01270522_electrognetic Desing of Pmg Axial Flux

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Electromagnetic design of axial-flux permanentmagnet machines

J.R. Bumby, R. Martin, M.A Mueller, E. Spooner, N.L. Brown and B.J. Chalmers

Abstract: A general, analytic approach to the calculation of magnet fields in a slottless, axial-flux,permanent magnet generator is presented. The basic building block is the vector potentialproduced by a current sheet situated between two, infinitely permeable, iron surfaces. By modellingthe magnet by currents at its periphery, and integrating over the magnet thickness, the vectorpotential and magnetic field due to the permanent magnets can be found. In contrast the armaturewinding is represented by a current sheet close to the stator iron surface. Magnetic field resultsproduced by the analytic equations have been compared with two-, and three-, dimensional finiteelement studies and found to produce results comparable to within 5%. In addition emf, flux andinductance measurements have been made on two generators and compared with both finiteelement and analytic results. The analytic model predicts the emf to within 5%. The end windinginductance of a toroidal, air-gap, armature winding, is shown to contribute significantly to theoverall inductance with the analytical model predicting the inductance to within 10% of themeasured values.

List of symbols

A vector potentialB flux density, TBrem permanent magnet remanence, Tc running clearance, mDm mean core diameter, mE emf, VH magnetic field strength, A/mi current, A

J current density, A/m2K linear current density, A/mKdn nth harmonic distribution factorkL,mag effective length ratio (see (26))Lm magnet radial length, mn harmonic numberNc number of turns per armature coilp number of pole pairsRi inner radius of the iron stator core, mRo outer radius of the stator iron core, mRm mean core radius, mtc core thickness, mun 2pn/l

w coil width at the mean radius, mYa armature thickness, mYm magnet thickness, mY1 position of current sheet, mY2 distance between rotor and stator iron surfaces, mY2eff effective gap, m

Z number of conductors per armature coilg displacement between armature coils, mm0 permeability of free space, H/mmrec magnet recoil permeabilityl wavelength l 2pRm/p, mtp pole pitch tp l/2, mtm magnet width, mse, sm coil spread, electrical or mechanical radiansF flux, WbC flux linkage, Wb

Subscriptsarm armaturecoil coiln harmonic numberph phaser radialz axial

1 Introduction

The wide availability and reducing cost of high-remanence,neodymium-iron-boron (NdFeB) permanent magnets havemade axial-flux machines a cost-effective alternative forlow- and medium-power motor and generator applications.In many situations the specific torque of an axial-fluxmachine is better than that of its radial-flux counterpart[1, 2] whilst its geometric proportions may be morecompatible with the general proportions of the machinerydriving, or being driven by, the electrical machine. The veryshort axial length required to accommodate the magneticand electric components can lead to designs that do notrequire separate bearings and the high moment of inertia ofthe rotor can serve a useful flywheel function. Particularexamples of the use of axial-flux machines are for directdrive wind generators [35], in compact engine-generatorsets, either for general applications [6] or in a hybrid electricvehicle [7], or as in-wheel electric motors [811].

J.R. Bumby, R. Martin, M. Mueller and E. Spooner are with the School ofEngineering, University of Durham, Science Site, South Road, Durham DH13LE, UK

N.L. Brown is with the Newage-AVK-SEG, Barnack Road, Stamford,Lincolnshire PE9 2NB, UK

B.J. Chalmers is with the Department of Electrical and Electronic Engineering,UMIST, PO Box 88, Sackville Street, Manchester M60 1QD, UK

r IEE, 2003

IEE Proceedings online no. 20031063

doi:10.1049/ip-epa:20031063

Paper first received 24th March 2003 and in revised form 17th September 2003

IEE Proc.-Electr. Power Appl., Vol. 151, No. 2, March 2004 151
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In general an axial-flux machine comprises of at least onerotor disc carrying axially polarised magnets and one statordisc carrying either a slotless or slotted winding. The statormay be magnetic or non-magnetic depending on themachine topology. Based on these concepts a large numberof axial-flux topologies are possible including single-sided[11], double-sided [7, 12] or multi-stage designs. We willconcentrate on the iron-cored slotless variety with a strip-wound stator core and toroidal coils forming the air-gapstator winding. Because of the winding structure this type of

machine is often referred to as a torus machine [12]. Thebasic cross-section of such a machine is shown in Fig. 1 andconsists of two rotor discs each with magnets arrangedcircumferentially around the rotor plates in a N-S-N-Sarrangement. A north magnet on one disc faces a northmagnet on the other disc so that magnetic flux travelsaxially across the air-gap and then turns circumferentiallyinto the strip-wound iron core before returning to the rotorsone pole pitch further on. Each armature coil is woundtoroidally round the strip-wound core. For a three-phasemachine there are three armature coils per pole each with amaximum spread of 601 (electrical). This machine topologyis used in the compact, variable-speed integrated generating(VSIG) set commercialised by Newage-AEG-SVK [6, 7].

