Axial Flux Permanent Magnet Brushless Machines

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Axial Flux Permanent MagnetBrushless Machines

byJACEK F. GIERASUnited Te chnolo gie s Re s e arch C ente 4East Harford, Connecticut, U.S.A.

RONG-JIE WANGUniv ersity of Stellenbosch,Stellenbosch, Western Cape, South Africa

and

MAARTEN J. KAMPERUniv e rsity of Ste llenbo s ch,Stellenbosch, Westem Cape, South Africa

W&wwKLUWER ACADEMIC PUBLISHERSDORDRECHT / BOSTON / LONDON

1 .

Contents

Preface

INTRODUCTION1.1 ScopeI.2 FeaturesI.3 Development of AFPM machinesI.4 Types of axial flux PM machines1.5 Topologies and geometries1.6 Axial magnetic field excited by PMsI.1 PM eddy-current brake as the simplest AFPM

brushless machine1.8 AFPM machines versus RFPM machinesI.9 Power limitation of AFPM machinesNumerical examples

PRINCIPLES OF AFPM MACHINES2.I Magnetic circuits

2.I.1 Single-sidedmachines2.I.2 Double-sided machines with internal PM disc rotor2.1.3 Double-sided machines with internal ring-shaped

core stator2.I.4 Double-sided machines with internal slotted stator2.1.5 Double-sided machines with internal coreless stator2.I.6 Multidiscmachines

2.2 Windings2.2.1 Three-phase windings distributed in slots2.2.2 Drum-type winding

xi

I1IaJ

2.

46

10

t3I6t9I927272727

293 l) L

3 Z

a aJ J

a aJ J

35

Vi AXIAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

2.2.3 Coreless stator winding2.2.4 Salient pole windings

2.3 Torque production2.4 Magnetic flux2.5 Electromagnetic torque and EMF2.6 Losses and efficiency

2.6.1 Stator winding losses2.6.2 Stator core losses2.6.3 Core loss finite element model2.6.4 Losses in permanent magnets2.6.5 Rotor core losses2.6.6 Eddy current losses in stator conductors2.6.7 Rotationallosses2.6.8 Losses for nonsinusoidal current2.6.9 EfficiencyPhasor diagramsSizing equationsArmature reactionAFPM motor2.10.1 Sine-wave motor2.10.2 Square-wave motorAFPM synchronous generator2.IIl Performance characteristics of a stand alone

generator2.II.2 Synchronization with utility grid

2.12.82.92.r0

353737394042424445454l48495050515457616 l6265

65666879797983848790909599

103

2 . l l

Numerical examples

3. MATERIALS AND FABRICATION3.1 Stator cores

3.1.1 Nonoriented electrical steels3.1.2 Amorphous ferromagnetic alloys3.1.3 Soft magnetic powder composites3.L4 Fabrication of stator cores

3.2 Rotor magnetic circuits3.2.1 PM materials3.2.2 Characteristics of PM materials3.2.3 Operating diagram3.2.4 Permeances for main and leakage fluxes

Contents

3.2.5 Calculation of magnetic circuits with PMs3.2.6 Fabrication of rotor masnetic circuits

3.3 Windings3.3.I Conductors3.3.2 Fabrication of slotted windings3.3.3 Fabrication of coreless windines

Numerical examples

4. AFPM MACHINES WITH IRON CORES 1254.I Geometries I254.2 Commercial AFPM machines with stator ferromagnetic cores 1264.3 Some features of iron-cored AFPM machines 1274.4 Magnetic flux density distribution in the air gap I284.5 Calculation of reactances 130

4.5.I Synchronous and armature reaction reactances 1304.5.2 Stator leakage reactance I3I

4.6 Performance characteristics I344.7 Performance calculation 136

4.7.I Sine-wave AFPM machine 1364.7.2 Synchronous generator 1384.7.3 Square-wave AFPM machine l4I

4.8 Finite element calculations I4lNumerical examples 144

5. AFPM MACHINES WITHOUT STATOR CORES5.1 Advantages and disadvantages5.2 Commercial coreless stator AFPM machines5.3 Performance calculation

5.3.1 Steady-state performance5.3.2 Dynamicperformance

5.4 Calculation of coreless winding inductances5.4.1 Classical approach5.4.2 FEM approach

5.5 Performance characteristics5.6 Eddy current losses in the stator windings

5.6.1 Eddy current loss resistance5-6.2 Reduction of eddy current losses5.6.3 Reduction of circulatins current losses

vll

107109It2l 12I12tt4116

153r53r53155155r57r59159160162r63r63r67168

vllt AXIAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

5.6.4 Measurement of eddy current losses5.7 Armature Reaction5.8 Mechanical design features

5.8.1 Mechanical strength analysis5.8.2 Imbalanced axial force on the stator

5.9 Thermal problemsNumerical examples

6. AFPM MACHINES WITHOUT STA|OR AND ROTOR CORES 1896.I Advantages and disadvantages 1896.2 Topology and construction 1896.3 Air gap magnetic flux density I926.4 Electromagnetic torque and EMF I936.5 Commercial coreless AFPM motors I946.6 Case study: low-speed AFPM coreless brushless motor 197

6.6.1 Performance characteristics 1976.6.2 Cost analysis 1986.6.3 Comparison with cylindrical motor with laminated

stator and rotor cores 1996.7 Case study: low-speed coreless AFPM brushless generator 2006.8 Characteristics of coreless AFPM machines 20INumerical examples 204

170n0173174t77179n9

7. CONTROL7 .I Control of trapezoidal AFPM machine

7.1.I Voltageequations7.1.2 Solid-stateconverter7.I.3 Current control7.1.4 Speed control7.1.5 High speed operation

7.2 Control of sinusoidal AFPM machine7.2.I Mathematical model and dq equivalent circuits7.2.2 Current control1.2.3 Speed control7.2.4 Hardware of sinusoidal AFPM machine drive

7.3 Sensorless position controlNumerical examples

2t32t3214216219222222223224229230234^ a aL J I

239

8 .

Contents

COOLING AND HEAT TRANSFER8.1 Importance of thermal analysis8.2 Heat transfer modes

8.2.1 Conduct ion8.2.2 Radiation8.2.3 Convect ion

8.3 Cooling of AFPM machines8.3.1 AFPM machines with self-ventilation8.3.2 AFPM machines with external ventilation

8.4 Lumped parameter thermal model8.4.1 Thermal equivalent circuit8.4.2 Conservation of energy

8.5 Machine duties8.5.1 Continuous duty8.5.2 Short-time duty8.5.3 Intermittentdutv

Numerical examples

APPLICATIONS9.1 Power generation

9.I.1 High speed generators9.I.2 Low speed generators

9.2 Electric vehicles9.2.1 Hybrid electric vehicles9.2.2 Battery electric vehicles

9.3 Ship propulsion9.3.1 Large AFPM motors9.3.2 Propulsion of unmanned submarines9.3.3 Counterrotating rotor marine propulsion system

9.4 Elecffomagnetic aircraft launch system9.5 Mobile drill rigs9.6 Elevators9.7 Miniature AFPM brushless motors9.8 Vibration motors9.9 Computer hard disc drivesNumerical examples

IX

24924924925025025r25525526426726826927027027r272272

28128128r282285287289291291292292295297299302304306307

9.

AXIAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

Symbols and Abbreviations

References

Index

3II

32rJ J I

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 particulady 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 builfin wheel motors the payoff isa simpler power train, higher effrciency and lower cost. Dual-function rotorsmay also appear in pumps, elevators, energy storages and other machinery,bringing added values and new levels ofperfonnance 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. Grpnas, R. WRIrtc AND M.J. Kavppn

Chapter 1

INTRODUCTION

1.1 ScopeThe term axialfl.ux permanent magnel (AFPM) machine in this book relates

only 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 FeaturesThe AFPM machine, also called the disc-type machine, is an attractive a1-

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.


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

Quantity S632D S634D S712F 5714F S8O2F S8O4F

Rated power, WRated torque, NmMaximumtorque, NmStandstilltorque, NmRated current, AMaximum current, AStandstill current. ARated speed, rpmMaximumspeed, rpmArmature constant,V/l000 rpmTorque constant,Nm/AResistance, QInductance, mHMoment of inertia,kgm2 x 10-3Mass, kgDiameterof frame, mmLengthof frame, mmPowerdensity, WkgTorquedensity, Nmlkg

6801 . 3

7

t . 74.02 l5 .3

5000

6000

23

0.352.5- ) -L

0.084.5

150

82

1 5 1 . 1

0.289

940t . 8

o

2.34.9256.3

5000

6000

25

0.39t . 82.8

0 . 1 25 .0

150

82

188 .0

0.16

9 1 02.9

I4

3 .54.9245 .9

3000

6000

^ a

0.642.45.4

0 .216.2

174

89

146.8

0.468

r2604 .0

t 8

4.76.6307.8

3000

6000

42

0.641 . 54.2

0.3o . o

t74

89

190 .9

0.606

1 8s05 .9

28

7.09.9471t.73000

6000

42

0.640.763 .0

0.69.7

210

103

t90.7

0.608

26708.5

40

10 .0I 1 .956

14.03000

6000

50

0.770.623 .0

1 . 010 .5

210

103

254.3

0.809

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 sensofs are thus a vital part of brushlessdisc motors.

