Axial Flux Permanent Magnet Brushless Machines

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Axial Flux PermanentMagnet BrushlessMachines by JACEK F. GIERAS United Technologies Research Cente4 East Harford, Connecticut, U.S.A.

RONG-JIEWANG University of Stellenbosch, Stellenbosch,Western Cape, South Africa and

MAARTEN J. KAMPER Univ ersity of Stellenbo sch, Stellenbosch,Westem Cape, South Africa

ww W&

KLUWERACADEMICPUBLISHERS DORDRECHT/ BOSTON/ LONDON

Contents

Preface

xi

1 . INTRODUCTION

I

1.1 Scope

1

I.2

Features

I

I.3

Development of AFPM machines

a J

I.4

Types of axial flux PM machines

4

1.5 Topologiesand geometries 1.6 Axial magneticfield excitedby PMs I.1

PM eddy-currentbrake as the simplest AFPM brushlessmachine

6 10 t3

1.8 AFPM machinesversusRFPM machines

I6

I.9

t9

Power limitation of AFPM machines

Numerical examples

2 . PRINCIPLES OF AFPM MACHINES 2.I

2.2

Magnetic circuits 2.I.1 Single-sidedmachines 2.I.2 Double-sidedmachineswith internalPM disc rotor 2.1.3 Double-sidedmachineswith internalring-shaped core stator 2.I.4 Double-sided machineswith internal slotted stator 2.1.5 Double-sidedmachineswith internal corelessstator 2.I.6 Multidiscmachines Windings 2.2.1 Three-phasewindings distributed in slots 2.2.2 Drum-type winding

I9 27 27 27 27 29 3l )L 3Z aa JJ aa JJ

35

MACHINES AXIAL FLUX PERMANENTMAGNETBRUSHLESS

Vi 2.2.3 2.2.4

Corelessstatorwinding Salientpole windings

2.3

Torque production

2.4

Magnetic flux

2.5

Electromagnetictorque and EMF

35 37 37 39 40

2.1 Phasordiagrams

42 42 44 45 45 4l 48 49 50 50 51

2.8 Sizing equations

54

2.9 Armature reaction

57 61 6l 62 65

2.6 Lossesand efficiency 2.6.1 Statorwinding losses 2.6.2 Statorcore losses 2.6.3 Core loss finite elementmodel 2.6.4 Lossesin permanentmagnets 2.6.5 Rotor core losses 2.6.6 Eddy currentlossesin statorconductors 2.6.7 Rotationallosses 2.6.8 Lossesfor nonsinusoidalcurrent 2.6.9 Efficiency

2 . r 0AFPM

motor 2.10.1 Sine-wavemotor 2.10.2 Square-wavemotor

2 . l l AFPM synchronousgenerator of a standalone 2.IIl Performancecharacteristics generator 2.II.2 Synchronizationwith utility grid Numerical examples 3. MATERIALS AND FABRICATION 3.1 Statorcores 3.1.1 Nonorientedelectricalsteels 3.1.2 Amorphousferromagneticalloys 3.1.3 Soft magneticpowder composites 3.L4 Fabricationof statorcores 3.2

Rotor magneticcircuits 3.2.1 PM materials 3.2.2 Characteristicsof PM materials 3.2.3 Operatingdiagram 3.2.4 Permeancesfor main and leakagefluxes

65 66 68 79 79 79 83 84 87 90 90 95 99 103

Contents

vll

3.2.5 3.2.6 3.3

Calculationof magneticcircuits with PMs Fabrication of rotor masnetic circuits

Windings 3.3.I Conductors 3.3.2 Fabricationof slottedwindings 3.3.3 Fabricationof corelesswindines

Numerical examples 4. AFPM MACHINES WITH IRON CORES

107 109 It2 l12 I12 tt4 116 125

4.I

Geometries

I25

4.2

Commercial AFPM machineswith stator ferromagnetic cores

126

4.3

Some featuresof iron-cored AFPM machines

127

4.4

Magnetic flux density distribution in the air gap

I28

4.5

Calculationof reactances 4.5.I Synchronousand armaturereactionreactances 4.5.2 Stator leakagereactance

130 130 I3I

4.6

Performancecharacteristics

I34

4.7

Performancecalculation 4.7.I Sine-waveAFPM machine 4.7.2 Synchronousgenerator 4.7.3 Square-waveAFPM machine

136 136 138 l4I

4.8

Finite elementcalculations

I4l

Numerical examples 5. AFPM MACHINES WITHOUT STATOR CORES 5.1

Advantagesand disadvantages

5.2

Commercial corelessstator AFPM machines

5.3

Performancecalculation 5.3.1 Steady-stateperformance 5.3.2 Dynamicperformance

5.4

Calculation of corelesswinding inductances 5.4.1 Classicalapproach 5.4.2 FEM approach

5.5


5.6

Eddy current lossesin the stator windings 5.6.1 Eddy currentloss resistance 5-6.2 Reductionof eddy currentlosses 5.6.3 Reductionof circulatins currentlosses

144

153

r53 r53 155 155

r57 r59 159 160 162

r63 r63 r67 168

AXIAL FLUX PERMANENTMAGNETBRUSHLESS MACHINES

vllt

5.6.4

Measurementof eddy currentlosses

170

5.7

Armature Reaction

n0

5.8

Mechanicaldesignfeatures 5.8.1 Mechanicalstrengthanalysis 5.8.2 Imbalanced axial force on the stator

173 174 t77

5.9

Thermal problems

179

Numerical examples 6. AFPM MACHINES WITHOUT STA|OR AND ROTOR CORES

n9 189

6.I


189

6.2

Topology and construction

189

6.3

Air gap magnetic flux density

I92

6.4


I93

6.5

Commercial corelessAFPM motors

I94

6.6

Casestudy: low-speedAFPM corelessbrushlessmotor 6.6.1 Performancecharacteristics 6.6.2 Cost analysis 6.6.3 Comparison with cylindrical motor with laminated stator and rotor cores

197 197 198

6.7

Casestudy: low-speedcorelessAFPM brushlessgenerator

200

6.8

Characteristicsof corelessAFPM machines

20I

Numerical examples

199

204

7 .I

Control of trapezoidalAFPM machine 7.1.I Voltageequations 7.1.2 Solid-stateconverter 7.I.3 Currentcontrol 7.1.4 Speedcontrol 7.1.5 High speedoperation

7.2

Control of sinusoidal AFPM machine 7.2.I Mathematical model and dq equivalent circuits 7.2.2 Currentcontrol 1.2.3 Speedcontrol 7.2.4 Hardwareof sinusoidalAFPM machinedrive

2t3 2t3 214 216 219 222 222 223 224 229 230 234

7.3

Sensorlessposition control

LJI

7. CONTROL

Numerical examples

âa

239

IX

Contents

8 . COOLING AND HEAT TRANSFER 8.1

Importance of thermal analysis

8.2

Heat transfermodes 8.2.1 Conduction 8.2.2 Radiation 8.2.3 Convection

8.3

Cooling of AFPM machines 8.3.1 AFPM machineswith self-ventilation 8.3.2 AFPM machineswith external ventilation

8.4

Lumped parameterthermal model 8.4.1 Thermal equivalentcircuit 8.4.2 Conservationof energy

8.5

Machine duties 8.5.1 Continuousduty 8.5.2 Short-timeduty 8.5.3 Intermittentdutv

Numerical examples

9 . APPLICATIONS 9.1

Power generation 9.I.1 High speedgenerators 9.I.2 Low speedgenerators

9.2

Electric vehicles 9.2.1 Hybrid electricvehicles 9.2.2 Battery electric vehicles

9.3

Ship propulsion 9.3.1 Large AFPM motors 9.3.2 Propulsion of unmannedsubmarines 9.3.3 Counterrotatingrotor marine propulsion system

9.4

Elecffomagnetic aircraft launch system

9.5

Mobile drill rigs

9.6

Elevators

9.7

Miniature AFPM brushlessmotors

9.8

Vibration motors

9.9

Computer hard disc drives

Numerical examples

249 249 249 250 250

25r 255 255 264 267 268 269 270 270 27r 272 272 281 281 28r 282 285 287 289 291 291 292 292 295 297 299 302 304 306 307

AXIAL FLUX PERMANENTMAGNETBRUSHLESS MACHINES Symbols and Abbreviations

3II

References

32r

Index

JJI

Preface

The drop in prices of rare-earthpermanentmagnet (PM) materials and progressin power electronicshave played an important role in the developmentof PM brushlessmachinesin the last threedecades.Thesemachineshaverecently becomemature and their high efficiency, power density and reliability has led to PM brushlessmachines successfully replacing d.c. commutator machines and cage induction machinesin many areas. The axial flux PM (AFPM) brushlessmachine,also called the disc-type machine, is an attractivealternativeto its cylindrical radial flux counterpartdue to the pancakeshape,compact construction and high torque density. AFPM motors are particulady suitable for electrical vehicles, pumps, valve control, centrifuges, fans, machine tools, hoists, robots and manufacturing. They have become widely used for low-torque servo and speedcontrol systems.The application of AFPM machinesas generatorsis justified in wind turbines, portable generator sets and road vehicles. The power range of AFPM brushlessmachines is now from a fraction of a watt to sub-MW. Disc-type rotors can be embeddedin power-transmissioncomponents or flywheels to optimize the volume, mass, number of parts, power transfer and assemblytime. For electric vehicles with builfin wheel motors the payoff is a simpler power train, higher effrciency and lower cost. Dual-function rotors may also appear in pumps, elevators, energy storagesand other machinery, bringing addedvalues and new levels ofperfonnance to theseproducts. The authorsbelieve that this first book in English devotedentirely to AFPM brushlessmachineswill serveas a textbook, useful referenceand designhandbook of AFPM machinesand will stimulate innovationsin this field. J.F. Grpnas, R. WRIrtcAND M.J. Kavppn

Chapter1 INTRODUCTION

1.1

Scope

The term axialfl.ux permanent magnel (AFPM) machine in this book relates only to permanentmagnet (PM) machineswith disc type rotors. Other AFPM machine topologies, e.g. transverseflux machines,have not been considered. In principle, the electromagneticdesign of AFPM machines is similar to its radial flux PM (RFPM) counterpartswith cylindrical rotors. However,the mechanical design,thermal analysisand assemblyprocessare more complex.

1,2

Features

The AFPM machine, also called the disc-type machine, is an attractive a1ternative to the cylindrical RFPM machine due to its pancakeshape,compact construction and high power density. AFPM motors are particularly suitable for electrical vehicles, pumps, fans, valve control, centrifuges,machine tools, robots and industrial equipment. The large diameter rotor with its high moment of inertia can be utilised as a flywheel. AFPM machinescan also operate as small to medium power generators. Since a large number of poles can be accommodated,these machines are ideal for low speed applications, as for example,electromechanicaltraction drives, hoists or wind generators. The unique disc-type profile of the rotor and stator of AFPM machines makes it possibleto generatediverse and interchangeabledesigns.AFPM machines can be designedas single air gap or multiple air gaps machines,with slotted, slotlessor eventotally ironless armature.Low power AFPM machines are frequently designedas machineswith slotlesswindings and surfacePMs. As the output power of the AFPM machines increases,the contact surface between the rotor and the shaft in proportion to the power becomes smaller.


Table 1.1. Specifications of double-sided disc type AFPM brushless servo motors manufactured by E Bautz GmbH, Weiterstadt,Germany

Quantity Rated power, W Rated torque, Nm Maximum torque, Nm Standstill torque, Nm Rated current, A Maximum current, A Standstill current. A Rated speed,rpm Maximum speed,rpm Armature constant, V/l000 rpm Torque constant, Nm/A Resistance,Q Inductance,mH Moment of inertia, kgm2 x 10-3 Mass,kg Diameter of frame, mm Length of frame, mm Power density, Wkg Torque density, Nmlkg

S632D

S634D

S712F

5714F

S8O2F

S8O4F

680 1.3

940 t.8

910 2.9

r260 4.0

18s0 5.9

2670 8.5

7

o

I4

t8

28

40

t.7 4.0 2l 5.3 5000

2.3 4.9 25 6.3 5000

3.5 4.9 24 5.9 3000

4.7 6.6 30 7.8 3000

7.0 9.9 47 1t.7 3000

10.0 I 1.9 56 14.0 3000

6000

6000

6000

6000

6000

6000

23

25

42

42

50

0.35 2.5

0.39 t.8 2.8

0.64 2.4 5.4

0.64 1.5 4.2

0.64 0.76 3.0

0.77 0.62 3.0

0.08 4.5

0.12 5.0

0.21 6.2

0.3 o.o

0.6 9.7

1.0 10.5

150

150

174

t74

210

210

82

82

89

89

103

103

-) -L

â

151.1

188.0

146.8

190.9

t90.7

254.3

0.289

0.16

0.468

0.606

0.608

0.809

Careful attentionmust be given to the design of the rotor-shaftmechanical joint as this is usuallythe causeof failuresof disc type machines. In some cases,rotors are embeddedin power-transmissioncomponentsto optimisethe numberof parts,volume,mass,powertransferand assemblytime. For electric vehicles (EVs) with built-in wheel motors the payoff is a simpler electromechanicaldrive system,higher efficiency and lower cost. Dualfunction rotors may also appear in pumps, elevators, fans and other types of machinery bringing new levelsof performanceto theseproducts.

Introduction

3

Most applicationsuse the AFPM machine as a d.c. brushlessmotor' Encoders,resolversor other rotor position sensofsare thus a vital part of brushless disc motors. Table 1.1 showsspecificationsof AFPM brushlessservomotorsratedup to 2.7 kW, manufacturedby E. Bautz GmbH, Weiterstadt,Germany.

1.3

Developmentof AFPM machines

The history of electrical machines reveals that the earliest machines were axial flux machines(M. Faraday,1831,anonymousinventorwith initials P.M., 1832.W. Ritchie, 1833,B. Jacobi,1834).However,shortly after T. Davenport (1837) claimedthe first patent [66] for a radial flux machine,conventionaltadial flux machineshave been widely acceptedas the mainstreamconfiguration for electricalmachines[30, 49]. The first primitive working prototype of an axial flux machine ever recorded was M. Faraday'sdisc (1831) - seeNumerical Example l.l. The disc type constructionof electricalmachinesalsoappearsin N. Tesla'spatents,e.g. U.S. patent No. 405 858 12251entitledElectro-Magnetic Motor and published in 1889 (Fig. l.l). The reasonsfor shelvingthe axial flux machinewere multifold and mav be summarisedas follows: strongaxial (normal) magneticattractionforce betweenthe statorand rotor; fabrication difficulties, such as cutting slots in laminated cores and other methodsof making slotted stator cores; high costsinvolved in manufacturingthe laminated stator cores; difficulties in assemblingthe machine and keeping the uniform aft gapAlthough, the first PM excitation systemwas applied to electrical machinesas early as the 1830s,the poor quality of hard magnetic materials soon discouragedtheir use. The invention of Alnico in I 93 1, barium ferrite in the 1950s and especiallythe rare-earthneodymium-iron-boron(NdFeB) material (announced in 1983)havemadea comebackof the PM excitationsystempossible. It is generally believed that the availability of high energy PM materials (especially rare earth PMs) is the main driving force for exploitation of novel PM machine topologies and has thus revived AFPM machines. Prices of rareearth PMs have been following a descendingcurve in the last decade of the 20th century with a sharp decline in the last three years. A recent market survey shows that the NdFeB PMs can now be purchasedin the Far East for less than U.S.$ 20 per kilogram. With the availability of more affordable PM materials,AFPM machinesmay play a more important role in the near future.

AXTALFLUX PERMANENTMAGNETBRUSHLESS MACHINES

Intvf:Tnn

Figure l.l. Electro-magneticmotor with disc rotor according to N. Tesla's patent No. 405 8 5 8 , 1 8 8 9[ 2 2 5 ] .

1.4

Types of axial flux PM machines

In principle, each type of a radial flux machine should have its corresponding axial flux (disc type) version.In practice,disc type machinesare limited to the following three types: r PM d.c. commutatormachines; r PM brushlessd.c. and synchronousmachines; r inductionmachines Similar to its RFPM counterpart, the AFPM d.c. commutator machine uses PMs to replace the electromagneticfield excitation system. The rotor (armature) can be designedas a wound rotor or printed winding rotor.

Introduction

Figure1.2. AFMPM 8-poled.c. commutatormotorwith printedrotorwinding:(a) statorwith PMs, (b) crosssection,(c) rotor (armature)windingsandbrushes,(d) constructionof 2p : g windingwith 145bars. I - rotorwith double-sided printedwinding,2 - pMs, 3 - brushes.

In the wound rotor, the almature winding is made of copper wires and moulded with resin. The commutator is similar to that of the conventional type, i.e. it can be either a cylindrical or radial commutator. The disc-fiipe printed armature winding motor is shown in Fig. 1.2. The rotor (armature)does not have a ferromagnetic core and its winding is similar to the wave winding of conventional d.c. commutator machines. The coils are stamped from pieces of sheet copper and then welded, forming a wave winding. When this motor was inventedby J. Henry Baudot [16], the armature was made using a similar method to that by which printed circuit boards are fabricated. Hence, this is called the printed winding motor. The magnetic flux of a d.c. printed winding commutator motor with a large air gap can be produced using cost effective Alnico magnetswith high remanence. AFPM d.c. commutator motors are still a versatile and economical choice for certain industrial, automotiveand domesticapplicationssuchas fans, blowers, small EVs, power tools, appliances,etc. Practically, d.c. brushlessand a.c. synchronousmachineshave almost the samestructure,though their theory and operationprinciples are somewhatdifferent [96, 112, 172]. The main difference is in the shape of the operation currentwaveform(Fig. 1.3),i.e.:

r the d.c. brushlessmachine generatesa trapezoidalEMF waveform and is operatedwith a rectangular line current waveform (also called a squarewqve mzchine):

MAGNETBRUSHLESS MACHTNES ANAL FLUX PERMANENT

?1 PhnseA

PlusE B

130 t80 240 300 16f {B}

0

t20 240 (b)

Figtre 1.3. Cunent waveforms for AFPM brushlessmachines: (a) square-wavemachine, (b) sinewavemachine.

r the a.c. synchronousmachinegeneratesa sinusoidalEMF waveformand is operatedwith sinewavecurrents(also called a sinewavemachine). It is difficult to manufacturea laminatedrotor with cagewinding for a disc-type inductionmachine11481.If the cagewinding is replacedwith a non-magnetic high conductivity(Cu or Al) homogenousdisc or steeldisc coatedwith copper layer, the performanceof the machinedrastically deteriorates.Therefore,there is little interestin disc type inductionmachinesso far 1148,238].

1.5

Topologiesand geometries

From a consfiuctionpoint of view, brushlessAFPM machinescan be designed as single-sidedor double-sided,with or without armatureslots, with or without armature core, with intemal or external PM rotors, with surface mounted or interior PMs and as single stageor multi-stage machines. In the case of double-sidedconfigurations, either the external stator or external rotor affangementcan be adopted.The first choice has the advantageof using fewer PMs at the expenseof poor winding utilisation while the second machinetopology [34]. The one is consideredas a particularly advantageous diversetopologiesof AFPM brushlessmachinesmay be classifiedas follows: r single-sidedAFPM machines with slottedstator(Fig. 1.4a) with slotlessstator with salient-polestator

Introduction

double-sided AFPM machines with internalstator(Fig. l.ab) x with slottedstator x with slotlessstator . with iron corestator . with corelessstator(Fig. l.4d) . without both rotor and stator cores

Figure 1.4. Basic topologies of AFPM machines: (a) single-sidedslotted machine, (b) doublesided slotless machines with internal stator and twin PM rotor, (c) double sided machine with slotted stator and internal PM rotor, (d) double-sided corelessmotor with intemal stator. I 5 -frame, 6-bearing, 7 shaft. statorcore,2 - statorwinding, 3 -rotor, 4-PM,

AXIAL FLUX PERMANENTMAGNETBRUSHLESS MACHII,{ES *

with salientpole stator(Fig. 1.5)

with internalrotor (Fig. 1.ac) x x

with slotted stator with slotlessstator

x

with salientpole stator(Fig. L6)

multi-stage(multidisc)AFPM machines(Fig. 1.7)

Figure 1.5. Double-sided AFPM brushlessmachine with internal salient-pole stator and twin extemal rotor [170]: (a) construction;(b) stator; (c) rotor. 1 PM,2 rotor backing steel disc, 3 - statorpole, 4 - statorcoi1.

The air gap of the slotted armatureAFPM machineis relatively small. The mean magnetic flux density in the air gap decreasesunder each slot opening due to increasein the reluctance.The changein the meanmagneticflux density causedby slot openingscorrespondsto a fictitious increasein the air gap I I I 1]. The relation between fictitious g' and physical air gapg is expressedwith the aid of Cartercoefficientkc ) I,i.e. g' : gkc

,t1

(1.1)

(r.2) 4*79

(*) +l*arc,an

(1.3)

Introduction

Figure 1.6. Double-sided AFPM brushlessmachine with three-phase,9-coil extemal salientpole stator and 8-pole intemal rotor. 1 - PM ,2 - statorbacking ferromagnetic disc, 3 stator pole,4-statorcoil.

where 11is the averageslot pitch and b1ais the width of slot opening. For AFPM machines with slotless windings the air gap is much larger and equal to the mechanical clearanceplus the thickness of all non-magneticmaterials (winding, insulation, potting, suppofting structure) that is passedby the main magnetic flux. Since there are no slots, Carter coefficient kc : I. Comparedto a conventionalslotted winding, the slotlessarrnaturewinding has advantagessuch as simple stator assembly,elimination of the cogging torque and reduction of rotor surface losses,magnetic safurationand acoustic noise. The disadvantagesinclude the use of more PM material, lower winding inductancessometimescausingproblems for inverter-fed motors and significant eddy current lossesin slotlessconductors[45]. In the double-sided, salient-pole AFPM brushlessmachine shown in Fig. 1.5, the stator coils with concentratedparametersare wound on axially laminated poles. To obtain a three-phaseself-starting motor, the number of the statorpoles should be different from the number of the rotor poles, e.g. 12 statorpolesand 8 rotor poles [60, 161, 170]. Fig. 1.6 showsa double-sided AFPM machine with external salient pole statorsand internal PM rotor. There are nine stator coils and eight rotor poles for a three-phaseAFPM machine. Depending on the application and operating environment, slotless stators may have ferromagnetic coresor be completely coreless.Corelessstator con-

l0


figurations eliminate any fenomagnetic material from the stator (armature) system,thus making the associatededdy cuffent and hysteresiscore losses nonexisting.This type of configurationalso eliminatesaxial magneticattraction forces between the stator and rotor at zero-currentstate. It is interesting that slotlessAFPM machinesare often classifiedaccordingto their winding alrangementsand coil shapes,namely, toroidal, trapezoidal and rhomboidal forms [34, 45,79J.

Figure I .7. Coreless multidisc AFPM machine with three corelessstators and four PM rotor uaits: 1 statorwinding, 2 rotorunit, 4-PM, 3 -frame, 4 bearing,5 shaft.

1.6

Axial magnetic field excited by PMs

A double-sidedAFPM machinewith twin PM rotor in ryz rectangularcoordinatesystemis shown in Fig. l.B. Assuming that the radius of curvature is higher than the pole pitch and the centre axes of the oppositerotor poles are shiftedby a linear distance16, the normal componentof the magneticflux densityon the surfaceofthe rotor can be describedin the stationaryrAz cootdinatesystemby the following equations: . atz :0.Ysd oo

B-,t(r.t) -

t

at z:

-0.5d

^\-Bo\,b,cos(";"!, d,r - ;) u-l "'2

(1.4)

Introduction

ll

pole Pole pitch Pitch

\

Figure1.8. Twin-statordouble-sided AFPM machinein Cartesiancoordinatesystem.

oo

B,n,z(r,t): Bo\U"cosl.,.,tT A,@ ,o) I

; |

(1.5)

-l Jl

where .86 is the value of the nornal componentin the center axis of the North pole, and ^17

lJu - I/*

(1.6)

]uu - BrrrDn

(r.7)

T

er:

.4

A_ u u--

r

- t (t)^ .o.r, t (*)' :o".,h *] fr ". l*sinha x sin@[)

.:l#"*'

"rn(r&)

(1.8) (p,b1) (;)' h -#] *,n*"i,(,;) cos

12


0.9

(a)

ll ii I iil l, I lli i lril r:

0.3 B m z t( x , r )

i \i\ / ,n\ \r/ t . l l i \ \ l','\'r llitli:llililifrLli r l--ll-i UJ-t lJ

B mzz( x.i)

l li

---.---.LI

I I

ll

I

ril

,

l

i i i l : l

0.1

0.l5

0.2

0.25

0.1

z/r

x (b) 0.9

B mz1(x,t)

09

r.Ylrl N,4riiM,i

iffl fft'l ' ,jft* \ \ l l \ \i/i\ r lir\l iliilillil M,1

llilirlr,ltlll,rll,

0.3

:B- 0

nrarx't)

-0.1

.i trr{-u, rtj i[i]'i [il

'0 9 -0.9 0

0.05

0.1

0

0.r5

0.2

0.25

0.3

zpr

x

Figure 1.9. Distribution of the normal components of the magnetic flux density according to e q n s( 1 . 4 ) ,( 1 . 5 )a n d ( 1 . 8 )f o r r - 0 . 0 4 , m , b , : 0 . 0 3 m , B o : 0 . 6 T , t : 0 , r o : 0 . 5 2 : ( a ) a : 0. (b) cv- 0.8.

T*b,

,

o 4;

"2

Q.

on

In the aboveeqns(1.4),(1.5),(1.6),(1.7),and (1.8) u : ?rris the linearspeed of the rotor in the z-direction, n - n D lu is the rotational speedin rev/s and the parametera dependson the shapeof the distribution of the normal component of the magneticflux density (Fig. 1.9). Forthe flat-toppedcurve a : 0 and for the concave curve (armature or eddy-current reaction) 0 < <-v< 1. The coefficientb, accordingto eqn (1.8) hasbeenderivedin [93]. The averagediameter D of the PM excitation system and corresponding averagepole pitch are: D :0.5(Dout *

Din)

rD )tt

( l e)

t3

Introduction

whereDin is theinnerdiameterof PMs,Do6 is theouterdiameterof PMsand 2p is thenumberof poles. The electromagnetic field analysisin AFPM brushlessmachineshasbeen in e.g.190,91,250,251). discussed

1.7

PM eddy-currentbrake asthe simplestAFPM brushlessmachine

A double-sided,PM excited eddy-currentbrake with high conductivity nonmagneticdisc-typerotor is one of the simplestbrushlessAFPM machines(Fig. 1.8). In an eddy-currentbrake the PM excitation system is stationary and the conductiverotor rotatesat the speedn. Eqns (l.a) to (1.9) are valid since the stationary PM excitation systemand rotating electric circuit (armature)are equivalentto the rotating PMs and the stationary electric circuit. It is assumedthat the eddy currents in the non-magnetic conductive disc flow only in the radial direction, i.e., in the gr-direction(Fig. 1.8). Thus, the magnetic vector potential ,4 in the disc is described by the following scalar equation(2D analysis):

^ -o, A^0, t* o, A^0, = : ()'AmY' aFar,

(1.10)

a, - t/ ja,Fo4ro : (I + j)k"

(1.11)

where

(1.12) In eqns(1.11) and (1.12) the electric conductivityo dependson the disc temperature. The relative magnetic permeability of paramagnetic(A1) or diamagnetic (Cu) materialsFr x I. The angularfrequencyforhigher spaceharmonics is accordingto eqn (1.7) or

u,:Ztrf,;

f"--uf;

u:I,3,5,...

( 1 .l 3 )

Generalsolutionto eqn( I .l0) canbe written,for example,as

A,na:i

sio(ur,t * g,r - T)lOr,exp(-n,z)* A',exp(n,z))(1.14)

l4


where

az"+03:(an,*ax,)k,

utlu

(l . l s )

( 1 .I 6 )

-

(1.17) Since the curents in the disc flow only in the radial direction U, E*, : 0 Ern" : 0 and Brno 0 for *0.5d I z ( 0.5d. Using the magneticvector _: potentialY x A : E andthe secondMaxwell's equationi E : *08 lAt, " the remaining electric and magnetic componentsin the disc can be found as E^y:

co ti-

jt"sin(-u,1 I 3,r

I u:1

-jarA*u

[)lot"

exp(-n,z) * Az, exp(n,z')l( 1.18)

12 - - flAma urnr 0, rc

\,2

u:l

[)l^r"

exp(- n,z) - Az, exp(n,z))

(1.19)

Brn":9+* OT :

I

p, rlos(a,t* p,r * I)(Or, exp(.*n,z) I Az, exp(n,z)) (1.20)

uI

The integrationconstants41, and 42, canbe found on the basisof equality of normal componentsof the magneticflux densityin the air and in the disc at z - A.Sdandz : -0.5d. i.e.

15

Introduction r a tz : 0 . 5 d p, cos(u,t * g,r - I)Vr, z-

exp(-n,d,f 2) -t Az, exp(n,d,l2))

fr\ : Bobrcos(wrt* 0rr -

t)

t a tz :

-0.5d

p, coslw,t+ g,@ - ro) Xllqr" exp(-n,d,f2) + 42, exp(n,cll2)l : Bob, coslu,t+ 0,(r - *ol -

[l

magneticfield in theair gapof aneddy-current Thereis onlybackward-rotating brake,so thatthetermsp,r and0"@ * 16) in eqns(1.4)and(1.5)arewith the * sign.Thus,

Atu: 42,:

1-,

1

s i n h ( r c , d " 1 21)1 - ,

nBob"?ffi#

:

z 6,nou,ffiG;I6sinh(rc"d) (r.21)

sinh(2e): 2 sinhr coshr. Puttingeqn(1.21) into(1.18),(1.19) and because (1.20),the particularsolutionto eqn(1.10)for E*0, B*, andB^" componentsare cco)brl,^77

Ems: -2t""TeCiqsin(u,tt

g,r -

(t.22) 7)cosh(rc,2)

B'n':f-""T^#rtrsin(u't

* 0'r- |)'i'n@'") (r'23)

Bn" :2uru"^"iOncos(a,t

* F,r - [) cosh(n,z)(r.24)

16

AXIAL FLUX PERMANENTMAGI,{ETBRUSHLESS MACHINES

The surface wave impedancefor the uth spaceharmonic is calculated on the basisof eqns(1.22')and(1.23)

-trotirl*,)z

zu - ru* jr,:

oi:rj: *:"ntt,{o,f;)

: l(Bp,Ap, - Bx,Ax,) + j(Bx,Aa, - Po,Ar,)l?

Q.25)

where a-,

tLtTu

-

sinh(ap"k"d) prk,d) - cos(ay,k,d) cosh(ct

(r.26)

- sin(ay"k"r|) cosh(ap, k, d) - cos(ay

(t.27)

"k "

d)

BR,:G#Te)

(r.28)

The impedanceof the disc for the yth spaceharmonic

v

Lu

-

0.5(Do.t - Dn) , -;Kzu r/u

a2, t 0.b1D.,,- -D i n ) . d. Kru cotlr\ti,ur)

au -

0 Ku

rlu

(r.29)

where Do6 is the outer diameterof PMs, D;n is the inner diameterof pMs, z is the averagepole pitch (1.9) and k"r rsthe impedanceincreasefactor due to circumferential cunents (in the r direction). The impedanceincreasefactor for the zth harmonicis [62]

k", :71

1.8

I u2 o.b(Dou,- Dm)

(1.30)

AFPM machinesversusRFPM machines

In pace with the application of new materials, innovation in manufacturing technologyand improvementsin cooling techniques,further increasein the power density(outputpower per massor volume) of the electricalmachinehas been madepossible. There is an inherent limit to this increasefor conventional radialflux PM (RFPM) machinesbecauseof 127, 49,96, 153,l77l:

L7

Introduction

r the bottle-neck feature for the flux path at the root of the rotor tooth in the case of induction and d.c. commutator machines or brushlessmachines with externalrotors (Fig. 1.10); r much of the rotor core around the shaft (rotor yoke) is hardly utilised as a magnetic circuit; r heat from the stator winding is transferredto the stator core and then to the frame - there is poor heat removal through the stator air gap, rotor and shaft without forced cooling arrangements. These limitations are inherently bound with radial flux structuresand cannot be removed easily unless a new topology is adopted. The AFPM machine, recognisedas having a higher power density than the RI'PM machine, is more compactthan its radial flux counterpartL26,49,96, 153].

(a) Figure 1.10.

0)

Topologies of (a) RFPM machine (b) AFPM machine.

Moreover. since the inner diameterof the core of an AFPM machine is usually much greaterthan the shaft diameter (see Fig. 1.4), better ventilation and cooling can be expected.In general,the specialpropertiesof AIPM machines, which are consideredadvantageousover RFPM machines in certain applications, can be summarisedas follows [48, 96]: r AFPM machines have much larser diameter to lensth ratio than RFPM machines; r AFPM machineshave a planar and somewhatadjustableak gap;

AXIAL FLUX PERMANENTMAG],{ETBRUSHLESSMACHINES

18

r capability of being designedto possessa higher power densitywith some saving in core material; r the topology of an AFPM machineis ideal to design a modular machine in which the number of the same modules is adjusted to power or torque requirements; r the larger the outer diameterof the core, the higher the number of poles that can be accommodated,making the AFPM machinesa suitablechoice for high frequencyor 1owspeedoperations. Consequently,AFPM type machines are particularly suitable for servo, traction, distributed generationand special-purposeapplicationswhere their properties offer distinct advantasesover their conventionalRFPM countemarts. 600

? soo

: .i .i :oo L 200 9 i * roo a 0.25

0.25

1l Odpdpowr,

o ,EEE2001

O IEEE2OOI

kv

ll5 Outd

Fwr,

kW

9m 80m 500 t

7000

I uooo

400

'6 ' 100

E 5oo() E i cmo

! :

?00

$ rooo

1S

&.* rm0 0 0.25

]

I

o IEEE2001

Figure I . I I .

Or&u

pow.

J kW

]O

0,25 cJlrEE t00l

Odpd powr, kW

Perfomance comparison of RFPM and AFPM machines [2 14].

The quantitativecomparisonbetweentraditional RFPM machineand AFPM machine is always difficult as it may raise the question of this comparison's fairness. Somepublished work dealt with quantitativeinvestigationsof RFPM and AFPM machine configurationsin terms of sizing and power density equations [8, 49, 121, 2a71. Fig. 1. 1I gives the performancecomparisonbetween a conventionalRFPM machine and a number of AFPM machinesof different

t9

Introduction

configurationsat five different power levels [214], which showsthat the AFPM machine has a smaller volume and less active material mass for a given power rating than the RFPM machine.

1.9

Power limitation of AFPM machines

The power range of AFPM disc-type brushlessmachines is now from a fraction of a Watt to sub-MW. As the output power of the AFPM machine increases,the contact surfacebetween the rotor and shaft becomessmaller in comparison with the rated power. It is more difficult to design a high mechanical integrity rotor-shaft mechanicaljoint in the higher range of the output power. A common solution to the improvement of the mechanicalintegrity of the rotor-shaftjoint is to designa multidisc (multi-stage)machine(Fig. 1.7). Since the scaling of the torque capability of the AFPM machine as the cube of the diameter (see eqn (2.9\) while the torque of a RFPM machinesscaleas the squareof the diameter times the length, the benefits associatedwith axial flux geometriesmay be lost as the power level or the geometric ratio of the length to diameter of the motor is increasedLlT4l. The transition occurs near the point where the radius equals twice the length of RFPM machine. This may be a limiting design considerationfor the power rating of a single-stage disc machine as the power level can always be increasedby simply stackingof disc machineson the sameshaft and in the sameenclosure.

Numerical example 1.1 A copper disc with its dimensionsas shown in Fig. 1.12 rotateswith the speedof 12 000 rpm betweenU-shapedlaminatedpole shoesof a PM. A sliding contact consisting of two brushesis usedto collect the electric current generatedby this primitive homopolar generator: one brush slides on the external diameter D,tut : 0.232 m and the secondbrush is located directly below one of the poles at the distanceof O.\Din : 0.03 m from the axis of the disc. The remanentmagnetic flux density of the NdFeB PM is B, : 7.25 T, coercivity is H. : 950 000 A/m and height 2hp1 :0.016 m. The thicknessof the disc is d : 0.005 m, one-sideda* gap is 9 : 0.0015 m and the width of the pole is bo : 20 mm. The relative magnetic permeability of the laminated core is F, : 1000 and the conductivityof the disc is o : 57 x 106 S/m at 20"C. The length of the flux path in the laminatedcore is lp" - 0.328 m. Find: (a) the magnetic flux density in the air gap; (b) the EMF betweenbrushes; (c) the current, if the line resistanceis El : Rr - 0.02O.

0.001 f, and load resistanceis

20

AXIALFLUX PERMANENTMAGNETBRUSHLESS MACHINES

Figure 1,12. Faraday's disc: a nonmagnetic conductive disc rotating in a stationary magnetic field according to Numerical examole I .I .

The magnetic flux fringing effect in the air gap, the variation of magnetic permeability with the magnetic field intensity in the laminated core and brush voltage drop are neglected. Solution This is a homopolartype d.c. generatorknown as Faraday'sdisc and canbe usedas a cunent source,e.g. for electrolysis. (a) Magnetic ffux density in the air gap The relative recoil magnetic permeability

IAB t'rt'e(

\

-

po LH

1.25-0

0.4r x lO-b 950000

0

r.047

The magnetic flux density in the air gap and saturationfactor of the magnetic circuit can be found on the basis of Kirchhoff's magnetic voltage law, i.e. 'B" '

"B" e

2 h n t-

l-LOFrrec

u'

f? 2hy + 1!2g + Hrrlr"

llollrrec

LtO

Bn

'ont :

Fjllrrec

R

2tttt + a 2 g k " o t

F}Frrec

P0

where the saturationfactor of the masnetic circuit I.

k s a t: 7

*

1T -- T

2p,(g+ 0.5d)

|

LF".

:7.042 2 x 1 0 0 0 ( 0 . 0 0*1 i0 . 5x 0 . 0 0 5 )

Introduction

2l

Thus,the air gapmagneticflux densityis 8,,

k

"

I + (g + 0.5d)k"otH,,",fh.11

t'25, :o.8o9T , 1 + ( 0 . 0 0 1 5 * 0 .x5 0 . 0 0 5 ) 1 . 0x4120 0 0 / 0 . 0 0 8" '

:

(b) EMF betweenbrushes Since Elt : ?- x En o, d,E : Bnudr where n : ,ur : Ztrrn is the linear speed,r is the radius ofthe disc and n is the rotationalspeedin rev/s, dE : Bs(2nrn)dr. Thlus p0.5Dea1

*-

12

-'ll'!'-e (n7".,- n?^\ n: I Bnl2rrn)d7 *'') Y\ \-our 4 Jo.sD;,,

: 4#ry

_ 0.006,) :6 83v (o.zsz,

(c) Current,if the line resistanceis El : 0.001 Q and load resistanceis /?1 :

0.020. Neglecting the current fringing effect in the disc, the resistanceof the disc for induced current at 20oC is 0 . 5 ( D o d- D " )

- 0.06) 0.5(0.232

n H +:0.0000151 -a x - - - # :odbo 57x 106x 0.005x 0.02

Q

Thecurrentis -I :

-: 3 2 5 . \ --: _ : -..,,,,,,6'8,3,, Ra-r RL]- Rt 0.0000151 + 0.001+ 0.02 "'

T h e t e r m i n a l v o l t a g e i s VI : x R y : 3 2 5 x 0 . 0 2 : 6 . 5 V a n d t h e l i n e voltagedropis LV : I x R1:325 x 0.001: 0.325V.

Numerical example 1.2 Find the impedanceof the aluminumdisc of a double-sidededdy cument brakeat ambienttemperature 20"C. The innerdiameterof the PMs is Drr, : 0.14m, the outerdiameterof the PMsrs Do6 : 0.242,thickness of the disc d : 3 mm, numberof poles2p : 16 and speedn : 3000rpm. Assume the conductivityof the aluminumo : 30 x 106S/m at2}oC and its relative magneticpermeability Fr = t.

MACHINES AATALFLUX PERMANENTMAGNETBRUSHLESS

22 Solution

The averagediameter of the disc D : 0 . 5 ( D m* D o , t ) :

0 . 5 ( 0 . 1 4+ 0 . 2 3 2 ) : 0 . 1 9 1m

The averagepole pitch rD t:2p:

n'0.191 -o'037m t6

The frequencyand angular frequencyof the curent in the disc for fundamental

u :7 spaceharmonic . f

^ 3 0 0 0: 4 0 0 H 2 p, - 8 irr

"';: 2r f : 21400: 2513.3t/s

The attenuationfactor for the fundamental spaceharmonic u : I of the electromagneticfield in the disc accordingto eqn ( 1.12) is

x 0 . 4 r 1 0 - 6x 1 . 0x 3 0 x 1 0 6 2513.3 : 277.661lm

t." -

The coefficientsaccordingto eqns (1.6), (1.16), (1.17), (1.26), (1.27) and ( 1.28) for u : 1, are respectively

't : #u

:84'865l1lYn

1.038

ux:

Ap

Aa

x 2 1 7 . 6 6x 0 . 0 0 3 ) sinh(l.038 x217.66 x 0.003)- cos(0.964x277.66 x 0.003) coslr(1.038 1.699

- sin(0.964 x 217.66x 0.003) x 277.66x 0.003) x 277.66x 0.003)- cos(O.964 cosh(1.038 -1.369

Introduction

L-)

. 0.002 2.lo-' 0.0015

n(v)

tr

o'oo1

s . t oa 00

1 3 5 7 9 1113t517r92r2325n2931333537394r 1v4l Figure I .I 3. Resistanceand reactanceof the copper disc for higher spaceharmonics z accordingto Numerical example 1.2.

BR:

0.964 - 0.961 0.5(1.0382 + 0.9642)

Bx :

1.038 : 1'035 o'5(10g82 + or9o42)

The surfacewave impedancefor u : I accordingto eqn (1.25)

- 1.0s5 x 1.699 x (-1.869)] r : [0.9G1 ffi

:2.2128x 10-5Q

r : [1.035x 1.699+ 0.961x (-r.369)]H,6:^ :3.2L6 x 10 6 f) "57 x 10o z : 2.2123x 10-5+ j3.216 x 10-6 Q Theimpedance increasefactordueto circumferentialcurrents(in ther- directo tion)according eqn(1.30)is

k,,:r+ I1 2==,=9'937 =r,, : 1.233 0 . 5 ( 0 . 2 4-2 0 . 1 4 ) The disc impedanceper pole for the fundamentalspaceharmonicu : 1 accordingto eqn(1.29)

AXIAL FLUX PERMANENTMAG],{ETBRUSHLESSMACHII,IES

1A

L+

l 0 5* f i . 2 1 6 xl 0 6 l 9 3 f f i t . t o u z - ( 2 . 2 1x2 3 : 5 . 2 3 1x 1 0 - 5+ j 7 . 6 0 5x 1 0 - 6o R : 5.231x 10-5 f)

X : 7.605x 10-6O

The disc resistanceR, andreactanceX, for higher spaceharmonics v ) 1 is plottedin Fig. 1.13.

Numericalexample1.3 Estimatethe cost of materialsof a 3-phase,250 kW 2300 rpm brushless AFPM machine. The internal stator does not have any ferromagnetic core. The twin-type externalrotor (Fig. L4d) consistsof sinteredNdFeB PMs (40 poles) gluedto two solid steeldiscs.The massof the statorcopperconductors rs'm6u: 18.7kg, massof the statorwinding insulation(includingencapsu1.87kg, massof PMs is rnpy : 32.1kg, massof the twin lation)is mirr: rotor core is mr": 50.8 kg (two discs)and massof the shaft is m"h : 48.5 kg. The cost of materialsin U.S. dollars per kilogram is: copper conductor cgu : 5.51, insulatingmaterialsc'irts: 7.00, sinteredNdFeB magnets cpM :20.00 and carbonsteelc"1""7: 0.65. The cost of frame, end discs (bells) and bearingsis C1 : $164.00. The cost of componentsthat are independentof the machinegeometry(encoder,terminal leads,terminal board, nameplate)is Cs : $140.00. Coefficientstaking into accountthe manufacturingand utilisation of steel for the shaftand rotor core (steeldiscs)are as follows: r total volume of the steelbar to the volume of the shaft,ku"h:

1.94

r coefHcienttaking into accountthe cost of machiningof the shaft, k-"7, : 2.15 r total volume of the steelplate to the volume of the rotor discs,kr, : !.1 . coefficienttaking into accountthe costof machiningof the rotor steeldiscs, k^, : \.8 Solution The cost of the copperwinding C- : nlgrcC,:

78.7x 5.51: $103.12

Introduction The cost of the insulationand encapsulation Cir" : Trlinsc'in":1.87x 7'00 : $13'10 The cost of PMs Cpt',r: rnpMcpM :32.7 x 20.00: $640.80 The cost of the rotor core (two steel discs) Cr.:

kurkrn mr.csteel:2.1 x 1.8 x 50.8 X 0.65 : $tZ4.B5

The cost ofthe shaft C"h - kr"7rk*rnr1"csteel: L.94x 2.I5 x 48.5 x 0.65 : $131.55 Total cost of materials

C : Cf t Co + C. + C,rnrl Cpm * Cr.l Crn : 1 6 4 . 0 0 *1 4 0 . 0 0 + 103.12+ 1 3 . 1 0 + 6 4 0 . 8 07+2 4 . 8 5 + 1 3 1 . 5 :5 $ t 3 t 7 . a 5

Chapter2 PRINCIPLES OF AFPM MACHINES

In this chapter the basic principles of the AFPM machine are explained in details. Considerable attention is given to the magnetic circuits, windings, torque production, losses,equivalentcircuits, sizing procedure,armaturereaction and performancecharacteristicsof AFPM machines.

2.1 2.1.1

Magneticcircuits Single-sided machines

The single-sided construction of an axial flux machine is simpler than the double-sidedone, but the torque production capacityis lower. Fig.2.1 shows typical constructionsof single-sidedAFPM brushlessmachines with surface PM rotors and laminated stators wound from electromechanicalsteel strips. The single-sidedmotor according to Fig. 2.lahas a standardframe and shaft. It can be used in industrial, traction and servo electromechanicaldrives. The motor for hoist applications shown in Fig. 2.1b is integrated with a sheave (drum for ropes) and brake (not shown). It is used in gearlesselevators[103].

2.1.2

Double-sidedmachineswith internal PM disc rotor

Inthe double-sidedmachinewith internal PM disc rotor, the armaturewinding is located on two stator cores. The disc with PMs rotates between two stators. An eight-pole configuration is shown in Fig. 2.2. P}ds are embedded or glued in a nonmagnetic rotor skeleton. The nonmagnetic air gap is large, i.e. the total air gap is equal to two mechanical clearancesplus the thickness of a PM with its relative magnetic permeability close to unity. A double-sided machine with parallel connectedstatorscan operateeven if one stator winding

28

MACH]NES AXIAL FLUX PERMANENTMAGNETBRUSHLESS

Figure 2.1. Single sided disc type machines: (a) for industrial and traction electromechanical drives, (b) for hoist applications. I - laminated stator,2 - PM, 3 - rotor, 4 - frame, 5 sheave. shaft,6

ROTOE

3

STATOR

/.

2

Figure 2.2. Configuration of double-sided AFPM brushlessmachine with internal disc rotor rotor, 2 - PM, 3 stator core, 4 - stator winding. 1

is broken. On the other hand, a series connection is preferred becauseit can provide equalbut opposingaxial attractiveforces. AFPM brushless 200 H2,3000 rpm, double-sided A practicalthree-phase, motor with built-in brake is shown in Fig. 2.3 [145]. The three-phasewinding

Principles of AFPM machines

29

AFPM brushlessservomotorwith built-inbrakeandencoder:1 Figure2.3. Double-sided statorwinding,2-statorcore,3 discrotorwithPMs,4-shaft,5-left frame,6 right brake,1l frame,7 - flange,8 - brakeshield,9 - brakeflange,10 - elecfromagnetic Electrical Bratislava and Technologt STU, of resolver. Courtesy of SlovakUniversity encoderor and ksting Institute,Nov| Dubnica,Slovakia. Research

is Y-connected,while phase windings of the two stators are in series. This motor is used as a flange-mounted servo motor. The ratio X"alX"q t 1.0 so the motor can be analysed as a cylindrical non-salient rotor synchronous machine[118,143,145].

2.1.3

Double-sidedmachineswith internal ring-shaped core stator

A double-sidedmachine with internal ring-shaped stator core has a polyphaseslotlessarmaturewinding (drum type) wound on the surfaceof the stator ferromagneticcore. 192,160,218, 2451.In this machine the ring-shapedstator core is formed either from a continuous steel tape or sintered powders. The total air gap is equal to the thickness of the stator winding with insulation, mechanical clearanceand the thickness of the PM in the axial direction. The double-sidedrotor simply called twin rotor with PMs is located at two sides of the stator. The configurations with intemal and external rotors are shown in Fig. 2.4. The three phasewinding arrangement,magnet polarities and flux paths in the magnetic circuit are shown in Fig. 2.5.

30

AXIAL FLUX PERMANEIVT MAGNETBRUSHLESS MACHINES

Figure 2.4. Double-sidedmachineswith one slotlessstator: (a) internal rotor, (b) external rotor. I -statorcore, 2 statorwinding, 3 -steel rotor, 4 PMs, 5 resin, 6 frame, 7 shaft.

Figure2.5. Three-phase winding,PM polaritiesandmagneticflux pathsof a double-sided disc machinewith one intemalslotiessstator.1 winding,2 PM, 3 - statoryoke,4 rotor yoke.

The AFPM machinesdesignedasshownin Fig. 2.4acanbeusedasa propulsion motor or combustion engine synchronousgenerator. The machine with externalrotor, as shownin Fig. 2.4b,hasbeen designedfor hoist applications. A similar machine can be designedas an electric car wheel propulsion motor. Additional magnetson cylindrical parts of the rotor are sometimesadded [ 160] or U-shapedmagnetscan be designed.Suchmagnetsembracethe arma-

3l

Principlesof AFPM machines

ture winding from three sidesand only the internal portion of the winding does not produce any electromagnetictorque. Owing to the large air gapthe maximum flux density does not exceed0.65 T. To produce this flux density a large volume of PMs is usually required. As the perrneancecomponent of the flux ripple associatedwith the slots is eliminated, the cogging torque is practically absent. The magnetic circuit is unsaturated(slotless stator core). On the other hand, the machine structure lacksthe necessaryrobustness[218]. Both buried magnetand surfacemagnet rotors can be used. There are a number of applications for medium and large power axial flux motors with external PM rotors, especially in electrical vehicles [92, 245]. Disc-type motors with extemal rotors have a particular advantagein low speed high torque applications,suchasbusesand shuttles,due to their large radius for torque production. For small electric cars the electric motor mounted directly into the wheel is recommendedl92l.

2.I.4

Double-sidedmachineswith internal slotted stator

The ring-type stator core can also be made with slots (Fig. 2.6). For this type of motor, slots ale proglessively notched into the steeltape as it is passed from one mandrel to another and the polyphasewinding is inserted 12451.ln the caseof the slotted stator the air gap is small (g < 1 mm) and the air gap magnetic flux density can increaseto 0.85 T 1921.The magnet volume is less than 50Yothat of the previous design, shown in Figs 2.4 and2.5.

Figure 2.6. Double-sided machine with one intemal slotted stator and buried PMs. I nonmagnetic rotor disc. core with slots, 2 - PM, 3 - mild steel core (pole), 4

stator

)L

AXIAL FLUX PERMANENTMAGNETBRL|SHLESS MACHINES

Figure 2.7. Double-stator,triple-rotorAFPM brushlessmotor with water cooling system[55] I -PM,2-statorcore, 3 statorwinding.

2.1.5

Double-sided machineswith internalcoreless stator

AFPM machines with corelessstators have the stator winding wound on a non-magnetic and non-conductive supporting structure or mould. The stator core losses,i.e. hysteresisand eddy current lossesdo not exist. The lossesin PMs and rotor solid steeldisc are negligible. This type of designoffers higher efficiency at zero cogging torque. In order to maintain a reasonablelevel of flux density in the air gap,a much larger volume of PMs in comparisonwith laminatedstatorcore AFPM machineis required.The statorwinding is placed in the air gap magnetic field generatedby the PMs mounted on two opposing rotor discs(Fig. l.ad). When operatingat relativelyhigh frequency,significant eddy current lossesin the stator winding conductorsmay ocun 12341.

2.1,6 Multidiscmachines There is a limit on the increaseof motor torque that can be achievedby enlargingthe motor diameter.Factorslimiting the singledisc designare: (a) axial force takenby bearings; (b) integrity of mechanicaljoint betweenthe disc and shaft; (c) disc stiffness. A more reasonablesolution for large torques aredouble or triple disc motors. Thereare severalconfigurationsof multidiscmotorsl5-7 , 55,73, 7 41.Large multidisc motors ratedat least300-kW havea water cooling system(Fig.2.7)

33


with radiatorsaround the winding end connections[55]. To minimize the winding lossesdue to skin effect, variable cross sectionconductorsmay be used so that the cross sectionof conductorsis bigger in the slot areathan in the end connectionregion. Using a variablecrosssectionmeansa gain of 40o/oin the rated power [55]. However, variable cross section of conductorsmeansan increasedcost of manufacturing. Owing to high mechanicalstressesa titanium alloy is recommendedfor disc rotors.

2.2 Windings 2.2.1 Three-phasewindingsdistributedin slots In a single-layerwinding, only one coil side is locatedin a single slot. The numberof all coils is s1f 2 andthe numberof coils per phaseis n. : s1I (2m1) where si is the number of stator slots and rn1 is the number of phases. In a double-layer winding two sides of different coils are accommodatedin each slot. The number of all coils is s1 and the number of coils per phaseis n. : stlmt.The numberof slotsper pole is

o'-h

(2.r)

where 2p is the number of poles. The number of slots per pole per phaseis Sl

qr:

(2.2)

2p*,

The number of conductorsper coil can be calculatedas r for a single-layerwinding AI-

Lrc

aoa-N1 -

T1'c

_

aoa*Nt s1l(2m1)

_ aoa-Nt PQt

(2.3)

r for a double-laverwindine A r --

ara-Nr

rrc

TLc

_ aoa-Nt stlmt

_ ara*Nt 2pqt

(2.4)

where ,Ay'lis the number ofturns in seriesper phase,oo is the number of parallel current paths and a- is the number of parallel conductors. The number of conductorsper slot is the samefor both single-layerand double-layerwindings, i.e.

1A

J+

AXIAL FLUX PERMANENTMAGNETBRUSHLESS MACHINES ctoa*Nt

ALt :

(2.s)

PQt

The full coil pitch measuredin terms of the numberof slots is y1 : Q1 where Q1 is accordingto eqn (2.1). The shorl coil pitch can be expressedas

at:YPet r\r

(2.6)

)

where tu"(r) is the coil pitch measuredin units of length at a given radius r and r(r) is the pole pitch measuredat the sameradius.The coil pitch-to-pole pitch ratio is independenton the radius, i.e. ut"\t') / \ T\7')

t'-

(.2.7)

The distribution factor of a polyphasewinding for the fundamentalspaceharmonic u : 1 is defined as the ratio of thephasor sum-to-arithmetic sum of EMFs induced in each coil and expressedby the following equation: 1 . , , --

tro |

sin(nl2m) qlsin}t l(2m1q1)l

(2.8)

The pitch factor for the fundamental spaceharmonic u : I is defined as the ratio of the phasor sum to-arithmetic sum of the EMFs per coil side and expressedas:

kpr: sin(Utr)

(2.e)

The winding factor for fundamental is the product of the distribution factor (eqn (2.8)) times pitch factor (eqn (2.9)), i.e. kur:

katkrt

(2.r0)

The anglein electricaldegreesbetweenneighbouringslots is

360" t--P

S1

(2.rt)

Fig. 2.8 showsa single layer winding distributedin s1 : 36 slots of a threePhase,2p:6 AFPM machine.

35


Figure 2.8.

SingleJayer winding of an AIPM machine with rnr :

3' 2p :

6, sr :

36'

At:Qt:6andQt-2.

2.2.2

Drum-typewinding

Drum-type stator windings are used in twin-rotor double-sidedAFPM machines (Fig. 2.q. The drum-type stator winding of a three-phase,six-pole AFPM machine with twin external rotor is shown in Fig. 2.9. Each phase of the winding has an equal number of coils connectedin opposition so as to cancel the possible flux circulation in the stator core. Those coils are evenly distributed along the stator core diametrically opposing each other so the only of the drum-type possiblenumberof poles are2,6,10, . . . etc. The advantages stator winding (sometimescalled toroidal stator winding 146,219]) are short end connection,simple stator core and easydesign of any number of phases'

2.2.3

Corelessstatorwinding

Coreless stator windings are used in twin-rotor double-sided AFPM machines (Fig. l. d). For the easeof construction, the stator winding normally consistsof a number of single layer ttapezoidally shapedcoils. The assembly of the statoris madepossibleby bending the endsof the coils by a certainangle, so that the active conductorslie evenly in the sameplane and the end windings nest closely together. The windings are held together in position by using a compositematerial of epoxy resin and hardener.Fig. 2.10 showsthe coreless stator winding of a three-phase,eight-pole AFPM machine. obviously, the relations used in the slotted stator windings can be directly used for coreless

36

AXIAL FL\]X PERMANE]VT MAGNETBRT]SHLESS MACHINES

Figure 2.9. Drum-type winding of a three-phase,six-pole, 18-coil AFpM machinewith fwin extemal rotor.

Figure 2.10. rotor.

Corelesswinding of a three-phase,eight-pole AFPM machine with twin external

trapezoidalstatorwinding with the exceptionthat the term "slot" is replaced by the "coil side". Another coil profi1ethat has been used in corelessstator AFPM machinesis the rhomboidal co1l. It has shorler end connectionsthan


Figure 2.11. machine.

3/

Connection diagram ofa three-phase,nine-coil winding ofan AFPM brushless

the trapezoidalcoils. The inclined arrangementof the coil's active sidesmakes it possible to place water cooling ducts inside the stator. The main drawback of rhomboidal coils is the reduction of the torque.

2.2.4

Salientpolewindings

A salientpole AFPM motor can be of single-sidedor double sided construction (Figs 1.5, 1.6,2.12). The statorwinding consistsof a numberof coils with concentratedparametersand separateferromagneticcores for each of the coil. In general, the number of stator poles (coils) is different than the number of rotor poles. Fig. 1.5bshowsa three-phase,12-coi1statorwinding of a doublesided AFPM machineswith 2p : 8 rotor poles. Figures 1.6, 2.ll and 2.12 show a three-phase,9-coil stator winding. The difference in the number of the stator and rotor poles is necessaryto provide the starting torque for motors and the reduction of torque pulsations.

2.3

Torqueproduction

Since the dimensions of AFPM machines are functions of the radius, the electromagnetictorque is produced over a continuum of radii, not just at a constantradius as in cylindrical machines. The pole pitch r(r) andpole width br(r) of an axial flux machine are functions of the radiusr, i.e.

38

MACHINES AXIAL FLUX PERMANEI{TMAGNETBRUSHLESS

Figure 2.12. Three-phase,nine-coil stator of a single-sidedAFPM brushlessmachine with West Lebanon,NH, U.S.A. salientpoles stator.Photo-courtesyof Mii Technologies,

2trr 2pp

oP, w, l r l : 0 ; T l r l :

m

();

(2.r2)

7f f'

( 2 .13 )

P

where a; is the ratio of the averageBorn to peak value B-,n of the magnetic ffux densityin the air gap.i.e. Boun

ai:

B*,

or

ai:

br ( ' ) r(r)

(2.14)

Both the pole pitch r(r) and pole width br(r) arefunctionsof the radiusr, the parametera; is notmally independentof the radius. The line current densiQ is also a function of the radius r. Thus the peak value of the line currentdensityis ^ , \

r Lmv

J

mtJ|Nrto p'r \t' )

mrfiNtlo 7Tr

(2.rs)

The tangentialforce acting on the disc can be calculated on the basis of Ampere'sequation

dh, - Io(rirx En1: A(r)(d-s, nn)

(2.r6)


39

where lod,; : AQ)d,3, AQ) : A*(ilt/2 accordingto eqn (2.15), di is tne radius element, d3 is the surface element and En is the vector of the normal component (perpendicularto the disc surface) of the magnetic flux density in the air gap. An AFPM disc-type machine provides Bn practically independent of the radiusr. Assuming the magnetic flux density in the air gap B,nn is independentof the radius r, dS : 2nrdr and Born : a,iBrns according to eqn (2.I4), the torqueon the basisof eqn (2.16) is electromagnetic

d,T6: rd,F, : rlk6A(r)Bousd,Sl : 2ratk-rA(r)B*nr2dr

(2.17)

The line current density ,4(r) is the electric loading per one stator active surface in the case of a typical stator winding distributed in slots (double-sided stator and internal rotor) or electric loading of the whole stator in the caseof an internal drum type or corelessstator.

2.4

Magnetic flux

For sinusoidal distribution of the magnetic flux density waveform excited by PMs, the averagemagneticflux density is Baug:

Trlp- 0

" Bnn ''t(pa)d.a: -* u *n *",o",] " lo"'o [i

: -]- B*olcos?r- cosol : ? u*o '71

(2.18)

7f

Since the surface element per pole is 2nrdrl(2p), the magnetic flux excited by PMs per pole for a nonsinusoidalmagnetic flux density waveform Baus : a;B*n is

oJ

: I:-

21

a i B * n- - j r d r : zp

7l

f

cl

Rout

a i B ^np l r - | ltl

- R?.) : onB,.n#@?,,,

t-l

l'

) no"

(2.re)

where B*n is the peak value of the magneticflux density in the air gap, p is the number of pole pairs, Ro'1 : O.5Doutis the outer radius of the PMs and Rh : 0.5D,;n is the inner radius of the PMs. It is convenientto use the inner-to-outer PM radius or inner-to-outer PM diameter ratio. i.e.

