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Axial Flux Permanent Magnet Brushless Machines

Axial Flux Permanent MagnetBrushless Machines

by

JACEK F. GIERASUnited Technologies Research Center,East Hartford, Connecticut, U.S.A.

RONG-JIE WANGUniversity of Stellenbosch,Stellenbosch, Western Cape, South Africa

and

MAARTEN J. KAMPERUniversity of Stellenbosch,Stellenbosch, Western Cape, South Africa

KLUWER ACADEMIC PUBLISHERSNEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: 1-4020-2720-6Print ISBN: 1-4020-2661-7

©2005 Springer Science + Business Media, Inc.

Print ©2004 Kluwer Academic Publishers

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Dordrecht

Contents

Preface

1. INTRODUCTION

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Scope

Features

Development of AFPM machines

Types of axial flux PM machines

Topologies and geometries

Axial magnetic field excited by PMs

PM eddy-current brake as the simplest AFPMbrushless machine

1.8

1.9

AFPM machines versus RFPM machines

Power limitation of AFPM machines

Numerical examples

2. PRINCIPLES OF AFPM MACHINES

2.1 Magnetic circuits2.1.12.1.22.1.3

Single-sided machinesDouble-sided machines with internal PM disc rotorDouble-sided machines with internal ring-shapedcore stator

2.1.42.1.52.1.6

Double-sided machines with internal slotted statorDouble-sided machines with internal coreless statorMultidisc machines

2.2 Windings2.2.12.2.2

Three-phase windings distributed in slotsDrum-type winding

xi

1

1

1

3

4

6

10

13

16

19

19

27

272727

29313232

333335

vi AXIAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

2.2.32.2.4

Coreless stator windingSalient pole windings

2.3

2.4

2.5

2.6

Torque production

Magnetic flux

Electromagnetic torque and EMF

Losses and efficiency

2.6.12.6.22.6.32.6.42.6.52.6.62.6.72.6.82.6.9

Stator winding lossesStator core lossesCore loss finite element modelLosses in permanent magnetsRotor core lossesEddy current losses in stator conductorsRotational lossesLosses for nonsinusoidal current

Efficiency

2.7

2.8

2.9

Phasor diagrams

Sizing equations

Armature reaction

2.10 AFPM motor

2.10.12.10.2

Sine-wave motorSquare-wave motor

2.11 AFPM synchronous generator2.11.1 Performance characteristics of a stand alone

generator

2.11.2 Synchronization with utility grid

Numerical examples

3. MATERIALS AND FABRICATION

3.1 Stator cores3.1.13.1.23.1.33.1.4

Nonoriented electrical steelsAmorphous ferromagnetic alloysSoft magnetic powder compositesFabrication of stator cores

3.2 Rotor magnetic circuits3.2.13.2.23.2.33.2.4

PM materialsCharacteristics of PM materialsOperating diagramPermeances for main and leakage fluxes

3537

37

39

40

42424445454748495050

51

54

57

616162

65

103

6566

68

79

7979838487

90909599

Contents vii

3.2.53.2.6

Calculation of magnetic circuits with PMsFabrication of rotor magnetic circuits

3.3 Windings3.3.13.3.23.3.3

ConductorsFabrication of slotted windingsFabrication of coreless windings

Numerical examples

4. AFPM MACHINES WITH IRON CORES

4.1

4.2

4.3

4.4

4.5

Geometries

Commercial AFPM machines with stator ferromagnetic cores

Some features of iron-cored AFPM machines

Magnetic flux density distribution in the air gap

Calculation of reactances4.5.14.5.2

Synchronous and armature reaction reactancesStator leakage reactance

4.6

4.7

Performance characteristics

Performance calculation4.7.14.7.24.7.3

Sine-wave AFPM machineSynchronous generatorSquare-wave AFPM machine

4.8 Finite element calculations

Numerical examples

5. AFPM MACHINES WITHOUT STATOR CORES

5.1

5.2

5.3

Advantages and disadvantages

Commercial coreless stator AFPM machines

Performance calculation5.3.15.3.2

Steady-state performanceDynamic performance

5.4 Calculation of coreless winding inductances5.4.15.4.2

Classical approachFEM approach

5.5

5.6

Performance characteristics

Eddy current losses in the stator windings5.6.15.6.25.6.3

Eddy current loss resistanceReduction of eddy current lossesReduction of circulating current losses