When the machine rotates air is drawn in at the axialcentre of the machine and natural pumping action forcesthis cooling air radially out across the armature winding.This pumping action is further enhanced by the magnetsthemselves acting as fan blades so ensuring excellent coolingof the armature winding. This direct air-cooling of thewinding allows current densities of up to about 20 A/mm2

to be used.

The flux inside the machine arises from the permanentmagnets and from the current passing through the statorcoils. To determine a machine design, and assess itsperformance, both flux components are required and areconveniently expressed in terms of the emf induced by the

magnet flux and by the reactance of the armature winding.Design expressions for both emf and inductance, based onmagnetic equivalent circuits, are readily evaluated and canbe used to design machines that perform as predicted [7].However, such design expressions do not readily takeaccount of leakage and fringe fields whilst in many designs[7, 12] the magnets extend radially beyond the stator toincrease armature flux linkage and emf. Analytical designexpressions that inherently account for these effects are nowpresented. The design expressions predict magnet flux

density, induced emf and armature inductance and theiraccuracy is supported by finite element and experimentalwork.

Marignetti and Scarano [10] used an analytic techniqueto predict magnet flux density in an axial-flux machine butrequired a finite element solution as part of the solutionprocess. Chalmers et al. [13] produced armature inductanceexpressions using a two-dimensional solution to Laplacesequation. We use a similar approach to Chalmers et al. [13]but now a unified 2-dimensional (2-D) approach is used tocalculate both flux from the magnets and the armaturereaction flux. Experimentally measured emf and armatureinductance values will be used in a comparison with boththe analytic approach and with finite element studies. The

finite element solutions in 2-D are computed using bothMEGA [14] and FEMLAB [15] and in 3-D by MEGA.

2 The analytical model

2.1 The modelLooking radially inwards onto the machine shown in Fig. 1,and ignoring curvature, allows the machine to be repre-sented as shown in Fig. 2 where the x-coordinate representsthe circumferential direction and the y-coordinate the axialdirection. The model assumes the radial direction to beinfinite.

Figure 3 illustrates the 2-D mathematical model used toanalyse the magnetic field in one of the air-gaps. Currentsheets are used to model both the magnet and the armaturecurrent so that computing the vector potential from ageneralised current sheet located between the stator androtor iron is the basic building block used in the analysis.Similar techniques have been used in [16, 17]. In the analysisthe stator and rotor are treated as infinitely permeableboundaries. The current sheet has a harmonic distributionof the form:

Knx ^KKn sin unx 1where un

2pn/l and l, the wavelength, is equal to twice

the pole pitch. The value of ^KKn depends on the actualcurrent distribution and is evaluated in Section 2.3 for themagnet and in Section 2.4 for the armature winding.

rotor disc

fixing ring

engine

shaft

armature

winding

stator core

magnets

cooling

hole

rotor shaft

Fig. 1 Cross-section of a bearingless axial-flux, toroidal generator

y

x

N

N

S

S

N

NS

S

Fig. 2 Simplified machine model

152 IEE Proc.-Electr. Power Appl., Vol. 151, No. 2, March 2004
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2.2 Vector potential and flux density from acurrent sheetThe vector potential and magnetic field are found above,and below, the winding by solving Laplaces equationr2A 0 subject to boundary conditions at the iron surfacesand at the current sheet:

y 0 Hx 0y Y1 DHxn Hx2n Hx1n Kn; Hycontinuousy Y2 Hx 0

The magnetic vector potential has a z-component only, sothe magnetic field is obtained from:

Hx 1

m0

@A

@y andHy 1

m0

@A

@x 2The solution of Laplaces equation gives the vectorpotential, and the normal component of magnetic field, inregions 1 and 2 as:

Region 1:

Azn1x ^KKnm0un

cosh unY2 Y1sinh unY2

cosh uny

sin unx3

Hyn1

x

^KKn

cosh unY2 Y1sinh unY2

cosh uny

cos unx 4

Region 2:

Azn2x ^KKnm0un

cosh unY1

sinh unY2cosh unY2 y

sin unx5

Hyn2x ^KKn cosh unY1sinh unY2

cosh unY2 y cosunx

6

2.3 The magnetic field from the magnetThe magnetic field from the magnet at the armature(region 2) is calculated from (5) and (6) by representing themagnet magnetisation by a large number of layers with eachlayer represented by an equivalent current sheet. The vectorpotential at the field point y is then found by integratingover the magnet thickness, Fig. 4. As the relative recoilpermeability of the permanent magnets is greater than onethis must be accounted for in the analysis. In formulatingthe problem the effect of recoil permeability appears in twoplaces; firstly in the magnitude of the current used to modelthe magnets and, secondly, as the relative permeability ofone of the regions in the field solution. The approach takenhere is to account for the effect of recoil permeability in themagnet modelling current whilst assuming mrec 1 in thefield solution. This allows (5) and (6) to be used directly. A

modifying factor to account for the effect of recoilpermeability on the reluctance of the magnet region isintroduced later by comparison with a magnetic circuit

solution. The accuracy and implications of this approachare discussed later in Section 4.