Table 1 .1 shows specifications of AFPM brushless servo motors rated up to2.7 kW, manufacturedby 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 ta-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 l.l. The disc typeconstruction of electrical machines also appears in N. Tesla's patents, e.g. U.S.patent No. 405 858 12251entitled Electro-Magnetic Motor and published in1889 (Fig. l.l). The reasons for shelving the axial flux machine were multi-fold and mav 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 aft 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 I 93 1, barium ferrite in the 1 950s 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 l.l. Electro-magnetic858, 1889 [225] .

AXTAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

I n t v f : T n n

motor with disc rotor according to N. Tesla's patent No. 405

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:

r PM d.c. commutator machines;r PM brushless d.c. and synchronous machines;

r 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

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 of 2p : gwinding with 145 bars. I - rotor with double-sided printed winding, 2 - pMs, 3 - brushes.

In the wound rotor, the almature 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-fiipe 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.:

r the d.c. brushless machine generates a trapezoidal EMF waveform and isoperated with a rectangular line current waveform (also called a square-wqve mzchine):

ANAL FLUX PERMANENT MAGNET BRUSHLESS MACHTNES

?1

Phnse A

PlusE B

130 t80 240 300 16f{ B }

0 t20 240(b)

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

r 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 11481. 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 1148, 238].

1.5 Topologies and geometriesFrom a consfiuction 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 intemal 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 affangement 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:

r single-sided AFPM machines

with slotted stator (Fig. 1.4a)with slotless statorwith salient-pole stator

Introduction

double-sided AFPM machines

with internal stator (Fig. l.ab)x with slotted statorx with slotless stator

. with iron core stator

. with coreless stator (Fig. l.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 intemal stator. Istatorcore, 2 - statorwinding, 3 -rotor, 4-PM, 5 -frame, 6-bearing, 7 shaft.

AXIAL FLUX PERMANENT MAGNET BRUSHLESS MACHII,{ES

* with salient pole stator (Fig. 1.5)with internal rotor (Fig. 1.ac)x with slotted statorx with slotless statorx with salient pole stator (Fig. L6)

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

Figure 1.5. Double-sided AFPM brushless machine with internal salient-pole stator and twinextemal rotor [170]: (a) construction; (b) stator; (c) rotor. 1 PM,2 rotor backing steeldisc, 3 - stator pole, 4 - stator coi1.

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 I I I 1 ].The relation between fictitious g' and physical air gap g is expressed with theaid of Carter coefficient kc ) I,i.e.

g' : gkc

, t 14 * 7 9

( 1 .1 )

( r .2)

+l*arc,an (*) (1 .3 )

Introduction

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

where 11 is the average slot pitch and b1a 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, suppofting structure) that is passed bythe main magnetic flux. Since there are no slots, Carter coefficient kc : I.Compared to a conventional slotted winding, the slotless arrnature winding hasadvantages such as simple stator assembly, elimination of the cogging torqueand reduction of rotor surface losses, magnetic safuration 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 [60, 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-

l 0 AXIAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

figurations eliminate any fenomagnetic material from the stator (armature)system, thus making the associated eddy cuffent 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 windingalrangements and coil shapes, namely, toroidal, trapezoidal and rhomboidalforms [34, 45,79J.

Figure I .7. Coreless multidisc AFPM machine with three coreless stators and four PM rotoruaits: 1 statorwinding, 2 rotorunit, 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 ryz rectangular co-

ordinate system is shown in Fig. l.B. Assuming that the radius of curvatureis higher than the pole pitch and the centre axes of the opposite rotor polesare shifted by a linear distance 16, the normal component of the magnetic fluxdensity on the surface ofthe rotor can be described in the stationary rAz coot-dinate system by the following equations:

. a t z :0 .Ysd

oo^ \ - -B - , t ( r . t ) - B o \ , b , c o s ( " ; " ! , d , r - ; )

" ' 2u - l

t a t z : - 0 . 5 d

(1 .4)

Introduction

\ Pole Pitch

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

l l

where .86 is the value of the nornal component in the center axis of the Northpole, and

(1 .6)

er: ]uu - BrrrDn ( r .7)

^ 1 7lJu - I/*

T

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

oo

B,n,z(r,t) : Bo\U"cosl.,.,tT A,@ ,o) ; | (1.5)I l l- J

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

"rn(r&)

. 4A _u u - -

r

pole pitch

( 1 .8 )

12 AXTAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

(a) 0.9

B m z t ( x , r )

B mzz ( x. i )

0.9 09

0.3B mz1 (x,t)

: - 0B n r a r x ' t )

-0.1

0.3 l l i i I i i l l, I l l i i lr i l r:i \i\ / ,n \ \r/ i \ \ l','\'rl l i t . l lllitli:llililifrLli---.---.L r l--ll-i UJ-t lJI I I l l I r i l , l

i i i l

: l

(b)

0.1 0 . l5 0 .2 0 .25 0 .1

x z/r

0.05 0 .1 0 . r5 0 .2 0 .25 0 .3x zpr

Q.o4 ;

on

rD) t t

ft* fft'l ifflr.Ylrl N,4ri iM,i M,1l l i l i r l r , l t l l l , r l l ,' ,j \ \ l l \ \i/i \ r lir \liliilillil.i trr{-u, rtj i[i]'i [il

'0 9 -0.9

00

Figure 1.9. Distribution of the normal components of the magnetic flux density according toeqns (1 .4 ) , (1 .5 ) and (1 .8 ) fo r r - 0 .04 ,m,b , :0 .03 m, Bo : 0 .6 T , t : 0 , ro : 0 .52 : (a )a : 0. (b) cv - 0.8.

In the above eqns (1.4), (1.5), (1.6), (1.7), and (1.8) u : ?rr is the linear speedof the rotor in the z-direction, n - n D lu is the rotational speed in rev/s and theparameter a depends on the shape of the distribution of the normal componentof the magnetic flux density (Fig. 1.9). Forthe flat-topped curve a : 0 andfor the concave curve (armature or eddy-current reaction) 0 <

Introduction

where Din is the inner diameter of PMs, Do6 is the outer diameter of PMs and2p is the number of poles.

The electromagnetic field analysis in AFPM brushless machines has beendiscussed in e.g. 190, 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 n. Eqns (l.a) 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 gr-direction (Fig. 1.8). Thus, themagnetic vector potential ,4 in the disc is described by the following scalarequation (2D analysis):

t3

o, A^0, *

o, A^0, =

^-

aF- t ar, : ()'AmY' ( 1 .10 )

where

a, - t/ ja,Fo4ro : (I + j)k" ( 1 .11 )

(1 .12 )

In eqns (1.11) and (1.12) the electric conductivity o depends on the disc tem-perature. The relative magnetic permeability of paramagnetic (A1) or diamag-netic (Cu) materials Fr x I. The angular frequency forhigher space harmonicsis according to eqn (1.7) or

u , :Z t r f , ; f " - - u f ; u : I , 3 , 5 , . . .

General solution to eqn ( I . l0) can be written, for example, as(1 . l 3 )

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

l 4

where


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

u t l u - ( 1 . I 6 )

(1 .17 )

Since the curents in the disc flow only in the radial direction U, E*, : 0Ern" : 0 and Brno

_: 0 for *0.5d I z ( 0.5d. Using the magnetic vector

potential Y x A : E andthe second Maxwell's equation i "

E : *08 lAt,the remaining electric and magnetic components in the disc can be found as

E^y: - jarA*u

coti-I j t "s in ( -u ,1 I 3 , ru:1

rc\,-2u : l

[)lot" exp(- n,z) * Az, exp(n,z')l ( 1. 18)

12 flAmaurnr - - 0,

[ ) l^ r " exp(- n,z) - Az, exp(n,z)) (1.19)

Brn":9+*O T

: I p, rlos(a,t * p,r * I)(Or, exp(.* n,z) I Az, exp(n,z)) (1 .20)u I

The integration constants 41, and 42, can be found on the basis of equalityof normal components of the magnetic flux density in the air and in the disc atz - A.Sd and z : -0.5d. i .e.