40

MACHINES AX]AL FLUX PERMANE]{TMAGNETBRUSHLESS ,

-

R'in

D';,

Rn,,t

Dr'rt

(2.20)

Thus, o '/ : o , B *"o' 2! [p(' 0 . 5 D , , , 1- 2( 0 . 5 D ; , ) :' ] o ,B * "n ! D I , , G- k ' i lQ . 2 l l 6p

The sameequation for a cylindrical type machine is [96] Q r : ?rLiB,no '77

(2.22)

"

where r is the pole pitch and Lt, is the effective length of the stack. T,hepermeanceof the air gap in the d-axis at the radius r is given by d,Gn :

uofiatffdror ,^

r

fR.ur

tt^

.,lr2lR""'

r c t-r # " , ; l ' r l G"n: * * , ' : I g' PLz)a,, g', pJR,"

I n - t t o ) a i * ( n : , ,- R 7 , ) g' tp

- R?,) : ^n*,+@2,,

Q.23)

where \s : lroI g' is the permeanceper unit surface andg' is the equivalentair gap'

2.5


The average electromagnetictorque developedby AFPM motor according to eqns(2.15)and (2.17)is d,Td: 2a im1 I oIY1k.1 B rnnrdr If the aboveequationis integratedfrom Dou1f2to Di,f 2 with respectto r, the averageelectromagnetictorque may be written as

Ta : :

),nn

rrloNrk-rB^s(D3.r - D?,)

)aimrt{&dB^sD'lut0

- k:i)h

(2.24)

where k; is accordingto eqn (2.20). Putting eqn (2.21) into eqn (2.24) the averagetorque is

41


(2.2s)

Ta:2L*rN1k*1Q sIo 1T

To obtain the rms torque for sinusoidal current and sinusoidal magnetic flux = 1.11,i.e. density,eqn(2.25) shouldbe multiplied by the coefficient "rDla

Ta: TlpNtk,te ylo- kTIo v'2

(2.26)

where the torque constant

k, :

(2.27)

ftrNtk*t{_f

In somepublications [84, 218] the electromagneticforce on the rotor is simply calculatedasthe product of the magneticand electric loading BoooA and active surfaceof PMs S : n(R\,, - R?,),i.e. F, : rBo,,nA(R\., J Rl.),where A is the rms line current density at the inner radius R4n. For a double-sided AFPM machine S : 2tr(R?-",- R?"). Thus, the averageelectromagnetic torque of a double-sidedAFPM machine is

Ta: F*&n:2trBounA(R?""r-R?)Rn, :2trBo,gAR2*(ka- tcll 1z.ztl Taking the first derivative of the electromagnetictorque 7a with respectto k4 and equatingit to zero. the maximum torqueis for ,ka = | I /i. lndustrial practice showsthat the maximum torque is for ka I t lt/9. The EMF at no load can be found by differentiating the first harmonic of the magneticflux waveform iDyr : O/ sinot and multiplying by ly'1k,,1,i.e. ey : Nlk-14*-!,t :2rf dt

N{c-1Qy cosolt

The magneticflux (D1is expressedby eqns(2.19) and (2.21). The rms value is obtainedby dividing the peak value2n f N1k.tQ f of the EMF by y8,, i.e. Et:

nrtf

N1k*1Q1: nt[2pNtk-rQfn":

kErL"

(2.29)

where the EMF constant (armatureconstant)is kB : ntf2pNyk-1Qy

(2.30)

The same form of eqn (2.29) can be obtained on the basis of the developed torqueTa : mrElI"l(2trn") in which 74 is accordingto eqn (2.26). For the drum type winding the winding factor k-1 :1.


42

2.6 Lossesand efficiency 2.6.1 Stator winding losses The stator (armature) winding resistanceper phasefor the d.c. current is

R7d.:

Nil.-,.,

(2.31)

a,pauosa

is the averagelength where l/1 is the number of armaturetums per phase,11.,,., of turn, c,ois the numberof parallel currentpaths,a- is the numberof parallel conductors,o1 is the electric conductivityof the armatureconductorat given temperature(for a copper conductorot r= 57 x 106 S/m at 20"C and o1 = 47 x 706 S/m at 75"C), and so is the conductorcross section. The average leneth of the armatureturn is 11o,,-- 2Lt + lun ! Iuut

(2.32)

in which l1;,-,is the length of the inner end connectionand l1ou7is the length of the outer end connection. For a.c. curuentand statorwinding distributedin slots (ferromagneticcore) the statorwinding resistanceshouldbe divided into the resistanceof bars (radial portions of conductors)R16 and resistanceof the end connectionsfir",

Rt : Rn *

Rr":

ly't aosa

(2L;ktn I ltn I luut) x k111R14" Q.33)

where k16 is the skin-effectcoefficientfor the stator(annature)resistance.For a double-layerwinding and ur.(r) : ,(r') or 13: 1 [149]:

k t r y , : p r ( € r )+

-(v.'";)'] t4# vr({r)

(2.34)

where sinh 26r * sirr2{r cos2{1

p r ( d r ): € r cosh2{1 i 'l r r, /\ F \ r,/ \

qinhF' - sin F, -''coshq, -cos{r

7rI F"ot

,

011

(2.3s) (2.36) (2.37)

43


nfid m4 is the number of conductors per slot arranged above each other in two layers (this must be an even number), 7 is the phase angle between the currentsof the two layers, / is the input frequency,bpon is the width of all the conductorsin a slot, b11is the slot width andh, is the height of a conductorin the slot. If there artetT"lconductorsside by side at the sameheight of the slot, they are taken as a single conductor carrying n";-times greatercurrent. In general,for a three-phasewinding ? : 60o and

:p,(€r) krR . (4f

+)vr(6r)

(2.38)

For a chordedwinding (w"(r) < r (r)) and ? : 600

- y'lr)-3,.] k* xer((r)*l*?'-- t - 3(r lri "'l v,re,t 3 L

e.3sl

The skin-effect coefficient k16 for hollow conductorsis given, for example, in

[r4el. (a) If m"1: 1 and 1 : 0, the skin-effectcoefficientkyp : pr({r) (the same as for a cagewinding). (b) Ifl

: 0, the currentsin all conductorsare equal and

ktn= q\t)

-2--1

+ -:!'_:Vr(€r)

(2.40)

J

For small motors with round armatureconductorsfed from power frequencies of 50 or 60H2, Rt x Rta.

(2.4r)

LPw : ruISRr = mlllRla.kyp

(2.42)

The armature winding lossesare

Since the skin effect is only in this part of the conductor which is located in the slot - eqn (2.33), the armaturewinding lossesshould be multiplied by the coefficient

kn * lu. I (2Li) * 11"6I (2L.i) 1-t lu"l(2Li) + tt*t l(2Li) rather than by k1a.

(2.43)

44

2.6.2

AXIAL FLUX PERMAI{ENTMAG]VETBRUSHLESS MACHINES

Statorcorelosses

The magnetic flux in the stator (armature) core is nonsinusoidal. The rotor PM excitation systemproducesatrapezoidalshapeof the magneticflux density waveform. The statorwindings are fed from switchedd.c. sourceswith PWM or square wave control. The applied voltage thus contains many harmonics which are seenin the stator flux. The eddy current lossescan be calculatedusing the following classicalformula ,m

LP,," : "^ "ja 1ztll.n,p.t + *-,,1 0 PF" n I "''le'-,, -2 - ';

^_

- B?,,,1q7, d2p,mp"lB2-,,

#f

(2.44)

where oF", dF", pp" andlmlle arethe electricconductivity,thickness,specific density and mass of laminationsrespectively,n are the odd time harmonics, Brnrn and B,nrn arethe harmonic componentsof the magnetic flux density in the r (tangential) and z (normal) directions and

(38*"2)'2 + (3B,nA)2

(58ry!:\2 +(!F-,i)2

B2r-r, + Blr,

DmrT

T

+

Dm.z1

Q.451

is the cofficient o.f distortion of the magneticfux density. For qa : 1, eqn (2.44)expresses the eddy currentlossesundersinusoidalmagneticflux density. In a similar way, the hysteresis losses can be expressedwith the aid of Richter'sformula, i.e.

LPnp" :

rr O O

r];r,rp"\rr2lal,", It,t,

+ 82,-",1

n_1

{ t - : - t r rt r , l b ' - , t * I J ' - r 1 ) r 1 ' 4 100

(2.46)

wheree,:7.2ro2.0mal(]Hkg) foranisotropiclaminationswith4o/oSi, e : 3.8 m4l1Hkg) for isotropic laminationswith2o/oSi and e : 4.4to 4.8 m4l(H kg) for isotropicsiliconlesslaminations. Eqns (2.44) and (2.46) exclude the excesslosses(due to magnetic anomaly) and lossesdue to metallurgicaland manufacturingprocesses.There is a poor correlationbetweenmeasuredcore lossesand thosecalculatedusing classical methods.The lossescalculatedaccordingto eqns(2.44) and (2.46) are lower than those obtainedfrom measurements.The coefficient of additional core

45

Principlesof AFPM machtnes

lossesko6 ) 7 can help to obtain a better agreementof predicted and measured core losses LPtp":

koa(LP"r" * LP1rp")

(2.47)

If the speciJiccore lossesare known, the stator core losses LPt p" can be calculatedon the basisof the specificcore lossesand massesof teeth and yoke, i.e. t r ,,4/3

LPtp"

: ar'z'o(jo)

-t lkoa1B! 1m11 k o,1,Bl om 1ol

(2.48)

where kad.t ), \ and ko6, ) 1 are the factors accounting for the lncreaseln lossesdue to metallurgical and manufacturingprocesses,L11lso is the specific core loss in Wkg at I T and 50H2,.B11is the magnetic flux density in a tooth, B1o is the magnetic flux density in the yoke, m11is the mass of the teeth, and m1, isthe mass of the yoke. For teeth ko6 : 1.7 to 2.0 and for the yoke k o d ,:s 2 . 4 t o 4 . 0 l l 4 7 l .

2.6.3

Core loss finite element model

The core losseswithin the stator and rotor are calculatedusing a set of finite element method (FEM) models, assuming constant rotor speedand balanced three-phasearmature currents. The eddy current and hysteresislosses,within the cores,in the 2D FEM including distorted flux waveforrnscan be expressed by eqns(2.44) and(2.46). The field distribution at several time intervals in the fundamental current cycle is neededto createthe magneticflux density waveforms. This is obtained by rotation of the rotor grid and phaseadvancementof the statorcurrents.From a field solution, for a particular rotor position, the magneticflux density at each element centroid is calculated. Three flux density componentsin an element can be obtained from a single FEM solution.

2.6.4

Lossesin permanent magnets

The electric conductivity of sinteredNdFeB magnetsis from 0.6 to 0.85 x 106 S/m. The electricconductivityof SmCo magnetsis from 1.1to 1.4 x 106 S/m. Since the electric conductivity of rare earth PMs is only 4 to 9 times lower than that of a copper conductor, the /ossesin conductive PMs due to higher harmonic magneticfields producedby the stator cannot be neglected. The most important losses in PMs are generatedby the fundamental frequency magnetic flux due to the stator slot openings. In practice, those losses

46


are only in AFPM machineswith slotted stator feromagnetic cores. The fundamental frequency of the magnetic flux density component due to the stator slot openingis fsl:

(2.4e)

stPn

where s1 is the number of statorslots,p is the number of pole pairs and n is the rotor speedin rev/s. The magneticflux densitycomponentdue to slot openingis [111] Bst : ct,11B"1,kgBo,,o

(2.s0)

where Boro is the mean magneticflux density overthe slot pitch - eqn(2.4), kg is Cartercoefficient(l.2\ and

o':*(otn

s i n ( 61 n r )

(2.51)

"#*) ,Jrr:05(1

l:b

-

.1I\

(2.s2)

/;-, tI vL |ti'/

htt

t1

( 2.s3)

g'

In the aboveequations(2.51), (2.52) and (2.53) b1ais the statorslot opening, gt : g * hm I ltrr"" is the equivalentair gap,tt is the slot pitch and h,r,ris the magnetthicknessper pole. Assuming that the relative recoil magnetic permeabllity 1.t,.,r,.r 1, the power lossesin PMs canbe expressedby the following equationobtainedfrom a 2D electromagneticfield distribution,i.e. tat2( B,t I LPptr -- irn,k-.; t-, z

lJ

\HUHrr''

\2 /

k ^ -rpn1 oP'\]

(2.s4)

whereap, is accordingto eqn (1.16)for u : 1, a : (t + j)k is accordingto eqn (l.l 1) for u : I, k is accordingto eqn (1.12)for u : 7, p ts accordingto eqn (1.6) for u : 7 and oppl is the electricconductivityof PMs. Eqn(2.5a) can also be usedto estimatethe reactivelossesin PMs if ap, is replacedwith uy, accordingto eqn (1.17). The coefficientfor including the circumferentialcomponentof currentsinducedin PMs can be found as

47


ru2-rt^

I)ou,t

-

(2.ss) l)in

where 0.5tr is the inducedcuffent loop spanand0.5(Do"t - Dn) is the radial length of the PM - seeeqn (1.30). The activesurfaceareaof all PMs (singlesidedmachines)is

SpM:

"onQ3,t

- D7")

(2.s6)

wherethe averagepole pitch r is accordingto eqn (1.9).

2.6.5

Rotor corelosses

The rotor core losses,i.e. lossesin backing solid steeldiscs supporting PMs are due to the pulsating flux produced by rapid changesin air gap reluctance as the rotor passesthe stator teeth. The magneticpermeability of a solid steeldisc varieswith the z axis (normal axis). To take into account the variable magnetic permeability and hysteresis lossesin solid ferromagneticdiscs, coefficientsa1u, axu accordingto eqns (1.16) and (1.17)must be replacedby the following coefficients: _1

p'lu

1 &RFe

,/,

l@_.a2p-a2y*

Fl

( 2.s7)

-t

axFe

p'l"

1

....:l

rt

k2 l I

(2.58)

L4 to 1.5,ax :0.8 to 0.9 accordingto Neyman[184],k" is wherean: accordingto eqn (1.12) and B is accordingto eqn (1.6) for u :1. The power lossesin solid ferromagneticdiscs can be expressedby a similar equationto eqn (2.54),i.e.

: *o*r"*"ry LPzp" ffi)' *t r.

(2.se)

where appu is accordingto eqn (2.57) for u : 1, c is accordingto eqn (1.11) for u : 1, k is accordingto eqn (1.12) for u : 1, B is accordingto eqn ( 1.6)for u : l,8"7 is accordingto eqn (2.50),p, is the relativemagneticpermeability

48

AXIAL FLUX PERMANENTMAGNET BRLTSHLESS MACHINES

and op. is the electric conductivity of the solid ferromagneticdisc. The frequencyin the attenuationcoefficientk is accordingto eqn (2.49). Eqn (2.59) can also be usedto estimatethe reactivelossesin the solid ferromagneticdisc if o,pp" is replacedwith a;p" accordingto eqn (2.58). The flux density B.r7 is accordingto eqns (2.50) to (2.53). The coefficientk" is accordingto eqn (2.55) and the surfaceof the disc is

sF.: 2.6.6

[{n3,, D?.)

(2.60)

Eddy current lossesin stator conductors

For slottedAFPM machinesthe eddycurrent lossesin the statorwinding are generally ignored as the magnetic flux penetratesthrough the teeth and yoke and only small leakageflux penetratesthrough the slot spacewith conductors. In slotlessand corelessmachines,the statorwinding is exposedto the air gap magneticfield. The motion of PMs relativeto the statorwinding producesan alternatingfield througheachconductorand induceseddy currents.In the case of a corelessAFPM machinewith a solid ferromagneticrotor discs, there is also a tangentialfield componentB,n, inaddition to the axial field component Brr". This can lead to seriousadditionaleddy current loss especiallyat high frequency. Neglecting the proximity effect, the eddy current loss in the stator winding may be calculatedby using a classicalequationsimilar to eqn (2.44) for calculationof the eddy cunent lossesin laminatedcores,i.e. r for round conductors[43]

Lp":

oc

-2*

+ B?,",] f rr2lB2^*, Titrar-.o,, ' n:7

: |\ 12a'rn,onlB'^,t + B?,,"rln\t +I)

Q.6l)

r for rectangularconductors[43] )

(Y)

AP" = ; ^f',r'^.o, I JP

n,2[82,n*, + B?r"r]

n-l

2^ '1fo

3-1 p

L-. f'o'*"orlB?rrt

+ g?'r."tl'l'o

(.2.62)


49

where d is the diameterof the conductor,a is the width of the conductor(parallel to the statorplane), o is the electric conductiviry p is specific mass density of the conductar, m"on is the mass of the stator conductorswithout end connectionsand insulation, / is the stator current frequency,Brr* andBrn" arcthe peak values of tangential and axial componentsof the magnetic flux density, respectively,andrla is the coefficientof distortionaccordingto eqn (2.45).

2.6.7

Rotationallosses

The rotational or mecltanical losses LProt consist of friction lossesAPy, in bearings,windage lossesLP*tnd. and ventilation lossesLPu.nt (if there is a forcedcooling system),i.e. LPr"t -

LPr, * LP,ttrd, * LPo"nt

(2.63)

There are many semi-empirical equationsfor calculating the rotational losses giving various degreesof accuracy. The friction losses in bearings of small machinescan be evaluatedusine the followine formula

A,Py,:0.06kya(m, * m"6)n

W

(2.64)

where kf,, : I to 3 m2ls2,rn, is the massof the rotor inkg, m"6 is the mass of the shaft in kg and n is the speedin rpm. The Reynoldsnumber for a rotating disc with its outer radius Roul is R":

pRou" I,L

:

2trnpRf;u,

(2.6s)

where p is the specific density of the cooling medium, u : 1)r - 2trRo",1nis the linear velocity at the outer radius Rout, fr is the rotational speedand p is the dynamic viscosity of the fluid. Most AFPM machines are air cooled. The air density at I atm and 20"C is 1.2 kg/m3. The dynamic viscosity of the air at 1 atm and 20"C is F : 1.8 x 10-5 Pa s. The cofficient of drag for turbulent flow can be found as (2.66) The windage losses for a rotating disc are

LP-rnd.:

)c

ypQnfit @u.,,- R\n)

(2.67)

50


where fi.r7,is the shaft diameter. AFPM machinesare usually designedwithout a cooling fan so that the ventilation lossesAP",unr: 0.

2.6.8

Lossesfor nonsinusoidalcurrent

Higher time harmonicsgeneratedby static convertersproduce additional losses.The higher time harmonicfrequencyin the statorwinding is n/ where fr:5,7,1I,.... The armaturewindinglosses,the corelosses,andthe stray lossesare frequency-dependent. The mechanicallossesdo not dependon the shapeof the input waveform. If the stator winding and stator core losseshave been calculatedfor fundamental frequency, thefrequency-dependentlossesof an invefter-fed motor or generatorloaded with a rectifier can be found as o stator (armature)windins losses: oc

APr., - \z . LPr*,:

*tD

linRy,,x m1R16,1LlZ,ttt*,

rt:l

(2.68)

n-l

o stator(armature)core losses co

L,P,r;-

f

Z-z

oo

AP, ,-- : lA,P,.--l--, \" )"-'

/J

/Vt,\2

t_l

\V, )

n-B'7

(2.69)

where k16,, is the a.c.armatureresistanceskin effect coefficientfor n,f , Io, is the higher harmonic rms armaturecurrent, [r, is the higher harmonic inverter outputvoltage,V1r.is the ratedvoltage,[APrp.,.]n:l are the statorcore losses for n : 1 and rated voltase.

2.6.9

Efficiency

Thetotalpowerlossesof anAFPM machineare AP : LP:.- * APrr'" * LPzp" + LPzM + LP" -f LProt

Q.70)

The fficiency is rD olrt

P,6 * L'P

(2.71)

where Po6 is the mechanicaloutput power for a motor and the electrical output power for a generator.

5l


UNDTRIXCITTD GTNERATOR RC LOAD o> Y >-90'

OVER[XCITTD CTNTRATOR RL LOAD

-l-

o
,9 il cr

g

trod:Io sinY

UNDTREXCITID MOTOR RL *gO'>Y >- 180' Figure 2.13.

2.7

OVTRIXCITID MOTOR pn

9 O ' < Y< 1 8 0 '

Location of the armature current l" in d-q

coordinate system.

Phasordiagrams

The synchronousreactanceof a synchronousmachine (sine-wavemachine) is defined asthe sum of the armaturereaction (mutual) reactanceXo4, Xon and stator (armature)leakagereactanceX1, i.e.

r in the d-axis Xsd,: Xo4 I X1

(2.72)

r in the q-axis X"q:

Xon * Xt

(2.13)

When drawingphasor diagrams of synchronousmachines,two arrow systems areused: (a) generatoralrow system,i.e. E.r : Vr i-IoRt * jloaX"a + jloqx"q

52

AXIAL FLUX PERMANENTMAGNETBRUSHLESS MACHINES : Vr f I"a(Rt * jX,a) *I"q(R4 * jX"q)

(2.74)

(b) consumer(motor) arrow system,i.e. Vr : E/ * Io-Rr * jIo6Xr,1+ jloqxr,l : E / - l T " a ( R t* j X " a ) * l o q ( . R 1 * . i X " q )

(2.7s)

where

(2.76)

Io-Io6,*l6q

and Ioa:

Iosin'U

1oq :

1o cos \!

(2.71)

The angle V is betweenthe q-axis and armaturecurrent 1o. When the current arows are in the oppositedirection,the phasorsIo, l,r,L aadl.aqare reversed by 180". The sameappliesto the voltagedrops. The location of the armature current Io with respect to the d- and q-axis for generatorand motor mode is shownin Fig. 2.13. Phasordiagramsfor synchronousgeneratorsare constructedusing the generator arrow system. The same system can be used for motors, however, the consumerarrow systemis more convenient. An overexcitedgenerator(Fig. 2.I4a) deliversboth active and reactivepower to the load or utility grid. An underexcitedmotor (Fig. 2.lab) drawsboth activeand reactivepower from the line. For example,the load culrent I" (Fig. 2.14b ) lagsthe voltagephasorV1 by the angle$. An overexcitedmotol consequently,draws a leadingcurrent from the circuit and delivers reactivepower to it. In the phasordiagramsaccordingto Fig. 2.141116] the statorcore losses have been neglected. This assumptionis justified only for power frequency machineswith unsaturatedarmaturecores. For an underexcitedsynchronousmotor (Fig. 2.lab) the input voltage I projectionson the d andq axesare 7rsin6:IuqX"n-IoaRt -I cos6 E.f t Io6X"4 t IoqRl

(2.78)


)J

jl"X,

Figure2.14. Phasordiagramsofsalient-polesynchronous generator machine:(a) overexcited (generator arrowsystem);(b) underexcited motor (consumerarrowsystem). (a)

jl"a4o

ill"o +--

<_

+

Figure 2. 15. Equivalent circuit per phaseof an AFPM synchronousmachine: (a) for generating mode; (b) for motoring mode. Stator core losseshave been neglected.

where d is the load angle betweenthe voltage V1 andEMF Ep (q-axis). For an overexcitedmotor V 1s i n S : I o n X " n * I o 6 R 1

I/r cos 6 : Ef - IodX"d, * IoqRt

(2.7e)

AXIAL FLUX PERMANENT MAGNETBRUSHLESSMACHI NES The currentsof an overexcitedmotor ',u u

V r ( X " o c o s d- R r s i nd ) - E I X " ,

t 1

xr6x"ns- R'2,

- EfRt

',' t ' l _ V r ( R 1 c o s 6 + X " 6 s i n d ) X"6X"qR21

(2.80)

(2.81)

are obtainedby solving the set of eqns(2.78). The rms armaturecurrentas a functionof V1,81, X"d, X",r, d, and R1 is

Vr I*:ffi+lfu- X"aX"q * f(X"n cos6

R?

- Rtsind) - Ef X,n1' + l(/?r cosrI * X"4 sin 6) E f Rr]2 (2.82)

The phasordiagram can also be used to find the input electric power. For a motor Pin : mlVylo cos @ - rnlVy(Io,n cos d - 1"a sin d)

(2.83)

Thus, the electromagnetic power for a motor mode is

P"Ln ';;;"ff;;;,ff)&",: x"q)t Lprp. eB4) On the basis of phasor diagrams(Fig. 2.14), equivalentcircuits of a AFPM synchronousmachinecan be drawn (Fig. 2.15).

2.8

Sizing equations

The main dimensionsof a double-sidedPM brushlessmotor with internal disc rotor can be determinedusing the following assumptions:(a) the electric and magnetic loadings are known, (b) the number of tums per phaseper one stator is Nr; (c) the phasearmaturecurrent in one stator winding is 1o; (d) the back EMF per phaseper one statorwinding is 81. The peak line current density at the averageradius per one statoris expressed by eqn (2.15) in which the radiusmay be replacedby an averagediameter


55

80000 100000 Figure 2.16. Outer diameter D6a1 zs a {tnction of the output power Po,1 and parameter ki7 for e : 0.9, k-1q cosQ : 0.84, n" - 1000 rpm: 16,67rev/s and B*nA^ : 26, 000 TA/m.

D :0.5(D^"t + Din) :0.5D^"t(7 -t ka)

(2.8s)

where Doul is the outer diameter, Dan is the inner diameter of the stator core and ka : DinlD^ft (according to eqn (2.20)). Thus, ^ tant:

A{2mtloNr "D^,^r

(2.86)

+ kd)

The EMF induced in the stator winding by the rotor excitation system, according to eqns (2.29) and(2.21) has the following form

: Ef : nrf2n"pN1k.yQ7 !rtn"Ntk,1B^nD2our$ n3)

e.87)

The apparent electromagnetic power in two stators is S.r,n :

:

m1(281)1": mtEf (21") _2

ik-1n"8*sA*DZdG

+ ki\

- k2d)

(2.88)

For series connection the EMF is equal to 2Ey and for parallel connection the current is equal to 2Io. For a multidisc motor the number "2" should be replacedby the number of stators.Putting

AXIAL FLUX PERMANENTMAGNETBRTJSHLESS MACHINES

56

1

,tr:

r ( 1+ k , i 1 - k : i )

( 2.8e)

the apparentelectromagneticpower is S elrn : n2 k 11k*1n

"B

rngArnDZut

(2.e0)

The apparentelectromagneticpower expressedin terms of active output power is thus ^

e\ , e t n t -

Pout f

-

,/ cos g

(2.er)

where thephase EMF-to-phase voltage ratio is Ey I/

(2.e2)

v1

Formotors e < 1 and for generatorse ) 1. In connectionwith eqns(2.90) and(2.91), the PM outer diameter(equalto the outer diameter of the stator core) is ,,

UOUI

-

cPout

,l ll

o,

\l r'2k pk *p, B ^nA -r7 coso

r? q3)

The outer diameterof PMs is the most importantdimensionof disc rotor PM motors. SinceDo,1 o, i/P""t the outer diameterincreasesrather slowly with the increaseof the output power (Fig. 2.16). This is why small power disc motors have relatively large diameters.The disc-type construction is preferred for medium and large power motors. Motors with output power over l0 kW have reasonablediameters. Also, disc constructionis recommendedfor a.c. servomotors fed with high frequencyvoltage. The electromagnetictorqueis proportionalrc D!,r, i.e. D" t e|m

"

)--

4 tt t Ls

,S"7- cOSV )on

: T kok -, D3ou, B ^n Arn cosv

(2.e4)

where P"1* is the active electromagneticpower and V is the anglebetweenthe stator current Io and EMF 81.

Prr'nciplesof AFPM nachines

2.9

57

Armature reaction

The magneticfluxes produced by the stator (armature) can be expressedin a similar way as the field excitationflux - eqn (2.19),i.e. r in the d-axis

t Qod:

aBrnod.t"

- nt L2o,l ),

7T ""**'p

R?,

(2.es)

2

r in the q axis )

aO'r-R?n

(2.e6)

Qoq: }"*on',;'"t

,

where Bmad1 is the peak value of the first harmonic of the stator (armature reaction) magnetic flux density in the d-axis and Brno4 is the peak value of the first harmonic of the stator magnetic flux density in the q-axis. The stator linkage fluxes are r in the d-axis

*o:

e.s7l

ry #*tk,oteaa: i*rk-t1B-,"ar|

r in the q axis

, I', o: *& \/2

k-te o s: l N rn -,,? o *-- ,!R?' "' iBmaqT--*f v,

- R?'

(2'98)

where l/r is the number of statorturns per phaseand k.,1 is the winding factor for the fundamentalspaceharmonic. Neglecting the magnetic saturation,the first harmonic of the statormagnetic flux density normal componentsare r in the d-axis

Bmailr: kydB,ood,: kya\aFoa:kro#*f

Yr"o

Q.gg)

MAGNET BRUSHLESS MACHINES AXTAL FLUXPERMANENT

58 r

in theg-axis

B - o q t - k l a B - o d- k l a A q F o o :t r t u * y Q ! ! ! , . n qaftP

(2.100)

where rn1 is the number of statorphasesand the perrneanceper unit surfacein the d- and q-axisare \ . - \-

/\t1

/\0

r

-lto

-

-

/\q

I

l'o

ea

(2.r0r)

In the aboveequations(2.99)and (2.100)theformfactors of armaturereaction are defined as the ratios of the first harmonic amplitude-to-maximum value of normal componentsof armature reaction magnetic flux density in the rJ- and q-axisrespectively,i.e. , t'Jd -

BôdL

B-oql ".lQ

Bôrl

(2.r02)

Bn,oq

The equivalent air gaps in the d- and q-axis for surface configuration of PMs are r for a stator with ferromasnetic core g' :

qkckrot I

nM

( 2 . 10 3 )

tt--

g'n: skck"atqlhxll

(2.104)

. e':2 [{n* o.tr-; H)

(2.los)

9n: 2l(g + 0.51,,)-t h'vl

Q.ra6)

r for a corelessstator

where f,., is the statorwinding axial thickness,h1,1is the axial height of the PM andp,rr"" is the relative recoil permeability of the PM. To take into accountthe

PrinciplesoJ'AFPMmachines

59

effect of slots, the air gap (mechanical clearance)for a slotted ferromagnetic core is increasedby Carter coefficientkc > l accordingto eqn (1.2). The saturation of the magnetic circuit can be included with the aid of saturation factors k"ot ) 1 in the d-axis and k"o4 ) 1 in the g-axis. For a coreless stator the effect of magnetic saturationof the rotor ferromagnetic discs (core) is negligible. The MMFs fo4 and fon in the d-axis and q-axis are m t t/'l 1\ 1K61 , tad '1T p

rad-

li

uT a q- 1 f p

rr

(2.101)

t

Tn1 V Z 1\1K,1tr1 'aq

(2.108)

where Io6 and Ion arethe d- and q-axis stator (armature)currentsrespectively. The armaturereaction (mutual) inductanceis calculatedas r in the d-axis

,

Lad:

vd.

1

mrpo-

lad

7f

(Yy)'@58-r,,

(2toe)

r in the o-axis T"aq

\trq

1

TnILLOlaQ

7T

(R3*:,R?,) r,n

(:F)'

(2110)

For p,,ec = 1 and surface configuration of PMs (kf a : kf n : 1), the d- and g-axis armaturereaction inductancesare equal, i.e.

r d - I L _a er - m t q \ i \ Lr a : _ La

' (n3,,- R?,)

l/l/r/c-r\ ,

( 2 . 1l 1 )

/

For surfaceconfigurationof PMs kfa: kfn : I 196]. For other configurations, equations for MMFs in the d- and q-axis will contain reaction factors kya* kfq196l. The armaturereaction EMFs in the d- and q-axis are Ead,: nt/2f Nrk*rQoa

(2.112)

60

Table 2.1.


Armature reaction equations for cvlindrical and disc-tvoe machines

Quantity Armature reaction magnetic flux in the d-axis Amature reactlon magnetic flux in the q-axis Pemeance ofthe air gap in the d-axis Permeance ofthe air gap

Cylindrical machine

Disc-typemachine

Qo4 :

1B-o41rLi

oua: lB,noatff(R\,, - R7,,)

Qo, :

z Brrru,rlrLi

Qou: ? Bn,untfi(R|*t - R7,,)

ô: #*fi( RZ- ,- R?^)

A ,' ,g ,:T E 9 Z r L ,

A

'

-

1Ls2 r

l,o2-r 9;, n

in the q-axis Permeanceof the air gap per surface ofone pole in the d-axis Permeanceof the air gap per surface areaof one pole

\ ,

l/J)

\-

c.9 sit

R?")

1p2

o'r"'"u'

T

in the q-axis

Armaturereaction feactance in the d-axis Amature reactlon

\' tr"rt

--

D",l ta

Eoa I a,t

Y,-

k,"t\2 4 , n , t r,n( Nl rU # r # tr 'I ,' , ,

z R2..

Nr Au,r

:2mtp,of

X,"r: 1-

Y "

, o q

k.ra

-Eaq ,oa

reactance \2

in the q-axis Armature reaction inductance in the d-axis

r uod

--

:2mttt,tf({*.t)'Wo,,

f"u: H

Vo,t I,n-

- 2 n r- ' r r , r 0 4 d ' t ' , , k. l r , t r'D

'

- m.rpo+/ A ' rA . , r \ 2 \

r

Armature reactron inductance in the q-axis

r,

-'2rrrrl,.uJ*{?f

.,,,

R f , , , - R 7 ,t,. ,

I

'!

ot

"ta

n.o

('' ,,')'Wr,,

: nL1lr(r+

61


(2.rr3)

Eoq : nt/2f N1k*1@,o

where the armaturefluxes Qo6 and Qon areaccordingto eqns(2.95) and(2.96). The armature reaction reactancescan be calculatedby dividing EMFs .tr"a arrdE aqaccordingto eqns(2.112) and(2.1I 3) by currentsI o4 andI on,i.e. r in the d-axis -

Xad.:2trfLoa

- 2m, u^, ( Ntkn\2 ''""\ p /

Eo,l, I od.

(R3,,- R?,),-

*Id

d

(2.1t4)

r in the q-axis

Xaq:2trfL"n :

:2mr - ' un, -" ( \

Nft-r\2 p /

Eon Jas

(R2,,- R7),

-.

-Kfn

g;

(2.rts)

Table 2.1compares the armature reaction equationsbetween conventional cylindrical machinesand disc-type machines.

2.10 AFPM motor 2.10.1 Sine-wavemotor The three-phasestator winding with distributed parametersproduces sinusoidal or quasi-sinusoidaldistribution of the MMF. In the caseof inverter operation all three solid stateswitchesconduct current at any instant of time. The shapeof the stator winding waveforms is shown in Fig. 1.3b. The sine-wave motor works as a PM synchronousmotor. For a sinusoidalexcitation (synchronousmachine) the excitation flux can be found on the basisof eqn (2.19) or (2.21),the EMF per phaseinducedby PM rotor can be found on the basis of eqn (2.29) and the electromagnetictorque on the basis of eqn (2.26). The EMF constant kB and torque constant k7 are expressedby eqns (2.30) and (2.27) respectively.

62

2.10.2

AXIAL FLUX PERMANE]{TMAGNETBRUSHLESS MACHINES

Square-wave motor

PM d.c. brushlessmotors with square-wavestator current (Fig.. 1..3a)are predominantlydesignedwith largeeflectivepole-arc coefficients,a)"q,, where *f,"n' : br,(r)I r (r) . Sometimesconcentratedstatorwindings and salientstator poles are used.For the Y-connectedwindings,as in Fig. 2.17, only two of the threemotor phasewindings conductat the sametime, i.e. i"as (TlT4), ioAC (T1T6), i"s6 (T3T6), i"pn (T3T2), i"sa (T5T2), i,,sB (T5T4), etc. At the on-time interval (120") for the phasewindings A and B, the solid state switchesT1 and T4 conduct(Fig.2.l7a). when Tl switchesoffthe current freewheelsthrough diode D2. For the off-time interval both switchesTl and T4 areturned-offand the diodesD2 and D3 conductthe armaturecurrent which chargesthe capacitorc. If the solid statedevicesare switched at relatively high frequency, the winding inductance keeps the on-off rectangular cunent waveforms smooth. For d.c. currentexcitationur * 0, eqn(2.75) is similar to that describinga steadystatecondition of a d.c. commutatormotor, i.e.

V4.: Elr r ]_2RrIGo)

(2.r16)

where 2fi1 is the sum of two-phaseresistances in series(for Y-connectedphase windings), and Eyr-t is the sum of two phase EMFs in series,V;" is the d.c. input voltage supplying the invefter and I["q) is the flat-topped value of the square-wavecur:rentequal to the inverter input curent. The solid switch voltage drops have been neglectedin eqn (2.116). The phasoranalysisdoes not apply to this type of operationsincethe armaturecurrentis nonsinusoidal. For a rectangular distribution of Brns : const with the pole shoe width bp { , being included,the excitationflux is

- R?.) *fn): ol"d u,*n#(R3,,

(2.117)

For a square-waveexcitation the EMF induced in a single turn (two conductors) is 2B*nLiu : ApnB,noL;r. Includingbo andfringing flux, the EMF for Nrk*t turns e.1 : 4pnNrk.urc'l"q)B,ns+,(R|,,, - R7) : 4pnN1k-yQt7'q). For the Y-connectionof the armaturewindings,as in Fig. 2.l7,two phasesare conductingat the sametime. The line-to,line EMF of a Y-connectedsquarewave motor is

Eyr-L : 2ef : 8pl/rk.r al'q)B,,n(nl2p)(R3*,- n?,)" : BpNlk-1Ffn',

: k E,t.tt,

(2.118)

63


I I IT

C

I I

+

f "

dc

I ! I

I I

lr" I

-

l-.>

U

* o '.t- f r .

\T

uzl .l

\

'4

I

__)

\'.1 ""il]lf

tl tl

F,

y.,'\

C

I o,,

(a)

Figure 2.17. Inverter currents in the stator winding ofa PM brushlessmotor: (a) currents ln phasesA and B for on*time and off-time intervals, (b) on-offrectangular current waveform.

where the EMF constant or armatureconstantks4" is k*d.:

lPN1k-1Q\fq)

(2.rre)

The electromagnetictorque developedby the motor is P"t* 2rn, ,1

Eyur,I["q) 2nn

,

= lply'r k-rotfn'If"o) : kra"I|"n)

(2.r20)

7f

where the torque constant of a square-wavemotor is

kTd":*:lnnlvrt-rofn)

(2.121)

64


and I;"'f is the flat-top value of the phasecurrent. The ratio Ta of a square-wavemotor-to-Ta qf a sinewavemotor is

,:"' :!C!i!['o' z.-;rtaf

I"

r;q) -^cefq) =''o*, L

(2.t22)

Assuming the same motor (*l"n' : ct6)and,the samevalues of air gap magnetic flux densities,the ratio of the square*wavemotor flux to sinewavemotor flux is 5- ( - s a ) q ^ (i " q )uof r l qtt-t , | J lQdl(R\", _

- R?,)

oi-ffi:1t,

1

t, ^t '.t^z^ r. )

where [96]

r, : *.u,(i.)

(2.124)

Q7 : @yr: aikSB,nn*rrO3,,- R?,)

(2.r2s)

and

on the basis of eqns (2.118) and (2]20) the torque*speedcharacteristiccan be expressedin the following simplified form fl

.

Io

:t-t-

no

Insh

T,l

.

Td"t

(2.126)

where the no-load speed, locked rotor armature current and stall torque are respectively

n o - W,. kE

I""n-i

vd,

Trlrt:

kr6.Io"h

(2.r27)

For half-wave operationi? : Rr while for full-wave operationR : 2Rt. Note that eqn (2.126) neglectsthe armaturereaction, rotational and switching losses. The torque-speedcharacteristicsare shown in Fig. 2.18. Eqns (2.126) and, (2.127)arevery approximateand cannotbe usedin calculationof performance characteristics of practicalPM d.c. brushlessmotors. Theoreticaltorque-speed characteristics(Fig. 2.18a) differ from practical characteristics(Fig. 2.18b). The continuous torque line is set by the maximum rated temperatureof the

65


(a)

m

() f

Y

o f g"

_

peak stalltorque peak rated torque

T

cont. stall torque cont. rated torque

raled speed

speed Figure 2.18. tical.

speed

Torque-speed characteristicsofa PM brushlessmotor: (a) theoretical, (b) prac-

motor. The intermittent duty operation zone is bounded by the peak torque line and the maximum input voltage. The rms stator current of a d.c. brushlessmotor for a 120o squarewave (T : 2r lw) is T-

LO.

-

+1,"'"

iz(t)dt:

?l::;"

/fotlzot

1 r ' \ - r(ss) to, 6;)

2.ll 2.ll.l

(2.t28)

AFPM synchronous generator Performance characteristics of a stand alone generator

An AFPM machine driven by a prime mover and connectedto an electric load operates as a stand alone synchronousgenerator. For the same synchronous reactancesin the d'and g-axis Xsd. : X"q : X" : aL", and load impedanceper phase .l

Zr,:Rr*jwLp-i

u a C^

(2.tze)

66

AXIAL FLUX PERMANENTMAGNETBRT]SHLESS MACHINES

the input current in the stator (armature)winding is ,-EI

(2.l 30)

The voltage acrossthe output terminals is

,r:r"W

(2.r3r)

characteristicsof EMF per phase t;, phase voltage vt, stator (load) current Io, output power Po6, inpttt power P;,r, efficiency q and power factor pf : cosl versusspeedn ofa standaloneAFPM synchronousgeneratorfor inductiveloadZy : Rr * jaLr areplottedinFtg.2.l9.

2.11.2

Synchronization with utility grid

An AFPM synchronousgeneratorcan be synchronizeddirectly (connected in parallel) with the utility grid. The number of pole pairs is usually minimum2p: 6, so that to synchronizea 6-polegeneratorwith a 50Hz power system(infinite bus), the speedof the prime mover must be n : 60(f lp) : 60(50/3) : 1000 rpm. For 2p : 72 and the samefrequency,the speedwill drop to n : 500 rpm. In general,direct parallelingof AFPM generatorsrequireslow speedprime movers.Before connectingthe generatorto the infinite bus, the incoming generatorand the infinite bus must have the same: r voltage r frequency r phasesequence r phase In power plants,thoseconditionsare checkedusing a synchroscope. AFPM synchronousgeneratorshave recently been used in distributed generator systemsas microturbine- driven high speed self-excited generators. A microturbineis a small, singleshaftgasturbine of which the rotor is integrated with a high speedelectricgenerator(up to 200 000 rpm), typically ratedfrom 30 to 200 kW. If an AFPM generatoris drivenby a microturbine,in most cases the speedexceeds20 000 rpm. The high frequency current of the generator must first be rectified and then inverted to obtain the same frequency as that of the power system. To minimize the higher harmoniccontentsin the stator windings,an activeratherthan passiverectifier is used(Fig. 2.20).

67


(b)

(a, Ef

t-

l1

v1

(c) Port P,n

Figure 2 .19. Characteristicsof a stand alone AFPM synchronousgenerator for inductive load (a) EMF .try perphase andphasevoltage l/r versusspeedn, (b) load cunent Zr, : Rr,l juLt 1o versus speedn, (c) output power Pout andinput power P;, versus speedn, (d) efficiency 4 and power factor pf: cos @versus speed z.

active rectifier

cl+

Figure 2.20.

Power circuit of a microturbine driven AFPM generator.

68

AXIAL FLUX PERMANEI,IT MAGNETBRUSHLESS MACHTNES

Numericalexample2.1 Find the armaturecurrent, torque, electromagneticpower and winding losses of the S802FAFPM d.c. brushlessservomotor (Table 1. 1) at n : 2500rpm and input voltage230 V (line-to-line). Solution The line to line EMF is

L . s t - L: t / J k ' e a:" nf i * : t ^ ^ " + P : 2 p , 2 4 Y l(X)t) ttO because theEMF constantkEd..: 42 V 11000 rpm is for thephaseEMF. Assumingthatthed.c.busvoltageis approximately equalto theinputvoltage,the amaturecurrentat2500rpm is TGq) -

ro

230, 278.24 : 7.74A 2 x 0.76

becausethe resistanceper phaseis 0.76 0. The shaft torque at2500 rpm is T : kra"Ilss) : 0.64 x 7.74: 4.95Nnr becausethe torque constantis k76": 0.64 NmiA. The electromagneticpower (only two phasesconduct current at the same time) is PeIm: E1r. rlf,"') :218.24 x 7.74t 1689W The windine lossesfor 2 x 0.76 O line-to-lineresistanceare LP,J : (2 x 0.76) x 7.742- 91 W

Numerical example 2.2 A three phase,Y-connected,2p : 6 pole, AFPM brushlessmotor has the surfacePM of inner radius Rtn : 0.06 m, outer radius Rout : 0.11 m and the pole shoe width-to-pole pitch ratio a.i : 0.84, the number of turns per phase-Aft : 222, the winding factor k-1 : 0.926 and the peak value of the air gapmagneticflux densityBms :0.65 T. Neglectingthe armaturereaction, find approximatevalues of the EMF, electromagnetictorque developedby the power at n : 1200rym andrms currentIo : 73.6 motor and electromagnetic A for:


69

(a) sinewaveoperation (V : 0o) (b) 120" squarewave operation Solution (a) sinewave operation at V : 0o The speedin rev/s n:

1200 :20lrev ls 60

The form factor of the excitation field

r, : lsinff : !,,i"9+ll : 1.233 The excitation flux according to eqn (2.19) is 2

oy: o;r :1r

7tb

1 . 2 3x80 . 6 b x ! x e . I f - 0 . 0 6 2:)0 . 0 0 1 1 4 W b

where B,nsl : kf B*n.The EMF constantaccordingto eqn (2.30) is kn : npt/iN1k,,o1Ly: 7Tx g x \/, x 222 x0.96 x 0.00114: 6.452Vs The EMF per phaseaccording to eqn (2.29) is Ef : knn:0.6.452 x 20 :129V The line-to-line EMF for Y connectionis

E1r,-L: JSnt : rt x L29: 223.5Y The torque constantsaccording to eqn (2.27) is kr:\pNtk-t@!: \/2-

x 3 x 2 2 2 x 0 . 9 6x 0 . 0 0 1 1 4 : 3 . 0 8 N m / A + \/2

The electromagnetictorque developedat 14 A according to eqn (2.25) is Ta: krlo:

3 . 0 8x 1 3 . 6: 4 1 . 9N m

The electromagneticpower is Perrn: mlBylacosV : 3 x 729 x 13.6 x 1 : 5264.5W

7O

MACHII,{ES AXIAL FLUX PERMANENTMAGNETBRUSHLESS

where iU is the angle betweenthe EMF Ey and stator current Io. (b) 120" square-waveoPeration The flat-toppedvalue of the phasecurrentaccordingto eqn (2.128)is t;

r;

: ,,1|r": A t te.e: 16.66 If,"o) /U to eqn(2.117)is Theexcitationflux according wb ofn) : 0.84x 0.65x 6n"x (0.tt2 - 0.062): 0.00122 flux to sinewaveflux The ratio of the square-wave

*fn) o;

: o.oor22:1.02 0.00114

Note that it hasbeenassumedthat ai : 2ln : 0'6366 for sinewavemode and oi the flux ratio is equalto al"q) - 0.84 for squarewave mode, For the s€rrne ! lkt. The EMF constantaccordingto eqn (2.119)is B x 3 x 222x0'96 x 0'00122:L2'43Ys

kE,t":BpN1k-1*fn':

The line-to-line EMF (two phasesin series)accordingto eqn (2.1l8) is

E yt-L :

l '2 .4x3

1200

ff

: 248.6 V

The torqueconstantaccordingto eqn (2.121)is krd,:

r:0" 2n

:12'43 2r

- 1.g7gNm/A

The electromagnetictorqueat ltss) : 16'66 A accordingto eqn (2'120) is Ta : kra"Il"q) : 1'978 x 16'66 : 32'95 Nm poweris The electromagnetic Petm: Ett-rlf,"q) :248'6 x 16'66: 4140'6W


7l

Numericalexample2.3 Find the main dimensions,approximatenumber of turns per phaseand approximate cross section of the stator slot for a three-phase,double-sided,doublestator disc rotor PM brushless motor with a laminated stator core rated at: Po't : 75 kW VrL :460 V (Y connection),.f : 100 Hz, n" : 1500 rpm. The stator windings are connectedin series. Solution For / : 100 Hz andn": 1500lpm : 25 revlsthe numberof polesis 2p : g. Assumingka : Dn/Dout : IlJ7 the parameterkT,laccordingto eqn (2.89) is kD:

:('.#) ['-

: 0.131

(*)']

For a 75 kW motor the product q cos$ x 0.9. The, phasecurrentfor series connectedstatorwindings is rf a- -

P^rt 75 000 : 104.6 A m 1 ( 2 V g r 1 c o s $ 3 x 2 6 5 . 6x 0 . 9

where 2V1 : 4601\/3 : 265.6 V. The electromagneticloading canbe assumedas Brnn: 0.65 T andArn: 40,000 A/m. The ratio e : Et lVr x 0.9 and the stator winding factor has been assumedk,,:r : 0.96. Thus, the stator outer diameter according to eqn (2.93) is Dout:

0.9 x 75 000 :0.452rn a'2x 0.131x 0.96x 25 x 0.65x 40 000x 0.9

The inner diameter according to eqns (2.20) rs Din:

D
\/3

0.452 :0.261m,

Jz

The magnetic flux according to eqrr(2.21\ is r ^/ )o x0.65 x --l' x 0 . 4 5 2 2 (t Q'r : r Ex4 \

1

\

^ I :0.00555Wb (r/J)'/

The number of stator turns per phaseper stator calculated on the basis of line current density according to eqn (2.86) is n, _ nD*t(t + ka)A* _ 7rx0.453x 1t +tlrt)x ",4*rrf2h 4yffi^1g43

40000 _ *, -"'

72

AX]AL FLUX PERMANENTMAGNETBRUSHLESS MACHINES

The numberof statorturns per phaseper statorcalculatedon the basisof eqn (2.87) and(2.92)is ^r "

eVt r{2f

k",1QI

0.9x 265.612

=50 nt/1 x100 x 0.96x 0.00555 A doublelayer winding can be located,say,in 16 slotsper phase,i.e. s1 - 4g slots for a three phasemachine. The number of tums should be roundedto 48. This is an approximatenumber of turns which can be calculatedexactly only after performingdetailedelectromagneticand thermalcalculationsof the machine. The numberof slotsper pole per phaseaccordingto eqn (2.2) is s1 ''-2p.r

- :4' )8 Bx3-'

The number of statorcoils (double-layerwinding) is the sameas the number of slots,i.e. 2pq1my: B x 2 x 3 :48. If the statorwinding is madeof four parallel conductorsau : 4, the numberof conductorsin a singlecoil for one parallel currentpath ar: 1 accordingto eqn (2.$ is o,,,lVr

"'' - W,r) A/

-

-

4 x 48

'' 14gJg7

1.)

The current density in the stator conductorcan be assumedJ, x 4.5 > 106 A/m2 (totally encloseda.c. machinesrated up to 100 kW). The cross section areaofthe statorconductoris Io

104.6

aru Jo.

4x4.5

DO-

:

).

D.6I1mn]-

The stator winding of a 75 kW machine is made of a copper conductorof rectangular cross section. The slot fill factor for rectangular conductors and low voltagemachinescan be assumedto be 0.6. The crosssectionof the stator slot should, approximately,be 5 . 8 1 1x 1 2 x 2

0.6

= 233 mm2

wherethenumberof conductors in a sinsleslotis 12 x 2 : 24.Theminimum statorslotpitchis

t'î'n:

ry

: -f:

m : 17'1 o'0171 rrrur

PrinciplesoJAFPM machines

73

The stator slot width can be chosento be ll.9 mm; this meansthat the stator slot depth is 23311L.9 = 20 mm, and the stator narrowest tooth width is crrnin : 17.1 - 11.9 : 5.2 mm. Magnetic flux densityin the narrowestpart of the stator tooth is B * s t r * i n - 0 ' 6 5 - x1 7 ' 1 :2'I4T B 7 t u n o-' CImin

5.'Z

This is a permissible value for the narrowestpart of the tooth and silicon electrical steelwith saturationmasnetic flux densitv of 2.2 T. The maximum stator slot pitch is tr,no,:

TDert

:

r x 0.453 48

;

: 0 . 0 2 9 6m :

2 9 . 6m m

The magnetic flux density in the widest part of the stator tooth is B7t'ni'' =

B*nt"no' "rrr"*

-- 0'65 x 29'6 -: 1'09 r'ur Tr 29.6- 1l-9

Numericalexample2.4 A 7.5-kg cylindrical core is wound of isotropic silicon steel ribbon. It can be assumedthat the magnetic field inside the core is uniform and the vector of the magnetic flux density is parallel to steel laminations. The magnitudes of magnetic flux density time harmonics are: BrnL - 1.7 T, B,n3 : 0.25 T, Brns : 0.20 T and B-7 : 0.05 T. Harmonicsn, > 7 are negligible. The electric conductivity of laminationsis op" : 3.3 x 106 S/m, specificmass densitypFe :7600 kg/m3, thicknessdF" :0.5 mm and Richter'scoefficient of hysteresislossese : 3.8. Find the core lossesat 50 Hz. Solution Eddy-current losses The coefficient of distortion of the magnetic flux density according to eqn (2.45\ is 0.25\ 2 L?)

0 . 2 0 \2 L7)

0.05\ 2 : 1.258 r.t

/

The eddy current lossesaccording to eqn (2.44) arc r specific lossesat sinusoidalmagnetic flux density

Lp""i,:

+

^=#

: \.29w/kg x 502x 0.0005 x 1.72

74

AXIAL FLIJX PERMANEI{TMAGNETBRUSHLESS MACHINES

r specificlossesat nonsinusoidal magneticflux density Lp.:

:2'04w/kg LP""i,n x n3:7'29 x 7'2582

r lossesat sinusoidal magneticflux density LP"ri.n- Lp""inrrtFe:1.29 x 7.5 : 9.68W r lossesat nonsinusoidalmagnetic flux density LP" :

Ap"*Fu : 2.04 x 7.5 : 15.32W

Hysteresislosses The hysteresislossesmay be calculatedaccordingto eqn (2.46), in which Richter'scoefficientof hysteresislossese : 3.8 r specific lossesat sinusoidalmagnetic flux density

Lryh,tn:3'8x

#

"

502x 1'72: 5'49wlkg

r specificlossesat nonsinusoidalmagneticflux density LPn:

LPlrin x 11]:5'49 x L2582: 8'69 Wlkg

r lossesat sinusoidalmagnetic flux density L P n " m : L p h " t n m p " , : 5 . 4 9 x 7 . 5 : 4 1 . 1 8W r lossesat nonsinusoidalmagneticflux density LPh: Lqnmp.:

8 . 6 9x 7 . 5 : 6 5 . 1 9 W

Specificeddy currentandhysteresislossesplottedagainstfrequencyare shown inFig.2.2I. Total losses The eddy cun:entand hysteresislossesat sinusoidalmagneticflux density are calculatedas

75


.t o- t 6 I

t4

AP "rin(0 Ap .(t)

50 I i/

l2

-:

l0 R

_ipii"ttr

o

_^r_{0

4

r /?'

:/./

t' l/*

.-/ / /- /

.r''

"

' :;-"'*'.....,-.

2 S.

//

' ./ '

,

",:

"""'

0l 0

100

r00

0 Figure 2.21. 2.4.

Specific eddy-currentandhysteresislossesversusfrequency.Numerical example

APF""I^ :

LP""tn -l LPn"tn: 9.68 + 41.18 : 50.86W

The eddy curent and hysteresislossesat nonsinusoidalmagnetic flux density afe

LPp.:

LP. * LP':

15.32+ 65.19: 80.51W

Theratio of hysteresis*to-total lossesis the samefor both sinusoidalandnonsinusoidalmagneticflux densiryi.e. 41.18

65.19 : 0.81

50.86 80.51

Numericalexample2.5 Find the power lossesin PMs and solid rotor backing steel disc of a singlesided AFPM with slotted stator at ambient temperature of 20oC. The inner diameterof PMs is D,;n: 0.14 m, the outer diameterof PMs is Doul : 0.242 m, number of poles 2p : 8, magnet width-to-pole pitch ratio is at : 0.75, height of surfacepermanentmagnet is hxy : 6 mm, air gap thicknessg : 1.2 mm, number of slots 5t : 36 and speed n : 3000 rpm. Assume the peak valueofthe air gapmagneticflux densityBms :0.71 conductivityof NdFeB PMs op- : 0.8 x 106 S/m at20oC, its relativerecoil magneticpermeability Frrec:1.05, conductivityof solid disc op" : 4.5 x 106S/m at20oCand its relativerecoil magneticpermeability1tr: 300.

76

AXIAL FLUX PERMANEI,{T MAGNETBRUSHLESS MACHTNES

Solution The averagemagneticflux density is Born : (2lr)B*n : (2ln) x 0.7 : 0.4661the average diameterD :0.5(Dout I D,n) :0.5(0.242 + 0.14) : 0.191m, the averagepole pitch is r : nD lQp) : r0.Igll}: 0.075m, the :0.017 m. The frequencyof averageslot pitch h : rDlsl : 7T0.191136 the fundamental harmonic of the slot component of the magnetic flux density accordingto eqn (2.49) is

f "t : stpn: 36x 4 t

3#

: 72oo Hz.

The correspondingangularfrequencyis c,r"l: 217200: 45 238.9 radls. The equivalentair gap in the d-axis for kct : 1 accordingto eqn (2.103)is

q q0=6 st :0.0012+ 1 . 0 5 : 0.006914rn The parameters rc : 0.434,0a:0.041, | : 0.18 anda,"1: 0.546havebeen found on the basis of eqns (2.53), (2.52) and (2.51) respectively.The Carler coefficientis k6 :1.012 - eqns(1.2) and (1.3). Thus, eqn (2.50)givesthe magneticffux densitycomponentdue to slot openingas B s r : 0 . 5 4 6 x 0 . 0 4 1x 1 . 0 1 2x 0 . 4 4 6 : 0 . 0 1 0 1 8T The parameters k : 139.283l/m for prrec:1.05, A : \lk - 0.0072m, o : 1 3 9 . 2 8 +3 j 1 3 9 . 2 8 3 1 l m , 0 : r l Q . S t t ) : r l ( 0 . 5 x 0 . 0 1 7 ): 3 7 7 . 2 4 1 llm, n - 386.668+ j50.962 llm, as, - 2.733 andaa, : 0.366 for u : 1. havebeenfoundon the basisofeqns (1.12),(1.11),(1.6),(1.12),(1.16)and (1.17) respectively.The transverseedgeeffect coefficientk" accordingto eqn (2.55)is 0 . 5x 0 . 0 1 7 A'z:r*olqz llq

l'u6l

The activesurfaceareaof PMs accordingto eqn (2.56) is x l(0.5 x 0.242)2- (0.5 x o.ta)21: 0.023rn2

Sp*r :035r

The activelossesin PMs accordingto eqn (2.51is LPpv:

1

t

2

+ j139.2831 x 2 .7 3 3 xr.o stx ( 1 1 3 9 . 2 8 3 377.24r

0.01018 " ( 0.4tr x 10 6 x 1.05)'

)'

: 1r 8.2w ,, -1 39?!t x 0.023

77

Principles of AFPM machines The reactive power losses in PMs can be estimated as

aX' x 118.2: 15.8/VAr LQpm A,Pp1u: 3* aR, 2.733 For calculation of power lossesin the solid steel disc of the rotor, the magnetic flux densitycomponentdue to slot openingsB"t : 0.01018T is the same.The relative magnetic permeability F, : 300 and conductivity of the disc op" : 4.5 x 106 S/m are different, so that the attenuationfactor (1.12) calculatedfor thesevalues will be k : 6194.6 1/m. The coefficientsaRFe : 1.451 and axFe -- 0.849 are calculatedusing eqns(2.57) and (2.58) for 471- 1.45 and (trx : 0.85. The remaining field parameters,i.e. a : 8982 -t j5265 llm, 13: n I (0.5tt) : r I (0.5 x 0.017: 377.241llm andrc: 8988* j5362 llm havebeenfound from eqns(1.11), (1.6) and (1.12)respectively. The surfacearea of the rotor disc according to eqn (2.60) is SF": \ x (0.2a22- 0.742): 0.031m2 4' The active power lossesin the rotor disc according to eqn (2.59) is

x 1.081 x ffgYi4Tlf)' LPr.z:1' 1.4b1 2 ZrZ.z+1 \

0.0101s ,. 1(o.4;ffi)

\2

6194.6

" *;lou

)

: l8"tw x o'031

The coefficients k, : 1.081is the sameas for calculationof lossesin PMs. Thereactivepowerlossesin therotor disccanbe estimatedas LQP.z-axFe'

^n

ffitor.z:

0'84q

lfr

: 10.8 x 118.2 VAr

Numerical example 2.6 Find the rotational loss of a 3000 rpm disc type machine with outer radius of the rotor Rout : 0.2 m, radius of shaft behind the rotor core ,R"7,: 0.025 m, mass of the rotor m, : 2.72 kg and mass of the shaft.m"7, : I.49 kg. The machine is naturally cooled with air. The ambient temperatureis 20" C. The coefficient of bearing friction is ky6 : 1.5. Solution The bearing friction lossesaccording to eqn (2.64) are

L,P1,:0.06 x 1.5x \2.72+ 1.49)

3000 60

1 8 . 9 6W

78


The Reynoldsnumberaccordingto eqn (2.65) is Re:1.2, ,

2n r-5|';r

x

3ooo x 0.22 : 8.378 x 105 ;;

where the air density is p : 1.2 kglm2 and dynamic viscosity of the air is /l : 1.8 x 10-5 Pa s. The coefficientof drag accordingto eqn (2.66) is 3.87

"/:75=ffi*:4.228x10*3 The windagelossesaccordingto eqn (2.67) ue / 1 3000\ LP*;,d -;x4.228x10-3xl.2x (Zn x = ) ( 0 . 2 5 - 0 . 0 2 5 t: )2 5 . 1 7 W 2 6Ul \ The rotationallossesare thus LP""t:

18.96+ 25.17: 44.13W

Chapter3 MATERIALS AND FABRICATION

3.1

Stator cores

Statorcoresof AFPM brushlessmachinesare made of laminatedsteelsor soft magneticpowder materials. Soft magneticpowders simplify the manufacturing processand reducethe cost of AFPM machines.

3.1.1

Nonorientedelectricalsteels

Most laminatedcoresfor statorsof AFPM brushlessmachinesare made of nonoriented (isotropic) silicon steel rlbbons with standardthickness from 0.12 to 0.64 mm. Nonorientedsteelsare Fe-Si alloys with a randomorientationof crystal cubes. Magnetic properties are practically the samein any direction in the plane of the sheetor ribbon. A secondaryrecrystallizationprocessis not neededand high temperatureannealing is not essential. Nonoriented grades containbetween0.5ohand3.25yoSi with up to 0.5% Al addition to increase the resistivity and lower the temperatureof the primary recrystallization. Nonorientedelectricalsteelsare availableasbothfully processedandsemiprocessed products. Fully processednonoriented electrical steels are completely processedby the steel manufacturer, ready for use without any additional processing required to achieve the desired magnetic quality. Semiprocessednonorientedelectrotechnicalsteelsare those which have not been given the full annealingtreatmentby the steel producer. In some cases,users prefer to develop the final magnetic quality and achieve relief of fabricating stressesin laminationsor assembledcoresfor small machines. The most universallyacceptedgradingof electricalsteelsby core lossesis the American Iron and Steellndustry (AISI) system(Table3.1), the so called "M-grading". For small and medium power electricalmachines(outputpower

80

Table3.1.