To compute the equivalent magnet linear current density

distribution ^KKn each magnet is represented by a current atits edges assumed to flow in a vanishingly small thickness ofangular spread 2d, Fig. 4b. The equivalent magnet currenthas a magnitude of:

Brem

momrecYm

and a corresponding current density:

JJ Bremmomrec

p

2dtp7

where tp is the pole pitch. The individual current densityharmonics can now be obtained by Fourier analysis to givethe equivalent current density distribution for the magnetas:

Jx Xnodd

^JJn sin npx

tp8

where:

^JJn 8p

JJd sin na 4tp

Brem

m0mrecsinnp

2

tm

tp

9

The set of magnets are now divided into a number ofcurrent sheets each of width dy with a linear current density

of:Knx ^JJndy sin unx 10

The vector potential at any point, y, in region 2 is thenfound by substituting (10) into (5) and integrating over themagnet thickness Ym to obtain:

Azn2x ^JJnm0u2n

sinh unYm

sinh unY2cosh unY2 y

sin unx11

with:

Byn2x ^JJnm0un

sinh unYmsinh unY2

cosh unY2 y cosunx

12

iron

iron

region 2

region 1

current sheet Kn(x) = Knsin unxy

x

Y1

Y2

Fig. 3 Current sheet model

iron

iron

y

x

Ym

Y2

Y1

field point(x, y)

(0.0)

2

2

I= BremYm

orecJ

m

p

b

a

Fig. 4 Magnetic field from the magnets in region 2a Model used to compute the vector potential

b Model used to compute the magnetic linear current density

distribution



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2.4 Armature magnetic fields and currentdistributionThe armature magnetic field can be computed using asimilar technique as above or it can simply be representedby a current sheet when (3) to (6) can be applied directly.The latter approach is used here. The armature current ineach air-gap can be represented by the current distributionof Fig. 5 where Z conductors per phase band each carry acurrent of iA. The single-layer winding shown is simple toconstruct as an air-gap winding whilst more complex

double-layer arrangements are readily assembled in slottedstators. Air-gap windings only are considered here.

For this single-layer winding the linear current density is:

K Ziw

ZismRm

p ZiRmse

13

where w is the winding width which, at the mean radiusRm (Ro+Ri)/2, can be written as w smRm where sm isthe winding spread in mechanical degrees. In electricaldegrees sepsm. The linear current density is:

Kx X

^KKn sin unx 14

with the peak linear current density being obtained byFourier analysis as:

^KKn 4Knp

sinnp2

sinnse

2 2pZpkdn

Rmsinnp2i 15

For a toroidally wound machine Z Nc, the number ofturns per armature coil.

3 Prototype generator details

A number of prototype axial-flux generators have been builtand used to benchmark the design equations. The details oftwo, three-phase, machines are tabulated in Table 1 withnominal ratings of 40kW, 4500 rpm and 20 kW, 3000 rpmrespectively. Load test results for the 20 kW machine have

been reported elsewhere (18). Earlier work on the 40 kWmachine was used in the commercial development of theVSIG set at Newage-AVK-SEG [6].

The 40 kW machine is shown in Fig. 6 and is supportedon its own bearings between two end plates. Figure 6ashows the generator on test when connected to a dieselengine whilst the stator and rotor discs are shown in Fig. 6b.The 20 kW generator is fixed directly to the end of an enginein place of the flywheel (see Fig. 1) and is therefore abearingless design. One of the main differences between the40 and 20 kW designs is the ratio of the magnet width topole pitch. In the 40 kW design this value is 0.66 whilst inthe 20 kW design it is 0.8. The increase in magnet width topole pitch increases output voltage and power output [19]but also changes the harmonic content of the inducedvoltage. Figure 7 shows the assembled 20 kW generator ontest.

4 Results and discussion

4.1 Magnet field and induced emf

4.1.1 Analytic and finite element analysismethods: The magnetic field produced by the perma-nent magnets varies circumferentially around and radiallyacross the armature with the flux entering the stator coreover one pole pitch determining the induced emf. Thecircumferential variation of the axial magnet field in the40kW generator, computed at the mean radius and 1 mmfrom the stator core, is shown in Fig. 8a. This Figure showsthe value predicted by (12), and computed by 3-D and 2-Dfinite elements. Assuming that mrec 1 in both the magnetmodelling current and in the field equation gives the fluxdensity plot in curve A. If recoil permeability is allowed for

in the modelling current only, curve B is obtained. A finalmodification is to change Y2 in the denominator of (11) and(12) to an effective gap value Y2eff where:

Y2;eff Ymmrec

Ya c

16

With this modification the flux density plot of curve C isobtained and is identical to the 2-D and 3-D finite elementresults. The justification for using an effective gap Y2eff inthe denominator of (11) and (12) is based on a comparisonwith the flux density equation obtained from a magneticequivalent circuit approach (see the Appendix) [7]. TheAppendix shows that the magnet recoil permeabilityappears in the numerator of the flux density equation,due to the magnet modelling current, and in thedenominator, due to the reluctance of the magnet region.In (11) and (12) the effect ofmrec on the modelling current

