magnet motor (10.3 kg magnet mass) and a cosine formed rotor surface with magnets

in V-position (12.5 kg magnet mass) at load.

The parameters of the prototype machine are shown in Table 4.1. Dimensions and winding

arrangements are shown in Appendix G.

Table 4.1. 45 kW prototype parameters

Magnet V-magnet

Slots-poles 12 - 10

Winding factor 0.933

Rated current (A) 97

Output power (kW) 45

Speed (rpm) 420

Efficiency (%) 91.5

Terminal voltage (V) 351

Winding turns per phase, Nph 104

Phase resistance (Ω) 0.114

Current density (A/mm2) 5.4

Air-gap (mm) 1.25

Mass of magnets (kg) 12.5

Air-gap flux density created by magnets (T) 0.935

The average length of the end winding of the prototype motor, shown in Fig. 4.2, was only 80

mm and the length in axial direction 41 mm. It was calculated that the end winding length

124

would be 120 mm, but, at the end, it became even shorter than expected. Thereby, the end

winding copper mass for the concentrated fractional slot q = 0.4 motor is 6 kg while for an

integer slot 10-poles motor with q = 1 the mass is estimated to be 25 kg. In this machine size the

copper in the end windings was about 70% less than for the corresponding conventional integer

slot (q = 1) windings.

Fig. 4.2. The end winding of the 12-slot-10-pole prototype motor with concentrated windings. The average

length of the end winding is 80 mm and the length in axial direction is 41 mm.

4.2. No-load test

A no-load test was carried out in the speed range of 50 to 600 rpm. The measurements were

performed at room temperature (rotor temperature about 20°C) and at rotor temperature about

100°C. The no-load losses at the rated (420 rpm) speed were 310 W and 374 W, as Fig. 4.3

illustrates. A no-load computation was carried out with the FEA and the iron losses at no-load

were 265 W.

41 mm

80 mm

125

50 100 150 200 250 300 350 400 450 500 550 6000

100

200

300

400

500

600

700

n [rpm]

Pno

load

[W]

Cold machineHot machine

Fig. 4.3. Measured no-load power as a function of speed. The measurements were performed at room

temperature (about 20°C) and at steady state (rotor temperature about 100°C).

The no-load voltage waveform has a shape that is similar to the FE computed voltage waveform

shown in Fig. 4.4. The measured no-load voltage RMS value was 200 V while the analytically

calculated value was 203 V. In the computations the 3rd, 5th, 7th and 9th harmonic components

appear.

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30

Time (ms)

Vol

tage

(V)

Measurement

Computation

Fig. 4.4. Measured and computed no-load voltage waveforms of the 45 kW prototype V-magnet motor.

T ≈ 20°C

T ≈ 100°C

126

4.3. Generator test

The 45 kW PM machine was loaded (resistive load) to achieve an output power 22.5 kW, which

is half of the rated power 45 kW. In this test a DC-motor was used to run the PM machine. The

temperature of the machine was measured with Pt-100 temperature sensors. The power

produced by the machine was measured with a Yokogawa PZ4000 power analyser and the

phase currents were measured with Strömberg Kore 05 current transformer (accuracy 0.5%).

A time stepping FEA computation was carried out to see, if the same values would be achieved

through computation. The motor winding was Wye connected. The speed was fixed to a rated

speed of 420 rpm. The waveforms of the phase voltage and the current at the end of the

measurement when the machine is at 22 kW load are given in Fig. 4.5.

0.14 0.15 0.16 0.17 0.18 0.19 0.2-250

-200

-150

-100

-50

0

50

100

150

200

250

t [s]

Uph

[V]

0.14 0.15 0.16 0.17 0.18 0.19 0.2

-80

-60

-40

-20

0

20

40

60

80

t [s]

I ph [A

]

Fig. 4.5. The waveforms of the phase voltages and currents at the end of the measurement at steady state

(rotor temperature about 100°C). The machine is at 22 kW load and at 420 rpm speed.

The computed 140 V voltage is less than the measured 143 V. This might be because the

magnet material used in the computations had a remanence flux density Br of 1.05 T. The finite

element analysis showed that the current, power and voltage values are similar to the

measurement results, as it is shown in Table 4.2. The efficiency of the generator in the

beginning of the test was 93.6% at 24.3 kW output power and at the end of the test 93.2% at

22.0 kW output power. With 2.8 Ω load resistance the input power at the shaft was 23.6 kW and

the output power was 22.0 kW, which caused 1.6 kW total losses. The copper losses at this

measurement are 0.9 kW, which means that the iron and the additional losses are about 0.7 kW.