107109

112112112114

116

125

125

126

127

128

130130131

134

136136138141

141

144

153

153

153

155155157

159159160

162

163163167168

viii AXIAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

5.6.4 Measurement of eddy current losses

5.7

5.8

Armature Reaction

Mechanical design features5.8.15.8.2

Mechanical strength analysisImbalanced axial force on the stator

5.9 Thermal problems

Numerical examples

6. AFPM MACHINES WITHOUT STATOR AND ROTOR CORES

6.1

6.2

6.3

6.4

6.5

6.6

Advantages and disadvantages

Topology and construction

Air gap magnetic flux density

Electromagnetic torque and EMF

Commercial coreless AFPM motors

Case study: low-speed AFPM coreless brushless motor6.6.16.6.26.6.3

Performance characteristicsCost analysisComparison with cylindrical motor with laminatedstator and rotor cores

6.7

6.8

Case study: low-speed coreless AFPM brushless generator

Characteristics of coreless AFPM machines

Numerical examples

7. CONTROL

7.1 Control of trapezoidal AFPM machine7.1.17.1.27.1.37.1.47.1.5

Voltage equationsSolid-state converterCurrent controlSpeed controlHigh speed operation

7.2 Control of sinusoidal AFPM machine7.2.17.2.27.2.37.2.4

Mathematical model and dq equivalent circuitsCurrent controlSpeed controlHardware of sinusoidal AFPM machine drive

7.3 Sensorless position control

Numerical examples

170

170

173174177

179

179

189

189

189

192

193

194

197197198

199

200

201

204

213

213214216219222222

223224229230234

237

239

Contents ix

8. COOLING AND HEAT TRANSFER

8.1

8.2

Importance of thermal analysis

Heat transfer modes8.2.18.2.28.2.3

ConductionRadiationConvection

8.3 Cooling of AFPM machines8.3.18.3.2

AFPM machines with self-ventilationAFPM machines with external ventilation

8.4 Lumped parameter thermal model8.4.18.4.2

Thermal equivalent circuitConservation of energy

8.5 Machine duties8.5.18.5.28.5.3

Continuous dutyShort-time dutyIntermittent duty

Numerical examples

9. APPLICATIONS

9.1 Power generation9.1.19.1.2

High speed generatorsLow speed generators

9.2 Electric vehicles9.2.19.2.2

Hybrid electric vehiclesBattery electric vehicles

9.3 Ship propulsion9.3.19.3.29.3.3

Large AFPM motorsPropulsion of unmanned submarinesCounterrotating rotor marine propulsion system

9.4

9.5

9.6

9.7

9.8

9.9

Electromagnetic aircraft launch system

Mobile drill rigs

Elevators

Miniature AFPM brushless motors

Vibration motors

Computer hard disc drives

Numerical examples

249

249

249250250251

255255264

267268269

270270271272

272

281

281281282

285287289

291291292292

295

297

299

302

304

306

307

x AXIAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

Symbols and Abbreviations

References

Index

311

321

337

Preface

The drop in prices of rare-earth permanent magnet (PM) materials and pro-gress in power electronics have played an important role in the development ofPM brushless machines in the last three decades. These machines have recentlybecome mature and their high efficiency, power density and reliability has ledto PM brushless machines successfully replacing d.c. commutator machinesand cage induction machines in many areas.

The axial flux PM (AFPM) brushless machine, also called the disc-type ma-chine, is an attractive alternative to its cylindrical radial flux counterpart due tothe pancake shape, compact construction and high torque density. AFPM mo-tors are particularly suitable for electrical vehicles, pumps, valve control, cen-trifuges, fans, machine tools, hoists, robots and manufacturing. They have be-come widely used for low-torque servo and speed control systems. The appli-cation of AFPM machines as generators is justified in wind turbines, portablegenerator sets and road vehicles. The power range of AFPM brushless ma-chines is now from a fraction of a watt to sub-MW.