The most important silicon steel designationsspecified by different standards

Europe rB.c404-8-4 (re86)

U.S.A AISI

250-35-A5 270-3s-A5 300-3s-A5 330-35-A5 2"10-50-45 290-50-A5 3 t 0-50-45 330-50-45 350-50-A5 400-s0-A5 470-50-A5 530-50-A.5 600-50-A5 700-50-A5 800-50-A5 350-65-A5 400-65-A5 470-65-A5 530-65-As 600-65-A5 700-65-A5 800-65-A5 1000-65-A5

M-15 M-19 M-22 M-36

Japan JIS2552 ( l 986) 354250 35/'210 35A300 501.270 504290 504310

M-l5 M-19 M-27 M-36 M43

50A3s0 50A400 501'470

M-45 50A600 50A700 50A800

M-4'7

Russia GOST21427 a-75 l+ 15 L+IL

24t1

2413 1A1a

2411 2312 2311 2212 2112 2111

M-19 M-27 M-43 M-45 65A800 654.1000

2312 2212 2211 2u2

lessthat 75 kW), the following gradescan be used:M-27, M-36, M-43,M-45 and M-47. Laminating a magneticcore is ineffective in keepingexcessiveeddy currents fiom circulating within the entire core unless the surfacesof laminations are adequately insulated. Types of surface insulation include the natural oxide surface,inorganic insulation, enamel,vamish or chemically treated surface. The thicknessof the insulationis expressedwith the aid of the stackingfactor: t-_

d+2L

(3.1)

g'hered is the thicknessof bare lamination and A is the thicknessof the insulation layer measuredon one side. The stacking factor ki is typically from 0.94to 0.97.

81

Materials andfa bricatio n

Table3.2. Specificcore lossesof Armco DI-MAX nonorientedelectricalsteelsM-27, M-36 and M-43 at 60 Hz Specificcore losses Magnetic flux density I

).20 1.50 ).70 1.00 t.30 t.50 t.60 t.70 1.80

wkg 0.36mm I o.+rmm o.o+mm I M - 2 71 M - 3 61 M - 2 71 M - 3 6I M - 4 31 M - 2 71 M - 3 61 M - 4 3 0.09 0.47 0.81 1.46 2.39 3.37 4.00 4.95

0.10 0.52 0.89 r.61 2.58 3.57 4.t9 4.74 s.14

0.10 0.53 0.92 1.6"/ 2 . 6/

3.68 4.30 4.85 5.23

0 . 1I 0.56 0.97 1.7 5 2.80 3.86

0 . 11 0.59 1.03 1.87 2.99 4.09

A <1

A1a

5.08 5.48

5.33 5.79

0.12 0.62 t.1l 2.06 -) - -)z+

4.56 5.34 5.99 6.52

0.12 0.64 1.14 2.12 3.46 4.70 5.48 6.15 6.68

0.13 0.66 t.t7 2.t9 3.56 4.83 5.60 6.28 6.84

Table3.3. Magnetizationcurvesof fully processed Armco DI-MAX nonorientedelectrical steelsM-27,M-36 andM-43 Magnetic flux density T

0.20 0.40 0.70 r.00 r.20 1.50 1.60 1.70 1.80 2.00 2.10 2.20

Magnetic field intensity A/m

M-27 1 36 50 74 116 175 859 2188 4759 8785 26 977 64 93s t37 203

M-36I 41 5'l 80 119 t74 785 2109 4727 8722 26 022 6s 492 t36 9'77

M-43 4'1 64 89 130 187 777 1981 4592 8682 2 68 1 8 66 925 t37 075

Core loss curvesof Armco DI-MAX nonorientedelectricalsteelsM-27, M36 and M-43 testedat 60 Hz are given in Table 3.2. Core losseswhen tested at 50 Hz would be approximate$ A.79times the core loss at 60 Hz. Magnetizationcurvesof the sameelectricalsteelsare given in Table3.3. The specific

82

AXIAL FL UX PERMANENT MAGNET BRUSHLESSMACHINES

Table 3.4. Specific core losses and d.c. rnagnetization curve ofnonoriented thin eiectrical steelsmanufacturedby Cogent Power Ltd., Newport, U.K.

Specificcore losses

wkg Magn. flux dens. T

NO 12 0.12mnr

NO 18 0 . 1 8m m

so l+ool 2.5

HzlHzlkIlz

0.10 0.20 0.30 0.40 0.50 0.60 0.'70 0.80 0.90 1.00 1.t0 1.20 1.30 t.40 1.50 1.60 1.70 1.80

0.02 0.08 0.16 0.26 0.37 0.48 0.62 0.76 0.32 1.09 L31 1.56 1.89 2.29 2.74 3.14 3.49 3.78

0.16 0.'71 1.55 2.57 3.75 5.05 6.49 8.09 9.84 11.8 14.1 16.7 19.9 24.0 28.5

Magn. field

1.65 6.83 15.2 2.54 3'1.7 52.0 66.1 8 3 .I 103.0 156

NO 20 0.2mm

50 laool 2.s

Ilzl

0.02 0.08 0.16 0.26 0.36 0.47 0.61 0.75 0.90 1.07 1.28 1.52 1.84 2.23 z.o/

3.06 3.40 3.69

LIz lkHz

0.18 0.73 1.50 2.54 3.86 5.22 6.77 8.47 10.4 12.3 14.9 1 81. 21.6 25.6 30.0

2.t8 10.6 19.1 3t.7 45.9 61.5 81.1 104.0 161.0 198.0

SffCN.

kA/m

5 0 1 4 0 0I 2 . s

HzlHzlkHz

0.02 0.07 0.t4 0.23 0.32 042 0.54 0.66 0.80 0.95 1.14

0.r7 0.72 1.49 2.50 3.80 5.17 6.70 8.36 10.3 12.2 14.8 I.JO 17.9 1 . 6 5 21.4 2.00 25.3 2.40 2 9 . 7 2.75 3.06 ).JZ

2.79 10.6 24.4 40.4 58.4 78.4 103.0 133.0 205.0 253,0

0.025 0.032 0.039 0.044 0.051 0.057 0.064 0.073 0.084 0.099 0.124 0.160 0.248 0.470 1.290 3.550 '1.070 IJ

massdensityof DI-MAX M-27.M-36andM-43 is 7650.7700and77 50kg/m3 respectively.The stackingfactor is k; : 0.95 to 0.96. The prefix DI-MAX, e.g. DI-MAX M27, designatesa registeredtrademarkwith a special stripannealedprocessthat maximizespunchability. DI-MAX gradeshave superior pemeability at high magnetic flux density, low core losses and good gauge uniformity. A smooth surface, excellent flatness and high stacking factor is obtainedas a result of cold finishing and strip annealing. For frequenciesexceedingthe power frequency of 50 or 60 Hz nonoriented laminationsthinnerthan 0.2 mm must be used.Table3.4 showsmagnetization and specificcore lossesof nonorientedgradesNO12,NO l8 andNO 20 capable of operating up to 2.5 kHz (Cogent Power ltd , Newport, UK). Typical chemicalcompositionis 3.0 % Si, 0.4 o A1,96.6% Fe. The standardthickness of an inorganic phosphatebasedinsulation is 1 pm (one side). The maximum continuous operating temperaturein the air is 230"C, maximum intermittent

83

Materisls and fabrication

Table3.5. Physical properties of iron based METGLAS amorphousalloy ribbons (Honeywell, Morristown,NJ. U.S.A.)

Saturation magnetic flux density, T Specificcore losses at 5OHz and I T, Wkg Specific density, kg/m3

2605CO

2605SA1

1.8

1.59annealed 1 . 5 7c a s t

lessthan0.28

about0.125

7 560

7200annealed 7190cast

0 . 8 1 3r 1 0 6S / m

0.769x 10bS/m

810

900

1 0 0 . .1. 10

1 0 0 . .1. 10

Stacking factor

lessthan0.75

lessthan0.79

Crystallization temperature,"C

430

507

Curie temperature,"C

4ts

392

Maximum service

125

150

Electric conductivity, S/m Hardnessin Vicker's scale E l a s t i c i t ryn o d u l u sG, N r m 2

fpnnarof.'rp

of

operatingtemperaturein an inert gas is 850oC,hardness180 HV and density 7650kg/m3.

3.1.2

Amorphous ferromagnetic alloys

To minimize core lossesat high frequencies,nonorientedelectrotechnical steelsshould be replacedwith amorphousmagnetic alloys (Tables3.5 and 3.6). Amoryhors fenomagneticalloys, in comparisonwith electrical steels with crystal structure,do not havearrangedin order, regular inner crystal structure (lattice). Amorphousalloy ribbonsbasedon alloys of iron, nickel and cobaltareproduced by rapid solidificationof molten metals at cooling rates of about 10tr or crystal"C/s. The alloys solidify beforethe atomshavea chanc€to segregate

84


Table3.6. Specific core lossesof iron basedMETGLAS amorphousalloy ribbons (.Honeyu*ell, Morristown.NJ. U.S.A.)

Magnetic flux density, B T

0.05 0 .1 0 0.20 0.40 0.60 0.80 1.00

Specificcorelosses, Ap, Wkg 2605CO 2605SA1 5 0H z I 60Hz 5 0H z | 60Hz 0.0024 0.003 0.0009 0.0012 0.0071 0.009 0.0027 0.0035 0.024 0.030 0.0063 0.008 0.063 0.080 0.016 0.02 4.12s 0 .1 6 0.032 0.04 0.196 0.25 0.063 0.08 0.274 0.35 0.125 0 .l 6

lize. The resultis a metal alloy with a glass-likestructure,i.e. a non-crystalline frozen liquid. Application of amorphousalloy ribbons to the massproductionof electrical machinesis limited by hardness,up to 1100 in Vicker's scale. Standard cutting methods like a guillotine or blank die are not suitable. The mechanically stressedamorphousmaterial cracks. Laser and electro-discharge machining (EDM) cutting methods melt the amorphous material and cause undesirablecrystallization. In addition, thesemethodsmake electricalcontacts between laminations which contribute to the increasededdy-currentand additional losses.In the early 1980schemicalmethodswere usedin GeneralElectric to cut amorphousmaterials but thesemethods were very slow and expensive ll76]. The problem of cutting hard amorphousribbons can be overcome by using a liquid 1et12201.This methodmakes it possibleto cut amorphous materialsin ambienttemperaturewithout cracking, melting, crystallization and electriccontactsbetweenisolatedribbons.

3.1.3

Softmagneticpowdercomposites

Powdermetallurgyis usedin the productionof ferromagneticcoresof small electricalmachinesor feromagnetic coreswith complicatedshapes.The componentsof soft magneticpowder compositesareiron powder, dielectric (epoxy resin) and filler (glass or carbon fibers) for mechanical strengthening.Powder compositesfor ferromagnetic cores of electrical machines and apparatuscan be divided into 1236): dielectromagnetics and magnetodielectrics, magneticsinters.

M(tterials and fabr icatio n

85

Table3.7. Magnetizationand specificcore loss characteristics ofnon-sinteredAccucore(TSC Ferr i te I nternational, Wadsworth, IL, U. S.A. )

Magnr lzatlon curve Magneticflux Magnetic field density,B intensity, 11 T A/m 0.r0 152 0.20 L J.) 0.30 312 0.40 400 0.50 498 0.60 otJ 0.70 749 0.80 909 0.90 1107 1.00 135'7 1.10 t677 1.20 2101 1.30 2687 1.40 3525 1.50 4763 1.60 6563 1.70 9035 1.7 5 r0,146

ipecific core loss curves

60Hz wkg 0 . 13 2 0.4t9 0.772 1.212 1.742 2.315 2.954 3.660 4.431 5.247 6.129 7.033 7.981 8.929 9.965 10.869 11.707 t2.125

1 0 0H z wkg 0.242 0.683 1.323 2.472 2.976 3.968 5.071 6.305 7.650 9.039 r0 . 5 8 2 12.2r4 13.845 15.565 17.394 19.048 20.63s 21.407

400Hz wkg 1.0s8 3.263 6.2t7 9.811 14.088 18.850 24.295 30.490 37.346 44.489 52.9r1 61.377 70.1 51 79.168 90.302 99.671 r09.880

Dielectromagneticsand magnetodielectricsare namesreferring to materials consistingof the samebasic components:ferromagnetic(mostly iron powder) and dielectric(mostly epoxy resin) material12361.The main tasksof the dielectricmaterialis insulationandbinding of ferromagneticparticles.In practice, compositescontainingup to 2o/o(of their mass)of dielectricmaterialsare consideredas dielectromagnetics.Those with a higher content of dielectric materialare consideredas magnetodielectrics f236). TSCInternational,Wadsworth,IL, U.S.A., has developeda new soft powder material, Accltcore,which is competitiveto traditional steel laminations [4]. The magnetizationcurve and specific core loss curves of the non-sintered Accucoreare given in Table 3.7. When sintered,Accucorehas higher saturation magneticflux densitythan the non-sinteredmaterial.The specificdensity is 7550to 7700kg/m3. Hdgancis,Hogands,Sweden,manufacturessoft magneticcomposite(SMC) powdersthat are surface-coated metal powderswith excellentcompressibility

86

AXIALFLUX PERMANENTMAG}{ETBRUSHLESS MACHINES

Table 3.8. Specific core lossesof Somaloy?A{500+0.S% Kenolube, 800 MPa, treated at 500"C for 30 min in the air, Hriganas, Hriganiis, Sweden

Magnetic flux l^-^:+-. ug[D rLyr

A/m

5 0 H zI 0.4 0.5 0.6 0.8 1.0 2.0

1.5 1.9 2.1 4.6 6.8 30

l 0 0 H z| 3 3.6 6 10 16 50

Specific losses wkg 3 0 0 H 2| s o o g , | t2 t/

l8 2'7

2l

.)+

)z

52 ii0 270

48 170

700H2 27 40 55 92 140 400

l000Hz 45 60 90 t20 180 570

Table 3.9. Magnetizationcurvesof Sornaloy?^t500 +0.5% Kenolube, treaiedat 500"C for 30 min in the air, Hdganai, Hrigands,Sweden

Magnetic field intensity H A/m

I 500 3 200 4 000 6 000 1 00 0 0 1 50 0 0 20 000 40 000 60 000 80 000 1 0 00 0 0

B at density 6690 kg/m3 T

0.7 0.8 0.91 1. 0 1 1.12 t.24 1.32 1.52 r.65 1.75 1.82

D

at density 7l 00 kgAn" T 0.83 1.13 1.22 1.32 1.42 t.52 1.59 1.78 1.89 t.97 2.02

B at density 7180kg/m'' T 0.87 1.19 t.28 1.38 l 5l l 6t 1.69 1.87 t.97 2.05 2.t0

1216).SontaloyTLI500(Tables3.8, 3.9) has been developedfor 3D magnetic circuits of electricalmachines,transformers,ignition systemsand sensors.

8l

Mater ials andJitbricatio n

Figure 3.1. Stator core segmentformed from lamination strip: 1- lamination strip, 2 groove,3 - folding,4 compressedsegment,5- finishedsegment.

3,1.4

Fabrication

of stator cores

Fabrication of laminated stator cores Normally, the stator cores are wound from electrotechnicalsteel strips and the slots are machined by shaping or planing. An altemative method is first to punch the slots with variabledistancesbetweenthem and then to wind the steelstrip into the form of the slottedtoroidal core (R & D Institute of Electrical MachinesV(IES in Brno, Republicof Czech).In addition,this manufacturing processallows for making skewedslots to minimize the cogging torque and the effect of slot harmonics. Each stator core has skewedslots in opposite directions. It is recommendedthat a wave statorwinding shouldbe made to obtain shorter end connectionsand more spacefor the shaft. An odd number of slots, e.g. 25 insteadof 24 canhelp to reducethe cogging torque ([/UES Brno). Another technique is to form the stator core using trapezoidal segments [228]. Each segmentcorrespondsto one slot pitch (Fig. 3.1). The lamination strip of constantwidth is folded at distancesproportional to the radius. To make folding easy,the strip hastransversegrooveson oppositesidesof the alternativesteps.The zigzaglaminatedsegmentis finally compressedand fixed using a tape or thennosetting,as shownin Fig. 3.T 12281. Fabrication of soft magnetic powder stator cores The laminated cores of axial flux machinesare much more difficult to fabricate than those of radial flux machines. SMC powders simplify the manufacturing processof stator coreswith complicatedshapes,in general,3D cores.

88

AXIAL FLUX PERMANEIVT MAGNETBRUSHLESS MACHINES

6800r 400

450

500

550 compacting

600

650

700

750

800

pressure, MPa

Figure 3.2. The effect of compacting pressure on the specific mass density of Hoganiis soft magneticcompositepowders.

Mass productionof AFPM machinesis much more cost effectiveif soft magnetic powder compositesare usedas materialsfor statorcores. Using SMC powders the stator core of an AFPM machine can be made as a slottedcore, slotlesscylindrical core and salientpole uorewith one coil per pole. The slottedand slotlesscylindrical coresfor AFPM machinescan be made in a powder metallurgy process using a ferromagnetic powder with a small amount of lubricants or binders. The powder metallurgy process generally consistsof four basic steps,namely: (1) powder manufacture,(2) mixing or blending, (3) compactingand (4) sintering. Most compactingis done with mechanical,hydraulic or pneumaticpressesand rigid tools. Compactingpressuresgenerallyrangebetween70 to 800 MPa with 150 to 500 MPa being the most common. The outer diameterof the core is limited by the presscapability. Frequently, the stator core must be divided into smaller segments. Most powder metallurgyproductsmust havecrosssectionsof lessthan 2000 mm2. If the press capacity is sufficient,sectionsup to 6500 mm2 can be pressed. Figure 3.2 showsthe effect of compactingpressureon the density of Hdgands SMC powders. For SomaloyTMS00the heat treatment temperature(sintering) is typically 500"C for 30 min. After heat treatmentthe compactedpowder has much less mechanicalstrengththan solid steel. The thermal expansionof conductorswithin the stator slots createsthermal expansionstressesin the stator teeth. The magnitude of thesestressesdepends upon the difference in the temperatureof the winding and core, difference in

Materials and fabricatio n

89

Figure3.3. Slottedstatorsfor smallsingle-sided disc-type motors[236].

Figure j.4. Powder salient pole statorsfor small single-sided AFPM motors. Courtesy of Mli Technologies, ZZC, West Lebanon,NH, U.S.A.

coefficientsof thermal expansionof both materialsand slot filI factor. This problem is more important in powder cores than in laminated cores since the tensile stressof powder coresis at least25 times lower and their modulus of elasticityis lessthan 100 GPa (versus200 GPa for steellaminations). Slotted stators for small disc-type motors fabricated from SMC powders are shown in Fig. 3.3 [148, 236]. SMC powder salient-polestatorsfor small single-sidedAFPM motorsmanufacturedbyMii kchnologies, LLC, Lebanon, NH, U.S.A. are shownin Fig. 3.4. The three-phasestatorhas9 poles. A single SMC powder salientpole manufacturedby Htiganrisis shown in Fig. 3.5.

90


(a)

Figtre 3.5, SMC powder salientpole for small single-sidedAFPM motors: (a) single SMC pole; (b) double-sidedAFPM motor. Courlesy of Hdganris, Hoganiis, Sweden.

3.2

Rotor magnetic circuits

Magnetic circuits of rotors consistof PMs and mild steelbacking rings or discs. Since the air gap is somewhatlarger that that in similar RIPM counterparls, high energy density PMs should be used. Normally, surfacemagnetsare glued to smooth backing rings or rings with cavities of the sameshapeas magnetswithout any additional mechanicalprotection against normal attractive forces. Epoxy, acrylic or silicon based adhesivesare used for gluing betweenmagnetsand backing rings or between magnets. The minimum requiredshearingstrengthof adhesivesis 20 x 106 Pa. There were attempts to develop interior PM rotor for AFPM machines. According to [201], rotor poles can only be fabricatedby using soft magnetic powders[201]. The main advantageof this configurationis the improvedflux weakeningperformance.However,the complexity and high cost of the rotor structurediscouragefuither commercializing development.

3.2.1

PM materials

A PM can produce magneticflux in an air gap with no exciting winding and no dissipationof electric power. As any other ferromagneticmaterial, a PM can be describedby its B-H hysteresisloop. PMs are also called hard magnetic materials, which meansferromagneticmaterialswith a wide hysteresisloop. The basis for the evaluationof a PM is the portion of its hysteresisloop located in the upper left-hand'quadrant,called the demagnetizationcunte (Fig. 3.6). If reversemagneticfield intensityis appliedto a previouslymagnetized, say,toroidal specimen,the magnetic flux density drops down to the magnitude determined by the point K. When the reversal magnetic flux density is removed, the flux density returns to the point tr according to a minor hysteresis loop. Thus, the applicationof a reversefield has reducedthe remanence,ot

91


Bsa Fr

r\-****----**; .K {afi )*ur)

w,H

& r** -*AaBE

Figure 3.6. ability.

Demagnetization curve, recoil loop, energy of a PM, and recoil magnetic pcrme-

remanentmagnetism. Reapplyingmagneticfield intensity will again reduce the flux density,completingthe minor hysteresisloop by retuming the core to approximatelythe samevalue of ffux densify at the point K as before. The minor hysteresisloop may usually be replacedwith little error by a straight line called the recoil line. This line has a slope called the recoil permeability lJr er.'

As long as the negativevalue of applied magneticfield intensity doesnot exceedthe maximum value correspondingto the point K, the PM may be regardedas being reasonablypermanent. If, however, greater negative field intensity fI is applied, the magneticflux density will be reducedto a value lower than that at point 1(. On the removalof H, a new and lower recoil line will be established. Remanentmagnetic.fluxdensityB,-, or remanence,is the magneticflux density correspondingto zero magneticfield intensity. Coercive.fieldstrength Hr, or coercivily, is the value of demagnetizingfield intensity necessaryto bring the magnetic flux density to zero in a material previouslymagnetized. Both B" andH" decreaseas the magnettemperatureincreases,i.e.

B,:B,2olr*ffi@pM_ 20)l Hc: Hczotl +

ffi

(t9v,x1- 2o)l

(3.2)

(3.3)

MACHINES AX]AL FLUX PERMANENTMAGNETBRUSHLESS

92

F

- 0.8 dl

0.6

-1200

-1000

-uo*0o,. -400

-800

-20o

0

", and Be-H demagnetization curves and their variations Figure 3.7. Comparison of B-H with the temperature for sinteredN48M NdFeB PMs. Courtesy of ShinEtstt, Takefu-shi, Fukui Prefecture,Japan.

where Spm is the temperatureof PM, Br2o and Ha2g are the remanentmagnetic flux density and coercive force at 20oC and ap ( 0 and all <1 0 ate temperaturecoefficientsfor B, andHcino/ol"C respectively.Thus, demagnetization curves are sensitiveto the temperature(Fig. 3.7). lntrinsic demagnetizationcuwe (Fig. 3.7) is the portion of the Bt. : f (H) hysteresisloop locatedin the upperleft-handquadrant,where Bt : B - FoH. For f1 : 0 the intrinsic magneticflux densityBt: Br. Intrinsic coerciviQ iH. is the magnetic field strength required to bring to zero the intrinsic magnetic flux density B; of a magnetic material describedby the Bi : f (H) curve. For PM materialstH, ) H.. saturation magneticflur densityB5a1conespondsto high valuesof the magnetic field intensiry when an increasein the appliedmagneticfield producesno further effect on the magnetic flux density. In the saturation region the alignment of all the magnetic momentsof domains is in the direction of the external appliedmagneticfield. Recoil magneticpermeability l1,..is the ratio of the magneticflux densityto-magnetic field intensity at any point on the demagnetizationcurve, i.e.

AB I-trec

-

ltolLrcec

-

A 1 ,:,H

(3.4)

MaterialsandJitbrication

93

where the relativerecoilpermeability ltrrrer:- 1 . . .4.5. Maximum magnetic energyper unit producedby a PM in the external space is equal to the maximum magnetic energy density per volume, i.e. (BH)*o,

un,or

, , ;t , Jl m -

2

(3.5)

where the product (BH),-", correspondsto the maximum energy density point on the demagnetizationcurve with coordinates B,no* and H,no, (Fig. 3.6). Form factor oJ'the demagnetizcttioncurve characterizesthe concave shape of the demasnetization curve.i.e. 'l:

(BH)n,o* BrH,

:

B*orH,no^,

B,H"

(3.6)

for a square demagnetizationcurve 1 : I and for a straisht line (rare-earth P M s )7 : 0 . 2 5 . The leakage flux causesthe magnetic flux to be distributed nonuniformly along the height 2lt.y of a PM, where h,y1is the height per pole. As a result, the MMF producedby the PM is not constant.The magneticflux is higher in the neutral cross section and lower at the ends, but the behavior of the MMF distributionis the opposite[96]. The PM surfaceis not equipotential.The magneticpotential at eachpoint on the surface is a function of the distance to the neutral zone. To simplify the calculation, the magnetic flux which is a function of the MMF distribution alongthe height hy per pole is replacedby an equivalentflux. This equivalent flux goesthrough the whole height hy andexits from the surfaceof the poles. To find the equivalent leakageflux and the whole flux of a PM, the equivalent magneticfield intensityneedsto be found, i.e.

H:j

1

rhv

l nnr Jr:

H,dr

fvr h,nr

(3.7)

where 11" is the magnetic field intensity at a distancer from the neutral cross sectionandf tr is the MMF of the PM per pole (MMF : 2Fru per pole pair). The equivalentmagneticfield intensity (3.7) allows the equivalentleakage flux of the PM to be found. i.e. Qu$ : Qm - Qo

(3.8)

where Ona is the full equivalentflux of the PM and On is the air gap magnetic flux. The cofficient of leakageflux of the PM,

94

AXIAL FLUX PERMANENTMAGNETBRUSHLESS MACHII\trES otnt :

(Dn,

Q, n,

-! Qs

| + -+:1 > l Qs

(3.e)

simply allows the air gap magneticflux to be expressedas (Dn: Qrulotu. The following leakageperrneanceexpressedin the flux O*MMF coordinate systemcorrespondsto the equivalentleakageflux of the PM:

Gmr-y

( 3 .l 0 )

.r A,t

An accurateestimationof the leakagepermeanceG61 is the most difficult task in analyticalcalculationof magneticcircuits with PMs. Using the field approach,e.g. the FEM, the leakagepermeancecan be found fairly accurately. The averageequivalent magnetic flux and equivalent MMF mean that the magneticflux densityand magneticfield intensityare assumedto be the same in the whole volume of a PM. The full energy produced by the magnet in the outer spaceis

w:Tv*

r

(3.11)

where I/,y is the volume of the PM or a systemof PMs. For a PM circuit with a rectangularcrosssection,single PM and two mild steel pole shoes,the magnetic flux density Bn in a given air gapvolume 7n : gumlu is directly proportionalto the squareroot of the magneticenergyproduct(B1yHx4) [96], i.e.

Bs :

: trroHs

u,$av1n*,

(3.r2)

vq

where 11p" is the magnetic field intensity in the mild steel yoke, fI, is the magneticfield intensity in the ak gap,VA,r: 2hpywxal7,1 is the magnetvolutae, LUlylis the PM width, 1,11 is the PM length and2lp". is the length of the magnetic flux path in two mild steelpole shoes.Following the trend to smaller packaging,smallermassandhigherefficiency,the materialresearchin the field of PMs hasfocusedon finding materialswith high valuesof the maximum energy product (BH),n"". The air gap magnetic flux density Bn can be estimatedanalytically on the basis of the demagnetizationcurve, air gap and leakageperrneancelines and

95

Materia ls andfab rication

recoil lines [96]. Approximately,it can be found on the basisof the balanceof the magneticvoltagedrops,i.e. B'

h t r :t L r t

1 1+ B ' J t

oilpermeabil ity). li r*pM (,";;". rec where F,,"" ls,n.,.,""111i1,* "*rri, Hence, B,hm

1) l)y

: Bo=3" hll Hr,".9 1+ y,',,"glhn "

(3.13)

The air gap magnetic flux density is proporlional to the remanent magnetic flux densityB, and decreasesas the air gapg increases.Eqn (3.13) can only be usedfor preliminary calculations. For rare-earthPMs the approximation of the demagnetizationcurve is simple due to practicallylinear demagnetizationcurve,i.e.

B(H):u,(r -

#)

(3.14)

cules (Alnico or The approximationof more complicateddemagnetizations ferrites)is given e.g. in [96]. The intersectionpoint of the above demagnetizationcurve (3.14) and the following line representingthe perrneanceof the air gap

B(H): 1,oL!!s

(3.l s)

gives a point called the operatingpoint. This point conespondsto the air gap magneticflux densityBn multiplied by the leakagecoefficientolM according to eqn (3.9).

3.2.2

Characteristicsof PM materials

Thereare threeclassesof PMs currentlyusedfor electricmotors: r Alnicos (Al, Ni, Co, Fe); r Ceramics(fenites), e.g. barium ferrite BaOx6FezO3and strontiumferrite SrOx 6Fe2O3; r Rare-earthmaterials,i.e. samarium-cobaltSmCo and neodymium-ironboron NdFeB. The demagnetizationcuwes of the above three types of permanent magnet materialsare given in Fig. 3.8.

96


\'

\J

= tl

14

^l V ^L : ^

1,0 I NEODYMIUM-IRON B0R0Nl*>/

I

'l \LNICO

(rorl*,u"1 COBALT

rERRrr2

*tt

"

tTl

0,6 0,4 0,2 0

900 800 700 600 500 400 300 200 100 -*Figure 3.8.

-H [xelm] Demagnetizationcuryes for diflerent PM materials.

Alnico Alnico magnets dominated the PM motor market in the range from a few watts to 150 kW betweenthe mid 1940sand the late 1960s.The main advantages of Alnico are its high magnetic remanentflux density and low temperature coefficients(Table 3. 10). The temperaturecoefficient of B, is -0.02ohl" C and maximum servicetemperatureis 520"C. Unfoftunately, the coercive force is very low and the demagnetizationcurve is extremely non-linear. Therefore, it is very easynot only to magnetizebut also to demagnetizeAlnico. Alnico has been used in PM d.c. commutatormotors of the disc type with relatively large air gaps. This results in a negligible armature reaction magnetic flux acting on the PMs. Sometimes,Alnico PMs are protected ffom the armature flux, and consequentlyfrom demagnetization,using additional mild steel pole shoes. Ferrites Barium and strontium ferrites produced by powder metallurgy were invented in the 1950s. Their chemical formulation may be expressedas MOx 6(Fe2O3),where M is Ba, Sr, or Pb. Fer:ritemagnetsare availablein isotropic and anisotropic grades. A ferrite has a higher coercive force than Alnico, but at the sametime has a lower remanentmagneticflux density(Table3.10). Temperaturecoefficients are relativelyhigh, i.e. the coefficientof B,- is -0.20%loc and the coefficient of H. is *0.27 to -0.4ohf"C. The maximum servicetemperatureis 450"C. The main advantagesof ferrites are their low cost and very high electric re-

9'7

Materials andfab ricatictn

Tsble3.10. Physicalpropertiesof a representative selectionof PM materialsfor small motors offeredby Magnaquench GnBH, Essen,Germany

Property Remanentflux density, -B", T Coercivity,H.,kAlm Intrinsic coercivity, i H,,, k A/m (B H)-,o,,kJlm3 Relative recoil magnetic permeability Temperaturecoefficient o s of 8,, at 20 to 100"c, %/"c Temperaturecoefficient a ;.rr of iH,. at20 to t00"c,o f c Curie temperature,"C Maximum continuous service temperature,"C Thermal conductivity, W(m "C) Specificmassdensity,ppv,kglms Electric conductiviry S/m Coefficient of thermal expanslon at20to 100"c, x10 6/"c Specific heat, J "C/kg

Alnico sintered Koerzit500 1.24 5l

Hardfenite bonded Koercxl2l22p 0.26 r80 225

Hardferrite sintered Koerox350 0.39 270 310 30

3 to 4.5

l.l

1.1

-0.02

-0.2

-0.2

* 0 . 0 3t o - 0 . 0 7 850

+0.4 4s0

+0.3 450

500 1 0t o 1 0 0 7300 1 . 4t o 2 . 5x 1 0 o

100to 200

11ro13 350to 500

20 to 50

200 4 4800 < 0.0001 I 2 parallel 8 vertical 800

4t.4

3400 < 0.0001

sistance,which meanspractically no eddy-currentlossesin the PM volume. Ferritemagnetsare most economicalin fractionalhorsepowermotors.Barium ferrite PMs are commonly used in small d.c. commutatormotors for automobiles (blowers,fans,windscreenwipers,pumps,etc.) and electrictoys. Rare-earth permanent magnets The first generationof rare-earthpermanentmognets,i.e. alloys basedon the compositionof SmCo5has been commerciallyproducedsince the early 1970s(inventedin the 1960s). SmCo5has the advantageof a high remanent flux densiry high coercive force, high energy product, a linear demagnetization curve and a Iow temperaturecoemcient (Table 3.11). The temperature coemcientof B, is 0.02 to -0.045%l"C and the temperaturecoemcientof 11" is -0.14 to -0.40%loC. Maximum servicetemperatureis 300 to 350oC. It is suitable for motors with low volumes and motors operating at increased temperatures, e.g. brushlessgeneratorsfor microturbines.Both Sm and Co are relatively expensivedue to their supply restrictions.

98

AXIAL FLUX PERMANENTMAGNETBRTJSHLESS MACHINES

Table 3. I l. Physical properlies of Vacomax sintered SmzCorz PM materials at room temperature 20"C manufacturedby VacuumschmelzeGmhH, Hanau. Germanv

Vacomax 240 HR

Property Remanentflux density, -B".,T Coercivity,H.,kAlm Intrinsic coercivity,i H., kAlm (BH)-.^",kIlm3 Relative recoil magneticpermeability Temperaturecoefficient clB of B ,. at 20 to 100"c , o/ol"c Temperaturecoefficient a;r1 of i,H,. at 20 to 100"c, o/ol"c Temperaturecoeffrcient or of B" at 20to 150"C.,o/of'c Temperaturecoefficient o;s of i H ,, at 20 to 1 50"C, o/ol"C Curie temperature,"C Maximum continuous servicetemperature,''C Thermal conductiviry W/(m "C) Specificmassdensity,pntt,kglm3 Electncconductivity,, | 0" S/m Coefficientol thermalexpansion at20to 100'c, x10-6/"c Young'smodulus, x 106 MPa Bending stress,MPa Vcker's hardness ,

^f

Vacomax 225 HR

Vacomax 240

1 . 0 5t o 1 . 1 2 600ro 730 640to 800 200to 240

1 . 0 3t o 1 . 1 0 720to 820 1590ro 2070 190to 225

0 . 9 8t o 1 . 0 5 580to 720 640to 800 1 8 0t o 2 1 0

1 . 2 2t o 1 . 3 9

I . 0 6t o 1 . 3 4

1 . 1 6t o 1 . 3 4

-0.030 0.15

-0.18

-0.15

-0.035 -0.16

-0.19 approximately 800

-0.16

300

350 approximately12 8400 1 . 1 8t o 1 . 3 3

300

10 0 . 15 0 9 0t o 1 5 0 approximately 640

With the discovery in the recent years of a secondgenerationof rare-earth magnetson the basis of inexpensiveneodymium (1.{d),remarkableprogress with regard to lowering raw material costs has been achieved.The new generation of rare-earthPMs basedon inexpensiveneodymium (Nd) was announced by SumitomoSpecialMetals,Iapan,in 1983 atthe29th Annual Conferenceof Magnetismand MagneticMaterialsheld in Pittsburgh,PA, U.S.A. The Nd is a much more abundantrare-earthelement than Sm. NdFeB magnets,which are now produced in increasing quantities have better magnetic properties (Table 3.12)thanthoseof SmCo,but unfortunatelyonly at room temperature.The demagnetizationcurves, especially the coercive force, are strongly temperature dependent.The temperaturecoefficientof B, is -0.09 to -0.15o/ol"C and the temperaturecoefficientof fI" is -0.40 to -0.80%1"C. The maximum service

Materialsandfabrication

99

temperatureis 250oCand Curie temperatureis 350oC.The NdFeB is also susceptible to corrosion. NdFeB magnetshave great potential for considerably improving the performance-to cost ratio for many applications. For this reason they will have a major impact on the developmentand applicationof PM machinesin the future. Chemicalreactivityof rare-earthmagnetsis similar to that of alkalineearth metals, e.g. magnesium. The reaction is acceleratedat increasedtemperature and humidity. The NdFeB alloy if exposedto hydrogen gas, usually at a slightly elevatedtemperatureand/or elevatedpressure,becomesbrittle and with very little effort, it can be crushed.Diffusion of hydrogeninto the alloy causesit literally to fall apart. Corrosionprotectivecoatingscan be divided into metallic and organic.For metallic coatings,e.g. nickel and tin, galvanicprocessesare used as a rule. Organic coatingsinclude powder coatingsapplied electrostatically,varnishes and resins. Nowadays, for the industrial production of rare-eafthPMs the powder metallurgical route is mainly used 1194]. Apart from some material specificparameters,this processingtechnologyis, in general,the samefor all rare-earth magnet materials. The alloys are produced by vacuum induction melting or by a calciothetmicreductionof the oxides. The material is then size-reduced by crushingand milling to a single crystallinepowder with particle sizesless than 10 trrm. In order to obtain anisotropicPMs with the highest possible (BH)-", value, the powdersare then aligned in an externalmagneticfield, pressedand densifiedto nearly theoreticaldensityby sintering.The most economical methodfor massproductionof simply shapedparts like blocks, rings or arc segmentsis die pressingof the powders in approximatelythe final shape. Researchersat General Motors, U.S.A., developeda fabrication method basedon the melt-spinningcastingsystemoriginally inventedfor the production of amorphousmetal alloys. In this technologya molten streamof NdFeCoB materialis first formed into ribbons 30 to 50-pm thick by rapid quenching, then cold pressed,extrudedand hot pressedinto bulk. Hot pressingand hot working are carried out while maintaining the fine grain to provide a high density close to 100% which eliminatesthe possibility of internal corrosion. epoxy resin coatingprovides excellentcorroThe standardelectro-deposited sion resistance. The prices of NdFeB magnetsorderedin large quantitiesare now below US$20 per kg. Owing to a large supply of NdFeB magnetsfrom China it is expectedthat the prices will fall further.

3.2.3

Operatingdiagram

The energyofa PM in the externalspaceonly existsifthe reluctanceofthe extemalmagneticcircuit is higher Ihan zero. If a previouslymagnetizedPM

100


Table3.12. Physicalproperliesof Hicorex-SupersinteredNdFeB PM materialsat room temperature 20"C manufacturedby Hitachi Metals,ltd, Tokyo, Japan

Property Remanent flux densiry .B., T Coercivity,H",kAlm Intrinsic coercivity, i,H,,, kNm (BH),,,o*,kJlm3 Relative recoil magnetic permeability Temperaturecoe{ficient as of 8,. at20 to l00"C,okl"C Temperaturecoefficient otr of i,H,.at20 to 100"c,yol"c Curie temperature,"C Maximum continuous sewice temperafure,"C Thermal conductivity, W(m" C) Speciflcmassdens.,pp M,kglms Electric conductivity, x 106 S/m Coefficient of thermal expanslon at20to 100"c,x10 6/"c Young'smodulus, x 106 MPa Bending stress,MPa Vicker's hardness Features

Hicorex-Super HS-38AV

Hicorex-Super HS-25EV

Hicorex-Super HS-47AH

1 . 2 0t o 1 . 3 0 875to 1035 m i n .1 1 1 4 278to 319

0 . 9 8t o 1 . 0 8 7 16to 844 min. 1989 183to 223

1 . 3 5t o 1 . 4 3 1 0 1 8t o I 1 2 3 min. 1I 14 342to 390

1 . 0 3t o 1 . 0 6 - 0 . 1I t o - 0 . l 3 0.65to -0.72 r 310 160

180

140

7500 x 0,67

High energy product

* 1.5 0 . 15 0 260 = 600 High temperature

Superhigh performance

is placedinside a closedideal fenomagneticcircuit, i.e. toroid, this PM does not show any magneticproperties in the external space,in spite of the fact that there is the masnetic flux Qr:BrS74:Brwx4lp1

( 3 . 16 )

coffespondingto the remanentflux densityE, insidethe PM. A PM previouslymagnetizedand placedalone in an open space,as in Fig. 3.9a, generatesa magnetic field. To sustain a magnetic flux in the external open space,an MMF developedby the magnetis necessary.The stateof the PM is characterizedby the point K on the demagnetizationcule (Fig. 3.10). The location of the point K is at the intersectionof the demagnetizationcurve with a straight line representingthe peffneanceof the externalmagnetic circuit (open space):

r0r

Materials and fabrication

Figure 3.9. Stabilizationof a PM: (a) PM alone, (b) PM with pole shoes,(c) PM inside an external magnetic circuit, (d) PM with a complete extemal amature system.

G"rt: Y, J-K

tan G"r; :

Q r cl Q ,

FxlF"

--

r c (: v e l r o, rr

(3.l 7)

The permeanceGs.r1coffespondsto the flux O-MMF coordinate systemand is referred to as MMF at the ends of the PM. In the O-MMF coordinate systemthe remanentflux O" is accordingto eqn (3.16) and the MMF correspondingto the coercivity FI" is f.:

(3.18)

H.hM

The magnetic energy per unit producedby the PM in the extemal spaceis rDK : BKHK 12. This energyis proporlionalto the rectanglelimited by the coordinate system and lines perpendicular to the @ and f coordinates projected from the point 1{. It is obvious that the maximum magneticenergyis for B1a : Brno, and for HK : Hrnor. If the poles are furnishedwith pole shoes(Fig. 3.9b) the permeanceof the external spaceincreases.The point which characterizesa new stateof the PM in Fig. 3.10 movesalong the recoil line from the point K to the point A. The recoil line KGu is the sameas the intemal permeanceof the PM, i.e. \t1y1:

.Snl

wttltrt 1 1r e c

rLNl

Ft".,

(3.1e)

nL,l

The point A is the intersectionof the recoil line KGm and the straight line OGa representingthe leakagepermeanceof the PM with pole shoes,i.e.

GA:y!, J-A

tana4 - ,o

F

*

(3.20)

102

AXIAL FLUX PERMANENTMAGIVETBRUSHLESS MACHINES

The energy produced by the PM in the external spacedecreasesas compared with the previouscase,i.e. utl: ffogo1r. The next stage is to place the PM in an extemal ferromagnetic circuit as shown in Fig. 3.9c. The resultantperneanceof this systemis 6-

Gp: i!, {P

tanap

F

",

*,

(3.2r)

which meetsthe conditionG p > G n ) G."t. For an externalmagneticcircuit without any electric circuit carrying the armaturecurrent, the magnetic stateof the PM is characterizedbythe point P (Fig. 3.10), i.e. the intersectionof the recoil line K G u and the perrneanceline OG p . When the external magnetic circuit is furnished with an armature winding and when this winding is fed with a currentwhich producesan MMF magnetizing the PM (Fig. 3.9d), the magneticflux in the PM increasesto the value O1,'. The d-axis }l4MF FL.r of the external (armature) field acting directly on the PM correspondsto O1,'.The magneticstateof the PM is describedby the point l/ located on the recoil line on the right-hand side of the origin of the coordinatesystem. To obtain this point it is necessaryto lay off the distance Of'"aandto draw aline G p from the point ft"6 inclined by the angle ap to the F-axis. The intersectionof the recoil line and the perrneanceline Gp givesthe point l/. If the exciting current inthe externalarmaturewinding is increased further, the point lt/ will move further along the recoil line to the right, up to the saturationof the PM.

Figure 3.10.

Diagram of a PM for finding the origin of the recoil line and operating point

103


When the excitation current is reversed,the externalatmaturemagneticfield will demagnetizethe PM. In this case it is necessaryto lay off the distance Of'"ofrom the origin of the coordinatesystemto the left (Fig. 3.10). The line the demagnetization Gp drawnfrom the point,ftro with the sloperr p intersects curve at the point K' . This point can be aboveor below the point K (for the PM alone in the open space).The point K/ is the origin of a new recoil line K'G'n.Now if the armatureexciting cur:rentdecreases,the operatingpoint will move along the new recoil line K'G'y to the right. If the armaturecurrent drops down to zero,the operatingpoint takesthe position P'(intersection of the new recoil line K'G'n with the permeanceline Gp drawn from the origin of the coordinatesystem). On the basisofFig. 3.10the energieslDpr : Bp, Hp, f2, u)p : BpHp f2, zndusp, I wp. The location of the origin of the recoil line, as well as the location of the operatingpoint, determinethe level of utilization of the energy produced by the PM. A PM behavesdffirently than a d.c. electromagnet:lhe energy of a PM is not constant if the perrneanceand exciting current of the external ar:rnaturechanges. The location of the origin of the recoil line is determinedby the minimum value of the permeanceof the external magnetic circuit or the demagnetization action ofthe externalfield. To obtain the propertiesof PMs more independentof the externalmagnetic fields,PMs needto be stabilized.Stabilizationmeansthe PM is demagnetized up to a valuewhich is slightly higherthanthe most dangerousdemagnetization field during the operationof a systemwhere the PM is installed. In magnetic circuits with stabilized PMs the operatingpoint describing the stateof the PM is locatedon the recoil line. More detailsabouthow to find the operatingpoint of a PM graphicallyand analyticallycan be found in [96].

3.2.4

Permeancesfor main and leakagefluxes

Permeancesof air gaps and permeancesfor leakage fluxes can be found analyticallyby dividing the magneticfield into simple solids. Petmeancesof simple solids shownin Fig. 3.11 can be found using the following fonnulae: (a) Rectangularprism (Fig. 3.1I a)

G: LnYYy H

(3.22)

104

AX]AL FLUX PERMAAIENTMAGNETBRUSHLESS MACHINES

,r,M]

rlwrl s {*rf-

o' N -'-l{;{;f

E

Figure 3.11. Simple solids: (a) rectanguiarprism, (b) cylindeq (c) half-cylinder, (d) onequarter of a cylinder, (e) half-ring, (0 one-quarter of a ring, (g) one-quarter of a sphere, (h) one-eighthofa sphere,(i) one-quarterofa shell, O one-eighthofa shell.

105

Materials andfabrication

(b) Cylinder(Fig.3.1Ib) (3.23)

G:lro*

(c) Half-cylinder(Fig.3.11c) G :0.26p,olnr where the average air gap gau : replacedby 0.322glLr ll3)

(3.24)

1.229 and,the surface wmltw should be

(d) One-quarterof a cylinder (Fig. 3.11d)

G :0.52p,o1u

(3.2s)

(e) Half-ring (Fig. 3.1le)

2lm Lr

l-Lt)

r(gltuM * 7)

(3.26)

For g { 3tuy,

G: ^Vm(t +'**\ 7T\.9/

(3.27)

(f) One-quarterof a ring (Fig. 3. I 10

r':l'ton@1"+w

(3.28)

r" (r + 9) G : po2tM g/

e.2s)

For g < 3c,

7T

\

(g) One-quarterof a sphere(Fig. 3.I I g) G : 0.077|,tog

(3.30)

106


(h) One-eighthof a sphere(Fig. 3.I th) G : 0.308pr,og

(3.31)

(i) One-quarter of a shell(Fig. 3.11i) G:po;

fi) One-eighthof a shell(Fig. 3.1lj)

G: t"o;

(3.33)

Fig. 3.12 showsa model of a flat electricalmachinewith smootharmaturecore (without slots) and surface PM excitation system. The armature is of steel laminations.The PMs are fixed to the mild steelyoke. The pole pitch is r, the width of eachPM is w16, andits length is /7yr.In an AFPM machine lu :0.5(Dout - Dm)

(3.34)

The spacebetweenthe pole face and the armaturecore is divided into a prism (l), four quartersofcylinders (2 and4), four quartersofrings (3 and 5), four piecesof 1/8 of a sphere(6) and four piecesof 1/B of a shell (7). Formulae for the perneance calculations have been found on the assumption that the perrneanceof a solid is equal to its averagecross sectionareato the average length of the flux line. If we ignore the fringing flux, the permeanceof a rectangularair gapper pole (prism I in Fig. 3.12) is

Ggt: prYff

( 3.3s)

The equivalent air gap g' is only equal to the nonmagnetic gap (mechanical clearance) g for a slotless and unsaturatedarmature. To take into account slots (if they exist) and magnetic saturation,the air gap g is increased to gt - gkck"ot, where kc > 7 is Carter'scoefficienttaking into accountslots (1.2), and k"at )> 1 is the saturationfactor of the magneticcircuit definedas

t07

Mq terials andJbbrication

(b)

(c)

,,i.,!-------i-------T; 1l lsl+

I

1l 1l

4

Itl lst lg Iti Itl

'10

x

;+i t

ft" lm

Figure3.12. Electricalmachinewith flat slotlessarmatureand flat PM excitationsystemfieldintosimplesolids:(a) longitudinal divisionof thespaceoccupied by themagnetic section, (b) air gapfield, (c) leakageficld (betweenthePM andsteelyoke).

the ratio of the MMF per pole pair to the air gap magneticvoltagedrop (MVD) takentwice 1961. To take into accountthe fringing flux it is necessaryto include all paths for the magnetic flux coming from the excitation systemthrough the air gap to the armaturesystem(Fig. 3.l2), i.e. Gn : G6 * 2(Gsz * Gsz * Gsa -t Gss)* 4(Gsa+ Gsz)

(3.36)

where Go1 is the air gap permeanceaccordingto eqn (3.35) and G sz to G sz are the atr gapperrneancesfor fringing fluxes. The permeancesG sz to G n5 can be found using eqns(3.25), (3.28), (3.3I ) and(3.32). In a similar way the resultantperrneancefor the leakageflux of the PM can be found.i.e.

GLu :2(GB * Grc)* AGns

(3.37)

where GB, Grc (one-quafterof a cylinder) and G110(one-eightof a sphere)are the permeancesfor leakagefluxes between the PM and rotor yoke according to Fig. 3.12c- eqns(3.25)and(3.31).

3.2.5

Calculationof magneticcircuitswith PMs

The equivalent magnetic circuit of a PM systemwith armature is shown in Fig. 3.13. The reluctancesof pole shoes(mild steel)and armaturestack(electrotechnical laminated steel) are much smaller than those of the air gap and

MACHINES AXIAL FLUX PERMANENTMAG]\IETBRUSHLESS

108

R,'m

'tprg

l*,,*nn,t

Figure 3. I 3.

.fi '00

''irta

Equivalent circuit (in the d-axis) of a PM system with armature.

PM and have been neglected. The "open circuit" MMF acting along the internal magnetpermeanceG M : Tlftpnt is Tmo : H Nrohx,Lthe d-axis armature the leakageflux reactionMMF is f oy,thetotal magneticflux of the PM is @,y1, flux of the On, leakage the the useful air gap magneticflux is of the PM is Q17,1, (01o, the flux in the d-axis produced by the atmaexternal armature system is ture is Ooa(demagnetizingor magnetizing),the reluctancefor the PM leakage flux is sf4,r.r: LlGtm,the airgapreluctanceis Sr, :7lGg, andthe external armatureleakagereluctanceis S,ro - llGgn.The following Kirchhoff's equations can be written on the basis of the equivalent circuit shown in Fig. 3 .l 3 ( D , r :1Q 1 g 1 l Q s + F ,o,o, !! QIo:

Vl1,,ta

: 0 Tuo * QnlWpr4 - Q11qTlp17r QtmTlttr - Q nBrn T To,t : 0 The solutionto the aboveequationsystemyields the air gap magneticflux: Qg:

I lftt"Tfactn

L

GnG't (Gg+ GtM)(Gar+ Gt^r)l Gn GnGtu -s*Ga,I I Gn*Gtu*Gyr

or rAIO

-d

t

Gt(Gnr * Gttr

f od,

GnGn'r

GnG* GtlGyr

(3.38)

109

Mnterials and fabrication

Figure 3.14. Shapesof PM rotors of disc-type machines: (a) trapezoidal,(b) circular, (c) semicircular.

where the total resultantperrneanceG1 for the flux of the PM is Gt:

( 3.3e)

Gs I Gw : ot11Gs

and the direct-axis armatureMMF acting directly on the PM is

F[a- ,"ocjb; 'lt*Urnt

: ro,t(' _,_ +) G, \

-'

: {-uo

)

otrr

(3.40)

The upper sign in eqn (3.38) is for the demagnetizingarmatureffux and the lower sign is for the magnetizing armatureflux. The coefficientof the PM leakageflux (3.9) can also be expressedin terms of permeances, i.e. olM:

, ,

ol,ri

,

Gttt

(3.41)

%

3.2.6

Fabrication of rotor magneticcircuits

Magnetic circuits of rotors of AFPM brushlessmachinesprovide the excitation flux and are designedas: r PMs glued to a ferromagneticring or disc which seryesas a backing magnetic circuit (yoke); r PMs arrangedinto Halbach array without any ferromagnetic core. Shapesof PMs are usually trapezoidal,circular or semicircular(Fig. 3.14). The shape of PMs affects the distribution of the air gap magnetic field and contentsof higher spaceharmonics.The output voltagequality (harmonicsof

110


Table 3.13. Magnetization cur.res of solid ferromagnetic materials: I (0.27%C).2- castiron

Magneticflux densiryB

carbon steel

Magnetic field intensity, 11 Mild carbon steel0.27o/oC Cast iron A/ur A/m

T 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.5 1.6 1.7

190 280 320 450 900 I 500 3000 1500 6600 r I,000

900 I 600 3000 51 5 0 9s00 18,000 28,000

the EMF) of AFPM generatorsdependson the PM geometry(circular, semicircular,trapezoidal)and distancebetweenadjacentmagnets[75]. Sincethe magneticflux in the rotor magneticcircuit is stationarymild steel (carbon steel)backing rings can be used. Rings can be cut from 4 to 6 mm mild steel sheets. Table 3.13 shows magnetizalioncharacteristicsB-11 of a mild carbonsteel and cast iron. Electrical conductivitiesof carbon steelsare from 4.5 r 106to 7.0 x 106Stmar 20oC. Halbach array Twin rotors of double-sidedcorelessAFPM machines(Fig. l.ad) may use PMs arranged in Halbach array I I 04- 106] . The key conceptof Halbach array is that the magnetizationvector of PMs shouldrotateas a function of distance

I

t

Figure 3.15.

+

CartesianHalbach array

I


l1 1

alongthe array (Fig. 3.15) [04-106]. Halbach anay hasthe following advantages: r the fundamental field is strongerby a factor of 1.4 than in a conventional PM array,and thus the power efficiencyof the machineis doubled; r the amay of PMs does not require any backing steel magnetic circuit and PMs can be bonded directly to a non-ferromagnetic suppofiing structure (aluminum,plastics); r the magneticfield is more sinusoidalthan that of a conventionalPM array; r Halbach array has very low back-sidefields. The peak value of the magnetic flux density at the active surface of Halbach array is

Bmo- B,l7 - exp(-ljhl,)l ,rjp

(3.42)

where B," is the remanentmagneticflux density of the magnet,13:2rlI" seealso eqn (1.6), lo is the spatialperiod (wavelength)of the arcayandnyl is the number of PM piecesper wavelength.For the array shown in Fig. 3.15 nM : 4. For example,assumingB, : 1.25 T, hat : 6 mm, lo : 24 mm, Dm : 4 (rectangular PMs), the peak magnetic flux density at the surface of Halbachanay B1q6: 0.891T. The tangential B, and normal B, componentsof Halbach arliayat the distancez from the surfaceof PMs are B"(r, z) : B,nocos(l3r)erp( l3t)

B

"(r,

z) : B,nosi.n(Pr)erp(- Pz)

(3.43)

(3.44)

For a double-sidedconfigurationof Halbach arrays, i.e. twin disc external rotors, the tangential and normal component of the magnetic flux density distribution in the snacebetween discs are

B,(r,

")

: B-.oicos(0r)-- ^Lsinh(pz)

B z) : Bôsin( e") Ai0r,cosh(Pz) "(r,

(3.45)

(3.46)

t12

AXIAL FLUX PERMAAIENTMAGNETBRUSHLESS MACHINES

where B,ns is according to eqn (3.42) and I is magnet-to-magnet distance betweentwo halvesof the disc. The origin of the \ry z coordinatesystemis as in Fig.1.8.

3.3 Windings 3.3.1 Conductors Stator (armature)windings of electric motors are made of solid copper conductor wires with round or rectangularcross sections. The electricconductiviQ at20oCof copperwires is 57 x 106 ) ozo ) 56 x 106 S/m. For aluminum wires o2ox 33 x 106 Sim. The electricconductivity is temperaturedependentand for 0 - 20" < 150"C can be expressedas o20

l+a(d-20")

(3.41)

where a is the temperaturecoefficient of electric resistance.For copper wires a : 0.003931/"C and for aluminumwires o : 0.00403l/"C. For I - 20" > 150oCeqn(3.47)containstwo temperaturecoefficientscr and p of the electric resistance. i.e. o20

1 * a(t9 - 2A) + p(0 - 20")2

(3.48)

The maximum temperaturerise for the windings of electrical machines is determinedby the temperaturelimits of insulatingmaterials.The maximum temperaturerise in Table3.14 assumesthat the temperatureof the cooling medium 8. < 40"C. The maximum temperatureof windings is actually Srno*:8.+ L8

(3.4e)

where Ari is the maximum allowabletemperaturerise accordingto Table3.14. A polyester-imideand polyamide-imide coat can provide an operatingtemperatureof 200"C. The highest operatingtemperatures(over 600oC) can be achievedasing nickel clad copper or palladium-silver conductor wires and ceramic insulation.

3.3.2

Fabrication of slotted windings

Statorwindings are usuallymadeof insulatedcopperconductors.The cross sectionof conductorscanbe circularor rectangular.For largeAFPM machines a direct watercooling systemand consequentlyhollow conductorscan be considered. It is difficult to make and form stator coils if the round conductor is thicker than 1.5 mm. If the current density is too high, parallel conductorwires of

113

Materials andfabrication

Table 3.14. Maximum temperaturerise Arl for armature windings of electrical machines according to IEC and NEMA (basedon 40"C ambient temperature)

Ratedpower ofmachines, length of core and voltage IEC a.c. machines< 5000 kVA (resistancemethod) IEC a.c. machines> 5000 kVA or length ofcore ) I m (embeddeddetector method) NEMA a.c.machines< 1500 hp (embeddeddetector method) NEMA a.c. machines> 1500 hp and < 7 kV (embeddeddetectormethod)

Insulationclass AEBFH "C "C

"C

60

75

80

100

125

60

70

80

100

125

70

90

115

140

65

85

I 10

135

OC

"C

smallerdiameterare recommendedratherthan one thicker wire. Statorwindings can also haveparallel currentpaths. The anrraturewindings can be either single-layer or double layer (Seclion 2.2). Af\er coils are wound, they must be securedin place, somehow,so as to avoid conductor movement. Two standardmethods are used to securethe conductorsof electricalmachinesin place: t dipping the whole component into a varnish-like material, and then baking off its solvent, t trickle impregnation method, which uses heat to cure a catalyzed resin which is drippedonto the component. Polyester,epoxyor silicon resinsareusedmost often asimpregnatingmaterials for treatmentof statorwindings. Silicon resinsof high thermal enduranceare able to withstand $n,', ) 225"C. Recently,a new method of conductor securingthat doesnot require any additionalmaterial,and usesvery low energyinput, hasemerged[63]. The solid conductorwire (usuallycopper)is coatedwith a heat and/or solventactivated adhesive.The adhesivewhich is usuallya polyvinyl butyral,utilizesa low temperaturethermoplasticresin [163]. This meansthat the bondedadhesivecan come aparl after a certain minimum temperatureis reached,or it again comes in contact with the solvent. Norrnally this temperatureis much lower than the

tt4

AXIAL FLUX PERMANENT MAGNET BRUSHLESSMACHINES

Figure 3.16. Disc-type corelesswinding assembledof coils of the same shape according to U.S. PatentNo. 5 744 896 !391: (a) single coil; (b) three adjacentcoils. I - coil side, 2 inner offsetting bend, 3 - outer offsetting bend.

thermal rating of the baseinsulation layer. The adhesiveis activatedby either passingthe wire through a solventwhile winding or heatingthe finished coil as a resultofpassing electriccurrentthrough it. The conductor wire with a heat activatedadhesiveovercoatcostsmore than the sameclassof non-bondableconductor. However,a less than two second culrent pulse is required to bond the heat activatedadhesivelayer and bonding machinerycostsabouthalf as much as trickle impregnationmachinery[63].

3.3.3

Fabrication of corelesswindings

Stator coreless windings of AFPM machines are fabricated as uniformly distributed coils on a disc-type cylindrical supporting structure (hub) made of nonmagneticand nonconductivematerial.Thereare two types of windings: (a) winding comprised of multi-turn coils wound with turns of insulated conductor ofround or rectangularcrosssection; (b) printed winding also calledy'lm coil winding. Coils are connectedin groups to form the phasewindings typically connected in staror delta. Coils or groupsof coils of the samephasecan be connectedin parallel to form parallel paths. To assemblethe winding of the samecoils and obtain high density packing, coils shouldbe formed with offsettingbends,as shownin Fig. 3. I 6. The space betweentwo sidesof the samecoil is filled with coil sidesfrom each of the adjacentcoils. Coils can be placedin a slottedstructureof the mould (Fig. 3.17). With all the coils in position, the winding (often with a supporting structure or hub) is

Materials and fabrication

I l5

(a)

Figure 3.17. Moulds for positioning the coils: (a) mould with guide slots; (b) mould with guide pins.

Figure j.18.

Film coils for AFPM micromotors.Courtesyof EMbest,Soeul,Korea

mouldedinto a mixture of epoxy resin and hardenerand then curedin a heated oven. Becauseof the difficulty of releasingthe cured stator from the slotted structureof the mould (Fig. 3. l7a), eachspacingblock that forms a guide slot consistsof severalremovablepins of different size (Fig. 3.17b). For very small AFPM machinesand micromachinesprinted circuit corelesswindings allow for automationof production.Printedcircuit windings for

MACHINES MAGNETBRUSHLESS AXIAL FLUX PERMAAIENT

I16

AFPM brushlessmachinesfabricatedin a similar way as printed circuit boards have not been commercializeddue to poor performance.A better perfotmance hasbeenachievedusing film coil windings madethrough the sameprocessas flexible printed circuits [85]. The coil patternis formed by etchingtwo copper films that are then attachedto both sides of a board made of insulating materials (Fig. 3.18). Compactcoil patternsare madepossibleby connectingboth sidesofcoil patternsthroughholes [85].

Numerical example3.1 A simple stationarymagneticcircuit is shown in Fig. 3.19. There are two Vacomax240 HR SmCoPMs (Table3.11) with B, : 7.10T, H, : 680 kA/m, temperafurecoefficients()B : -0.03%l"C and as : -0.15o/ol"CaI 20 I 'Bpm { 100"C. The height of the PM per pole is hy : 6 mm and the air gap thicknessg : 7 mm. The U-shapedand I-shaped(top) ferromagneticcores are made of a laminated electrotechnicalsteel. The width of the magnetsand coresis 17 mm. Calculatethe air gap magneticflux density,air gap magnetic field strength, the useful energy of PMs and normal attractive force per two polesat: (a) 8pu: 20oCand (b) ,5ptr :100oC' The MVD in the laminated core, leakageand fringing magnetic flux can be neglected. Solution: (a) Magnet temperatureI pm : 20"C The relativerecoil magneticpermeabilityaccordingto eqn (3.4) for a straight curveis line demasnetization

1AB Itrrrec:

tto,LH

:

M;

1.10-0 1 x - 0 L.29 Y 104 680,0U0

The air gap magneticflux densityaccordingto eqn (3.13) is Dug-

1.10 : 0.906T L+I.29 x 1.0/6.0

The air gap magneticfield strengthaccordingto eqn (3.14) in which H : Hs andB: Bg is /8.