+ ++ ++ ++ + Z, iK

w,

2 2x

Fig. 5 Armature current sheet

Table 1: Axial-flux machine parameters

Parameter 40 kW

generator

20kW

generator

Number of poles per disc 16 12

Turns per armature coil 4 9

Magnet material NdFeB NdFeB

Magnet shape trapezoidal trapezoidal

Brem 1.2 1.2

mrec 1.05 1.05

Magnet thickness, mm 6 6

Magnet radial length, mm 75 72.5

Magnet width: OD/ID, mm 51/31 62/32

Magnet overhang: OD, mm 6 5

Magnet overhang: ID, mm 6.5 5

Pole pitch, mm/deg 62/22.51 60/301

Magnet width: mean diam, mm 41 48

Magnet width/pole pitch 0.66 0.8

Gap (winding+mech.clear), mm 8 9

g+tm, mm 14 15

Rotor disc thickness, mm 10 10

Rotor OD, mm 415 314

Core OD, mm 378 290

Core ID, mm 253 165

Core radial depth, mm 62.5 62.5

Mean diameter, mm 316 227.5

Core thickness, mm 28.4 22

Overall machine axial length, mm 74 73

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can be accounted for directly by setting mrec 1.05 in (9)whilst its effect on the reluctance can be include by using aneffective gap value Y2eff in the denominator in place of Y2.The results of Fig. 8 demonstrate the validity of thisapproach.

The results in Fig. 8a show that including the effect of themagnet recoil permeability mrec in the calculation reducesthe flux density. Although the magnet permeability reducesthe reluctance of the gap between the rotor and stator iron(denominator of (38)) it also reduces the magnitude of themagnet mmf (numerator of (38)). As the numerator has thestronger effect the flux density reduces.

Shown in Fig. 8b is the vector potential calculated by(11), the values computed by the 2-D finite element analyses(FEA) are indistinguishable from this. In 2-D the differencein the vector potential between two points is the magnetic

flux/per metre so, by evaluating (11) at y Y2, the fluxentering the stator core/per metre over one pole pitch can beevaluated by:

FFpole;n fAznt=2 Aznt=2gyY2 R0 RikL;mag

17

where (R0Ri) is the core radial length and kL,mag is theeffective length ratio yet to be determined. This is ageneral equation that can be evaluated numericallyin a design spreadsheet and used directly in the emfequation.

Equation (17) can be evaluated algebraically using (11)and (12) and noting that at y

Y2:

^BBn m0^JJn

an

sinh unYs

sinh unY2effandAzn

^BBn

ansin unx 18

Fig. 6 The 40 kW generatora The generator on test

b Rotor and stator discs

Fig. 7 The 20 kW generator on test

0.6

0.4

0.2

0

0.2

0.4

0.6

0.80.10 0.08 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08 0.10

circumferential position, m

curve C

curve B

curve A

normalfluxdensity,

T

0.010

0.008

0.006

0.004

0.0020

0.002

0.004

0.006

0.008

0.0100.10 0.08 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08 0.10

tangential position, m

vectorpoten

tial

b

a

Fig. 8 Magnetic field and vector potential in the 40 kW generator.Circumferentially around the stator core; mean radius; 1 mm axially

from the core. x-coordinate is circumferential position from the polecentre (pole pitch is 0.062 m)a Axial flux density: for curve A mrec 1; for curve B mrec 1.05magnet current only; for curve C mrec 1.05 in magnet current andeffective gap. For curve C (middle line) the analytic results and 2-D

and 3-D FEA indistinguishable

b Vector potential



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Substituting into (16) and noting that for a toroidal

machine: tp l2 pDm2p

FFpole;n ^BBn

npDmkL;magRo Ri sin np

219

In the stator arrangement shown in Fig. 1 the magnetic fluxsplits in half to the link the two coils but is reinforced byflux from the other side so that:

FFcore;n

FFpole;n

20

For different axial-flux toroidal configurations the fluxlinkage might be different e.g. single rotor, double stator thecore flux per coil would be half the flux per pole.

In general, the emf/coil root-mean-square voltage is givenby:

Ecoil;n ffiffiffi

2p

pnfFFcore;nkdnNcoil 21The total coil emf, taking account of all harmonics, is then:

Ecoil ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2coil;1 E2coil;3 E2coil;n

q22

For a 2p pole machine with all coils connected in series:

Eph 2pEcoil 23The main problem in applying (17) or (19) is in determiningthe effective length ratio kL,mag. This factor must take intoaccount the fringing due to the length of the magnets. Thislength may be different to the length of the iron core andoften the magnets are made radially longer than the core tocapitalise on this effect. Figure 9 shows the fringing flux inquestion while Fig. 10 shows schematically the ideal andactual vector potential and the axial-flux density distribu-tion radially across the stator core. The radial variation ofthe vector potential and the axial-flux density can beobtained from (11) and (12) by setting the magnet widthequal to the magnet radial length with the pole pitch

significantly longer than this; typically by about a factor of100. It is also important to include a large number of spaceharmonics in the computation to guarantee convergence.