Disc-type rotors can be embedded in power-transmission components orflywheels to optimize the volume, mass, number of parts, power transfer andassembly time. For electric vehicles with built-in wheel motors the payoff isa simpler power train, higher efficiency and lower cost. Dual-function rotorsmay also appear in pumps, elevators, energy storages and other machinery,bringing added values and new levels of performance to these products.

The authors believe that this first book in English devoted entirely to AFPMbrushless machines will serve as a textbook, useful reference and design hand-book of AFPM machines and will stimulate innovations in this field.

J.F. GIERAS, R. WANG AND M.J. KAMPER

Chapter 1

INTRODUCTION

1.1 Scope

The term axial flux permanent magnet (AFPM) machine in this book relatesonly to permanent magnet (PM) machines with disc type rotors. Other AFPMmachine topologies, e.g. transverse flux machines, have not been considered.In principle, the electromagnetic design of AFPM machines is similar to itsradial flux PM (RFPM) counterparts with cylindrical rotors. However, the me-chanical design, thermal analysis and assembly process are more complex.

1.2 Features

The AFPM machine, also called the disc-type machine, is an attractive al-ternative to the cylindrical RFPM machine due to its pancake shape, compactconstruction and high power density. AFPM motors are particularly suitablefor electrical vehicles, pumps, fans, valve control, centrifuges, machine tools,robots and industrial equipment. The large diameter rotor with its high mo-ment of inertia can be utilised as a flywheel. AFPM machines can also operateas small to medium power generators. Since a large number of poles can beaccommodated, these machines are ideal for low speed applications, as forexample, electromechanical traction drives, hoists or wind generators.

The unique disc-type profile of the rotor and stator of AFPM machinesmakes it possible to generate diverse and interchangeable designs. AFPM ma-chines can be designed as single air gap or multiple air gaps machines, withslotted, slotless or even totally ironless armature. Low power AFPM machinesare frequently designed as machines with slotless windings and surface PMs.

As the output power of the AFPM machines increases, the contact surfacebetween the rotor and the shaft in proportion to the power becomes smaller.

2 AXIAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

Careful attention must be given to the design of the rotor-shaft mechanicaljoint as this is usually the cause of failures of disc type machines.

In some cases, rotors are embedded in power-transmission components tooptimise the number of parts, volume, mass, power transfer and assembly time.For electric vehicles (EVs) with built-in wheel motors the payoff is a sim-pler electromechanical drive system, higher efficiency and lower cost. Dual-function rotors may also appear in pumps, elevators, fans and other types ofmachinery, bringing new levels of performance to these products.

Introduction 3

Most applications use the AFPM machine as a d.c. brushless motor. En-coders, resolvers or other rotor position sensors are thus a vital part of brushlessdisc motors.

Table 1.1 shows specifications of AFPM brushless servo motors rated up to2.7 kW, manufactured by E. Bautz GmbH, Weiterstadt, Germany.

1.3 Development of AFPM machinesThe history of electrical machines reveals that the earliest machines were

axial flux machines (M. Faraday, 1831, anonymous inventor with initials P.M.,1832, W. Ritchie, 1833, B. Jacobi, 1834). However, shortly after T. Davenport(1837) claimed the first patent [66] for a radial flux machine, conventional ra-dial flux machines have been widely accepted as the mainstream configurationfor electrical machines [30, 49].

The first primitive working prototype of an axial flux machine ever recordedwas M. Faraday’s disc (1831) - see Numerical Example 1.1. The disc typeconstruction of electrical machines also appears in N. Tesla’s patents, e.g. U.S.patent No. 405 858 [225] entitled Electro-Magnetic Motor and published in1889 (Fig. 1.1). The reasons for shelving the axial flux machine were multi-fold and may be summarised as follows:

strong axial (normal) magnetic attraction force between the stator and rotor;

fabrication difficulties, such as cutting slots in laminated cores and othermethods of making slotted stator cores;

high costs involved in manufacturing the laminated stator cores;

difficulties in assembling the machine and keeping the uniform air gap.

