Fig. 1 Structure of rotor loss calculation

Approximate Methods for Calculating Rotor Lossesin Permanent-Magnet Brushless Machines

T. J. E. Miller, M.I. McGilp K.W. Klontz SPEED Laboratory Advanced MotorTech LLC

University of Glasgow 9117 Park Blvd Glasgow G12 8LT, U.K. Largo, FL 33777-4133

Abstract — This paper describes approximate methods for calculatingthe rotor losses due to induced currents in rotors of permanent-magnetbrushless machines, that is, in the magnets, the shaft, and in any retainingcan. Different methods are developed for surface-magnet and interior-magnet machines. The methods include segmentation of the magnets andthe rotor can in both the axial and circumferential directions. They canbe extended via the frequency-response method to calculate thesubtransient reactance and time-constant needed for fault calculations.

I. INTRODUCTION

In an ideal synchronous machine the field rotates insynchronism with the rotor, and the flux-density is time-invariantthroughout the rotor cross-section. There is no tendency foreddy-currents to flow anywhere in the rotor. There are no lossesin the magnets, the rotor body, or the shaft. Such ideal conditionswould exist at constant speed in a machine with smoothcylindrical surfaces (no slotting); with sine-distributed windings;and with balanced polyphase sinusoidal currents.

Eddy-currents are induced in practice by imperfections ordepartures from the ideal, some of which are described with codeletters as follows:

h1 Space-harmonics in the stator ampere-conductor distributionand time-harmonics in the stator current waveform produceasynchronous field components that rotate forwards orbackwards relative to the rotor. For example in motors withsquarewave drive, the stator ampere-conductor distributionremains fixed in space for successive intervals, typically of60E duration. As the rotor moves relative to the fixed“armature reaction” field, EMFs and eddy-currents may beinduced in it. Again, commutation of the current from onephase to another is, in effect, a voltage step at the statorterminals, inducing a “transformer” EMF in closed paths inthe rotor. The resulting eddy-currents are transient, butexcited repetitively, resulting in a steady-state average loss.

h2 PWM-frequency components of the stator current produceasynchronous fields.

h3 Even in motors with sinewave drive, imbalance between thephase currents results in a negative-sequence component ofthe armature-reaction field, which rotates backwards andinduces EMFs in the rotor at twice the fundamental frequency.

m1 The overall permeance of the magnetic circuit may bemodulated at the slot-passing frequency. The permeanceharmonics produce cogging torque, “slot ripple” in the EMF

waveform, and eddy-currents in the rotor.

m2 Stator slot-openings create “dips” in the airgap flux-distribution which rotate backwards at synchronous speedrelative to the rotor. They produce pools of motion-inducededdy-currents that remain stationary in space, roughlyopposite the slot openings.

All of the h-type losses can be analyzed by solving theelectromagnetic field excited by time-harmonics in the currentwaveform acting together with the space-harmonics in thewinding distribution. If the induced currents are assumed not toaffect the stator current waveform, the field and circuit analysescan be separated, simplifying matters greatly. This solution iscalled “current-forced”.

For surface-magnet machines this solution for h-type lossescan be developed by solving the complex diffusion equation in amulti-layer cylindrical structure, including the shaft and anyretaining sleeve [1]. A complete simulation of the machine anddrive is required to obtain the time-harmonics of the currentwaveforms, and a harmonic analysis of the winding is requiredto obtain the space harmonics of the ampere-conductordistribution [2].

Approximate modifications are applied to the basic 2-dimensional result, to deal with finite length and/or the

1978-1-4244-4252-2/09/$25.00 ©2009 IEEE

Ae j(m2 T t) (10)

d 2A

dr 2

1r

dAdr

p 2

r 2jTµ F A ' 0 . (1)

W '2B2T2F

r2

r1

AA(rdr W/mz (5)T ' ( mN ± n )Te rad/s , (11)

A ' c1 Ip ( j1/2 r/d ) c2 Kp ( j1/2 r/d ) . (2)

J ' FE ' jTFA. (4)

Akf/bmn(N,t) '1

2rS

mIkn Fkm cos [m2 (mKnP)T0 t ] . (8)

Akm(N,t) '1rS

mIkn Fkm cos (mN $km) cos (nTet "kn) . (7)

A ' c3 r p c4 r p . (3)

S ' (m K nP)T0 /m rad/s . (9)

W '12

Re jTAH(

2

r2

r1

W/m 2 . (6)

Fig. 2 Cross-section of surface-magnet machine

segmentation of magnets in the circumferential and axialdirections. Several original methods are described. The methodis also extended to deal with m1-type losses by means of anequivalent harmonic current sheet [3,8].