The basic 2-D analysis assumes that the flux density isconstant over the radial length of the stator core. As Figs. 9and 10 imply this is not the case and account must be takenof the fringing flux. This is normally allowed for byintroducing an effective length factor kL,mag. In the past thisfactor has been evaluated geometrically depending on theratio of the magnet length to the core radial length [7]. Analternative approach is to base the effective length ratio onthe vector potential (magnetic flux) calculated analyticallyusing the 2-D analytic model to account for the actuallength of the magnet in the radial direction. Unfortunately

the 2-D analytic model does not account for the finitelength of the iron core (it is assumed infinite) butnevertheless an acceptable effective length ratio can beobtained.

If fringing flux is neglected, Fig. 9a, the basic analysis

assumes ^BBn constant over the core length (RoRi) and, asBn@A/@x (the normal is in the y-direction):

DAbasic ^BBnDr ^BBnRo Ri 24The actual change in magnetic potential (and hence flux)allowing for fringing flux is greater and is evaluated close tothe stator core over the radial length of the magnet by:

DALLm fAzLm=2 AzLm=2gyY2 25As the purpose of the effective length ratio is to account forthe difference between the ideal and actual flux density

distributions the effective length ratio is given by:

kL;mag DALLmDAbasic

26

To check the validity of this approach Fig. 11 plots thevariation of vector potential and axial flux density radiallyacross the surface of the iron stator core for the 40kWgenerator. Flux density plots are obtained using the analyticexpressions and 2-D and 3-D FEA computations whereasanalytic and 2-D FEA are used for the vector potential.The accuracy between the different models is acceptableconsidering that the finite element models account for thefinite length of the core and rotor discs in the radialdirection whereas the analytic expressions assume these tobe infinite.

aa

magnet

core

(RoRi)

b

a

Fig. 9 Magnet fringing fluxa Schematic

b Flux density vectors for 3-D FEA

coreAz

Bn

Bn

r

(Ro Ri)

magnet

Fig. 10 Schematic of the ideal and actual vector potential and fluxdensity distributions

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4.1.2 Test results: Results obtained from the analyticanalysis described above, FEA and measurement on the testgenerators are presented in Table 2.

Measured results are obtained during testing using acalibrated Voltec Powermeter. All the results show goodagreement with the measured and calculated emfs compar-ing to within 5%. The ability of the analytic method toindicate the magnitude of the harmonic emfs is also goodwith the effect of changing the magnet pitch/pole pitch ratiofrom 0.66 in the 40 kW to 0.8 in the 20 kW constructionbeing clearly indicated.

For a three-phase machine the winding spread is typically601 (electrical). For the axial-flux toroidal machine thisspread only occurs at the inner radius of the stator. As thewinding is rectangular its angular spread reduces towardsthe outer radius and the winding spread factor changes.Generally a value taken at the mean radius will suffice andthis is used in the analytic calculations of the emf. In boththe 20 and 40 kW designs the stator core is supported fromthe outer diameter so that a 601 spread at the inner radius ispossible, Figs. 6 and 7. In some applications the stator coreis supported from the inner diameter when the windingspread will be reduced from the 601 maximum.

4.2 Armature inductance

4.2.1 Analytic considerations: Accurately knownarmature inductance values are important in order to

predict machine performance. The analysis of Section 2 isused to compute the armature inductance directly from thevector potential in a general way whilst FEA is used toshow the importance of end-winding leakage.

The armature winding is represented by a current sheet sothat (3) and (4) for the vector potential and the flux densityrespectively can be used directly. With the armature windinglocated at Y1 Ya/2 the flux produced by the armaturewinding over one pole pitch in region 1 of the radial activeregion of the machine is:

FFn;active fAznt=2 Aznt=2gy0;Y1ta=2 R0 Ri

27

where Azn is obtained from (3). As it is the field at thearmature surface that is of interest, the field point is set toy 0. (R0Ri) is the radial length of the core. Equation (27)is most easily evaluated numerically directly in the designspreadsheet. As the flux calculated by (27) splits in half as itpasses into the stator core only half of this flux links anarmature coil. However, this flux is reinforced by the sameamount from the current on the other side of the core sothat (27) gives the value of the core flux. Equation (27) onlyaccounts for armature flux produced by the armaturecurrent as it passes radially across the core surface; it takesno account of the flux produced by the current as it passesaxially along the core ends at the inner and outer radii. Thisleakage flux can be calculated in a similar way as for themain flux in (27) but now with the iron surface representingthe rotor removed (i.e. increase Y2 to a large value) and withthe active length changed to the core thickness i.e.:

FFn;leakage

fAznt=2 Aznt=2gy0;Y1ta=2;Y2!1 tc28

0.040

0.035

0.030

0.025

0.020

0.015

0.010

0.005

00.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04

vectorpotential

radial position from mean core radius, m

radial position from mean core radius, m

a

0.6

0.5

0.4

0.3

0.2

0.1

0

0.10.06 0.04 0.02 0 0.02 0.04 0.06

b

normalfluxd

ensity,

T

analytic

FEA

analytic

FEA

Fig. 11 Variation of vector potential and axial flux density withradius; 40 kW generator 1 mm from the iron corea Vector potential

b Axial-flux density

Table 2: Magnet flux and emf results

Parameter 40 kW

generator

20kW

generator

Bz, T analytic 0.493 0.46

2-D FEA 0.495 0.46

3-D FEA 0.494 0.46

kL Equation eq (29) 1.11 1.07

2-D FEA 1.09 1.08

Core flux, mWb Analytic 1.37 1.36

3-D FEA 1.37 1.36

Measured 1.3 Not measured

Emf

Vrms/1000 rpm Analytic 52.1 61.9

3-D FEA 51.9 62.2

Measured 49.6 62.2

Emf

harmonics first analytic V 52 61.6

first measured 49.6 61.6

third analytic 1.4% 9.7%

third measured F 12.4%

fifth analytic 4.0% F

fifth measured 4.4% 0.3%

seventh analytic 0.5% 0.2%

seveth measured 0.6% 0.7%



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The total armature flux is obtained from the sum of (27)and (28) as:

FFn;arm FFn;active FFn;leakage 29The armature inductance can be evaluated by the followingprocedure:

Step 1: Evaluate the flux per metre from (29) using thecurrent distribution of (15).

Step 2: Compute the flux linkage CCcoil;n FFn;armNckdn.Step 3: Compute the coil inductance by dividing by the

armature current.Step 4: Computation of mutual inductance follows steps

1 to 3 with flux computed from (29) but with the coildisplacement g from the exciting coil introduced into theequation i.e. evaluate between (gt/2) and (g+t/2).

If required, algebraic expressions for armature flux andinductance can be evaluated from (27), (28) and (29). Forexample, if the armature current sheet is assumed at thestator iron surface, then:

FFn 2^KKnm0un

R0 Ri coth unY2 tc sin np2

30

Substituting for Kn from (15), introducing the distribution

factor kdn and noting that l 2t 2pRm/p and Z Nc thecoil flux linkage is:

CCcoil X 4m0

npN2c k

2bniRo Ri coth unY2 tc 31

Finally the coil inductance is obtained from Lcoil;n CCcoil;n=ito give:

Lcoil;n 4m0npN2c k

2dnRo Ri coth unY2 tc 32

The coil mutual inductance is computed in the same waybut now with the coil displacement g included to give:

Mcoil;n

4m0

npN2

ck2

dnR

o R

icoth u

nY

2 t

c cosung

33

The actual phase inductance depends on the coil connectionand for all 2p coils connected in series is 2p times the abovevalues while the phase effective inductance is:

Leff;n 2pLcoil;n Mcoil;n 34

Equations (32) and (33) are identical to those derived byChalmers et al. [13]. A progressive number of simplificationscan be made to the above inductance expressions. If the air-gap is assumed to be small compared to the pole pitch,Y2{tp, and the effect of the end-winding leakage flux isneglected then, considering the first harmonic only, gives:

Lcoil 2pp

moN2coilk

2d1Dm

Ro RiY2

35

with the effective inductance being:

Ld 6pmoN

2coilk

2d1Dm

Ro RiY2

36

These are identical to the values obtained from themagnetic equivalent circuit approach described in [7].

Alternatively if the air-gap Y2 is assumed to be smallcompared to the pole pitch, the effect of the end-windingleakage flux is neglected and the coils are assumedconcentrated then the self-inductance expression in (32)reduces to that originally used by Spooner et al. [20].

4.2.2 Analytic and FEA comparisons: The ac-curacy to which (4) predicts the armature magnetic field isexamined in Fig. 12 by plotting the circumferential variationof the axial component of the armature magnet field at thesurface of the stator core over one pole pair at the meanradius. The excitation current is 5 A and one-phase only ofthe 40 kW machine is excited. Both 3-D FEA and analyticalresults are shown. Both methods give very similar resultsbeing the same midway between the coils with a differenceof about 10% at the coil edges due to edge-effects being

taken into account by 3-D FEA but not by the analyticmethod.

The way in which the axial-flux density changes radiallyacross the active region was calculated by 3-D FEA and isshown in Fig. 13. The flux density stays sensibly constantover most of the active region with peaks at the edge of theiron core.