Although, the first PM excitation system was applied to electrical machines asearly as the 1830s, the poor quality of hard magnetic materials soon discour-aged their use. The invention of Alnico in 1931, barium ferrite in the 1950s andespecially the rare-earth neodymium-iron-boron (NdFeB) material (announcedin 1983) have made a comeback of the PM excitation system possible.

It is generally believed that the availability of high energy PM materials(especially rare earth PMs) is the main driving force for exploitation of novelPM machine topologies and has thus revived AFPM machines. Prices of rare-earth PMs have been following a descending curve in the last decade of the20th century with a sharp decline in the last three years. A recent marketsurvey shows that the NdFeB PMs can now be purchased in the Far East forless than U.S.$ 20 per kilogram. With the availability of more affordable PMmaterials, AFPM machines may play a more important role in the near future.


Figure 1.1. Electro-magnetic motor with disc rotor according to N. Tesla’s patent No. 405858, 1889 [225].

1.4 Types of axial flux PM machinesIn principle, each type of a radial flux machine should have its correspond-

ing axial flux (disc type) version. In practice, disc type machines are limited tothe following three types:

PM d.c. commutator machines;

PM brushless d.c. and synchronous machines;

induction machines

Similar to its RFPM counterpart, the AFPM d.c. commutator machine usesPMs to replace the electromagnetic field excitation system. The rotor (arma-ture) can be designed as a wound rotor or printed winding rotor.

Introduction 5

Figure 1.2. AFMPM 8-pole d.c. commutator motor with printed rotor winding: (a) stator withPMs, (b) cross section, (c) rotor (armature) windings and brushes, (d) construction ofwinding with 145 bars. 1 — rotor with double-sided printed winding, 2 — PMs, 3 — brushes.

In the wound rotor, the armature winding is made of copper wires andmoulded with resin. The commutator is similar to that of the conventionaltype, i.e. it can be either a cylindrical or radial commutator.

The disc-type printed armature winding motor is shown in Fig. 1.2. Therotor (armature) does not have a ferromagnetic core and its winding is similarto the wave winding of conventional d.c. commutator machines. The coilsare stamped from pieces of sheet copper and then welded, forming a wavewinding. When this motor was invented by J. Henry Baudot [16], the armaturewas made using a similar method to that by which printed circuit boards arefabricated. Hence, this is called the printed winding motor. The magneticflux of a d.c. printed winding commutator motor with a large air gap can beproduced using cost effective Alnico magnets with high remanence.

AFPM d.c. commutator motors are still a versatile and economical choicefor certain industrial, automotive and domestic applications such as fans, blow-ers, small EVs, power tools, appliances, etc.

Practically, d.c. brushless and a.c. synchronous machines have almost thesame structure, though their theory and operation principles are somewhat dif-ferent [96, 112, 172]. The main difference is in the shape of the operationcurrent waveform (Fig. 1.3), i.e.:

the d.c. brushless machine generates a trapezoidal EMF waveform and isoperated with a rectangular line current waveform (also called a square-wave machine);


Figure 1.3. Current waveforms for AFPM brushless machines: (a) square-wave machine, (b)sinewave machine.

the a.c. synchronous machine generates a sinusoidal EMF waveform and isoperated with sinewave currents (also called a sinewave machine).

It is difficult to manufacture a laminated rotor with cage winding for a disc-typeinduction machine [148]. If the cage winding is replaced with a non-magnetichigh conductivity (Cu or Al) homogenous disc or steel disc coated with copperlayer, the performance of the machine drastically deteriorates. Therefore, thereis little interest in disc type induction machines so far [148, 238].

1.5 Topologies and geometriesFrom a construction point of view, brushless AFPM machines can be de-

signed as single-sided or double-sided, with or without armature slots, withor without armature core, with internal or external PM rotors, with surfacemounted or interior PMs and as single stage or multi-stage machines.