For interior-magnet machines the analytical solution of thecomplex diffusion equation is too difficult and so a completelydifferent approach is used, based on the calculated frequency-response of the complex synchronous inductance in the d-axis.An important by-product of this analysis is the calculation of thesubtransient reactance and time-constant, which are needed forcalculating sudden short-circuit faults. The frequency-responsemethod is also developed for surface-magnet machines from thecomplex diffusion equation.[4]

The calculation of slotting effects (m2-type losses) by the oldmethod of “flux-dip-sweeping” [7] is not included: it can be usedfor thin screens but is inadequate for magnet losses.

The calculation structure is shown in Fig. 1,which also formsthe structure of the paper. Iron losses are not covered in detail asthey have been reported earlier [5].

II. THEORY

A. Surface-magnet Machines

The basic calculation is the solution of the complex diffusionequation for vector potential A in Fig. 2,

where the conductivity F is zero in non-conducting regions. Thegeneral solution is [1]

Ip and Kp are modified Bessel functions of the first and secondkind; c1 and c2 are complex constants; and d = 1//TµF, whichis 1//2 times the conventional “skin depth”. In non-conductingregions

where c1 and c2 are complex constants. In conducting regions the eddy-current density is

The power loss in conducting regions can be determined byintegrating J2/F or EJ* over the region volume. For a continuousconducting cylinder with radii r1 and r2, this can be shown to be

Alternatively the radial component of Poynting's vector* can be used to give the difference between the averageE × H

power losses over the inner and outer surfaces of the cylinder,

The sign of W indicates whether the power flow is radiallyinwards or outwards.

The solution contains two complex constants c for each region.These are determined from the boundary conditions betweenregions, [1]. The method is commonplace and not repeated here,but it is noted that the current-sheet excitation at the surface ofthe stator is introduced by the boundary condition on H2 at radiusrS. For zero-frequency conditions the solution degenerates to thesolution given in [6], providing a useful check.

The current-sheet excitation is an ampere-conductordistribution given for a single harmonic in phase k by

where N is the azimuthal coordinate, m is the mechanical orderof the space-harmonic of the normalized winding MMF F, n isthe order of the time-harmonic in the current waveform Ik, Te isthe fundamental radian line frequency, and "kn and $km are phasereference angles which are set to zero for simplicity. The A-waveis resolved into forward and backward components and thentransformed into a reference frame synchronous with the rotor,using 2 = N T0t where T0 = Te/P is the rotor angular velocityand 2P is the number of poles. The result is

Relative to the rotor, the angular velocities of the forward- andbackward ampere-conductor distributions are

It is required to express the current-sheet in the form

and this is obtained by the substitution A = Ak f/b mn and

where mN = m/P is the electrical order of the space-harmonic.Tables I and II show the rotor frequencies normalized to Te forelectrical space-harmonics mNand time harmonics n up to the 37th

for one phase. For balanced 3-phase operation the number ofharmonics decreases from 703 to 222, and if triplen harmonicsare precluded, the total number reduces to 156, a saving of nearly80% in the number of field solutions required.