127

Table 4.2 Generator resistive load test results compared to the FEA computations

Measured

at room temperature about 20°

Measured at rotor temperature

about 100° Computed

Current, I (A) 54 52 52 Voltage, U (V) 150 143 140 Power, P (kW) 24.3 22.0 22.0 Shaft torque, T (Nm) 555 510 504 Frequency, f (Hz) 35 35 35 Speed, n (rpm) 420 420 420 Load resistance (Ω) 2.8 2.8 2.7 Efficiency, η (%) 93.6 93.2 -

0

0.1

0.2

0.3

0.4

0.5

0.6

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Time (s)

Shaf

t tor

que

(p.u

.) ΨPM

ePM

us

Ψs

isqXd

Ψδ

q d

is

isdXq

Fig. 4.6. Torque as a function of time obtained from a FEA computation of the generator test with resistive

load, output power 22.0 kW. A space vector diagram for the generator test is drawn, assuming that Ris = 0.

4.3.1. Temperature rise test

The machine was driven as a generator and the load resistor was diminished until the load phase

resistance was 1.46 ohms, Wye connected. After adjusting the load phase resistance to 0.92 Ω

further temperature measurements were carried out. In this measurement the motor is a little hot

at the start situation. The results are shown in Table 4.3 and Fig. 4.7. Fig. 4.7 shows the

measured temperatures with a) a 63 A phase current (at the end of the measurement) and b)

128

with a 67 A phase current. At the end of the measurement with 0.92 Ω load resistance the

output power was 12.8 kW and the input power at the shaft was 15.1 kW, which caused 2.3 kW

total losses. The copper losses are approximately 1.5 kW, which means that the iron losses,

friction losses and additional losses are about 0.8 kW. The end windings reach an F-class

temperature of 150°C when phase current is 67 A. The machine is not able to cool down

enough without an external fan, because the rated phase current is 97 A. According to

measurements, 2.4 kW power losses can be removed from the machine without external blower

if ambient temperature is around 20°C. From the loss values it can be estimated that the

efficiency of the prototype motor at rated load is about 91.5%.

Table 4.3. Generator test results

Measured At start

Measured At end

Measured At end

Current, I (A) 68 63 67 Voltage, U (V) 99 92 64 Power, P (kW) 20.3 17.2 12.8 Frequency, f (Hz) 35 35 35 Speed, n (rpm) 420 420 420 Load resistance (Ω) 1.46 1.46 0.92

Efficiency, η (%) 90.4 89.2 85

0 60 120 180 240 3000

30

60

90

120

150

t (min)

T (°

C)

End winding at D-end

at center

Slot center, 50 mm from the D-end

Stator back, D-end

Frame center

0 60 120 1800

30

60

90

120

150

t(min)

T (°

C)

End winding at D-end

Stator back, D-end

Slot center, 50 mmfrom the D-end

Frame center

at center

a) b)

Fig. 4.7. The results of the heat load test the machine being used as a generator at 420 rpm speed.

Measurement results with a) a 63 A phase current at the end of the measurement and b) with a 67 A phase

current.

129

4.3.2. Vibration measurement

During the no-load measurements it was noticed that a mechanical resonance arises in the speed

range of 320…340 rpm. A vibration measurement was carried out by an acceleration probe at

300 rpm speed to estimate the frequency of the vibration. The probe is attached to the stator

yoke. Fig. 4.8 a) shows the signal of the acceleration at 300 rpm speed and b) presents the fast

Fourier transform spectrum of the signal showing the harmonic content of the signal. The

measurement time was selected to be 0.2 seconds, which corresponds to one whole cycle at

300 rpm speed. According to the measurement, the frequency of the signal is 10 times the rated

speed. This is exactly the tooth frequency of the machine. During one mechanical rotation, there

will be 10 poles passing one teeth. Thicker tooth tips may have decreased the noise effect.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-1.5

-1

-0.5

0

0.5

1

1.5

Acc

eler

atio

n (m

/s-2)

Time (s)5 10 15 20 25 30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Harmonic order number

Acc

eler

atio

n (m

/s-2)

a) b)

Fig. 4.8. The acceleration probe showing a) a signal during one revolution of the rotor and b) the fast

Fourier transform spectrum of the signal.

4.4. Cogging torque measurement

The torque ripple, especially at a low torque ripple level, is difficult to measure, because the

mechanics will produce their own resonances in the measurement. Fig 4.9 shows a FEA

computed and a measured cogging torque for the prototype motor at 420 rpm speed.

130

-0.002

-0.001

0.000

0.001

0.002

0.00 0.01 0.02Time (s)

Cog

ging

torq

ue (p

.u.)