Introduction 15

r a t z : 0 . 5 d

p, cos(u,t * g,r - I)Vr, exp(-n,d,f 2) -t Az, exp(n,d,l2))z -

: Bobrcos(wrt * 0rr -

t a t z : - 0 . 5 d

p, coslw,t + g,@ - ro) - Xllqr" exp(-n,d,f 2) + 42, exp(n,cll2)l

: Bob, coslu,t + 0,(r - *ol - [lThere is only backward-rotating magnetic field in the air gap of an eddy-currentbrake, so that the terms p,r and 0"@ * 16) in eqns (1.4) and (1.5) are withthe * sign. Thus,

1 - , s i nh ( r c ,d "12 ) 11 - , 1Atu: 42,: nBob"?ffi#

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

because sinh(2e) : 2 sinh r cosh r. Putting eqn (1.2 1) into (1. 1 8), (1. 1 9) and(1.20), the particular solution to eqn (1.10) for E*0, B*, and B^" compo-nents are

c c o ) b r l , ^ 7 7Ems: -2t""TeCiqsin(u,t t g,r - 7)cosh(rc,2) ( t .22)

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)

f r \t)

16 AXIAL FLUX PERMANENT MAGI,{ET BRUSHLESS MACHINESThe surface wave impedance for the uth space harmonic is calculated on thebasis of eqns (1.22') and (1.23)

zu - r u * j r , : - t r o t i r l * , ) z o i : r j : * : " n t t , { o , f ; )

: l(Bp,Ap, - Bx,Ax,) + j(Bx,Aa, - Po,Ar,) l? Q.25)

where

a - ,t L t T u -sinh(ap"k"d)

cosh(ct prk,d) - cos(ay,k,d) (r .26)

( t .27)

(r.28)

- sin(ay"k"r|)cosh(a p, k, d) - cos(a y

"k " d)

The impedance of the disc for the yth space harmonic

BR,:G#Te)

0.5(Do.t - Dn) ,

- -;- Kzur / u

vL u - a u

(r .29)where Do6 is the outer diameter of PMs, D;n is the inner diameter of pMs,z is the average pole pitch (1.9) and k"r rsthe impedance increase factor dueto circumferential cunents (in the r direction). The impedance increase factorfor the zth harmonic is [62]

k " , : 71 I (1 .30 )u2 o.b(Dou, - Dm)

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 127 , 49,96, 153, l77l:

a2, t 0.b1D.,, -0 Ku r lu

Din ) .

d .- Kru cotlr\ti,ur)

Introduction

r 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);

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

r 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 RI'PM machine, is morecompact than its radial flux counterpart L26,49,96, 153].

(a) 0)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 AIPM machines,which are considered advantageous over RFPM machines in certain applica-tions, can be summarised as follows [48, 96]:r AFPM machines have much larser diameter to lensth ratio than RFPM

machines;

r AFPM machines have a planar and somewhat adjustable ak gap;

L7

18 AXIAL FLUX PERMANENT MAG],{ET BRUSHLESS MACHINES

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

r 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;

r 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 1ow 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 advantases over their conventional RFPM countemarts.

600

? soo:.i.i :oo

L 2009i* roo

a

9m

80m

- 7000

I uoooE 5oo()Ei cmo

$ rooo&.*

rm00

500

t 400

'6' 100

!: ?00

1S

0.25

o ,EEE 2001

1 l

Odpdpowr, kv

l l 5

Outd Fwr, kW

0.25

O IEEE 2OOI

0,25cJ l rEE t00l

0.25o IEEE 2001

I ] J ] O

Or&u pow. kW Odpd powr, kW

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

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, 2a71. Fig. 1 . 1 I gives the performance comparison betweena conventional RFPM machine and a number of AFPM machines of different

Introduction

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 (2.9\) 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 increasedLlT4l. 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 D,tut : 0.232 m and the second brush is located directly below oneof the poles at the distance of O.\Din : 0.03 m from the axis of the disc. Theremanent magnetic flux density of the NdFeB PM is B, : 7.25 T, coercivityis H. : 950 000 A/m and height 2hp1 :0.016 m. The thickness of the discis d : 0.005 m, one-sided a* gap is 9 : 0.0015 m and the width of the poleis bo : 20 mm. The relative magnetic permeability of the laminated core isF, : 1000 and the conductivity of the disc is o : 57 x 106 S/m at 20"C. Thelength of the flux path in the laminated core is lp" - 0.328 m. Find:

(a) the magnetic flux density in the air gap;

(b) the EMF between brushes;

(c) the current, if the line resistance is El : 0.001 f, and load resistance isRr - 0.02 O.

t9

20 AXIAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

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

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 cunent source, e.g. for electrolysis.

(a) Magnetic ffux density in the air gapThe relative recoil magnetic permeability

IAB \ 1 .25 -0t ' r t 'e( po LH

-

0.4r x lO-b 950000 0r.047

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.

u' 'ont : Bn 2tttt +

Fjllrrec F}Frrecwhere the saturation factor of the masnetic circuit

B" B" f?' ' 2 h n t - " e 2 h y + 1 ! 2 g + H r r l r "

l-LO Frrec llollrrec LtO

Ra2gk"otP0

L F".I .k s a t : 7 * 2p, (g + 0 .5d)

- 1 |- T T

2 x 1000(0.001i * 0 .5 x 0.005):7 .042

Introduction

Thus, the air gap magnetic flux density is

2 l

8,,k

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

: t ' 25 , , :o .8o9T1+(0 .0015*0 .5 x 0 .005)1 .042 x 1000/0 .008 " '(b) EMF between brushes

Since Elt : ?- x En o, d,E : Bnudr where n : ,ur : Ztrrn is thelinear speed, r is the radius ofthe disc and n is the rotational speed in rev/s,dE : Bs(2nrn)dr. Thlus

p0.5Dea1 *- 12n : I Bnl2rrn)d7 -' l l ' ! '-e (n7"., - n?^\J o . s D ; , , Y \ 4 \ - o u r * ' ' )

: 4#ry (o.zsz, _ 0.006,) :6 83v(c) Current, if the line resistance is El : 0.001 Q and load resistance is /?1 :0.02 0.

Neglecting the current fringing effect in the disc, the resistance of the discfor induced current at 20oC is

n 0 .5 (Dod - D" ) 0 .5 (0 .232 - 0 .06)Hax- - -# :+ :0 .0000151 Q- odbo 57 x 106 x 0.005 x 0.02The current is

I : - - : _ : - . . , , , , , , 6 ' 8 ,3 , , : 325 . \- Ra-r RL ]- Rt 0.0000151 + 0.001 + 0.02 - " 'The te rm ina l vo l t age i sV : I x Ry :325 x 0 .02 :6 .5Vand the l i nevoltage drop is LV : I x R1 :325 x 0.001 : 0.325 V.

Numerical example 1.2Find the impedance of the aluminum disc of a double-sided eddy cument

brake at ambient temperature 20"C. The inner diameter of the PMs is Drr, :0.14 m, the outer diameter of the PMs rs Do6 : 0.242, thickness of the discd : 3 mm, number of poles 2p : 16 and speed n : 3000 rpm. Assumethe conductivity of the aluminum o : 30 x 106 S/m at2}oC and its relativemagnetic permeability Fr = t.

22 AATAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

Solution

The average diameter of the disc

D : 0 .5 (Dm * Do, t ) : 0 .5 (0 .14 + 0 .232) : 0 .191 mThe average pole pitch

rD n'0.191t : 2 p :

t 6 - o ' 0 3 7 m

The frequency and angular frequency of the curent in the disc for fundamentalspace harmonic u :7

. ^3000f p , - 8 i r r : 400H2

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

The attenuation factor for the fundamental space harmonic u : I of the elec-tromagnetic field in the disc according to eqn ( 1 .12) is

t." - : 277.66 1lm

The coefficients according to 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 :

s inh( l .038 x217.66 x 0.003)coslr(1.038 x217.66 x 0.003) - cos(0.964 x277.66 x 0.003)1 .699

- sin(0.964 x 217.66 x 0.003)cosh(1.038 x 277.66 x 0.003) - cos(O.964 x 277.66 x 0.003)-1 .369

Ap

Aa

2513.3 x 0.4r10-6 x 1.0 x 30 x 106

Introduction

. 0.0022. lo- '

0.0015

n(v)

tr o'oo1

s . to a

0 0

L-)

1 v 4 lFigure I . I 3 . Resistance and reactance of the copper disc for higher space harmonics z accord-ingto Numerical example 1.2.

0.964B R : - 0.961

0.5(1.0382 + 0.9642)

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

The surface wave impedance for u : I according to eqn (1.25)

r : [0.9G1 x 1.699 - 1.0s5 x (-1.869)] f f i :2.2128 x 10-5 Q

r : [1.035 x 1.699 + 0.961 x (-r.369)] H,6:^ :3.2L6 x 10 6 f)"57 x 10o

z : 2.2123 x 10-5 + j3.216 x 10-6 QThe impedance increase factor due to circumferential currents (in the r- direc-tion) according to eqn (1.30) is

k,, : r+ I = =,= 9 '937 = r , , : 1.23312 0.5(0.242 - 0 .14)The disc impedance per pole for the fundamental space harmonic u : 1 ac-cording to eqn (1.29)

1 3 5 7 9 1113 t517 r92 r2325n2931333537394 r

1 AL+ AXIAL FLUX PERMANENT MAG],{ET BRUSHLESS MACHII,IES

z - (2.2123 x l0 5 * f i .216x l0 6 l93f f i t . tou

: 5 .231 x 10-5 + j7 .605 x 10-6 o

R : 5.231 x 10-5 f) X : 7.605 x 10-6 OThe disc resistance R, and reactance X, for higher space harmonics v ) 1 isplotted in Fig. 1.13.

