686 IEEE Transactions on Energy Conversion, Vol. 14, No. 3, September 1999

Slotless Brushless Permanent Magnet Machines: Influence of Design Parameters

Y.S. Chen, Z.Q. Zhu, Member IEEE, and D. Howe Department of Electronic and Elcctrical Enginecring, University 01' Shcfl'icld,

Mappin Street. Shcll'icld SI 3JD. UK

Abstract: The airgap field distribution, hack-emf wavelorm, winding inductances, and iron loss are predicted analytically from a 2-D polar co-ordinate model of a slotless brushless permancnt magnet machine. The influence of pole number, magnet pole-arc and thickness, and magnetisation on the operational performance is then investigated, and a guide to design optimisation is presented.

Keywords: permanent magnet, brushless, slotless. design optimisation, magnetic field.

I. INTRODUCTION

For some applications, which may require ripple-free torque and/or a very high rotational speed, for example. slotless brushless permanent magnet machines are often appropriate. Fig.1 shows the cross-section of a 2-pole, 3- phase slotless machine, which, in general, will have a higher electrical loading and a lower magnetic loading than an equivalent slotted machine.

A 2-D polar co-ordinate analytical model of slotless machines has been developed and validated for the prediction of the airgap field distribution, the hack-emf waveform, the winding inductances, and the iron loss [I]- [3]. In this paper, the model is used to investigate the influence of the pole numher. the magnet pole-arc and thickness, and the magnetisation, with particular rclkrcnce to efficiency, emf harmonic content. and magnet voIumc. In addition, the results of design studies for minimum lo compared with corresponding findings deduced from ii

simple I-D modcl.

11.2-D ANALYTICAL MODEL

The developed 2-D analytical model is described in detail in [1]-[3]. It enables the 2-D distribution of both the open- circuit airgap field and the armature reaction field to he obtained from the solution of the Laplacian/quasi-Poissonian field equations in the airgap/magnet/winding regions. and differs from the model given in [4] in that it accounts ior

PE-1348-EC-0-2-1998 A paper recommended and approved by the IEEE Electric Machinery Committee of the IEEE Power Engineering Society for publication in the IEEE Transactions on Energy Conversion. Manuscript submitted August 27, 1997; made available for printing February 18, 1998.

both surface and v~ lu inc current densities and thc relative recoil pcrmeahility of the magnets.

Fig. I. Typical ?-pole slolless hriisliless PM i i ix l i i i ic ,

Fig. 2 compares analytically calculated and liiiitc elciiieni predictcd distrihutions of radial and circumf'crcntial components of open-circuit airgap llux dcnsity for a n 8-polc motor cquippcd with parallel magnetised pcrmancnt magnets. The motor design paramctcrs arc given in tlic Appendix. The analytical model assumes inlinitcly permeable hack-iron, whcrcas thc FE analysis accounts for the non-linear magnctisatioii characteristic. However. cxccllcnt agreement is cihtained, since thc maximum tlux density in the hack-iron is only I .4 T. Whcn iron saturation is likely to he significant. the analytical inodel can he incorporated i n a non-linear lumped circuit analysis of a motor. In gencral, howcvcr. saturation is not usually significiuit i n slotlcss inachincs.

Thc phase hack-cinf wavcforin is ohtaincd hy integrating open-circuit ficld distribution ovcr the cross-scctioii of thc appropriate coils. Fig. 3 shows a comparison of the analytically calculated, linitc clcmcnt predicted and measured hack-emf waveforins lor the X-pole motor. whilst Tahle I compares analytically and I'initc elcmcnt prcdictcd scll- and mutual- inductances with mcasurcd values.

The iron loss i n tlic stator hack-iron is also estimated analytically 131. and for the X-pole motor is predicted to hc 9.0 W at 3000 rpm, which agrces wcll with a finite cleinetit predicted loss of9.6 W.

111. INFLUENCE OF MOTOR DESIGN PARAMETERS

Section I1 has dcmonstratcd thc ut i l i ty of' thc analytical inodel, which will now l)c used to investigate thc inllucncc 01'

0885-8969/99/$10.00 0 1998 IEEE

687

leading design parameters. The investigation is again based around the 8-pole motor whose design details are given i n the Appendix. However, as design parameters are varied, others are changed so as to achieve the rated torque and speed.