FEA

Measurement

Fig 4.9 A FEA computed cogging torque and a measured cogging torque for the prototype motor at

420 rpm speed.

4.5. Measured values compared to the computed values

The torque was measured with different load angles at 45 kW power. Measurement is done at

rated stator flux linkage, at 240 rpm speed and the motor was supplied with DTC-inverter. The

measured torque as a function of the load angle curve is almost identical to the FEA result, as it

shown in Fig. 4.10. This means that also the other FE analysis might be reliable.

0

0.2

0.4

0.6

0.8

1

1.2

0 45 90 135 180Load angle (deg)

Torq

ue (p

.u.)

Measurement

FEA points

FEA

Fig. 4.10. Measured torque as a function of the load angle compared to the FEA computed torque (points).

The dashed line represents the developed torque according to the torque equation.

131

4.6. Comments and suggestions

The main parameters of the prototype motor were selected in an early phase of the research. It

may be stated that the performance of the prototype machine does not at all meet the

recommended criteria. According to the results of the work done, instead of the prototype

machine designed for this study it could be suggested e.g. a prototype with 24 slots and 20

poles. However, the prototype motor served well to verify the calculation methods developed

during the research. Some concerns came up, as the designing process of the prototype

fractional slot wound permanent magnet machine was started. In particular, the power of the

machine, as high as 45 kW, set the designer a demanding task. Fractional slot machines have a

high air-gap flux density harmonic content, which may lead to increased torque oscillations and

extra heating of the machine, compared to the rotating field machines. In the embedded

permanent magnet structure there may be high flux density harmonics in the rotor laminations

causing high iron losses. Despite of the sinusoidal terminal quantities – voltages and currents –

another cause of concern appeared to be the supplying converter; it had to be solved what kind

of a vector control could be used. Usually, the load angle of a permanent magnet motor at start

up should be exactly known, but, in the case of a fractional slot winding, the concept of the load

angle remains, to a certain degree, obscure. A calculatory load angle may be determined only by

using the terminal quantities; no accurate load angle by the measurements of the rotor position

may be done, since the load angle is an average value of all the different poles in the machine. It

was decided to use a direct torque controlled converter drive to obtain a 35 Hz supply

frequency. This proved to be a successful selection and no problems with the sensorless direct

torque control were met. The prototype machine worked exactly like a rotating field machine,

which is, of course, very encouraging, since the future operating field for this machine type is

considered to be that of an industrial motor.

Also the mechanical construction of the prototype motor should be improved. According to the

vibration measurements performed in the laboratory some noise occurred at speeds of 320 – 340

rpm and the noise frequency was analysed to coincide with the tooth frequency of the machine.

The tooth tips and also the teeth of the stator were analysed to be mechanically too weak and,

for this reason, they may have caused some noise effect. Also the stator stack fitting to the

stator frame was too loose, which made it possible for the whole stator stack to vibrate. It was

aimed for a machine with high torque per volume. Therefore, a high amount of copper

conductors were needed in the slots and the slots had to be large since the 12-slot-10-pole

132

solution was chosen. The cooling area of the few large slots is small compared to that of a

machine with a higher number of small size slots. The heat transfer properties of the machine

were thus also far from optimal.

The slot filling factor, however, became high due to the high-quality hand made winding. This

also allowed the end windings to become shorter than it was calculated. So, as a matter of fact,

enough space is left for the same frame to enclose a considerably longer stator and also a

considerably larger output torque. In the case of the prototype, the core length is the same that

may be used for an integer slot machine.

It was shown that the FEA results are similar to the computed values and both methods may

thus be used to analyse these machines. The loss values of the machine were in close correlation

to the computed values. As a high pull-out torque is required, a 12-slot-10-pole machine with

surface magnet structure should have been a far better alternative. The embedded V-magnet

structure is not at all as suitable for a fractional slot machine as it is for an integer slot machine.

The machine type also seems to need an exactly correct geometry. In the 12-slot-10-pole

machine the dimensions of the poles seemed to be too large and far better results would have

been reached by doubling the amount of poles and slots.

133

5. CONCLUSION

The results of this study offer new information on the performance characteristics of fractional

slot machines (with q < 1) and some guiding criteria for choosing the proper and suitable slot-

pole combination to be used for the application concerned. This study offers also criteria for the

selection of motor design variables.