Although the core thickness is small in most axial-fluxmachines, end winding-flux can still be significant. Forexample in the 40 kW design, Table 1, the stator core is29mm thick compared to an active radial core length of62.5 mm and (30) implies that the core ends account forabout 23% of the fundamental component of flux. Toexamine the effect of the core ends in more detail Fig. 14plots the circumferential variation of the radial flux densityacross the inner and outer core ends computed by the 3-DFEA method. Also plotted is the variation predicted by (4).The field point is taken just above the core surface at themean thickness. The two FEA calculations take account ofthe machine radius so that the flux at the inner radius isrestricted to a smaller area thereby resulting in a larger flux

0.0015

0.0010

0.0005

0

0.0005

0.0010

0.0015

0 5 10 15 20 25 30 35 40 45

angle, degrees

fluxdensity,

T

3D FEA Analy tic

Fig. 12 Three-dimensional FEA and analytic plots of thecircumferential variation of the axial armature flux density 1 mmabove the core surface and at the mean core diameter: 5 Aexcitation, 40 kW machine

0

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0 0.05 0.10 0.15 0.20 0.25

radial distance, m

fluxdensity,

T

core boundaries

winding thickness

Fig. 13 Three-dimensional FEA plot of the radial variation of theaxial armature flux density 1 mm above the core surface mid-way

between two armature coils: 5 A excitation, 40 kW machine

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density than at the outer radius. As plotted, the analyticmodel does not account for machine curvature and assumesthat the coil pitch is defined at the mean core radius. Tocomplete the flux density plots Fig. 15 shows how the radialflux density varies axially across the core ends as computedby the 3-D FEA model.

4.2.3 Test Results: High pole-number, air-gapwound, axial-flux machines such as those described heregenerally have a low inductance so that measuring itaccurately is difficult. Standard short-circuit tests can not beused as these can only be carried out at very low speedswhen the armature reactance is negligible and is totallyswamped by resistance effects. If a standard inductance

bridge is used eddy-currents are induced in the rotor and themagnets and the inductance measured is not the requiredone. Once the machine is built this is probably the best thatcan be achieved. This problem can be overcome if thearmature inductance is measured before the generator isassembled by using other strip-wound armature cores, inplace of the two rotor discs, placed above and below theactual armature at a distance equal to the separation ofthe stator and rotor discs in the final machine assembly. Thestrip-wound rotors do not allow eddy-currents to beinduced in them and the inductance measurements (by aninductance bridge) are frequency independent. The resultspresented in Table 3 were obtained by this means at atypical stator frequency and is the average across the phases(equivalent to a running speed of 1500 rpm).

The 2-D analytic inductance results are calculatedassuming that the current sheet representing the armature

winding is situated on the stator core surface, as experiencehas shown that this consistently produces results compar-able with measured data. These inductance calculations alsotake account of the winding distribution through the use ofthe harmonic distribution factor calculated at the meanwinding radius. This is an approximation as the windingspread varies from a maximum at the inner radius to aminimum at the outer radius. The analytic approachcalculates the end-winding flux separate to the flux fromthe active region and, for the 40kW machine, thesecalculations show that the end-winding flux is 1.24mWb(for a 5 A excitation) and thereby constitutes 24% of thetotal flux and is an important component in assessing thewinding inductance.

5 Conclusions

We have shown how a unified approach can be adopted tothe analytic calculation of the magnetic field in a permanentmagnet, axial-flux generator. A comparison of the resultsobtained using FEA and measurements on two prototypelaboratory generators has demonstrated the validity of theapproach. Armature reactance is particularly difficult tomeasure using conventional methods as, in these types ofair-gap wound machines, it is very small and, in manyinstances, armature resistance dominates. However, asimple, reliable, method of measuring armature inductance

before the machine is assembled has been described. Theanalytic approach is ideally suited for use in designspreadsheets.

6 Acknowledgments

Most of this work was completed under a DTI/EPSRCForesight Link Project and the financial support of EPSRCand Newage-AVK-SEG is gratefully acknowledged.

7 References

1 Huang, S., Aydin, M., and Lipo, T.A.: A direct approach to electricalmachine performance evaluation: Torque density assessment andsizing optimisation. Presented at the Int. Conf. on Electricalmachines, Bruges, Belgium, Aug. 2002

2 Brown, N., Haydock, L., and Bumby, J.R.: An idealised geometricapproach to electromagnetically comparing axial and radial PM

0.0015

0.0010

0.0005

0

0.0005

0.0010

0.0015

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0

angle, degrees

fluxdensity,

T

outer inner analyt ic

Fig. 14 Three-dimensional FEA and analytic plots of thecircumferential variation of the radial armature flux density half-way across the core end and at the core surface. 5 A excitation,40 kW machine

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.03 0.02 0.01 0.00 0.01 0.02 0.03

axial distance from centre of core, m

fluxdensity,

T

outer inner

Core boundaries

Winding thickness

Fig. 15 Three-dimensional FEA plot of the axial variation of theradial armature flux density across the inner and outer core ends,1 mm above the core surface: 5 A excitation, 40 kW machine

Table 3: Armature flux and inductance results; 5 A excita-tion in the armature coils

