Permanent-magnet linear eddy-current brake with a non-magnetic reaction plate

I S 1

I N 1

J.D.Edwards, B.V.Jayawant, W.R.C.Dawson and D.T.Wright

N 1 1 S I secondary

s 1 1 N I

Abstract: A new two-dimensional analytical model is presented for a double-sided permanent-magnet brake with a non-magnetic reaction plate, based on an equivalent volume current representation of the permanent magnets. An exact solution of the field equations for an infinitely long machine, together with an approximate treatment of end effects, leads to simple expressions for the brakmg and normal forces as functions of the secondary plate speed. These expressions clearly show the effects of the brake geometric parameters and the plate conductivity on the performance. The model has been validated by experimental measurement of the braking force as a function of speed, and by finite- element calculation of the flux density and normal force at standstill.

1 Introduction

Eddy-current braking, which depends on the motion of a conductor in a non-umform magnetic field, can be an effec- tive alternative or supplement to mechanical friction brak- ing in linear-motion systems. If the field source is a permanent magnet, and the brake is designed to avoid demagnetisation, an eddy-current brake cannot fail to oper- ate.

Fig. 1 shows the structure of a double-sided linear eddy- current brake with a non-magnetic secondary reaction plate. Each part of the primary comprises an array of sur- face-mounted permanent-magnet blocks on an iron back- ing. Magnetic materials such as neodymium iron boron (NdFeB) enable a large airgap to be used. There is no nor- mal force on the secondary, and the brake characteristics can be varied over a wide range by selection of the plate thickness and resistivity.

Fig. 1 Double-sided pernument-magnet linear e+-cwrentplrrte

Laithwaite [I] has used a one-dimensional theory of the linear induction motor to derive an expression for the brak- ing force on a conducting plate that is consistent with Riidenberg's frequently cited treatment [2], but this is not applicable to large-entrefer permanent-magnet brakes of the form shown in Fig. 1. The same objection applies to the one-dimensional analysis by Poloujadoff [3], and to the more recent work by Wouterse [4] for brakes with widely spaced poles.

0 IEE, 1999 ZEE Proceedings online no. 19990574 DOL 10.1049/ipepa:19990574 P a p first received 8th M b e r 1998 and in revised form 1st June 1999 The authors are With the School of Engineering, University of Sussex, Falmer, Brighton, East Sussex, BN1 9QT, UK

Brakes with magnetic reaction members and small air- gaps have been widely studied, for example by Davies [5], Venkataratnam and Kadir [6, 71 and Wang and Chiueh [SI, but this work is not applicable to the brake considered here. Poloujadoff [3] gives a two-dimensional analysis of brakes with large airgaps, neglecting the eddycurrent reac- tion field, and so the analysis it is only applicable to low- speed operation. Nagaya and others [9, 101 have used Fourier methods to analyse brakes with surface-mounted rare-earth magnets and non-magnetic reaction plates. These studies also neglect the eddy-current reaction field, and the results are very complex.

The paper presents a new two-dimensional analytical the- ory of this type of brake, which includes the eddy-current reaction field, leading to simple expressions for the braking force and the normal force between the two halves of the primary.

2 Steady-state model

2.1 Model structure A general model for the permanent-magnet linear eddy- current brake (J'MLB) is developed for the single-sided structure shown in Fig. 2, where the secondary is a non- magnetic conducting plate on an iron backing. A double- sided PMLB can be analysed as a pair of single-sided PMLBs, each having half the airgap length and conduct- ing-plate thckness of the double-sided machine.

5 secondary c'

Fig. 2 Smgie-sikdpermanent-magnet linear eddy-ament brake

The analysis in this paper is related to that of Ooi and W t e for the linear induction motor [l 11, but, in place of a current-sheet representation of a primary winding, it uses an equivalent volume current representation of the perma- nent-magnet region. This leads to an exact solution of the airgap field equations, from whch the normal and tractive forces can be calculated.