2

TABLE I ROTOR FREQUENCIES FOR FORWARD FIELD COMPONENT IN UNBALANCED MACHINE

mN 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

3 !2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

5 !4 !2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

7 !6 !4 !2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

9 !8 !6 !4 !2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

11 !1 !8 !6 !4 !2 0 2 4 6 8 10 12 14 16 18 20 22 24 26

13 !1 !1 !8 !6 !4 !2 0 2 4 6 8 10 12 14 16 18 20 22 24

15 !1 !1 !1 !8 !6 !4 !2 0 2 4 6 8 10 12 14 16 18 20 22

17 !1 !1 !1 !1 !8 !6 !4 !2 0 2 4 6 8 10 12 14 16 18 20

19 !1 !1 !1 !1 !1 !8 !6 !4 !2 0 2 4 6 8 10 12 14 16 18

21 !2 !1 !1 !1 !1 !1 !8 !6 !4 !2 0 2 4 6 8 10 12 14 16

23 !2 !2 !1 !1 !1 !1 !1 !8 !6 !4 !2 0 2 4 6 8 10 12 14

25 !2 !2 !2 !1 !1 !1 !1 !1 !8 !6 !4 !2 0 2 4 6 8 10 12

27 !2 !2 !2 !2 !1 !1 !1 !1 !1 !8 !6 !4 !2 0 2 4 6 8 10

29 !2 !2 !2 !2 !2 !1 !1 !1 !1 !1 !8 !6 !4 !2 0 2 4 6 8

31 !3 !2 !2 !2 !2 !2 !1 !1 !1 !1 !1 !8 !6 !4 !2 0 2 4 6

33 !3 !3 !2 !2 !2 !2 !2 !1 !1 !1 !1 !1 !8 !6 !4 !2 0 2 4

35 !3 !3 !3 !2 !2 !2 !2 !2 !1 !1 !1 !1 !1 !8 !6 !4 !2 0 2

37 !3 !3 !3 !3 !2 !2 !2 !2 !2 !1 !1 !1 !1 !1 !8 !6 !4 !2 0

Table II ROTOR FREQUENCIES FOR BACKWARD FIELD COMPONENT IN UNBALANCED MACHINE

mN 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

1 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38

3 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40

5 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42

7 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44

9 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46

11 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48

13 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50

15 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52

17 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54

19 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56

21 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58

23 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60

25 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62

27 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62 -64

29 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62 -64 -66

31 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62 -64 -66 -68

33 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62 -64 -66 -68 -70

35 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62 -64 -66 -68 -70 -72

37 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62 -64 -66 -68 -70 -72 -74

3

22

21

J1 sin2d2 (12)

22

21

(J0 J1 sin2 )d2 ' 0 , (13)

J0 ' J1

cos 22 cos 21

22 21

' J1 sin > sin ($ /2)$ /2

. (14)

7 ' J12

22

21

(q sin 2 )2 d2

' J12 (q 2 1

2)$ 4q sin $

2sin > 1

2cos2>sin$ .

(15)

7

70

'2$

(q 2 12

)$ 4q sin $2

sin > 12

cos 2> sin $ (16)

Fig. 3 Residual current suppression

Circumferential segmentation; residual suppression—Fig. 3shows a magnet of width $ in the circumferential direction, witha segment J1 of the current-density wave that would be inducedby a harmonic field rotating relative to the magnet, if the magnetwas a complete ring. J1 is obtained from the earlier fieldsolution. Fig . 3 is drawn for one instant in time. The position ofthe magnet relative to the J-wave is defined as the phase > of thecentre-line of the magnet relative to the J-wave: thus > = (21 +22)/2 and $ = (22 21); or 21 = > $/2 and 22 = > + $/2. Theangles 21, 22, > and $ are all measured in electrical radiansrelative to the harmonic wave length 8. Thus if $m is the magnetwidth in actual mechanical radians, $ = $m × 2B/8.

If the J-wave is represented as J1 sin 2, the integral

represents the net current that would flow in a section of acomplete ring of magnet between the angles 21 and 22. Thisintegral is not in general zero. However, the net current in amagnet block of finite width $ must be zero, because there is noconnection by which it can find a return path in another magnet.The net current must therefore return through the magnet blockitself. The distribution of return current-density across the cross-section of the magnet block is not known a priori; but it can besupposed to be uniform on the grounds that any non-uniformitywould imply that there were loops in the return current pathcontaining induced EMFs, when the induced EMFs have in factalready been completely accounted for by the solution of the fieldequations leading to the J-wave. (In finite-element programs theimposition of zero net current is achieved by a related process —often without proof — in which the end-faces of the conductingregions are shorted together. Thus with a uniformly-distributedreturn current density J0 we have

from which

The loss in the magnet block is proportional to (J0 + J1 sin 2)2

integrated over $ : thus if q = J0/J1 we can write a loss function

When $ ' 360kE (for integer k), the magnet width is an integralmultiple of the wavelength 8 of the harmonic field component,and (13) is automatically satisfied, with J0 = 0, regardless of thephase or position of the J-wave relative to the magnet. Theharmonic loss in a magnet whose width is equal to an integralmultiple of the harmonic wavelength is therefore unaffected bysegmentation. The integral 7 in (15) then degenerates to 70 'J1