Parameter 40 kW

generator

20kW

generator

Bz, active region, mT analytic 0.90 1.97

3-D FEA 0.90 1.89

Core flux, mWb analytic 5.23 10.5

3-D FEA 5.12 10.6measured 5.48 10.0

Lph, mH analytic 62.9 212

3D FEA 65.5 229

measured 70.8 226

Mph, mH analytic 18.5 62.6

3-D FEA 23.6 82.1

Measured 24.7 78.6



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machines. Presented at the Int. Conf. on Electrical machines, Bruges,Belgium, Aug. 2002

3 http:// www.jeumont-framatome.com, Jeumont J48, 750 kW,Direct drive discoidal wind turbine generator (last accessedOctober 2002)

4 Muljadi, E., Butterfield, C.P., and Wan, Y.: Axial-flux modularpermanent magnet generator with a toroidal winding for wind-turbineapplications, IEEE Trans. Ind. Appl., July/August 1999, 35, (4), pp.831836

5 Brown, N., Scott, K., Lye, E., Bumby, J.R., and Spooner, E.: Acomparison of iron-cored and ironless axial-flux PM machines.Presented at the 36th Universities power engineering conference,Swansea, September 2001

6 Brown, N., and Haydock, L.: Full integration of an axial flux

machine for reciprocating engine variable speed generating sets. Proc.IEE Seminar on axial air-gap machines, London, UK, May 2001, pp.6/16/10

7 Brown, N., Haydock, L., and Bumby, J.R.: Foresight Vehicle: AToroidal, Axial Flux Generator for Hybrid IC Engine/Battery ElectricVehicle Applications. Proc. SAE Conf. paper 2002-01-0829, Detroit,USA, March 2002

8 Ramsden, V.S., Mecrow, B.C., Lovatt, H.C., and Gwan, P.: A highefficiency in-wheel drive motor for a solar-powered vehicle. Proc. IEEColloqium on Electrical machine design for all-electric andhybrid-electric vehicles, Savoy Place, London, October 1999,pp. 3/13/6

9 Carricchi, F., and Crescimbini, F.: Design and construction of awheel-directly coupled axial-flux PM motor for EVs. Proc. IEEEIndustrial Applications Society Conf., New York, USA, 1994, pp.254261

10 Marignetti, F., and Scarano, M.: Mathematical modelling of an axial-flux PM motor wheel. Proc. Int. Conf. on Electrical machines,Helsinki, August 2000, pp. 12751279

11 Patterson, D., and Spee, R.: The design and development of an axialflux permanent magnet brushless DC motor for wheel drive in a solarpowered vehicle. Proc. IEEE Industrial Applications Society Conf.,Denver, 1994, Vol. 1, pp. 188195

12 Spooner, E., and Chalmers, B.J.: TORUS: A slotless, toroidal-stator,permanent-magnet generator, IEE Proc., Electr. Power Appl.,November 1992, 139, (6), pp. 497506

13 Chalmers, B.J., Green, A.M., Reece, A.B.J., and Al-Badi, A.H.:Modelling and simulation of the TORUS generator, IEE Proc.Electr. Power Appl., November 1997, 144, (6), pp. 446452

14 MEGA version 6.29m (University of Bath, Applied ElectromagneticResearch Centre, Bath, UK, 2003)

15 FEMLAB, Users Guide and Introduction, Version 2.2, COMSOLAB, http://www.comsol.com (last accessed October 2002)

16 Smith, A.C., Williamson, S., Benhama, A., Counter, L., andPapadopolous, J.M.: Magnetic Drive Couplings. Proc. IEE Conf.on power electronics machines and drives, London, UK, September1999, pp. 232236

17 Wallace, A., von Jouanne, A., Williamson, S., and Smith, A.:Performance prediction and test of adjustable permanent magnet loadtransmission systems. Presented at the IEEE Industrial Proc.Applications Society, Chicago, IL, USA, 2001

18 Brooking, P., and Bumby, J.R.: An integrated engine-generator setwith power electronic interface for hybrid electric vehicle applications.

Proc. IEE Conf. on Power electronics machines and drives, Bath, UK,2002, pp. 15315819 Brown, N., Haydock, L., Spooner, E.S., and Bumby, J.R.:

Optimistation of Axial Flux Generators by Pre-Computed 3D-FEAModelling. Proc. Int. Conf. on Electrical machines, Helsinki, Finland,2000, pp. 14711474

20 Spooner, E., Chalmers, B.J., El-Missiry, M.M., and Kitzmann,I.: The design of an axial-flux, slottless, toroidal-stator,permanent-magnet machine for starter alternator applications.Proc. Universities power engineering Conf., Aberdeen, 1990,pp. 171174

8 Appendix

Assuming all the flux from the magnets crosses the air-gapto the stator core then applying Amperes law to Fig. 4 gives:

HairYa c HmagYm B

remY

mm0mrec

37

but since Bm0Hairm0mrecHmag, then:B BremYm=mrecYm=mrec Ya c

38


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