The main objective of this work was to compare different pole and slot combinations applied to

a machine, which has a fixed air-gap diameter, a 225 frame size and a 45 kW output power. The

performance analysis was done for machines with concentrated windings, the coil of which is

around the tooth and with q is equal or less than 0.5. Different slot-pole (Qs - p) combinations

for fractional slot (q ≤ 0.5) motors were analysed to find out, which slot-pole combinations

have a high pull-out torque. Also the torque quality of machines producing a high pull-out

torque was studied. Therefore, the cogging torque and torque ripple were also analysed.

The winding factors of fractional slot machines were closely examined, because the winding

factor is usually an important parameter for the designing of a motor. It was, however,

discovered that, in the case of fractional slot machines, the fundamental winding factor does not

necessarily indicate the amount of pull-out torque. It was also noticed that some winding

arrangements have unwanted properties, which may be, e.g. when the number of slots is odd

and especially when Qs ≈ 2p, an unbalanced magnetic pull.

It was discovered that the method used to estimate the cogging of brushless DC machines, may

be appropriately applied to certain fractional slot PM motors. The cogging torques appears to

behave as expected, producing a curve with minima. It depends on the slot opening width where

the minima do appear. For each machine a factor kw can be calculated to estimate the minimum

points. The cogging torque values of the analysed fractional slot motor types can be less than

1% of the rated torque, in the case of multi-pole machines a cogging torque as low as 0.05%

could be estimated.

It was discovered, that when a low cogging torque is required, the least common multiplier

LCM appears to be a useful and also easily available parameter. The proper procedure to obtain

a low cogging torque and low torque ripple is suggested to be the selecting of a high value for

the LCM.

134

The effect of the slot opening was studied closer with machines under load, as the torque ripple

is under load usually higher than at no-load. A comparing study is done for 13 fractional slot

machine constructions of which the relative magnet widths and the relative slot opening widths

were varied. The lowest ripple values (% of the rated torque, peak-to-peak values) obtained for

different machines are presented as a function of the relative magnet width. Ripple values even

less than 0.5% were achieved.

For machines with semi-closed slots there seems to be several relative magnet widths that,

when used, make it possible to achieve a minimum for the torque ripple. It may be stated that

the value 0.75 for the relative magnet width is to be recommended if the stator includes open

slots and a fractional slotted structure. Machines with open slots and Qs ≈ 2p (q close to 0.33)

produce torque ripples that are small compared to the other q values. These are, unfortunately,

also the machines, which suffer from the unbalanced magnetic pull effect. In most of the

analysed cases, the torque ripple average level remains lower when the machine has open slots

than when it has semi-closed slots.

According to the results obtained for a particular slot number, the peak-to-peak value of the

torque ripple grows as the number of poles increases. The results given can be used to estimate

the ripple of the motor.

When comparing the pull-out torque of the motors belonging to the same frame size category,

some differences are to be mentioned. The highest torque obtained is 2.1 p.u. and the lowest

1.0 p.u. The results show that for a certain slot number the highest torque is usually achieved

then when the machine has a small pole number. High pull-out torques of 2.1 p.u. and 2.0 p.u.

were obtained with the 18-slot-12-pole and 24-slot-16-pole machines for both of which q is

equal to 0.5.

Increasing the pole number and keeping the slot number constant reduces the developed pull-

out torque in most of the analysed cases, as the magnet material and the machine size (and air-

gap diameter) were kept practically constant. Further increasing of the slot number and keeping

the pole number constant increases the developed pull-out torque.

135

The performance of the surface magnet motor was compared to the embedded magnet motor.

When the slot and pole numbers are low, the surface magnet structure produces higher pull-out

torques than the corresponding embedded magnet motor of the same frame size. This is due to

the high armature reaction effect occurring in the embedded magnet machine. When the slot

and pole numbers are high, the pull-out torque may be similar for both the surface and

embedded structures.

To verify the computations a 45 kW prototype motor, being a 12-slot-10-pole embedded

magnet machine, was manufactured. It was shown that the values for the pull-out torque and

losses obtained with the FEA are similar to the computed values.

136

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140

APPENDIX A Winding arrangements

Winding arrangements of some fractional slot machines, two-layer windings:

12-slot-8-pole 21

34212

2=

⋅⋅===

pmQ

nzq s

1 2 3 4 5 6C- A- B- C- A- B- upper layerA+ B+ C+ A+ B+ C+ lower layer

12-slot-10-pole 52

35212

2=

⋅⋅===

pmQ

nzq s

1 2 3 4 5 6 7 8 9 10 11 12A+ A- B- B+ C+ C- A- A+ B+ B- C- C+ upper layerA+ B+ B- C- C+ A+ A- B- B+ C+ C- A- lower layer