2$/2, so we can characterize the effect of circumferentialsegmentation of the magnets by the loss reduction factor

In Fig. 3 the magnet width is $ = 90E relative to the harmonicwavelength, and its phase position is > = 80E. Thus 21 = 80 90/2 = 35E and 22 = 80 + 90/2 = 125E. Equation (14) gives q =J0/J1 = 0@88664, and with $ = B/2 radians (16) gives the lossreduction factor 7/70 = 0@0207. While this seems a verysubstantial reduction, it needs further comment.

First, this value applies only at one instant corresponding to thephase angle > = 80E. As the cycle progresses and the harmonicwave moves relative to the magnet, 7/70 varies and there areinstants when it is equal to 1. Therefore we can expect theaverage loss reduction factor to be greater than 0@0207.

Secondly, a magnet width of $ = 90Eis one-quarter of theharmonic wavelength. Consider the 5th space-harmonic of thestator winding, whose wavelength at fundamental frequency is1/5th of a pole-pitch. $ = 90E for this harmonic implies that themagnet is segmented into 20 blocks per pole — rather a largenumber. A magnet with 180E arc not segmented would have $ =5 × 90 ' 450E, and it can be verified from (16) that the lossreduction factor does not fall much below 1. A magnet with 80%pole-arc (144E) would have $ = 360E relative to the 5th space-harmonic, and in this case the loss reduction factor would be 1.Of course the magnet “region” is only 80% filled with magnet, sothere is a reduction of 0@8 relative to a full-ring magnet; but thereis no reduction due to the “residual current suppression” effect.

The upper graph J1 in Fig. 3 is the “original” currentdistribution and the lower graph J is the net current distributionwith “residual suppression”. Also shown are the squares ofthese two graphs. At this phase position the reduction in squaredcurrent-density is noticeable.

The arrows in Fig. 3 show the direction and magnitude of thecurrent density in the magnet block, “seen from above”. At theinstant > = 80E it is asymmetrical. When $ is an integral multipleof the harmonic wavelength, the distribution is symmetrical, andas > advances the arrows at one end leave the block and re-appearat the other end, while J0 remains zero at all times.

4

EDJ z

' 1 myd

' 1 my

h b ym

' 1 m 2 ymh (b y) (24)

u ' 1 sin $$

2$$a 2 4a sin $

2sin $ sin2 > . (17)

J y )z ' J z )y . (19)

J z 'E /D

1 m 2 y / [mh (b y)]with J y ' mJ z . (25)

U '1

2B

2B

0

u (>)d> ' 1 sin($ /2)$ /2

2

. (18)

)y)z

' tan ( ' m , (20)

w ' D (J z2 d)y J y

2 y)z ) (26)

J y ' mJ z . (21)

w

DJ z2' d)y m 2 y)z ' (mh (b y) m 2 y )y

m (27)

d ' h a ' h b ym

. (22)

w 'E2

D

[mh (b y) ]2

m [mh (b y) m 2 y ])y . (28)

Ed ' D J z d J yy (23)

w0 'E2

Dh )y . (29)

8 (y ) 'ww0

'[mh (b y)]2

mh [mh (b y) m 2 y ]. (30)

8 (y ) 'ww0

'[mh (1 y)]2

mh [mh (1 y) m 2 y ]. (31)

8 (y ) 'ww0

'[h (1 y)]2

h [h 1 2y ]. (32)

Fig. 4 Estimation of finite-length effect

The loss reduction factor (16) is specific to one field harmonic.This means that simplified methods of allowing for segmentation(such as equivalent resistivity) are hardly likely to be successful.For similar reasons, (16) cannot be used to introduce the effect ofsegmentation into the results of a 2-dimensional finite-elementcalculation.