In the case of double-sided configurations, either the external stator or ex-ternal rotor arrangement can be adopted. The first choice has the advantage ofusing fewer PMs at the expense of poor winding utilisation while the secondone is considered as a particularly advantageous machine topology [34]. Thediverse topologies of AFPM brushless machines may be classified as follows:

single-sided AFPM machines

with slotted stator (Fig. 1.4a)

with slotless stator

with salient-pole stator

Introduction 7

double-sided AFPM machines

with internal stator (Fig. 1.4b)

with slotted statorwith slotless stator

with iron core statorwith coreless stator (Fig. 1.4d)without both rotor and stator cores

Figure 1.4. Basic topologies of AFPM machines: (a) single-sided slotted machine, (b) double-sided slotless machines with internal stator and twin PM rotor, (c) double sided machine withslotted stator and internal PM rotor, (d) double-sided coreless motor with internal stator. 1 —stator core, 2 — stator winding, 3 — rotor, 4 — PM, 5 — frame, 6 — bearing, 7 — shaft.


with salient pole stator (Fig. 1.5)

with internal rotor (Fig. 1.4c)

with slotted statorwith slotless statorwith salient pole stator (Fig. 1.6)

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

Figure 1.5. Double-sided AFPM brushless machine with internal salient-pole stator and twinexternal rotor [170]: (a) construction; (b) stator; (c) rotor. 1 — PM, 2 — rotor backing steeldisc, 3 — stator pole, 4 — stator coil.

The air gap of the slotted armature AFPM machine is relatively small. Themean magnetic flux density in the air gap decreases under each slot openingdue to increase in the reluctance. The change in the mean magnetic flux densitycaused by slot openings corresponds to a fictitious increase in the air gap [111].The relation between fictitious and physical air gap is expressed with theaid of Carter coefficient i.e.

Introduction 9

Figure 1.6. Double-sided AFPM brushless machine with three-phase, 9-coil external salient-pole stator and 8-pole internal rotor. 1 — PM, 2 — stator backing ferromagnetic disc, 3 — statorpole, 4 — stator coil.

where is the average slot pitch and is the width of slot opening.For AFPM machines with slotless windings the air gap is much larger and

equal to the mechanical clearance plus the thickness of all non-magnetic ma-terials (winding, insulation, potting, supporting structure) that is passed bythe main magnetic flux. Since there are no slots, Carter coefficientCompared to a conventional slotted winding, the slotless armature winding hasadvantages such as simple stator assembly, elimination of the cogging torqueand reduction of rotor surface losses, magnetic saturation and acoustic noise.The disadvantages include the use of more PM material, lower winding in-ductances sometimes causing problems for inverter-fed motors and significanteddy current losses in slotless conductors [45].

In the double-sided, salient-pole AFPM brushless machine shown in Fig.1.5, the stator coils with concentrated parameters are wound on axially lam-inated poles. To obtain a three-phase self-starting motor, the number of thestator poles should be different from the number of the rotor poles, e.g. 12stator poles and 8 rotor poles [160, 161, 170]. Fig. 1.6 shows a double-sidedAFPM machine with external salient pole stators and internal PM rotor. Thereare nine stator coils and eight rotor poles for a three-phase AFPM machine.

Depending on the application and operating environment, slotless statorsmay have ferromagnetic cores or be completely coreless. Coreless stator con-


figurations eliminate any ferromagnetic material from the stator (armature)system, thus making the associated eddy current and hysteresis core lossesnonexisting. This type of configuration also eliminates axial magnetic attrac-tion forces between the stator and rotor at zero-current state. It is interestingthat slotless AFPM machines are often classified according to their windingarrangements and coil shapes, namely, toroidal, trapezoidal and rhomboidalforms [34, 45, 79].

Figure 1.7. Coreless multidisc AFPM machine with three coreless stators and four PM rotorunits: 1 — stator winding, 2 — rotor unit, 4 — PM, 3 — frame, 4 — bearing, 5 — shaft.