12-slot-14-pole 72

37212

2=

⋅⋅===

pmQ

nzq s

1 2 3 4 5 6 7 8 9 10 11 12A+ A- C- C+ B+ B- A- A+ C+ C- B- B+ upper layerB- A- A+ C+ C- B- B+ A+ A- C- C+ B+ lower layer

12-slot-16-pole 41

38212

2=

⋅⋅===

pmQ

nzq s

1 2 3 4 5 6B- A- C- B- A- C- upper layerA+ C+ B+ A+ C+ B+ lower layer

18-slot-14-pole 73

37218

2=

⋅⋅===

pmQ

nzq s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18A+ A- B- C- C+ A+ B+ B- C- A- A+ B+ C+ C- A- B- B+ C+ upper layerA+ B+ C+ C- A- B- B+ C+ A+ A- B- C- C+ A+ B+ B- C- A- lower layer

24-slot-22-pole 114

311224

2=

⋅⋅===

pmQ

nzq s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24A+ A- B- B+ B- B+ C+ C- C+ C- A- A+ A- A+ B+ B- B+ B- C- C+ C- C+ A+ A-A+ B+ B- B+ B- C- C+ C- C+ A+ A- A+ A- B- B+ B- B+ C+ C- C+ C- A- A+ A-

24-slot-26-pole 134

313224

2=

⋅⋅===

pmQ

nzq s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24B- A- A+ A- A+ C+ C- C+ C- B- B+ B- B+ A+ A- A+ A- C- C+ C- C+ B+ B- B+A+ A- A+ A- C- C+ C- C+ B+ B- B+ B- A- A+ A- A+ C+ C- C+ C- B- B+ B- B+

141

APPENDIX B Periodical behaviour of harmonics

The winding factors can be organized in tables or series according to their order numbers. 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. An example of the periodical

behaviour is given in table B.1 for a winding with z = 9 and n = 1, 2, 4 and 5.

Table B.1. Slot harmonics for windings q = 9, 9/2, 9/4 and 9/5. (Tüxen, 1941)

n =1, q = 9 n =2, q = 9/2 n =4, q = 9/4 n =5, q = 9/5

1st slot harmonics ν =2⋅3⋅9⋅-1+1=-53 ν =2⋅3⋅9⋅1+1=55

1st slot harmonics ν =2⋅3⋅9/2⋅-1+1=-26 ν =2⋅3⋅9/2⋅1+1=28

1st slot harmonics ν =-25/2 and 29/2

1st slot harmonics ν = -49/5 and 59/5

ν =2mqg+1 2nd slot harmonics ν =2⋅3⋅9/2⋅-2+1=53 ν =2⋅3⋅9/2⋅2+1=55

2nd slot harmonics ν =-26 and 28

2nd slot harmonics ν =-103/5 and 113/5

3rd slot harmonics ν = -79/2 and 83/2

3rd slot harmonics ν =-157/5 and 167/5

4th slot harmonics ν = -53 and 55

4th slot harmonics ν =-211/5 and 221/5

5th slot harmonics ν =-53 and 55

From Table B.1 it can be noticed that the slot harmonics are periodical. The 1st slot harmonic

pair (-53 and 55) of an integer slot winding q = 9 will be found also from the fractional slot

winding, but now the order number is changed to the nth slot harmonic pair. In a fractional slot

winding, there appear also (n-1) fractional slot harmonic pairs between the slot harmonic pairs

created by the fundamental wave. Table B.2 gives the harmonic order numbers up to 79 and

their winding factors for a winding with q = 9/4. The waves, shown in the same row, have the

same winding factors. The first row gives the fundamental and the slot harmonic waves with the

same winding factor ξ = 0.958.

From all harmonic order numbers and their winding factors of windings with q = 9, 9/2, 9/4 and

9/5 a diagram was drawn, shown in Fig. B.1. According to the staples shown in Fig. B.1, there

are series of harmonic groups developed by the fractional slot windings. For a certain z, there

will be harmonic groups depending on the value of n (in this example z = 9 and n = 1, 2 and 4).