Numerical example 1.3Estimate the cost of materials of a 3-phase, 250 kW 2300 rpm brushless

AFPM machine. The internal stator does not have any ferromagnetic core.The twin-type external rotor (Fig. L4d) consists of sintered NdFeB PMs (40poles) glued to two solid steel discs. The mass of the stator copper conductorsrs'm6u: 18.7 kg, mass of the stator winding insulation (including encapsu-lation) is mirr: 1.87 kg, mass of PMs is rnpy : 32.1kg, mass of the twinrotor core is mr": 50.8 kg (two discs) and mass of the shaft is m"h : 48.5kg. The cost of materials in U.S. dollars per kilogram is: copper conduc-tor cgu : 5.51, insulating materials c'irts : 7.00, sintered NdFeB magnetscpM :20.00 and carbon steel c"1""7 : 0.65. The cost of frame, end discs(bells) and bearings is C1 : $164.00. The cost of components that are in-dependent of the machine geometry (encoder, terminal leads, terminal board,nameplate) is Cs : $140.00.

Coefficients taking into account the manufacturing and utilisation of steelfor the shaft and rotor core (steel discs) are as follows:r total volume of the steel bar to the volume of the shaft, ku"h: 1.94

r coefHcient taking into account the cost of machining of the shaft, k-"7, :2 . 1 5

r total volume of the steel plate to the volume of the rotor discs, kr, : !.1

. coefficient taking into account the cost of machining of the rotor steel discs,k^, : \ .8

Solution

The cost of the copper winding

C- : nlgrcC,: 78.7 x 5.51 : $103.12

Introduction

The cost of the insulation and encapsulation

Cir" : Tr l insc' in": 1.87 x 7'00 : $13'10The cost of PMs

Cpt',r : rnpMcpM :32.7 x 20.00 : $640.80The 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.94 x 2.I5 x 48.5 x 0.65 : $131.55Total cost of materials

C : Cf t Co + C. + C,rnr l Cpm * Cr.l Crn

: 164.00* 140.00+ 103.12+ 13.10+640.80+ 724.85+ 131.55 : $ t3t7.a5

Chapter 2

PRINCIPLES OF AFPM MACHINES

In this chapter the basic principles of the AFPM machine are explained indetails. Considerable attention is given to the magnetic circuits, windings,torque production, losses, equivalent circuits, sizing procedure, armature reac-tion and performance characteristics of AFPM machines.

2.1 Magnetic circuits2.1.1 Single-sided machines

The single-sided construction of an axial flux machine is simpler than thedouble-sided one, but the torque production capacity is lower. Fig.2.1 showstypical constructions of single-sided AFPM brushless machines with surfacePM rotors and laminated stators wound from electromechanical steel strips.The single-sided motor according to Fig. 2.lahas a standard frame and shaft.It can be used in industrial, traction and servo electromechanical drives. Themotor for hoist applications shown in Fig. 2.1b is integrated with a sheave(drum for ropes) and brake (not shown). It is used in gearless elevators [103].

2.1.2 Double-sided machines with internalPM disc rotor

Inthe double-sided machine with internal PM disc rotor, the armature wind-ing is located on two stator cores. The disc with PMs rotates between twostators. An eight-pole configuration is shown in Fig. 2.2. P}ds are embeddedor glued in a nonmagnetic rotor skeleton. The nonmagnetic air gap is large,i.e. the total air gap is equal to two mechanical clearances plus the thicknessof a PM with its relative magnetic permeability close to unity. A double-sidedmachine with parallel connected stators can operate even if one stator winding

28 AXIAL FLUX PERMANENT MAGNET BRUSHLESS MACH]NES

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

ROTOE STATOR

Figure 2.2. Configuration of double-sided AFPM brushless machine with internal disc rotor1 rotor, 2 - PM, 3 stator core, 4 - stator winding.

is broken. On the other hand, a series connection is preferred because it canprovide equal but opposing axial attractive forces.

A practical three-phase, 200 H2,3000 rpm, double-sided AFPM brushlessmotor with built-in brake is shown in Fig. 2.3 [145]. The three-phase winding

/.3

2

Principles of AFPM machines

Figure 2.3. Double-sided AFPM brushless servo motor with built-in brake and encoder: 1 -statorwinding,2- statorcore,3 discrotorwithPMs,4-shaft,5 -left frame,6 rightframe, 7 - flange, 8 - brake shield, 9 - brake flange, 10 - elecfromagnetic brake, 1l -encoder or resolver. Courtesy of SlovakUniversity of Technologt STU, Bratislava and ElectricalResearch and ksting Institute,Nov| Dubnica, Slovakia.

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

2.1.3 Double-sided machines with internal ring-shapedcore stator

A double-sided machine with internal ring-shaped stator core has a poly-phase slotless armature winding (drum type) wound on the surface of the statorferromagnetic core. 192, 160,218, 2451. In this machine the ring-shaped statorcore is formed either from a continuous steel tape or sintered powders. Thetotal air gap is equal to the thickness of the stator winding with insulation,mechanical clearance and the thickness of the PM in the axial direction. Thedouble-sided rotor simply called twin rotor with PMs is located at two sidesof the stator. The configurations with intemal and external rotors are shownin Fig. 2.4. The three phase winding arrangement, magnet polarities and fluxpaths in the magnetic circuit are shown in Fig. 2.5.

29

30 AXIAL FLUX PERMANEIVT MAGNET BRUSHLESS MACHINES

Figure 2.4. Double-sided machines with one slotless stator: (a) internal rotor, (b) externalrotor. I -statorcore, 2 statorwinding, 3 -steel rotor, 4 PMs, 5 resin, 6 frame,7 - shaft.

Figure 2.5. Three-phase winding, PM polarities and magnetic flux paths of a double-sided discmachine with one intemal slotiess stator. 1 winding, 2 PM, 3 - stator yoke, 4 rotoryoke.

The AFPM machines designed as shown in Fig. 2.4acanbe used as a propul-sion motor or combustion engine synchronous generator. The machine withexternal rotor, as shown in Fig. 2.4b,has been designed for hoist applications.A similar machine can be designed as an electric car wheel propulsion mo-tor. Additional magnets on cylindrical parts of the rotor are sometimes added[ 160] or U-shaped magnets can be designed. Such magnets embrace the arma-

Principles of AFPM machines 3l

ture winding from three sides and only the internal portion of the winding doesnot produce any electromagnetic torque.

Owing to the large air gap the maximum flux density does not exceed 0.65T. To produce this flux density a large volume of PMs is usually required.As the perrneance component of the flux ripple associated with the slots iseliminated, the cogging torque is practically absent. The magnetic circuit isunsaturated (slotless stator core). On the other hand, the machine structurelacks the necessary robustness [218]. Both buried magnet and surface magnetrotors can be used.

There are a number of applications for medium and large power axial fluxmotors with external PM rotors, especially in electrical vehicles [92, 245].Disc-type motors with extemal rotors have a particular advantage in low speedhigh torque applications, such as buses and shuttles, due to their large radius fortorque production. For small electric cars the electric motor mounted directlyinto the wheel is recommendedl92l.

2.I.4 Double-sided machines with internal slotted statorThe ring-type stator core can also be made with slots (Fig. 2.6). For this

type of motor, slots ale proglessively notched into the steel tape as it is passedfrom one mandrel to another and the polyphase winding is inserted 12451. lnthe case of the slotted stator the air gap is small (g < 1 mm) and the air gapmagnetic flux density can increase to 0.85 T 1921. The magnet volume is lessthan 50Yo that 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 - statorcore with slots, 2 - PM, 3 - mild steel core (pole), 4 - nonmagnetic rotor disc.

) L AXIAL FLUX PERMANENT MAGNET BRL|SHLESS MACHINES

Figure 2.7. Double-stator, triple-rotor AFPM brushless motor with water cooling system [55]I -PM,2-statorcore, 3 statorwinding.

2.1.5 Double-sided machines with internal coreless statorAFPM machines with coreless stators have the stator winding wound on a

non-magnetic and non-conductive supporting structure or mould. The statorcore losses, i.e. hysteresis and eddy current losses do not exist. The losses inPMs and rotor solid steel disc are negligible. This type of design offers higherefficiency at zero cogging torque. In order to maintain a reasonable level offlux density in the air gap, a much larger volume of PMs in comparison withlaminated stator core AFPM machine is required. The stator winding is placedin the air gap magnetic field generated by the PMs mounted on two opposingrotor discs (Fig. l.ad). When operating at relatively high frequency, significanteddy current losses in the stator winding conductors may ocun 12341.