1.5 " " " " " '

E

x 3 !r

1 0

0 5

0

-0 5

-1 0

-1.5' 1 0 60 120 180 240 300 360

Angular position (elec.deg.)

Fig. 2. Comparison of analyticnl and finite elenient predicted field dist~ihutions ( ~ 4 0 . 5 Inin).

100

80

60

40

- 20 z U 0 I

-20

-40

-80

-100 0 60 120 180 240 300 360

Angular position (elec.deg.)

Fig. 3. Back-emif waveforins (5% rpln)

TABLE 1

COMPARISON O F WINDING INDUCTANCES

Se1 f- 2.166* 2.760'~ 3. I Inductance (inH) Analytical Finite element Measured

Mutual- -0.969 -0.56X -0.44 Includes I .R iiiH end-component o f inductance. predicted using colivciitioiinl

theory [SI.

A. Magnet Thickness

The magnet thickness k,,, is a significant parameter lor all permanent magnet motors, since i t alfects the efficiency, tlie

dcmagnctisation withstand and the cost. For the values of R,, R, and R,,, given i n tlic appendix, and rated torque. there is an optimal magnet thickness which results i n miniintiin copper loss, a s shown i n Fig. 4. which also shows the variation o l the iron loss and tlie iota1 loss. As will he sccn. at rated load the copper loss is dominant. as is usually the case in slotless inotors. However. due to iron loss, the optimal magnet thickness for minimum total loss is slightly lower than that for minilnuin copper loss, especially ior high pole numbers. Both thin and thick inagncts lead to a Iowcr efficiency. as well a s a higher harmonic content i n the hack- cm1 wavelorin, sincc thin inagncts result in a I O W airgap Ilux density, a poor radial t'icld orientation and high Ilux-leakage. whilst thick inagncts result i n a high airgap flux density and reduced winding space, and. thcrelorc, ii high copper loss.

H. M ~ i g i i ~ f Po/P-Ar('

Fig. 5 shows the cllCct o i varying the inagnct pole-arc on tlic total loss and harmonic content 0 1 Ihc induced emf. Sincc the copper loss is tlie doininant loss coinponcnt. tlic iota1 loss decreases with an increase 0 1 inagnct pole-arc.

The line induced emf wavelorin cannot conlain triplcn harinonics. Hcncc the harmonic content (dclincd as \IC E,,?/E,' 100 % ) is IOW, and is always less than

3.5% for the motor designs which have been considercd. However, a pole-arc o l around 150" might he preferred so

as to obtain a sinusoidal emf waveform and lower the cost. alhcit reducing tlic efficiency, although a pole-arc 0 1 180" would result an increased clt'icicncy. alhcit the utilisation 0 1 tlic magnets inay hc Iowcr due to increased inter-polc leakage.

C. Po/(, N~iniher

,U=?

For a given speed, the Iundamcntal frcqucncy increases. as tlie numhcr of poles is increased. whilst the outsiclc diameter 0 1 the motor inay he rcdt~ced.

El 1'' N, ' [O,,,

The hack-emf constant. Kc,, dcfincd as K, =

where /J is the numhcr of pole-pairs, N, is the nuinher or turns pcr pole-pair per phase. iind U,,, is the inccliiinical angular velocity, corresponds to Ihc average Ilux-linkage per turn. Due to inter-pole leakage, K,. varies with the pole number, as shown in Fig. 60. Parallcl inagnctiscd inagncts result in a higher hack-cml than radial magnetised inagncts. except in the case nl a 2-pole motor, lor which the fundamental cinl is Iowcr lor pmi l lc l inagnctisiition Further. parallel inagnctisation r c s t ~ l t s in csscnl ial ly sinusoidal airgap llux and cm1 wavclorms. whereas riiiliiil

magnetisation results in irqxzoidal waveforms.

688

20 -

i 10 1

,... A .-- -. 2 poles I A F .

h," (mm)

(a) iron loss and copper loss

20 - a- - -. 2 pales *--a 4 poles

o - - - - o 8 poles *----Q 6 pOl0S

0 5 i o 15 20

h," (mm)

(b) total loss

Fig. 4. Variatioti of copper', iron and total loss with magnet thickness.

If the stator back-iron thickness is varied with the pole number, so as to maintain a constant maximum flux density in the hack-iron, the iron loss density(W/kg) increases as the pole nuinber is increased, whilst the volume of iron decreases. As a result, the iron loss remains essentially constant for the motor under consideration, as shown in Fig. 6b.