142

Table B.2. Harmonic waves and their winding factors ξν for a q = 9/4 winding (Tüxen, 1941) ν ξ ν

-1 -25/2 29/2 -26 28 -79/2 83/2 -53 55 0.958 (3) (21/2) (33/2) (24) (30) (75/2) (87/2) (51) (57) 0.638 -5 17/2 -37/2 22 -32 71/2 -91/2 49 -59 0.193 7 -13/2 41/2 -20 34 -67/2 95/2 -47 61 -0.140

(9) (9/2) (45/2) (18) (36) (63/2) (99/2) (45) (63) 0.222 -11 5/2 -49/2 16 -38 59/2 -103/2 43 -65 -0.093 -13 -1/2 53/2 -14 40 -55/2 107/2 -41 67 0.081 (15) (3/2) (57/2) (12) (42) (51/2) (111/2) (39) (69) 0.145 -17 7/2 -61/2 10 -44 47/2 -115/2 37 -71 0.066 -19 21/2 65/2 -8 46 -43/2 119/2 -35 73 -0.062 (21) (15/2) (69/2) (6) (48) (39/2) (123/2) (33) (75) 0.118 -23 -19/2 -73/2 4 -50 35/2 -127/2 31 -77 -0.057 -25 23/2 77/2 -2 52 -31/2 131/2 -29 79 0.056

(27/2) (27) (81/2) (54) 0.111

0.0

0.5

1.0

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73

0.0

0.5

1.0

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73

0.0

0.5

1.0

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73

odd

even

fractional

harmonic order number

ξν

q = 9

q = 92

q = 94

integer, even ν = 2, 4, 6, 8, ...integer, odd ν = 1, 3, 5, 7, 11 ...

fractional ν = 1/2, 3/2, 5/2, ...

odd

ξν

ξν

Fig. B.1. Winding factors of windings q = 9, 9/2 and 9/4. In case of q = 9 there exist harmonic order

numbers which are all integer ν = 1, 5, 7, 11, 13 … shown as white bars. In case of q = 9/2 there exist also

harmonic order numbers which are even integer numbers ν = 2, 4, 8, 10, … shown as grey bars. In case of

q = 9/4 there exist also fractional harmonic order numbers ν = 1/2, 5/2, 7/2, … shown as black bars.

143

APPENDIX C Winding factors

Harmonics ν and their winding factors ξν for fractional slot concentrated windings

Qs – 2p 12 – 8 q = 1/2

Qs – 2p 12 – 10 q = 2/5

Qs – 2p 12 – 14 q = 2/7

Qs – 2p 12 – 16 q = 1/4

ν ξν ν ξν ν ξν ν ξν 1 2 4 5 7 8

10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44 46 47 49 50 52 53 55 56 58 59 61 62 64 65 67 68 70 71 73 74 76

0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866

0.2 1

1.4 2.2 2.6 3.4 3.8 4.6 5

5.8 6.2 7

7.4 8.2 8.6 9.4 9.8

10.6 11

11.8 12.2 13

13.4 14.2 14.6 15.4 15.8 16.6 17

17.8 18.2 19

19.4 20.2 20.6 21.4 21.8 22.6 23

23.8 24.2 25

25.4 26.2 26.6 27.4 27.8 28.6 29

29.8 30.2

0.067 0.933 0.933 0.067 0.067 0.933 0.933 0.067 0.067 0.933 0.933 0.067 0.067 0.933 0.933 0.067 0.067 0.933 0.933 0.067 0.067 0.933 0.933 0.067 0.067 0.933 0.933 0.067 0.067 0.933 0.933 0.067 0.067 0.933 0.933 0.067 0.067 0.933 0.933 0.067 0.067 0.933 0.933 0.067 0.067 0.933 0.933 0.067 0.067 0.933 0.933

0.14 0.29 0.43 0.57 0.71 0.86

1 1.14 1.29 1.43 1.57 1.71 1.86

2 2.14 2.29 2.43 2.57 2.71 2.86

3 3.14 3.29 3.43 3.57 3.71 3.86

4 4.14 4.29 4.43 4.57 4.71 4.86

5 5.14 5.29 5.43 5.57 5.71 5.86 6.00 6.14 6.29 6.43 6.57 6.71 6.86

7 7.14 7.29

0.25 0.5

0.707 0.866 0.933

0 0.933 0.866 0.707 0.5

0.25 0

0.25 0.5

0.707 0.866 0.933

0 0.933 0.866 0.707 0.5

0.25 0

0.25 0.5

0.707 0.866 0.933

0 0.933 0.866 0.707 0.5

0.25 0

0.25 0.5

0.707 0.866 0.933

0 0.933 0.866 0.707 0.5

0.25 0

0.25 0.5

0.707

0.5 1 2

2.5 3.5 4 5

5.5 6.5 7 8

8.5 9.5 10 11

11.5 12.5 13 14

14.5 15.5 16 17

17.5 18.5 19 20

20.5 21.5 22 23

23.5 24.5 25 26

26.5 27.5 28 29

29.5 30.5 31 32

32.5 33.5 34 35

35.5 36.5 37 38

0.5 0.866 0.866 0.5 0.5

0.866 0.866 0.5 0.5

0.866 0.866 0.5 0.5

0.866 0.866 0.5 0.5

0.866 0.866 0.5 0.5

0.866 0.866 0.5 0.5

0.866 0.866 0.5 0.5

0.866 0.866 0.5 0.5

0.866 0.866 0.5 0.5

0.866 0.866 0.5 0.5

0.866 0.866 0.5 0.5

0.866 0.866 0.5 0.5

0.866 0.866

144

Harmonics ν and their winding factors ξν for fractional slot concentrated windings presented in

stables diagram.