Ho: H.lt

"u',): ouux ro3(r

'r:,t1)

x r03A/rn r20.r2

The useful energyper magnetvolume accordingto eqn (3.5) is

-,: u-p :W#e:54

J/m3 3e5.8

117

M aterial s and.fabric atio n

17

m ffi Figure 3.19.

A simple stationary magnetic circuit with PMs and air gap. Numerical example

3.1. The useful energyper pole pair is Wg:wgVIV:

5 4 , 3 9 5 . 8 ( x2 6 x 1 5 x 1 7 x 1 0 n ) : 0 . 1 6 6 J

The normal attractiveforce per 2 poles is Bl , - ,fr(2s.lr)

I o€\a'2

(b) Magnet temperature,3pnt -- 100oC The remanentmagneticflux densityand coercivityat l00oC accordingto eqns (.3.2)and(3.3) is

B,: r ro +#(roo-,o)] ror4r [r. r/" : 680x tor fr + **troo L

100

- 20)l: 5e8.4 x 103A/m j

At'Bpu : 100"C the demagnetization curveis nonlinear.lts linearpart is only between0.5 T and B, parallelto the demagnetization curveat 20oC.Thus,the permeabthty gap relative recoil magnetic magnetic field strength 1.t,.,,..and air Ho at l00oC are approximately the sameas those at room temperature. The air gap magneticflux densityaccordingto (3.13) is

Bs*

r.074

: 0.884T

7 + 7 . 2 9x 1 . 0 / 6 . 0


118

The useful energyper magnetvolume is ugx

0.884x 720,120: 53090.3Um3

Theusefulenergyperpolepair is e) W g : 5 3 0 9 0 . 3 (x2 6 x 1 5x 1 7 x 1 0 : 0 . 1 6 2J forceper2 polesis Thenormalattractive

r:

N RR42

*ffi(r5

N x 17x to 6): 158.6

Numerical example3.2 A single-sided,8-poleAFPM machinewith slottedferromagneticstatorhas the PM outer diameterDo6 : 0.22 m and inner diametet Din : 0.12 m. The air gap including the effect of slotting (Carter coefficient) is I : 1.9 1.15 T and /1. : 900 mm. TrapezoidalsinteredNdFeB magnetshave B,: o/ol"C and kA/m at 20oC.The temperaturecoefficientfor B" is as : *0.15 -0'64 o/ol"C' The coefficientof coefficientfor H.t's as: the temperature PM leakageflux is otv,r : 1.15 and pole width-to-pole pitch coefficient is a;:0.72' Find PM dimensionsto obtainthe air gap magneticflux densitYBs : 0.64 T at no-loadand temperature8pv : B0oC.Sketchoperatingdiagramsin the B-H and O-I,t M F coordinatesystemat no load. Assumethat the magnetic circuit is unsaturated. Solution: The remanenceand coercivity at Spxn : (3.3)is

B,: r.tb . [r *#

80"C accordingto eqns (3.2) and

( fp m* r o ) ] 1 . 0 4r 6

- 2o)l: bb4.4kAln H. : eooft* *9(dp.,rr ruu I L accordingto eqn (3'14) at 20oCand Approximationof demagnetizationcurves B0"C respectively /H\ B t o ( H -) r . r i [ , -

oo*oo]

/H\ B(H):1.046t1--l 5544001 "


119

The relativerecoil magneticpermeabilityaccordingto eqn (3.a) is lf,,ec:

1. 0 4 6 g4r. ,a 1g-o * ttn n*

: -t'ir

The axial heightofthe PM perpole accordingto eqn (3.13)is otAIBn htr: !,,." Ufffire:

,

l.l$vnA,a

t.S**ffia0.0019

- 0.0068m

The equivalent air gap is

s " q :e - # :

l'e I

L:

:6'4mrn

The averagediameter,pole pitch (1.9), length (3.34) and width of the magnet are respectively

D : 0.'o(0.22 + 0.72):0.17m I,r.r- 0.5(0.22-0.I2): 0.05m

"

: y::

ry

: 0.0668 m

ux,1: crir :0.72x0.0668: 0.048m

Thepermeance of the air gapaccording to eqn(2.23)is

Go:1roiY\oi,, - D?,) o8n

0'72r : 0.4n10-6--1rc.222- 0.122): 1.59x 10-6H U.00198x4\"'"" OI

tL'711174 .^ 6 0.048x0.05 bffi 1.59x tu 6 H Cu - t-ro::L - 0.4x n'10

The total penneancefor magneticflux including leakagepermeancesis G 1 : o y n G g: 1 . 1 5x 1 . 5 9x 1 0 6 : 1 . 8 2 8x 1 0 6 H Approximation of the total perneance (air gap and leakage)line can be expressedas a linear function of 1{ h,, \ t ( H )- r , f f i H

0 .;0; 0 69 6 od ;G : t . B 2xE.rô o H : b . 1 4 6v i o 6 H

The magneticfield intensity correspondingto the operatingpoint of the magnet is calculatedas

I2O


(a)

,)

ll

l u z o( l t ) B(H)

0.8 0.6

lrr*, 0.4 0.2

I

(b) o.ot)l

1ou 8.10' 1000000

-4 .t0-

6.10'

-2.1{)'

o ool 0.0025

o( F)

o.oo2

O .' ( F )

-Ty"

0 . 0 0t 5

oool 5 ro-' -4000

-3s00

-3000

-2500

-2000

-r500

-r000

-500

- 4000

0

Figure 3.20. Operating diagram of PM at no load: (a) in B-H I,I L,I F coordinate system. Numerical example 3.2.

HM-

-

0

coordinate system; (b) in O-

B,

Gfty1fQt1rtru)* B,lH.

1'046 : 148 795 Alrn "" '-' x 10-6 5.146 + 7.0461554,400

The air gap magnetic flux density as obtainedfrom the operating diagram (Fig. 3.20a)is h n, 0.0068 6 B e- : G - Y^ w _ !; !r!lL; r 1 1 n , : 1 . 5 9r l 0 _ - l _ " " " ! - - l 4 B 7 g b : 0 . 6 6 6T 0.048x0.05 To plot the operating diagram in the Q-M M F coordinatesystem,eqns (3.16) and (3.18) are usedto find the magneticflux O, correspondingtoBr and the MMF -A per pole correspondingtoH", i.e.

Wb O, : 1.046x 0.048x 0.05: 0.00252


t2l : 3750.6 554400x U.0068 A

f,:

Approximation of themagneticflux line is , O ( F ) : O - ,1\ T11- :

,\

:ooo252(r-#)

r c/

\

The MMF F11 coffespondingto the operatingpoint of the magnetin the OM M F coordinatesystem(Fig. 3.20b)is F".

*

o_ Gt I Q,lf"

0.00252 : 1006.6A 1.828x 10-6+ 0.002521375A.6

The air gap flux line (Fig. 3.20b)is O q ( . F ): G g F :

1 . 5 9> 1 0 6 F

The total flux line (Fig. 3.20b)is Or(f) -GLf:LB2Bx10-6f The air gap magneticflux conespondingto the operatingpoint is Qs(FM) : 0.0016 Wb. The air gap magneticflux densityis Bn : 0.0016/(0.048x : 0.666T. 0.011)

Numerical example3.3 The magneticfield in the air gap of an AFPM machinewith corelessstator is excitedby sinteredNdFeB magnetsarrangedin Halbacharray.The extemal twin rotors do not have any backing steel discs (Fig. l.ad). The remanent magneticflux density is B, - 7.25 T, the height of the PM is hxn : 6 mm, the wavelength at the averagediameter is lo - 2r : 48 mm and the magnetto-magnetdistanceis f - 10 mm. Find the distributionof the magneticflux densityin the spacebetweenmagnets for the 90-degreeHalbacharray,i.e. nNr :4. Estimate,how the number n11 of PMs per wavelengthaffects the magnetic flux density at the active surface of PMs. Solution The peak value of the magneticflux density at the active surfaceof Halbach array accordingto eqn (3.42) is

* exp(-130.e Bmo:1.2511 x 0.006)1"#P: 0.62 T

122

MACHINES AXIALFLLTXPERMANENTMAGNETBRUSHLESS

(a)

08

0.8 0.6 0.4

e

" 7*x,:j. "r r'[';j

roo 0.2 - 0.2 - 0.4 - 0.6 0 . 8- 0 . 8

0

0.01

0.03

0.02

0.04

0

(b)

,))r

0.8

0.6 04

B x(0,2) 100 ----7 , B - l 0 . 2 5I ^ " z l

0.2 0 *0.4 -0.6

o 8 -o.s

-0.004

-0.002

0

0.002

0.004

zg

T

;

Figure3.2L Distributionof B* andB, components: (a) in the z direction; (b) in the z direction.Numericalexample3.3.

where0 : 2 x r 10.048: 130.9lim. The distribution of the tangential componentB, in the spacebetweenmagnets is describedby eqn (3.45) and the distributionof the normal component B, is describedby eqn (3.46). Both componentsB" andBz areplotted in Fig.

3.2r.

0.8

0.8

B m0lim

0.6

s ,oo(nN4 )

re

0.4 0.2

0

0

2 2

10

8 nM

t2

14

16 16

Figw'e 3.22. Peak value of magnetic flux density 8,,, as a function of number n,ry of PMs per wavelength. Numerical example 3.3.


123

For the 90-degreeHalbach array(ny1 : 4) the peak value of the magnetic flux densityat the activesurfaceof PMs is -8,r,6: 0.42 T. Similarly,using eqn (.3.42),the peak value B-s can be calculatedfor otherHalbachconfigurations. For 60-degreeHalbach arcay(n,y : 6) B-o : 0.649 T and for 45-degree Halbach array(nx1 : B) B-o : 0.663 T. In general,

- \h'm)1ry# B,l1-exP(1 ,#T""Brno: : 1 . 2 5 [ 1- e x p ( 1 3 0 . 9 x 0 . 0 0 6 ) ]x 1 : 0 . 6 8 T sincelim"*ssinrf r : 1. The peak value Br,rg?s a function of number n,y of PMs per wavelengthis shown inFig.3.22.

Chapter4 AFPM MACHINES WITH IRON CORES

In Chapter2, principlesof operation,topologiesand fundamentalequations of a broad family of AFPM machineswere discussed.In this chapter,the focus is on those types of AFPM machinesthat make use of the stator and rotor ferromagneticcores. The AFPM machineswith ferromagneticcoresare designed both as single-sidedand double-sided.The statorcore can be fabricatedusing either laminated steelsor SMC powders. Generalequationsgiven in Chapter2 for the performancecalculations will be developedfurther and adjustedto the construction of AFPM machineswith ferromagnetic cores. Application of the FEM analysisto performancecalculationsis also emphasized.

4,1

Geometries

Single-sidedAFPM machineswith stator ferromagnetic cores have a single PM rotor disc oppositeto a singlestatorunit consistingof a polyphasewinding and ferromagneticcore(Fig. 2.1). The statorferromagneticcorescanbe slotted or slotless.The statorwinding is alwaysmade of flat wound coils (Fig. 2.8). The PMs can be mounted on the surface of the rotor or embedded(buried) in the rotor disc. In the caseof a slotlessstatorthe magnetsare almost always surface mounted, while in the case of a slotted stator with a small air gap between the rotor and stator core, the magnetscan be either surfacemounted on the disc (Fig. 2.1) or buriedin the rotor disc {Fig.2.6). Largeaxial magnetic forceson bearingsarethe main drawbackof single-sidedAFPM machineswith ferromagnetic stator cores. ln double-sidedAFPM machineswith ideal mechanicaland magneticsymmetry the axial magnetic forces are balanced. Double-sidedAFPM machines with stator ferromagnetic cores have either a single PM rotor disc with ironcored statorson both sidesof the disc (Figs l.4c and 2.3) or outer PM rotor discswith iron-coredstatorfixed in the middle (Figs 2.4,2.5 and2.6').As with

126

AXIAL FLUX PERMANENTMAGNETBRUSHLESS MACHTNES

single-sidedAFPM machinesthe statorferromagneticcorescan be slottedor slotless,and the rotor magnetscan be surfacemounted,embeddedor buried (Fig. 2.6). Again, in the caseof a slotlessstatorwith a large air gap between the rotor and stator core the magnetsare almost always surfacemounted. The statorwindings of double-sidedAFPM machinescan be flat wound (slottedor slotless)as shownin Fig. 2.8 or toroidally wound (normally slotless)as shown in Fig. 2.9.

4.2

Commercial AFPM machineswith stator ferromagnetic cores

An example of a commercial double-sidedAFPM servo motor with ferromagneticcore is shown in Figs.4.1 and 4.2. Extemal statorshave slotted ring-shapedcoresmade of nonorientedelectrotechnicalsteelribbon. The inner rotor does not have any ferromagnetic material. PMs are mounted on a nonmagneticrotatingdisc.

Figure 4. I . Double-sided AFPM brushlessservo motor with the stator f'erromagneticcore and built-in brake. Courtesyof Mavilor Motors, 5.A., Barcelona,Spain.

Fig. 4.3 showsa double-sidedAFPM synchronousgeneratorwith the stator core wound of amorphousalloy ribbon manufacturedbyLE Incorporated,Indianapolis,IN, U.S.A. The volume of lE AFPM generatorsis approximately 60o%lower than that of classical synchronousgeneratorsof the samerating.

AFPM machineswith iron cores

r27

Figure 4.2. Longitudinal section of the double-sided AFPM brushlessservo motor shown in Fig.4.1. Courtesyof Mavilor Motors,,S.l., Barcelona,Spain.

'v

:.

*l:::::,::::::,:,:::,,:,& wlillrlll', .:llllllll:t l

Lt Figure 4.3. Gensrnart:I AFPM synchronous generator with amorphous alloy stator core. Courtesyof LE Incorporated,lndianapolis,IN, U.S.A.

4.3

Somefeaturesof iron-cored AFPM machines

Iron-coredAFPM machinesare distinguishedin two ways from coreless (air-cored)AFPM machines,namely: (i) iron-coredmachineshavecore losses while corelessmachinesdo not and (ii) the per unit valuesof the synchronous reactancesof iron-coredmachinesare much hieher than thoseof corelessmachines.

128


The core lossesare functions of amongstother things the frequencyand density of the magneticflux in the ferromagneticcore - see eqn (2.a8). The frequency of the ffux variation in the core is in turn determinedby the speed and number of pole pairs. As AFPM brushlessmachinestend to have a high numberofpoles (minimum 2p : 6), the speedsof iron-coredAFPM machines are limited to keep the frequency,and hencethe core lossesof the machine, within limits; unlessthe machinesare designedto have low flux densitiesin ferromagnetic cores. Flux frequenciesare normally kept below, say 100 Hz, for laminated ferromagnetic cores. If higher frequencies are required, then silicon steellaminationswith a thicknessof lessthan0.2 mm, amorphousalloy ribbonsor SMC powdersmust be used. The higher synchronousinductanceof iron-coredAFPM machinesaffects negatively the voltage regulation of the machine in generator mode, which might be consideredas a disadvantage.However, for solid-stateconverter-fed applicationsthe higher synchronousinductanceof iron-cored AFPM motors is an important advantageas it helps reduce the current-ripple due to converter switching.Thus the relativelylow inductanceof corelessAFPM machinescan be a significantdisadvantagein switchedsolid stateconverterapplications.

4.4

Magneticflux densitydistributionin the air gap

AFPM brushless machines with ferromagnetic stator and rotor cores can produce strong magnetic flux density in the air gap with a minimum volume of PMs. The distribution of the normal component of the magnetic flux density in the r-direction (circumferential) excited by PMs without armaturereaction at the radius correspondingto the averagepole pitch r canbe describedby the following equation 1

BnQ):. "K6

B,pu(r)*8"1\r)

(4.1)

where the PM excitation flux density for smooth stator core is

and the magnetic flux density componentdue to stator slots is

Ba(n):x"@)#B"pnr(r)

(4.3)

The peakvaluesof the higherharmonicsof the magneticflux densitydistribution of eqn (4.2) arc B*s, : (Bnf o1y)b".

t29


t

1.0

0.8 0.6 0.4

B J9 o.z B - ' ( x )- 0 _ 2 "' -0.4 -0.6 -0.8 -1o

-t

0

0.02 0.04 0.06 0.08

0.1

0

0.12 0.14 0.16 0.18

0.2

x

0.22 2.pt

Figure 4.1. Magnetic flux density distribution in the air gap of an AFPM machine with iron slottedcore obtainedon the basisof eqn (4.1) forp : 3, s1 : 18, r : 36.7 mm, a; : 0.78, b p : d t r , B , : 1 . 2 T , p , , . " : 1 . 0 6 1g, : 1 . 5 m m , b r a : 3 . 5 m m , h , x 1: 5 m m , o : 0 a n d otv : 1.!5.

In the aboveequations(4.1), (4.2) and(4.3) kc is Cafier's coefficientaccording to eqn (1.2), b, is accordingto eqn (1.8) usually for a : 0, r is the averagepole pitch accordingto eqn (1.9), g' is the equivalentair gap in the daxis accordingto eqn (2.103),Bo is the flat-toppedvalue of the magneticflux

- 3.l0j 3.10) s 2.5 105

2.10" p,(x) t.s t05 l'10' 5'10' 0

0 0

0.02 0.04 0.06 0.08

0.1 x

0.12 0.14 0.16 0.18

0.2

0.22 2.p.r

Figure 4.5. Magnetic pressure distribution in an AFPM machine with iron slotted core obtainedonthebasis ofeqn (4.7) forp : 3, s1 - 18, r : 36.7 mm, ar : 0.78, b, : a.r, B , : 7 . 2 T , p 1 , , " , ,: 1 . 0 6 1 ,g : 7 . 5 m m , b 1 a : 3 . 5 m m , h m : 5 m m , a : 0 a n d otu : 1.15.

130


densityaccordingto eqn (3.13), o1x1isthe coefficientof the PM leakageflux accordingto eqn (3.9) and .\"t is the relativeslot leakageperrneancegiven by

/2r\ lu xl \tr /

(4.4)

in which the averageslot pitch h : 2prls1, s1 is the number of statorslots and the amplitude of slot harmonics as derived by W. Weber 164, I 1I I is Bln)l4(^.

u2l2

a u: tto 'n ," ' + ,; (O.s 'i B _ tFpj

\

sin(t.6utr r )

( 4.5)

The remainingquantities0(n), rc and I are expressedby the following equations: 1 /I\

13(") 2

t-l

\-

bu

I

\E -E )'

9*hu/tt,,,.'

(46)

f-= 1:1

The distribution of the magnetic flux density in the air gap obtained on the basisof eqn (4.1) is shownin Fig. 4.4. The magneticpressureon the statorand rotor activesurfacescan be found on the basisof the normal component(4.1) of the magneticflux densitydistributionin the air gap, i.e. Hzy*, -

7 Bo(t)'2 2

(4.7)

lrl

The distributionof the magneticpressure(.7) is visualizedin Fig. 4.5. The magneticpressureresolvedinto Fourier seriesyields harmonicsof magnetic forcesacting on the stator.It is necessaryto calculatethoseharmonics,e.g. in the noise analysisof electromagneticorigin radiatedby the AFPM machine.

4.5 Calculationof reactances 4.5.1 Synchronous and armaturereactionreactances Calculations of the stator current, other machine parametersand characteristics require knowledge of the synchronousreactanceper phase which is expressedby eqns(2.12) and(2.73). The synchronousreactanceX"4 andX", is the sum of the armature reaction (mutual) reactanceX,rd, Xoq and the stator winding leakage reactance X1 . The atmature reaction reactancesXod",Xoq and armaturereactioninductancesLod,,Loq can be calculatedwith the aid of equationsgiven in Section2.9 and Table2.1. The analyticalapproachto calculation of the statorleakagereactanceX1 is discussedin Section4.5.2.

131

AFPM machines with irutncrtres

(c)

(b)

h

h "14

h."

h,,o h.^

frtz

n ''12

h "12

fl,,.,

h ' '11

h,,

fira

'11

-11

Or=

9p

Figure 4.6. Stator slots of AFPM brushless machines: (a) rectangular semi-open slot; (b) rectangular open slot; (c) oval semi-open slot.

4.5.2

Stator leakagereactance

The armatureleakagereactanceis the sum of the slot leakagereactance X1.r, the end connectionleakagereactanceXp, and the differential leakage reactanceX14 (for higher spaceharmonics),i.e. Xt:

Xt" I Xrc * lra

, : nnf,.tJ#(Ar"krx + 1ii,, . * Iro..,t^,,"',r * )ra) i; t^,.,,

(4.8)

where l/1 is the number of turns per phase,k1;; is the skin-effectcoefficient is the for leakagereactance,p is the number of pole pairs. q1 : "tl(2prn,1) numberof statorslotss1 per pole per phase(2.2),11i"is the lengthof the stator winding inner end connection,l1,,rr1is the lengthof the statorwinding outerend connection,)r" is the coefficientof the slot leakagepenneance(slot-specific permeance) , \r",in is the coefficientof the leakageperrneanceof the inner end connection,\reout is the coefficientof the leakageperneanceof the outer end connectionand )1,7is the coefficientof the differentialleakage. of the slots shownin Fig. 4.6 are: The coefficientsof leakageperrneances r for a rectangularsemi-openslot (Fig. 4.6a): ,

At^

h1

hn

2ltt:r

ht+

3 b rr

b rr

bll * b11

brt

-

(4.e)

132

AXIAL FLTIXPERMANEAIT MAGNETBRUSHLESS MACHINES

r for a rectangularopen slot (Fig. 4.6b): ,

htt

hn*h'm*hM

-

il't-

(4.10)

, OtI

Joi I

r for an oval semi-openslot (Fig.4.6c):

) r . = 0 . t 4 2 4 1l f * ! ' ' t + 0 . 5 a r c s i n l Jbrr bn

(bulbn)2)+ t #

u14

(4.1r)

The coefficientsof leakagepelmeancesfor other shapesof slots than those shown in Fig. 4.6 are given, e.g. in [147]. The abovespecific-slotpenneances (4.9), (4.10) and (4.11) are for single-layerwindings. To obtain the specific perrneances of slots containingdouble-layerwindings, it is necessaryto multiply eqns(4.9), (4.10) and (4. I l) by the factor

30+t 4 where p is according to eqn (2.7). Such an approach is justified if 213

(4.r2)

<{3<

1.0. The specificperrneanceof the end connection(overhang)is estimatedon the basisof experiments.For double-layer,low-vo1tage,small and medium-power machinesthe specificpermeancesof the inner and outer end connectionsare: r inner end connection

:W) ) k i , , x 0 . 7 7 q 1-A

(4.13)

iff;l ) 1 o o=, 10 . 1 7 q 1-O

(4.14)

r outerend connection

where l1;'-,is the length of the inner end connection, l1ou1, is the length of the outer end connection, ul",;r.is the inner coil span, and u).orr1is the outer coil span.The speciflcpeffneanceof two end connectionsis a sum of permeances (4.13)and(4.14),i.e.

133

AFPM machines with iron cores

)1u :

)10,i,,* )l"ort :

* tu"outlrin\ 0.34qt (t - lt'u"i'Ibut ?r \ hinlbu,t )

@'15)

For wrir, : ucout : 'tltc &fid lyin - lrou,t - 1i1"the speciflc permeance (see eqn (4,15)) of two end connections has the same form as that for a cylindrical stator. i.e.

)r.:o.34qr(r-?g)

(4.r6)

71'lf,,/

\

Putting ut.fl1" : 0.64, eqn (4.16) also gives satisfactoryresults for singlelayer windings, i.e. the specificperrneanceof two end connectionsbecomes

(4.17)

)1" ru 0.2q1

For double-layer,high-voltagewindings the specificperneancesof the inner and outerend connections are: . inner end connection

?ffilni, x 0.27q11_ A7"in

(4.18)

r outer end connection

t r e o u=ro . 2 1 q 1- :r#t r n ' - ,

(4.re)

where the statorwinding factor k-1 for the fundamentalspacehatmonic z : 1 is accordingto eqn (2.10). The specificperneanceof two end connections

) r " : ) r , i n r ) t to u t: o . a z q 1( t \

l at'\tLl}lr't!t\ n

llintlout

nt,

(4.20)

/

In general,the following approximaterelation may be used for most of the windinss: )1" = 0.3q1

(4.21)

134

AXIAL FLUX PERMANENT MAGNETBRUSHLESS MACHTNES

The specific penneanceof the differential leakageflux is , Atd:

mtqtrk'|,,t Tat *

(4.22)

T'9' Ksat

whereg/ is the equivalentair gap accordingto eqns(2.103)or (2.104) andkrol is the saturationfactor of the masnetic circuit. The differential leakase factor Tdr is

rat-J-f1l-r1z 'b"' t r t l

(4.23)

t I

yS\

The curvesofthe differentialleakagefactor16 are given inpublications dealing with designof inductionmotors,e.g. 196,| 11, 1471. In practicalcalculationsit is convenientto usethe following formula

tot

-

n2(toql + z) . / 3 0 ' \ sllrl-l-1 27

(4.24)

\Qrl

The specificperrneancebetweenthe headsof teeth:

Str=ffi

(4.2s)

should be added to the differential specificpeffneance)1a of slotted stator windings.

4.6


Specificationsof Mavilor AFPM motors (Figs. 4.1 and 4.2) are shown in Table4.1. Torque-speedcharacteristicsof Mavilor AFPM motors are plotted in Fig. 4.7. AFPM servomotors MA-3 to MA-30 are designedwith ClassF insulationand IP-65 protection. Table 4.2 showsspecificationsof brakesfor MA-3 to MA-30 servomotors. Table 4.3 shows specificationsof GenSmartTMAFPM synchronousgenerators with amorphousalloy stator cores manufacturedby LE Incorporated, Indianopolis,IN, U.S.A. GenSmar{M AFPM synchronousgeneratorscan be used both as stationary and mobile generators.

135


Tnrqur iNmi 18 lr!

tul4-![

14 1! 1*

MA.-30

s fi 4 7. fi 0

_ MA-1fi

.MK.E - MA"3

5flf

1nflfi

'}*{Jt} 'l 35flCI 4Dnfl 5flfi ?0fifl 25*fi $peecl irpnti

Figure 4.7. Torque-speedcurves of AFPM brushlessservo motors manufactured by Mavilor, Barcelona,Spain.

Table 4.1.

AFPM brushlessservomotorsmanufacturedbv Mcrvilor. Barcelona" Spain

Maximum speed,rpm Stall torque,Nm Stall current, A Peak torque, Nm Torque density, Nm/kg EMF constant,Vs/rad Torque constant,Nm/A Cogging torque, Nm Winding resistance,{) Winding inductance,mH Rotor moment of inertia,kgm2 x 10-3 Mechanical time constant,ms E,lectricaltime constant,ms Thermal time constant,s Thermalresistance,"C/W Mass, kg Radial load, N

MA-3 9000 1.3 2.2 5.2 0.1 0.3 0.6 < 0.2 10.2 25.0 0.04 2.0 2.5 1500 1.1

MA-6 6000 3.6 4.2 28.5 0.8 0.5 0.9 < 0.1 5.3 11.6 0.30 3.8 2.2 1500 0.6

MA-10 6000 5.8 6.8 40.7 Ll 0.5 0.9 < 0.1 2.2 0.4 2.10 3.6 7.5 1800 0.5

2r8

225

225

MA.2O 6000 10.0 10.3 69.8 1.2 0.6 1.0 < 0.2 1.4 7.0 0.80 2.1 5.0 1500 0.4 34 390

MA-30 6000 16.0 16.5 96.0 1.6 0.6 1.0 < 0.3 0.5 4.0 1.60 1.6 7.5 1500 0.4 36 390

r36

AXIALFLUX PERMAI{ENTMAGNETBRUSHLESS MACHINES

Table 4.2. Speciflcationsof built-in brakes for AFPM brushlessseryomotorsmanufacturedby Mavilor, Barcelona, Spain

Holding torque, Nm d.c. voltage,V Input power, W Momentof inertia,kgm2 x 10-3 Disengagementresponsetime, ms Engagementresponsetime, ms Mass, kg

MA-3 I 4.8 5 0.08 7 5 0.3

MA6 4 24 22 0.3 30 7 0.8

MA-10 4 24 22 0.3 30 7 0.8

MA-20 88 11

Ma-30 1/

22 0.3 30 77

22 0.3 30

0.u

0.8

Table 4.3. GenSmartT^r AFPM synchronousgeneratorsmanufactured by LE, Inc., Indianopolis. IN. U.S.A.

28-G22 Rated output poweq kW Rated speed,rpm Maximum speed,rpm Voltage, V Stator insulation Efficiency, To Frame extemal diameter, mm Frame extemal length, mm Stator core material Cooling Ambient temperature,"C Maximum allowabie temperature,"C Enclosure Mounting Overtemperaturecontrol Mass,kg Power densiry kWkg

90-G32 90

28 3600 4240

92.2 216 150

I.JJJ

480or 208 ClassH 94.9 315 186 Amorphous alloy liquid or air cooled -50to 60 125(risefrom40"C) Totallyenclosed Flangeor foot Embedded thermistor 44 2.045

t20-G49 t20 2s00 3000

95.3 485 200

115 1.043

4.7 Performance calculation 4.7.1 Sine-waveAFPM machine The electromagneticpower of a sine-wave(synchronous)AFPM motor is expressedby eqn (2.84)which can also be broughtto the following form:

t37

AFPM machineswith iron cores rll + * (t", - X,n)I?,sin 2V] Perr, : mllloE J lcos 2\

(4.26)

Hence,the developedtorque is

, ['s^ :

:

P"l^ 2nn"

t#

llrr

1 k u ,11I o o < sv * ) t r " o - L " s ) I : , i znv l @ . 2 7 1

For buried and interior PM rotors X",t t X"n and L",t ) ,L"n, which implies that an additional reluctancepower and torque component is developed by the machine. For surface PM rotors the reluctancedependentcomponents of power and torque are almost not existent as X"a = Xsa and L",1.= L"n. In eqn(4.26)the EMF E y canbe calculatedby meansof eqn (2.29). The current Io and angle V in eqns (4.26) and (4.27) are either specified,or determined accordingto Section2.7 using voltage phasorV1 projectionson the d, and q axes. The reactancesand inductancesare calculatedas describedin Sections 2.9 and 4.5.2. Hence,with all the parametersknown, the performanceof the AFPM brushlessmotor in terms of the electromagneticpower and developed torquecanbe calculated.Note that, for the generatingmode,the secondterm in eqns(4.26) and(4.27) (dependenton the differencebetweenthe d- and q-axis synchronousinductances)is written with the minus ( ) sign. The electric power available,after losses,of an AFPM machinewith the statorferromagneticcore is calculatedas r the input power for motoring mode

Pr.n: P.t,n * LPt- * LPrp" * LPzp" * APppy + AP"

(4.28)

r the output power for generatingmode

Pout : P"t^ - LPu, - LPtt"",

LPzp" * A.Ppm - LP",

(4.29)

where P"1^ is accordingto eqn (4.26), LPr, are the stator winding losses accordingto eqn (2.42), LPtpu are the stator core lossesaccordingto eqn (2.47), LPzp" are the rotor core lossesaccordingto eqn (2.59), LPpyl are

138

AXIAL FLUX PERMANENTMAGNETBRL]SHLESS MACHINES

the lossesin the PMs accordingto eqn (2.54) and AP" are the eddy current lossesin the stator conductorsaccordingto eqns (2.61) or (2.62) (only for slotlessstators). The shaft power P"7..(input power Pi, for generators,output power Po6 for motors) is calculatedby adding to the electromagneticpower P"1- (generators)or subtractingfrom the electromagneticpower (motors) the rotationallossesLProt accordingto eqn (2.63). The shaft torque is r for a motor (developedby the motor itself) Por,t

-t , s n

2rn

(4.30)

r for a generator(developedby the prime mover) P,L ,_, -_ ", 27r,,

(4.3r)

and the efficiency is P

L

't t

4.7.2

OUf

(4.32)

p 1,n,

Synchronousgenerator

The phasordiagramof a salientpole synchronousgeneratorwith RL load is shownin Fig. 4.8. The input voltageprojectionson the d andq axesaccording to Fig. 4.8 are Vrsin6:IonX"n-IoaRt

l/r cos6 : Ef * IodX",t,* IoqRt

(4.33)

and

14 sin 6 : IoaRr

IoqXr

I cos6 : IonRl * IoaXr

G}4)

139


jl"oX, jl"oXL

l"oR,

o I

Figure 4.8.

Phasordiagram of an overexcited salient pole synchronousgeneratorfor RL load.

where Zr - Rr * jXr, is the load impedanceperphase acrossthe outpul tetminals.The d- and q-axiscurents are

r from eqns(4.33) r,-

'ad,

-

E.fX", - Vt(X"qcos d F R1 sind)

X"aX"q * R,'4

' ta-q

(4.3s)

Vr(X"a sin d - fir cosd) + tf Rt X"aX"q* R?

(4.36)

W(Rr sind * X; cosd) R2r_ + x2,

(4.37)

r from eqns(4.34)

r,-

tq.a

-

140

AXIAL FLUX PERMANENTMAGNETBRUSHLESS MACHINES Vt (.Rr cosli t,o r : W

Xl sin d)

(4.38)

The load angle6 betweenthe voltageVl andEMF Ei can be determined,e.g. from the first eqn (4.33),i.e. I"dRL - I"qXL\ d : arcsin (

\I/r

/

(4.3e)

Combining eqns(4.33) and (4.34), the d- and q-axis currentsare independent of the load angled, i.e.

r.-

Lad

"n

-

-

Ef (X'q a xr) (X"a* xr)(X"qaxr) +(/?rlRr)' ES(Rt * Rr,)

(x"o* xy)(x"n a xr) + (i?r* Rr)2

(4.40)

(4.4r)

The angle V between the current Io and g-axis and the angle $ between the current Io and voltage Vy arc respectively,

irr: arccos (?)d: arccos (+):

arccos (m) arccos (X)

(4.42)

(4.43)

where I" : lia * Iln @eealso eqn (2.82)). The output electrical power on I the basisof phasordiagram(Fig. 4.8) and eqn (a.33) is Pout : myVllocos @ - m1V1(Ia,rcos d * Ioa sin d) : mllEylo,

- Io6,Ior(X"a - X"q) - IiA,l

6.44)

Including only the statorwinding losses,the internalelectromagneticpower of the eeneratoris Petrn: Pout -f LP:'- : mllEylon - Io6Ion(X"a - X"q))

(4.45)

AFPM machines with iron cores

4.7.3

t4l

Square-waveAFPM machine

The square(trapezoidal) wave AFPM brushlessmachine has already been introducedin Section2.10.2. Square-wavemachinesarecharacterized by their trapezoidal or quasi-squareback EMF waveforms with the conduction angle from 100 to 150 electricaldegrees,dependingon the construction.The trapezoidalback EMF waveformscanbe obtainedby properdesignof the magnetic and electric circuits. Both slofied and slotlessAFPM machinescan be designedso as to obtain sinusoidalor trapezoidalback EMF waveforms,except the toroidal AFPM machineof Figs 2.4 and2.5.The toroidalmachineis almost always designedto have atrapezotdalback EMF waveform. The amplitude of the back EMF can be calculatedby meansof eqn (2.118) for a three phase Y-connectedmachinewith two phasewindings active. The square-waveAFPM machine is always fed by meansof a solid state inverter.The invertersuppliesd.c. PWM voltagesto the machinephasewindings during the conductionintervals. The current squarewaveforms(seeFig. 1.3a)are describedby the flat-toppedvalueof the current4"q). The developed torque Ta of Ihe square-waveAFPM brushlessmachine can be calculated by meansof eqn (2.120). Hence,the electromagneticpower Pr1- of the machine can be found. Using the same loss and power balanceequationsas for the sine-wavemachine,the input and output power can be calculatedwith the aid of eqn {4.28) and the ouQut power Pottt : Putr, - LProt. Note that LPn, must be calculated by using the true rms value of the current given by eqn (2.t28).

4.8

Finite elementcalculations

Insteadof using analyticalor semi-empiricalformulae, someof the equivalent circuit parametersand lossesof the AFPM machine can be calculated accuratelyby meansof the FEM analysis.The 2D and 3D FEM electromagnetic software packagesare nowadays available at affordable prices and the field solution time has become attractivelyshort. For example,most of the 2D FEM magnetostaticproblemscan be solvedin secondson today'spersonal computers. The synchronousinductances-Lr6 and Lrn and the EMF Ef of an AFPM brushlessmachineare the critical parametersusedin the performancecalculations. The first step is to specify on a 2D plane the outline, dimensionsand materials of the AFPM machine. Due to the symmetry of an AFPM machine, it is usually sufficientto model only one pole pitch (or sometimeseven half a pole pitch) of the machineby applying periodic boundaryconditions[233]. The cross section is taken at the averageradius of the active part of the stator winding, i.e. at 0.2s(Dort + D,"). Dependingon the machinetopology

142


-''\,\

Air Gap F'lux 0 Density

(r)

\

,!

\---:+-

900 Rotor angle

lg0 "

lg00

=iliilii

t'

..t--j--___= /'--'--z;:*;

!:

t,',";-E

Air (iap Flux g Density

(r)

0

(b) Figure 4.9. Magnetic flux plot and air gap flux density with only PM excitation of (a) slotless and (b) slotteddisc type machinewith surfacePMs.

and material propefties,either Neumannor Dirichlet conditionmay be imposed on the remainingboundaries. The length of the winding, which is in the radial direction, is taken as the active length of the stator winding, i.e. Lt: ltr :0.5(Dout Dm) - eqn (3.34).After assigningdifferentmaterial propefiiesto various regions of the model, the defined structurecan then be meshed. The secondstep is to specif,uthe phasecurrents,i.q,,ia andig, in the machine accordingto the position of the rotor. To calculatethe parameterL"4 ot L", the phasecumentsmust be assignedin such a way that the statorMMF can be alignedwith the d-axis or q-axis respectively.In both casesthe PMs haveto be kept inactive.In the caseof calculationof the EMF Ef equalto the no load voltage in generatingmode, the phasecurrentsare set to zero and only PMs are active. The third stepis to run the FEM solverand obtainfor eachof the threecases describedabove a solution for the air gap magneticflux density in the axial directionof the machine.The fundamentalcomponentof the air gap magnetic

t43

AFPM machineswith iron cctres

\/

- r . ' - - - - - = - 1 ' \ ' /

J ' - - - - .

ffi#a+ji)ArAf-,5.$

mnii.ltitif--li| |ii-litiiiilt i lt -+;

it[.;tf-]riltL:lIiilj illif {--l

0.1 Magnet Flux Density

l1

(r) o ' * / \ _ , -0.1

:7

\r

Aluminium Rotor angle

ffi=='

Ltffi

0.24 Magnet Flux Density

(r)

g00

(b)

I 80-

Rotorangle

Figure 4.10. Magnetic ffux plot and air gap flux density of interior AFPM machine with (a) only d-axis and (b) only g-axis statorwinding excitation.

flux densityis determinedby Fourierexpansionof the axial componentof the air gap flux density. Hence, the peak values of the fundamental air gap flux densitiescan be determinedfor eachof the casesabove: (1) B,n"at with only the d-axis MMF and thus only the d-axis cuffent I"a, (2) Brroql with only the g-axisMMF and the q-axis cunent lor,and(3) Bn,s with only the PMs active. Step four is to calculateby meansof analytical equationsthe inductances and induced EMF. With Brnc,,1rand B,nonl known from the FEM analysis,the flux linkagesVa and Ilrncanbe calculatedaccordingto eqns(2.97)and (2.98). The inductancesl",y and L"o can be calculatedas describedin Section5.4.2. With B-e known from the FEM analysis,the back EMF Ey can be found by first calculatingthe magneticflux per pole Oy accordingto eqn (.2.22)and then .O1according to eqn (2.29). Fig. 4.9 showsthe FEM flux plots and air gap flux densitiesdue to PM excitation alone for two single-sidedAFPM machine configurationswith slotted and slotlessstators.Fig. 4.10 showsthe FEM flux plots and air gap flux den-

144


sities for a single-sidedinterior AFPM machine with only the d- and q-axis statorwindingexcitation.

Numerical example4.1 A three-phase, 2.2-kW, 50-Hz, 380-V (line-to-line),Y-connected,750-rpm, : 7870, cos T @ : 0.83, double-sideddisc PM synchronousmotor has the following dimensionsof the magneticcircuit: PM outerdiameterDout : 0.28 m, PM inner diameterD;r: 0.16 m, thicknessof the rotor (PMs) 2h7,1: g mm, single-sidedmechanicalclearanceg - 1.5 mm. A surfaceconfiguration of uniformly distributed PMs has been used. The rotor does not have any soft ferromagnetic material. The rotor outer and inner diameters correspond to the outer and inner outline of PMs and the statorstack. The dimensionsof rectangularsemi-closedslots(Fig. 4.6b) are: hn : 11 mm, hr2 :0.5 mm, hs : 1 mm, hu - 1 mm, bn : 13 mm andbla : 3 mm. The number of statorslots (one unit) is sr :24, the number of armaturetuins of a single statorper phaseis ly'r : 456, the diameterof the stator copperconductoris 0.5 mm (without insulation),the numberof statorparallelwires is a- : 2 and the air gap magneticflux density is B,nn - 0.65 T. The rotational lossesare LP,ot: 80 W and the coreand PM lossesarc L,P1p. + LP1M :0.\5Pout. There are two winding layers in each stator slot. The twin-stator Y-connected windings are fed in parallel. Find the motor performanceat the load angled : 11o. Comparethe calculationsobtainedfrom the circuital approachwith the FEM results. Solution The phasevoltage is I,z1: 3801/3 : 220 V. The number of pole pairs is p : f ln: 50 x 601750: 4 and 2p : B. The minimum slotpitch is ,

. tm?,n

rD,i,n

zr x 0.16

51

24,

-

: 0.0209rn = 21 mnr

The width of the slot is b12 : 13 mm, that meansthe narrowesttooth width clrnin : trnir, - btz : 2I - 13 : 8 mm. The magneticffux density in the narrowestpart of the stator tooth Brtnro, -

0 . 6 5x 2 1

B*nt\'ni' cLrnirt

:7.7 T

which is rather low. The averagediameterand averagepole pitch accordingto eqn (1.9) are

D : 0 . 5 ( 0 . 2 +8 0 . 1 6:)0 . 2 2 m n x 0.22

: 0.0864rn


145

Because thenumberof slotsperpoleperphase(2.2) qI

sl

24

zp-t-g"J

the winding factorkr,1 : k6kpt: a n d( 2 . 1 0 ) .

I

1 x 1 : 1 asexpressedby eqns(2.8),(2.9)

The magneticflux accordingto eqn (2.22) and EMF inducedby the rotor excitationsystemaccordingto eqns(2.29) are respectively ) e r : 1 x 0 . 6 b { [ ( 0 . 5x 0 . 2 q 2 - ( 0 5 x 0 . 1 6 ) 2:] 0 . 0 0 2 1 4w5 b " Tt 8"

: 2rT.z\l Ef : nt/250 x 456x 1 x o.oo2r4b It hasbeenassumedthat B,',ny - Brns. Now it is necessaryto check the electric loading, current density and fill (space)factor of the statorslot. For two parallel wires a- : 2,the numberof conductorsper coil of a double-layerphasewinding accordingto eqn (2.3) is orly'r 2 x 456 ,, n Ar "" - (", (2413) lr,,) Thus the number of conductorsin a single slot l/"1 is equal to the (number of layers) x (numberof conductorsper coil N") : 2 x 774 : 228 or according to eqn (2.5) | ',2 x 456 ,\',i:__*-:22x The rated input current in a single stator is I -

Pout 2200 " 2 m 1 V 1 r 7 c o s @2 x 3 x 2 2 0 x 0 . 7 8

-r.57_\ > U.83

The statorline currentdensity(peakvalue) on the basisof eqn (2.86) is 4r/2 *tloNr 3 \ / r 4 5 6x 2 . s 7 ^ - -A- : _r;-l-l -152.2A/rn t,\uout + Di") 0.0864x 4 which is rather a low value even for small PM a.c. motors. The crosssection of the stator (armature)conductor

,^ o o -:

ndT-- ro'52:o.1gzmm2 ,t /

MACHINES AXIAL FLUX PERMANENTMAGI'{ETBRUSHLESS

146

gives the following current density under rated conditions ,la

2.57 :6.57 Af nln2 2 x 0.797

-

This is an acceptablevalue of the current density for disc-typea.c. machines rated from I to l0 kW. For the class F enamel insulation of the armature conductors,the diameter of the wire with insulationis 0.548 mm. Hence.the total crosssectionalarea ofall conductorsin the statorslot is ^^ . n0.548'

22s#

= 54 mnr2

h nbn : 11x 13 : 143 Thecrosssectionareaof asingleslotis approximately mm2. The slot fill factor54f143 : 0.38 showsthatthe statorcanbe easily wound,sincethe averageslot filI factorfor low voltagemachineswith round is about0.4. statorconductors for a discrotora.c.machine Theaverage lengthof thestatorendconnection to eqn(2.32) lengthof thestatorturnaccording is 11"= 0.154m. Theaverage is l t o , :, 2 ( L t . + l t " ) : 2 ( 0 . 0 6+ 0 . 1 5 4:) 0 . 4 2 8m - Dn) : 0.5(0.28- 0.16) : 0.06m. The stator whereLt : O.5(Do,ut per phaseat a temperature of 75"C (hotmotor)according windingresistance to eqn(2.33)is

Rr

lfr lrou

456x 0.428

: 1 0 . 5 7O

a1n6so, 47 x106 x 2 x 0.1965

Carter'scoefficientis calculatedon the basisof eqns(1.2)and (l .3),i.e. roo-l

1 : +[*

28.8 - \ I2 : 1.004 2 8 . 8 0 . 0 0 5 2x6 1 7 /

arctan (#")

-'"

: 000526 *f] V[ (--?__,,,

whereh : rDlsr : '1Tx 0.22124: 0.0288m: 2B.Bmm. The nonferromagneticair gap in calculationof Carter'scoefficientis g' = 29 l2h'y : 2 x 7.5 * 8 : 11 mm. Sincetherearetwo slottedsurfacesof twin statorcores, Carter'scoefficienthasbeensquared. The stator (one unit) leakagereactancehasbeen calculatedaccordingto eqn ( 4 . 8 )i n w h i c hl y ; , A 1 . ; nLf; - l l r o u t \ t , o uLr f; = i 1 " )t " f L , . i . e .


147

t i'm (o.T7g+ x t - r x 0 . 4 rv I 0 6 r , . 5 g a 5 6 ' ffio.zra + 0 . 2 2 9 7+ 0 . 9 3 2 2 ): 6 . 1 5 8C I where r the coefficientof slot leakagereactance- eqn (4.9) is Ar":

11 o'5 2x1 l + + *5 :o'ttn 3x 13 rs re+g

* eqns(4.16)in which r the coefficientof end connectionleakageperrneance the averagecoil spanwc: T is

) r " = 0 . 3 x4 t ( t - ? * r y ) \

/f u.t)+ /

: 0.218

- eqns(4.22) and(4.24) is r the coefficientof differentialleakageperrneance

:o22e. oe66

)r'r:#i#?##o r(tr 4t#-,i,,$

_ r o.oe6ei

- eqn(4.25)is r thecoefficient peflneance of tooth-topleakage 5 x7713

. rr:5 A *o;fi-0.9322 Accordingto eqns(2.114)and (2.115)in which kf a: kf n:1, the armature reaction reactancesfor surfacetvne PM rotors and unsaturatedmachinesare X o ' t: f , a a

*

2x3 x0.4x rx

10 6 t so ({9t1)' \4/

- (o .sx o .to )2 ( o .rx o .zs)2 _ 5.856 CI 1 .0 0 x4 0 .0 1 1

where the air gap, in the denominator,for the armatureflux should be equal to g' x 2 x 1.5t8 - 11 ffiffi(F,,.. = 1). The synchronous reactances according to eqns(2.12)and(2.73)are

148


Figure 4.11. Magnetic flux plots in the double sided AFPM motor: (a) zero armature curent, (b) rated armature current. Numerical example 4.1.

X s d:, X " q : 6 . 1 5 8 + 5 . 8 5 6: 1 2 . 0 1O The armaturecurrentsare calculatedon the basis of eqns (2.80), (2.81) and (2.82). For d : 11" (cosd : 0.982,sind : 0.191)the currentcomponents z r e 'f. o 4 : 1 . 8 2A , 1 o q: 1 . 8 8A a n dI o : 2 . 6 2 A . The input power absorbedby one stator is expressedby eqn (2.83). The input power accordingto eqn (2.83)absorbedby two statorsin parallelis twice as much, i.e. Pin -2

x 3 x 2 2 0 ( 1 . 8 8x 0 . 9 8 2- ( - 1 . 8 2 ) 0 . 1 9 1:) 2 8 9 2 . 4 W

The input apparentpower absorbedby two statorsis


149

I

(a)

0.E 0.6 Magoetic

nllli, o.n' (r) 0.2 0 *q.2

-90'

-450

0" Rotor angl*

4$E

I

(b) 0.8 0.6 Magnetic

nll'li, o.r {r) 4.2 0 *o.2 -s0q

*45"

00 Rntor angle

Figure4.12. Magneticflux densitydistributionalongthepolepitch: (a) zeroarmaturecunent, (b) ratedarmaturecurrent.Numericalexa.mnle 4.I .

5;,, : 2 x 3 x 220 x 2.62 : 3458.4VA The power factor is ,aa, ,4 ';rr*:0.836 cosE-

Thelossesin two statorwindingsaccording to eqn(2.42),in which kyp rt I, are calculatedas LPt,, : 2 x 3 y 2.622x 10.57:

435.2W

The outputpowerassumingthat LP1p. + LP.4:

0.05Porris

71 P u u t- 7 1 ; ( P ; , , - A p r . r .uit

+ J 5 . 2 B 0 . t t )= 2 2 6 4 W

Lp,or):

.;-(2g92.4

l.Ulr

150

A,YIALFLUX PERMANENTMAGNETBRL]SHLESS MACH]NES I 35 30 Reluctance torgue €-Total torque -*-

25 Torque 20 Nm 15

10 o

0 *5 -154

Figtrre 4. I 3.

0,

150 Rotor angle

304

4go

Torque as a function of the disc rotor position. Numerical example 4. I

Thus

LPrr. + LPPM -- 0.05x 2264: 113.2W Themotorefficiencyis 2264.0 2892.4

: 0.783

or

rl :78.37o

The shaft torque is ))(\A

- 28'83 Nm 2o\7,.,1V60)

["n The electromagneticpower is Petm: P,n- LPt--

LPtr"'"- LPpm - 2892.4-435.2-713.2 :2344W

The electromagnetictorquedevelopedby the motor is ry1 , _ rd,

2344 : 29.84Nm 2r x 750160

The resultsof the FEM analysisare shownin Figs 4.11,4.12 and4.13. The FEM gives higher valuesofthe averagedevelopedtorque than the analytical approach.Eqns(2.114),(2.115)and (a.8) do not give accuratevaluesof X,6, X"n andX1. The electromagnetictorqueplotted in Fig. 4.13 againstthe rotor position has a significantcogging effect componentwith the period equal to doubleslotpitch, i.e. 30".

AFPM machtneswith iron cores

l5l

Numericalexample4.2 A three-phase,Y-connected,50 Hz, 5.5-kW AFPM synchronousgenerator with slotted stator has the following parametersof the steady-stateequivalent circuit: fir : 0.1 Q, X"d, : 2.3 Q and X"n : 2.2 Q. The generatoris loaded with a resistance of Rt, : 2.2 Q and inductanceof L7: 0.0007H connected in series. The EMF per phaseat no load inducedby the rotor PM excitation systemis Ey : 100 V and rotationallossesare LProT: 90 W. Find the statorcurrents,electromagneticpowet output power and efficiency. Statorcore lossesand lossesin PMs havebeenisnored. Solution The load reactanceand module of load impedanceare XL:2tr.f Lr:2r

x 50x 0.0007:0.22{)

Z 7 : / 2 2 2 + 0 2 2 2: 2 . 2 1 1 Q The statorcurrentsaccordingto eqns(4.40),(4.41) and(2.82) are ,

"n

:

'L^ u q-

: \ 4 . . r- r 0 . 2 2 ) ( 2 . 2+ 0 . 2 2 )+ ( 0 . 1+ 2 . 2 ) 2

21'2ltA

Loo(o'l- 2'2) t{) 2()A 0 . 2 2 ) ( 2 . 2 + o . 2+ 2 )( 0 . 1+ 2 . 2 ) 2 23 Io:

J212b2+ 2020, : 2g.32A

Theterminalvoltage(across theloadimpedance) is V : I u Z r : 2 9 . 3 2 x 2 . 2 7 I: 6 4 . 8 2 Y V t L : n E V t : \ / 5 6 4 . 8 2: 1 1 2 . 3 V The powerfactor,loadangled andangleV accordingto eqns(4.43),(4.39) and(4.42)respectively are cos@-

zg'??\-z'z:0.9e5 64.82

v : arccos (ffi)

4:5.77'

:46.4b"

152

AXIALFLTJXPERMANENTMAGT)ETBRUSHLESS MACHINES

x 2'2-- 20'20x 0'22 d : arcsin(27'25 ) : 40.24o 64.82 \ I d : V * 4,: 46.450- 5.77": 40.740 Theelectromagnetic poweraccording to eqn(a.45)is - 2.2)l: 5930.3 Petrn:31100 x 20.20 2I.25x 20.20(2.3 W Thestatorwindinglossesaccording to eqn(2.42)is L P t - * : 3 x 2 9 3 2 2x 0 . 1 : 2 5 7 . 8 W Theoutputpoweris : rrllVllecosd - 3 x 64.82x 29.32x 0.995: 5672.5W Pou,t Pout: P"t^ * APw:

5930.3- 257'8 :5672.4

The inputpoweris Pin : P.h, i LProt: 5930.3* 90 : 6020.3W Theefficiencyis 5672.5 4:6020-3:o'942

Chapter5 AFPM MACHINES WITHOUT STATOR CORES

5.1

Advantages and disadvantages

Dependingon the applicationand operatingenvironment,statorsof AFPM machinesmay haveferromagneticcoresor be completelycoreless.A coreless stator AFPM machine has an internal stator and twin external PM rotor (Fig. L4d). PMs can be glued to the rotor backing steeldiscsor nonmagneticsupporting structures.In the secondcasePMs are arrangedin Halbachanay (Fig. 3.15) and the machineis completelycoreless.The electromagnetictorque developedby a corelessAFPM brushlessmachineis producedby the open space current-carrying conductor-PM interaction (Lor entz force theorem). Corelessconfigurationseliminateany ferromagneticmaterial,i.e. steellaminationsor SMC powdersfrom the stator(armature),thus eliminating the associatededdy current and hysteresiscore losses. Becauseof the absenceof core losses,a corelessstatorAFPM machinecan operateat higher efficiency than conventionalmachines.On the other hand, owing to the increasednonmagneticak gap, such a machineusesmore PM material than an equivalent machine with a ferromagnetic stator core. Typical coil shapesused in the winding of a corelessstator are shown in Figs3.16 and3.17. In this chapterAFPM brushlessmachineswith corelessstatorand conventional PM excitation,i.e. PM fixed to backing steeldiscs,will be discussed.

5.2

Commercial corelessstator AFPM machines

Bodine Electric Company,Chicago,IL, U.S.A. manufactures178-mm (7inch) and 356-mm (I4-inch) diameter e-TORQTMAFPM brushlessmotors with corelessstatorwindings and twin externalPM rotors with steelbacking discs (Fig. 5.1a). The corelessstatordesigneliminatesthe so called cogging

t54

AXIAL FLUX PERMANEI{TMAGNETBRUSHLESS MACHINES

Figure 5.1. AFPM brushlesse-TORQT\I motor with corelessstator windings: (a) general view; (b) motor integratedwith wheel of a solar powered car. Photo coudesy of Bodine Electric Company, Chicago,IL, USA, www.Bodine-Electric.com.

(detent) torque, improves low speedcontrol, yields linear torque-cunent characteristicsdue to the absenceof magnetic saturationand provides peak torque up to ten times the rated torque. Motors can run smoothly at extremely low speeds,evenwhen poweredby a standardsolid stateconverler.In addition, the high peak torque capability can allow, in certain applications, the elimination ofcostly gearboxesand reducethe risk oflubricant leaks. The 356-mm diametere-TORQTMmotors have been used successfullyby studentsof North Dakota StateUniversity for direct propulsion of a solar powered car participatingin2003 American Solar Challenge(Fig. 5.lb). A well designed solar-poweredvehicle needs a very effrcient and very light electric motor to convert the maximum amount of solar energy into mechanicalenergy at minimum rolling resistance.CorelessAFPM brushlessmotors satisfythese requirements. Small ironless motors may haveprinted circuit stator windings or film coil windings. The film coil statorwinding has many coil layerswhile the printed circuit winding has one or two coil layers.Fig. 5.2 showsan ironlessbrushless motor with film coil statorwinding manufacturedbyEmBesl,Soeul,SouthKorea. This motor has single-sidedPM excitationsystemat one side of the stator and backingsteeldisc at the other sideof the stator.Small film coil motorscan be usedin computerperipherals,computerhard disc drives(HDDs) 1128,129), mobile phones,pagers,flight recorders,cardreaders,copiers,printers,plotters, micrometers,labelingmachines,video recordersand medicalequipment.

AFPM MACHTNESWITHOT]TSTATORCORES

H <;c|H

9d

155

*

Ug *S.rr

.*HA: 'U:

,:p,

' ];,rr:,l,lli' -" 16 14

6: w:

3 , *v) b fr 2zP+

gs

f

H;

5

Figure 5.2. Explodedview of the AFPM brushlessmotor with film coil corelessstatorwinding and single-sidedrotor PM excitationsystem.Courtesyof Embest,Soeul,South Korea.

5.3 Performancecalculation 5.3.1 Steady-stateperformance To calculatethe steady-state performanceof a corelessstatorAFPM brushlessmachineit is essentialto considerthe equivalentcircuits. (a)

jl"oX"o jl"oX"o <__

-+ +,l

t"

jl"oX"o jl"oX"t x1

<__

<_

'"{

Figure 5.3. Per phaseequivalentcircuit ofan AFPM machinewith a corelessstator: (a) generator arow system; (b) consumer(motor) arrow system. Eddy current lossesare accountedfor bv fhc shuntrc5istance n, .

The steady-stateper phase eqLdvalentcircuit of a corelessstator AFPM brushlessmachinemay be representedby the circuit shown in Fig. 5.3, where R1 is the statorresistaoce,X1 is the statorleakagereactance,E1 is the EMF

156


induced in the stator winding by the rotor PM excitation system,Ei isthe rms value of the internal phasevoltage, V1 is the terminal voltage and Io is the rrns statorcurrent. The shuntresistance-Ruis the statoreddy current loss resistance, which is definedin the sameway as the core loss resistance[116] for slotted PM brushlessmotors. The statorcurrentslo,t",Ioq andIo for a given load angle d (correspondingto the slip s in induction machines)can be calculatedon the basisof eqns(2.80), (2.81) and (2.82) respectively. If loa : 0, all the stator current Io : Io,t produces the electromagnetictorque and the load angle 5 : Q. The angle / between the stator current 1o and terminal voltage % is determinedby the power factor cos@. If we ignore the lossesin PMs and lossesin rotor backing steeldiscs,the input power can then be calculatedas follows r for the motoring mode (electrical power) Pi,:

P"hn * LPt- + LP"-

( s1 )

r for the generatingmode (mechanical shaft power)

(.s.21

Pin:P"t*,*LProt

where P"1,nis the electromagneticpower accordingto eqn (2.84) in which LPtr" : 0, LP1, is the statorwinding loss accordingto eqn (2.42),A,P" are the eddy currentlossesin the statorconductorsaccordingto eqn (2.61) or eqn (2.62) and A,P,o1arethe rotationallossesaccordingto eqn (2.63). Similarly, the output power is: r for the motoring mode (shaftpower)

(s.3 )

Pout:Pulnr-LProt

r

for the generating mode (electrical power) Pout:

P"I*, -

LPt-

-

LP",

(s.4)

The shaft torque is given by eqn (4.30) for motoring mode or eqn (4.31) for generatingmode,and the efficiencyis expressedby eqn (4.32).

157

AFPM MACHTNESWITHOUTSTATORCORES

5.3,2

Dynamicperformance

For a salient pole synchronousmachine without any rotor winding the voltage equationsfor the stator circuit are - duq

utd: Rti"o +*

(s.s)

dL

(s.6)

utq:R1i'o+**aad AL

in which the linkage fluxes are defined as

tba: (L"a* Lv)i"a* ths: L'aioa* 4;f

,!q :

(Loq -t Ly)i,"n :

L"qioq

(s.7)

(s.B)

ln the aboveequations(5.5) to (5.8) tr14and'u1nared- andq-axis components of the terminal voltage,r! S is the maximum flux linkage per phaseproduced by the excitationsystem,Br is the armaturewinding resistance,Lo6, Lon are the d- and g-axis componentsof the armatureself-inductance,r,r : 2n f is the angular frequency of the armature current, iod, ioq are the cl- and q-axis componentsofthe armaturecurrent. The resultantarmatureinductancesLsd,: Lo,j I L1 andLrq : Loq t Ly arereferredto as synchronousinductances.In a three-phasemachineLod : Gl\Lt*,j and Lo, : (3f 2)Lt"q where Ltooand L'on are self-inductancesof a single phasemachine. The excitation linkage flux /7 : Lyaly where Lya is the maximum value of the mutual inductance between the armature and field winding. In the case of a PM excitation, the fictitious field currentis 1; : H.hu. Puttingeqns(5.7) and (5.8) into eqns(5.5) and (5.6),the statorvoltageequations in the d and q-axiscan be written as u1,1: R1,io6 * 4#f dt

,u1n:

"o

- aL"nion

f"n I aLr6io4 a alt t R1,ion - + *o,1;

(s.e) (s.10)

:0,Io:Io4l : (dI d,t)L For the steadystateoperation (dI dt) L "qi"q "d,iacr jl,rq,Yt : Va I jVtq,loy : f2Iod, i,s.q: f2lo, , u14- J2Vra, rrn :

r58


,fiVtn, Ey : wLyalt lt/2 : ,rltf lt/2 [86]. Thequantities aL,4 and, uL,, areknownasthe d- andq-axissynchronous reactances respectively. Theinstantaneous powerto themotorinputterminalsis [86,96] pi, :

[email protected],i.