Fig.6h shows, however, that a 2-pole motor has the highest total loss, due largely to having the longest end- windings, and that 4, 6 and %pole motors have more or less the same efficiency.

IV. DESIGN OPTIMISATION

It will be evident from Section 111 that copper loss is usually the dominant loss component in slotless machines, and that there exist optimal motor designs for minilnuin copper loss. Whilst such designs can be identificd by incorporating the developed 2-D analytical model in an appropriate objective function and employing an automatic optiinisation technique, such as simulated annealing or a

genetic algorithm, i t is possible to deduce optimal dimensional ratios directly from the analytical inodal.

In this section, it will be shown that the usual I-D airgap field analysis is unacceptablc, and that a 2-D field analysis is necessary.

40

20 90

I 120 150 160

Magnet arc (e1ec.deg.l

(a) t(lti1l loss

Magnet arc (elec.deg.)

(h) emf I~ariiionic contcnt

Fig. S. Varicition 01 total 10s and einfliarinonic contenl with mngnrt pole-arc.

2 4 6 8 10

Pals number

(a) emf cciiistuit

689

total 103s

2 4 6 8 10

Pole number

(b) iron loss and total lnss Fig. 6. Variation olernfconstant, iron loss. and total loss with pole nutnhu

atid magnetisation (h,,,= IOinin).

A. I-D Opimi.su/fo/7

For an ni-phase slotless permanent magnet motor, the copper loss P,, can be written as:

where p. i s the resistivity of copper, 2, = coil pitch; I,, = active axial length; K,.,,.= end-winding factor; K/, = winding packing factor.

The back-emf can be calculated from:

where wis the electricill angular velocity.

i n the airgap/winding region tnay be estimated l'roin: Assuming a 1-D airgap field distribution, tlie llux density

wliere /I = g +/I, +/I,,, = I R , ~ - R,.I , and B, is tile magnet

remanence. By neglecting the end-windings, equation ( I) can be rewritten, using equations (2) and (3). as:

For a motor with a specified output power (mEI), and electrical angular velocity U, and specified values of I,, , Rs, Rr and g , the optiinal value of 11,. lor minimum I><. can he

obtained by setting - = 0 , which yields: d/lNz,

2(h - g ) + 3R,$ - .\/4(/1 - g )' + 9 R,, ' 6

/ I , . = ( 5 1

The variation of the optimal ratio olinagnel thickiicss. /I<,,. to winding thickness, / I , , with tlic ratio R,/R, i s shown in Fig; 7. It wil l he notcd that the optiinal value of h J 1 , dctluccd from a I-D analysis does not depend on the pole nuinher. and that il' curvature is neglected, i.e. K , + m the optiinal value of/1,J/1~-+2, which agrccs with the result given i n r6l.

U. 2 - 0 Optiiiii,sLitio/1

The 2-D analytical cxprcssions lor tlic airgap f ield distrihution and the various machine parainclcrs have hccn incorporakc1 into a CAD prograin which eiiahlcs inotor designs to he synthcsiscd very quickly, to inecl a spccificd torque-speed requirement, a s the leading design paramctcrs arc incrcinciired hctwccn prcscrihcd l im i t s . By way ol cxamplc, each point i n Fig. X rcprcsents a lciisiblc motor design lor a given torquc/spced specification, a large nuinher of designs being gcncrittcd as tlic inagnct thickness and pole- arc arc viiricd limn 5 inin to I 7 nnn and 120" to I XO". respectively ( in incremcnts o l I inin and lo"), and R, and K , arc varied lroin 46 nnn lo SO inin and lroin 23 iniiii to 17 inin. rcspcclively, (both i n iiicrcincnts o l I inin). For ii givcn inagnct wcight tlicrc is clcarly one optiinal design which leads to ininitnuin total loss. By generating similar graphs which show the varialion (11' other selccted pcrli)rinance paramctcrs. an optiinal motor design can he idcntificd which best satisfies a coinhina~ion ol tCcIitio-cc~~ii(~inic ohjcctivcs.