0.0

0.2

0.4

0.6

0.8

1.0

1 7 13 19 25 31 37 43 49 55 61 67 73

harmonic order number

win

ding

fact

or

12-slot-8-pole

0.0

0.2

0.4

0.6

0.8

1.0

0.2 2.6 5.0 7.4 9.8 12.2 14.6 17.0 19.4 21.8 24.2 26.6 29.0

harmonic order number

win

ding

fact

or

12-slot-10-pole

0.0

0.2

0.4

0.6

0.8

1.0

0.14 0.71 1.29 1.86 2.43 3.00 3.57 4.14 4.71 5.29 5.86 6.43 7.00

harmonic order number

win

ding

fact

or

12-slot-14-pole

0.0

0.2

0.4

0.6

0.8

1.0

0.5 4 8 11.5 15.5 19 23 26.5 30.5 34 38

harmonic order number

win

ding

fact

or

12-slot-16-pole

145

APPENDIX D Calculation example of inductances

Calculation example of inductances for a 24-slot-22-pole fractional slot wound motor

The slot dimensions are:

Slot width x1 = 25 mm,

Slot opening width x4 = 3 mm,

Slot height y1 = 32 mm,

y2 = 0.5 mm, y3 = 0.5 mm, y4 = 0.5 mm, y5 = 0.5 mm.

(Thickness of insulation material ≈ 0.5 mm)

Air-gap length (radial) δ = 1.25 mm.

xk = x1 - x4 = 22 mm Fig. D1. Slot dimensions

Physical length of the stator core, L = 270 mm

Equivalent air-gap δeff = (δ + hm)kC. Magnet height hm is 7.43 mm and Carter’s factor kC is

1.032.

The magnetizing reactance Xmd is based on Eq. (2.25) in page 50.

( )2ph1i

effs

s0md

πµ4 NL

mQ

fmX s ξδ

τ

= . (D.1)

[ ] [ ][ ]

[ ]( ) [ ]Ω=⋅⋅⋅

⋅

⋅⋅= 42.02104949.0m27.0

m108.96π324

m033.0Hz73.333AmVs7-π10434

3-

Leakage reactance:

The leakage inductance factor λz is defined using Fig. 2.11 in page 57. Because x4/δ is 2.4, λz

gets the value of 0.05. For a 24-slot 22-pole machine τs/τp ≈ 1 and the factors k1 = 1 and k2 = 1

were selected (shown in Fig. 2.11). Leakage factor λns according to Eq. (2.49), in page 57, may

be defined as

1

5z

4

4

k

3

1

22

1

11ns 43 x

yxy

xy

xy

kxy

k +

++++= λλ (D.2)

025.040005.005.0

003.00005.0

023.00005.0

026.00005.01

026.03032.01

⋅+

++++

⋅⋅= = 0.691.

x4

slot pitch, τs

x1

y1

y12

y11

y3 y4

y5

y2

146

Slot reactance of both layers is computed as (based on Eq. (2.42))

ns2phi0s

sn µ242 λπ NLf

QmX = (D.3)

[ ] [ ] [ ]Ω=⋅⋅⋅

⋅⋅⋅⋅

⋅= − 169.1691.0104m27.0

AmVs10π4Hz73.3332π

24342 27

End winding reactance is computed as (based on Eq. (2.43) in page 55)

s2phi0s

sw µπ24 λNLfq

QmX = . (D.4)

[ ] [ ] 029.0138.0m033.0m518.0m023.022 -1-1wbebs =⋅+⋅⋅=+= λλλ hb (D.5)

Factor hb is the height and bb is the width of the end winding. There are several methods

available to estimate the values for the reactance factors for the end windings λe and λw, as e.g.

given by Richter (1963) and Jokinen (1973). It was used the reactance factors λe = 0.518 [m-1]

and λw = 0.138 [m-1], which are defined for synchronous machines by Richter (1963).