IEE Proc-Electr. Power Appl.. Vol. 146, No. 6. November 1999 621

Fig. 3 is a four-region model of the PMLB shown in Fig. 2. Point 0 is the origin of a right-handed co-ordinate system, with the z-axis perpendicular to the plane of the drawing. A surface current sheet KT represents the second- ary conductor, which is valid if skin effect in the secondary can be neglected; this assumption will be justified in Section 2.5.

y=e K5 I region 3 air

y=d

y=o

Fig. 3 Four-region model

2.2 Representation of permanent magnets W i t h each primary magnet block, the relationshp between the magnetic quantities B and H can be expressed in terms of a magnetisation vector M

= P O ( N + M ) = PO(PLH+ MO) (1) where is the recoil relative permeability and MO is a con- stant magnetisation independent of H. The direction of MO alternates between the positive and negative y-directions in successive magnet blocks, and its magnitude is given by

where Br,, is the remanence. The layer of magnet blocks is replaced by an equivalent

continuous layer of material, in which a Fourier series rep- resents the y-component of magnetisation

CO

M i = > M p , c o s r k x . ( r = 1 , 3 , 5 ,...) (3) ,=I

where MPr is the amplitude of the harmonic, k = dz, z is the pole pitch, and x is the distance from the mid-line of a magnet block. The harmonic coefficients are given by

4MO . r7ra MPT = - sin - r7r 2

( r = 1 ,3 ,5 , . . .) (4)

where a is the ratio of the magnet pole length I, to the pole pitch z. The rth harmonic is given by

M i , = MpT cos rkx (5) It is convenient to express eqn. 5 in terms of a complex quantity Myr

M;, = Re(MyT) = Re(M,,expjrkz) (6) where

My, = MpT expj rkx (7 ) Stratton [12] shows that an equivalent volume and surface current distribution can replace MO

J = V x MO K = MO x n ( 8 ) where n is the unit normal vector at the surface of the material. For an infinitely long machme, K = 0, and the volume current density in the magnet region is

dMY, J , = J,, = - = j r kMpT expj rkx (9) dX

2.3 Airgap field The magnetic flux density B and the magnetic intensity H in the model will satisfy the Maxwell equations

V . B = O (10)

V x H = J (11) As the secondary current sheet has negligible thickness, in region 3 eqn. 11 reduces to

V x H = Q (12) At the interfaces between regions, the following boundary condition applies [I 21:

n x ( H + - H - ) = K (13) where n is the unit normal vector and K is the density of any current sheet at the interface. In addition, there is the constitutive equation

In region 2, the relative permeabhty will vary from the recoil value ,u'~ in the permanent-magnet blocks to 1.0 in the spaces between the blocks. We therefore take a mean value given by

where a is the ratio of the magnet pole length to the pole pitch. In region 3, pr = 1.

Motion of the secondary material induces the secondary current density K,, which is determined as follows:

B = PH = pOpTH (14)

pr = ~ P L (15)

KST = = g d S C U [ B Z 3 T ] t J = e (16) where CT is the effective conductivity of the plate (see Sec- tion 2.6), d,, is the plate thckness, Er is the induced electric field in the plate, U is the secondary speed, Bx3r is the x component of flux density in region 3, and e = d + g is the total gap, or entrefer, between the iron regions.

For materials such as NdFeB, the recoil permeability ,U> is approximately 1.05 [13]. As the value of a in eqn. 15 is typically 0.5, the value of in region 2 wdl be close to 1.0. Eqns. 10-12 can be solved for the field in regions 2 and 3, and the solution is greatly simplified if the value of 4 is set to 1.0. The components of flux density in region 3 at the secondary surface are then given by

-vp0Mp, sinh rkd sinh rke - .iv cosh rke BSZ!T = [B13T]y=e = exp j rkx

(17)

poMpT sinhrkd sinh rke - iv cosh rke B S y T = [By3T]y=e = exp j r k x

(18)

(19)

(20)

where v is the fractional speed defined by U

' U = - U0

1 and the speed U,, is

U0 = ~

P o d C The total flux density components at the secondary surface are

M ..

B,, = B,,, (r = i , 3 , 5 , . . .) (21) T = l

03

B,, = B~~~ (r = 1 , 3 , 5 , . . .) (22) T = l

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These series converge rapidly, and it is usually necessary to take only the first five terms (up to r = 9) for an accurate representation of the field.

2.4 Braking and normal forces The contributions of the rth harmonic to the tangential (braking) and normal components of force on the second- ary are given by the Maxwell stress expressions [l 11

where wp is the width of the primary, p is the number of pole pairs, and t is the pole pitch. In a double-sided PMLB, the total normal force on the secondary is zero, eqn. 24 gives the normal force on each side of the primary, and eqn. 23 gives half of the total braking force.