-l uhioq)

( s 1. 1 )

The input power of a motor (5.11) is equivalentto the output power of a generator.The electromagnetic power of a three-phase machineis [86, 96] * (L"a - L")i'oa)zoq

:l"bl,t Petrrt

$.12)

The electromagnetictorque of a three phasemotor with p pole pairs is

ro: p&# :')nltr -t (L"a- L"n)i,oali,on

(s.r3)

The relationshipsbetween,io4,,ioo andphasecurrentsi,o4,ioB and'iog ate o uao, -

. ioq:

, O

' i o 4 c o s u t * i 4 6 c o s l/w t

\

2r\ 2^\l l * i,og r:os ( " , + 3/ T))

/ 2 / -32 . \ sinc,:/* io6 sin (r, | rnr-sin (ut + ; Lt,o ) 2l

2"\l

T))

( s.14)

(s.l 5)

The reverserelations,obtainedby simultaneoussolution of eqns (5.14) and (5.15)in conjunctionwith iot * i'e t ioc - 0, are daA :'io4cos ut - ion sin at

/

2r\

Za,B :

?o4 COS

(.''- T)-

r ^ ^s i n

Q'-T)

LaC,' :

?,adCOS

2r\ / [uf+i)-

ri-^sin

Q'*T)

(5.I 6)

159

AFPM MACHINESWITHOUTSTATORCORES

5.4 Calculation of corelesswinding inductances 5.4.1 Classicalapproach The synchronousinductanceL" consistsof the atmaturereaction(mutual) inductance Lo and the leakageinductance -L1. For a machine with magnetic asymmetry,i.e. with a differencein reluctancesin the d and q axes,the synarewritten as sums chronousinductancesin the d* and q-axis, L and L "n, "6 of the armaturereaction inductances(mutual inductances),Loa and Lo,,, and tf . i.e. leakageinductance LsrL:

Loil * Lt

( 5 .l 7 )

L"rr: Lo, * Lt

The armaturereactioninductancesare expressedby eqns(2.109),(2.110)and (2.111), in which air gaps in the d- and q-axis are given by eqns (2.105) and(2.106). The armaturereactionreactancesare given by eqns Q.fl4) and (2.115). Table 2.1 comparesarmaturereactionequationsfor cylindrical and disc-typemachines. The leakageinductanceis expressedanalyticallyas a sum of three components,i.e.

Lt : Lr, * Lt" * Lta: 2LroY()r, +

E^r,+

)ra)

(s.1 B)

where Lt : O.S(Dout- Dm) is the active length of a coil equalto the radial length ly of the PM, qr is the numberof coil sidesper pole per phase(equivalent to the number of slots) accordingto eqn (2.2), ly" : 0.5(1u,, -f luut) is the averagelength of the single-sidedend connection,Ly" and .\7" are the inductanceand specificperrneancefor the leakageflux about radial portions of conductors(correspondingto slot leakagein classicalmachines)respectively, Lp and.\1s ore the inductance and specific perrneancefor the leakage flux about the end connectionsrespectively, and L6 and ).y1 are the inductance and specific perrneancefor differential leakageflux (due to higher spaceharmonics)respectively. It is difficult to derive an accurateanalytical expressionfor ),1" for a corelesselectricalmachine. The specificpenneances.\1" and )1" can roughly be estimatedfrom the following semi-analyticalequation: ,\15!,\1.=0.39r

(s.1e)

The specific permeancefor the differential leakageflux can be found in a similar way as for an inductionmachineI I I I ], using eqn(4.22)in which ks,,lx I

160

AXIAL FLUX PERMANENTMAGNET BRUSHLESS MACHINES

andgt x 2l(g +0.5t-) +hml. The thicknessof the statorwinding is f- andthe distancefrom the stator disc surfaceto the pM active surfaceis g (mechanical clearance).It is not difficult to show that L1a : L,,ray flll).

5.4.2

FEM approach

unlike conventionalslottedAFPM machines,thereis no cleardeflnitionfor main and leakageinductancesin a corelessor slotlessmachine as discussed in U2,77, 1221.using the 2D FEM analysisboth mutual and leakagefluxes can be taken into account. The only remaining part is the end winding leakage flux. with the 2D finite element solution the magnetic vector potential I t as only a z component,i.e. A : A@,il.a" where a, is the unit vector in z direction (axial direction). The total stator flux of a phasewinding th,sBc that excludesthe end-windingflux leakagecan be readily calculatedby using Stokes'theorem, i.e.

aARCl"E.{s:l"t"A a3-t',Aa1

(s.20)

As an approximationthe flux linkage of a phase coil can be calculatedby working out the differencebetweenmaximum magneticvector potential values of each coil side. In the casethat the coil is not very thin, magneticvector potentialvariesin the coil cross-sectionarea.Therefore,the averagemagnetic vector potential values should be used. For first-order triangular elements,the flux linkageof a coil with N1 turns,area,g and length I is given by [133]

$:N,i?tf* o,,f

(s.21)

where Aii is the nodal value of the magnetic vector potential of the triangular elementj, e : *1 or ( : -1 indicatesthe directionof integrationeitherinto the plane or out of the plane, A, is the areaof the triangularelementj, andn is the total numberof elementsof the in-going and out-goingareasof the coil. It follows that for an AFPM machinewith only one pole modelled,the total flux linkageof a phasewinding is

,b.qec:'+y,r [T f..^,,7

(s.22)

whereu is the total numberof elementsof the meshedcoil areasof the phasein the pole region and ao is the number of parallel circuits (parallel cul:rentpaths) of the statorwindins.

16I


From a machine design perspective,it is important to find the fundamental componentsof the total flux linkages. For a corelessstator AFPM machine with usually unsaturatedrotor yoke, the flux linkage harmonics due to iron stator slots and magnetic saturationare absent.Owing to the large air gap, the statorwinding MMF spaceharmonicsare negligible in most cases.The most important harmonics neededto account for are those due to the flat-shaped PMs. Given theseconsiderations,the ffux linkage wave of an AFPM machineis nearly sinusoidal,though for a concentratedparameter(non-distributed)winding, an appreciable3'd and less significant 5th and Tth harmonicsare still prisent in the total flux linkage waveform. If the 5th, 7Lh andhigher harmonics are ignored,the fundamentaltotal phaseffux linkagescan be calculatedby usingthe techniquegivenin [133],i.e.

frl,anct) = lrl,ancl - lQaBcs)

( s.23)

wherethe co-phasal3'd harmonicflux linkage,includingthe higherordertriple harmonics,can be obtainedfrom:

tbas: ,!tt : tbczx

|{r,,

(s.24)

* 4;s +,!s)

with the fundamentaltotal phaseflux linkages and rotor position known, the d and q-axisflux linkagesare calculatedusing Park's transformationas follows [86]:

(s.2s)

Itaqol: lKoll,l'asct) where [86]

, I cosI

cos(o

cos(g+

,u, ,, I

!) {) Kr--;l -'i"e -s\n(|r-*) *sin(e+#) lt7t

With a constant rotor angular velocity cuthe angle 0 : wt' In the 2D FEM model the d-axis and g-axis synchronousinductancesdo not includethe componentdue to end connectionleakageflux [96], i.e. L'",1,: Loa * L\ : Lod.* L* i Lu

(s.27)

L o n* L t " I L u

( s.28)

L ' " q : L o st L \ :

162


The inductancestr'", andLt"oresultfrom eqns(5.7) and (5.8), i.e.

ancl L'"r: ;Vq

L'"d: *#

(s.2e)

"aq

The end winding inductanceL1" e&nbe calculated from numerical evaluation of the energy storedin the end connections1169]or simply by using an approximateequationresultingfrom eqn (5.18), i.e. r trle

:

n ZpO-

^T2l lYl [1c

. Ale

( s.30)

PQT

Finally, Lsd : Lt"r1* Lp

5.5

and

Lsd, :

L'",7* Lrc

( s.31)


Specificationsof e-TOROTN{AFptrrtbrushlessmotorsmanufacture d,by Bodine Electric Company,Chicago,IL, U.S.A. are given in Table5.1. The EMF constant,torque constant,winding resistanceand winding inductanceare lineto-line quantities. Motors are designedfor a maximum winding continuous temperatureof 130"c. Steady-state performancecharacteristicsof a 356-mm, l-kW, 170-V e-TORQrra motor are shown in Fig. 5.4. Fig. 5.5 shows an air-cooled 160-kW AFPM brushlessgeneratorwith a corelessstator built at the university of Stellenbosch,South Africa 12321.The stator winding consistsof sixty single-layertrapezoidal-shapecoils, which have the advantageofbeing easy to fabricate and have relatively short overhangs (Fig.3.17). The winding coils are held togetherto form a disc-type statorby using compositematerialof epoxy resin and hardener.SinteredNdFeB PMs with B, = 1.16 T and maximum allowable working temperature around130oChavebeenused.The detaileddesigndataare given in Table5.1. The output power and phase current at different speedsare shown in Fig. 5.6. Owing to very low statorwinding inductanceper phase,the outputvoltage variesalmostlinearly with the load cunent. It has been found from both experimental tests and calculations that for a typical sine-waveAFPM machinewith a corelessstator,the ratio of the d- and q-axisinductancesof the phasewinding is nearunity, i.e. Lral Lr, = 1. Thus, the analysisof the AFPM brushlessmachinewith a corelessstatormay be done in a similar way as that of the three-phasecylindrical machinewith surfaced P M s [ 1 1 8 ,1 4 5 ] .

AFPM MACHII,IESWITHOUTSTATORCORES

163

Tabte5.1. Speciflcationsof e-TORQTNTAFPM motors with corelessstatorsmanufacturedby Bodine Electric Companr,Chicaso, IL, U.S.A.

Parameter Output power, kW Number of poles d.c. bus voltage,V Speed,rym Peak speed,rpm Torque, Nm Peak torque, Nm Current, A Peak current, A Efficiency, % Torque constant,Nm/A EMF constant,V/krpm Winding resistance,Q Winding inductance,mH Viscous friction, Nm/rad/s Static friction, Nm Electrical time constant,ms Mechanical time constant,ms Moment of inerlia, kgm2 Mass of activepafts,kg Power density, W/kg

356-mmAFPMmotor 178-mmAFPM motor Low EMF High EMF Low EMF HighEMF constant constant constant constant 0.26 0.57 1.0 0.7 l6 l6 8 8 170 300 170 300 70 1500 300 3000 2200 700 400 3500 2.83 31.1 33.9 2.26 84.4 t3.67 t52.1 22.6 2 13 5 5 t2.5 10.5 64 50 77 75 84 8l 6.814 1.299 2.38 0.4859 249 713 137 50 43 t4.3 1.33 2.2 29.4 10.5 3.6 t.4 0.012 0.00019 0.00669 9 . 9x 1 0 - 5 0.1442 0.0378 0.02357 0.00728 2.'11 0.684 0.734 a.$64 17.78 4.5 4.026 4.538 0.21 0.00525 0.21 0.00525 30.87 6.17 30.87 6.17 32.39 8.42 11 3 . 5 92.4

5.6 Eddy current lossesin the stator windings 5.6.1 Eddy current lossresistance For an AFPM machinewith a corelessstator,associatedstatoriron losses are absent. The rotor discs rotate at the same speed as the main magnetic field, thus the core lossesin the rotor discs due to the fundamental harmonic of the statorfield also do not exist. Howevel the eddy culrent lossesin the stator winding are significant due to the fact that the machine is designedwith 2p > 6 to operateat relatively high frequenciesf = pn. Eddy current losses AP" in stator conductors are calculated according to eqn (2.61) or eqn (2.62). A more detailedmethod of the eddy current losses computationis discussedinl234l. Eddy cunent lossesin the statorconductors can be accountedfor in the sameway as core lossesin the statorstack [l16]. The eddy-loss cunent Iu and its d- and q-axis components1",1and I"n shown in Fig. 5.3 is in phasewith the internalphasevoltage t7 acrossthe shuntre-

164


Table5.2. Specifications of a three-phase, 160-kw, 1950-rpmAFpM brushlesssynchronous generatorwith ironlessstatorcorebuilt at the (Jniversityofstellenbosch.SouthAfrica

OutputpowerP,"t, kW Speedn, rpm Number of phasesm1 Rated line voltage, V Rated phase current, A Frequency /, Hz Number of stator modules Number of pole pairs p Number of stator coils (3 phases) Number of turns per phase Wire diameter,mm Number of parailel wires ri,Axial height of PM per pole h,y, mm Axial thicknessof the winding f,,,, mm Air gap (one side) 9, mm Air gap magnetic flux density B-n under load, T Current density, Almm2 PM outer diameterDou1.,mm ka : Dtn/Do,,i r&tio Class of insulation Winding temperaturerise, "C Cooling system Type of windi

160 1950 3 (Wye) 435V 215A 100 I 20 60 51 0.42 12 10.7 t5.7 2.75 0.58 7.1 720 0.69 F 56 Selfair-cooled le layer trapezoidal

sistanceBu. The internal phase voltage -oi is generatedby the resultant air gap flux and usually referred to as the air gap voltage [86]. consequently, the shunt resistancein the equivalent circuit representingthe eddy current loss is expressedas

(s.32) where

(5.33) on the basisof equivalentcircuits(Fig. 5.3), the following equationscan be written in the phasor form:

AFPM MACHINESWITHOI]T STATORCORES

165

;oo (a) fi00

3 3 soo o I

g 4oo o- 300 E

d zoo g. 100

0 100

150 troque, Nm

(b) 80

",

-70 x j

.9 o50

{

E*'

0

50

100

150

200

250

300

torque, Nm

Figure5.4. Steadystatecharacteristicsofof356-mm, l-kW 170-ve-ToReTNiAFpMmotor: (a) outputpowerPo.u,1 andspeedn versusshafttorque?"6; (b) phasecurrentandefficiencyversusshafttorqueT"6. Datacapturedat22"C for totallyenclosednon-ventilated motor.Courtesv of BodineElectricCompany,Chicago,IL, U.S.A. r

for generators

E"r: Er I jTo6Xo6+ jloqxo,,t

I'u:Io*I"

( s.34)

(s.3s)

166

AATALFLUX PERMAI,IENT MAGI{ETBRUSHLESS MACHTNES

Figure 5.5. Air-cooled single-stagesynchronousAFPM machine: (a) rotor disc with surface mountedPM segrnents,(b) corelessstatorwith busbars,and (c) the assembledmachine. Courtesy of the University o.fStellenbosch,South Africa.

0

250

500

750

1000 1250 Speed(rpm)

1500

1750

2000

Figure 5.6. Output power and phasecurrent versusspeedfor generatingmode ofa 160-kW AFPM brushlessgenerator.Specificationsare given in Table5.2.


t67

r for motors Ef :E1-

jToaXoa- jloqXoq

I'o:Io-

I.

(5.36)

( s.37)

where Il is the stator current with eddy curents being accountedfor and Iu is the stator current with eddy currentsbeing ignored.

5.6.2

Reduction of eddy current losses

In an AFPM machinewith a corelessstatorthe winding is directly exposed to the air gap magneticfield (seeFig. 5.7). Motion of PMs over the coreless winding produces an alternating field through each conductor inducing eddy currents. The loss due to eddy currents in the conductorsdependson both the geometry of the wire cross sectionand the amplitude and waveform of the flux density.In orderto minimize the eddy currentloss in the conductors,the stator winding shouldbe designedin one of the following ways using: r parallelwires with smallercrosssectionsinsteadof a one thick conductor; r strandedconductors(Litz wires); r coils madeof copperor aluminumribbon (foil winding). In an AFPM machinewith an ironlesswinding alrangement(as shown in Fig. 5.7a),in additionto its normal component,the air gap magneticfield has a tangentialcomponent,which can lead to seriousadditionaleddy curent loss (Fig, 5.7b). The existenceof a tangentialfield componentin the air gap discouragesthe use of ribbon conductorsas a low cost arrangement.Litz wires allow significantreductionof eddy current 1oss,but they are more expensive and have fairly poor filling factors. As a cost effective solution, a bunch of parallel thin wires can be used. However,this may createa new problem, i.e. unlessa completebalanceof inducedEMFs amongthe individualconductingpathsis achieved,a circulating curent betweenany of theseparallelpaths[49, 206,234] may occuras shown in Fig. 5.7c,causingcirculatingeddy currentlosses. When operatingat relatively high frequencymagneticfields, theseeddy current effectsmay causea significantincreaseof winding losses,which are intensified if there are circulating currents among the parallel circuits. These

168


Axial Flux TangentialFlux (b) LocalEddyLosses

Airgapfield

Circulating Eddy Losses (a)

(c)

Figwre5.7. Eddy curents in the corelessstatorwindingof an AFPM machine:(a) magnetic field distributionin a corelessstator;(b) eddy currentsin a conductor;(c) circulatingeddy currentsamongparallelconnected conductors.

losseswill deterioratethe performanceof the AFPM brushlessmachine.Predicting the winding eddy current losseswith a good accuracyis thereforevery important at the early stageof design of such machines. Eddy current lossesmay be resistancelimited when the flux producedby the eddy currentshas a negligible influence on the total field 1206].In this case the conductordimensions(diameterof thickness)are small when comparedto the equivalent depth of penetration A : I lk" of the electromagneticfield e q n( 1 . 1 2 ) .

5.6.3

Reduction of circulating current losses

To minimise the circulating current in a coil made of parallel conductors, the normal practice is to twist or transposethe wires in such a fashion that each parallel conductoroccupiesall possiblelayer positionsfor the samelength of the coil. The effect is to equalizethe induced EMFs in all parallel conductors, and to allow them to be paralleled at the ends without producing eddy circulating currentsbetweenthe parallel conductors. Figure 5.8 illustratesthe effectivenessof suppressingcirculatingeddy current by wire twisting. Four coils are made with the only differencethat the par-

AFPMMACHINESWITHOUTSTATORCORES

t69

fl (a) Ch 1:2 NDiv 2.5 ms/Div

(b) Ch 1: 100 mA/Div 2.5 ms/Div

i' lt" ' 'M" tq""irl "i l,ll : : 'n il t: il:l

I ll '; :1 " lur

:! 1l ilr { . l, : t t ' l : , ' h ' i l l '

,L;,1 I

.l

It

t:,/{'.h' !!.'h..

,il.i'U

I

..1-r. : i I : _ . l t.i. u . _: i l . . . , \ 1 . ., ,: -..i1l

;e*

i4

:J

:

lt::::

(c) Ch 1: 100 mA/Div 2.5 ms/Div (d) Ch 1: 100 mA/Div 1 mslDiv Figure 5.8. Measuredcirculatingcumentof (a) non-twistedcoii, (b) slightly twisted coil, (c) moderatelytwisted coil, (d) heavily twisted coil.

allel wires of eachcoil are: (a) non-twisted,(b) slightly twisted (10 to l5 turns per metre),(c) moderatelytwisted (25 to 30 turns per metre),and (d) heavily twisted (45 to 50 tums per metre)respectively.All the coils havebeenusedto form a portion of an experimentalstator,which is placedin the middle of the two opposingPM rotor discs. The machineoperatesat a constantspeed(400 rpm in this case). The circulating currentsbetweentwo parallel conductors havebeen measuredand logged on a storageoscilloscope.It can be seenthat the inducedcirculatingcurrent is greatly reducedevenwith a slightly twisted coil and can generallybe ignoredin a heavily twisted coil. Thesetwisted wires can easily be manufacturedin a cost effective way. The fill (space)factorsfor the non-twistedand the heavily twisted coils are estimatedas 0.545 and 0.5 respectively,which is slightly lessthan that of the Litz wires (typically from 0.55 to 0.6). However,the savingin manufacturing costsis usuallymore important. It shouldbe notedthat due to the low impedanceof a corelessstatorwinding, circulating current could also exist among coil groups connectedin parallel (parallel current paths) if a perfect symmetry of coils cannot be guaranteed.

170

5.6.4

AXIAL FLUX PERMANENTMAGNETBRUSHLESS MACH]NES

Measurement of eddy current losses

The resistancelimited eddy loss in the statorof an AFPM machinemay be experimentally determinedby measuringthe difference in input shaft powers of the AFPM machine at the same speed,first with the stator in, and then by replacing it with a dummy stator (no conductors). The dummy stator has the samedimensionsand surfacefinish as the real statorand is meantto keep the windagelossesthe same. A schematicrepresentationof the experimentaltest setup for measuring eddy current lossesin stator conductorsis shown in Fig. 5.9. The shaft of the tested prototype and the shaft of the prime mover (driving machine) are coupledtogethervia a torque meter. The statoris positionedin the middle of the two rotor discswith the outer end ring (seeFig. 5.9a)mountedon the outside supportingframe. Temperaturesensorsare also attachedto the conductors in order to take into accounttemperaturefactor in the measurement. Initially, the prototype(with the corelessstatorplacedin it) was driven by a variable speedmotor for a number of different speeds.The corresponding torque measurementswere taken. Replacing the real stator with a dummy one, the tests were repeatedfor the same speeds. Both the torque and the temperaturevalueswere recorded.The differenceof the torquesmultiplied by the speedgivesthe eddy currentloss. The eddy lossesdue to eddy-circulating currentin the windings can also be determinedby measuringthe differenceof input shaft powers,first with all the parallel circuits connectedand then with disconnectedparallel circuits. Fig. 5.I 0 showsthe measuredand calculatedresistancelimited eddy losses of the prototype machine. It can be seenthat the calculatededdy lossesobtained by using the standardanalyticalformula - eqns (2.61) and (2.62) with values(43% only the B-, componenttakeninto accountyield underestimated less). The discrepancybetweenthe measuredand calculatedresultsbecomes large at high speeds.Better accuracymay be achievedby including both the normal B,n" and tangential B-" componentsof the magnetic flux density and implementingeqn (2.61)or eqn (2.62)into the 2D or 3D FEM [235].

5.7

Armature Reaction

The 3-D FEM analysishasbeenappliedto the modelling of a corelessstator AFPM machinein [10]. Both no-loadand load operationshavebeenmodelled. It hasbeen shown that good accuracycan be expectedevenwhen the first order linear FEM solutionis appliedto the AFPM machineif the magneticcircuit is not saturated(large air gap). It has also been found that the armaturereaction for an AFPM machinewith a corelessstatoris generallynegligible. Using a 40-pole,766 Hz corelessstatorAFPM machineas an example,the simulated effects of armaturereaction to the air gap flux distribution has been


171

(a) Thermometer t,'''_'l

u't' ,"',{l '

HryTf

Supplv\ q .q . ,

AFPM Machine

Speedometer _l tr--.. L:I

l

Torque Sensor

Figure5.9. Laboratorysetformeasuringeddycurrentlossesin the statorwinding:(a) speciallydesigned stator;(b) experimental (c) schematic machine; of experimental set-up.

demonstratedin Figs 5.1I and 5.12. lt can be seenthat the plateauof the axial field plot is somewhattilted due to the interactionbetweenthe flux of the PMs and the armatureffux generatedby the ratedculrent (Fig. 5.11). Similarly,the modified tangentialfield plot is comparedwith the original one in Fig. 5.12. As the atmaturereactionflux is small in corelessAFPM machineswith large air gaps,the air gap magnetic flux density maintains its maximum value very close to that at no load. The influenceof the armaturereactionon the eddy current lossesis usually insignificantthough it can be easily accountedfor when the FEM modelling is used. The harmonic contentsof the air gap flux densityobtainedwith and without armaturereactionarecomparedin Table5.3. It is evidentthat, in this case,the changeof the field harmoniccompositiondue to the armaturereactionis negligible.

172


900 800 700 ,

600

soo E o I- 4nn t|J 3oo 200 100 0 600

800

1000

1200

Speed (rpm) Figure 5.10.

Comparison of calculated eddy loss with measurements.

0 Rotor angle (degrees)

Figure 5.1I.

Air gap axial field component at noload and full load conditions

AFPM MACHINESWTTHOUT STATORCORES

173

04 0.3 0.2

01 F e0

m -0.1 -02

---o---Zeroarmature current -03

--+- Ratedarmature current -0.4 Rotor angle{degrees) Figure 5. I 2.

Table 5.3. action Harmonics I 2rtd

5'',t 7'h gtlt lLth 1 2Lh t rth l7th lgth

5.8

Air gap tangential field component at no-load and full load conditions.

Hatmonics content of the axial air-gap flur density with and without armature re-

Rated armature current

0.60346784 0.0374565 1 0.0409906 I 0.03502689 0.01231397 0.00304573 0.00559543 0.00314908 0.00083 159 0.00307828

Zero armaturecunent 0.60246033 0.03745759 0.04070525 14609 0.035 0.01230032 0.00262309 0.005'79739 0.00312t79 I2 0.000868 4.00272939

Mechanical design features

In the mechanicaldesignof an AFPM brushlessmachine,obtaininga uniform air gap betweenthe rotor disc and the stator is important. Therefore, the methodsof fixing the rotor discs onto the shaft and the statoronto the enclosure (frame)arevery important.Impropermethodsof fixing, or misalignmentin the

t74

AXIAL FLLIX PERMANENTMAGNETBRUSHLESSMACHINES

assemblingof the statorand rotor will causea nonuniform air gap, vibration, noise,torquepulsationand deteriorationof electricalperformance. With regardto cooling in an air-cooledAFPM machine,the entry lossesof the air flow can be quite high if the machine-air-inletis poorly designed.It is importantto reducetheselosseswithout weakeningthe mechanicalstructure for bettercooling. To summarise,attentionshouldbe paid to the following aspectsof the mechanicaldesign: r Shaft. The load torque, the first critical speedand the shaft dynamics shouldbe taken into accountin the shaft design. r Rotor. (i) The deflectionof the rotor disc due to the strong magneticattraction force, (ii) the meansof mounting and securingthe magnetson the rotor discs to counteractthe strong centrifugal force especiallyfor high speedapplications,and (iii) the balancingof the rotor discs. r Stator. (i) The strengthandrigidity ofthe resinreinforcedstatorand frame, and (ii) the positioningand spacingof the coils to ensureperfectsymmetry. r Cooling. For air cooled AFPM machines,the air inlet and air flow paths throughthe machineshouldbe carefullydesignedin orderto ensurea better massflow rate and thereforebetter cooling. r Assembly. An effectivetool to facilitatethe assemblingand dismantlingof the machinefor maintenance.

5.8.1

Mechanicalstrengthanalysis

The deflectionof the rotor discsdue to the strongmagneticpull may have the following undesirableefTectson the operationand conditionof AFPM machines: closingthe running clearancebetweenthe rotor disc and the stator: looseor broken PMs: r reducingair-flow dischargingareathus deterioratingthe cooling capacity; r nonuniform air gap causing a drift in electrical performancefrom the optimum. For a double-sidedAFPM machinewith an internal corelessstator,the rotor discsaccountfor roughly 50% of the total activemassof an AFPM machine. Hence,the optimal designof the rotor discsis of greatimportanceto realisea designof high power-to-massratio. All theseaspectsrequirethe mechanical stressanalysisof the rotor disc.

AFPMMACHINES WITHOUT STATOR CORES

175

Attraction forcebetweenrotor discs The attractionforcebetweentwo parallelrotor discscanbe calculated by usinethevirtualwork method.i.e. t-

'

dW

LW

Wz-Wt

dz

L,z

z2-zr

(5.38)

where tr4/is the total magnetic energy storedin the machine and Az is the small variation of the air gap length. The accurateprediction of the attraction force is a prerequisitefor the mechanicalstressanalysis.Hence,the respective magnetic stored energiesW1 andW2 for air gap lengths z1 and ,z2are usually calculatedby using the FEM. Analytically, the normal attractiveforce between two parallel discs with PMs can be expressedas

," = ttlLnsr*

(s.3e)

wherethe activeareaof PMs is accordingto eqn (2.56). Optimum design of rotor discs The structureof the rotor steeldiscs may be optimisedwith the aid of an FEM structural program. It is important that the deflection of the rotor steel discs,due to the axial magneticpull force, is not too dangerousfor small running clearancebetweenthe corelessstatorand PMs. Two importantconstraints shouldbe considered,i.e. (i) the maximum allowabledeflection,and (ii) the maximum mechanical strengthof the material used to fabricate the disc. When choosingallowabledeflection,one needsto make surethat the PMs do not experienceany excessiveforces due to bending of the disc that can potentially peel them off from the backing steel disc. Owing to the cyclical symmetry of the disc structures,it is sufficient to model only a section of the disc with symmetry boundary conditions applied. In the FEM analysis the axial magnetic force can be applied in the form of a constantpressureload over the total areathat the PMs occupy. Usually, the axial-symmetric elements are preferred for modelling a relatively thick disc. However, it has been shown in [167] that the FEM modelling of rotor discs using both axial-symmetricelementsand shell-elementsgivesvery closeresults. Fig. 5.13 showsthe finite elementmodel of the analysedrotor disc using4with symmetricalboundaryconditionsapplied.The axial node shell-elements, magneticattractionforce has been calculatedas 14.7 kN and applied in the form of a constant69.8 kPa pressureload. Specificationof the investigated AFPM machineare given in Table 5.2. The stiffnessprovidedby the magnets

r76


hasnot beenincluded,to keepthe designon the conservativeside.As the rotor discs are mounted on the centre support hub, additional boundary constraints have been definedso that there is no axial displacementin the vicinity of the mounting bolts and contact area. To find the suitablethicknessof the rotor disc, which satisfiesthe critical strengthrequirementsof the rotor disc with a 1owsteelcontent,the linear FEM static analysiswas performed for different thicknessesof the rotor disc.

kPa masnetic force

Supportboundaryconditions Figtre 5.I j.

FEM model for analysing the mechanical stressof the rotor disc.

Basedon the analysis,the rotor disc thicknesswas chosenas 17 mm with a maximum deflectionof 0.145 mm. Fig. 5.14 showsthe deffection(blown-up) and the von Mises stressdistribution of the laboratory prototype of the 17-mm thick disc. The maximum stressof 35.6 MPa is much lower than the typical yield strengthof mild steel,that is in the region of 300 MPa. It has been shown in literature ]67lthat the bending of the rotor disc decreasestowards its outer periphery. The rotor disc may be machined in such a way that the disc becomesthinner towards the outer periphery. As shown in Table 5.4, the tapereddisc usesapproximatelyl0o/olesssteelthanthe straightdisc. The maximum deflectionincreasesby only 0.021 mm with the tapereddisc, which is negligible.This can effectivelyreducethe activemassof the machinewithout compromising the mechanicalshength,but doesnot bring down the cost of the steel. If manufacturing costsare taken into accountfor small production volumes, it is betterto usea steeldisc with uniform thickness.Obviouslv.a uniform disc

AFPM MACHINESWITHOUTSTATORCORAS

177

re

35.58 m 3327 w 30.95',(z 28.64 , ' l 26.33

24o1Ji 217 | 1, 1 e 3 e II

noTJ 1476i :

z

12.451 | t: 10131

I

7818w

X Output Set: MSCA,IASTRAN Case I Deibmed [0. 166]: Total Translation Contour: Plate Top VonMises Stress

Figure 5.14,

Tip deflection resulting fi'om 147 I 5N magneticforce : 0. 166 mm

5504m 31e1u

Deflection (blown-up) and von Mises stressdistribution of the rotor disc.

meansa heavierrotor disc. It can,howevel be arguedthat the extramachining work neededfor producinga taperedrotor disc may be too costly to justifii the profit due to improvement in the dynamic perfomance. As long as the added masscan be tolerated,this option is viable for small productionvolumesand for laboratory prototypes.

Table5.4.

Comparison of different designs of rotor disc

Parameter

Straightdisc

Tapereddisc

Mass,kg

3 9 .l 8 4

31.296

Maximumdeflection, mm

0 . 14 5

0 . 16 6

Maximumvon Misesstress, MPa

35.6

33.4

5.8.2

Imtlalanced axial force on the stator

As a result of the interaction of alternating currents in conductors and the tangential componentof the magnetic field, there is an axial magnetic force f " on eachhalf of the coil as shownin Fis. 5.15. When the statoris locatedin the

r78

AXIAL FLUX PERMANENTMAGNETBRUSHLESSMACHINES

a s Figure 5.15.

z.

Schematic diagram showing the axial force exerted on stator.

zv

9

Fo 3 <

-zv

Electrical angle (degrees)

Fisure 5.16. Unbalanced axial force exerted on the stator.

middle of the air gap, the forces on each side of the stator should cancel each other. Assumingthat the corelessstatorof the AFPM machineis slightly off centre, the axial forceson eachsideof the stator(/1 and f2) as shownin Fig. 5.16 will not be the same,resulting in an unbalancedforce, A,f : lh - fzl,being exertedon the stator.This unbalancedforce may causeextra vibration and thus have an adverseeffect on the mechanicalstrengthof the epoxy reinforced stator.

AFPM MACHINESWITHOIJTSTATORCORES

5.9

179

Thermal problems

Owing to the excessiveheat generatedin the statorwinding and the resultant thermal expansion,an epoxy encapsulatedstator is subject to certain deformation. When the deformationis significant,it may causephysicalcontact betweenthe stator and the magnetsresulting in seriousdamageto the stator winding and the magnets. 0.6 0.5 ts

E oa . 9 ^- - . (E

g 02 o o 0.1

)

40

60

80

100

Temperature(degreeC) Figure5.17. Thermalexpansionof the statorfor differenttemperatures.

Fig.5.l7 shows the thermal expansionof the stator at different temperatures.The ratedcurrent(16 A) hasbeenconductedthrougha coil in the epoxy encapsulatedstator. A thermal coupler and a roundout meter have been used to measurethe coil temperatureand surface deformation respectively. lt has been found that the relationship between the stator deformation and its temperatureis almost linear. On average,the deflectionof the stator surfaceis about 0.0056 mm/oC, resulting in a 0.5 mm deffectionat the temperatureof I77"C. The remainingair contentsin the epoxy can also contributeto the heat transferdeteriorationand temperatureincrease.To solvethis problem,a more dedicatedproductionprocessis suggested.The running clearancebetweenthe stator and the rotor should be kept reasonablylarge.

Numerical example5.1 A three-phase, Y-connected,3000 rpm PM disc motor has a corelessstator and twin externalrotor with surfacePMs and backing steel discs. Sintered NdFeB PMs with B, : 7.2 T and prrec: 1.045havebeenused. The nonmagneticdistancebetweenoppositePMs is t - 1.1mm,the winding thickness is f- : B mm and the height of PMs (in axial direction) is h71 : 6 mm.

180


The outer diameter of PMs equal to the outer diameter of the stator conductors is Dort : 0.22 m and the parameterka : 71J5. The number of poles is 2yt : 6, the number of single layer coil sides(equivalentto the number of slots) is sr : 54, the number of turns per phaseis I'I1 : 234,the number of parallel conductorsau : 2, the diameterof wire d- : 0.912 mm (AWG 19) and the coil pitch is ur" : 7 coil sides. Find the motor steadystateperformance,i.e. output power, torque, efficiency and power factor assumingthat the total armature current Io : 8.2 A is torqueproducing(Ioa:0). The saturationfactor of the magneticcircuit is ksc,t: 1.02,the motor is fed with sinusoidalvoltage,the massof the twin rotor TrLy:3.4 kg, the massof shaftTilsh:0.64 kg, the radiusof shaftftsh : 15 mm, the conductivity of copper conductorso : 47 x 106 S/m at 75"C, the specific density of copper conductorsp : 8800 kg/m3, the density of PMs ppttr : 7700kg1m3,the air density p : 1.2 kg/m3, the dynamic viscosity of air pr,: 1.8 x 10-s Pa s, the coefficientof bearingfriction kfu = 1.5 and the coefficientof distortionof the magneticflux density\a:1.15. Lossesin PMs, lossesin steelrotor discsand the tangentialcomponentof the magnetic flux density in the air gap arenegligible. Solution The numberof coils per phasefor a singlelayerwinding is n" : s t I (2rnt) : 54lQ x 3) : 9. The numberof tums per coil is ly'., : l{tln, : 23419 : 26. The number of coil sidesper pole per phase(equivalentto the number of slotsper pole per phase)qt : stl(2p*t) : 5 l$ x 3) : 3. The air gap (mechanical is 9 : 0.5(t - t-) :0.5(11 - B) : 1.S clearance) mm and the pole pitch measuredin coil sidesis rr : sl f (2p) : 5416 : L The input frequency at 3000 rpm is

. r -,!sy-

3000 60

x3:150H2

The magnetic voltage drop equationper pole pair is

: nB#l#+ (o+|'-)n"") nffinû Hence Brng

7 I Ut,,."(g+ 0.5t",)f hulk"o, 7.2

I + f1 .0 4 5 (1+0 .5.5 x s)/01x I.02

: 0.607T


t8l

The magneticflux accordingto eqn (2.21) is

?[o.oor - f+)'l : oooru32 o,, : 1 x 0.222[r wb BrJ-'\V:,t] t-

Thewindingfactoraccording to eqns(2.8),(2.9)and(2.10)is sinrlQx3) , rr,_ffi_

r

/t

: 0.9397 kor: sin ( Or / 1

\

1t)" T ^ 0.ebe8:O__T_,

kut : 0.9598x 0.9397: 0.9019

The EMF constantaccordingto eqn(2.30)andtorqueconstantaccordingto eqn(2.27)arerespectively k n : n t / 2 x 3 x 2 3 4 x 0 . 9 0 1 9 x 0 . 0 0 1 6 342. b : 9 r V / ( r e v / s ): 0 . 0 7 6V 5/rprn

:2.192Nm/A kr : kr3: '27T 4.b91;1 2r

The EMF at 3000rpm is E f : k a n : 0 . 0 7 6 5 x 3 0 0 0- 2 2 9 . 5 V The electromagnetic torqueat Io, :1oq : 8.2 A is T a : k r l u : 2 . 1 9 2 x 8 . 2: 1 7 . 9 7N m The electromagnetic poweris Petm: 2rrtT,1. * 6: 0 2n-9

x \7.97:

5646.8W

: 0.221\n : 0.127 m, the average The inner diameterD.i, : Doutll! diameterD :0.5(Dout* Dm) :0.5(0.22+0.727): 0.1735m, theaverage pole pitch r : z-0.173516:0.091 m, the lengthof conductor(equalto the radial length ly1 of the PM) Li : Iy : 0.5(Do6 * Dm) : 0.5(0.220.I27) : 0.0465m, the length of shorlerend connectionwithout inner bends I7e.-in: w. f r,,)x D n I Qp) : (7I 9)n x 0.727I 6 : 0.052m and the length of the longerend connectionwithout outerbendsl7".rno,: 0.052x 0.2210.727: 0.0896m.

MACH]NES AXIAL FLUX PEKMANENTMAGNETBRI.]SHLESS

182

The averagelength of the statorturn with 15-mmbends(Fig. 3.16) is lroo x 2Lt * h"rnin* lt"rno*+ 4 x 0.015

:2 x 0.0465 m + 0.052 + 0.0896 + 0.06: 0.2943 The statorwinding resistanceat75oC accordingto eqn (2.33) is rDr , I -

234 x 0.2943 1 ltro x70 s)214 4 7 x 1 0 6x 2 x r x ( 0 . 9 1 2

The maximum width of the coil at the diameterDinisluw : rD;rf s1 : n0.I27 154: 0.0074 m : 7.4 mm. The thicknessof the coil is fr, : 8 mm. The number of conductorsper coil is Af : &y x l{6 : 2 x 26 : 52. The maximum value of the coil packing factor is at Din, i.e.

klrri,q -:

tt?-x N" :o o}?l:r" t-u*

:0.732

8x7.4

The statorcurrentdensityis

-Jl-:6'28 t": = 2 x n0.912214

Af nm2

The statorwinding lossesaccordingto eqn (2.42) are A , P 1 u , : 3 x 8 . 2 2x I . I 2 2 : 2 2 6 . 2 W The eddy curent lossesin stator round conductorsaccording to eqn (2.61) ate n2 4T x 106 Lp":;#1b02

x 0 . 0 0 0 9 1 2x2 0 . 2 5x 0 . C 0 7 x2 1 . 1 5 2: 9 0 . 2 W

where the mass of stator conductors(radial parts) is N1 ( *\ mcon: p6uTrllau, \41

Ot,,

: 0.75ks : 8800x 3 x 2 . zz+(4 !0991'2)rr,, 0.0465) \4/

The friction lossesin bearingsaccordingto eqn (2.64) are

W . 40 . 6 4 ) # : 1 8 . 2 L P y , : 0 . 0 6x 1 . 5 ( 3 +

AFPM MACHII{ESWITHOUTSTATORCORES

183

The windagelossesaccordingto eqn (2.67) are . 1 . / 300{)\:l A P , , . i n , l :l 7 . 8 5 3 x l 0 " x 1 . 2( r " : : ) lto.sx0.l15)s-(05x0.01b)51 z

6u/

\

r2.8W where x 0.5Do6+ 0.005: 0.5 x 0.22+ 0.005: 0.115m R'o,,t,

:2.77 x ror' R": L22:(:oo?!60=) x 0.11b2 1 . 8x 1 0 - b

" 'r :

-,-j.L. : 2 . 3 5 3x 1 o - 3 J2.77x 7os

The rotational(mechanical)lossesaccordingto eqn (2.63) are LP,ot:

IB.2 * 2.8 : 21 W

The output power is Pout: P".tr,- LProt: 5646.8- 21 :5625.8W The shaft torque is 5625'8 : 17.g1Nm T"r. : "n- 2r x3000/60 The input power is Pin : P"hn * LP.LP":

5646.8+ 226.2+ 90.2 : 5963.2W

The efficiencyis 'l :

5625.8 o'943 sg6:,2:

The leakagereactancecan be approximatelycalculatedtaking into accountthe radial part ofconductors,end connectionand differentialleakagefluxes,i.e. ) r n = 0 . 3 q 1: 0 . 3 x 3 : 0 . 9

)r" ry)r":0.9

AXIALFLUXPERMANENT MAGNET BRUSHLESS MACHINES

I84

, : )r,l

3x3x0.091 x0.90192

where accordingto eqn (4.23),the differentialleakagefactor16: 51u<997.

0.0059for

is The statorleakagereactance

xr :4trx0.4nx10 6,. ,rn?L|$out

(o n Jr|ul, x 0.e- 0.m6)

: 1 . 5 4 30 lengthof oneendconnection is 11": 0.5(h",n;.r*lt.,no*): wheretheaverage : 0 . 5 ( 0 . 0 5+2 0 . 0 8 9 6 ) 0 . 0 7 1m . to eqn(2.105)is Theequivalent air gapin thed axisaccording .t :22.7 rnm : 2 1 ( 1 . 5 + 0 . 5x 8/) 1 . 0 *2 . : l e "' 1.0451 L. to eqn(2.106)is Theequivalent air gapin theg axisaccording s L : 2 1 ( 1 . 5 l *0 . 5x s ) + 6 l : 2 3r r r n to eqns(2.114)and(2.115)are Armaturereactionreactances according x o , t : 2 x 3 x 0 . 4 x t r x 1 0 - 6 x 1 5'f-

/ 2 3 4x 0 ' 9 0 1 9 \ 2 1

:

) otnn

x [(0.5x 0.22f - (0.5x 0.727)2): 1.e8e() ))7

Xaq X",t

',,.n:

1.963o

of PMs[96]. wherekf a: kf n : 1 for surfaceconfiguration to eqns(2.72)and(2.73)arerespectively reactances according Synchronous X " , t , : 1 . 5 4 3+ 1 . 9 8 9: 3 . 5 3 2o : 3 . 5 0 6f ) x"q:1.543 + 1.963

AFPM MACHINES WITHOUTSTATORCORES

r85

The input phasevoltageis Vt:

J@t-r

I o R l \ 2+ 1 I " X , n 1 z

: I (229.5 :240.by * 8.2x 7.722)2 + (8.2x 3.500)2 Theline-to-linevoltage Vtr,-r : J3 x 240.b: 416.5V. The power factor is E1 I IoRr

cos@: T:

-229.5+ 8.2 x 1.122: 0 . 9 9 3= 1 . 0 2405

Numerical example5.2 For the corelessstatorAFPM brushlessmotor describedin Numerical example -I.1 find the rotor moment of inefiia, mechanicaland electromagnetic time constantsand axial magnetic attractive force between the backing steel discsof the twin rotor. Solution The following input data and resultsof calculationof Numerical example 5.I arc necessary: r inputphasevoltageV : 240.5Y r input frequency f : 750 Hz r aft gapmagneticflux densityBrrs : 0.607 T r Statorwinding resistanceper phaseRt : I.I22 Q r d-axissynchronousreactance Xscl:3.532 Q : q-axis synchronousreactancef sa

3.500 0

r torqueconstantkr :2.792 Nm/A r Electromagnetictorque7a : 19.97Nm . speedn : 3000 rpm r power factor cosd : 0.993 r PM outer diameterDo6 : 0.22 m r PM innerdiameterDin :0.727 m

MACHINES AXIAL FLUX PERMANENTMAG]{ET BRLTSHLESS

I86

r shaftdiameterD"h - 2R"h :0.03 m r numberofpole pairsp : 3 r polewidth to-polepitchratioo ; :2lr r axial height of PM (onepole) h,u :6

mm

r massof twin rotor without shaftm, : 3.4k9 r massof shaft wlsh : 0.64 kg r specificmassdensityof PMs {)pL,t: 7700 kglms r specificmassdensityof mild steelpp" : 7850 kg/m3 1. Rotor moment of inertia The activesurfaceareaof PMs (one side)accordingto eqn (2.56) is

m2 sp v : ? !p .zz2 - 0 .1,27\:0.01614 1T4

The massof all PMs is 'ttLp1,1 :2

x 7700x 0.01614x 0.006: 1.49kg

The massof backing steeldiscsis m F e : m r - T n P A :, 3f . 4 - I . 4 9 - 1 . 9 1k g The shaft moment of inefiia is

J"h: **+:

0 . 6 4 0 ' : 3: ' o . o 7 2 x1 o - 3k g m 2

Themomentof inertiaof PMs is -

D 7 , ,+, D ? ^

J P n:r * r n , " #

^ . ^ 0 . 2 2-2 0 . 1 2 7 :2

, nnT

o ' o t k2 s m 2

The moment of inertia of backing steeldiscsis D2

JF, ''r'ot#:

t D2,

0.222-r 0.032

t'91ff

o ' o l l 'k g u r 2

The resultantmoment of inertia of the rotor is J, : J"h I Jv'p1I Jpo,: 0.072x 10-3 + 0.012+ 0.0118: 0.02386kgni2

AFPM MACHINESWITHOL]TSTATORCORES

187

2. Mechanical and electromagnetictime constants Since the d-axis current f ad.: 0, the angle betweenthe q axis and stator cuffent t! : 0. For the power factor cosd : 0.993 the angle betweenstator currentand terminalvoltaged : 6.78". The load angleis 6:d-,$:6.78_0:6.78" At the first instantof starlingthe EMF Ef : 0. Assuming the load angle at startingis the sameas for nominal operation,the stator current accordingto eqns(2.80),(2.81)and(2.82)is r

--

r0shd. -

240.5(3.50 x 60 . 9 9 3- 7 . I 2 2x 0 . 1 1 8-) 0 x 3 . 5 0 6 3.532x3.506+L1222

r

' oassnn qq-

- r J59.04 "'t , A

240.5(1.r22 x 0.993+ 3.532x 0.118)- 0 x 7.122 3 . 5 3 2x 3 . b 0 6 + r . 1 , 2 2 2

n vA atn

1 ,," r,:/b g n |, n P :6 4 .g2L where sin/ : 0.118 (cosd : accordingto eqn (2.26) is

0.993). The electromagnetic startingtorque

TrJrt: krlo"t"s : 2.192x 27.A: 59.2Nm The no load speedassuminglineartorque-speed curveaccordingto eqn (2.126) is ns -

3000 : 4526.9rplrl : 75.44 revf s I _ I'LW l*g2

The mechanicaltime constantis

T-,,h r,'rU: 0.02386'"Jl#: o.[9]s The d- and q-axis synchronousinductancesare

Lsd- -4t -:r.bil22r7bo: o.oo375 H t1T .I

-

L"q

x"2;i

3 . 5 0 f ' 2 n 1 5 0= 0 . 0 0 3 7 2H

188

MACHINES ANAL FLUX PERMANENTMAGNETBRUSHLESS

The electromagnetictime constantof the stator winding is T,z l n :

L"d'

:

6

0'00375 : o.oo334s In2

The mechanical-to-electromagnetictime constantratio is T*""h - 0.191 - b 7. . 2 T.t,n 0.00334 3. Axial attractive force tretween backing steel discs of the twin rotor Assuming that the two twin backing steel discs are perfectly parallel, the axial magnetic attractiveforce betweenthem can be found on the basis of eqn

(s.39) 0" ' "FtOGz -" F "' 0 . 0 1 6 1 4- - 2 3 6 5 . 5N 2uo

The magneticpressureis F, r).: Sptt

2365.5

- 1 4 66 0 1 . 2P a = 1 4 6 . 6k P a 0'01614

The backing steeldisc thicknessis ,

v r 'l -

TfLFe

d -

2pp,r(D!,,

1.91 - 0.7272) x 2 7850r(0.22'2 I4

D?")14 0.0033m

3.3rnm

and outer diameterDout : 0.22 m can provide adequatestiffnessof the disc under the action of 146.6kPa magnetic pressurein the axial direction.

Chapter6

AFPM MACHINES WITHOUT STATOR AND ROTOR CORES

6.1


With the availability of high energymagnetsthe statorsand rotors of AFPM brushlessmachinescan be fabricatedwithout ferromagneticcores 197, 138, 139,2031.A completelycorelessdesignreduces the massand increasesthe efficiency of the machineas comparedwith a conventionaldesign.Besidesthis, a corelessAFPM brushlessmachine does not produce any normal attractive force between the stator and rotor. It also does not produce torque pulsations at zero current state. There is a limit to the increase of the electromagnetictorque that can be achievedby enlargingthe machinediameter. Factorslimiting the single disc designare listed in Section2.1.6. A reasonablesolution for largertorquesare doubleor triple-discmachines. The disc-type PM brushlessmachines without stator and rotor cores were first manufacturedcommercially in the late 1990sfor servomechanismsand industrial electromechanicaldrives [138], solar powered electrical vehicles 1203]as well as micromotorsfor computerperipheralsand vibration motors for mobilephones[85].

6.2

Topology and construction

The AFPM brushlessmachine without any ferromagnetic core is shown in Fig. 6.1. The machineconsistsof a twin rotor (3) with rare earthPMs (2) and nonmagneticsupporting structure. The steel-freestator (armature)winding (1) is located betweentwo identical parts of the rotor. The stator polyphasewinding fixed to the frame (6) is assembledas "flower petals" (Fig. 3.16) !391. Multi-turn coils are arranged in overlapping layers around the shaft-axis of the machine. The whole windine is then embeddedin a hich mechanicalin-

190

Figure 6.1. 2-PMs,3


Disc type corelessPM brushlessmachine: I - corelessstator (armature)winding, twinrotor,4-shaft,5 bearing,6- frame.

Figure 6.2. Segmentalconstntction of a disc type corelessPM brushlessmachine: (a) single module (segment),(b) three-moduleassembly.

tegrity plastic or resin. Sincethe topology shown in Fig 6.1 doesnot use any feruomagneticcore with slots, the machine is free of cogging (detent) torque and core losses.The only eddy currentlossesare lossesin the statorwinding conductorsand metallic parts (if they exist) that reinforcethe stator coreless winding.

,ffi

AFPM machineswithout stqtor and rotor cores

(a)

191

machines with PMs ar8-poleAFPM brushless Figure 6.3. Constructionof a three-phase, rangedin Halbacharray:(a) PM ring; (b) statorwinding;(c) onehalf ol the twin rotor; (d) statorwindingandcompletetwin rotor.

Figure 6.1. 3D magnetic flux density distribution excited by an 8-po1etwin rotor with PMs arrangedin Halbach array.

The corelessmachinedesignedas a segmental(modular)machineis shown in Fig. 6.2. The output shaft power can easily be adjustedto desiredlevel by addinsmore modules.

192

AXIAL FLUX PERMANENTMAGNET BRUSHLESS MACHINES {a} 900

(b)

(c)

Figure 6.5.

45o

Arrangement of the twin rotor PMs in 90", 60", and 4b" Halbach array.

To obtain a high power (or high torque) density motor, the magnetic flux density in the air gap should be as high as possible. This can be achieved by using PMs arrangedin "Halbach array" (Figs 3.15, 6.3 and,Fig 6.4). The magnetic flux density distribution excited by Halbach arrays is describedby eqns(3.42) to (3.46).ln practice,the anglebetweenmagnetizationvectorsof adjacentmagnetsis 90", 60" or 45o (Fig. 6.5).

6.3

Air gap magneticflux density

Fig. 6.6 showsthe resultsof the 2D FEM modelling of the magneticfield in the air gap of a corelessAFPM brushlessmachine. NdFeB magnetswith remanentflux density B, :7.2 T and coercivity H.: 950 kA/m havebeen considered.The thicknessof eachPM has been assumed6 mm, the coreless statorwinding thicknessis l0 mm and one-sidedair gap thicknessequalsto 1 mm. With the aid of Halbach anay a high peak value (over 0.6 T) of the normal componentof the magneticflux density has been excited. This value is sufficient to obtain a high electromagnetictorque. The peak value of the flux densitycanbe evenhigher for the optimizedmagneticcircuit of an AFpM ma-

193

AFPM machineswithout stator and rotor cores

1,m

(b)

-Nonlal +'fancenl

,^

t5 6

I

I q

\ \

F

\

\

1 E

d

Figure 6.6. Normal and tangential components of the magnetic flux density in the centre of the air gap of a double-sidedcorelessAFPM brushelssmachine excited by Halbach arraysof PMs: (a) 90"; (b) 45".

chine. In pmctice,60" and 45" Halbacharraysproducesimilar peak magnetic flux density(Fig.3.22). The peak value of the nolmal componentof the magnetic flux density is higher than that in a standardarangement of surfacePMs. The backing ferromagneticdiscsaddedto the double-sidedPM structurewith aTargenonmagneticgap do not increasethe flux densityas much as Halbach array excitationdoes.

6.4

Electromagnetic torque and EMF

The electromagnetictorque of a corelessAFPM brushlessmachineswith Halbach array PM excitation can be calculatedon the basis of eqns (2.25), (2.26), (2.27), (2.120) and (2.121). Similarly, the EMF can be calculatedaccordingto eqns(2.29),(2.30),(2.I 18) and (2.119).

194


Figure 6.7. Exploded view of the SEMA AFPM brushlessmachine. Courtesy of Lynx Motion Technologlt,Greenville,IN, U.S.A.

6.5

Commercial corelessAFPM motors

CorelessAFPM machine architectureknown assegmentedelectro-magnetic array (SEMA) hasbeenproposedbyLynx Motion Technology,Greenville,IN, U.S.A. for electric motor, actuatorand generatorapplicationsrequiring high power density and high efficiency !a01. Applications that require an electromechanical drive with high power density, high efficiency and low torque ripple include precisionmotion control, naval propulsion systemsand acoustically sensitiveapplications.SEMA technologymay also find applicationsin distributed generation systemsand energy storage systems, such asflywheel motor generator systems. Motors with high torque and efficiency are demandedfor gearlesselectromechanical drives. The use of direct drive motors without speedreducerseliminatesgearnoise,oil leaks,positioningerrorsfrom backlashand lack oftorsional stiffness. The statorcorelessand toothlessdesign (Fig. 6.7) not only eliminatesthe cogging torque but also increasesthe area availablefor conductors. It also boostspeak torque capability and allows PMs to be more effectively utilized lla0l. The statorcoils are fully pottedin high-strength,thermally conductive epoxy resin. Suchconstructiongivesthe machinestructuralintegrity and effectively damps the high frequency vibration when the motor is PWM invertedfed. Machines with corelessstator winding perform excellently at constant speed,variable speedand at reversals.

195

AFPM machines without stator and rotor cores

Figure 6.8. In-wheel motor, PM ring and stator winding of a corelessAFPM btushlessmotor for solar-poweredvehicles. Courtesy of CSIRO, Lindfield, NSW, Australia.

, s4 : &

s.iq

!'

3:.

;5 ls gE

r3

., 6

i

,4. ot t9

az {.1 g:

Figure 6.9. Explodedview of the AFPM brushlessmotor with film coil corelessstatorwinding and twin PM rotor. Courtesv of EmBest^Soeul, South Korea.

196

AXTALFLUX PERMAI,{ENTMAGNETBRUSHLESS MACHINES

Table 6.L Design data and parameters of a three-phase, 10-kw, 750-rpm disc type AFpM brushlessmachinewith ironlessstatorcore Designdata Output power Pot,W Speedn, rpm Number of phasesm1 Input current 1, A Input frequencyf ,Hz Number of stators(modules) Number of pole pairs p Number of coils (3 phases) Number of turns per phase Coil pitch Wire diameter, mm Axial thicknessof PM hx1,mm Axial thickness of the winding f-, mm Air gap (one side) g, mm Air gap magnetic flux density B,,n under load, T Current densiry Almmz ka,: D,;,,1Do"1ratio [5] Rotor outer diameter,o1lr, mm Winding packing factor at r :0.SD.tn Winding temperature,oC Cooling system Classof insulation

10,000 '7 50 3 (Wye) 28.4 100 2 8 24 100(onemodule) I slot 6x7.2 6 10 1 0.58 4.175 tr,16

t/yJ

360 0.65 75 natural f

CSIRO klecommunications and Industrial Physics, Lindfield, NSW, Australia manufacturesin-wheel corelessAFPM brushlessmotors for solar poweredvehicles1203].solar-poweredvehiclescompetein the world Solar challenge solar car race organizedevery two years in Australia between Darwin and Adelaide,which hasbecomea well-known intemationalevent. cslRo provides in-wheel AFPM brushlessmotors both with surfacepMs glued to steeldiscs and with PMs arrangedin Halbach array. The motor structure, PM disc and statorwinding are shownin Fig. 6.8 [203]. Fig. 6.9 showsa corelessbrushlessmotor with film coil statorwinding and twin PM rotor manufacturedby EmBest, Soeul, South Korea.The coreless statorhas a foil winding at both sides. The 8-pole PM rotor is designedas a twin extemal rotor.

197

AFPM machines without stqtor and rotor cores

(a)

aoo

(b)

4oo

l.0i

T,6(1) r00

I

.1. tf"' ,\p u(l)

| (l) -

2oo

AP

0.9

_'1l

ily"' 100

0.8 0.75

l0

olrozoro40 0140

010201040 0140 current. A

cunent. A

Figure 6.10. Calculated performance characteristics of a 10 kW coreless AFPM brushless motor accordingto Table 6.1: (a) shaft torque 7"1, line-to-line voltage [7., winding losses AP,,,,additionallossesLPoaa,andmechanicallossesAP",,versusstatorcurrentatn:750 rpm - const; (b) efficiency 4 and power factor pJ versus stato. cunent at n : 75A rpm : const.

6.6

AFPM coreless Casestudy:low-speed brushlessmotor

A low speed,three-phase,Y-connectedAFPM brushlessmotor rated at 10 kW 750 rpm, 28.5 A has been designedand investigated.The rotor doesnot have any ferromagnetic core and consists of trapezoidal coils embeddedin a high mechanicalintegrity resin. Averagequality sinteredNdFeB PMs with B, : I.2 T and H, : 950 kA/m have been used. The double-discconstructionis similar to that shown in Fig. 6.2b. The designdata and calculated parametersare given in Table6.1. All parametersexceptthe air gap magnetic flux density B-n and synchronousreactanceX"d : Xsq X, have been calculated analytically. The magnetic flux density and synchronousreactance has been calculatedusing the 2D FEM. The resistanceR1 : 0.175 0 and synchronousreactanceX" : 0.609 O perphase are for two stator modules connectedelectricallyin series.The EMF constantkB : 5.013 Vs and torque constantkr : 2.394Nm/A havebeencalculatedfor one statorsegment.

6.6.1


The currentdensity4.175 Almm2 in the statorwinding at rated load is low and a totally enclosedmotor can even operatewithout any forced air cooling. The recommendedcuffent density for encloseda.c. motors with class F of insulationratedup to l0 kW is from 4.5 to'/ R/mm2 [63]. The calculatedcharacteristics are shown in Figs. 6.10 and 6.11. It can be seenfrom Fig. 6.11 that the maximum-to-rated current ratio for torquespeedcharacteristicis 1.3. In Fig. 6.11 the voltage increaseslinearly up to approximately rated speedand then is kept constant.

198

MACHTNES AXIAL FLI]X PERMANENTMAGNETBRUSHLESS

200

116Gp)

r0il

;i;; z

200

1

150

v1

100

50

0

2

4

6

8

l0

t2

.14

0nk14 speed,rev/s Figure 6. I L Shaft torque, cuffent and phasevoltage versus speedof a 10 kW corelessAFPM brushlessmotor accordingto Table6.1.

6.6,2

Costanalysis

The cost model hasbeenpresentedin [96]. Averagepricesof materialsfor electricalmachineconstructionare as follows (in US dollars: $): silicon sheet steelcp" : 1.25 $/kg, isolatedcopperconductor(magnetwire) c6, : 5.51 : $/kg, sinteredNdFeB PMs cp,y : t5 $/kg and steel bar for the shaft cste,et PMs. The total 2.6 $/kg. The activematerialsare conductors,sheetsteeland volume of the steelbar to the volume of shaft is 1.94. The coefficienttaking into accountthe cost of machining of the shaft is 2.15. The cost of labour involved in the assemblyof active parts has not been estimated.

Table 6.2. Mass and cost of components of a disc type corelessPM brushlessmotor rated at 10-kW.750-mrn

ltem Winding Statorplasticmaterial PMs Shaft Frame,end bells, and bearings Components independentof the machine shape Total

Mass,kg

1 0 .31 1.01 12.38 5.48

22.51(activeparts)

Cost,$

55.84 7.09 185.66 59.4s 165.00 140.00 613.05


t99

Table6.3. Mass and cost of componentsof a cylindrical PM brushlessmotor with laminated core rated at 10-kW 750-rpm

Item Winding Core PMs Shaft Frame,end bells, and bearings Componentsindependentof the machineshape Total

Mass,kg 8.51 30.98 1.57 5.12

41.06(activeparts)

Cost,$ 46.9 72.49 23.55 62.07 i 65.00 r40.00 51 0 . 0 1

Table6.2lists the massand costof componentsof the 10 kW 750 rpm corelessmotor of disc construction.Table6.3 lists the massand costof components of the equivalentcylindricalmotor with laminatedstatorand rotor cores.Componentsindependentof the machineshapeare the encoder,terminal leads,terminal board,andnameplate. The 10 kW corelessAFPM brushlessmotor costs $6 I 3.05 (PMs cost $ 185.66)while a classicalPM brushlessmotor of cylindrical constructionwith the sameratingscostsonly $510.01(PMs cost $23.55). The contributionof PMs to the total costsof a corelessmotor is predominant.

6.6.3

Comparison with cylindrical motor with laminated stator and rotor cores

Table6.4 comparesthe designdata and perfomance of a l0 kW, 750 rpm corelessAFPM brushlessmotor and the equivalent10 kW, 750 rpm PM brushlessmotor with laminated statorand rotor cylindrical cores. In both motors the numberof phasesis 3, numberof polesis2yt:16, and input frequencyis 100 Hz. The analysis and the FEM simulation have shown the advantagesof the corelessdisc type PM brushlessmotors in terms of performanceand massof active materials. They have higher efficiency and power density when comparedwith conventionalPM machineswith laminatedcores. The efficiencyof the l0-kW AFPM brushlessmotor is l.2o/ohigher (0.925 versus 0.914) and power density (output power to mass of active materials) is 82o/ohigher (444.25Wkg versus243.55Wkg) becausethe massof active materials is 45o/olower than that of an equivalent PM brushless motor with cylindrical ferromagneticcore. Atzerc current statethe normal forces between the stator and rotor do not exist. However.the normal forces betweenthe rotor PM discsare large.

