Torque in Permanent Magnet Couplings : Comparison

of Uniform and Radial Magnetization

Romain Ravaud, Guy Lemarquand, Valerie Lemarquand, Claude Depollier

To cite this version:

Romain Ravaud, Guy Lemarquand, Valerie Lemarquand, Claude Depollier. Torque in Perma-nent Magnet Couplings : Comparison of Uniform and Radial Magnetization. Journal of AppliedPhysics, American Institute of Physics (AIP), 2009, 105, pp.053904. .

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1Torque in PM Couplings: Comparison of

Uniform and Radial Magnetization

R. Ravaud, G. Lemarquand, V. Lemarquand and C. Depollier

Abstract1

We present a three-dimensional study of the torque transmitted between tile permanent magnets2

uniformly magnetized. All this study is based on the Coulombian model. The torque is calculated semi-3

analytically by considering all the surface densities that appear on the tiles. In addition, no simplifying4

assumptions are done in the expressions given in this paper. Consequently, the evaluation of the torque is5

very accurate and allows us to show the drawbacks of using tile permanent magnets uniformly magnetized6

instead of using tile permanent magnets radially magnetized. Such an approach is useful because it allows7

us to realize easily parametric studies.8

Index Terms9

Magnetic coupling, Tile permanent magnet, Torque, Uniform magnetization, Radial magnetization,10

Three-dimensional calculation.11

I. INTRODUCTION12

MAGNETIC couplings are often realized with tile permanent magnets radially or uniformly13

magnetized. Tile permanent magnets radially magnetized allow us to obtain great couplings14

but they are difficult to fabricate. Consequently, an alternative experimental method consists in using15

tiles uniformly magnetized. Indeed, they are simpler to fabricate than tiles radially magnetized and thus16

cheaper. Unfortunately, they are also less efficient and can lower the quality of transmission between tiles17

located on the stator and tiles located on the rotor. Therefore, it is interesting to predict theoretically the18

Manuscript Received October 25,revised December 8, 2008

The authors are with the Laboratoire dAcoustique de lUniversite du Maine UMR CNRS 6613, Avenue Olivier Messiaen, 72085Le Mans Cedex 9, France

2way the torque is transmitted between tile permanent magnets uniformly magnetized. Such a model allows19

us to realize easily parametric studies and thus to know when the use of tile permanent magnets radially20

magnetized is necessary or not. Indeed, the angular width has a great influence on the radial field created by21

a tile permanent magnet uniformly magnetized. As this radial field is not perfectly symmetrical, the torque22

transmitted is modified. Therefore, the aim of this paper is to determine precisely with a three-dimensional23

model how the torque transmitted between tiles changes according to the way the tiles are magnetized.24

Many authors have studied magnetic couplings [1]-[4] and structures using permanent magnets [5]-[10].25

Historically, we can say that the first models used to study magnetic couplings were the two-dimensional26

models [11]-[19]. The main reason lies in the fact that a two-dimensional approach is fully analytical and27

allows us to make an easy parametric optimization of the permanent magnet dimensions. However, this28

2D-approach is not valid when we determine the magnetic field far from one magnet [20].29

Tree-dimensional approaches seem to be more difficult to realize parametric studies because they are30

not fully analytical. In fact, the calculation of the radial and axial magnetic field components created by31

tile permanent magnets is not strictly analytical but is necessary based on special functions [21]- [40].32

More generally, all the semi-analytical or analytical approaches used by several authors allow manu-33

facturers to optimize devices using permanent magnets [41]-[48].34

This paper presents a semi-analytical approach based on the Coulombian model for calculating pre-35

cisely the torque transmitted between tile permanent magnets uniformly magnetized. This semi-analytical36

approach is three-dimensional. We explain in the second section how this problem can be solved. Then,37

we present a semi-analytical expression of the torque transmitted between two tile permanent magnets38

uniformly magnetized. Eventually, we present the main drawbacks of tile permanent magnets uniformly39

magnetized.40

II. MODELING TILE PERMANENT MAGNETS WITH THE COULOMBIAN MODEL41

This section presents the geometry studied and the model used for the modeling of the torque transmitted42

between two tile permanent magnets uniformly magnetized.43

3b

1

2

3

4

u

uu

r

z

r rr

r in1 out1out2in2

0

z

z

c

d

za

z

Fig. 1. Representation of the geometry considered in three-dimensions : two tile permanent magnets uniformly magnetized