Substituting for B,,, and Bsyr in eqns. 23 and 24 gives the following results for the tangential and normal components of force:

FtT = 0.51,wPpoM~, sinh’ rkd

(25) V

X sinh’ rke + v2 cosh’ rke

F,, = ~ . 2 5 1 , w P p ~ M ~ , sinh2 rkd

(26) 1 -U’

sinh’ rke + v2 cosh2 rke X

where l p is the length of the primary. Eqn. 25 gives the con- tribution of the rth harmonic to the braking force, whch has a maximum value when

v = U, = tanhrke (27) and the maximum force is given by

1 Ft,, = 0.51,w,p.0M~~ sinh2 rkd

2 sinh rke cosh rke

Thus the fractional speed for maximum bralung force depends on the harmonic number and the pole pitch. The fundamental component is dominant, and so the fractional speed for maximum total braking force is close to the value given by eqn. 27 with r = 1. The plate speed for this condi- tion is given by

(29) . tanhke

U , = U O V , = uo tanhke = ~

Po& As k = nit, the maximum force condition depends on the ratio of entrefer length e to pole pitch t.

Eqn. 26 shows that the normal force will change from attraction to repulsion when / V I > 1, which occurs when the plate speed U exceeds ~ 0 . Ths condition does not depend on the harmonic number or the pole pitch.

The total tangential and normal force components are given by

00

Ft = Ft, (r = 1,3,5, . . .) (30) T = l

00

F, = F,, (rL= 1,3 ,5 , . . .) (31)

With a magnet configuration simdar to that described in Section 3.1, acceptable results are obtained using only the

T = l

_ I

IEE Proc -Electr Power A p p l , Vol 146, No 6, November 1999

fundamental term (r = 1). However, test results confirm that including up to the first five terms (r = 9) gives a sig- nificant increase in accuracy.

2.5 Secondary skin effect The fundamental frequency of secondary induced currents is given by

(32) U f = - 23-

and the skin depth in a non-magnetic secondary conductor is given by

Practical eddy-current brakes will be designed for speeds that are not greatly in excess of the value of U, given by eqn. 29. At twice this speed, the skin depth is given by

8 = / X I p 0 n - m ~ ~ = /r T tanh ke (34)

For the experimental brake described in Section 4, the value of 6 is approximately 8.4mm. As the plate half-thick- ness d, is only 4.8mm, skin effect is insignificant. With geo- metrically s d a r brakes operating at speeds of 2um, eqn. 34 shows that the skin depth will be a constant multi- ple of the plate thckness, and so it is generally valid to neglect skin effect. For operation at speeds much hgher than 2um, layer theory [14] could be used to extend the model to include skin effect.

2.6 Finite secondary width The finite width of the secondary can be taken into account by reducing the conductivity from the physical value aP to an effective value CI = k,op, where k, is the Russell and Norsworthy factor [I51 given by

(35) tanh E

~ ( 1 + tanh E tanh 4) k , = l -

where TW

27

where wp is the width of the primary, w,, is the width of the secondary conductor, and z is the pole pitch.

~ 2.7 Primary end effects Distortion of the airgap field at the ends of the primary can be represented by forming the Fourier series of an infinite sequence of magnet sections separated by non-magnetic gaps, but the effect is insignificant if the primary has four or more poles. As with short-primary linear induction motors, however, there will be additional forward and backward travelling-field components associated with the entry and exit ends of the primary [l, 3, 161. As the effect is small in the PMLB, an approximate representation will be sufficient, and the following treatment is based on Yama- mura’s one-dimensional analysis [16]. Let Kp(x) be an equivalent current sheet representing the permanent-mag- net region, and let K,(x) be a current sheet representing the secondary plate. Ampkre’s circuital law gives

where e is the entrefer length. K, is given by

K, = d,, Js = d,,uaB (39)

629

giving the result dB v B dx e - + - = -p&p

where v is the fractional speed given by eqn. 19. The end- effect field is given by the solution of the homogeneous equation

d B v B - + - = o dx e

which is

Be = Boexp(-vx /e ) (42) and the corresponding current density in the plate is given by eqn. 39. If the total field is completely suppressed at the entry edge, where x = 0, then Bo is equal and opposite to the field of the first magnet block. From eqn. 18, the result is

00 p0Mpr sinhrkd rar Bo=-C cos - (43)

sinh rke 2 r=l

The braking force on the plate from the entry-edge field is given by

1,

F t e = wp 1 BeKsedx

= w p lp d,,uaB: exp(-2vx/e)dx

(44) wpeBi

2PO - - [I - exp(-2vZp/e)]

Thls w d add to the braking force given by eqn. 30.