200

AXIAL FLLTXPERMANENTMAGNETBRT]SHLESS MACHINES

Table 6.4.

Comparison of parametersand performance of 10-kW, 750-rpm PM brushlessmo-

tOIS

Parameter Input voltage(line-to-line),V Shaft torque, Nm Efficiency, % Power factor Air gap magnetic flux density, T Stator winding current density, Nmm2 Statorwinding resistanceper phaseat 75"C, Q Synchronousreactance,Q Mass of PMs, kg Mass of active rnaterials,kg Power density (only active materials), W&g Cost excluding labour,U.S.$

Coreless disctype 22',7.6 131 92.s 0.991 0 . 58 4 . 1 57 0.1'73 0.609 t2.38 22.51 444.25 613.05

Cylindrical with laminatedcore 220.0 133 91.4 0.96 0.78 I

1Aa

0.037 0.75u 1.57 41.06 243.55 5l 0.01

Thesenew types of machinesare coggingtorque free. Owing to the lack of ferromagneticcoresthe hysteresisand eddy cunent lossesin thoseparts do not exist. The costof materialsand componentsof the disc tlpe l0 kW corelessmotor is 20oh higher than that of a cylindrical motor with laminated core. On the otherhand,the cost of tooling will be much lower as a corelessmotor doesnot require stamping and stacking tools. The high cost of PMs limits commercial applicationsof corelessAFPM brushlessmotors to small size machines,special servo drives, airborne apparatus (lightweight construction)and electromechanicaldrives where the cogging torqueripple and normal forcesmust be reducedto zero.

6.7

Casestudy:low-speed coreless AFPM brushless generator

The AFPM brushlessmachine(Table6.1) hasbeenanalysedasa standalone a.c. generator. The open circuit characteristic,i.e. EMF (line-to-line) as a function of speedis shownin Fig. 6.12. This characteristicis linear. The AFPM generatorhasbeenconnectedto the inductiveload Zr : R,r * j Xr : 7 + j0.628 O. The curent-speedcharacteristicfor generatingmode is shownin Fig. 6.13. The line voltageversusspeedis shown in Fig.6.14,output and input active power in Fig. 6.15,and efficiencyandpower factor in Fig. 6.16. The voltageis


201

250 200

nn(nJ

150 100

l0 0

ni speed, rev/s

t5 15

Figure 6.12. Open circuit characteristic:line-to-line E,MFversusspeed.

50

-t ot

t",, ('

') | l0

l5

u

r) .oeed::v/s

Figure 6.13. Currentversusspeedfor generatingmode.

a nonlinearfunction of speedand dependson the value of the load impedance zl and power factor. For inductive load (RL) the power factor decreasesas the speedincreases.

6.8

Characteristicsof corelessAFPM machines

Table6.5 showsspecificationsof corelessAFPM brushlessmotorsfor solar poweredvehicleswith surfacePMs glued to a steel yoke and PMs arranged in Halbach anay (CSIRO, Australia) respectively. The steady-stareefficiency curyesfor surfacePMs and Halbach array areplotted in Fig. 6.17. The maxi-

202

MACHINES AXIAL FLI]X PERMANEI:IT MAGNETBRUSHLESS

1s0 .t-, 9^ lv_,rn

\| iJI roo I

50

l0

5

l5 15

I

speed,rev/s

Figure 6.14. Line voltage versus speedfor generatingmode.

1 .1 0 4 I 0000 ,"r/

8000

,/

Pgo,, (ni)

e,

,/

l-r",) 'gtn

/

l0 o

ni

l5 l5

speed,rym Figure 6.15. Output power Psout. and input power Psn, for generatingmode.

mum efficiencyis obtainedwhen the inverterappliesa sinusoidalvoltagesynchronizedto the rotor position. The inductanceof the statorwinding is low and most inverlers require additional seriesinductance. The 8225 corelessAFPM motor (Table 6.6) has been developedby Lynx Motion kchnology, Greenville,IN, U.S.A. for air-handlingapplications,in which systemefficiencyis a high priority. The family of Lynx 225-mm diameter motor designs offers a broad range of possible system configurations for speedand position servoapplicationsrequiring both high efficiencyand high torque. Performanceratingsof theE225 motor are basedon air-cooled105"C winding at25oC ambientairtemperature.The EB13corelessAFPM machines

203


r.05 o

rts(n i) p r g( n i )

0.5

I

F

5

l0 n

I

l5 15

rpm speed, Figure 6.I 6.

Efficiency 4n and power factor pf n for generating mode.

;e

'o

0

200 400 *t

1400160018002000 ::,'r,i;fi"],'ff

Figure 6.17. Efficiency as a function ofoutput power ofa 3-phase,1.8-kW corelessAFPM brushlessmotors for solar-poweredvehicles: (a) rotor with surfacePMs; (b) rotor with PMs arrangedin Halbach array. Courtesy of CSIRO, Lindfleld, NSW, Australia.

has been designedfor power generationapplications(distributedgeneration systems)where high efficiencyof the systemis important. Performanceratings of the E813 machine are basedon 130"C winding at 40oCambient air temperature.Both 8225 andE8l3 machineshaveextremelylinear torqueconstantk7 independentof speed.Other applicationsof Lynx corelessmachines include servo applications,precision robotics, marine propulsion,generators and weaponstunets.

204


Table 6.5. Comparison of three-phase1.8-kW corelessAFPM brushlessmotors with surface and Halbach PM excitation manufactured bv CSIRO. Australia

Parameter Output power, W Efficiency, % Number of poles Nominal speed,rpm Maximum speed,rpm Nominal torque, Nm Maximum continuous torque at 1060 rpm and maximum winding temperature,Nm Absolute maximum torque, Nm Torque constantper phase at sinusoidal excitatir:n Nominal phaseEMF, V EMF constantper phase at sinusoidal current, Vs/rad Nominal phasecurrent, A Winding losses,W Eddy current lossesin the stator winding, W Windagelosses,W Total losses,W Winding temperaturerise, "C Overload winding temperaturerise, "C Maximum winding temperature,oC Air gap at each side, mm Resistanceper phase, Q Mass of complete motor, kg Mass of framelessmotor, kg

SurfacePMs andsteelyoke I 800 97.4 40 l 060 286s 16.2

JI

50.2 0.39 *J

Halbach arfay

1800 98.2 40 1060 2865 16.2

39 s0.2 0.56 62

0.39 13.9 43.9

0.56 9.6 27.6

2.6 2.1 48.6 22 64 110 2 0.0757

2.7 2.1 32.4 14 40 110 2 0.0997

to.z

tJ.z

r0.7

7.7

When coupled with a three-phaseH-bridge inverler, Lynx motors offer a highly efficient electromechanicaldrive with the high peak torque capability requiredfor the most demandinghigh-transientapplications,e.g. fuel-cell air compressors.

Numerical example6.1 The corelessAFPM motor with the samedimensionsas that describedin NumericalExample 5.1 has been redesignedto obtain the speedof 1000rpm zt Io : 8.2 A (I"d : 0). The number of pole pairs has been increasedto


205

Table 6.6. Specificationsof corelessAFPM motors manufacturedby L1,nxMotion kchnologt, Greenville"IN" U.S.A.

Parameter Number of poles d.c. bus voltage,V Terminal voltage, V

Speed,rpm Torque, Nm Peak torque, Nm Current, A Peak curent, A Output power, kW Torque constant,Nm/A EMF constantper phase,V/krpm Electrical time constant,/rs Winding resistanceat 100"C, f,) Winding inductance, g,H Rotormomenlof inenia.kgm2 Mass of motor, kg External diameter, m

E225AFPM motor l2 155 t17 rmsphasevoltage H-bridge 6000 9.9 97.4 18.7 180 6.2 0.53 l9 0.348 0.204 71 0.0184 8.4 0.225

8813AFPMmotor 28 851) 571 peakline-to-line 2750 450 900 ll9 238 130 3.78 209 1.66 0.0265 44 9.16 295 0.813

p : 12, the number of coil sidesto st : 72 and PMs have been arranged in 90o Halbach affay with nLr : 4 PM piecesper wavelength. The number of statorwinding turns per phaseis l,h : 240, the number of parallel wires on, : B, the diameterof wire d* : 0.455 mm (AWG 25) and the coil pitch measuredin coils sidesa'. - 3. Sinceboth statorand rotor do not have any steelmagneticcircuit, the magneticsaturationdoesnot exist and ksat : 1. All other parametersand assumptionsare the sameas those in Numerical example 5 . 1. Find the motor steadystateperformance,i.e. output power, torque and efficiencyat ratedcurrentIo : Iorr: 8.2 A. Solution The following calculatedparametersare the sameasthosein NumericalexIat :46.5 Di,n:0.727 ffi, D -- 0.1735r'rr,Li: ample5.1: g: 1.5 rtfi11, : 0.75 mm, Rtorrl: 0.115m (for windagelosses),R"h : 0.015m andn'LcorL kg.

206

AXIAL FLUX PERMANENTMAGNET BRUSHLESS MACHINES

The numberof coils per phasefor a singlelayer windingis n,..: stl(2rn) :72lQx3) : 12.Thenumberofturnspercoil is Af,, : Ntln,:240172 : 20. The number of coil sidesper pole per phase(equivalentto the number of slotsper pole per phase)qt : stl(2p*t') - 721Q4 x 3) - 1. The pole pitch measuredin coil sidesis r, : slf (2p) : 72124: 3 and the averagepole pitch r : nD I Qp) : r0.7735124 : 0.023m The input frequency at 1000 rpm f" : n"p:

1000 x 12 - 200}J2 60

The wavelengthIo at the averagediameter and constantp for Halbach array is

l o : 2 r : 2 x 0 . 0 2 :3 0 . 0 4m 6

1 3 8 .13/ m I J: + 1,,,: ] I - : 0.046

The magneticflux densityin the air gap accordingto eqn (3.42) is B , , g N B m o : 1 . 2 1 r * e x p ( - 1 3 8 . 3 x 0 . 0 0 6 ) ] s r n ( t l 4 : 0 . 6 0 9T 7 T/ 4

The magneticflux accordingto eqn (.2.21)is

- (a) I : o o,' : 1B?lr urox 0222[r " "oo,,nr " - ' rru r12 L

\/3/l

The winding factor according to eqns (2.8), (2.9) and(2.10) is kd1:;sinn/(2x3)

t s i n n / ( 2x 3 x 1 )

kpr:si'(3;)

: r

: l1i

n

':

uJc, 7

i.:

g

k,or:1 x 1:1

The EMF constantaccordingto eqn(2.30)andtorqueconstantaccordingto eqn(2.27)is kn:nJ2x12x240

x r x 0 . 0 0 0 4-1b . 2 4 , y 1 $ e v l s ): 0 . 0 B Z 3 V / r p m

k r : k / 'Z7T + : b . 2'zir 4!:2.5 Nm/A TheEMF at 1000rpm is Ef : kan,: 0.0873 x 1 0 0 0: 8 7 . 3 4V


201

The electromagnetic torque Lt Io - Ioq : 8.2 A is

Ta : krlo : 2.5 x 8.2 : 20.5Nm poweris The electromagnetic

t99o : 2rn x 20.5: 2148.6 w Put,n ,o tt0 "7,1: The length of the statorshorterend connectionwithout inner bendslr.min : (w"f r.)rDi"l(2p) - (313)r x0.127124: 0.017m andthe lengthof the sta0.017x0.22f 0.I27 : tor longerendconnectionwithoutouterbendslkôr: (Fig. 3.16). 0.0288m The averagelength of the statorturn with 15-mmbendsis llooN2Lt'*ll"rn'i, lLl.rro, +4 x 0'015 :2

x 0 . 0 4 6 5+ 0 . 0 1 7+ 0 . 0 2 8 8+ 0 . 0 6: 0 . 1 9 8 4m

The statorwinding resistanceat 75"C accordingto eqn (2.33) is t) f Ll -

240 x 0.1984 : 0.7963O 47 x 706x B x n-x (0.45* t}-s1'z14

The maximum width of the coil at the diameterD;' is uuu:: rDinf sy : of the coil is l- : B mm. 10.127172: 0.0055m: 7.4 mm. The thickness : rtryx l{s1: 8 x 20: 160. The is Al, Thenumberofconductorspercoil maximum value of the coil packing factor is at Din, i.e.

k1'nin:'4*: l,,,LD,u

nnui.l : o'731 lon 6 x C.D

The stator cur:rentdensity is

8.2 Jr,,:

B x n0.452 l4

:6.45 Af mm2

The statorwinding lossesaccordingto eqn (2.42) are AP,, : 3 x 8.22x 0.7963: 160.6W The eddy cuffent lossesin statorround conductorsaccordingto eqn (2.61) are A,p. arr- :

n2 47 ^ 106 20G x0.000452x 0.75 x 0.60g2x 1.152: 3g.3 W 4 Sftoo

208

AXIAL FLUX PERMANENTMAG]{ET BRLISHLESS MACHINES

The friction lossesin bearingsaccordingto eqn (2.64) arc

: 6.1\[' L P y,:0 .0 6x 1 .5 (3 .4 + 0.0+1400 tlu

The windagelossesaccordingto eqn (2.67) are 1 r 1OOOr3 L P , i n d - - 0 . 0 1 3 r 1 . 2( z o ' " 1 1 1 fLr\o . sx 0 . i l 5 ) 5 - ( o . sx 0 . 0 1 5 ) 5 1 2 60 / \

r 0.18\[r where

n": r.2ffffi 'f :

x 0.1152: x 1oa e.233 3.87

,,/g.2',r,.Iff:0'013

The rotational(mechanical)lossesaccordingto eqn {2.63) are L P r o t : 6 . 1* 0 . 1 8: 6 . 2 8W The output power is Pout : P.trn - LProt : 2748.6* 6.28 : 2742.3W The shaft torque is 2142'3 T"r" : "n-2rx1000/60

: 20.40 Nm

The input power is Pi,:

P.t^ * LP* * LP":2148.6 + 160.6+ 39.3: 2348.5W

The efficiencyis 2142.3

n : 2 3 4 & s: o' 912


209

Numericalexample6.2 A three-phase,15-kW,3600-rpmcorelessAFPM motor with Halbacharray PM twin rotor hasthe statorwinding resistanceper phaseRr - 0.24 f), stator winding leakagereactanceX1 : 0.92 O and armaturereaction reactances Xod. : Xoq : 0.BBO. The input phasevoltage is [ : 220V, power factor cos4! : 0.96, angle betweenthe stator current and q axis \F : 5", EMF constantkr : 3.363V/(rev/s),eddycurrentlossesin statorconductorsLPe,: 184 W and rotationallossesLP,ot: 79 W. Find the internal voltage fl excited by the resultant air gap flux and the stator d- and q-axis current adjustedfor eddy current losses. Solution The phasordiagram showing the EMFs Ey and E; is drawn in Fig. 6.18. The EMF E7 excited by the rotor magnetic flux

H I : k e, : 3 . 3 6 3 t

#

: 2 1 0 . 7 8V

The synchronous reactance is Xsd,: Xsq:

Xoa* Xt : Xo,t * Xr : 0.BB+ 0.92 : 1.80

The angle betweenthe stator current and input voltage r/ - arcsin 0.96 : 16.26". The load angle betweenthe voltage and EMF Ey is 6 : rf W: : 76.26 5 7I.26". The statorcurrentsaccordingto eqns(2.80), (2.81) and (2.82\ are:

r.- -

'ad

r-

'aq

- 0.24sin11.26o) - 201.78x 1.8 220(I.Bcos 11.260 :4.5A 1 . 8x 1 . 8+ 0 . 2 4 2 2 2 0 ( 1 .c8o s1 1 . 2 6*0 1 . 8s i nI L 2 6 " ) - 2 0 L 7 8x 0 . 2 4 :24.5 A 1 . 8x 1 . 8+ 0 . 2 4 2 I o : 1 / L b 2+ A l r 2 : 2 4 . g A

Thestatorwindinglossesaccording to eqn(2.42)are LPt',: 3 x 24.92x 0.24:445'6 W The inputpoweris - 3 x 220x 24.9x 0.96: 15763.0W Pin : rnlVlIacos@

210

ANAL FLUX PERMANENTMAGNETBRUSHLESS MACHINES

blgure 6.18. Phasordiagram of u .o..t.r, AFPM synchronousmotor (eddy current 1" is in phasewith the intemal voltage Ei). Numerical example 6.2.

The electromagneticpower accordingto eqn (5.1) is Petrn:15 763

4 4 5 ' 6 1 8 4: 1 5 1 3 3 . 3W

The outputpower accordingto eqn (5.3) is Pout: 15 133.3- 79 : 15 054.3W The efficiencyis ?:

1 50 5 4 . 3 : o'955 15 z63J

The internal voltage E; excited by the resultant air gap flux according to the phasordiagram(Fig. 6.18)is

En:@ :

V 1fQOt.fA* 4.5 x 0.88)2+ (24.5x 0.88)2: 206.87

AFPMmachines withoutstatorandrotor cores

2Il

The angle d; betweenthe EMF Ei and g-axis is

24si*%*) - arcr,n d;: ar"r (ro, , ",,(o!;i*",) to4Aod \f .t t /

- , or"

The eddy currentshuntresistanceaccordingto eqn (5.32) is )nG R72

-6e7.75Q

R":3?

The currentin the vertical branchof the equivalentcircuit (Fig. 5.3) found on the basisof the phasordiagram(Fig. 6.18) Ei.c I." a : - t s i n d ; :

t", :

*cos

d,.:

tlo+ t!,n

I.'

= Ei ft:

- 206.87 -0.031A rr,^sin5.97":

#

A 5.97": 0.295 rryrcos

(-0.031)2+ 0.2952: 0.296A

246'8T

,* ^cos

: 0.296 A 5.97"

The stator current adjusted for eddy current lossesfound on the basis of the phasordiagram(Fig. 6.18)

IL.a:

EisinSt

Xo,]

R"-

206.87cos5.972- 2 0 1 . 7 8

2 0 6 . 8 7s i n 5 . 9 7 2

0.88

697.75

E; sin d; 'aq

Ezcos6i - Ef

v -

Aae

ro, -

E; cos dr _ 206.8Tsin 5.972 R" O.BB

r4.5A

206.87 cos5.972 = 2 4 . 8A 697.75

(I'"a),+ (IL)2 : \f Lb2+rLB2 x 25.2A

Chapter7 CONTROL

In the previous chaptersit was shown that there is a direct relationship between the rotor speed and the frequency of the phase voltages and curents of the AFPM machine, It was also shown that the magnitude of the induced phasevoltagesis directly relatedto the frequencyand hencethe rotor-speed of the AFPM machine. Therefore, for variable speedoperation of the AFPM machine both the frequency and magnitude of the supply voltage must be adjustable.This requiresa solid-stateconvefterto be usedbetweenthe flxed a.c. or d.c. supplyand the terminalsof the AFPM machineas shownin Fig. 7.1. To have good position andlor speedcontrol of the converter-fedAFPM machine drive, the torque and thereforethe currentof the machinemust be controlled. For currentcontrol, informationof the phasecurent as well as the position of the rotor is necessary.Thus, both the currentand rotor position of the AFPM machine is sensedand fed back to the controller as shown in Fig. 7.1. The controller in its turn controls the supply voltage and frequency of the AFPM machine via the solid-stateconvefier (inverter). This chapterfocuseson all the control aspectsof the AFPM machinedrive of Fig. 7.1. The control of the two types of AFPM machinesconsideredin previous chapters,namely the ttapezoidal and sinusoidal AFPM machines,is dealt with separately.Finally, position sensorlesscontrol is briefly discussed and somenumerical examplesare given.

7.1

Control of trapezoidalAFPM machine

TrapezoidalAFPM machinesare surface-mountedPM machinesand are characterisedby their trapezoidal or quasi-squareEMF waveforms. Example of theseEMF waveforns are shownin Figs 7.2a and7.2b,which are the open circuit phasevoltagewaveformsinducedby PM flux of a small 1 kW, slotted trapezoidalAFPM machineat half and rated speedrespectively.It can be ob-

214

Fixed AC or DC supply

AXIALFLUX PERMAIVENT MAGNETBRUSHLESS MACH]NES

Solid-StateConverter (SSC)

Figure 7.1. Converter-fedAFPM machine drive.

served,firstly, that at rated speedthe frequency and the magnitude of the EMF are double what they are at half rated speed. Secondly,and very important, it can be seenthat the shapeof the EMF waveforrn is not sinusoidal,but trapezoidal with a flat-toppedvoltage of about 120 electricaldegrees.It is during this period of 120 electricaldegreesthat the phaseis excitedby meansof the solid-stateconverter. This period is therefore called the conductionperiod, as cunent is flowing in that phaseduring that period as shown in Fig. 7.2c. Note that the time of conduction or the time of occurrenceof the flat-topped voltage is directly relatedto the position of the rotor; the time of conductioncan thus exactlybe determinedby meansof a simple,low resolutionposition sensor. During the conductionperiod onecanconsiderthe inducedback-EMFphase voltage as a d.c. quantity due to the flat voltage waveform. With a three-phase machinethe phasevoltagesare 120electricaldegreesout of phase,which implies that the conductionperiods of two phasesare always overlapping(see Fig. 7.2) and that two phasesare alwaysactive. In the caseof AFPM machine drives with 180 electricaldegreeconductionperiods,e.g. [31], then all three phasesare active. A different convefiershould be used in this case. In this sectiononly thosetrapezoidalAFPM machinedrives that have 120 electrical degreeconductionperiodsare consideredas they are commonly used.

7,1.1 Voltageequations With two phasesof the Y-connectedthree-phasemachine always active (the third phase is switched off by means of the solid-state converter), the power circuit of the trapezoidalAFPM brushlessmachinecan be representedby the electric circuit of Fig. 7.3a. Here, phasesA and B are activeand phaseC is switchedoff(shown in dottedlines). In this circuit R1 and L" arerespectively the phaseresistanceand phase synchronousinductance. The synchronousinductancesin the d and q-axis are equal, i.e. L"4 - L.sq- L". The voltages efA, efB and eyc are the EMFs per phase. As phasesA and B are in their conductionperiods,both e1e and ey s are at their flat-toppedvalues. Hence,

215

Control

20 15 10 od

:

-10

-20 0 . 0 0 8 0 0 1 2 0.0'16 0.02

(a)

0.024

Time (s) 40 30

20 > 16

= -10 s -24

-30 40

(b)

0.008 0.012 0.016 a.a2 Time (s)

0.024 0.028

0.032

(c) Figure 7.2. Open-circuit induced phasevoltage waveforms of a l-kW trapezoidal AFPM machine obtained from measurements:(a) at half speed (30 Hz); (b) at rated speed (60 Hz); (c:) theoretical EMF and phase current waveforms.

efA: El!'' unaeJB: -Ef') wherenl!'l t" the flat-toppedvalue of the trapezoidal voltagewaveform(fig. Z.Zc1.Combiningthe resistances, inductancesandinducedvoltagesof Fig.7.3aresultsin theequivalent circuitof Fig. 7.3b inwhich Eyr-r is takenas a d.c.voltageequalto E1r,-L : 2Ef'), R.p : 2R1, Lo : 2L", ia : ioA voltage equation of this circuit is

-ioB

and uo :

urL-L.

The dynamic

216

AXIAL FLUX PERMANENTMAG]{ET BRUSHLESS MACHINES

" ll-l.

Figure 7.j. TrapezoidalAFPM machine connectedto the solid-stateconverterwith two phases active: (a) electricalcircuit; (b) combinedequivalentcircuit for the analysis.

u,p

2R1ioa+ 2L"q9! a[

:

+2Ey')

R' p i oI L-ro+, t r L l t - t

(7.1)

Applying a d.c. voltage up : V, acrossterminals A and B by meansof the solid-stateconverterwill imply that ideally in the steadystate,a d.c. current io : IL"q)will flow throughthe two phasewindingsaccordingto the following steady-state voltageequationderivedfrom eqn (7.1'),i.e.

Vp: R'pIL"q) + Eyr-r

(7.2)

power The inducedvoltageE yr-L is givenby eqn (2.I l8), the electromagnetic of the machine is P.1,rn: E f r, ;I["q) andthe developedtorque is given by eqn (2.120).Note that da: I\sq) andEyr I are instantquantitiesthat can change with time, similar to a brush-typed.c. machine.

7.1.2

Solid-stateconverter

The completepower circuit connectedto the three-phasetrapezoidalAFPM brushlessmachine is shown in Fig. 7.4. In this circuit the fixed three-phase a.c. supply is first rectifiedby meansof a three-phasediode rectifier to obtain a fixed d.c. voltage. The d.c. voltage is then inverted to a three-phasea.c. trapezoidalvoltage. With the diode rectifier no power can flow back to the fixed a.c. supply if the AFPM machine is operating in generatormode. This means that any generatedpower from the machine must be dissipated at the d.c. link of the converter.This is done by switching a braking resistanceat the d.c. link as shown in Fig. 7.4. lf it is requiredthat the generatedpower must

Control

217

flow back to the fixed a.c. supply,then use must be made of a so-calledactive rectifier.In the rest of this sectiononly the switchingand control of the d.c. to a.c. inverterof Fig. 7.4 is studied,assuminga fixed d.c.-link voltage. SSC

AC supply

Command

feedback

Figure 7.4. Solid-state a.c.-a.c. converter and equivalent three-phaseAFPM machine connection.

The three-phaseinvefter of Fig. 7.4 has six power electronictransistorswitches. Nowadays Insulated-GateBipolar Transistors(IGBTs) or MoS field effect transistors(MoSFETs) are very popular for use as switches for the inverter. The power transistorswitchesare switchedon and off at a relatively high switching frequency(typically from I to 20kHz or evenhigher). Either the top transistorof an inverter-legis switched on and the bottom transistor is switchedoff, or vice versa,or both transistorsare switchedoff. It was explainedin the previoussectionsthat only two phasesof the converter-fedtrapezoidal AFPM machine drive are active at any time. The third phaseis switched off by switching offthe top and bottom transistorsof the inverter-legconnected to that phasewinding. The three-phaseinverterof Fig. 7.4 alsohas six antiparallel diodes to make current flow possible when a switch is turned off (this is due to inductive stored energy) and also to make the generator (braking) operationof the drive possible. The circuit diagramsof Figs 7.3 and 7.4 show that the trapezoidalAFPM machine drive can be analysedby consideringonly the two active phasesof the machine and the two active switching legs of the inverter. The circuit then reducesto the generalcircuit of Fig. 7.5. One method of controlling the power semiconductorswitchesof the inverler shown in Fig. 7.5 is explainedas follows. consider in the forward speeddirection that the voltage Eyr*r and the average current io@us) : I[uq) are positive,then the averagesupplyvoltageup@rg)also hasto be positiveaccording to the steady-stateequation,eqn (7.2), i.e. up(ood : Vp ) Eyr_r. A positive averagesupply voltage can be obtainedby switching on transistor Tu

2t8

AXIAL FLUX PERMAI{ENTMAGI{ET BRUSHLESSMACHINES

,,-l

+ *:-) la

V,

"-]

Figure 7.5. Equivalent circuit ofconverter-fed trapezoidalAFPM machine (two phasesactive)

and pulse-width-switching(PWM-switching)transistorsTu and7l of Fig. 7.5 at duty cycles D and (1 D) respectively,such that in the steady-state DV4: up(ous)

(7.3)

where

D:bL

ts

-

+ c of n . t s

(7.4)

andto,.is the switching-ontime, f" is the switchingperiod and,f is the switch" ing frequency of the transistors. This switching and the switching waveform of the supplyvoltage?rpacrossthe terminalsof the two phasesof the machine are shown in Fig. 7.6(a). As is shown up@,g)is the switchingperiod average of the volta1e up. Note that in Fig. 7.6 the drive is not in the steady-stateas the instant averagecurrent io@us)and the averagevoltage up@u!t)changewith time. Also note that the drive systemonly reactson the averageof the supply voltage and not on the high frequency componentsthereof, as the switching frequency is relatively high and the time constantof the mechanical systemis relatively large. Let us consider next the flow of cuffent in the converter and the machine in motoring mode. When transistorsTu and To of Fig. 7.5 ate switched on, cuffent flows from the d.c. supply through Tu andthroughthe machinewinding andTr, and then back to the supply. If ?l, and T, arc switched on, the curent then free-wheelsthrough diode D, and through the machine andTr. In this caseno current is flowing from or to the d.c. supply. An example of the phase

2t9

Control

winding and d.c.-link current waveformsin motoring mode is shown in Fig. 7.6a. As mentionedbefore io@r,fi is the instant averageof the phasecurent i,o.

In generating(braking) mode with E I r L positive, the duty cycle D of eqn (7.3) is adjustedin sucha way that upGog) < E yr _ L to allow i o@,ry) to become negativeas shown in Fig. 7.6b. ln this caseDo is alwaysconductingand the current either flows through 71,when switched on, or through D, to the supply when switchedoff. An exampleof the d.c.-link curent waveformin generating mode is shownin Fig. 7.6b. When operatingthe machinein the reversespeeddirection, in which case Eyr,L is negativewith respectto the polarity given in Fig. 7.5, transistor2,, is switchedon during the conductioncycle and transistorsI and T, arepWMswitched.In this casethe supplyvoltageun(auo)is negativeand the curent can be controlledagainto be positiveor negativein the sameway as in the forward speedmode. The advantageof the above switching method is that only one inverter-leg is switchedat a high frequencyat any time, keepingthe switching lossesof the invefier low Which inverter-legof Fig. 7.5 shouldbe selectedto be PWMswitchedis simply a logic decisionthat dependson the sign (positiveor negative) of the supply voltageu,@,s). Note that l)p(ouilis controlleddirectly by the controllerof the drive (Fig. 7.1).

7.1.3

Current control

For good speedcontrol of any drtvea.fasttorque control is necessary.According to eqn (2.120)the torqueof the trapezoidalAFPM machinedrive can be controlled directly by controlling the stator current io and hence the instant averagecurrent io@oti explainedin Fig. 7.6. Close examinationof the currentsof Fig. 7.6 revealsthat the required information regardingthe curtent 'io1oon1 is also presentin the averaged.c.-link current i,I1ors1,i.e. if the sample-measurement of current i,4 is at the right instant. The current of all the phasesof the machinecanbe controlledby just controllingthe d.c.-link current i46us1.Thus,only one currentsensoris necessary.One aspectof this method of currentsensing,however,must be kept in mind. If the machineis operating, e.g. in forward motoring mode or reversemotoring mode, then in both cases the cumenti,t1o,,s1 will be positive.This createsa problem from a controlpoint of view as the referencecurrent io,(,r,,s)* will be, say,positivefor forward motoring mode, but negativefor reversemotoring mode. Howeveq this problem can easily be solvedin the controllerby alwaysmultiplying the d.c.-link current signalwith the sign (positiveor negative)of the controliedsupplyvoltage up(oug)* signal. A block diagram of the single sensorcurrent controller is shown in Fig. 7.7. The currentcontrolleractson the comparisonbetweenthe desiredcurrent

220

MACHINES AXIAL FLUX PERMANENTMAGNETBRTJSHLESS

t,

'o ',

I p(^9,

vd l

D"i

---1 T:

lo

0

Id

0

(a)

(b)

Figure 7.6. Voltage and cuffent waveforrns of a converter-fedtrapezoidal AFPM machine: (a) motoring mode; (b) generatingmode (seeFig. 7.5).

andthen controlsthe supply io1oud*andthe measuredd.c.-link cuffent i,J(aua), ou@utvoltage by controlling the signal uolorsy*.The PWM generatoroutputs the PWM signal in terms of a logic I or 0 to the programmablelogic device (PLD). The sign (positiveor negative)of up6,,s)* is also an logic input (1 or 0) to the logic device.Threelogic position signalssn, s6 arrdsc are also inputs to the PLD. These three signals are received from three Hall-effect sensors mounted in the machine. The signals supply information about which two phasesmust be active according to the rotor position. The outputs of the PLD directly control the switchingof the six power semiconductorswitches.

221

Control

Two aspectsin the block diagram of Fig. 7 .7 have not been referred to yet. The first is that a speed voltage signal is added to the output of the cuffent regulator. This is to decouple the current regulator from the effects of speed variation. The gain, 1{i, representsthe gain of the solid-state invefter. The secondaspectis that the d.c.-link currentis sampledat certaintimes to get the averagevalte'i61oon)as shownin Fig. 7.6.

Figure7.7. Blockdiagram of singlesensor currentcontroller fortrapezoidal AFPM.

To determine the parametersof the current regulator the transfer function of the current control systemmust be obtained. According to Fig. 7.3b and eqn (7.1), the dynamic voltage equationof the machine in terms of average componentsis up(oud :

n, Rpioloos) * L,

,lti,t',6

* E7t-r

(7.s)

where up : up(o"-g) * Itr,'io : io(oug)i i,r.,andu, andi," arethe ripple voltage and ripple current respectively(seeFig. 7.6). ln Laplacetransformeqn (7.5) becomes : (Eo * Lrs)i.o1ouel(s) up(at,s;(s) + Eyr-r(s)

(1.6)

With the use of the decoupling speedvoltage signal in the current control systemof Fig. 7.7, the speedvoltageterm E f r r(s) of eqn (7.6) hasno effect on the current regulator'sresponse.Hence,the control block diagram of the currentcontrol systemcan be represented by the block diagramshown in Fig. 7.8. In this block diagram the current regulatorcan be, e.g. a gain or a PIregulator. The gain 1{,; representsthe voltage gain of the solid-stateconverter

222


(inverter) asmentionedbefore. The transfer function of the current control system can be determinedfrom the block diagram of Fig. 7.8 and the parameters ofthe currentregulatorcan be designedaccordingto classicalcontrol system methods.

Figure 7.8. Block diagram of decoupled cuffent control system.

7.1.4

Speedcontrol

With the cuffent control systemin place, as explainedin section7.1.3, a speedcontrol loop can be appliedaroundthe currentcontrol loop. The electrical and mechanicaltorque balanceequationsfor the systemare as follows: Talous1ft) : k7,1.io1ore; (s)

T a l o u s ; ( r-)f r ( " )

:(B+

J.s)c,:(s)

(7."1)

(7.8)

is the load torque,B is the equivalenttotal dampingofthe system, where ";(s) J is the equivalent total moment of inertia of the system and a is the rotor speedin radls. For a two-pole machinethe angularmechanicalspeedis equal to the angularfrequencyof the statorcurrents.Note that eqns (7.7) and (7.8) refer to the instant averagecomponentsof current, torque and speed,and not to the high frequency ripple componentsdue to the inverter switching. From equations(7.7) and(7.8) and from Fig. 7.8, the speedcontrolblock diagramof the systemcanbe represented asin Fig. 7.9. Hence,the transferfunctionof the speedsystemcan be determinedand the parametersof the speedregulator can be designedto obtaina certainspeedresponse.A speedsignalcan be obtained electronicallyfrom the signalsof the threeHall-effectpositionsensorsinstalled in the machine.

7.1.5

High speedoperation

Equations(2.118),(7.2) and (7.3) show that as the speedof the machine increases,the EMF Eyr-t, the voltage urdorlt)and the duty cycle D will in-

Control

zz)

Figure 7.9. Block diagram of speedcontrol system.

creasetill the speedis reachedwhere D becomesunity. The speedwhere , becomesunity is simultaneouslythe speedwherethe outputvoltageof the converter(inverter)reachesa maximum. This speedis often called the basespeed of the drive. Field or flux weakeningin the high speedregion of the machine surface to keep E yt,- r : constant is not possiblewith high-energy-product mountedPMs. Therefore,at speedshigher than the basespeed,Eyr-r ) V,t, and control of the current and torque of the machine is lost. In fact, the current and torque rapidly drop to zero at speedsslightly abovethe basespeed.This implies that the speed-rangeof the trapezoidalAFPM machineusing surface mountedPMs is very limited. Techniquesto increasethe speedsubstantially beyondbase speedare: (a) changesmoothly from a 120" current conduction period, as discussedearlier in this chapter,to a 1B0osix-stepconductionperiod [17, 89, 208] and (b) advancethe current wave with respectto the EMF wave [125]; notethat during sub-basespeedoperationthe currentwave is controlled to be in-phasewith the EMF waveform as shown in Fig. "1.2c. The disadvantages of thesetechniquesmay be the following: (i) an increasein machine torque ripple, (ii) dangerof converterswitchesbeing exposedto high DC-1ink voltages if for some reason the converter switches are switched-off at high speeds(at this condition the high EMFs are rectifiedby the six diodes of the inverterto a high d.c.-link voltage)and (iii) a higher resolutionposition sensoris necessaryfor the smooth advancingof the current wave as the speed increases. A technique to obtain high speedoperation that avoid the disadvantagesof the above techniquesand obtain true flux weakening is to increasemechanically the air gap of the AFPM machinewith increasein speed;this technique is not possiblein commercialRFPM machines.However,it is questionableif this techniqueis economicallyviable.

7.2

Control of sinusoidalAFPM machine

SinusoidalAFPM machinesare designedto have sinusoidalor near sinusoidal EMF waveforms. Therefore, in variable speeddrive applicationswhere

224

AXIAL FLUX PERMANENTMAGNETBRT]SHLESS MACHINES

a solid-stateconverteris usedto supplythe machine(Fig. 7.i), the outputvoltageof the convertermust be sinusoidalor sinusoidalPWM modulated.Hence, the current in the machine is sinusoidalor else the current regulatorsof the drive will force the phasecurrentsto be sinusoidal. This sectionfocuseson the control aspectsof the sinusoidalAFPM machine drive. First, the modelling and equivalentcircuits of the sinusoidalAFPM machine are described.Secondly,the current and speedregulatorsof the drive are dealt with. Finally, the hardwareof the sinusoidalAFPM machinedrive is considered.

7.2.1

Mathematical model and dg equivalent circuits

The phasevoltageequationsof any three-phasea.c. electricalmachinecan be written as .u.l A B ( ' :

,,

nItaABC ,

dt

(7.e)

where tb.qsc is the total stator magnetic flux linkage of phaseA, B or C * eqn (5.20). Eqn (7.9) is expressedin the so-calledABC stationaryreferenceframe where the circuit variables (voltage, current and flux linkage) are expressed in a reference frame that is fixed to the stationary stator. For synchronous machines,like the AFPM machine, it is very convenientto kansform or refer the stationary stator variables to a reference frame that is fixed to the rotor. In this transformation the stationary ABC statorwindings of the machine are replacedby fictitious dgOwindings that rotate with the rotor. To explain this, let us considerthe cross-sectionof a two-pole AFPM machine as shown in Fig.7.10a (side view) and Fig. 7.10b (front view). Fig. 7.10ashowsthat the PMs are embeddedinto the rotor steelyoke, but the magnets in the sinusoidal AFPM machine may also be on the surface of the yoke like in atrapezoidal AFPM machine. With the magnetsembedded,the AFPM machine is not a pure PM machine anymore, but a combined PM-reluctance machinebecausethe electromagnetictorque has two components:PM synchronoustorque and reluctancetorque. Fig. 7.10 explains the ABC and dq winding layout of the machine. The rotating d and q-axis windings are in quadratureof each other and their magnetic axes are aligned with the d and q-axis of the rotor. In this case the d-axis of the rotor is chosento be alignedwith the centrelines of the PMs and the g-axis betweenthe magnets. The position of the rotor and dg-windings with respectto the stationaryABC windings is definedin Fig. 7.10 as the angle betweenthe rotatingd-axis andthe phase, -axis. Note, that in Fig. 7.10b the direction of the flux (O) is into (or out ofl the paper (z*direction), while the current is in the ry-plane.

225

Control

d - axis

I L

o,

iru--{a/

--"t, '

nttt iotr .'

(b) Figure 7.10. StationaryABC androtatingdq windings of a simple two-pole AFPM machine: (a) side view; (b) front view (PMs are shown in dotted lines).

To transfer the circuit variables from the stationary ABC refercnceframe to the dqOreferenceframe, or vice versa,the Park transformation

lfano): lKpllf ascl

(7.l 0)

226


and inversePark transformation

t lf anc): lKp) lfonol

( 7 .l l )

are used [86], where llfpl is accordingto eqn {5.26) and lkpl-i is the inverse matrix of lKp). In the above eqns (7.10) and (7.11), faqo and f asg are the dq} and ABC vanablesof voltage, current or flux linkage respectively. In eqn (5.26) the angle 0 is the electricalangledefinedin Fig. 7.10. With a constant rotor angular velocity cuthe angle 0 : at. For a balanced three-phase AFPM machine it is only necessaryto considerthe d and q-axis circuits as the zero-sequence circuit is playing no role in the performanceof the machine. Applying eqn (7.10) to the voltageeqn (7.9) leadsto the dq voltageequations (5.5) and (5.6) of the balancedAFPM machine (omitting the zero-sequence component).In eqns(5.5) and (5.6) u is the electricalspeedof the rotor, and tlta and tftnarc the d and q-axis flux linkages. Let us considerthe following three casesof operatingconditionsof the AFPM machine: (I) The first case is when the machine is open circuit and is mechanically driven at a certain speed cu. The dq stator currents, thus, are zero and only the magnetsare active. From Fig. 7.10 rt is clear that the magnet flux will only link the d-axis winding and not the q-axis winding. In this casetf,tn: 0 and rha : tbf , and from eqns(5.5) and (5.6) ura : 0 and u4: a$1' (ff

The secondcase is when both the magnetsand the d-axis winding are not active, but d.c. (steady-state)current is flowing in the q-axis stator winding. The dq stator windings and the rotor are at a rotating speedof u.,. Clearly with no magnetsand no d-axis current,lba : 0. With positived.c. q-axis current,i.e. ion: t/2loq, then ,ltn: t/lL"nloowhere ;,", is the q-axis winding synchronousinductance. Note that the relation between $q and ion is nonlinear due to magnetic saturation;L"o wlll thus vary with saturation. From eqns (5.9) and (5.10) it follows in this casethat ,rn : Rfi,r,, and u1g: -wL"ndor,

(III) In the third caseonly the d-axis winding is active in terms of a positive d.c.currentflowingof io,1: uEloa. Thereforerlrq:0asthemagnetsare not active anddon: 0, and da - J2L where.L"4is the synchronous "41o,1 inductanceof the d-axis winding. Again, the relationbetweenda andi,oa is nonlinear and L"4 will vary with saturation.In this casefrom eqn (5.9) and (5.10)u1n: aLr4'io4andu6 : Rliod. If the magnetsand the dq stator windings are active, then the above three casescan be summarized.by eqns(5.7) and (5.8). From eqns(5.5) to (5.10)it follows that the electromagneticpower of a three-phasemachine is given by

227

Control

Perrn:

ioaw'{q) :

|O"nrr1to

}r(.rbo'i"n

- |)qi"a)

(7.r2)

The factor 3l2has been explainedin Section5.13. The developedtorque of the machineis expressedby eqn (5.13). The first term in eqn (5.13) expresses the torque componentdue to PM excitation,while the secondterm expresses the reluctancetorque component. For surfacemounted AFPM machinesit can be assumedthat L"6: L"q, hencethe torque is directly proportionalto ior. In the caseof an embeddedAFPM machinelike in Fig. 7.10a it is clear that L"d 1 -L"n,which implies that i,o4in eqn (5.13) must be negativein order to generatea positivereluctancetorque. Fig. 7.1la showsthe phasordiagramsof the motor for i,o4: 0 while Fig. 7.1lb showsthe phasordiagramof the motor at unity power factor. q-4"\rs

q-axls

- aL,riu,

(b)

(a)

Figure 7.1L Phasor diagrams of and overexcited AFPM motor: (a) for i."a : 0; (b) at unity power factor.

The inductancesin the d and q-axis windings are expressedby eqns(5.17) and the magnetflux linkage T/1is further expressedas

4't

:

L"aif

(7.13)

where Lo4is the armaturereaction(mutual) inductancein the d-axis and zy is a fictitious magnetisingcurrent representingthe PM excitation. The flux linkage and mutual inductance between the d-axis winding and the fictitious permanentmagnet (field) winding need further explanation. For

228


the d-axis winding we have the following equation(ignoring crosscoupling effectswith the q-axis winding and ignoring any statorslottingeffects):

dr!a dt

,

dio,l , , dlt t Lfd , dl "'dl

:

L s"d-- - - l l -

:

( L r I L o a ) 4 : , 0* t , o * dt

(7.14)

d,r

where Loa is given by eqn (2.109) and L1 is a leakageinductance.17 is the fictitious magnet(field) winding current. L y4isthe mutual inductancebetween the d-axis winding and the fictitious magnet (field) winding given by r .*k,1NyQa1 _ - k,rN1Ola "JdI* t,

(7.1 s)

where k-f NJ and fo.rl/r are the effective turns in series of the fictitious magnet-field and d-axis windings respectively. o4,r and @yaare the proportions of flux producedby the d*axis and magnet-fieldwindings respectively that link with the oppositewinding. Note that in this caseo/d is equalto oy of eqn (2.19). From eqn (7.15) the flux linkage of the d*axis winding due to the field winding, Ty';,is given by $f : k-t l/rOfa : Lyaly

( 7 . 16 )

However,this flux linkageis alsoexpressedby eqn (7. l3) and is representedin the d-axis equivalentcircuit of Fig. 7.12b. Hence,from eqns(i .13) and(7.16i) L,, ir - ,'"II "

(7.171

Lod.

Furthermore,the d-axis inductanceLo4 is given by eqn(2.109)as

r. , -*

k.*1N1Qu,1

ua,o

Ioa

( 7r.8 )

where Qo4 is the d-axis flux passingthe air gap, linking the rotor. The total flux of the d-axis winding, (D4,is given by iDa :

Ol -f Qod,

(7.1e)

whereQ7is the leakageflux. Now, if O;i of eqn (7.15) [this is the proportionof the flux produced by the d-axis winding that links the field windingl is equal

229

Control

to kQo6,i.e. @ay: kQad,where0 < k < 1 (i.e. not all the d-axis flux linking the rotor necessarilylinks the fictitious magnet-field winding), then from eqns (7.15)and(7.l8) it followsthat

Lad:l##fr)',,

(7.20)

and fuilhermore from eqn (7 .17) that . 'r:

lk*1Nykl -

| 1rffiltr

(7.21)

Hence,if the effectiveturns of the d-axis and field windings arethe sameand k : l, then Loa is equal to L ya and i,1 is equal to Iy. Using eqns(5.7), (5.8) and (7.13) and assumingz1 is not varying with time, eqns(5.5) and (5.6) can be written as rtvl - Rfio4 + (Lr * t"o)T

uvn: Rf io, * (Lt + L")*

* u(h

- a(Lt I Loq)ioq

(7.22')

-l aLoaiy I Lo,1)zoa

(7.23)

On the basisof eqns(1.22) and(7.23) the completedq equivalentcircuits of the sinusoidalAFPM machinecan be drawn as shown in Fig. 7.12. It should be noted that the modelling accordingto eqns (7.22) and (1.23) is basedon assumptionsthat (a) thereare no core losses; (b) the magnetic circuit is linear, i.e. there is no saturationof the magnetic circuit; (c) there is no cross magnetisation or mutual coupling between the d- and q-axis circuits; (d) there are no stator slotting effects (smooth stator core).

7.2.2

Current control

The equivalentcircuits of the sinusoidalAFPM machineshown inFig. T.I2 can be usedto designthe current regulatorsfor controlling the d and g-axis currentsof the machine. Applying the principle of speedvoltage decoupling,

230

AXIAL FLUX PERMANENTMAGAIETBRUSHLESS MACHINES

(a) Figure 7.12.

(b)

The d and q-axis equivalent circuits of the sinusoidalAFPM brushlessmachine.

as for the trapezoidalAFPM machine (section7.1.3), the dq culrent control systemof the sinusoidalAFPM machinecan be representedby the block diagram of Fig . 7.13. Here, the d arrd q-axis current regulators act on the enors betweenthe desiredand actualdq currentsof the machine.The speedvoltage terms are added(or subtracted)to the outputs of the regulatorsto determinethe dg machine voltages. Within the machine model the speedvoltages are subtracted (or added) again,hencethe regulatorsact only on the RL*parts of the equivalent circuits. The block diagrams of the decoupled current controllers can thereforebe simplified to those shown in Fig. 7.14. With the equivalent circuit parametersof the machine known, the transfer functions of the d and q-axis cuffent controllers can be determinedand the d and q-axis current regulatorscanbe designedaccordinglyto obtaintherequired dqcurrentresponse.

7.2.3

Speedcontrol

It is always good in terms of efficiency to control the drive at maximum torque per ampere. For the surfacemounted (nonsalient)AFPM machine, whereit canbe takenthat Lsd,: L"n, it is clearfrom eqn (5.l3) that maximum torque per amperefor this machine is obtainedby keeping iod : 0. However, for embedded(salient-pole)AFPM machineswhere L"a * -L"n,maximum torque per ampere is obtained by making io,1negative,that is by controlling the current phasor at some current angle V as shown in Fig. 7.15 (see also Fig. 7.11b). Note from Fig. 7.15 that negativetorque is obtainedby keeping both'ir,1, and 'ion negative, or in the case of the nonsalient machine, only ,io, is negative. In the sub base speedregion (a < ,) of the drive, where there is enough voltage available so that the current regulators do not saturate,the machine is controlled to operateat maximum torque per ampere. For the nonsalient AFPM machine this implies that the current angle V : 0, while for the salient-polemachineiF : Vo (seeFig. 7.15), which is an (average)optimum anglefor all load conditions.

231

Control

dq Cument controller

Lu,l

i +

-t aLri,,

@L,qiq

Figure 7.13. Block diagramofd and q-axis currentcontrol systemofthe AFPM drive irnplementing speedvoltagedecoupling.

R,+ lus

(b) Figure 7.14. Decoupledd and q-axis cuffent control systems

However,in the high speedregion (a > wu),the EMF voltagewLo4i,y : at!y of eqn (7.23)(seealsoFig. 7.11b)becomeslargerthanthe possibleoutput voltage of the inverter. The current regulators therefore start to saturate,

232

AXIAL FLUX PERMANENTMAGNETBRUSHLESS MACHI],{ES

culrent control is lost and the torque of the machine rapidly drops as the speed increases.To avoid this torque decreaseand to ensurea wider high-speedregion one has to deviate from the maximum torque per amperecontrol and start to inject a largernegatived*axis currentto reducethe g*axis terminalvoltage (Fig. 7.1lb) as the speedincreases.This implies that the currentangle v must be increasedas shownin Fig. 7.15,keepingthe currentamplitudeequalor less than the rated curent amplitude dependingon the load. At a maximum current angle,V-, the speedrangecan be extendedmuch more if the currentangleis held constantat tlr : v- and the amplitude of the phasecurrent is reducedas shownin Fig. 7.15. Note from Fig. 7.1lb that as the curent angle rlr increases the apparentpower of the machine decreasesand the power factor improves. The above current control strategy can be used for both salient and nonsalientAFPM machines.It must be noted,however,that the speedrange obtained with the above control method dependsvery much on the value of the stator phase inductance of the machine (see Numerical example 7.2). If the phaseinductanceis relatively small (this implies a low per unit statorphase reactance),as is the casewith slotlessand air-coredAFPM machines,the high speedrangeof the machineis very limited. From above,thus, the current of the machine can be controlled by controlling the magnitude of the current and keeping the current angle constantat an optimum value,and advancingthe currentangleat higher speeds.To simplifz this control method, the current angle \Ir is defined as a positive angle between 0 to 90". The current angle is also taken as a function of the absolute value of the speed,lo'1.Furlhermore,a currenti7 is defined,which controlsthe torque of the machine and is proporlional to the current amplitude of the machine. Hence.

i o,t : iaq

-

l17'lsin tl.t ri7 cos V for

iI/ > 0

('7.24)

Substitutingeqn (7.24) into eqn (5.13) and noting that Lra{L"o, the electromagnetictorque is

Ta - 3ptb f i7 cos(V)+ f,nQ"n - L",t)ir.lirl sirr(2V)

(7.2s)

It can be seenfrom the first term of eqn (7.25) that there is a linear relation betweenthe PM torque componentand the cumenti7 for a given current angle V. The secondterm of eqn(7 .25) is the reluctancecomponentof the torque. It is found for reluctancemachines[32] that the inductancedifferenceA,L : Lsct L"a is not a constant at a ceftain current angle, but varies inversely with load current z7 due to saturation and cross magnetization. The cross

a1a

Control

Figure 7.15. Currentphasorcontrol ofthe AFPM drive in the low and high speedregionsof the machine.

magnetizationor crosscouplingis the magneticcouplingbetweenthe fictitious d and q-axis windings of the machine. It is thus a fairly good approximation to assumefor this componentof the torque a linear relation betweenreluctance torqueand currenti7 for a given cuffent angle V [134]. Thus, eqn(7.25) can be simplified as Ta x kyli,y * k7y.1i7 x k7i1.

fbr

V : consta?Lt

(7.26)

where lryl and k7r"1 zte the PM torque and reluctancetorque constants,respectively, and k7 : krf * kr,d. Note from eqn (7.26) that to generate a negativetorque, ri7 must be negative;V is taken in the control as a positive quantity between0 to 90". ln the casewhere V : 0, then k7r.1 : Q, kr : Kr f : 3prl,r 12 and,i,7 - ias. With the current control strategyin place and with the torque as a function of current as given by eqn (7.26), a speed controller for the AFPM machine drive can be implementedaroundthe currentcontrolleras shown in Fig. 7.16.

234


The block diagramof Fig. 7.16 describesthe basicprinciple of the currentand speed control of the AFPM machine drive. The speed regulator acts on the error betweenthe desiredand measuredspeedof the systemand controlsthe current i7 of eqn (7 .26). The maximum curuentof the machine is limited by the limiter as shownin Fig. 7.16. The currentangleV is, as a positivequantity, a function of the absolutevalue of the speedof the machine,lu.'1.as shown in Fig. 7.16. In the low speedregion the angle IF is kept constant.In the speed region abovethe base speed(a > ai the angle V is increasedwith speed. In the very high speedregion the angle I[ is kept constant again. With the current and curuentangle V known, the desired dq cunents, ,io,1*and ior* , are determinedby eqn (7 .24). With the measuredspeedand measureddq currents as the other inputs, the current controller outputs the necessarydq voltages accordingto the block diagram of Fig. 7.13. Thesevoltagesare transformed to ABC voltagesusing the inversetransformationgiven by eqn (7.11), and are then put through the PWM generatorto control the inverter and hence the terminal voltagesof the machine. It must be notedthat the speedcontrol methodof Fig. 7.16 is only one way of controllingthe speedof the PM brushlessmotor and is recognizedas a basic method. Severalother techniquesand algorithms for wide-range speedcontrol ofPM motor driveshavebeenproposed122,53,126, 152,164,249Jand can be appliedto the speedcontrol of the sinusoidalAFPM brushlessmotor drive. The important aspectis the preventionof saturationof the dq cunerftregulators in the high speedregion of the motor. To designthe speedregulatorof the systemof Fig. 7.16 a few approximations can be made to simplifz the model of the speedcontrol drive. Firstly, the mechanicaltime constantof drive systemsis in general much longer than the electrical time constant of the drive so that the latter can be ignored. Togetherwith this, a good approximationis to assumethat the currentcontrollers are forcing the actual currents equal to the command (desired) curents, i.e. iod": iod*, ioq : ioq*, andiy : i,r*. Applying further eqn (7.26),the speed control systemcanbe simplifiedto that shownin Fig. 7.17. From the simplified control system the approximate transfer function of the system can be determined and the speed regulator can be designedto obtain the required speed response.

7.2.4

Hardware of sinusoidal AFPM machine drive

For the sinusoidalAFPM brushlessmotor the samesolid-stateconverteras for the trapezoidalAFPM brushlessmotor can be used(Fig. 7.4). A more detailed diagram of the hardwareof the sinusoidalAFPM brushlessdrive system is shown in Fig. 7.18. The fixed a.c. supply voltage is rectified by meansof a diode rectifier or voltage-sourcecontrolled(active)PWM rectifier to obtain a fixed d.c. voltage. The d.c. voltage is then invertedby meansof a voltage-

235

Control

LR

s;

.i:

ili fll

-f^ ia

>:-

236

AXIAL FLUX PERMAI,IENT MAGNETBRUSHLESS MACHINES

Figure 7.17. Biock diagram of simplified speedcontrol system.

source PWM inverter to a three-phasea.c. voltage of variable frequency and magnitude. The difference between the trapezoidal and sinusoidal drives lies in the PWM switching-control of the inverler bridge. In the caseof the trapezoidal AFPM motor drive, quasi-squarePWM voltages are applied, whereas in the case of the sinusoidalAFPM motor drive, sinusoidalPWM voltages are applied and the phase currents of the stator are sinusoidally regulated as describedin the previoussections. The peak value of the fundamental frequency voltage component of the invefter-legsof Fig. 7.18 with respectto the neutralpoint l/ is clearly jV6. ns the fundamental frequency voltage componentsof the inverter-legs are 120" out of phase,the maximlrm rrns line-to-line voltage of the inverter at the fundamentalfrequency with sinusoidal PWM is VrL-Lô"

:--

'/5vo: 0 . 6 1 2 V a Jt2

(7.27)

This result showsthe limitation on the maximum rms outputvoltageof the inverter. This affects,amongstother things, the maximum speedat which rated torque can be delivered by the drive (seeNumerical example 7.2). The currentsin the threephasesof the AFPM machine drive are sinusoidally regulated. There is, however, a ripple in the phase currents due to the PWM switching. It can be shown that the peak-to-peak ripple in the phase curent, L,'ip-p, is directly proportionalto

Lir-, -

=Vo,

t rhls

('7.28)

where /" is the switching frequencyof the inverter and Ly,nthetotal load inductanceper phaseofthe inverter. The ripple current is thus inverselyproportional to Lon and,f ,. Note that Lph can be increasedby the addition of an extemal inductor in serieswith the stator (armature) of the AFPM machine as shown

af?

Control

LJI

in Fig. 7.18. Careful considerationshouldbe given to the selectionof Lrt" and As the air gaps of AFPM machines,specificallythose AFPM machines "f". with corelessand slotlessiron-coredstators,are relatively large, the internal phaseinductancasof thesemachinesare relatively small. Thus to keep the ripple current in thesemachineswithin limits, the switching frequency/" is requiredto be high. With power MOSFETs the switching frequencycan be madehigh (say /" < 50 kHz), but with IGBTs in the medium-to-largepower rangethe switchingfrequencyis very much limited to l, < 10 kHz. In sucha casethe phaseinductancemust be increased.The high ripple currentthat can occur in most of the AFPM machinedrivesis often overlooked.A high ripple currenthas seriousconsequences on the curent rating ofthe inverterand the efficiency,statorwinding temperature,torquequality and acousticnoiseof the AFPM brushlessmotor. For rotor position feedbackof the sinusoidalAFPM drive a position sensor with high resolutionis necessary.Resolversare often usedfor this as they are robustand can be usedin harshenvironments.Integratedand analogcircuitry arenecessaryto interfacethe resolverwith the digital controller(seeFig. 7.18). Digital interfaceoutputsof up to 16 bit resolutionare possible. Normally 12 bit resolutiongivesmore than enoughaccuracyfor position feedback. A11the measuredinformation of voltage, current and rotor position are fed to the digital controller as shown in Fig. 7.18. The controller outputs PWM signalsvia fibre optic links to the inverterbridge. Digital signal processors (DSPs) are nowadaysfrequently used as drive controllers [227]. Examples of dedicatedfixed point DSPs from kxes Instrumentsinclude TMS320F240 (20 MHz), TMS320LF2407A(40 MHz), TMS320F2B12(150 MHz). These DSPshave 12 or 16 PWM channels,16 A/D channels( l0 or l2 bit resolution) and operateat frequenciesof between20 to 150 MHz. ln anotherrecentdevelopmenton drive controllersInternationalRectifierintroducedtheir motion control chipsetIRACO2l0 [99]. This chipsetusesa high speedconfigurable field programmablegate array (FPGA), tightly coupled to gate driver and current senseinterface chips. In this way flexible drive control algorithms are implemented in FPGA hardware,rather than in software,

7.3

position control Sensorless

It is obvious that information of the rotor position of the AFPM brushless motor is absolutelynecessaryfor proper cuffent and speedcontrol. For trapezoidal AFPM motors a low resolution position sensor may be used, but for sinusoidalAFPM motors a high resolutionposition sensor is necessary.The useof a position sensorthat is mechanicallyconnectedto the machineand elecdrive system trically connectedto the controllerreducesthe electromechanical reliability and increasesits cost. Moteover, in some applicationsinsufficient po,sitionconspacemay makethe use of a position sensordifficult. Sensorless

238

AXIAL FLLIX PERMANENTMAGNETBRUSHLESS MACHINES

I vd

{

I

Figure 7.18. Hardwareof sinusoidalAFPM drive system.

trol techntques,in which an accurateinformation of the rotor position can be obtainedwithout the needof a mechanicalsensor,are superior. Variousmethodsand techniqueshavebeenproposedthe last two decadesfor position control of PM brushlessmotors. Most of thesemethods the sensorless are based on the measurementof the voltage andlor current of the machine. Thus, insteadof using a mechanicalsensor,electricalsensors(not mountedin the machine but placed at the solid-stateconverter)are usedto obtain the rotor position signal. ln this way, the position of the rotor is determinedindirectly, as measuredelectricalquantitiestogetherwith the model of the machineare used. Stateobservers,filters and other calculationsare used to estimatethe mechanicalposition of the rotor. The methodsof sensorless controlcanbe classifiedinto two types: (i) methods that arebasedon the back EMF estimationl2l, 82,242f, and(ii) methods that makeuseof the magneticsaliencypresentin the machinell27 , 154, 155, 209]. The EMF measurementis not suitablefor low andzero speedoperation of the motor as the induced voltage is proportional to the rotor speed.At stand still, the induced voltage is zero and there is no position-information anymore in the stator quantities. The secondmethod, which makes use of the magnetic saliency,is more suitablefor the detectionof the rotor position at low speedsand standstill. With this method either test voltagepulsesare applied to the machine or additional high-frequency signals superimposedon the fundamental voltages (or cunents) are injected. The injection of the revolving high-frequencyvoltage vector is commonly used. All thesemethodscan be used successfullyfor the position sensorlesscontrol of AFPM motor drives

239

Control

[187]. However,it is still questionableif the high-frequencyinjection sensorlesscontrol methodcan be appliedto corelesstype AFPM brushlessmotors as thesemachineshavepracticallyno magneticsaliency. Fig.7 .19 explainsthe basicprinciple of the highfrequencyvoltageinjection sensorlessposition control technique. High frequency voltage signals are injected to the systemby adding them to the dq voltage signalsu14 and u1,7of the drive. The currentresponseof the drive systemto thesevoltage signalsis monitoredby bandpassfiltering (BPF) the dq cut:rents,so asto obtainthe high frequency current componentsi,o41,and ion7r. These high frequency current componentscontain information about the rotor position as they are affected by the magnetic saliencyof the machine. These currents,togetherwith the high frequencyinjected signalsare usedby an observeror position estimator to estimatethe rotor position 0^. A low passfilter (LPF) is furthermore usedto extract from the dq currents the fundamental current componentsios and ion, which are necessaryfor the fundamental current controller of the drive (Fig. 7.13). The frequencyof the injectedvoltagesignalsmust be much higherthan the fundamentalfrequency,but also much lower than the switching frequency of the inverter,and is typically between500 Hz and2k}{z.