However, as an aid to design opliinisation tlic pcrlormancc can a lso be calculatcd l i ir spccilicd ratios n l K, /K , . K,,JR, (or R,/R,. /i,,Jli, ctc. j, g/R< , 1,/2K,. a / , , and lllc

polc-pair nuinher. The open-circuit airgap held distribution can he cxprcsscrl

a s [ 11:

where. a / , i s tlie inagnct pdc-arc 10 polc-pilch ratio. Hence.

cquiitions ( I ) and (2) can hc coinhinctl 10 give:

whcre,

which i s diincnsionlcss and. lroin eq~iation (7). shnws that lor givcn inotor diincnsions, R , and I,, , and oup i t powcr. !&I, the copper loss P, dcpcnds only on thc ti inc~ion ,/,,,

690

._ = 10.

Hence, the minimum copper loss corresponds to the ininiinuin value of f p , .

a ' ,,# A' A-- . p

>z*z*je;b;bF&*+* _ _ -* -0-d-

7-

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

R!Rs

(a) parallel magnetisation

E 1 0 -

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

R P , (b) radial magnetisation

Fig. I . Optimal dimensioiial ratios deduced from 1-11 and 2-D iinnlysrs

motor design

40 " " " " " " "

0.5 1.0 1.5 8.0

Magnelwslghl (Kg)

Fig. 8. Typical CAD generated results for %pole 1110t01

Fig. 7 shows that the optimal ratio 01' /?,& for in in i~nu~n J';,,., and therefore copper loss, now depends on the pole numher, and again varies with the ratio R,/R,$. It also shows that the optiinal ratios are allnost identical for both parallel and radially magnetised magnets, the largest difference being for a 2-pole machine.

Fig. 9 shows that, for ininiinuin copper loss, f;,. decreases with a reduction in the ratio R,/R,. However. a smaller R,/R, ratio will necessitate a thicker magnet. Hence, i t is also appropriate to consider the magnet volume, V,,M,

Fig. IO shows the optimal relationship between thc magnet VP,,

n4-4 volume V P M , (expressed as the ratio - ), and f p c for

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

R/Rs

Fig. 9. Voriatioli ( ~ I i i i i i i i i i i i ~ i i i ~ ~ , with I<,//?

- ~ - rn lopoles - - - e 8 poles v - - 6 poles D ~ - - o 4poles * -- il 2 poles

, -.ipL--i 01 0 0.2 0.4 0.6 0.8

Normalised magnet volume

Fig. IO. Variarion of ~iii~ii i i iu~iif,~,. with tlic ~i~ri i i i i l iset l iiinpnct \~11l11111e

The preccding optimisation study iissumcd g/R< to tic constant at 0.01 and /,JZR,% to he 1 .O. although thc ratios do not significantly affecl the findings, whilst the inagncr polc- arc was 180"(parallel inagnctisation). Radial inagnetisctl magnets yield similar rcsults LO h s c lor parallcl

691

magnetisation. Howevcr, a slight difference is that the copper loss does not always decrease with a reduction i n R,/R,T. When R,/R,v is reduced below 0.5, the copper loss begins to increase slightly due to the fact that a reduction of R,, and an increase in h,,,, does not always lcad to an increase in airgap flux for radial magnetisation, as descrihcd in 171.

V. CONCLUSIONS

Slotless brushless PM motors arc preferred for various applications, particularly when a I O W torque ripplc is required. Additionally, howevcr, duc to their low iron loss and low winding inductancc, they are likely to have significant potential for high speed applications.

An analysis has been devcloped to aid tlic design optimisation of slotless brushless pcrinanent magnet motors. It is based on the 2-dimensional calculation of the inagnctic fields, and the associated emf waveform and winding inductances, and copper and iron losses. Thc cffect of various design parameters on perlorinance factors, such as efficiency, back-emf harmonic content. and inagnct voluinc, has been investigated. It has been shown that thc optiinal ratio of h,JhC depends on the pole number. and is > 2 Ibr motors having an internal rotor. It will he noted that the 2-D analytical model can also be applied to external rotor motors, for which the optimal ratio of h,,JhC has been found to be < 2, shown in Fig. I I for parallel inagnctised inagncts.

VI. APPENDIX

The parameters of the 8 pole brushless ac motor which is used in the investigation arc: 1.3kW, 380 V, 3000 rpm. K,.=30 inm, /?,,,=IO min. g=l inin, R,!=48 inin, K,,,=S3 inin, 1,=6S nnn, magnet pole arc= I XO", Transi1300

laminations. sintercd NdFcB magnets with parallel tnagnetisation.

VII. REFERENCES

VIII. BIOGRAPHIES