1.6 Axial magnetic field excited by PMsA double-sided AFPM machine with twin PM rotor in xyz rectangular co-

ordinate system is shown in Fig. 1.8. Assuming that the radius of curvatureis higher than the pole pitch and the centre axes of the opposite rotor polesarc shifted by a linear distance the normal component of the magnetic fluxdensity on the surface of the rotor can be described in the stationary xyz coor-dinate system by the following equations:

at

at

Introduction 11

Figure 1.8. Twin-stator double-sided AFPM machine in Cartesian coordinate system.

where is the value of the normal component in the center axis of the Northpole, and


Figure 1.9. Distribution of the normal components of the magnetic flux density according toeqns (1.4), (1.5) and (1.8) for (a)

(b)

In the above eqns (1.4), (1.5), (1.6), (1.7), and (1.8) is the linear speedof the rotor in the is the rotational speed in rev/s and theparameter depends on the shape of the distribution of the normal componentof the magnetic flux density (Fig. 1.9). For the flat-topped curve andfor the concave curve (armature or eddy-current reaction) Thecoefficient according to eqn (1.8) has been derived in [93].

The average diameter D of the PM excitation system and correspondingaverage pole pitch are:

Introduction 13

where is the inner diameter of PMs, is the outer diameter of PMs andis the number of poles.The electromagnetic field analysis in AFPM brushless machines has been

discussed in e.g. [90, 91, 250, 251].

1.7 PM eddy-current brake as the simplest AFPMbrushless machine

A double-sided, PM excited eddy-current brake with high conductivity non-magnetic disc-type rotor is one of the simplest brushless AFPM machines (Fig.1.8). In an eddy-current brake the PM excitation system is stationary and theconductive rotor rotates at the speed Eqns (1.4) to (1.9) are valid sincethe stationary PM excitation system and rotating electric circuit (armature) areequivalent to the rotating PMs and the stationary electric circuit.

It is assumed that the eddy currents in the non-magnetic conductive discflow only in the radial direction, i.e., in the (Fig. 1.8). Thus, themagnetic vector potential in the disc is described by the following scalarequation (2D analysis):

where

In eqns (1.11) and (1.12) the electric conductivity depends on the disc tem-perature. The relative magnetic permeability of paramagnetic (Al) or diamag-netic (Cu) materials The angular frequency for higher space harmonicsis according to eqn (1.7) or

General solution to eqn (1.10) can be written, for example, as


where

Since the currents in the disc flow only in the radial directionand for Using the magnetic vector

potential and the second Maxwell’s equationthe remaining electric and magnetic components in the disc can be found as

The integration constants and can be found on the basis of equalityof normal components of the magnetic flux density in the air and in the disc at

and i.e.

Introduction 15

at

at

There is only backward-rotating magnetic field in the air gap of an eddy-currentbrake, so that the terms and in eqns (1.4) and (1.5) are withthe + sign. Thus,

because Putting eqn (1.21) into (1.18), (1.19) and(1.20), the particular solution to eqn (1.10) for and compo-nents are


The surface wave impedance for the space harmonic is calculated on thebasis of eqns (1.22) and (1.23)

where

The impedance of the disc for the space harmonic

where is the outer diameter of PMs, is the inner diameter of PMs,is the average pole pitch (1.9) and is the impedance increase factor due

to circumferential currents (in the direction). The impedance increase factorfor the harmonic is [62]

1.8 AFPM machines versus RFPM machinesIn pace with the application of new materials, innovation in manufacturing

technology and improvements in cooling techniques, further increase in thepower density (output power per mass or volume) of the electrical machine hasbeen made possible. There is an inherent limit to this increase for conventionalradial flux PM (RFPM) machines because of [27, 49, 96, 153, 177]:

Introduction 17

the bottle-neck feature for the flux path at the root of the rotor tooth in thecase of induction and d.c. commutator machines or brushless machineswith external rotors (Fig. 1.10);

much of the rotor core around the shaft (rotor yoke) is hardly utilised as amagnetic circuit;

heat from the stator winding is transferred to the stator core and then to theframe — there is poor heat removal through the stator air gap, rotor andshaft without forced cooling arrangements.

These limitations are inherently bound with radial flux structures and cannotbe removed easily unless a new topology is adopted. The AFPM machine,recognised as having a higher power density than the RFPM machine, is morecompact than its radial flux counterpart [26, 49, 96, 153].

Figure 1.10. Topologies of (a) RFPM machine (b) AFPM machine.