High liequency vof tag on -j ",rr1""t rotor position cstirnaior

positioncontrol. voltageinjectionsensoriess Fipre 7.19. Highfrequency

Numerical example7.1 A small, 35-W 4300 r/min, 12 pole, Y-connectedair-cored,trapezoidal AFPM brushlessmotor is usedas part of a reactionwheel for a micro-satellite (seeFig. 7.20). Reactionor momentumwheels are used on satellitesto keep the satellitesteadyand to orientateit. Orientationcontrol is necessaryas, e.g. the photovoltaicpanelsmust be pointed at all times to the sun for maximum

240

AXIAL FLUX PERMANENTMAGI{ETBRUSHLESS MACHINES

power generation.The motor has a statorphaseinductanceof -L" : 3.2 Lr,Hand a phaseresistanceof fi1 : 22 rn0. The motor is fed by a 3-phaseMOSFET inverter withVa: 14 V and is under square-wavecufrent control as described in Section7.1.3. Assumingthe voltagedrop acrossa MOSFET switch during switching-onas I V, calculate: (a) the switching frequency /" of the invefter; (b) the extemal phaseinductancerequired for a maximum peak-to*peak ripple currentlessthan 0.5 A if f" :40kHz.

Figure 7.20. Small corelessAFPM brushlessmotor. Photo courtesy of the Lrniversib,of Stellenbosch, South Africa. Numerical example 7.l.

Solution First, an approximate equation for the maximum peak-to-peak phase current ripple of the motor must be derived. For the circuit shown in Fig. 7.5 during switching-on,i.e. with the transistorsfl, and Q switchedon, assuming ideal transistorswitchesand assumingthe drive is in the steady-state, the voltageequation(7.l) canbe written as

v4 :

- E.rt L RrIu- R,pi,+ tod',i d,t

(7.2e)

where the phasecurrent 'io : Io + ir, Io is the steady-stateaveragecurrent and z, is the ripple current.Putting eqn (7.3) and neglectingthe voltage dropRri,, eqn (7.29) can be brought to di, Va - DVa rt , "P ,rt

(7.30)

241

Control From eqns (7.4) and (7.30) the peak to-peak ripple current Ai, is

n;' _Va(l

Aa,

-

- D)for, - D)D _Va(l

Lo

LrI"

(7.31)

The maximum peak-to-peak ripple curent will occur at a duty cycle of D 0.5, hence

Li,ô, = J!-

(7.32)

4LPJ "

(a) Switching frequency.Using eqn (7 .32) and taking into accountthe voltage drop acrossthe transistor switches,the switching frequencyfor a maximum ripple currentof 0.5 A is

w -2

"

+N Js -

;-; *4L pl\zrmox

t4-2

-

4.x(2x3.2x

10 6) x0.5

: 937.5 k}fz

(b) Externalinductance.Eqn(7.32) givesthe following additionalphaseinductance required to keep the ripple current lessthan 0.5 A at f u : 40 kHz: _1

Lph(atid)

Va-2

x ,

4f ,Li,,,o"

f_

14-2

4 x 40 000x 0.5

-3.2x10-o : 71.8pH (7.33)

Theseresults show the low phase inductanceproblem of corelessAFPM brushlessmotors when fed from solid-stateinvefters. Either the switching frequency of the inverter must be high or relatively large extemal inductors must be addedin serieswith the phasewindings of the machineto keep the ripple currentwithin limits. Usually,externalinductorsare used,as a high switching frequencycausesthe power lossesin the inverterto be high.

Numericalexample7.2 Considera 75-kW,460-V,105-A,Y-connected, 700-H2,1500r/min, 8-pole AFPM motor with single PM rotor disc and double slottedstatorsconnected in series. The equivalentcircuit parametersand flux linkage of this machine are:R1: 0.034d), L"d - Lsq : 2.13 mH,'$f : 0.538Wb. The machineis under dq cunent control by using, amongst other things, a three-phaseIGBT voltage-sourceinverterwith d.c.-link voltageVa - 755 V. Find: (a) the maximum speedand the developedpower of the drive with the machine at rated current and the machinecontrolled at maximum torque per ampere;

1Aa

AXIAL FLUX PERMANENT MAGNET BRUSHLE SSMACHINES

LAL

(b) the per unit high speed of the drive developing the same power as in (a) with the drive at rated current, rated voltage and using an advancedcuffent angle.

Solution (a) Maximum speedand electromagneticpower With a surface-mountedAFPM machine maximum torque per ampere is obtainedwith i,oa: 0 A. Thus, at ratedstatorcurrentioq : t/2 x 105 : 148.5 A. To obtain maximum speedthe output voltage of the invefier must be at a maximum,i.e. from eqn(1.27)thermsphasevoltageVl :0.612x 7551\/3 : 266.8 V. Thus, the peak phasevoltage, V1,, : 1/1 x 266.8 : 377.3 V and T note that Vr^ : tluu2 I ,to'. By consideringthe drive in the steadystate v

and ignoring the resistive voltage drop, from eqn (7.22), eqn (7.23) and Fig. 7.11we have : Vrrnsin d = -a L"nion u1-d"

(1.34)

(7.3s)

u1n: Vlpcos6 x wtf;y From this .

ubot

Land =

tlo

'v.f

-2.13x10*3x148.5 0.538

and d = -30.45"

Using eqn (7 .34) the angular frequencyis V1- sirrd w-_

-

Lsqx,q

3 7 7 . 2v s i n ( - 3 0 . 4 5 )

: - 2 . 7 3x 1 0 - ' x 1 4 8 . 5 604.5racl/s

and the mechanical speedO : S\alptr : 7443 rpm. The maximum speed with maximum torque per amperecontrol is thus just less than the rated speed of 1500r/min of the machine.Using eqn (7.34), an approximateequationfor the steady-stateelectromagneticpower (7 .I2) can be found as follows:

P,r* era Iu1 o;"n=irW)sind .) Z

x sin(-30.45)r 72.45kW

(7.36)

243 which is alsojust lessthan the ratedpower of 75 kW of the machine. (b) Per unit high speed The speedof the AFPM brushlessmotor canbe increasedbeyondbasespeed by increasingthe current angle !I, as shown in Fig. 7.1l(b), keepingthe peak value of the sinusoidalphase current Io : consto,nt. Note that lorn : V

f. ).. t Lad' + Laq'.

At ratedconditionslorn :148.5 A andVlr, : 377.i|V. The samepower as in (a), i.e. 72.45kW must be obtained.Hence,it is clear from eqn (.7.36)that d will be the sameas in (a), i.e. d : -30.45". With the machineconsideredas 100%efficient,the electromagneticpower is equalto the input power,i.e. P,L,nx

|Vrn,In- "or

6

where cos@!

72 450

: 0.862

(3lz)x377.3x148.5

and

Q x 30.45" (leading)

Theq-axis currenti:oncannowbe calculated as ioq: Io,ncos(ldl + 4) * cos(30.45 + 30.45): 72.2A Usingeqn(7.35)thatis still valid,theangularfrequency is 7r- sin d

,r aY -

- L sqlaq

377.3x sin(*30.45) : - 2 . 7 3 x 1 0 - 3x 7 2 . 2 1243radls

and the mechanicalspeedO : S\alptr : 2967 rpm. Taking the mechanical speedof 1443 rpm in (a) as the basespeed,the per unit speedts 297211443: 2.0 p.u. Hence,the samepower can be developedby the drive at two times basespeedby advancingthe currentangle V frorn zero to almost 610. It must be notedthat this increasein speedis very much dependenton the q-axis synchronousinductanceLrn as can be seenclearly from all the equationsused above. If .L"nbecomesrelatively smal1,the speedincreasewill be much less. The per unit phasereactanceof the aboveiron-cored,slottedAFPM machine is r" - 0.53 p.u. For slotlessand corelessAFPM machinesthe per unit phase reactanceis much less(typically 0.1 p.u.) and the speedincreasewill be little.

Numerical example7.3 Simulatethe dq andABC currentresponseofthe drive systemof l{umerical example7.2. Assumethat

244


.t fU

I IJ

R

;; i1" ,"1, 1 r t-

r'-l'

fi

\

+ *-?

k

4^ .E6

BT Od

or

s

Control

245

(a) the speedis constantat 1000 rlmin1' (b) the drive is operating at maximum torque per ampere; (c) the switchingfrequencyof the inverteris 1.5kHz; (d) the dq currentregulatorsconsistonly of a proportional gainK":

3 ViA.

Solution Various software, such as PSpiceand Simplorer, can be used for simulating converler-fedelectricalmachinedrives.Assumingthe switchingofthe inverter asideal,Matlab-Simulinkhasbeenusedin this exampleto simulatethe curent responseof the drive.

200 150 100 50

g 0 o

-50 -100

-200 Time (s)

(a) 200 175 150 125 100 I

75

50

d-axis

E

-

25 0 0 Y 25+

0.005

.50 l

(b) Figure 7.22. Simulated current response for step input in the q-axis current ioo: (a) phase currentsi.A,iatj andi,,c:; (b) dq currentsiaa andi,,r. Numericalexample7.1.

246


A block diagramof the simulationset-upis shown in Fig. 7.21. The simulation is done as "continuous",i.e. without any samplingand so forlh like in a digital control system.The currentcontrol is done inthe dq referenceframe using the currentcontrollerof Fig. 7.13. For both the d and q-axis regulators of Fig. 7.13 a gainK" is usedas shownin Fig. 7.21. The decouplingis doneas shownin Fig. 7.13 with the actualdq currentsand speedas inputs.For simulation of the invefter,the drl voltagesaretransformedtoABC voltagesby using eqn (7.11). To generatethe PWM phasevoltages,the ABC vollagesare comparedwith a 1.5kHz trianglewave (amplitudeof Val2), by using a subtractor (sum) and comparator(relay). The relay outputs+V,t 12. The generatedPWM voltages,uABCN,are with respectto the d.c.-link midpoint l/ of the invefier (seeFig. 7.18), and haveto be convertedto true phasevoltages,u7165r",, with respectto the floating neutralpoint n, of the machineas follows:

uAn: nlln

:

lJc,n -

]{rror '5{rr"* t5lrr" **

uBN

ucN)

uC,^/- uA-^/)

(7.37)

u/.^\r- uB,^r)

The next stepis to either solvethe machineequationsinthe ABC or in the dq referenceframe. Using the dq referenceframe, the true PWM phasevoltages must be convertedback to dq voltagesas shown in Fig. 7.21. Using the machine model of Fig. 7.13, the 'i6,4an:dz,rncurrentscan be determinedand fed back to be comparedwith the input command currents as shown. To obtain information of the actual phasecurrents,the ioy andio,ncunents are transformed to ABC currents. The whole block diagram of Fig. 7.21 can be set up in Matlab-Simulink using the Simulink blocks. For the machine model, eqns(7.22) and (7.23) can directly be set-upin Simulink using integrators.To simulatethe inverter switching,the relay block can be used.Al1 the other sourceand mathematical function blocks are availablein Simulink. The simulatedresultscan be output in various ways, in this caseto the Matlab workspace. All the machine parametersare known from Numerical example 7.2. The mechanicalspeedis taken as constantat 4I9 radls (1000 rpm). For maximum torque per amperethe dq commandcurrents'iod.*: 0 and'ioo* : 148.5 A are given as step inputs to the system (Fig. 7.21). The d,q and ABC ct;rrerf. responsesof the drive for a step input in i,or* are shown in Fig. 7.22. The effect of the switching of the inverter on the current waveforms is clear. In this case,the q-axis cur:rentbuilds up to the rated cumentin lessthan 5 ms. This dependsamongstotherson the value of the proporlional gain I{". The effect

Control

247

of the decouplingon the currentresponsecan be self-studiedby removing the decouplingsignalsin the simulation.

Chapter8 COOLING AND HEAT TRANSFER

8.1

Importance of thermal analysis

During the operationof an electricalmachine,heatis generateddueto power lossesin electricand magneticcircuits and mechanical(rotational)losses.To ensurea long operationallife for the machine,theselossesmust be removedas far aspossiblefrom the machineso that the temperaturelimitationsestablished for the machinematerials,suchas insulatingmaterials,lubricantsand pMs are complied with. In addition to the considerationof the machine'soperational life, a lower operatingtemperaturereducesextra winding lossesintroducedby the temperaturecoefficientof the electricresistance- eqn (3.47). whereas extensiveresearchhas beendevotedto the thermal studiesof conventionalelectricalmachines,AFPM machineshavereceivedvery little attention f 118,279,231,2321. Owing to the fact that AFPM machinespossessa relatively large air gap volume and quite often have multi-gaps, the general perception is that AFPM machineshave better ventilation capacitythan their radial field counterparts[48, 96]. Sincethe externaldiameterincreasesratherslowly with the increaseof output power,i.e. Do,,1x iE""t 196],the existingheatdissipationcapacity may be insufficient to cope with excessiveheat at certain power ratings, so that more effectivemeansof cooling haveto be enforced.Thus, quantitativestudies of the heat dissipationpotential of AFPM machineswith vastly different topologiesis important.

8.2

lleat transfer modes

Heat transfer is a complex phenomenonpresenting formidable analytical difficulties. Heat is removed from an electrical machine by a combination

250

MACHINES AXIAL FLUX PERMANENTMAGNETBRLTSHLESS

of condttction, radiation and convectionprocessesto the ambient air and surroundings.

8.2.1

Conduction

When a temperaturegradient exists in a solid body, such as in the copper, steel,PMs or the insulationof an electricalmachine,heat is transferredfrom the hightemperature region of temperature86o7to the low-temperatureregion of temperature8"o6 accordingto Fourier's law,which is given as:

LP": -nou#:

- 8"om) (onot

Y

(8.1)

where AP" is the rate of heat conduction,A is the areaof the flow path, I is the length of the flow path and ,k is the thermal conductivity of the material. The latter is experimentally determinedand is relatively insensitiveto temperature changes.Thermalpropertiesof typical materialsusedfor AFPM machinesare givenin TableB.l .

Table 8.1.

Selectedthermal properties of materials Grade

Material

(20"c)

p

c.p

k

Kg/m"

Ji(kg "C)

W(m "C) 0.0267

1.1'77

Water

1000

1005 4184

Mica

3000

813

Epoxy resin

1400

1700

0.5

8950

380

360

903

z)l

Alloy (cast)

2700 2790

1% Carbon Silicon

Air

Copper Pure

Aluminum Steel

Permanentmagnet

8.2.2

SinteredNdFeB

0.63 0.33

168

7850

883 450

7700

490

20-30

7600to 7"100

420

9

52

Radiation

The net radiant energyinterchangebetweentwo surfaceswith a temperature difference is a function of the absolute temperature, the emissivity and the geometry of each surface. If heat is transferredby radiation between two gray surfaces of finite size, 41 and 42, and temperature, t?1 and tSz (in Celsius degree),the rate of heat transfer,A,Pr, may be written as

Cooling and Heat Transfer

^ D 4rr '

251 ( t 9 , + )L 7t a?tt 1

( ,9.,

_\vt -u-

I

€l

\u2-.to) I

)7'l\4

!-3 l-'.

(8.2)

.tAt'AtFtz'ezAz

where o is the stefan-Boltzmann constant,F12is the shapefactor which takes into accountthe relative orientation of the two surfacesand e1 and,e2 are their respectiveemissivitieswhich dependon the surfacesandtheir treatment.Some selectedemissivitiesrelatedto AFPM machinesare siven in Tableg.2.

Table8.2.

Selectedemissivitiesrelevantto AFPM machines

Material

Surfacecondition

Ernissivity,a

Copper

Polished

0.025

Epoxy

Black

0.87

White Mild steel Cast iron

Oxidized

Stainlesssteel

0.57 0.2-0.7

Permanentmagnet, NdFeB

8.2.3

0 .u 5 0.2-0.3

Uncoated

0.9

Convection

convection is the term describingheat transferfrom a surfaceto a moving fluid. The rateof convectiveheattransfer,APr, is given accordingtoNewton's law of cooling as: LP,, :

hA(9 7,o1- 8,r,rd,)

(8.3)

where /z is the convectionheat transfercoefficient,which is a rather complex function of the surfacefinish and orientation,fluid properties,velocity and temperature,and is usually experimentallydetermined.The coefficienth increaseswith the velocity of the cooling medium relativeto the cooledsurface. For a surfacewith forced ventilation,the following empirical relation may be used[147],i.e. hy : h,"(l * r:6.y61

(8.4)

where h'y and.h,,, arethe coefficientsof heat transfer for the forced and natural convectionrespectively,t' is the linear velocity of cooling medium and cp.x 0.5 to 1.3is an empiricalcoefficient.

252

AXIAL T'LUXPERMANENTMAGNETBRUSHLESS MACHINES

Someimportant formulae for evaluatingconvectiveheattransfercoefficients of AFPM machinesare discussedin the followins sections. Convection heat transfer in disc systems The rotating disc system plays a major role in the cooling and ventilation of the AFPM machine. Accurately determining the convection heat transfer coefficientsneedsthorough theoretical and experimentalinvestigationbecause of the complexityof ffow regimes. In this section the convection heat transfer coefficients in different parts of the AFPM machine are evaluated,exploiting a number of existing models.

turbulent

laminar

/

laminar

(b)

(a) Figure 8.1. flow

Free rotating disc: (a) in laminar flow, (b) transition from laminar to turbulent

Freerotatingdisc The averageheat transfer coefficient at the outside surfaceof a rotating disc may be evaluatedusing the formula developedfor a free rotating disc [241], i.e. kh-:Nu

(8.s)

where -R is the radius of the disc and the averageNusselt number ly'u is given accordingto the differentflow conditionsas follows: (i) For combinedeffectsof free convectionand rotation in laminar flow (Fig. 8.ta) [241]

253


A/u:

) ^ l

r{n".*Grla

,_r, u2

(8.6)

(8.7)

where fie is the Reynoldsnumberaccordingto eqn (2.65), p is the coemcient of thermal expansion,z is the kinematic viscosity of the fluid and Atl is the temperaturedifference betweenthe disc surfaceand surrounding air. (ii) For a combination of laminar and turbulent flow with the transition at a radiusr" (Fig. 8.lb) [241]

l y ' u: 0 . 0 l 5 R e" s- t 00(!!P .R,

(8.8)

r.:(2.5x105u1Q)r/2

(8.e)

where

The angularspeedQ : 2trn when n is the rotationalspeedin rev/s. It is instructiveto comparethe heat transfer capabilitiesbetweena rotation disc and a stationarydisc. If we considera steeldisc, which has a diameterof 0.4 m and rotatesat 1260rpm., the convectionheattransfercoefficientmay be calculated as 41 W/(m' "C), which is aboutten timesthat of the samedisc at standstill. Altematively, one can say that the effectiveheat dissipationareaof the same disc can be increasedby a factor of 10 when the disc rotatesat the specific speed. Rotor radial peripheral edge The heat transferconelationsfor the radial peripheryof the rotor disc are similar to those of a rotating cylinder in air. In this case the averageheat transfercoefficientis givenas

he: (.kfD",)t{u

( 8 . 10 )

where Do6 is the outer diameterof the rotor disc, the averageNusseltnumber is given by ,) /a

1 vu : 0 . 1 3 3R e T rP r t / 3

(8.il)

254

AXIAL FLUX PERMANEI{TMAGNETBRUSHLESS MACHIAIES

andReynoldsnumberat the discperipheryis Re71: QDf;,rlu

(8.12)

Note that a uniform temperaturedistribution in the cylinder is normally assumedwhen eqn (8.10) is used. SinceDo is proportionalto the angularspeed 0, it may be concludedthat the rotor periphery plays an increasinglyimportant role in the heatdissipationas f,) increases. Rotor-statorsystem As seenin Fig. 8.2, an AFPM machineconsistsof a numberof rotatingand stationarydiscs. The heat transfer relations betweena rotating and a stationary disc are of paramountimportance in the thermal calculations. Due to centrifugal effects,there is a forced flow betweenthe two discs,which increasesthe local heat transfer rate comparedwith that of a free disc. The relative increase will dependon the gap ratio, G : g I R, where g is the clearancebetweenthe rotor and the stator and -R is the radius of the disc, the mass flow rate and the rotationalspeedof the system[91]. Having radial channelsand thick impellers,an air-cooledAFPM machine may be regardedas a poorly designedfan from a fluid flow perspective. Its tangentialvelocity componentis much larger than the radial component. Thus, the heat transfer rate near the rotating disc shows more dependenceon the rotationalReynoldsnumber, Rer, givenby eqn (2.65). Owen [89] provided an approximatesolution for the flow betweena rotating and a stationary disc, which relates the averageNusselt number to the moment coefficientof the stator-siderotor face, Cr.o, by the following equation:

l/z:

Re,C,nofn

' C n , o R e ' r ': 0 . 3 3 3 ) r where ,\7 is a twrbulence pqrameter given as a rate, Q, as follows \r

:

A-4 :;.Rer

s

( 8 .l 3 )

functionof volumetricflow

(8.14)

By replacing )7 in eqn (8.13) with eqn (8.14), the averageNusselt number becomes


255

n ly'tr : 0.333 * ruR,

(8.1 5)

As discussed in [190], it hasbeenshownthat for a smailgap ratio (G < 0.1) the flow in the air-gap spacebetween the rotor and stator can be treated as a boundary layer. Whilst it is not absolutelytrue that the convective heattransfer coefficientfrom the statorto the air flow is close to that of the air flow to a rotatingdisc, the sameheattransfercoefficientmay be assumedin the thermal circuit simulation.

8.3

Cooling of AFpM machines

Dependingon the size of the machineand the type of enclosures, different arrangementsfor cooling may be used. From a cooling perspective, AFPM machinesmay be classifiedinto two categoriesas follows: t machineswith se(-ventilation, in which cooling air is generatedby a rotating disc, PM channelsor other fan-alike devicesincorporated with the rotatingpart of the machine,and t machineswitlt external ventilation, in which the cooling medium is circu_ latedwith the aid of externaldevices,for an example,a fan or a pump.

8.3.1

AFPM machineswith self-ventilation

The majority of AFpM machinesare air-cooled. compared with conventional electrical machines,a pafiicularly advantageousieature of disc-type AFPM machinesfrom a cooling perspectiveis that they possessinherent selfventilalion capability. Fig. 8.2 showsthe layout and active components of a typical AFPM machine.A closeexaminationof the machinestructure reveals that an air streamwill be drawn through the air inlet holes into the machine and then forced outwards into the radial channel as the rotor discs rotate. .fhe PMs in fact actas impeller blades.The fluid behaviourof the AFpM machine is much like that of a centrifugalfan or compressor. The ideal radial channel According to the theory of an ideal impeller,a numberof assumptions have to be madeto establishthe one-dimensionalmodel of the ideal radial channel [7 1, 207]: (a) thereare no tangentialcomponentsin the flow throughthe channel; (b) the velocity variationacrossthe width or depthof the channel is zero; (c) the inlet flow is radial, which means that air entersthe impeller without ore-whirl:

256


2 Figure 8.2. Exploded view of an AFPM machine: 1 PM, and 4 - epoxy core.

rotor disc, 2 -

statorwinding, 3

(d) the pressureacrossthe bladescan be replacedby tangentialforces acting on the fluid; (e) the flow is treated as incompressibleand frictionless. Figure 8.3 showsa radial channelwith the velocity triangles drawn at the inlet and the outlet. It can be observedthat the pressuresat the inlet p1 and the outlet p2, and friction tri, make no contribution to the sum of the momentum, D Mo. If gravity is ignored, the generalrepresentationof conselation of momentum takes the following forrn 12371: z) J

Arr l 1 "o = ; L / Q i x ul r' ) p d V I lt | ( r ' * i ) p l u ' . r i ) d A dt'J.u'

(8.16)

J,."

where r- is the position vector from 0 to the elemental control volume dV and u-is the velocity of the element. For steady-state,one-dimensionalair flowing betweenthe entranceand exit of the channel,eqn (8.16) may be simplified as: : To : (Eo,t x u'z)mz- (En, x u*1)rn1 Y, Un

(8.17)

where rn2 : rnt : pQ, ut : QRm andu2 : ARout.The input shaft power P;r, is then given by:

hn: ToO: pee2(R7,,_ Ri")

( 8 . 18 )


257

Re-arrangingthe aboveequationgives:

( 8.1 e) Basedon the principle of conservationof energy,the input shaftpower may be given as:

:,h(P-?# . Pin,

ry

* zz- zr* r;z- u)

(8.20)

If the potential (.22- zl) and internal energy (uz - L[) (friction) are ignored, eqn (8.20) may be written in the sameunits as eqn (g.19) as:

Pnn-,^ u||-ul * \ r p-'n- tU't Ptl g

(8.21)

If equations(8.19) and (8.21) are equatedand noting that w1 : ef A1 and w2 : Q f 42, whereAy and42 arc the cross-section areasof the inlet and outlet of the channelrespectively,the pressuredifferenceAp betweenthe entrance and exit of the radial channel(shownin Fig. 8.3) may be expressedas:

258

AXIAL FLUX PERMANENTMAGAIETBRUSHLESS MACHINES

- R?,, Lp - pz- pt : pQ2(R3,, |rh

#rrr,

(8.22)

Eqn (8.22) may be termed the ideal equation describing the air flow through the radial channel.

,.

S]

'..i

I{elativeetlciy

->c\ .j,tz\-)

;71i':" ' , i ,' ' i

..

t

'l

,

r, '/-;;

i

r'f.', '"- ";/r' '-. ''il

il

"..

i;..''.';.ii i,;-t-':'4 Figure 8.4.

The relative eddy in the PM channel.

The actual radial channel The actual characteristicsof a hydraulic machine differ from the ideal case owing to two reasons:(i) the unevenspatial distribution of velocities in the blade passages,and (ii) the leakageand recirculationof flow and hydraulic lossessuchas friction and shock losses.Theseare completelydifferent issues lTll and shall be dealt with separately. Slip factor As a resultof the unbalancedvelocity distributionof the leadingand trailing edgesofa PM channeland the rotationeffects[2A7l,thereexists,accordingto Stodola1182,2471,a relativeeddy within the bladepassageshownin Fig. 8.3. This resultsin the reductionof the tangentialvelocity componentsand is called s/rp, which is usually accounted for using a slip factor. For approximately radial blades,theStanitzslip factor k" (B0o < l3z < 90") is k":1-

0.63tr1n1,

(8.23)

259


where B2 is the blade angle at exit and n6 is the number of the blades. When applying a slip factor,the pressurerelation (8.22) becomes

- R?,) (k"RZ., Lp: pe2 . irh -

(8.24)

leakageflow Q/

Total flow

Figure 8,5.

hrn'

Leakage flow in an AFPM machine (not to scale)

Shock, leakageand friction Energylossesdue to friction, separationofthe boundarylayer (shockloss) and leakageshould also be consideredin the flow analysis. As illustratedin Fig. 8.5, if the total volumetric flow rate through the PM channelis Q1, the pressuredifference betweenthe PM exit and the entrancewill causea leakage or recirculation of a volume of fluid Q1,thus reducing the ffow rate at outlet to Q : Qt-Q1. The Ql is a functionof massflow rateand dischargeand leakage path resistances.The leakageflow reachesits maximum when the main outlet flow is shut. Theselossescan be accountedfor by introducinga pressureloss term A,p1in eqn (8.24) as follows 12071

L p * p Q 2 ( k , R 3 ,- , R ? , )- r

p,I ;t \^ 1 , 1 2

-

hrn'

* Lpt

(82s)

260

AXTALFLUXPERMANENT MAG]NET BRUSHLESS MACHINES

Systemlosses As theair passes throughtheAFPM machine,thesystempressure lossdue to frictionmustbe takeninto account.Thesumof theselossesis eivenbv: ^/-r2

rL

L

Lpr,: +>,+

(8.26)

4.,fl;

z:l

where ki and A,; are the loss coefficient and the cross section area of the flow path e respectively. D1

@

Abrupt expansion

-1 _:

--1 -_l

@

u

Tapering enlargement

t

,_)

Elbow

\

t\

\

+

Roundbend

*]: /;\

[= Short tube

vl

aflI

LJ

Pipe to annulus bend

Figure8.6. Systemlossesof AFPM machine.

There are a number of sectionsthroughwhich the air flows in the AFpM machine (seeFig. 8.6). They are: ( I ) entry into the rotor air inlet holes; (2) passagethrough rotation short tube; (3) bendingof air from pipe to annulus(90o); (4) round bend air passage(90');

CoolingandHeat Transfer

261

(5) contractionofair in the taperingpassage; (6) bending of air through the 90o elbow; (7) entry to the permanentmagnet channels; (8) expansionof air throughthe taperingenlargement; (9) abruptexpansionofthe air on the exit ofthe channel; (10) expansionas the air leavesthe openingofthe parallelrotor discs. The loss coeffrcientsassociatedwith each sectionin eqn (8.26) are given in 157, 173, 182]. When the sectionis not circular,use is made of the hydraulic diameterto characterisethe crosssection. The hydraulic diameteris defined as Dn : 4Alp where -,4is the cross-sectionalareaof the flow path and p is the wetted perimeter. The loss coefficientfor a pipe is given by ),L ld where ,\ is a friction factor obtained as a function of Reynolds number R,e and surfaceroughnessfrom a Moody diagram [ 175]. To facilitate numeric calculations,the Moody diagram may be represented by [57]:

^

:

B {(B /8 e )1+2(x + r;-8 1 -!

X

:

s .zr11n} - 1} 16 { 2 .4 5 7rn {(7 l R e )o+o

Y:

(8.27)

{37ffirJlRe}1(,

^n- r-\ where Re : 41f and where7 is the equivalentsandgrain roughness[57].

Characteristics It is now possibleto relatethe theoreticalpredictionobtainedfrom the ideal flow model to the actual characteristicby accountingfor the various losses discussedabove. Assuming that the AFPM machine (shown in Fig. 8.1) operatesat a constant speedof 1200 rpm, the ideal developedpressurecharacteristicfor a radial channelis a function describedby eqn (8.22) as shown in Fig. 8.6. After introducing the slip factor, the resultant curve is shown as a dotted line as eqn (8.24). It was not possibleto obtain a suitablecorrelationin the literature[222] for the pressureloss due to shockand leakageas was the casefor the slip. The calculatedcharacteristiccurve without consideringshock and leakage losses,i.e. eqn (8.24) Apy" shown in Fig. 8.7, is significantlyhigher than the experimentalone. The shadedareain Fig. 8.7 represents the shockand leakage losses. It can be seenthat at low flow ratesthe shock and leakagelossesare greaterbut tend to zero at the maximum flow rate. This hasbeen discussedand experimentally validated tn [232].

262

AXIAL FLUX PERMANENT MAGNET BRUSHLESS MACHTNES

500 ideal curve {eqn(S.22))

4tx) Ap

300

lPal I { e q nt 8 . 2 6 t )

240

eqn (8.24)*APy .".,. experimrntalcrrre /' /

0 Figure 8.7.

0.006

Q (m3ls)

t).012

0.{.}18

Lossesand characteristiccurves at 1200 rpm

The derived characteristicsdescribesthe pressurerelationsof a single rotating PM disc facing a stator. For a double-sidedAFPM machinewith two identical coaxial rotating discs (Fig. 8.2) operating on the same stator, the characteristiccurve presentedin Fig. 8.7 representsonly half of the AFPM machine. The characteristiccurve of the whole machinemay be obtainedby adding flow rate at the samepressure,which is similar to two identical fans in parallel. Flow and pressure measurements Due to the nature and complexity of thermofluid analysis, the form of the systemcharacteristicscurve canat best be establishedby test. Dependingon the machine topologies and size, the measurementsmay be taken either at the air inlet or outlet. The AFPM machine under test is normally driven by another motor. Fig. 8.8 showsthe experimentalamangements of the flow measurementsat the machine outlet, in which a dischargeduct is set up to provide good conditions for observingthe flow. Along one side of the duct, severaltapping points were made for measuringthe static pressurewith a pressulegauge (manometer).Near the outlet of the duct, provision was made for measuring velocity using a hot-wire anemometerprobe. To vary the flow rate, the test duct was fitted at its outer end with an obstruction. The test was startedwith no obstruction at the end of the discharseduct.

263


Figure 8.8. The experimentalset up. 1 - manometer,2 * AFPM machine,3 - discharge duct, 4 prime mover (drive machine),5 - pressuretapping point, 6 - wind speedprobe. Photo courtesy of the University o/'Stellenboscl, South Africa.

The only resistancewas then the duct friction, which was small and could be readily computedout of results.As the end of the duct was obstructedprogressively,the ffow was reducedand the staticpressureincreasedto a maximum at zero volumetric flow rate. The staticpressuredifferenceAp is measuredas a function of volumetricflow rateQ : A' u for differentmotor speeds,where u is the linear speed. The air flow-rate measurementcan also be carried out by measuringinlet air pressuredifferenceAp, which is then used for calculatingmassflow-rate rn accordingto the following equation:

;n:

,/2pA4tA4

(8.28)

where ,46 is the crosssectionareaof the inlet duct. Fig. 8.9 showsthe setupof the flow measurement.A speciallydesignedinlet duct with a bell mouth was mountedto the inlet (hub) of the AFPM machine.A few tapping points were madeon the inlet duct for pressuremeasurements. The pressuredrop through the bell mouth and inlet duct may often be assumednegligible.

264

AXIAL FLUX PERMAINENT MAGNETBRUSHLESS MACHINES

Figure 8.9. Flow measurementat the air intake of AFPM machine. 1 Prime mover (drive machine),2 - inlet duct, 3 bell mouth, 4 manometer,and 5 pressuretapping point. Photo courtesy of the University of Stellenbosclr,South Africa.

8.3.2

AFPM machineswith external ventilation

For medium to large power AFPM machines,the loss per unit heat dissipation areaincreasesalmost linearly with the power ratings. Thus the forcedcooling with the aid of external devicesmay be necessary.Some common techniquesare describedas follows.

AFPM machine air discharge duct with fan

air flow direction

Figure 8.10. AFPM machinewith externalair cooling

Cooling and Heat Tran.sfer

265

External fans LargeAFPM machinesmay requirea substantialamountof air flow per unit time in orderto bring out the heatgeneratedin the statorwindings. Depending on the operationconditionsobtainedon site, eitheran air-blastor a suctionfan may be usedas shownin Fig. 8.10. In both cases,intakeandiordischargeducts are neededto direct and conditionthe air flow. Sincethe inlet air temperature, for a given volumetric flow rate, has a significant effect on the machine temperature,this cooling arrangementcan also help preventrecirculationof hot air shouldthe machineoperatein a confinedspace(e.g. small machineroom). For high speedAFPM machines,a shaft-integralfan may be a good option. Fig. 8.11 showsthe assemblyof a large power AFPM machinedevelopedin the Departrnentof Electrical and ElectronicEngineeringat the University of Stellenbosch,SouthAfrica, in which the rotor hub part seryesas both cooling fan and supporling shucture for the rotor discs. It can be seenthat the "blades" of the hub are not curved as the machine may operate in both directions of rotation. stahn'

Figure 8.1I.

right rotor disc

Configuration of the AFPM machine with shaft-integral fan.

Heat pipes The conceptofa passivetwo-phaseheattransferdevicecapableoftransferring large amount of heat with a minimal temperaturedrop was introduced by R.S. Gauglerof the GeneralMotors Corporationin 1942.This devicereceived little attentionuntil 1964,when Grover and his colleaguesat Los Alamos National Laboratory,NM, U.S.A., publishedthe resultsof an independentinvestigation and first used the term heat pipe. Since then, heat pipes have been employedin many applicationsranging from temperaturecontrol of the permafrost layer under the Alaska pipeline to the thermal control of optical surfacesin spacecraft. A typical heatpipe consistsof a sealedcontainerwith wicking material.The containeris evacuatedand filled with just enoughliquid to fully saturatethe wick. A heat pipe has three different regions,namely (1)an evaporator ot heat

266

AX]AL FLUX PERMANENTMAGNETBRUSHLESS MACH]NES

addition region ofthe container,(ii) a condenseror heatrejectionregion, and (111) an adiabatic or isothermalregion. lf the evaporatorregion is exposedto a high temperature,heat is addedand the working fluid in the wicking sffucture is heateduntil it evaporates.The high temperatureand the correspondinghigh pressurein this region causethe vapourto flow to the cooler condenserregion, where the vapour condenses,dissipatingits latent heat of vaporisation. The capillary forces existing in the wicking structurethen pump the liquid back to the evaporator.The wick structurethus ensuresthat the heat pipe can transfer heatwhetherthe heat sourceis below the cooledend or abovethe cooledend. finned surface

Figure 8.12. AFPM machinecooledby heatpipes

A heat pipe presentsan alternative means for removing the heat from the AFPM machine. The heat pipe in an AFPM machinemay be configuredas shown in Fig. 8.12. Heat is transferredinto the atmospherethroughthe finned surface.The finned surfaceis cooledby air moving over the fins. The heatloss removedby the heatpipe, LPno, is given by 12l0]: *A_p ,n p

thot - 0,:rrl,L 1r1 E;",,4t^-

-r

E..r,t,^""td

I

*

(8.2e)

il;il;AI-

where $7ro7is the averagetemperatureof the elementsthat surround the heat pipe in the stator,t9.op1isthe averagetemperatureof the air cooling the finned surface,h6o7is the convectiveheattransfercoefficienton the insidewall of the heat pipe in the statoE46o1 is the exposedarea of the heat pipe in the stator,

CoolingandHeat Transfer

267

h"o6 is the convectionheat transfercoefficienton the inside wall of the heat pipe in the finned area,A.opl is the exposedarea of the heat pipe at the flnned surface,\f t , is the efficiency of the finned surface,hln is the convectionheat transfer coefficient on the surfaceof the fins and As1n is the total exposedarea ofthe finned surface. Direct water cooling Dependingon the conditionsat the site of operation,it is often necessaryto use forced water circulationto cool the statorwindings directly,especiallyfor large power AFPM machines.An externalwater pump is requiredfor forcing water circulation. A longitudinalsectionof a water cooleddouble-discmotor hasbeenshownin Fig. 2.7. For corelesswinding AFPM machines,the winding coils may have a rhomboidal shapeso that the spacebetweenthe two active sidesof eachcoil may be utilisedforplacing a coolingwaterduct [41]. The heat removed through cooling pipes can be calculatedby using eqn (8.29). However,the heat transfercoefficients,h,6o1and h,"o14, arecalculated using the following relationships: (i) for laminar flow, i.e. Rea - ff < ZOOO. where u is the flow velocity and d is the diameterof the water'pipe, the Nusselt number may be obtained using following empiricalrelation [114]

: r.B6(Re6Prll IYu,1 f*li th)o'n

(8.30)

whereIo is the lengthof the waterpipe, trr,andpt* arethe dynamicviscosity of water at inlet and wall temperaturerespectively. (ii) for turbulent flow. i.e. Re6 : c a l c u l a t eads [ 2 3 ]

#

,

2000, the Nusselt number may be

l{ua - 0.02gRe,to'8 Prn

(8.3 r)

where n

8.4

_ ! 0.4 L 0.3

for heating of the water, for cooling of the water.

Lumped parameter thermal model

Lumped-parameter circuits,consistingof a network of thermalresistances, thermal capacitances,nodal temperaturesand heat sources,have been usedextensively to representthe complex distributed thermal parametersof electrical machines176, 147,2191.

268

8.4.1

AXIAL FLUX PERMAI,IENT MAGNETBRI]SHLESS MACHINES

Thermal equivalentcircuit

Athermal equivalentcircuit is essentiallyan analogyof an electricalcircuit, in which the heat (analogousto current) ffowing in each path of the circuit is given by a temperaturedifference (analogousto voltage) divided by a thermal resistance(analogousto electrical resistance).For conduction,the thermal resistancedependson the thermal conductivity of the material, k, andthe length, l, and cross-sectionalarea,A6, of the heatflow path and may be expressedas

nu: |-

(8.32)

AdK

Thermalresistancesfor convectionis definedas:

R.:h

(8.33)

where A. is the surface area of convectiveheat transfer between two regions and h is the convectioncoefficient. The thermalresistancefor radiationbetweentwo surfacesis

R,:

-_f €1A1

1l AtIt>

T €t A)

o[('sr+ 273)+ (82+ 273)11("r + 273)2+ (Oz+ 273)21

(8.34)

It is seenthat the radiation thermal resistancein eqn (8.34) dependson the difference of the third power of the temperature,the surface spectralproperty e and the surfaceorientation taken into accountby a form factor F. The thermal circuit in the steady state consists of thermal resistancesand heat sourcesconnectedbetweenmotor componentnodes. For transientanalysis, the thermal capacitances are usedadditionallyto accountfor the change of internal energy in the various parts of the machine with time. The heat capacitanceis definedas: g : pVct,: mcu

(8.35)

where c., is the heat capacity of the material, p is the density, and V and m, are the volume and massof the materialrespectively.Fig. 8.13ashowsa sectional view of an AFPM machinewith a corelessstator. [t can be observed that the AFPM stator is symmetrical from a heat transfer perspectiveand each half of the machine from the centre line mirrors the other half. It is therefore reasonableto model only half of the machineas shown in Fig. 8.13b. The heatsourcetetms AP1- , LP., LPpnt andL,P,ol standfor winding losses (2.42), eddy cur:rentlossesin one half of the statorwinding (2.61),lossesin PMs (2.54) and rotational losses(2.63) per one rotor disc respectively. C",

269


tr

I

(b)

(a)

Figure 8.13. The thermalresistancecircuit ofan AFPM brushless machinewith coreless stator.

C^ and C, are the thermal capacitancesof stator, PMs and rotor steel disc respectively.The heatresistancesusedin the circuit aredescribedin Table8.3.

Table8.3. Definition of thermal resistances

Definition

Symbols R.t 11 c2

R.z Ru.t Rcs R.a Rrt R,Z

/i,: R"t Rat

8.4.2

Convection resistancefrom stator end-winding to open air Convection resistancefrorn stator to air-gap Convection resistancefrom air-gap to permanentmagnets Convection resistancefrom air-gap to rotor disc plate Convectionresistancefrom rotor disc to open air Convection resistancefrom rotor radial periphery to open air Radiation resistancefrom stator end-winding to environment Radiation resistancefrom stator to permanent magnets Radiation resistancefrom stator to rotor disc Radiation resistancefrom rotor radial periphery to environment Conduction resistancefrom PMs to rotor disc

Conservationof energy

If conservationof energy is applied, the rate of internal energychangein eachpart of a machine(also calledcontrol volume)may be written as follows:

270

AXIAL FLUX PERMANENTMAGNETBRUSHLESS MACHINES LI Af

-

C4o : A,P,,

L P o u Ir m ; n i ; , - t i t o u l i o , l

(8.36)

where L/ is the internal energy,riz is the mass flow rate and z is the enthalpy andC is the thermal capacitanceof a control volume. For steady-stateconditionr, # 0 :

L1,.

-

: 0 and therefore, LPout *'rit,in:iin, - it,o,,1io,,t

(8.37)

Theseequationsare appliedto eachpart (the statoqair gap,PM and rotor disc) of the AFPM machineto obtaina setof equationswith the temperaturesof the parls being the only unknowns. This set of equationsis rathercomplexbut is readily solvedusing for example,the Gauss-Seideliteration It shouldbe noted that the tenn (m,iriin-rnouyior,7)in the aboveequationsrepresentsinternalheat removed due to the air flow pumped through the machine. This air flow is of paramountimportanceto the cooling of the machine.The determinationof the air gaptemperaturein the thermalequivalentcircuit is only possibleif the mass flow ratethroughthe air gap can be somehowpredicted,hencethe necessityof usingthe fluid-ffow model describedin section8.3.1.

8.5

Machineduties

Dependingon the load conditions,thereare mainly threetypes of dutiesor operationmodesfor all the electricalmachines,i.e. continuousduty, short-time duty and intermittent duty.

8.5.1

Continuousduty

When an electricalmachineoperatesfor a long period so that the temperatures of various parls of the machine reach steady-stateat a given ambient temperature,this operationmode is called continuousdue. Due to the different physical properties,the final stablisedtemperaturesin variousparts of the machinemay vary greatly. The machinecan be continuouslyoperatedfor an infinitely long time without exceedingthe temperaturelimits specifiedfor eachcomponentof the machine.When only the solid partsof the machineare considered,the air flow term in the eqn (8.36) is ignored.The temperaturerise versus time relationship in a control volume of the machine may be derived basedon the theory of solid body heatingas [47]:

8 , : L P n ( 1-

+ 0 oe - l "-i )

(8.38)

where B is the thermalresistance,r : RC is the thermaltime constant,C is the thermal capacitance,AP is the heat loss flow, and 8o is the initial temper-

271


aturerise of the control volume. In a casewhere t?. : 0" the aboveequationis simply: d.:

AP R(l -

"-!)

( 8.3e)

According to the propertiesof an exponentialfunction, the final temperature rise r9.; : A,P . /?. Fig. 8.14ashowsthe typical transienttemperatureresponse of an AFPM machine operatedcontinuously.

S,+90

I

(b)

Figure 8.14. Characteristic temperature curves of AFPM machines for different operation modes: (a) continuous duty, (b) intermittent duty, (c) shorl{ime duty.

8.5.2

Short-time duty

Short-time duty meansthat machine operatesonly within a short specified period followed by a long period of rest or no-load condition. The operation time is so short that the machine does not attain its steady-statetemperature, and the machine practically retulxs to its cold state after a long period of rest. Given the operation time f", the temperaturerise of the machine can be found by usingeqn (8.39)as

272

AXIAL FLUX PERMANENTMAGI{ETBRUSHLESS MACHINES

f " : A P R ( . 1 -e + ) : 8 q ( . 1 -

"-?)

(8.40)

Comparedwith continuousduty, it is obviousthat,r9..< 8.f .This implies that the permissibleload for the samemachine in short time duty can be 1l 0 ti*", greaterthan that in continuousduty. Fig. 8.14cshowsthe typical "-?1 temperaturerise curve for short time duty machines.

8.5.3

Intermittent duty

The intermittent duty is characterisedby shorl-time operations alternated with shorl-timepauseintervals.Supposea machineoperatesfor a shofi period of tonandthenstopsforaperiod of toyy,thecyclel",isthentqt:ton*tof f . The duty cycle d"o may be defined as:

n t .c,q

tuon,

(8.41)

-

Lon t

tofJ

The temperafurerise of the machine r9i during the to,nperiod can be calculated by using eqn (8.38) provided that thermaltime constantand the steadystate temperaturerise for continuous operation duty are known. During the tosy period, the machine loss AP : 0 and the machine'stemperaturedecreasesaccordingto an exponentialfunction,i.e. r?;e t,,ttl' 6s1s1ethe second cycle setsin. After many subsequent cycles,the machine'stemperaturevariationbecomes uniform and tendsto be within a ceftain limited range(seeFig. 8.14b). Under the sameload and cooling condition,the maximum stablisedtemperatureof a machine in intermittent operation duty is smaller than that in continuous duty. Hence, similar to shorl{ime operation duty, a machine operatedin intermittent duty hasoverloadcapacity.

Numerical example8.1 A self-cooled8-pole, 16-kW AFPM generatorwith an ironless stator as shown in Fig. 8.2 will be considered.The outer diameteris Do6 : 0.4 m and the inner diameteris Din - 0.23 m. The magnetwidth*to-pole pitch ratio a.4: 0.8 and thicknessof a rotor disc d : 0.014 m. The measuredflow characteristicculvesare shownin Fig. B.15.At ratedspeed1260rpm, the total Iossesare 1569 W, of which (i) rotationallosses LP,ot : 106 W; (ii) eddy current lossesin the stator A.Pr,: 23 W; (iii) stator winding losses(rated) LPru:7440 W. Find: (a) Convectiveheattransfercoefficientsof the disc system (b) Steady-statetemperaturesat different parts of the machine.


ZIJ

Solution (a) Convectiveheat transfer coefficientsofthe disc system The dynamicviscosity,densityand thermalconductivityof air are assumedto b e p : 1 . 8 4 6 7x [ U - ; P as . p : L 7 7 7k g / r n 3 a n dk : 0 . 0 2 6 2 4 W / ( m " C ) respectivelyin the following convectivecoefficientscalculations. Convectioncoefficient:outsiderotor disc surface At ratedspeed,the Reynoldsnumberaccordingto eqn (2.65) is Re

oD2 p----our : 4p

1 . 1 7 7x

2nx1260160x0.42 : 336384.7 4x1.846x10-5

According to eqn (8.9) the transitionbetweenlaminar and turbulentflow takes place at

The averageNusseltnumberof the disc accordingto eqn (8.8) is

l y'u: 0 .0 t5x R , i - 1 0 0x ( ?

f

Uout

: 0 . 0 1 5 x 3 3 6 3 8 4 . 7- *1 0 0* ( 2 ' = 0 , 1 7 2 ) ' : r x . n \

tJ.4

/

The averageheat transfer coefficient at the outer surfaceof the disc according to eqn(8.5)is o 0)614 k ; x l y ' u: : h-rr'|- : x 3 2 1 . 9: 4 2 . 2 W / ( m ' zo C ) Doa 12 0.412 Convection coefficient: rotor disc peripheral edge The Prandtlnumberis takenas Pr :0.7 (Atmosphericpressure,25 oC). The Reynoldsnumberat disc peripheryaccordingto eqn (8.12) is nZuL ReD-t ":

2n x 1260

,,

It a2

,--fr

: 134553u.d , s

The averageNusseltnumberaccordingto eqn (8.11) is ,^y'u: 0.133x Rer? x pr*:

x O.7i : 1439.3 0.133x 1345538.8?

274

AXIAL FLUX PERMANENTMAGIVETBRUSHLESS MACHINES

The averageheattransfercoemcientaroundthe radialperipheryaccordingto eqn(8.10)is k

,

Er:

i*

-

no'rain|

x N,

x 1 4 3 9 . :39 4 . 4 W / ( m 2 " C )

1?0 l{1{}rpn.r .llX.lrprn + {i(0 rptn .J 8fii.)rynn 1iii}t}rprr 120ilrpm A 14{}0qxrr {}

1{}l 8{l f], (t,l )

tltr

4{l 21) i

{

'' 1r

{ , -______________

tll __,:

l

."o I

Figure 8.15. Measuredcharacteristiccurvesof the AFPM machine.Numerical example8.1.

Convection coefficient: rotor-stator system The volumetric flow rate of the machine at rated speedcan be looked up fiom Fig. 8.15. Assumingequalflow on both sidesof stator,the equivalentflow rate is taken as Q : 0.013 m3/s. The averageNusseltnumber accordingto eqn ( 8 . 1 5 i)s

- o ss:r ,,r,u ,;m:

0 333x

0.013 n 1 . 5 6 6 x 1 05 x Q . a l 2 )

:440

heattransfercoefficient two discsaccording Theaverage between to eqn(8.5)is ; n'":

2k Dn*.

Nu :2

x 0.02624 x 440- bT.Tw/(m2 "c; 0.4


275

(b) Steady-statetemperatures at different parts of the machine If the radiation from the rotor disc to ambient, the convection from the stator overhang to air flow, and the conduction resistancebetween magnetsand rotor are ignored,the genericthermal equivalentcircuit given in Fig. 8.13 is simplifiedas shownin Fig. 8.16.

hg,sou,

+ fr

'

,/ halfstator

Figure 8. I 6.

R.

.'"'

rotor disc

Simplified thermal equivaient circuit. Numerical example [l.I

Controlvolume | (hal.fof the stator) The convection heat transfer resistancebetween the stator and air flow in the air gap is Rc1.:

E,"x[(D!.r-D?n)

"C/W / o 7 . 7 x X Q . + ' - 0 . 2 3:0.223 2)

The radiationheattransferresistancebetweenthe statorand rotor discsis Erl :

l-r' 1 -+-.-+etAt'AtFtz'ezAz

1

of(dr+ 273)+ (82+ 273))i('9r + 2nY + (82+ 27:t)21

in which the areasof both discs can be taken as the same,i.e. A1 - Az : D?") - 0.084 m2, the shapefactorFp - 1, the Stefan-Boltzmann i@2", constanto :5.67 x 10-8 W lQn2 K4),the emissivityof epoxyencapsulated

276


statore1 : 0.85. Sincepart of the rotor disc is coveredwith PMs, the emissivity of the rotor disc is defined based on the proporlion of different materials, i.e. - t , ) : 0 . 3 x 0 . 8 + 0 . 9 x ( 1 - 0 . 8 ): 9 . 4 2 € 2 : € f e a , Ii € . y . - ( 1 ApparentlyRrr is a function of r91and 82, &niterativeapproachhasto be used to solve R"1 for different temperatures. According to conservationofenergy,the steady-state energyequationfor control volume 1 can be written as I

r9'

;'2( A P r .

r9.'

"" I AP.) - 1'n ll"r

t9 " ''=

,9.

"'=

R,t

(8.42)

0

Control volume 2 (.airgap) The convection heat transfer resistancefrom air gap to the rotor disc may be takenasthe sameas that from statorto air gap, i.e. R"2 : R.c7.Themassflow rateir, : pQ : 7.177 x 0.013 : 0.0153kg. Assumethat the air temperature at the machine inlet is ambient temperature,i.e. 8;, : 8o and the air gap averagetemperature'82 : j($out * t\;r). The heat dissipateddue to the air flow is houtiout - rnin'iin: h cp (.$out- 0.-) :2rn c, (02 - 8a) : 2 x 0 . 0 1 5 3 x 1 0 0 5x ( r l z - 2 0 ) The steady-stateenergy equation for control volume 2 is 8t-8z

8z*r1z -2x0'0153x1005x(82-20)

:0

T

(8'43)

Controlvolurme3 (rotor disc) The convectionheattransferresistanceat the outsidesurfaceofthe disc is Rd - :-1rtlrrDoul2

42.4x n x0.4'2

:0.1877 "C/W

The convectionheattransferresistanceat the peripheryofthe disc is

_11 R*P:

i,

"

D.rt

:

oc/w g 4 4, n v g 4 v g 1 y 4 : o ' 6 0 2


277

The steady-state energy equation for control volume 3 is

b#.Wni^*.,-

'Bs- ,9u, Rct)

8s-8o

n

Rc:lp

(8.44)

Having establishedthe energyequations(8.42),(8.43) and (8.44) for eachpart of the machine,the steady-statetemperaturescan be found by solving these equations.Due to the temperaturedependencyof Rr1, a simple computerprogram using GaussSeidel iteration has been createdto find the solutions of the equations.The resultsare given in Table8.4.

Table8.4.

Predicted temperaturerises

Machineparts

Temperaturerise, "C

Statorwinding Rotor disc Air-gap flow

114.9 18.32 21.35

Numerical example8.2 A totally enclosedAFPM brushlessmachinehasa power lossof 2500 w in the statorwinding at continuousduty. The machine'souterand inner diametersare Dout : 0.72 m and D;n: 0.5 m respectively.To removethe heat from the stator,use is made of heat pipes for direct cooling of the motor as shown in Fig. 8.12. The convectionheattransfercoefficientson the inside wall of heat pipes in the statorand in the finned areaareassumedto be 1000 W/(m2 "C). The averageconvectiveheat transfer coefficient on the fin surface is taken as 50 W/(m2 "C). The finned surfacehas an overall area of 1.8 m2 and an efficiencyof g2%. The length of heat pipe embeddedin the finned surfaceis ly,in:1.5 m. Find: (a) Steady-statetemperatureof the stator winding if the heat pipe with a diameterD1,r,: 9 mm is placedalong the averageradiusof the stator (b) Steady-state temperaturesof the statorwinding if the heat pipe is replaced by a 9 mm water cooling pipe, in which water (with an averagetemperature of 60"C) ffowsat 0.5 m/s.

278


Solution (a) Steady-state temperatureof the statorwinding if the heatpipe with a diameterDTro: 9 rnnt is placedalong the averageradiusof the stator Assuming the outsidewall of the heat pipe is in full contactwith the stator winding and finned surface,the exposedareaof the heat pipe in the stator will be A7ro1: rD1rr.,

r(Dout + Dm)

: 0.009nx

n ( 0 . 7 2+ 0 . 5 )

:0.0542 rrz

The exposedareaof the heatpipe embeddedin the finned surfacewill be Acold.: rD6plf in:

0'0092 x 1'5 : 0'0424m2

Assuming the temperatureof cooling air over the finned area to be 8ro6 : 30"C, the steady-statetemperatureof the stator may be obtainedby using eqn (8.29)

f statc,,r: fcord+ LPr"e(h#t,.t*

,#;*

: 30*2500. ( r*#o*r+moo*olCIa**x*

Ty;,hyaAym

,. *)

oc : 165'3

(b) Steady-state temperaturesof the statorwinding if the heatpipe is replaced by a water cooling pipe of diameter d : I mm, in which water flows at 0.5 rn/s. The Reynolds number is first calculatedto determinethe flow regime. The propertiesof water at 60"C ?r€ p : 983.3 kg/mt. ro :4179 J/(kg "C), pas, k : 0.654W(m"C), pr : ltcplk : 4.7 x 10 a x F : 4.7 x 10-a x 10-r : 4 1 7 9 1 0 . 6 5 4 : 3 , R e d : p u d l ( 2 S t ): 9 8 3 . 3x 0 . 5 x 0 . 0 0 9 1 4 . 7 9474,5 > 2000, so that the flow is turbulent.Thus,the Nusseltnumberfor the heatingof the water is Nu61,: 0.\BRel'8 Pr0'4 :0.023 x 9414.50'8 x 304 : 53.9 while the Nusseltnumberfor the cooling of the water is - 0.023Re0u'8 IVn"6" P"0 3 - 0.023 x 9474.50'8 30::t: 48.3 " The convectionheattransfercoefficientsof the waterpipe insidethe statorh7.,o1, and in the finned areah,,,r,1r1, are calculatedas

279


k Nu61, -ffi , -f : 0.654 x h'hot:

53.9

: 3916'7 w/(m2"c)

0.654x 48.3 , : -k Nu,1.: -ffi : 3509'8 hcoLd, w/(m2 "C) f The steady-statetemperaturein the stator can be calculated as in (a) with the exceptionthat the heat transfer coefficientsare different : Scotd+ LPheG-++ ,gstator "n'hhotAhot

: 3 0 + 2 5 o o x(' : BB,B "C

r

--],

+ --,L,

h.o4A.old,0f,inhf.i,rtA.fm'

1

)

+ ) 3916.7x 0.0542 3509.8x 0.0424 0.92x 50 x 1.8'

Chapter9 APPLICATIONS

9.1

Power generation

BrushlessPM electrical machinesare the primary generatorsfor distributed generationsystems.They are compact,high efficient and reliable self-excited generators.The distributedgenerationis any electric power productiontechnology that is integratedwithin a distributionsystem. Distributedgeneration technologiesarecategorizedasrenewableandnonrenewable.Renewabletechnologiesinclude sola1 photovoltaic,thermal, wind, geothermaland oceanas sourcesof energy. Nonrenewabletechnologiesinclude internal combustion engines,combinedcycles,combustionturbines,microturbinesand fuel cells. AFPM brushlessgeneratorscan be usedboth as high speedand low speed generators.Their advantagesare high power density,modular construction, high efficiency and easy integration with other mechanical components like turbine rotors or flywheels. The output power is usually rectified and then invertedto matchthe utility grid frequencyor only rectified.

9.1.1

High speedgenerators

Although the minimization of windage lossesof high speedgeneratorsrequires rotors with small diameters,a multidisc designhas the advantagesof modular and compactconstruction,low synchronousreactance,good voltage regulationand very high efficiencyin the caseof corelessstators. A multidisc AFPM high speedgenerator is usually driven by a gas microturbine. The turbine rotor and PM rotors are mounted on the sameshaft. Such a generatorhascompactconstruction,low massand very high efficiency. Accordingto TurboGensetCompany,U.K., a 100-kW,60 000 rpm multidisc generatorhasan outerdiameterof 180mm, lengthof 300 mm and weighsonly

282

Figure 9. L U.K.

MACHINES AXIAL FLLTXPERMANEI{TMAGNETBRUSHLESS

rhoGen'set,London, I 00-kW 8-disc AFPM synchronousgenerator.Courtesy of Zr.r

l2kg (Fig. 9.t)r. the high frequencyoutput is rectified to d.c. and then invertedto 50, 60 or 400 Hz a.c. (Fig.2.20). The generatoris totally air cooled. Armed forces are interestedin applicationsof microturbine driven PM synchronous generatorsto battery chargersfor soldiers. The low energy density of portable bagpack batteriesimposesa major constraint and challengeto future infantry operations.With recentadvancementin PM brushlessmachine technologies,a lightweightminiaturegeneratorset canminimize the massof heavy batteries and charge them during field operations. A miniature generator can deliver electric power for a long period of time limited only by the fuel availability. Easy and rapid refueling can be done in the field by using, e.g. kerosene.Fig. 9.2 showsa tiny microturbineintegratedwith a miniature AFPM synchronousgenerator.At speed150 000 to 250 000 rpm and outer diameter of PM ring Do,rl = 50 mm, the generator can deliver about I kW electricpower.

9.1.2

Low speedgenerators

A low speed AFPM generator is usually driven by a wind turbine. Wilh wind power rapidly becoming one of the most desirablealternativeenergy sourcesworld-wide, AFPM generatorsoffer the ultimate low cost solution as comparedwith e.g. solarpanels.Table 9.1 showsspecificationsof five-phase

lULEV-TAP Newsletter,No. 2, 2000, wwwulev-tap.org

283

Applications

AFPM synchronousgeneratorsmanufacturedby Kestrel Wind Turbines (Pty) Ltd, Garteng, South Africa. A1l three types of Kestrel generatorsuse laminatedcores. The Kestrel 2000 is a double-sidedAFPM generatorwith twin rotors. The Kestrel 600 generatorrated at 400 W is in fact capableof deliv-

Figure 9.2. High speedAFPM synchronousgeneratorintegratedwith the rotor of a microturbine. 1 SmCo PM, 2 - backing steelring, 3 - rotor of microturbine,4 - nonmagnetic retaining ring, 5 - stator winding, 6 stator core.

_cL

3 ,roo t

uJ

3 I rooo

o o

-z:....

iI.......... N

o

b

Figure 9.3. Output power versuswind speedofKestrel AFPM synchronousgenerators.Courtesy of Kestrel fiiind Turbines. South Africa.

284

Table 9.1.