2.1,6 MultidiscmachinesThere is a limit on the increase of motor torque that can be achieved by

enlarging the motor diameter. Factors limiting the single disc design are:

(a) axial force taken by bearings;(b) integrity of mechanical joint between the disc and shaft;(c) disc stiffness.A more reasonable solution for large torques are double or triple disc motors.

There are several configurations of multidisc motors l5-7 , 55,73, 7 41. Largemultidisc motors rated at least 300-kW have a water cooling system (Fig.2.7)

Principles of AFPM machines 33

with radiators around the winding end connections [55]. To minimize thewinding losses due to skin effect, variable cross section conductors may beused so that the cross section of conductors is bigger in the slot area than in theend connection region. Using a variable cross section means a gain of 40o/o inthe rated power [55]. However, variable cross section of conductors means anincreased cost of manufacturing. Owing to high mechanical stresses a titaniumalloy is recommended for disc rotors.

2.2 Windings2.2.1 Three-phase windings distributed in slots

In a single-layer winding, only one coil side is located in a single slot. Thenumber of all coils is s1 f 2 and the number of coils per phase is n. : s 1 I (2m1)where si is the number of stator slots and rn1 is the number of phases. In adouble-layer winding two sides of different coils are accommodated in eachslot. The number of all coils is s1 and the number of coils per phase is n. :stlmt.The number of slots per pole is

(2.r )

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

o'- hS lq r :

2 p * ,

The number of conductors per coil can be calculated as

r for a single-layer winding

(2.2)

(2.3)

(2.4)

A I -L r c -

r for a double-laver windine

A r -r r c -

_

aoa*Nt _

aoa-Nts1l (2m1) PQt

aoa-Nt _

ara*Ntstlmt 2pqt

aoa-N1T1'c

ara-Nr _

TLc

where ,Ay'l is the number ofturns in series per phase, oo is the number of parallelcurrent paths and a- is the number of parallel conductors. The number ofconductors per slot is the same for both single-layer and double-layer windings,i .e.

1 AJ + AXIAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

ALt : ctoa*Nt

PQt(2.s)

The full coil pitch measured in terms of the number of slots is y1 : Q1 whereQ1 is according to eqn (2.1). The shorl coil pitch can be expressed as

(2.6)

where tu"(r) is the coil pitch measured in units of length at a given radius rand r(r) is the pole pitch measured at the same radius. The coil pitch-to-polepitch ratio is independent on the radius, i.e.

at:YPetr \ r )

-

u t " \ t ' )t ' - / \

T \7 ' )

kpr : sin (Utr)

The distribution factor of a polyphase winding for the fundamental space har-monic u : 1 is defined as the ratio of the phasor sum-to-arithmetic sum ofEMFs induced in each coil and expressed by the following equation:

1 . , , -t r o | -sin(nl2m) (2.8)

qlsin}t l(2m1q1)lThe pitch factor for the fundamental space harmonic u : I is defined as theratio of the phasor sum to-arithmetic sum of the EMFs per coil side andexpressed as:

(.2.7)

(2.rt)

(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 : ka tk r t

The angle in electrical degrees between neighbouring slots is

(2.r0)

360"t - - P S 1

Fig. 2.8 shows a single layer winding distributed in s1 : 36 slots of a three-Phase, 2p:6 AFPM machine.


Figure 2.8. SingleJayer winding of an AIPM machine with rnr : 3' 2p : 6, sr : 36'A t : Q t : 6 a n d Q t - 2 .

2.2.2 Drum-type windingDrum-type stator windings are used in twin-rotor double-sided AFPM ma-

chines (Fig. 2.q. The drum-type stator winding of a three-phase, six-poleAFPM machine with twin external rotor is shown in Fig. 2.9. Each phaseof the winding has an equal number of coils connected in opposition so as tocancel the possible flux circulation in the stator core. Those coils are evenlydistributed along the stator core diametrically opposing each other so the onlypossible number of poles are 2,6,10, . . . etc. The advantages of the drum-typestator winding (sometimes called toroidal stator winding 146,219]) are shortend connection, simple stator core and easy design of any number of phases'

2.2.3 Coreless stator windingCoreless stator windings are used in twin-rotor double-sided AFPM ma-

chines (Fig. l. d). For the ease of construction, the stator winding normallyconsists of a number of single layer ttapezoidally shaped coils. The assemblyof the stator is made possible by bending the ends of the coils by a certain angle,so that the active conductors lie evenly in the same plane and the end windingsnest closely together. The windings are held together in position by using acomposite material of epoxy resin and hardener. Fig. 2.10 shows the corelessstator winding of a three-phase, eight-pole AFPM machine. obviously, therelations used in the slotted stator windings can be directly used for coreless

35

36 AXIAL FL\]X PERMANE]VT MAGNET BRT]SHLESS MACHINES

Figure 2.9. Drum-type winding of a three-phase, six-pole, 18-coil AFpM machine with fwinextemal rotor.

Figure 2.10. Coreless winding of a three-phase, eight-pole AFPM machine with twin externalrotor.

trapezoidal stator winding with the exception that the term "slot" is replacedby the "coil side". Another coil profi1e that has been used in coreless statorAFPM machines is the rhomboidal co1l. It has shorler end connections than


Figure 2.11. Connection diagram ofa three-phase, nine-coil winding ofan AFPM brushlessmachine.

the trapezoidal coils. The inclined arrangement of the coil's active sides makesit possible to place water cooling ducts inside the stator. The main drawbackof rhomboidal coils is the reduction of the torque.

2.2.4 Salient pole windingsA salient pole AFPM motor can be of single-sided or double sided construc-

tion (Figs 1.5, 1.6, 2.12). The stator winding consists of a number of coils withconcentrated parameters and separate ferromagnetic cores for each of the coil.In general, the number of stator poles (coils) is different than the number ofrotor poles. Fig. 1.5b shows a three-phase, 12-coi1 stator winding of a double-sided AFPM machines with 2p : 8 rotor poles. Figures 1.6, 2.ll and 2.12show a three-phase, 9-coil stator winding. The difference in the number of thestator and rotor poles is necessary to provide the starting torque for motors andthe reduction of torque pulsations.

2.3 Torque productionSince the dimensions of AFPM machines are functions of the radius, the

electromagnetic torque is produced over a continuum of radii, not just at aconstant radius as in cylindrical machines.

The pole pitch r(r) and pole width br(r) of an axial flux machine are func-tions of the radius r, i.e.

3 /

38 AXIAL FLUX PERMANEI{T MAGNET BRUSHLESS MACHINES

Figure 2.12. Three-phase, nine-coil stator of a single-sided AFPM brushless machine withsalient poles stator. Photo-courtesy of Mii Technologies, West Lebanon, NH, U.S.A.

(2.r2)2trr m2 p p

where a; is the ratio of the average Born to peak value B-,n of the magneticf fux density in the air gap. i .e.

7f f'o , , l r l : 0 ; T l r l : ( ) ;P w

P

Bo un b r ( ' )a i :

B * , o r a i :

r ( r )

^ , \ m t J | N r t o m r f i N t l or L m v J p'r \t ' ) 7T r

(2 . 1 3 )

(2.14)

Both the pole pitch r(r) and pole width br(r) are functions of the radius r, theparameter a; is notmally independent of the radius.

The line current densiQ is also a function of the radius r. Thus the peakvalue of the line current density is

(2.rs)

The tangential force acting on the disc can be calculated on the basis of Am-pere's equation

dh, - Io(rir x En1 : A(r)(d-s, nn) (2.r6)


where lod,; : AQ)d,3, AQ) : A*(ilt/2 according to eqn (2.15), di is tneradius element, d3 is the surface element and En is the vector of the normalcomponent (perpendicular to the disc surface) of the magnetic flux density inthe air gap. An AFPM disc-type machine provides Bn practically independentof the radius r.

Assuming the magnetic flux density in the air gap B,nn is independent ofthe radius r, dS : 2nrdr and Born : a,iBrns according to eqn (2.I4), theelectromagnetic torque on the basis of eqn (2.16) is

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 sur-face in the case of a typical stator winding distributed in slots (double-sidedstator and internal rotor) or electric loading of the whole stator in the case ofan internal drum type or coreless stator.

2.4 Magnetic fluxFor sinusoidal distribution of the magnetic flux density waveform excited

by PMs, the average magneticflux density is

(2 .18)

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

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

: -]- B*olcos ?r - cos ol : ? u*o7f '71

21 7la iB*n- j rd r : a iB

- zp ^np

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

B a u g :Tr lp - 0 lo"'o

f c l R o u t

l r - |t - ll t ll ' ) n o "

: I:-oJ(2.re)

where B*n is the peak value of the magnetic flux density in the air gap, p isthe number of pole pairs, Ro'1 : O.5Dout is the outer radius of the PMs andRh : 0.5D,;n is the inner radius of the PMs.