[ ] [ ] [ ]Ω=⋅⋅⋅

⋅⋅⋅⋅⋅

⋅= − 033.0029.0104m27.0

AmVs10π4Hz73.3332π3636.0

2434 27

wX (D.6)

The air-gap leakage reactance (based on Eq. (2.27) in page 52) is calculated as

δph

iδ0

δ

2

πµ σ

δω ⋅

⋅⋅=

pN

LDmX (D.7)

[ ]( ) [ ] [ ] [ ] [ ]Ω=⋅

⋅⋅⋅

= 027.001.0

2

11104m27.0m254.0

m00125.03Hz73.3332π

AmVs

π

-7π104 ,

where the leakage factor δσ was computed using Eq. (2.28).

The leakage reactance is

Ω=++= 24.1wδns XXXX σ . (D.8)

The synchronous reactance is (in d-direction)

Ω=+= 7.1sσmdd XXX . (D.9)

The synchronous inductance is (in d-direction)

[ ] [ ] H0037.0)Hz33.73π2/(7.1d =⋅Ω=L . (D.10)

147

APPENDIX E B/H-curves for Neorem 495a

B/H-curves for Neorem 495a magnet material

Br 1.1 T 11.0 kG

Coercivity

BH0 830 kA/m 10.5 kOe

JH0 2400 kA/m 30.2 kOe

(BH)max 230 kJ/m3 29 MGOe

Nominal values at 20°C

148

APPENDIX F Torque ripples results from FEA

Torque ripples (% of average) peak-to-peak values of fractional motors from current driven FEA model are presented as a function of relative magnet width. Semi-closed slot openings and open slot structures were studied.

0

10

20

30

0.5 0.6 0.7 0.8 0.9 1.0

Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

Semi-closed

Open slot

0

10

20

30

0.5 0.6 0.7 0.8 0.9 1.0

Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

Semi-closed

Open slot

Fig. F.1. Torque ripples (% of average) peak-to-peak values of 24-slot-16-pole (q = 0.5) motor. Semi-closed slot openings of 0.09 and open slot structure were studied.

Fig. F.2. Torque ripples (% of average) peak-to-peak values of 36-slot-24-pole (q = 0.5) motor. Semi-closed slot openings of 0.09 and open slot structure were studied.

0

2

4

6

8

10

0.5 0.6 0.7 0.8 0.9 1.0

Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

Semi-closed 0.07

Semi-closed 0.14

Open slot

0

2

4

6

8

10

0.5 0.6 0.7 0.8 0.9 1.0

Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

Semi-closed 0.3

Semi-closed 0.08

Open slot

Fig. F.3. Torque ripples (% of average) peak-to-peak values of 18-slot-14-pole (q = 0.429) motor. Semi-closed slot openings of 0.07 and 0.14, and open slot structure of 0.43 were studied.

Fig. F.4. Torque ripples (% of average) peak-to-peak values of 12-slot-10-pole (q = 0.4) motor. Semi-closed slot openings of 0.08 and 0.3, and open slot structure were studied.

149

Torque ripples (% of average) peak-to-peak values of fractional motors from current driven FEA model are presented as a function of relative magnet width. Semi-closed slot openings and open slot structures were studied.

0

1

2

3

4

0.5 0.6 0.7 0.8 0.9 1.0Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

Semi-closed

Open slots

0

1

2

3

4

0.5 0.6 0.7 0.8 0.9 1.0Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

Semi-closed

Open slots

Fig. F.5. Torque ripples (% of average) peak-to-peak values of 36-slot-30-pole (q = 0.4). Semi-closed slot openings of 0.09 and open slot structure were studied.

Fig. F.6. Torque ripples (% of average) peak-to-peak values of 36-slot-42-pole (q = 0.286). Semi-closed slot openings of 0.09 and open slot structure were studied.

0

2

4

6

8

10

0.5 0.6 0.7 0.8 0.9 1.0

Relative magnet width

Torq

ue ri

pple

% o

f ave

rage

Semi-closed 0.09

Semi-closed 0.3

Open slot

Fig. F.7. Torque ripples (% of average) peak-to-peak values of 24-slot-28-pole (q = 0.286) motor. Semi-closed slot openings of 0.09 and open slot structure were studied.

150

APPENDIX G Prototype motor data

25

45.32

6.6

50.4 38

57.5

Fig. G.1. 12-slot-10-pole prototype motor dimensions [mm]

A+

A-A-

A+

C-

C+

C+

C-

B+

B-B-

B+A-

A+A+

A-C+

C-

C-

C+

B-

B+B+

B-

Fig. G.2. 12-slot-10-pole prototype motor wiring diagram

Stator outer diameter 364 mm Air-gap diameter, Dδ = 249 mm δ = 1.2 mm

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