The loss reduction factor u = 7/7o in (16) must be integratedover one harmonic cycle to get the average value: thus

where a = (2/$) sin($/2) Then the average value of u = 7/7o is

..a remarkably simple function. For the example consideredearlier with $ ' 90E or B/2, we get U = 0@18943. When $ ' 2Bor a multiple thereof, U = 1 as expected.

Axial segmentation — Fig. 4 shows half a block of magnetrolled out in rectangular coordinates. The direction of rotation isin the y direction and z is the axial coordinate. The axial lengthof the block is 2h and its width is 2b, which can be taken as thewavelength of the exciting harmonic.

Consider a filamentary loop of eddy-current. If the block wasinfinitely long in the axial direction, the current density in thefilament would be Jz0 = FE = jTFA, where A is the solution ofthe 2-dimensional field at the circumferential position y. The cutedge of the magnet at z = h forces all the current to veer into thecircumferential direction and rejoin its return path. In Fig. 4 thecurrent is assumed to be symmetrical about the centre-line of theblock. It is further assumed that the current in the filament )yturns abruptly into the circumferential direction and flows in acircumferential filament of width )z such that

The filament widths are further assumed to be related by

where ( is an arbitrary angle defining the slope of the “break”line shown dashed in Fig. 4. Hence

Considering just one quadrant of the loop, the driving EMF canbe identified as Ed. In the case where the magnet is infinitelylong in the z direction we have Ed = DJz0h, where D = 1/F is theresistivity. When the length is finite, the filament acquires asecond segment of length y in the y-direction, while the segmentin the z-direction is shortened from h to d, where

The shortening of the axial length reduces the EMFproportionally, while the addition of the second segmentincreases the total resistance. This is expressed by the equation

which can be expanded by writing

and so

The current-density Jz is decreased by the factor in brackets.However, the losses are proportional to

and this can be expanded by writing

Substituting for Jz from (25), and simplifying, we get

If there is no end-effect we have simply

Hence it is possible to define a loss reduction factor 8 for thefilament at y :

If we normalize h and y to b by writing h = h/b and y = y/b, thiscan be written

If ( = 45E, m = 1 and the expression simplifies further:

A magnet which is long in the axial direction has h >> 1, so that8(y) ÷ 1 for all values of y. For long magnets ( is not critical, butin short magnets ( will have a minimum value.

5

V d ( jT) ' [Rd jTLd ( jT ) ] Id ( jT ) (36)

Ld ( jT) ' LdR jLdX , (35)

Vd ' (Rd TLdX) jTLdR I d . (37)

Wm(d) ' TLdX Id2 , (38)

Fig. 5 End-effect factor: 1 proposed method; 2 Russell andNorsworthy's method [7] Fig. 6 Integration of slot-modulated airgap flux

7 '11

1

0

8(y )dy . (33)

7 (h) ' (AC B 1/C) ln (1 C) (B 1/2)C 1

C 2 k (k 1)(34)

For example a “square” magnet has h = 1 or h = b. In this case( must not be less than 45E, giving m > 1. A magnet for which h= 0@5 (or h = b/2) restricts ( to values greater than arc tan(2) =63@4E. These limits of course have no physical basis, but areconstraints imposed by the modelling assumptions.

The loss reduction factor for the whole block is obtained as theaverage of 8(y) for all the filaments, or

This can be integrated numerically or formally, giving

where k = mh, A = (k 1)2; B = 2 (k 1); and C = (1+m2)/(k 1).Fig. 5 shows examples of 7(h) calculated by (34) for various

values of the magnet length/width ratio, h. The parameter k isarbitrary and should be adjusted to match test data.

The overall factor 7 in (34) is so far simply an end-effectfactor for the losses in magnets of different length/width ratio h.Now suppose we start with a full-length magnet with a certainvalue of length/width ratio h1 and end-effect factor 71. If thismagnet is divided into n segments in the axial direction, the end-effect factor for each segment becomes 7n, calculated with hn =h1/n. Although 7n operates on only (1/n) of the losses, there arenow n segments, so the overall effect is that the end-effect factoris 7n for the whole array, instead of 71.