Specifications offive phaseKestrel AFPM synchronousgenerators

Typeofgenerator Rated power, W Maximum poweq W Number of poles 2p Generation steps Rated wind speed,m/s Cut in wind speed,m/s Rated rotational speed,rpm Rotor diameter,m Number of blades Type ofblade Tower top weight, kg Lateral thrust, N Speedcontrol Standardrectified d.c. voltage, V Protection

Kestrel 600 400at 12.5m/s 600at 14.5m/s 48 224 t2.5 2.5 11 0 0 1.2

Kestrel 800 800at 11.5m/s 850at 12mls 48

Kestrel 2000 2000at 10.5m/s 2200at I lm/s 200

11 . 5 2.5 1010

10,5 2.2 925 3.6 3

_/.. I

3

o

Full aerofoil

Basic aerofoil

23 2s0

35 300

80 450

Dynamic tail

Self stall 12,24, 36, 48

48

IP55

ffi Figure 9.4. Stator and rotor of a medium power AFPM generator with inner coreless stator and twin external rotor: (a) stator coils, (b) rotor disc with NdFeB PMs.

ering power in excessof 600 W. Controlledby a shuntregulatorthe maxrmum chargingculrent is 50 A tnto a 12 V d.c. batterybank. At wind speedsof 52.2 km/h, which translatesinto 14.5mls (28.2knots)androtationalspeedof I100 rpm, the poweroutputis 600 W (Fig. 9.3).

285

Applications '1t0

t

100

o 3

gso 5

j60 o 4""

5 o '; c " ' "n o

ti u

=

20 10 0 speed,rpm

Figure 9.5. Steady-stateperformancecharacteristicsof a 10-kW AFPM generatorshown in Fig. 9.4.

Fig.9.4 showsa medium-powerAFPM generatorwith twin externalrotor and inner corelessstator for a 5-m blade diameterwind turbine. The stator systemconsistsof l5 coils, one coil per phase. The thicknessof coil is 12 mm and diameterof wire 0.6 mm. Each coil is individually rectified to d.c. to reducethe cogging effect and provide better control over the voltage. The rotor has 2p: 16 poles. Perfonnancecharacteristics are shown in Fig. 9.5.

9.2

Electric vehicles

Electric vehicles are divided into two generalcategories:hybrid electric vehicles(HEYs) andbatteryelectricvehicles(EVs). The switch from gasolineto HEVs and EVs would reduce the total primary energy consumedfor personal transportation.Traction electric motors for HEVs and EVs should meet the following requirements : r power rating: high instantpower,high power density; torque-speed characteristics: high torque at low speed for starling and climbing, high speedat low torquefor cruising,wide speedrangeincluding constanttorqueregion and constantpower region,fast torqueresponse; high efficiencyover wide speedand torqueranges;

286

AXIAL FLUX PERMANENTMAGNETBRUSHLESS MACHT],,IES

Figure 9.6. Hybrid electric gasoline car: I - gasoline combustion engine, 2 - integratedmotorgenerator,3 - cranking clutch, 4 - gearbox, 5 - inverter, 6 - battery.

700

600

500

! aoo at

5 300

---t-**-F"-* electric molor to.que

200 enginetorque

.

s

100

0 2000

3000

4000

speed,rpm

Figure9.7. Torque-speedcharacteristicsofanelectricmotorandgasolineengine. Theelectric motor assiststhe gasolineengineat low speeds.

r high reliability and robustnessunder various operatingconditions,e.g. at high and low temperature,rain, snow,vibration, etc.; r low cost.

Applications

9.2.1

287

Hybrid electricvehicles

HEVs are now at the forefront of transportationtechnology development. HEVs combinethe internalcombustionengineof a conventionalvehicle with the electric motor of an EV, resulting in twice the fuel economy of conventional vehicles. The electric motor/generatoris usually located betweenthe combustionengineand gearbox. One end of the rotor shaftof the electricmotor/generatoris boltedto the combustionenginecrankshaft,while the opposite end can be bolted to the flywheel or gearboxvia the clutch (Fig. 9.6). The electricmotor servesa numberof functions.i.e.: assistingin vehicle propulsionwhen needed,allowing the use of a smaller internalcombustionengine; operatingasa generator,allowing excessenergy(during braking) to be used to rechargethe battery; replacingthe conventionalalternator,providing energythatultimately feeds the conventionallow voltage,e.g. 12 V electricalsystem; startingthe internal combustionenginevery quickly and quietly. This allows the internalcombustionengineto be tumed offwhen not needed,without anv delav in restartins on demand: r dampingcrankshaftspeedvariations,leadingto smootheridle. In a hybrid electricgasolinecar the electricmotor assiststhe gasolineenginein the low speedrangeby utilizing the high torque of the electric motor, as shown in Fig. 9.7. Currentlymanufacturedhybrid electricgasolinecars(Fig. 9.6) are equippedeither with cage induction motors or PM brushlessmotors. A PM brushlessmotor can increasethe overall torque by more than 500/o.In most applications,the ratedpower of electricmotors is from l0 to 75 kW. Because of limited spacebetweenthe combustionengine and gear box as well as the need to increasethe flywheel effect, electric motors for HEVs are shorl and havelargediameters.AFPM brushlessmachinesarepancaketypehigh torque densitymotors and fit perfectly the HEV requirements.An AFPM brushless machinecanbe liquid cooledand integratedwith a powerelectronicsconverter. integrated In the HEV shown in Fig. 9.8 the energyis storedin a fl).rnrheel with the AFPM machine. The vehicle may operatewithout an electrochemical battery. lf the electromagneticconversionprocessis includedwithin the flywheel, there is no external shaft end, becausethe power flow to and from the machine is via electric connectionsto the stationaryarmature[3]. Thus, complexvacuumsealsare avoided.

288


d.c.bus

Figure 9.8. HEV with flywheel energy storage: I - gasoline combustion engine, 2 - brushless generator, 3 - integrated flywheel-motor generator, 4 solid state convertet 5 d.c. bus,6 - electricalmotorizedwheel.

The moment of inertia and tensilestressare respectively r for a rotatins disc

J:

T,^Rf,,,,

pQ2RZut

(e.1)

r for a rotating ring

t:f,m@?,+ RZ,,),

{a?, I

R,inR66 + R?",r) (:9.2)

289

Applications

where p is the specificmassdensityof the rotating body, m is the mass,f) : 2trn is the angularspeed,Ro,1 is the outer radius and Ri, is the inner radius. The kinetic energyis r for a rotating disc

nr:

f,lrf

: n3prlRt,tn2 J

(e3)

r for a rotating ring

Er: 13pd(Rtt

(e.4)

Rt,)r,'

where d is the thicknessof the flywheel (disc or ring). Combiningtogetherpairs of eqns (9.1), (9.3) and (9.2), (9.4), the stored energydensityin a flywheel can be expressedas r for a rotating disc 87. €1r:-': m

3o a+p

(e.s)

Jlkg

r for a rotating ring R, rrL

R7,,,+ R?o,,, o _ - 3 4 R7.* R;nRo6* R7,,p

J/ke

(e.6)

Fora veryhighstrength steel,e.g.alloysteel(3%CrMoC)with o ry 1000MPa andp :7800 kglm:r[3], the storedenergydensityof a disctypeflywheelis ep : 96.I kJ/m3.If we needto storea kineticenergyEn : 500kJ, themass of thediscwill be 5.2kg. Battery electric vehicles 9.2.2 batteries BatteryEVs arevehiclesthatusesecondary batteries(rechargeable or storagebatteries)as their only sourceof energy.An EV powertrain can convedenergystoredin a batteryinto vehiclemotion and it canalsoreverse directionand convertthe kinetic energyof the vehicleback into the storage batterythroughregenerative braking.

290


Figure 9.9.

Single-sidedAFPM brushlessmotor fitted to spokedwheel ll95l.

AFPM brushlessmotors are usedin EVs as in-wheel motors (Figs 6.8 and 9.9) [ 95,2031.The pancakeshapeof an AFPM motor allows for designinga compactmotorizedwheel. With the aid ofthe AFPM brushlessmotorthe differentialmechanismcanbe replacedby an electronicdifferentialsystem[92]. The configurationshownin Fig. 9.I 0a illustratesthe useof a pair of electricmotorsmountedon the chassis to drive a pair of wheels through drive shaftswhich incorporateconstantvelocity joints. In the configurationshownin Fig. 9.10b,the motors forming the electronicdifferential are mounteddirectly in the wheels of the vehicle. The electromechanical drive systemis considerablysimplified, when the motor is mounted in the wheel, becausethe drive shafts and constantvelocity joints are now no longer needed. However, the resultant "unsprung" wheel mass of the vehicle is increasedby the mass of the motor. Wheel motors of this direct drive configurationalso suffer overloadingbecausethe speedofthe rotor is lower than would be the casewith a gearedalrangement.This leadsto an increaseof the activematerialsvolume requiredin the motor. The disadvantages of conventionalwheel motorscan be overcomeby using the arrangementshown in Fig. 9.11. The two statorsare directly attachedto the vehiclebody whilst the PM rotor is free to move in radial directions.It can be observedthat in this casethe wheel and disc rotor form the unsprungmass, whilst the statorsof the motor becomesprungmass supportedon the chassis f q ? tl '

L"

291

Applications

t0l

Figure 9.1(). Alternative forms of "electronic differential" drive schemes: (a) onboard motor, ft) wheel mounted motor.

,'l

Figure 9.11. Schematicrepresentationof the disc-typemachinewith lower unsprungmass: -wheel, 2 chassis. damper,6-spring, 7 shaft,5 discrotor, 3 -stator, 4

9.3 Ship propulsion 9.3.1 LargeAFPM motors Statorsof large AFPM brushlessmotors with disc type rotors usually have threebasicparts [44]: r aluminum cold plate; r bolted ferromagneticcore; r polyphasewinding.

292

AXIAL FLUX PERMANENTMAGNETBRI]SHLESS MACHTNES

13 1rl

1s t6 I

Figure 9.12. Large power double disc AFPM brushlessmotor. 1 - pMs, 2 - stator assembly, 3 -housing,4 shocksnubbeq5 -shockmount,6-rotorshaft, T-rotordisc clanrp,g shaft seal assembly, 9 - bearing retainer, 10 - stator segment, 1l centre frame housing, 12 - spacerhousing, 13 rotor disc, 14 - bearingassembly,15 - rotor seal runner, 16 rotor sealassembly.Courtesyof Kaman Aerospace,EDC. Hudson.MA, U.S.A.

The cold plate is a part of the frame and transfers heat from the stator to the heat exchangesurface. The slots are machinedinto a laminated core wound of steelribbon in a continuousspiral in the circumferentialdirection.The copper winding, frequentlyaLitzwke, is placedin slotsand then impregnatedwith a potting compound.The constructionof a doubledisc AFpM motor developed by Kaman Aerospace,EDC, Hudson,MA, U.S.A. is shown in Fig. 9.121441. Specificationsof large axial flux motorsmanufacturedbyKaman are given in Table9.2.

9.3.2

Propulsion of unmanned submarines

An electric propulsion systemfor submarines requires high output power, high efficiency,very salientand compactmotors[59, 180]. Disctype brushless motors can meet these requirements and run for over 100 000 h without a failure, cooled only by ambient seawater. Thesemotors are virtually silent and operatewith minimum vibration level. The output power at rated operating conditionscan exceed2.2kwlkg and torque density 5.5 Nm/kg. The typical rotor linear speedof largemarinepropulsionmotors is 20 to 30 m/s [lS0].

9.3.3

Counterrotatingrotor marine propulsionsystem

An AFPM brushlessmotor can be designedwith the counterrotationof two rotors [35]. This machinetopology can find applicationsin marinepropulsion

293

Applications

Table 9.2. Design data of large poweq three phase AFPM brushlessmotors manufactured by Kaman Aerospace,EDC, Hudson,MA, U.S.A.

Quantity Number of poles, 2p Number of windings per phase Output power P",r, kW Peakphasevoltage,V Rated speed,rpm Maximum speed,rpm Efficiency at rated speed Torque at rated speed,Nm Stall torque, Nm Continuous cunent (six step waveform), A Maximum current, A Peak EMF constant per phase,V/rpm Winding resistance perphaseat 500 Hz, 0 Winding inductance per phaseat 500 Hz, pH Moment of inertia,kgm2 Cooling Maximum allowable motor temperature,"C Mass, kg Power densiry kWkg Torque density, Nm/kg Diameter of frame, m Length of frame, m Application

PA44-5W-002

PA44-5W-001

28

PA57-2W-001

336 700 2860 3600

28 2 445 530 s200 6000

746 73s 3600 4000

0.95

0.96

0.96

tt20 1621

822 I2 8 8

r980 2712

370 500

370 500

290 365

0.24

0 .1 0

0.20

0.044

0.022

0.030

120 0.9

195 1.723 5.743 0.648 0.224

60 0.9 Waterandglycol mixture 150 195 2.282 4.215 0.648 0.224

Traction, Drilling indushy

36

100 2.065

340 2.194 s.823 0.787 0.259 General purpose

systemswhich use an additional counter-rotatingpropeller in order to recover energyfrom the rotationalflow of the main propeller stream. In this casethe use of an AFPM motor having counter-rotatingrotors allows the elimination of the motion reversalepicyclicalgear. The stator winding coils have a rectangularshapewhich dependson the cross sectionof the toroidal core [35] (also see Fig.2.q. Each coil has two

294


Figure 9.13. Exploded view of the axial flux PM motor with counter-rotatingrotor: I - main propeller,2-counter-rotatingpropeller,3-radialbearing,4-outershaft,5-PMroto16 motor bearing, 7 - assemblyring, 8 - stator, 9 inner shaft.

Table9.3. EMALS requirements [72] End speed,m/s Marimum peak-to-mean thrust force ratio Cycle time, s End speedvariation, m/s Maximum acceleration,m/s2 Launch energy,MJ Maximum thrust, MN Maximum poweq MW 3

volume, mMass, kg

103 1.05 45 1.5 56 r22 l.J

133 < 425 < 225000

active surfacesand each coil surface interacts with the facing PM rotor. In orderto achievethe oppositemotion of the two rotors,the statorwinding coils have to be arrangedin such a manner that counter-rotatingmagnetic fields are producedin the machine'sannularair gaps. The statoris positionedbetween two rotors which consist of mild steel discs and axially magnetizedNdFeB PMs. The magnets are mounted on the disc's surface from the stator sides. Each rotor has its own shaft which drives a propeller,i.e. the motor has two coaxial shafts which are separatedby a radial bearing. The anangement is shownin Fig. 9.13[35].

Applications

9.4

295

Electromagnetic aircraft launch system

Military aircrafts are catapultedfrom the decks of aircraft carriers with the aid of steamcatapults.A steamcatapultusestwo parallelrows of slottedcylinders with pistons connectedto the shuttlethat tows the aircraft. The pistons under the steampressureacceleratethe shuttleuntil the aircraft will takeoff. Steamcatapultshavemany drawbacks172],i.e.: r operation without feedbackcontrol; r largevolume(over 1100m3; andmass(up to 500 t); r occupationof the "prime" real estateon the canier and negativeeffect on the stability of the ship; r low efficiency(4 to 6%); r operationalenergy limit, approximately 95 MJ; r need for frequent maintenance. The electromagnetic aircraft launch system (EMALS) is free of the drawbackslisted above.The EMALS technologyemployslinear inductionor synchronous motors fed from disc-type altemators through cycloconverters.The electricalenergyobtainedfrom the carrierpower plant is storedkinetically in

stator Figure 9.14. AFPM synchronousmachine for EMALS. I - PM rotor assembly,2 brake,T-shaftencoder. assembly,3-bearing,4-enclosure,5 -mountingflange,6

296

Table9.4.

A,Y]ALFLUX PERMANEI{TMAGNETBRUSHLESS MACHINES

Specificationsof AFPM synchronousmachine for EMALS [72]

Number of phases Number of poles Number of active stator slots Type of stator Number ol stator windings Generatorwinding Motor winding Maximum speed,rpm Maximum frequency,Hz Ouput power at maximum speed,MW Maximum line-to-lineEMF for motoring, V Maximum line-to-line peak voltage, V Maximum phase peak current, A Power losses,kW Efficiency Resistanceper phase,mf) Inductanceper phase,pH Statorcooling Coolant Flow rate, l/min Average temperatureof winding, "C Average temperafureof stator core, "C Type of PMs Remanenceat 40"C, T Air gap magnetic flux density, T Tooth magnetic flux density, T Maximum energy storagecapacity, MJ Energy density, kJ/kg Mass,kg

fl

2'p: 4o s\ :24:0 double-sided slotted 2 air gapside bottomof slots 6400 2133 8 1. 6 1122 1700 6400 127 0.893 8,6 10.4 cold plate waterandgiycol mixture 151 84 6l sintered NdFeB 1.05 0.976 1.7 t21 1 81. 6685

the rotors ofAFPM synchronousgenerators.The energyis then releasedas a few-secondpulsesto accelerateand launchan aircraft.The cycloconverterbetween the AFPM generatorand linear motor raisesthe voltage and frequency. The EMALS operatesin a "real time" closedloop control [72]. The requirementsare given in Table9.3. Specificationsof the AFPM synchronousmachine shown in Fig. 9.74 are given in Table 9.4 [721. The EMALS usesfour AFPM machinesmounted in a torque frame and grouped in counter-rotatingpairs to reduce the torque and gyroscopiceffect. The rotor of the AFPM machineservesboth as the kinetic energystoragewhen operatingas a motor and a field excitationsystemwhen operating as a generator. The electric power from the on-board generatorsis

297

Applications

(b)

(a) ll fr a lnl

adi

!!i Tr, .drrd X

t 1i n . 4 -i4 ill

T O PO F TO{}LJ.ltNT

Figure 9.15. ECI drilling system: (a) general view, (b) power unit with AFPM synchronous motors, (c) PA-44 Kaman Aerospace AFPM motor. Courtesy of TescoCorporation, Calgary, Alberta, Canada.

fed to the AFPM machine via rectifier-inverters. There are two separatestator windings for motoring and generatingmode. The motor winding is locatedat the bottom of slotsto providebetterthermalconductivitybetweenthe winding and enclosure.

9.5

Motrile drill rigs

Drill rigs vary from small electric motor driven tools to large oil field rigs. The basic elementsof an abovegrounddrill rig are the power unit or motor, pump or air compressorfor circulatingdrilling fluid to the bit and cleaningthe borehole,drill head, hoisting drums and cables,derrick, mounting platform or deck and assortedequipment such as hammers for driving and removing casing,portablemud pit, racks for stackingthe drill rods and samples,small tools for coupling or uncouplingand hoisting the drill string, etc. The power unit (motor) performs the following functions:

298

AXIAL FLUX PERMANE]VTMAGNETBRUSHLESS MACHINES

Table 9.5. Specifications of ECI drilling systemswith AFPM synchronous motors manufactured by TbscoCorporation, Calgary,Alberta, Canada

ECE drilling system

670

1007

5,909

6,270

4.34

4.34 9,490 11,389 7,860 187

Drilline unit withoutswivel Mass, kg Operating length, m (including 2.8-m links and elevators) Make-up Torque (lpical), Nm Breakout torque, Nm Max. Drill Torque, Nm Max. Speed,rpm Power system (mechanical module) Approximate mass, kg Length, m Width, m Power system (power module) Approximatemass,kg Length, m width, m

5 q65

7,592 5,098 187 3,340 6.4 2.3

3,730 6.4

1 545

4 1)(l

6 . 3 5m m 2.311

a.-)

6.30 2.30

r operatesa drive weight mechanismfor percussionor churn drilling or provides rotary motion to tum augersand coring equipment for rotary drilling operations; r operatesa winch for raising and lowering the drilling and samplingequipment; r providesdownwardpressurefor pushingboring and samplingequipment, or lifting and dropping a hammer to drive casing or sampling equipment. For most drilling and samplingoperations,the power sourceis the power takeoff from the truck motor on which the drilling machineis mountedor from a separateenginewhich is assignedor attachedas an integralcomponentof the drilling rig. It is estimatedthat approximately 90% of the motors are gasoline or dieselenginesand 10ohare compressedair or electricmotors. A drive train which consistsof gearsor hydraulicpumps is usedto convertthe power supply to speedand torque for hoisting and rotating the drilling equipment. Most units have a transmissionwhich allows 4 to B speedsfor hoisting and drilling. ln general,the hoisting capacityof the drill rig govems the depth of the borehole. A rule of thumb for selecting the power source is that the power which is required to hoist the drill rods should be about three times the power which

Applications

299

is requiredto turn the drill string. For high elevations,the power loss is about 3ohfor each300 m. Largepower AFPM brushlessmotors (Table9.2) areespeciallysuitablefor portabledrilling equipmentbecauseof their compactdesign,light weight,precise speedcontrol, high efficiencyand high reliability. ln the ECI lightweight top drive drilling system(Fig. 9.15) manufactured by kscct Corporation, Calgary, Alberta, Canada,the traditional induction or brushtype d.c. motorshavebeenreplacedwith high performanceliquid cooled AFPM synchronousmotors. This systemis compatiblewith most 600 V a.c. rig power systems.The producthighlightsinclude: s Kaman PA44 AFPM synchronousmotors that (Table 9.2) have been successfullytestedto 609 triaxial loading, making them the ideal choice for rough drilling environments; r AFPM synchronousmotors which allow a high level of precisionin speed and torque control not availablewith other motors; r a modulardesignthat allows drilling to continueat reducedpower with one motor; r direct connectionto the rigs a.c. bus and interfacewith the existingpower supply with minimal degradationto the rig powen r the entire ECI systemcan be transpoftedin three standard6-m seacontainers. Specificationsof the ECI drilling systemare given in Table9.5.

9.6

Elevators

The concept of gearless electromechanicaldrive Jbr elevators was introduced in 1992 by Kone Corporation in Hyvinkiiii, Finland [103]. With the aid of a disc type low speedcompactAFPM brushlessmotor (Table9.7'),the penthousemachinery room can be replacedby a space-savingdirect electromechanicaldrive. In comparisonwith a low speedaxial flux cageinductionmotor of similar diameter,the AFPM brushlessmotor has a doubledefficiencyand a three times higher power factor. Fig. 9.16a shows the propulsion systemof the elevatorwhile Fig. 9.16b showshow the AFPM brushlessmotor is installedbetweenthe guide rails of the car and the hoistwaywall. Table 9.6 containskey parametersfor the comparisonof different hoisting technologies[103]. The disc type AFPM brushlessmotor is a clear winner. Specificationsof single-sidedAFPM brushlessmotors for gearlesspassenger elevatorsare given in Table 9.7 [103]. Laminatedstatorshave from 96 to

300

AXIAL FLUX PERMANEA|T MAGNETBRUSHLESS MACHINES

Figure 9.16. MonoSpace7nl elevator:(a) elevatorpropulsionsystem;(b) EcoDiscTnl motor Courtesy of Kone, Hyvinkziii, Finland.

Figure 9.17. Double-disc AFPM brushlessmotor for gearlesselevators. Courtesyof Kone, Hyvinkdd, Finland.

301

Applications

120 slots with threephaseshort pitch winding and classF insulation. For example,the MX05 motor ratedat 2.8 kW 280 V 18.7Hz hasthe statorwinding resistance Xt : 10 Q,2p :20, sheave Er : 3.5 f), statorwinding reactance diameter340 mm and weighs180kg.

Tahle9.6. Comparisonofhoisting technologiesfor 630 kg elevators[03]

Quantity Elevator speed,m/s Motor shaft power, kW Speedof motor, rpm Motor fuses,A Annual energy consumption, kWh Hoisting efficiency Oil requirements,I Mass,kg Noise ievel, dB(A)

Warm gear elevator

Direct PM brushless motor elevator

0.63 I 1.0 1500 50

1.0 5.5 I 500 35

1.0

7200 0.3 200 350 60..65

6000 0.4 3.5 430 7 0 . . .57

3000 0.6 0 t70 50...55

Hydraulic elevator

95 to

Table9.7. Specifications of single-sided AFPM bmshless motorsfor gearless elevators manufacturedby Kone,Hy",rinkiiii, Finland

Quantity

MXO5

MXO6

Rated output power, kW Rated torque, Nm Rated speed,rpm Rated current, A Efficiency Power factor Cooling Diameter of sheave,m Elevator load, kg Elevator speed,m/s

2.8 240 113 7.7 0.83 0.9 natural 0.34 480 I

360 96 10 0.85 0.9 natural 0.40 630 1

800 80 18 0.86 0.91 natural 0.48 1000 i

hoistway

hoistway

hoistway

Location

)- t

MXlO

MX18

o./

46.0 l 800 ItJ

138 0.92 0.92 forced 0.65 1800 4 machine room

302

AXIAL FLLTXPERMANENTMAGNETBRI]SHLESS MACHIAIES

Table 9.8. Specificationsof double-disc AFPM brushlessmotors manufacturedbv Kone, Hyvinkiiri, Finland

Quantity Rated output power, kW Rated torque, Nm Rated speed,rpm Rated current, A Efficiency Power factor Elevatorload, kg Elevator speed,m/s

MX32

MX4O

MXlOO

58 3600 153 122 0.92 0.93 I 600

92 5700 153

315 14,000 214 1060 0.95 0.96 4500 l3.5

o

zoz

0.93 0.93 2000 8

r&

Figure 9.18. Penny-motor.Photo courtesyof Mymotctrs& Actuator.sGmbH, Wendelsheim, Germany.

A doublediscAFPMbrushless motorfor gearless elevators is shownin Fig. 9.17[03]. Table9.8listsspecifications dataof doublediscAFPMbrushless motorsratedfrom58to 315kW [03]. 9.7 Miniature AFPM brushless motors Ultra-flatPM micromotor, the so calledpenny-motor is shownin Figs9.18 and9.19[ a2]. Thethickness is 1.4to 3.0mm,outerdiameter about12 mm, torqueconstantup to 0.4 p,Nm/mAand speedup to 60,000rym. A 400pm eightpole PM anda three-strand, 110-pmdisc shaped,lithographically producedstatorwindinghavebeenused[42]. PlasticboundNdFeBmagnetsarea costeffectivesolution.However,themaximumtorqueis achieved with sinteredNdFeBmasnets.A miniatureball bearinshasa diameterof 3

303

Applications

ffi Figure 9.19. Construction ofpenny-motor: 1 - shaft, 2 - soft steel yoke cover, 3 ring, 4 ball bearing, 5 - stator winding, 6 flange, 7 bottom steel yoke.

PM

Figure 9.20. Two-phase,four-coil, single-sidedl5-mm AFPM brushlessmotor. Photo courtesy of Moving Magnet kchnologies,,Sl, Besancon,France.

mm. Penny-motorsfind applicationsin miniaturizedhard disc drives,cellular phonesas vibration motors,mobile scannersand consumerelectronics. Moving Magnet kchnologies (MMf) two or three phase,miniature AFPM brnshlessmotors(Fig. 9.20) are well adaptedto high volume low costproduction. In order to obtain rigidity the stator coils are overmoulded. The stator is mounted on a single-sidedprinted circuit board. Each part can be manufacturedusing standardmoulding, stampingtechniquesand automaticcoil winding with a specificdesign for simple and efficient assembly.Three position sensingmethodsfor the closed-loopcontrol are used: (a) digital Hall probeslocatedin the stator,(b) EMF signal(sensorless mode) and (c) absolute analogueposition sensor(position sensormode). MMT AFPM miniaturerotary actuators(Fig. 9.21) havebeendesignedfor automotive applicationsto provide an efficient, contactless,direct drive rotary motion on a limited stroke. Ring, disc and tile shapedPMs have been used. The main featuresof this family of actuatorsare:

304

AXIAL FLUX PERMANENTMAG]VETBRUSHLESS MACHIAIES

Figure 9.21. Miniature two-phase,four-coil, 4-po1e,single-sidedAFpM rotary actuator. Photocourlesyof MovingMagnetTechnologies. 57, Besancon, France.

r contactlessactuationprinciple; r constanttorque independentof the angular position for a given cunent; r linear torque*currentcharacteristic; r two directionsof rotation; r high torque density. Additional functions,suchas a magneticreturn spring or analoguecontactless position sensingcan alsobe implemented.Owing to the linearity of the torquecurrentcharacteristicand its independence ofposition, it is possibleto operate actuatorsin an open-loopagainsta spring or in a simple closed-loopmodeby using a position sensor.

9.8

Vibration motors

Advancesin cellular telecommunicationhave made mobile phones to be highly popular communicationtool in modern society. Sincea mobile phone is now a gadgetas necessaryas a watch or wallet, small vibration motorswith diametersfrom 6 to 14 mm are manufacturedin unbelievablelarge quantities (Fi5.9.22). Thetrendsinvibrationmotors formobilephonesincludereduction of mass and size, minimization of energy consumption and guaranteedstable vibration alarmingin any circumstances[58]. There are two types of brushlessvibration motors for mobile phones:cylindrical or RFPM motor andcoin type or AFPM motor (Fig. 9.23). The unbalanced excitingforce generatedby an eccentric rotor is F :

me{l :2trrnert2

(e.7)

30s

Applications

E mobrle Dhone demand

1999

Figure 9.22.

2000

2001

A aoplications of vibration motors

2002

2003

2AA4

2AO5

Worldwide demand estimate for mobile phones and vibration motors

(b)

Figure9.23. CointypeAFPMvibrationmotorfor mobilephones[58]:(a) two coil motor;(b) rnulticoilmotor. 1 - phasecoil, 2 - PM (mechanically unbalanced system),3 * ferromagneticyoke,4 cover,5-baseplate, 6 shaft,7-bearing, 8-detentiron [58].

where m, e and n denote the rotor mass, eccentricity and rotational speed respectively. The rotational speed is the most effective design parameter to increase the unbalanced exciting force [58]. Moving Magnet Technologies single-phase AFPM brushless vibration motors provide a strong vibration feeling and silent alert with a very simple design (Fig. 9.2q. Fabrication of MMT vibration motors is cost effective due to contactless design, scalable size and slim shape.

306

AXIAL FLUX PERMANENTMAG},{ETBRUSHLESS MACHINES

Figure 9.24. Cost effective AFPM brushlessvibration motor. Photo courtesy of Moving Magnet kchnologies, Sl, Besancon,France.

9.9

Computerhard discdrives

The data storage capacity of a computer hard disc drive (HDD) is determined by the aerial recordingdensityand numberof discs. It is expectedthat the aerialdensitywill soon increasefrom 6 Gb/cm2- 38.7Gb/in2to 15.5 Gblcm2 - 100 Gb/in2. The massof the rotor, moment of inertia and vibration increasewith the numberof discs. Special design features of computer HDD motors are their high starting torque, limited current supply, low vibration and noise, physical constraints on volume and shape,protectionagainstcontaminationand scalingproblems. A high startingtorque, 10 to 20 times the running torqueis required,sincethe readlwrite head tends to stick to the disc when not moving. The starling current is limited by the computer power supply unit, which in turns, can severely limit the starting torque. AFPM brushlessmotors (Fig. 9.25) can develophigher startingtorquethan their radial field counterparts.The additional advantageis zero cogging torque. The drawbackof the single-sidedAFPM HDD motor shown in Fig. 9.25a is the high normal attractive force between the stator with ferromagnetic yoke and rotor mountedPMs. In double-sidedHDD AFPM motors the statordoes not have any ferromagnetic core and no normal attractive force is produced at zero-curent state.The statorhas a threephasewinding with corelesscoils fabricated with the aid of a litographic method. In a typical design, there are two coils per phasefor 2p: 8 PM polesand threecoils per phasefor 2'p: 12 PM poles. To reduce the air gap and increasethe magnetic flux density, the so called pole fingers are created in the lower part of the hub and the rotor

307

Applications

1

U

I

o

v

10

Figure 9.25. Constructionof computerHDDs with AFPM brushlessmotors: (a) single-sided PM, 3 - shalt, 4 -beanng. 5 - hub, motor; (b) double-sidedmotor. 1 - statorcoil, 2 6 rotor ferromagnetic yoke, 7 stator feromagnetic yoke, B - baseplate, 9 - pole fingers, 10 nonmagneticring.

magneticcircuit is bendedby pressingit towardsthe statorcoil centres(Fig. 9.25b). According to [129], for a HDD AFPM brushlessmotor with a 5l-mm stator outer diameter, r^/r : 180 turns per phase and torque constant k7 6.59 x 10-3 Nm/A the currentconsuptionis 0.09 A at 13 900 rpm and no load condition. The acousticnoise of HDD brushlessmotors with ball bearingsis usually below 30 dB(A) and projectedmean time betweenfailuresMTBF : 100 000 hours. HDD spindlemotors are now changingfrom ball bearingto fluid dynamic bearing (FDB) motors. Contact-freeFDBs produceless noise and are serviceablefor an extendedperiod of time.

Numerical example9.1 Find the kinetic energyandtensilestressof the ring with dimensionsRo,t : 0.142m, Rtn - 0.082m,thicknessd - 0.022mand densityP : 7800kg/mjl rotatingat the speedof 30 000 rym.

308

AXIALFLUXPERMAI\|IENT MAGNET BRUSHLESS MACHINES

Solution Themassof thering is rn : rpcl(R3,,- R7):

:7.25kg n7800x 0.022(0.1 422- 0.0822)

Themomentof inertiaaccording to eqn(9.2)is I :

rnr.zstl.7422 + 0.0822): 0.092kgm2

The rotationalspeedn : 30 000/60 : 500 rev/sand angularspeedO : 2tr500: 3141.6rad/s.Thusthekineticenergyaccording to eqn(9.a)is - 0.0824)x b002: 480.74 En: n37800 x 0.022(0.1424 kJ Thetensilestressaccording to eqn(9.2)is 7800x 3141.(2 o _ ______J (0.0822 + 0.082x 0.142+ 0.1.422):988.6Mpa 3 Theenergydensityaccording to eqn(9.6)is Lli

^

-

480740 : 66.3kJ/kg 7.25

The shapefactorof the ring [3] is

: : kst,. "rI eosooff$1su 0.b23 Numerical example9.2 Find the main dimensionsof a three-phase,1200 W, 4-pole, 200 000 rym AFPM synchronousgeneratorfor a mobile battery charger.The total nonmagnetic air gap including magnetretainingring (Fig. 9.2) shouldbe g : 2 mm, numberof slots sy: 12, air gapmagneticflux densityBs : 0.4 T, line cument density A- : 16 000 A/m, PM inner-to-outer diameterratio k,1: 0.5, PM temperature350"C. Vacomax225 SmCo PMs (Vaccumschmelze, Hanau,Germany) with Br2s : 1.04 T and H" = 760 kAlm at 20"C arc recommended. -0.035 o l"C,leakage flux coefThe temperaturecoefficientfor B, is a6: ficient opv : 1.3, EMF-to-voltage ratio e : 1.3 (Ef > V) andpole shoe width-to*pole pitch ratio a,; - 0.72 Solution The winding coefficients(2.8), (2.9) and (2.10) arc k41 : 7, k1t : 1 and k-7 - 1 since qr : 121(4 x 3) : 1. The coefficientkp accordingto eqn (2.20)is

309

Applications

t r , : 1 ( 1 + 0 . 5 ) ( -1 0 . 5 2:) 0 . 1 4 1 6

The outer diameterof PMs accordingto eqn (2.93) is Dout

1 . 3x 1 2 0 0 : 0.055m r 2 0 . 7 4 I x 1 x ( 2 0 0 0 0 0 / 6 0x) 0 . 4x 1 6 0 0 0x 0 . 8x 0 . 8

The remanentmagneticflux densityat 350"C accordingto eqns(3.2) is

-0'035 Ii, : 7' '^ t u Ll'r - r ttu' 350

2o)l : 0.973 7 l

Since demagnetizationcurves for different temperaturesare parallel lines, the relative recoil magnetic permeability will be approximately the same as for 20"C (linear magnetizationcule), i.e. 1.04 - 1.089 0.4trx 10 6x760000

fL t'r'e c

The PM thicknesswith the leakagecoefficiento1- included,can be calculated on the basisof eqns(3.13)and (3.9),i.e. oI1[B*o

, rt F,,"" hnr U;?ffig

1.3 x 0.4 : 1'08909f3_ - 0.0028 rn 1j x 040.002

The innerdiameterofPMs is Din: k,tDout: 0.5 x 55:27.5 mm, radial length of PM l,r,r : 0.5(55.0 - 25.0) : 1"3.7mm, averagediameterof PMs D : 0.5(55.0 + 27.5) : 41 mm, average polepitchr : r4ll4: 32.3and averagePM circumferentialwidth bo : e..ir : 0.72 x 32.3: 23.3 mm.

Numerical example9.3 A 10-kW 2200-rpm electric motor operatesat almost constantspeedaccording to the torque profile given in Fig.9.26. The overloadcapacityfactor ko"f : Trurr lTrh, : 1.8. Find the thermalutilisationcoefficientof the motor. Solution The required rated shaft torque is

.a r. s,n r

-

Port

10,000

2rn

2r x (2200160)

: 43.4Nm

The rms torquebasedon the given duty cycle is

T)|,,,"(tv+t2+. . .+f,) : T?tt+:rltz+...+Tlt. or T:^"Lto:\rlti

310

AXIAL FLUX PERMANENTMAGNETBRUSHLESSMACHINES

Thus

Trrn"

:

l-r?t V Ito 252x 3 t 402x8 + 652x 5 + 382x 30 + 152x 10

3+B+5+30+10 3 8 . 1N m The maximum torque in Fig. 9.26 cannotexceedthe rated shaft torque times the overloadcapacityfactorko4Trhr : 1.8 x 43.4:78.72 Nm. Also, the required 7"7." should be greaterthan or equal to fl--". The coefficientof thermalutilisation of the motor is

:#' ?ff " Trrovo

: B7'8% fioYo

Figure 9.26. Torque profile of the elechic motor accordingto Numerical example 9.3

Symbolsand Abbreviations

A

magneticvector potential

A

line currentdensity;cross-sectionarea

o

number of parallel current paths of the stator (armature)winding

E

vector magneticflux density

B

magneticflux density;dampingof the system

b

instantaneousvalue of the magnetic flux density; width of slot

be

pole shoewidth

Cy

cost of frame

Ci*

costof insulation

Cs

cost of all other componentsindependentof the shapeof the machine

Cpm

cost of PMs

Cr.

cost of the rotor core

C"h

cost ofshaft

C-

cost of winding

ccu

cost ofcopper conductorperkg

cE

armatureconstant(EMF constant)

cFe

cost of ferromagneticcore per kg

cins

cost of insulationper kg

cp

specific heat at constantpressure

cptt

cost of PMs per kg

csteet cost ofsteel per kg cu

heat capacity

312

AXIAL FLUX PERMANENTMAG}{ETBRUSHLESS MACHINES

D

diameter;duty cycle of power semiconductorswitches

Dn

inner diameterof PMs equalto the inner diameterof statorbars

D,t,t.

outer diameter of PMs equal to the outer diameter of stator bars

E

EMF (rnzs value);electricfield intensity

Ef

EMF per phaseinducedby the rotor without armaturereaction

Ei

internal EMF per phase

e

instantaneous EMF: eccentricitv

F

force

Ftz

shapefactor of two surfacesinvolved in radiation

f

spaceand/or time distribution of the MMF

fo

armaturereaction MMF

F"r"

MMF of the rotor excitation system

f G

frequency; friction factor perrneance;gap ratio g f R

g

air gap (mechanicalclearance);gravitational acceleration

Gr

Grashof number

g'

equivalentair gap

fr

vector magnetic field intensity

H

magnetic field intensity

h

height; heat transfer coefficient

hm

height of the PM

1

electric current

Io '

stator (armature)current instantaneousvalue of current; enthalpy vector electric current density moment of inertia cumentdensity in the stator (armature)winding currentregulatorgain inverter gain coefficient,generalsymbol; thermalconductivity skin effect coefficient for the stator conductor resistance reaction factor in d-axis reaclionfactorin o-axis

SymbolsandAbbreviations

a l l

JIJ

kg

Carter'scoefficient

l*,1

inner-to-outerdiameterratio ka:

kat

distribution factor for fundamental

ks

EMF constantkg : 6;6Q,

ky

form factor of the fie1dexcitation ky : B*,gf B*n

h

stacking factor of laminations

koc.f

overload capacity factor ko.y : Trro* /7"h, pitch factor for fundamental

kpt

DinlDaut

k"ot

saturationfactor of the magnetic circuit due to the main (linkage)magneticflux

ky klrt

torque constantky : c7Q 1 winding factor k-1 : k4ykpt for fundamental

L

inductance;length

lrc

length ofthe one-sidedend connection

Li

armaturestack effective length

174

axial length of PM

L|l

mutual inductance

M'

momentum

m

numberof phases;mass

m

massflow rate

rna,

amplitude modulation ratio

mf

frequency modulation ratio

N

numberof turns per phase;numberof machines

Nu

Nusseltnumber

n

rotational speedin rpm; independentvariables

no

no-loadspeed

P

active power

P"lrn

electromagneticpower

Pr"

input power

Pout

output power

Pr

Prandtl number

LP

activepower losses

LPtp" statorcore losses

314 LPt-

AXIAL FLUX PERMANENTMAGI,{ETBRUSHLESS MACHINES statorwinding losses

LPzp" rotor core losses LP".

eddy current lossesin stator conductors

LPp

friction losses

LPpm lossesin PMs LP,ot rotational(mechanical)losses LP.,tnd windagelosses A,p

specificcore loss

p

numberof pole pairs;pressure

p,

radial force per unit area

p

wetted perimeter

a

electric charge;reactivepower; volumetric flow rate

Q".n, enclosedelectriccharge radius;resistance R Rt

armaturewinding resistanceof a.c. motors

Rt,

inner radius of PMs equal to the inner radius of stator bars

Rout

outer radius of PMs equal to the outer radius of stator bars

Re

Reynolds number

ffirn

air gap reluctance

$tpto

external armatureleakagereluctance

ffip,vl Permanentmagnetreluctance apparentpower; surface S Sw

crosssectionareaof PM; SM : uuLnr or,9,x1-- boLnt

s

crosssectionareaofstator conductor

sr

number of stator slots equal to the number of statorteeth

T

torque

Ta

electromagnetictorque developedby the machine

Trt,"I

reluctancetorque

Td'u* synchronousor synchronizing torque T*,

mechanicaltime constant

Trt

shaft torque (output or load torque)

t

time; slot pitch

U

internal energy

315

Symbols and Abbreviations

u

tangentialvelocity

V

electricvoltage;volume

rr

instantaneous value of electricvoltage;linear velocity

W

energy produced in outer space of PM; rate of change of the air gap energy

W,

storedmagneticenergy

I-u 'It)M

energyper volume, J/m3;radial velocity width of PM

X

reactance

Xy

stator winding leakagereactance

Xod,

d-axis armaturereaction (mutual) reactance

f aa

q-axisamature reaction(mutual)reactance

X",j,

d-axis synchronousreactance;Xsd: Xt * Xo.a

Xss

q-axissynchronousreactance;X"q:

Z

i m p e d a nZc: e R + j X ; l Z 1 - - Z : \ / R + T

a

complex attenuationconstantof electromagneticfield

a.i

effectivepole arc coefficienta;:

7 d

fotm factor of demagnetizationcurue of PM material power (load) angle

6;

inner torque angle

€

eccentricity

s

emissivity; surfacespectralproperty

\

efficiency

1 d

equivalentsandgrain roughness rotor angularposition for brushlessmotors

?y

temperature;angle betweenIo and Io4

l

coefficientof leakageperrneance(specificleakagepermeance)

),r

turbulent parameter

p

dynamicviscosity

lto

magneticpermeabilityof free spacepo :0.4r

Fr

relativemagneticpermeability

Xt I Xoq

brf r

l-L,rec recoilmagneticpermeability Itrrrec relativerecoil penrreabilityltr,ec : lrr""l Fo

< 10-6 H/m

316 u

AXIAL FLUX PERMANENTMAGNETBRUSHLESSMACHINES number of the stator zth harmonic; kinematic viscosity reducedheight ofthe stator conductor

p

specificmassdensity electric conductivity ; Stefan-Bol tzmann constant

of

form factor to include the saturationeffect output coefficient

or

radiation factor

T

averagepole pitch; thermal time constant

o

magnetic flux

Qy

excitation magnetic flux

O1

leakageflux

6

power factor angle

v

flux linkage V : l/O; anglebetweenIo and E1

4,

flux linkage

a

angularspeedO :2rn

u)

angularfrequencya :2trf

Subscripts a

armature (stator)

au[J

average

c

conduction

cu

control volume

Cu

copper

d

direct axis; differential;developed

e

end connection;eddy-current

eIm

electromagnetic

eq

equivalent

erc

excitation

e.rt

external

Fe

ferromagnetic

f

field; forced

.f r

friction; free

Svmbolsand Abbreviations

aff gap h

hydraulic; hysteresis

'l,n

inner

I

leakage

M

magnet

'tn

peak value (amplitude)

frrt

normal and tangentialcomponents

out

outpui, outer

o

quadratureaxis

T'

rated; remanent;radiation; rotor

r , 0 , z cylindrical coordinatesystem reI

reluctance

'rot

rotational

5

slot; synchronous;system;stator

sctt

saturation

sh

shaft

st

starting

syn

synchronous or synchronizing

t

teeth; total

u

useful

,I)

convection

uent

ventilation

uind

windage

v

yoke

T , U , Z cartesiancoordinatesystem 1

stator; fundamentalharmonic; inlet

2

rotor; exit

Superscripts inc

incremental

(sq)

squarewave

(tt

ftapezoidal

317

318

AXTALFL\.]XPERMANEAIT MAGNETBRUSHLESS MACHI],{ES

Abbreviations A/D

analogto digital

AFPM

axial flux permanentmagnet

AIFI

American Iron and Steel Industry

a.c.

alternating current

BPF

band passfiltering

CAD

computer-aideddesign

CPU

centralprocessorunit

DSP

digital signalprocessor

d.c.

direct current

EDM

electro-discharge machining

EMALS

electro-magneticaircraftlaunchsystem

EMF

electromotiveforce

EMI

electromagneticinterference

EV

electricvehicle

FDB

fluid dynamicbearing

FEM

finite elementmethod

FPGA

field programmable gateanay

HDD

hard disk drive

HEV

hybrid electricvehicle

IC

integratedcircuit

IGBT

insulated-gate bipolar transistor

ISG

integratedstarter-generator

LPF

low passfilter

MMF

magnetomotiveforce

MMT

moving magnettechnologies

MOSFET

metal oxide semiconductor(MOS) field effecttransistor

MVD

magneticvoltagedrop

NdFeB

neodymium iron boron

PFM

pulse frequencymodulation

PLD

programmablelogic device

PM

permanentmagnet

Symbolsand Abbreviat ions

PWM

pulse width modulation

RFI

radio frequency interference

RFPM

radial flux permanentmagnet

SEMA

segmented electro-magnetic array

SMC

soft magneticcomposite

SmCo

samariumcobalt

SSC

solid stateconverter

3r9

References

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Index

Acoustic noise,9, 237, 307 Active rectifier, 66, 2 17 Air conrpressor,204, 297 Air-cooled, 49, | 62, l1 4, 202, 254, 255, 282 Applicationsof AFPM machines Computer hard drsc drive (HDD), 152, 306 Counterrotating marine propeller, 292 Electric vehicle, 2, 285, 287, 289 Electromagnetic aircraft launch system (EMALS),295 Electronicdifferential,290 G e a r l e s s e l e v a t o r s , 2 T , 2 9 93 0 2 High speedgenerators,281 Hybrid electricvehicle,285, 287 Low speedgenerators,281, 282 Microturbine, 66, 97, 281, 282 Mobile drill rigs, 297 Mobile phone, I 54, I 89, 304, 305 Power generation, 203, 240, 28 | Precisionrobotics,203 Propulsion system for submarine,292 Ship propulsion,29l Mbration motors, I 8l , 303 , 305 Wirrd generator,1,282 - 285 Armature constant, (see EMF constant) Armature reaction.27, 57, 58, 64,96, 128, 110, 111

Fotm factor, 58, 69 lnductance, 130, 159 Reactance,61, 130, 147, I 59, I 84, 209 Armature winding resistance,42, 157 Attenuation coefficient, 48 Axial flux PM machines with Corelessstator,7, 32, l2l , 153, I 94 Double-sidedconsruction, 6, 7, 10, 27 3 5 , 4 l , 5 4 , 1 1 1 ,1 2 5 ,t 2 6 , 1 7 4 ,1 9 3 , 262,283 Intemal PM ro1or,8, I Intemal stator,7,24, 153

Iron core, 7, 125. 121, 128,237 M u l t i d i s cc o n s t r u c t i o n8,, 1 9 ,3 2 , 5 5 , 2 8 1 Printed winding rotor, 4, 5, 114 Sine-waveexcitation,51, 61, 136, 162 Single-sidedconstruct'ioll,6, 27, 37, 89, 1 2 5 ,1 4 3 , 2 9 9 , 3 0 3 , 3 0 6 Square-waveexcttation, 62, 14 1 Wound rotor,4, 5 Battery electricvehicle,285, 289 Blowers,5, 97 Bottle-neck feature, I 7 Centrifugalforce,174 Coefficient of Additional core losses,44, 45 Differential leakage, I 3 I Distortion,44,49 Drag,49,78 Leakage perrneances,131, 112 Skin effec1,42, 43 Thermal expansion,89, 253 C o i l s h a p e s1, 0 , 1 5 3 Rhomboidal, 10, 36, 3'7,267 T o r o i d a l ,1 0 , 3 5 T r a p e z o i d a l1, 0 ,3 5 - 3 7 , l 4 l , 1 6 2 , 1 9 7 Computer hard disc drive (HDD), 154, 306 HDD spindlemotol 307 Pole fingers,306 Control,213 Conduction period, 214, 223 Currentangle,230,232 234,242 Current control, 219, 221, 229, 230, 232 _ )11 )10 Digital signalprocessor,237 High frequencyvoltage injection,239 Sensorlesspositioncontrol, 237 -239 Single sensorcurrent controller, 2 I 9 SinusoidalAFPM machine,223,224

338


Speedcontrol, 154, 222, 230, 234, 237 TrapezoidalAFPM machine,213, 214 Convefier-fed AFPM machine &ive, 213 Cooling of AFPM machine,255 Direct water cooling, 112,267 Extemal fan,265 Extemal ventilation,255, 264 H e a tp i p e s ,2 6 5 , 2 7 7 Self-ventilation, 255 Cost model, 198 Current Armature (stator),45, 50, 52, 54, 59,62, 64, 102,103,1s7 d-axis, 143,226,232 Density3 , 8 , 3 9 , 4 1 , 5 4 , 7 2 , 1 1 2 ,1 9 7 Fictitious field, 157,227,228 Instant average,218, 219 q - a x i s ,I 3 9 , 1 4 0 , 1 4 3 ,2 2 6 , 2 2 9 ,2 4 6 R i p p l e ,12 8 , 2 2 1 , 2 2 2 , 2 3 6 , 2 3 7 Starting, 306 versus speedcharacteristic,200 Cycloconverters, 295, 296 D i s c m o t o r s ,3 , 3 2 , 5 6 Distributed generation system, 194, 203, 281 Flywheel motor-generator systems, I 94 Nonrenewable,28 I Renewable,28l Duties (operation) Continuous duty, 270 Drty cycle,272 Intermittent duty, 65, 270, 272 Short{ime ilrty. 270. 27 1 Dynamic v t scosity, 49, 267 Eddy cunent brake, I 3, 2 I Eddy current loss resistance, 156, I 63 Efficiency,32, 42, 50,66, 138, 156, 194, 199, 202,285,295,299 Elevators,2, 27, 299, 302 EMF Armature reaction, 59 S i n e - w a v em o t o r , 4 1 ,6 1 , 1 3 7 Sinusoida6 l , , 1 4 1, 2 2 3 Square-wavefioro1 62, 14l Trapezoidal, 5, 141, 213 Waveform,5, 6, 141, 213, 214, 223 EMF constant,41, 63 Fabrication of C o r e l e s sw i n d i n g s ,1 1 4 1 1 6 Laminated siator cores, 87 Rotor magnetic circuits, 109 - I I I Slottedwindings, 112 - 114 Soft magnetic powder cores, 87 89 W i n d i n g s1, 1 2 1 1 6 Factor

Accounting for increasein losses,45 Carter factor, 8, 9, 46, 59, 106, 129 Distributron factor, 34 Form lactor ofdemagnetization curve, 93 of impedanceincrease,16, 23 Pitch factor, 34 Saturationfactor,20, 59, 106, 134, 180 Stacking factol 80, 82 Failure of rotor-shaft mechanicaljoint, 2, I 9 F an, l, 2, 5, 50, 9'7,252, 254, 255, 262, 265, 282 Faraday'sdisc, 3, 20 F i l m c o i l w i n d i n g ,1 1 4 , 1 1 6 , 1 5 4 F i n i t ee l e m e n m t e t h o d ,4 5 , 1 4 1 , 1 6 0 , 1 7 5 Axial-symrnetricelement,I 75 Boundaryconditions,141, 175 Dirichlet boundaryconditions,142 Neumann boundary conditions, I 42 Periodic boundary conditions, 14i Shell element,175 So'lver,142 Triangular element, 160 Virlual work method, 175 Fourier expansion,143 G a p r a t i o ,2 5 4 , 2 5 5 Gauss-Serdel iteration,270, 2'77 Halbach array, 109 lll, 121, 123, 153, 192, 193,196,201 Hall effects,220,222 Heat transfer, 179,249 Conduction,250,268 Convection,250 - 253,267,268 Emissivity, 250 Radiation,250,268 Shapefactor,25l Thermal conductivity, 250, 268 High speedgenerators,281 Microturbine,66, 97, 28 1, 282 Miniature generator set, 282 Turbocenset,28l Hydraulic diameter, 261 Inductance,9, 62, 137, 143, 159, 160, 162, 202, 2 2 6 , 2 2 7, 2 3 1 , 2 3 2 , 2 3 7, 2 4 0 Armature reaction, 59, 130, 159,227 End winding, 162 Leakage,159,228 Mutual, 157, 159, 227, 228 Self,157 Synchronous, 128, 137, 141, 157, 159,

160,2t4,215,227 Inductionmachine,4, 6, 156, 159 Differential leakage factol 134 Differentialleakageflux, 134,159 Disc-type,4, 6

339

INDEX Kinematic viscosityof fluid, 253 Krnetic energy, 289, 296

Saturation, 73,85,92 Magneticpermeability, 20,47 Magneticsaturation, 9, 57, 106,154,161.205,

Laplace transform, 221 Large AFPM motors,29l Basic pafis. 29 I Cold plate, 291,292 For ship propulsion, 291 294 Water cooling, 32, 3"1, 112,267 Laws Fourier'slaw, 250 Kirchhoff's magnetic voltage law, 20 Newton's law of cooling, 251 Line currentdensity,38, 39, 41, 54 Litz wires, 167, 169 Load angle,53, 140, 156 Lorentz force theorem, 153 Losses Armature (stator) winding, 42, 43, 50, 1 3 1 , 1 4 0 ,t 5 6 , 1 6 ' . 7 , 2 4 9 C o r e ,1 0 , 3 2 , 4 4 , 4 5 , 4 7 , 5 0 , 5 2 , ' 1 9 , 8 l 8 3 , 8 5 , 1 2 7 ,1 2 8 , 1 3 7 , 1 5 3 ,1 5 6 , 1 6 3 ,t 9 0 , 2 2 9 E d d y - c u r r e n t9, , 3 2 , 4 4 , 4 5 , 4 8 , 7 4 , ' 7 5 , 8 0 , 8 4 , 9 7 , 1 3 81 , 5 3 ,1 5 6 ,1 6 3 ,1 6 4 , l 6 7 , I 6 8 , 1 7 0 , 1 71 , 1 9 0 ,2 0 0 , 2 6 8 Excess,44 For nonsinuso'idalcunent, 50 Frequencydependent,50 Friction,49, 258 H y s t e r e s i s4,4 , 4 5 , 1 7 In PMs,32, 45, 46, 156,268 Minimization of, 281 Rotational,49, 138,249, 268, 272 Ventilation,49 W i n d a g e ,4 9 , 7 & , 1 7 0 , 1 8 3 ,2 0 8 ,2 8 1 Low speedgenerators,28 l, 282 Performancecharacteristics,285 Wind turbrne generator,282, 283, 285

zzo Magnetic vectorpotential,13, 14, 160 Miniature AFPM brushlessmotor, 302, 303 Generator set, 282 Moment of inertia, | , 222, 288,306 Moody diagram,261 Mutual inductance, (see Armature reaction inductance)

Magneticcircuit, I 7, 2'.7-32, 90,94, 101, 109 lll Calculation,94,107 109 F a b r i c a t i o n , 1 0 9l l l Magnetic flux density Air gap,32, 38, 39. 64, 68, 94,95, I I 6, 117, 142, 143, 192, 197 At the surface ofHalbach anay, I I 1 Average,38,39,76 Average-tG-peak ratio, 38 Coefficientof distoftion, 44, 49,73, 180 d-axis, 57 Distribution in the air gap, 128, 130 P e a kv a l u e ,3 8 , 3 9 , 6 8 , 7 5 , 1 1 l , 1 2 1 q-axis, 58 R e m a n e n t1, 9 , 9 1 , 9 2 , 9 5 - 9 7 , 1 0 0 , 111 , 1 1 7 .l 2 t . 1 9 2

N o i s e ,9 , 1 3 0 , 1 7 4 , 1 9 4 , 2 3 ' 7 , 3 0 63, 0 7 Nusseltnumber,252 254,267 Park's transformation, | 6 l, 225 Penny-motor,302, 303 Permanentmagnet,l, 90, 95 A l n i c o ,3 , 5 , 9 5 , 9 6 Classes,95 Coefficientofleakage flux, 93 C o e r c i v i t y9, 1 , l 0 l , 1 9 2 Demagnetizationcurve, 90, 92, 93

98,

100,103 F e n i t e , 3 , 99 57 Intrinsic coercivity,92 L e a k a g ef l u x , 9 3 , 9 4 , 1 0 3 , 1 0 7 , 1 0 8 , 1 0 9 , 130 M a g n e t i cc i r c u i t ,8 6 , 9 0 , 9 4 , 9 9 , 1 0 0 , 1 0 2 , 1 0 3 ,1 0 6 ,1 0 7 ,1 0 9- r r l Maximum magneticenergy,93, 101 N d F e B ,3 , 4 5 , 9 5 , 9 8 , 9 9 , 1 9 2 , 1 9 7 ,1 9 8 , 294,302 Operatingdiagram,99 Operatingpoint, 95, 103 Rare-earth,3, 93, 95, 97 99 R e c o i l i n e , 9 l , 9 5 , 1 0 1- 1 0 3 Recoil permeability,58, 91, 93, 95 Remanence,5, 90, 9l SmCo, 45, 95, 97, 98 Stabilization,I 03 Volume of, 94, 128 P e m r e a n c el 0, l - 1 0 3 , 1 0 6 Air gap,40, 95, I 03, I 07 Differential leakage,134, 159 Dividing the magnetic field into simple solids, 103 - 106 End connection leakage, 132, 133, 14'1. 159 Fringing flur, 107, S l o tl e a k a g e1 , 3 0- 1 3 2 ,1 5 9 Pole pitch, 10, 12, 16, 34, 3'1, 40, 47, 106, 129, t4l Power Apparent,148,232

340

MACHII{ES AXIAL FLUX PERMANENTMAGNETBRUSHLESS

Electromagnetic , 54 - 56, 136 - 138, 140, l 4 l , 1 5 0 ,1 5 6 ,1 5 8 ,2 1 6 , 2 2 6 Input, 66, 137, 138, 156, 158 O u t p u t .l , 1 9 , 5 0 , 5 6 , 6 6 , 7 9 , 1 3 7 , 1 3 8 , 1 4 1 , 1 5 6 ,1 5 8 , 1 6 2 ,2 4 9 , 2 8 1 ,2 9 2 Pulse width modulation (PWM), 44, 141,194, 218 - 220, 224, 234, 236, 237, 246 Pumps, l, 2, 97, 255, 266, 29'7,298 Reactance,130 Armaturereact'ion,5l , 61, 130, 159 Differential leakage, l3 I End connection leakage, 13 1 L e a k a g e5, 1 , 1 3 0 ,1 3 1 ,1 5 5 Synchronous,51, 65, 121, 130, 197 Synchronousd-axis, 158, 185 Synchronousq-axis, 158, 185 Relative recoil permeability, 58, 93, 95 Reynolds number, 49, 253, 254, 261 Rotary actuator, 303 S a l i e npt o l e ,6 , 8 , 9 , 3 1 , 8 8 , 8 9 , 1 3 8 ,1 5 7 , 2 3 0 S a l u r a t i o nf a c t o r , 5 9 ,1 0 6 ,l J 4 Shapesof PMs, 109 Shock, leakageand friction, 259 S i l i c o ns t e e l ,7 j , ' 7 9 , 1 2 8 Sinusoidalexcitation,6 I Sizing equations,54 56 Skewedslot, 87 Skin effect coeficient, 42, 43, 50, 131 Slip factor, 258, 259, 261 Soft magneticcomposite,84, 85, 87, 88 Solid-stateconvefter, 128, 213, 214, 216, 221, 224,234,238 Stacking factor, 80, 82 Stefan-Boltzmann constant, 25 I Stokes'theorem,160 Synchronization with utility grid, 66 Infinite bus, 66 Powercircuit, 6'7,214, 216 Synchroscope,66 Systemlosses,260,261 Thermal equivalent circuit, 268, 270, 2'/ 5 Conservation of energy, 25'7,269 Control volume, 256,269 -271 Thermal capacitance,267 - 210 Thermal resistance,26'7, 268,27{) Time constant Electrical,205,234 Electromagnetic,185, 187, 188

Mechanical,18"1,218, 234 Thermal,270,272 Transverseflux machines, i Topologies,l, 3, 6, 125,249, 262 Torque,37 39 Cogging,9, 31, 32, 87, 194,200, 306 Control. 219,299 Developed,41, 137, 150, 216, 227 Electromagnetic,3\, 37, 39, 40, 41, 56, 61, 63, 153, 156, 158, 189,192, 193,224,232 s h a f r ,1 3 8 , 1 5 0 , 1 5 6 Torque-curent characteristics, | 54. 304 Torque-speed characteristics, 64, 134, 197,285 Torqueconstant,4I, 61, 63, 162, 197,203, 233, 302,307 Toys,97 Turbulence parametet, 254 TypesofAFPM machines,4,125,213 Induction machines,4, 6, 156 PM brushlessd.c. machines,4 PM d.c. commutatormachines,4 Synchronousmachines,4, 5, 5l ,224 Unbalancedforce, 178, 304, 306 Vibration,174, 178, 189, 194,286,292, 303, 304, 306 Vibration motor, 189, 303 - 305 Coil-type motor, 304 Cylindrical, 304 Voltagegain, 221 Wind turbine, 282, 283, 285 Winding Coil pitch to pole pitch ratio, 34 Coreless stator winding, 35, 153, 169, 192,194 Distribution factor, 34 DoubleJayerwinding, 42,72, 132 Drum-type stator winding, 35 Pitch factor, 34 Printed, (seealso Film coil winding). 4, 5,

114-1r6,154,303 Salientpole winding, 37 S i n g l e - l a y ewr i n d i n g ,3 3 , 3 5 , 1 3 2 ,1 3 3 S l o t l e s sw i n d i n g s ,1 , 9 Winding factor, 34, 41, 57, 68,'71, 133, 145,181,206 Windscreen wipers, 97

Axial Flux Permanent Magnet Brushless Machines

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