Moreover, since the inner diameter of the core of an AFPM machine is usu-ally much greater than the shaft diameter (see Fig. 1.4), better ventilation andcooling can be expected. In general, the special properties of AFPM machines,which are considered advantageous over RFPM machines in certain applica-tions, can be summarised as follows [48, 96]:

AFPM machines have much larger diameter to length ratio than RFPMmachines;

AFPM machines have a planar and somewhat adjustable air gap;


capability of being designed to possess a higher power density with somesaving in core material;

the topology of an AFPM machine is ideal to design a modular machinein which the number of the same modules is adjusted to power or torquerequirements;

the larger the outer diameter of the core, the higher the number of poles thatcan be accommodated, making the AFPM machines a suitable choice forhigh frequency or low speed operations.

Consequently, AFPM type machines are particularly suitable for servo, trac-tion, distributed generation and special-purpose applications where their prop-erties offer distinct advantages over their conventional RFPM counterparts.

Figure 1.11. Performance comparison of RFPM and AFPM machines [214].

The quantitative comparison between traditional RFPM machine and AFPMmachine is always difficult as it may raise the question of this comparison’sfairness. Some published work dealt with quantitative investigations of RFPMand AFPM machine configurations in terms of sizing and power density equa-tions [8, 49, 121, 247]. Fig. 1.11 gives the performance comparison betweena conventional RFPM machine and a number of AFPM machines of different

Introduction 19

configurations at five different power levels [214], which shows that the AFPMmachine has a smaller volume and less active material mass for a given powerrating than the RFPM machine.

1.9 Power limitation of AFPM machinesThe power range of AFPM disc-type brushless machines is now from a

fraction of a Watt to sub-MW. As the output power of the AFPM machineincreases, the contact surface between the rotor and shaft becomes smaller incomparison with the rated power. It is more difficult to design a high me-chanical integrity rotor-shaft mechanical joint in the higher range of the outputpower. A common solution to the improvement of the mechanical integrity ofthe rotor-shaft joint is to design a multidisc (multi-stage) machine (Fig. 1.7).

Since the scaling of the torque capability of the AFPM machine as the cubeof the diameter (see eqn (2.94)) while the torque of a RFPM machines scale asthe square of the diameter times the length, the benefits associated with axialflux geometries may be lost as the power level or the geometric ratio of thelength to diameter of the motor is increased [174]. The transition occurs nearthe point where the radius equals twice the length of RFPM machine. Thismay be a limiting design consideration for the power rating of a single-stagedisc machine as the power level can always be increased by simply stacking ofdisc machines on the same shaft and in the same enclosure.

Numerical example 1.1A copper disc with its dimensions as shown in Fig. 1.12 rotates with the

speed of 12 000 rpm between U-shaped laminated pole shoes of a PM. A slid-ing contact consisting of two brushes is used to collect the electric current gen-erated by this primitive homopolar generator: one brush slides on the externaldiameter and the second brush is located directly below oneof the poles at the distance of from the axis of the disc. Theremanent magnetic flux density of the NdFeB PM is coercivityis and height The thickness of the discis one-sided air gap is and the width of the poleis The relative magnetic permeability of the laminated core is

and the conductivity of the disc is at 20°C. Thelength of the flux path in the laminated core is Find:

(a)

(b)

(c)

the magnetic flux density in the air gap;

the EMF between brushes;

the current, if the line resistance is and load resistance is


Figure 1.12. Faraday’s disc: a nonmagnetic conductive disc rotating in a stationary magneticfield according to Numerical example 1.1.

The magnetic flux fringing effect in the air gap, the variation of magnetic per-meability with the magnetic field intensity in the laminated core and brushvoltage drop are neglected.

Solution

This is a homopolar type d.c. generator known as Faraday’s disc and can beused as a current source, e.g. for electrolysis.

(a) Magnetic flux density in the air gap

The relative recoil magnetic permeability

The magnetic flux density in the air gap and saturation factor of the magneticcircuit can be found on the basis of Kirchhoff ’s magnetic voltage law, i.e.

where the saturation factor of the magnetic circuit

axial flux permanent magnet brushless machines · generator sets and road vehicles. the power range...

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