3 Model validation

3. I Experimental brake An experimental brake of the form shown in Fig. 1 has been constructed, using NdFeB magnets and an aluminium alloy reaction plate, with the parameters given below. The conductivity of the plate is the measured value at 20°C. number of poles 2p = 4 pole pitch t = 5 O m

air gap length g = 12.7mm primary width wp = 48mm magnet depth d = 2 0 m magnet length I,,, = 25mm magnet remanence B,,, = 1.2T plate width wp = 152.4mm plate thickness 2d, = 9.5mm plate conductivity ap = 34.5MSm-I The brakmg force has been measured by combining a lin- ear induction motor with the linear brake. The aluminium alloy plate was accelerated by the induction motor primary and then brought to rest in the brake section, where a piezo-electric force transducer measured the reaction force on the brake primary. An optical sensor attached to the plate generated pulses with a frequency. proportional to speed as it moved past a stationary 'strobe track' of equally spaced marker bars. Outputs from the force transducer and the speed sensor were captured with a high-speed data- acquisition processor.

Fig. 4 compares the experimental measurements with two different predicted curves for the braking force: the basic force values from eqn. 30 (broken line); and those

630

values including the end-effect term from eqn. 44 (solid line). The scatter of the experimental points in the range 8- 13ms-' can be attributed to mechanical resonance in the brake assembly, excited by the sudden entry of the plate, and so the results in this range must be regarded as unrelia- ble. Outside this range, the experimental results are within about 10% of the predicted values.

0 5.0 10.0 15.0 20.0

speed, ms 1

Fig. 4 ---- Without end effect ~ With end effect A Experimental measurement

Grnparison ofpredicted and m e m r e d brakmg force

An approximate transient analysis, based on the dynamic bralung theory for conventional induction motors [ 11, shows that transient effects are not likely to be significant in these tests.

Fig. 5 Finite-elementflwcplot

-1 .o ' I I I 0 20 40 60 80 100

x, mm Fig.6 Normalcomponent of airgapflux density _ _ _ _ Flnite element ~ Predicted

3.2 Finite-element analysis Fig. 5 shows a finite-element flux plot for the static mag- netic field in the experimental brake. The graph in Fig. 6 compares the finite-element result for By in the middle of the airgap with values computed from eqn. 22 with v = 0.

IEE Proc-Electr. Power AppL, Vol. 146, No. 6, November 1999

Evidently, the analytical model correctly predicts the nature of the variation of By with x, but the maximum value is 7% higher. When the plate is stationary, the finite-element solu- tion gives a force of attraction of 817N between the two halves of the primary. The corresponding predicted result from eqn. 31 is 938N, which is 13% higher: consistent with the higher predicted value of By. It was not possible to measure the force of attraction in the experimental brake.

4 Conclusions

The results of experimental measurement of the braking force and finite-element calculation of the flwc density and normal force at zero speed are acceptably close to predic- tions from the new analytical model. As well as predicting the braking and normal forces as functions of speed, ths model shows clearly the effects of the brake geometric parameters and the plate conductivity on the performance, making it a useful design tool for ths form of eddy-current brake.

5 Acknowledgments

The authors are grateful to PRT Systems and Arrow Dynamics for support of the work, and to J. Curtis and Dr. R.J. Whorlow for assistance with the experimental measurements.

6 References

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4 WOUTERSE, J.R.: ‘Critical torque and speed of eddy current brake with widely separated soft iron poles’, IEE Pruc., B, 1991, 138, (4), pp. 153-158 DAVIES, E.J.: ‘General theory of eddycurrent couplings and brakes’, Proc. IEE, 1966, 113, (9, pp. 825837 VENKATARATNAM, K., and KADIR, M.S.A.: ‘Analysis and per- formance of eddycurrent brakes with ferromagnetic loss drums, 1 - non-salient-pole brakes’, IEE Pruc., B, 1982, 129, (3), pp. 125-131 VENKATARATNAM, K., and KADIR, M.S.A.: ‘Analysis and per- formance of eddycurrent brakes with ferromagnetic loss drums, 2 - salient-pole brakes’, IEE Pruc., B, 1982, 129, (3), pp. 132-142

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5

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12 STRAlTON, J.A.: ‘Electromagnetic theory’ (McGraw-Hill, 1941) 13 McCAIG, M., and CLEGG, A.G.: ‘Permanent magnets in theory and

practice’ (Pentech Press, 1987, 2nd edn.) 14 FREEMAN, E.M.: ‘Travelling waves in induction machines, input

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