It is convenient to use the inner-to-outer PM radius or inner-to-outer PMdiameter ratio. i.e.

40 AX]AL FLUX PERMANE]{T MAGNET BRUSHLESS MACHINES

Thus,

o / : o ,B*o ! [ (0 .5D, , ,12 - (0 .5D; , ) ' ] : o , B*n ! DI , ,G - k ' i l Q .2 l l' " ' 2p ' " 6p

The same equation for a cylindrical type machine is [96]

Q r : ?rLiB,no" '77

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

uofiatffdr or

, ^ r f R . u r t t ^ . , l r 2 l R " " ' I nGn : * * , ' : I rc t r - #" , ; l ' r l - t to )a i * (n : , , - R7, )" g ' , p J R , " g ' P L z ) a , , g ' t p

: ^n*,+@2,, - R?,) Q.23)where \s : lro I g' is the permeance per unit surface and g' is the equivalent airgap'

2.5 Electromagnetic torque and EMFThe average electromagnetic torque developed by AFPM motor according

to eqns (2.15) and (2.17) is

d,Td : 2 a im1 I o IY 1k.1 B rnnr dr

If the above equation is integrated from Dou1f 2to Di,f 2 with respect to r, theaverage electromagnetic torque may be written as

, R'in D';,

- Rn,,t Dr'rt

Ta : ),nn rrloNrk-rB^s(D3.r - D?,): )aimrt{&dB^sD'lut0

- k:i)h

(2.20)

(2.22)

(2.24)where k; is according to eqn (2.20). Putting eqn (2.21) into eqn (2.24) theaverage torque is


T a : 2 L * r N 1 k * 1 Q s I o1T

To obtain the rms torque for sinusoidal current and sinusoidal magnetic fluxdensity, eqn (2.25) should be multiplied by the coefficient

"rDla = 1.11, i.e.

Ta : TlpNtk,te ylo - kTIov'2

where the torque constant

k, : ftrNtk*t{_f (2.27)In some publications [84, 218] the electromagnetic force on the rotor is simplycalculated as the product of the magnetic and electric loading BoooA and activesurface of PMs S : n(R\,, - R?,),i.e. F, : rBo,,nA(R\., J Rl.),whereA is the rms line current density at the inner radius R4n. For a double-sidedAFPM machine S : 2tr(R?-", - R?"). Thus, the average electromagnetictorque of a double-sided AFPM machine is

Ta: F*&n:2trBounA(R?""r- R?)Rn, :2trBo,gAR2*(ka - tcl l 1z.zt l

Taking the first derivative of the electromagnetic torque 7a with respect to k4and equating it to zero. the maximum torque is for ,ka

= | I /i. lndustrial

practice shows that 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 magnetic flux waveform iDyr : O/ sinot and multiplying by ly'1k,,1, i.e.

ey : Nlk-14*-! , t :2rf N{c-1Qy cosoltdt

The magnetic flux (D1 is expressed by eqns (2.19) and (2.21). The rms valueis obtained by dividing the peak value 2n f N1k.tQ f of the EMF by y8,, i.e.

Et: nrt f N1k*1Q1: nt[2pNtk-rQfn": kErL" (2.29)

where the EMF constant (armature constant) is

kB : ntf2pNyk-1Qy (2.30)

The same form of eqn (2.29) can be obtained on the basis of the developedtorque Ta : mrElI"l(2trn") in which 74 is according to eqn (2.26). For thedrum type winding the winding factor k-1 :1.

41

(2.2s)

(2.26)


2.6 Losses and efficiency2.6.1 Stator winding losses

The stator (armature) winding resistance per phase for the d.c. current is

R7d.: Nil.-,.,

a,pauosa

where l/1 is the number of armature tums per phase, 11.,,., is the average lengthof turn, c,o is the number of parallel current paths, a- is the number of parallelconductors, o1 is the electric conductivity of the armature conductor at giventemperature (for a copper conductor ot r= 57 x 106 S/m at 20"C and o1 =47 x 706 S/m at 75"C), and so is the conductor cross section. The averageleneth of the armature turn is

11o,, -- 2Lt + lun ! Iuut (2.32)

in which l1;,-, is the length of the inner end connection and l1ou7 is the lengthof the outer end connection.

For a.c. curuent and stator winding distributed in slots (ferromagnetic core)the stator winding resistance should be divided into the resistance of bars (ra-dial portions of conductors) R16 and resistance of the end connections fir",

Rt : Rn * Rr": ly ' t (2L;ktn I l tn I luut) x k111R14" Q.33)

aosa

where k16 is the skin-effect coefficient for the stator (annature) resistance. Fora double-layer winding and ur.(r) : ,(r') or 13 : 1 [149]:

k t r y , : p r ( r ) + vr ( { r ) (2.34)

where

t4# -(v.'";)']

(2.31)

(2.3s)

(2.36)

(2.37)

pr(dr) : r

sinh 26r * sirr 2{rcosh 2{1 cos 2{1

q i n h F ' - s i n F ,i l r , / F , \' r \ \ r / - ' ' c o s h q ,

- c o s { r

7r I F"ot ,0 1 1


nfid m4 is the number of conductors per slot arranged above each other intwo layers (this must be an even number), 7 is the phase angle between thecurrents of the two layers, / is the input frequency, bpon is the width of all theconductors in a slot, b11 is the slot width and h, is the height of a conductor inthe slot. If there arte tT"l conductors side by side at the same height of the slot,they are taken as a single conductor carrying n";-times greater current.

In general, for a three-phase winding ? : 60o and

(2.38)

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

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

The skin-effect coefficient k16 for hollow conductors is given, for example, in[r4el.

(a) If m"1: 1 and 1 : 0, the skin-effect coefficient kyp : pr({r) (the sameas for a cage winding).

(b) Ifl : 0, the currents in all conductors are equal and

krR :p,(r) . (4f +)vr(6r)

- 2 - - 1k tn= q \ t ) + - : ! ' _ :V r ( r )

J(2.40)

For small motors with round armature conductors fed from power frequenciesof 50 or 60H2,

Rt x Rta. (2.4r)The armature winding losses are

LPw : ruISRr = mlllRla.kyp (2.42)

Since the skin effect is only in this part of the conductor which is located inthe slot - eqn (2.33), the armature winding losses should be multiplied by thecoefficient

kn * lu. I (2Li) * 11"6 I (2L.i)

rather than by k1a.1 - t lu" l (2L i ) + t t * t l (2L i )

(2.43)

44 AXIAL FLUX PERMAI{ENT MAG]VET BRUSHLESS MACHINES

2.6.2 Stator core lossesThe magnetic flux in the stator (armature) core is nonsinusoidal. The rotor

PM excitation system produces atrapezoidal shape of the magnetic flux densitywaveform. The stator windings are fed from switched d.c. sources with PWMor square wave control. The applied voltage thus contains many harmonicswhich are seen in the stator flux.

The eddy current losses can be calculated using the following classical for-mula

, m

LP, ," : "^ "ja 1ztll.n,p.t "' ' le'-,, + *-,,10 P F " n I

- 2 ^ _-

' ; #f d2p,mp"lB2-,,

- B?,,,1q7,

where oF", dF", pp" andlmlle are the electric conductivity, thickness, specificdensity and mass of laminations respectively, n are the odd time harmonics,Brnrn and B,nrn are the harmonic components of the magnetic flux density inthe r (tangential) and z (normal) directions and

(38*"2)'2 + (3B,nA)2B2r-r, + Blr,

(58 ry ! : \ 2 + ( !F - , i ) 2 + Q .451

D m r T T D m . z 1

(2.44)

(2.46)

is the cofficient o.f distortion of the magnetic fux density. For qa : 1, eqn(2.44) expresses the eddy current losses under sinusoidal magnetic flux density.

In a similar way, the hysteresis losses can be expressed with the aid ofRichter's formula, i.e.

r O OrLPnp" : r] ;r,rp"\rr2lal,", + 82,-",1

I t , t , n _ 1

{t - : - t r r t r , lb ' - , t * I J ' - r1 ) r1 '4100

where e, :7.2ro2.0mal(]Hkg) foranisotropic laminations with4o/o Si, e : 3.8m4l1H kg) for isotropic laminations with2o/o Si and e : 4.4to 4.8 m4l(H kg)for isotropic siliconless laminations.

Eqns (2.44) and (2.46) exclude the excess losses (due to magnetic anomaly)and losses due to metallurgical and manufacturing processes. There is a poorcorrelation between measured core losses and those calculated using classicalmethods. The losses calculated according to eqns (2.44) and (2.46) are lowerthan those obtained from measurements. The coefficient of additional core

Principles of AFPM machtnes 45

losses ko6 ) 7 can help to obtain a better agreement of predicted and measuredcore losses

LPtp": koa(LP"r" * LP1rp") (2.47)

If the speciJic core losses are known, the stator core losses LPt p" can be cal-culated on the basis of the specific core losses and masses of teeth and yoke,i .e.