As an example, suppose we start with a full-length magnethaving h = 3, and divide it into 3 segments. The end-effect factorwith ( = 60E is 0@699 for the undivided magnet, and 0@2935 forthe divided magnet. Thus the division into 3 segments reduces theloss by a factor of 0@2935/0@699 = 0@419.

It should be said that this end-effect analysis is a rough-and-ready estimate. It is compared in Fig. 5 with Russell andNorsworthy's method, which is somewhat more “analytical” thanthe present method, and was tested experimentally on aninduction motor. Without detailed tests and finite-elementcalculations, it is impossible to say which is better for PMmachines, but the two methods support each other very roughly.

B. Interior-magnet Machines (IPM)

Losses caused by time-harmonics in the stator current — TheIPM presents a problem in that the magnet region inside the rotoris not a plain cylinder and so it does not conform to the earliersolution of the diffusion equation. However, an approximateestimate of harmonic losses can be made with the aid of theequivalent-circuit model and the frequency-dependentsynchronous reactance Ld(jT), which for the moment is assumedknown, [4]. We know that the phase angle of Ld(jT) is alwaysnegative, so we can write

where the real and imaginary components LdR and LdX are bothfunctions of frequency T.

A d-axis current Id(jT) which alternates at the radian frequencyT produces a voltage

where Rd is the armature resistance. Substituting the real andimaginary components of Ld(jT),

The term TLdX represents the resistance of the conductiveelements on the rotor, referred to the d-axis circuit of the stator.Generally these elements are just the magnets. The losses in themagnets are therefore given by

where Id is the RMS current at the harmonic frequency T. In a balanced 3-phase machine the simplest cases of such a

current arise from the (6k ± 1)th time-harmonics interacting withthe fundamental electrical space-harmonic of the windingdistribution. For example the 5th harmonic produces a rotatingampere-conductor distribution rotating backwards relative to therotor at six times the fundamental synchronous speed, and thiscan be resolved into d- and q-axis components which arestationary with respect to the rotor, but which pulsate at six timesthe fundamental frequency. (See Tables I and II.). The analysisfor higher-order space-harmonics is more complex, but theseshould be attenuated relative to the fundamental.

6

Z( jT ) 'LR

Td0NN(1 jTTd0NN) . (40)

I( jT) 'jTQm( jT )

Z( jT )(41)

Wm[slot mod] 'I 2 LR

Td0NN. (42)

Mm(>) ' RLstk

> $/2

> $/2

Bg(2)d2 . (39)

Fig. 7 Incorrect finite-element solution: 4 blocks conducting

Fig. 8 Approximate finite-element solution: 1 block conducting

JdS ' 0 (43)

Fig. 9 Approximate finite-element solution: 2 blocks conducting

Losses of this type can in principle also occur in the q-axis,but here they are ignored on the grounds that the q-axis armature-reaction flux does not pass through the magnets, and in any casethe circuits formed by induced currents in the magnets will be farless effective in the q-axis.

Losses caused by flux pulsations (slotting) — The flux Mmthrough the magnet can be seen as the integral of the airgap fluxmodulated by slotting, as shown in Fig. 6. Quite accurate slot-modulation functions can be calculated by Weber's method asreported in [11] and extended by Zhu et al [10]. Suppose the d-axis is at an angle > relative to a fixed point on the stator (such asthe axis of phase 1). If the pole-arc is $, then the limits ofintegration are 21 = > $/2 and 22 = > + $/2. Ignoring leakage,and taking R as the mean airgap radius and Lstk as the axiallength, we have

The fundamental time-harmonic component of Mm(Tt) can berepresented as a phasor Mm(jT) if we write > = Tt, where T/2Bis the slot-passing frequency; the harmonics can be treatedlikewise. The harmonic flux per pole Mm(jT) is now assumed tolink the fictitious N-turn coil wrapped around each magnet, andif all the poles are assumed to be in series it will produce a totalflux-linkage per phase equal to Qm(jT) = 2pN Mm(jT). Theinduced voltage will be jTQm(jT) at the harmonic frequency, andthis is “applied” to a rotor circuit whose inductance is denoted LRfor the moment. The resistance of this circuit is equal to LR/TdoNN,where Td0NN is the open-circuit subtransient time-constant, so theimpedance at the harmonic frequency is

The current is

and the associated losses are

Segmentation of the magnets in both the circumferential and axialdirections must be taken into account in the calculation of Td0NN.

For surface-magnet machines the fundamental component ofripple flux Mm(Tt) can be used to find the amplitude andharmonic frequency of an ampere-conductor distribution thatproduces the same harmonic field. This can then be used as thesource of excitation for the classical solution of the diffusionequation described earlier. In this way the losses caused by slot-modulation can be treated in the same way as those due to thespace- and time-harmonics in Tables I and II. This method wasoriginally described by Lawrenson, Reece and Ralph [3], and ittakes advantage of advanced conformal transformation methodsmentioned above to determine the slot-modulation, [10,11].

III. RESIDUAL CURRENT SUPPRESSION AND FINITE ELEMENTS

The finite-element method is held to produce results of assuredaccuracy, but it is also the slowest and most expensive method interms of engineering time. 3D calculations may take days, yetwhat is often needed is adequate accuracy in seconds or minutes.As mentioned earlier, 2D finite-elements are sometimes used ineddy-current problems, in which the zero net current condition

is imposed (usually without proof) individually across eachconducting region by means of a fictional external circuitconnection with a high resistance shorting the two ends.

7

This technique requires a voltage-driven solution and a specialformulation in which the circuit equations are incorporated in thefield-solution matrix, [12]. Notwithstanding the excellentmathematical literature on the finite-element method, a review often books on finite-elements in electrical engineering fails toprovide any practical reassurance of the validity of this method.

Fig. 7 shows the problem of calculating eddy-currents with aconventional 2D solver using current excitation (and no externalcircuit connection). The problem is to calculate the no-load eddy-current losses in the magnets at high speed. The particularexample in Fig. 7 uses the time-stepping algorithm of Crank-Nicholson, and predicts a total loss of 2@06 W, but this result isincorrect because the “infinite length” assumption permitsinduced current in magnet A to return in magnet B, and likewisecurrent in magnet C to return in magnet D. Since there is noelectrical connection between the magnets, this is a false result.(It is not the only example of the possibility of a false resultobtained from the finite-element method by the unwary user).

The simple finite-element formulation satisfies (43) only overthe whole solution domain, and not individually in each magnetregion. An improved result can be obtained by suppressing theconductivity of, say, magnets B, C, and D, and simply calculatingthe loss in magnet A. In a situation where the losses must beequal in all magnet blocks, the final result is obtained bymultiplying the result for one block by 8. When this is done inFig. 8, the total calculated loss is 0@97 W when scaled up toinclude all magnets. This is slightly less than half the incorrectfirst estimate. The method can be described as a crude form ofresidual current suppression without using external circuits.

This method of single-region eddy-current calculationobviously relies on the assumption that the eddy-currents in anymagnet do not affect the eddy-currents in any other magnet. Onecould assume that this is characteristic of “resistance-limited”eddy-currents, but it is more correct to describe it as the neglectof proximity effect, which in other situations is known to bedangerous. Nevertheless, engineers in all disciplines have longrelied on simplified calculations combined with other means ofverification than simply throwing time and money at the problemthrough very expensive calculating tools; and if this approach isstill permitted, the method is surely worth trying.

A “refinement”, if it can be so described, is to suppress theconductivity judiciously in pairs or patterns of magnets, and Fig.9 shows an example where magnets A and C are treated asconductive while magnets B and D are not. The total losscalculated in this case is 1@02 W, again scaled up to include allmagnets. This is reasonably close to the 0@97 W calculated usingthe single-block approximation, and the difference may givesome idea of the accuracy of the method. The only way to besure is to measure it, although some confidence might beobtained through more sophisticated calculation tools.