LPtp" lko a1 B! 1m 1 1 -t k o,1, Bl om 1 ol (2.48)

where kad.t ), \ and ko6, ) 1 are the factors accounting for the lncrease lnlosses due to metallurgical and manufacturing processes, L11lso is the specificcore loss in Wkg at I T and 50H2,.B11 is the magnetic flux density in a tooth,B1o is the magnetic flux density in the yoke, m11is the mass of the teeth, andm1, isthe mass of the yoke. For teeth ko6 : 1.7 to 2.0 and for the yokekod,s : 2 .4 to 4 .0 l l47 l .

2.6.3 Core loss finite element modelThe core losses within the stator and rotor are calculated using a set of finite

element method (FEM) models, assuming constant rotor speed and balancedthree-phase armature currents. The eddy current and hysteresis losses, withinthe cores, in the 2D FEM including distorted flux waveforrns can be expressedby eqns (2.44) and (2.46).

The field distribution at several time intervals in the fundamental currentcycle is needed to create the magnetic flux density waveforms. This is obtainedby rotation of the rotor grid and phase advancement of the stator currents. Froma field solution, for a particular rotor position, the magnetic flux density at eachelement centroid is calculated. Three flux density components in an elementcan be obtained from a single FEM solution.

2.6.4 Losses in permanent magnetsThe electric conductivity of sintered NdFeB magnets is from 0.6 to 0.85 x

106 S/m. The electric conductivity of SmCo magnets is from 1.1 to 1.4 x 106S/m. Since the electric conductivity of rare earth PMs is only 4 to 9 timeslower than that of a copper conductor, the /osses in conductive PMs due tohigher harmonic magnetic fields produced by the stator cannot be neglected.

The most important losses in PMs are generated by the fundamental fre-quency magnetic flux due to the stator slot openings. In practice, those losses

t r , , 4 / 3: ar'z'o (jo)


are only in AFPM machines with slotted stator feromagnetic cores. The fun-damental frequency of the magnetic flux density component due to the statorslot opening is

f s l : s t P n (2.4e)

where s1 is the number of stator slots, p is the number of pole pairs and n isthe rotor speed in rev/s.

The magnetic flux density component due to slot opening is [111]Bst : ct,11B"1,kgBo,,o (2.s0)

where Boro is the mean magnetic flux density overthe slot pitch - eqn(2.4),kg is Carter coefficient (l.2\ and

o ' : * (o tn "#* )

s in(1 6nr )

1 \. I

/ ; - , t Iv L | t i ' /

h t tg'

, J r r : 0 5 ( 1 -

(2 .51 )

(2.s2)

(2.s3)l :bt 1

In the above equations (2.51), (2.52) and (2.53) b1a is the stator slot opening,gt : g * hm I ltrr"" is the equivalent air gap, tt is the slot pitch and h,r,r is themagnet thickness per pole.

Assuming that the relative recoil magnetic permeabllity 1.t,.,r,. r 1, thepower losses in PMs can be expressed by the following equation obtained froma 2D electromagnetic field distribution, i.e.

(2.s4)

where ap, is according to eqn (1.16) for u : 1, a : (t + j)k is according toeqn (l.l 1) for u : I, k is according to eqn (1.12) for u : 7, p ts according toeqn (1.6) for u : 7 and oppl is the electric conductivity of PMs. Eqn(2.5a)can also be used to estimate the reactive losses in PMs if ap, is replaced withuy, according to eqn (1.17).

The coefficient for including the circumferential component of currents in-duced in PMs can be found as

I ta t2 ( B , t \2 k ^L P p t r - - i r n , k - . ; t - , - r p n 1z lJ \HUHrr ' ' / oP ' \ ]

where 0.5tr is the induced cuffent loop span and0.5(Do"t - Dn) is the radiallength of the PM - see eqn (1.30). The active surface area of all PMs (single-sided machines) is


&RFe

r u 2 - r t ^

I)ou,t - l)in

SpM : "o nQ3,t

- D7")

where the average pole pitch r is according to eqn (1.9).

l@_.a2p-a2y*

47

(2.ss)

(2.s6)

(2.s7)

2.6.5 Rotor core lossesThe rotor core losses, i.e. losses in backing solid steel discs supporting PMs

are due to the pulsating flux produced by rapid changes in air gap reluctanceas the rotor passes the stator teeth.

The magnetic permeability of a solid steel disc varies with the z axis (normalaxis). To take into account the variable magnetic permeability and hysteresislosses in solid ferromagnetic discs, coefficients a1u, axu according to eqns(1.16) and (1.17) must be replaced by the following coefficients:

_ 1p' luFl1

,/,

- tp' l". . . . : l

k2 lI

1rtaxFe (2.58)

where an: L4 to 1.5, ax :0.8 to 0.9 according to Neyman [184], k" isaccording to eqn (1.12) and B is according to eqn (1.6) for u :1.

The power losses in solid ferromagnetic discs can be expressed by a similarequation to eqn (2.54), i.e.

LPzp" : *o*r"*"ry ffi)' *t r. (2.se)

where appu is according to eqn (2.57) for u : 1, c is according to eqn (1.11)for u : 1, k is according to eqn (1. 1 2) for u : 1, B is according to eqn ( 1.6) foru : l,8"7 is according to eqn (2.50), p, is the relative magnetic permeability

48 AXIAL FLUX PERMANENT MAGNET BRLTSHLESS MACHINES

and op. is the electric conductivity of the solid ferromagnetic disc. The fre-quency in the attenuation coefficient k is according to eqn (2.49). Eqn (2.59)can also be used to estimate the reactive losses in the solid ferromagnetic discif o,pp" is replaced with a;p" according to eqn (2.58). The flux density B.r7is according to eqns (2.50) to (2.53). The coefficient k" is according to eqn(2.55) and the surface of the disc is

sF.: [{n3,, - D?.) (2.60)

2.6.6 Eddy current losses in stator conductorsFor slotted AFPM machines the eddy current losses in the stator winding are

generally ignored as the magnetic flux penetrates through the teeth and yokeand only small leakage flux penetrates through the slot space with conductors.

In slotless and coreless machines, the stator winding is exposed to the air gapmagnetic field. The motion of PMs relative to the stator winding produces analternating field through each conductor and induces eddy currents. In the caseof a coreless AFPM machine with a solid ferromagnetic rotor discs, there isalso a tangential field component B,n, inaddition to the axial field componentBrr". This can lead to serious additional eddy current loss especially at highfrequency. Neglecting the proximity effect, the eddy current loss in the statorwinding may be calculated by using a classical equation similar to eqn (2.44)for calculation of the eddy cunent losses in laminated cores, i.e.

r for round conductors [43]- 2 * o c

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

: | \ 12a'rn,onlB'^,t + B?,,"r ln\t Q.6l)+ I )

r for rectangular conductors [43]) (Y)

AP" = ; ^f',r'^.o, I n,2[82,n*, + B?r"r]J P n - l

2 ^

-1 f'o'*"orlB?rrtL-. + g?'1f- o

3p'r."tl'l'o (.2.62)


where d is the diameter of the conductor, a is the width of the conductor (paral-lel to the stator plane), o is the electric conductiviry p is specific mass densityof the conductar, m"on is the mass of the stator conductors without end con-nections and insulation, / is the stator current frequency, Brr* and Brn" arcthepeak values of tangential and axial components of the magnetic flux density,respectively, andrla is the coefficient of distortion according to eqn (2.45).

2.6.7 Rotational lossesThe rotational or mecltanical losses LProt consist of friction losses APy,

in bearings, windage losses LP*tnd. and ventilation losses LPu.nt (if there isa forced cooling system), i.e.

LPr"t - LPr, * LP,ttrd, * LPo"nt (2.63)

There are many semi-empirical equations for calculating the rotational lossesgiving various degrees of accuracy. The friction losses in bearings of smallmachines can be evaluated usine the followine formula

A,Py, :0.06kya(m, * m"6)n W (2.64)

where kf,, : I to 3 m2ls2, rn, is the mass of the rotor inkg, m"6 is the massof the shaft in kg and n is the speed in rpm.

The Reynolds number for a rotating disc with its outer radius Roul is

2trnpRf;u, (2.6s)

where p is the specific density of the cooling medium, u : 1)r - 2trRo",1n isthe linear velocity at the outer radius Rout, fr is the rotational speed and p isthe dynamic viscosity of the fluid. Most AFPM machines are air cooled. Theair density at I atm and 20"C is 1.2 kg/m3. The dynamic viscosity of the air at1 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

R": pRou" :I,L

LP-rnd. : )c y pQnfit @u.,, - R\n) (2.67)


where fi.r7, is the shaft diameter.AFPM machines are usually designed without a cooling fan so that the ven-

tilation losses AP",unr : 0.