An interesting observation in Figs. 7,8 and 9 is the “lateral”diffusion of the eddy-currents towards the edges of the magnets,with little variation in current density in a direction parallel to theflux. This property is used in [4] in the formulation of simplifiedloss calculations for the IPM, and the related calculation of thefrequency-dependent synchronous inductance mentioned earlier.

IV. RESULTS AND CONCLUSION

The methods presented in this paper are intended for fastcalculation of rotor losses in permanent-magnet brushlessmachines. Together with iron-loss calculations [5], they are partof a complete set of calculating methods proposed to deal withthis very complex calculation, and combined with the necessarysimulation tools [2] required to handle the external circuit andcontrols. Validation of the 2-dimensional solution of the complexdiffusion equation is not an issue, because it is a widely used andlong established method [1]. Validation of the separate methodsfor analysing the effect of magnet segmentation is almostimpossible by practical measurement because of the difficulty ofsegregating several coincident phenomena, but the total lossescan of course be compared with test data. Comparison withfinite-element calculations may also give some reassurance but isnot a satisfactory substitute for physical testing. The methodsdeveloped here are used in the companion paper [4] to helpcalculate subtransient parameters for short-circuit analysis.

ACKNOWLEDGMENT

We would like to thank M. Olaru and the companies of theSPEED Consortium for support and discussions.

REFERENCES

[1] T.J.E. Miller and P.J. Lawrenson, Penetration of Transient Magnetic FieldsThrough Conducting Cylindrical Structures, with Particular Reference toSuperconducting AC Machines, Proc. IEE, Vol. 123, No. 5, 1976, 437-443

[2] T.J.E. Miller, SPEED's Electric Machines 2009, available from theSPEED Laboratory

[3] P.J. Lawrenson, P. Reece and M.C. Ralph, Tooth-ripple losses in solidpoles, Proc. IEE., Vol. 113, No. 4, April 1966, pp. 657-662

[4] (in press) K.W. Klontz, T.J.E. Miller, H. Karmaker and P. Zhong, Short-circuit Analysis of a Permanent-Magnet Generator, IEMDC 2009

[5] D.M. Ionel, M. Popescu, M.I. McGilp,T.J.E.Miller,S.J. Dellinger, and R.J.Heidemann, Computation of Core Losses in Electrical Machines UsingImproved Models for Laminated Steel, IEEE Trans. Ind. Appl. vol. 43, No.6, Nov/Dec 2007, pp. 1554-1564.

[6] A. Hughes and T.J.E. Miller, Analysis of fields and inductances in air-coredand iron-cored synchronous machines, Proc. IEE, Vol. 124, No. 2, February1977, pp. 121-126

[7] R.L. Russell and K.H. Norsworthy, Eddy Currents and Wall Losses inScreened-Rotor Induction Motors, Proc. IEE, Vol. 105A, pp. 163-175, 1958

[8] K. Oberretl, Eddy Current Losses in Solid Poles of Synchronous Machinesat No-Load and On Load, IEEE Trans., Vol. PAS-91, 1972, pp. 152-160

[9] N.T. Irenji, S.M.Abu-Sharkh, and M.R. Harris, Effect of rotor sleeveconductivity on rotor eddy-current loss in high-speed PM machines, ICEM2000, Espoo, Finland, August 2000, pp. 645-648

[10]Z.Q. Zhu, K. Ng, N. Schofield and D. Howe, IEE Proc.-Electr. Power Appl.,Vol. 151, No. 6, November 2004, pp. 641-650

[11]B. Heller and V. Hamata, Harmonic effects in induction machines, ElsevierScientific Publishing Company, Amsterdam, 1977

[12]S.J. Salon, Finite-Element Analysis of Electrical Machines, KluwerAcademic Publishers, 1995

[13]Jung Jae-Woo et al, Optimum Design for Eddy Current Reduction inPermanent Magnet to Prevent Irreversible Demagnetization, ICEMS 2007,Oct. 8~11, Seoul, Korea, pp. 949-954

[14 N. Boules, Impact of Slot Harmonics on Losses of High-Speed PermanentMagnet Machines with a Magnet Retaining Ring, Electric Machines andElectromechanics, Nov. 1981, pp. 527-539.

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