2.6.8 Losses for nonsinusoidal currentHigher time harmonics generated by static converters produce additional

losses. The higher time harmonic frequency in the stator winding is n/ wherefr :5,7,1I, . . . . The armature winding losses, the core losses, andthe straylosses are frequency-dependent. The mechanical losses do not depend on theshape of the input waveform.

If the stator winding and stator core losses have been calculated for funda-mental frequency, the frequency-dependent losses of an invefter-fed motor orgenerator loaded with a rectifier can be found as

o stator (armature) windins losses:

APr., - LPr*,: *tD l inRy,, x m1R16,1L lZ,tt t*, (2.68)r t : l n- l

o stator (armature) core losses

co oo

L,P, r;- f AP, ,-- : lA,P,.-- l--, \-Z - z " ) " - ' / J

n-B'7 (2.69)

where k16,, is the a.c. armature resistance skin effect coefficient for n,f , Io, isthe higher harmonic rms armature current, [r, is the higher harmonic inverteroutput voltage, V1r. is the rated voltage, [APrp.,.]n:l are the stator core lossesfor n : 1 and rated voltase.

2.6.9 EfficiencyThe total power losses of an AFPM machine are

AP : LP:.- * APrr'" * LPzp" + LPzM + LP" -f LProt Q.70)

The fficiency isDr olrt (2.71)

P,6 * L'P

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

oc

\-z .

/ V t , \ 2t _ l\V , )


UNDTRIXCITTDGTNERATOR

RC LOADo> Y > -90 '

UNDTREXCITIDMOTOR

RL*gO'>Y >- 180 '

OVER[XCITTDCTNTRATOR

RL LOADo< Y


: Vr f I"a(Rt * jX,a) *I"q(R4 * jX"q)

(b) consumer (motor) arrow system, i.e.

Vr : E/ * Io-Rr * jIo6Xr,1+ jloqxr,l

: E / - l T "a(Rt * jX"a) * loq( .R1* . iX"q)

(2 .74)

(2.7s)

and

where

I o - I o 6 , * l 6 q (2.76)

Ioa: Ios in 'U 1oq : 1o cos \! (2.71)

The angle V is between the q-axis and armature current 1o. When the currentarows are in the opposite direction, the phasors Io, l,r,L aadl.aq are reversedby 180". The same applies to the voltage drops. The location of the armaturecurrent Io with respect to the d- and q-axis for generator and motor mode isshown in Fig. 2.13.

Phasor diagrams for synchronous generators are constructed using the gen-erator arrow system. The same system can be used for motors, however, theconsumer arrow system is more convenient. An overexcited generator (Fig.2.I4a) delivers both active and reactive power to the load or utility grid. Anunderexcited motor (Fig. 2.lab) draws both active and reactive power from theline. For example, the load culrent I" (Fig. 2.14b ) lags the voltage phasor V1by the angle $. An overexcited motol consequently, draws a leading currentfrom the circuit and delivers reactive power to it.

In the phasor diagrams according to Fig. 2.141116] the stator core losseshave been neglected. This assumption is justified only for power frequencymachines with unsaturated armature cores.

For an underexcited synchronous motor (Fig. 2.lab) the input voltage Iprojections on the d and q axes are

7 r s i n 6 : I u q X " n - I o a R t

I cos 6 -- E.f t Io6X"4 t IoqRl (2.78)


j l "X,

Figure 2.14. Phasor diagrams ofsalient-pole synchronous machine: (a) overexcited generator(generator arrow system); (b) underexcited motor (consumer arrow system).

) J

(a) jl"a4o

AXI A L F LUX P E RMAN E NT MA GNET B RU SH L E SS M AC H I N E S

The currents of an overexcited motor

, Vr (X"ocosd - Rr s in d ) - E IX" ,

t 1' uu

x r6x "ns - R '2 ,

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

X " 6 X " q - R 2 1

(2.80)

(2 .81 )

are obtained by solving the set of eqns (2.78). The rms armature current as afunction of V1,81, X"d, X",r, d, and R1 is

I*:ffi+lfu- X"aX"q * R?

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

The phasor diagram can also be used to find the input electric power. For amotor

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), equivalent circuits of a AFPMsynchronous machine can be drawn (Fig. 2.15).

2.8 Sizing equationsThe main dimensions of a double-sided PM brushless motor with internal

disc rotor can be determined using the following assumptions: (a) the electricand magnetic loadings are known, (b) the number of tums per phase per onestator is Nr; (c) the phase armature current in one stator winding is 1o; (d) theback EMF per phase per one stator winding is 81.

The peak line current density at the average radius per one stator is expressedby eqn (2.15) in which the radius may be replaced by an average diameter

Vr


8 0 0 0 01 0 0 0 0 0

Figure 2.16. Outer diameter D6a1 zs a {tnction of the output power Po,1 and parameter ki7for e : 0.9, k-1q cos Q : 0.84, n" - 1000 rpm: 16,67 rev/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 coreand ka : DinlD^ft (according to eqn (2.20)). Thus,

55

^ A{2mtloNrtant: "D^,^r

+ kd)

The EMF induced in the stator winding by the rotor excitation system, ac-cording 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

(2.86)

(2.88)S.r,n : m1(281)1": mtEf (21")

_2:

ik-1n"8*sA*DZdG + ki\ - k2d)

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

56 AXIAL FLUX PERMANENT MAGNET BRTJSHLESS MACHINES

S elrn : n2 k 11k*1n "B rng

ArnDZut (2.e0)

The apparent electromagnetic power expressed in terms of active output poweris thus

1, t r :

r (1 +k, i1-k: i )

the apparent electromagnetic power is

^ Pout\ , f -ee tn t -

, / cos g

where the phase EMF-to-phase voltage ratio is

E yI /v 1

D "t e | m" ) - -4 t t t L s

(2.8e)

(2.er)

(2.e2)

Formotors e < 1 and for generators e ) 1.In connection with eqns (2.90) and (2.91), the PM outer diameter (equal to

the outer diameter of the stator core) is

r? q3)

The outer diameter of PMs is the most important dimension of disc rotor PMmotors. Since Do,1 o, i/P""t the outer diameter increases rather slowly withthe increase of the output power (Fig. 2.16). This is why small power discmotors have relatively large diameters. The disc-type construction is preferredfor medium and large power motors. Motors with output power over l0 kWhave reasonable diameters. Also, disc construction is recommended for a.c.servo motors fed with high frequency voltage.

The electromagnetic torque is proportional rc D!,r, i.e.

, l c Pout, ,U O U I - l l o ,

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

,S"7- cOS V) o n

: T k ok -, D3ou, B ^ n Arn cos v (2.e4)

where P"1* is the active electromagnetic power and V is the angle between thestator current Io and EMF 81.

Prr'nciples of AFPM nachines 57

2.9 Armature reactionThe magnetic fluxes produced by the stator (armature) can be expressed in

a similar way as the field excitation flux - eqn (2.19), i.e.r in the d-axis

t - n2 , , R? ,Qod : aB rnod . t "

t Lo l )7T " " * * ' p 2

(2.es)

r in the q axis

(2.e6)

where Bmad1 is the peak value of the first harmonic of the stator (armaturereaction) magnetic flux density in the d-axis and Brno4 is the peak value ofthe first harmonic of the stator magnetic flux density in the q-axis. The statorlinkage fluxes are

r in the d-axis

) a O ' r - R ? nQoq:

} " *on ' , ; ' " t ,

*o: #*tk,oteaa: i*rk-t1B-,"ar| ry e.s7lr in the q axis

, I ,o : *& k-teos: lNrn-, ,?o*-- , !R?'" '

- R?'' \/2 v , iBmaqT--*f

(2'98)

where l/r is the number of stator turns per phase and k.,1 is the winding factorfor the fundamental space harmonic.

Neglecting the magnetic saturation, the first harmonic of the stator magneticflux density normal components are

r in the d-axis

Bmailr: kydB,ood,: kya\aFoa: kro#*f Yr"o Q.gg)

58

r

AXTAL FLUX PERMANENT MAGNET BRUSHLESS MACHINES

in the g-axis

B-oqt - k laB-od - k laAqFoo: t r tu*yQ! ! ! , .n (2 .100)qa f tP

, B ^ o d L B - o q l

t ' J d - B ^ o r l " . l Q B n , o q

where rn1 is the number of stator phases and the perrneance per unit surface inthe d- and q-axis are

\ . - \ - l t o/ \ t 1 - / \ 0 -I

r l'o/ \ q -

ea(2.r0r)

In the above equations (2.99) and (2.100) theformfactors of armature reactionare defined as the ratios of the first harmonic amplitude-to-maximum value ofnormal

Axial Flux Permanent Magnet Brushless Machines

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