A. Notation and geometry44

The geometry considered and the related parameters are shown in Fig 1. A two-dimensional repre-45

sentation of the geometry is shown in Fig 2. We consider here two tile permanent magnets uniformly46

magnetized. The inner radius of the outer tile is rin1 and its outer one is rout1. Its height is zb za and47its angular width is 2 1. The inner radius of the inner tile is rin2 and its outer one is rout2. Its height48is zdzc and its angular width is 43. The two magnetic polarization vectors ~J1 and ~J2 are expressed49as follows:50

~J1 = cos(2 + 12

)~ur sin(2 + 12

)~u (1)

51

~J2 = cos(4 + 32

)~ur sin(4 + 32

)~u (2)

The magnetic field created by tile permanent magnets can be obtained by using the Coulombian Model.52

Indeed, a permanent magnet can be represented by a magnetic pole surface density that is located on53

the faces of the magnet and a magnetic pole volume density that is located inside the magnet. In the54

configuration presented in Fig 1, the magnetic pole volume density is 0 because the magnetic polarizations55

of the magnets are uniform. Consequently, each tile permanent magnet is modelled by its magnetic pole56

surface density, which is determined as follows: by denoting ~nij , the unit normal vector of the face i of57

42

u

u

r0

1

2

3

4

n

n

n

n

n

nn

n

1

2

3

4

5

67

8

J J1

Fig. 2. Representation of the geometry considered in two dimensions : two tile permanent magnets uniformly magnetized

Surface densities Scalar Product Expression

11~J1.~n1 J1 cos( 1+22 )

12~J1.~n2 J1 sin(

212 )

13~J1.~n3 J1 cos( 1+22 )

14~J1.~n4 J1 sin(

212 )

25~J2.~n5 J2 cos( 3+42 )

26~J2.~n6 J2 sin(

432 )

27~J2.~n7 J2 cos( 3+42 )

28~J2.~n8 J2 sin(

432 )

TABLE I

DEFINITION OF THE MAGNETIC POLE SURFACE DENSITIES LOCATED ON THE MAGNETS

the magnet j, the corresponding surface density ij is determined with (3).58

ij = ~Jj . ~nij (3)

Thus, four faces of the two magnets are charged with a magnetic pole surface density. All the surface59

density calculations are represented in Table I. The torque transmitted between two tile permanent magnets60

can be determined in two steps. The first step consists in calculating the magnetic field created by one tile61

permanent magnet. The second step consists in integrating the magnetic field created by the first tile on62

5the second one. Let us first consider the tile permanent magnet located on the right of Fig 1. By using the63

Coulombian approach, we can write that the azimuthal field created by this tile is expressed as follows:64

H(r, , z) =J1

40

21

zbza

cos( 1 + 22

)~u(rin1)

|~u(rin1)|3 rin1ddz

J140

21

zbza

cos( 1 + 22

)~u(rout1)

|~u(rout1)|3 rout1ddz

+J1

40

rout1rin1

zbza

sin(2 1

2)~v(2)

|~v(2)|3 drdz

+J1

40

rout1rin1

zbza

sin(2 1

2)~v(1)

|~v(1)|3 drdz

(4)

where65

~u(x) = (r x cos( ))~ur x sin( )~u + (z z)~uz (5)

and66

~v(y) = (r r cos( y)~ur r sin( y)~u + (z z)~uz (6)

The next step is thus to express the torque transmitted to the second tile permanent magnet uniformly67

magnetized (as shown in Fig 1). By using (4), the torque T can be determined as follows:68

T =J1J2

40

43

zdzc

rin2H(rin2,, z)rin2 cos(

3 + 4

2)d

dz

J1J240

43

zdzc

rout2H(rout2,, z)rout2 cos(

3 + 4

2)d

dz

+J1J2

40

rout2rin2

zdzc

rH(r, 4, z) sin(4 3

2)drdz

+J1J2

40

rout2rin2

zdzc

rH(r, 3, z) sin(4 3

2)drdz

(7)

B. Semi-analytical Expression of the Torque69

The torque transmitted between two tile permanent magnets can be written as follows:70

T =J1J2

40

( 21

43

dT(1) + T

(2) +

zbza

dT(3) +

21

dT(4)

)(8)

60 0.25 0.5 0.75 1 1.25 1.5Angle @radD

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Torq

ue@N

.mD

Fig. 3. Representation of the torque transmitted between two tiles ; thick line: the magnetization is uniform, dashed lines:the magnetization is radial; 4 3=2 1=

pi

12, rin1 = 0.025m, rout1 = 0.028m, rin2 = 0.021m, rout2 = 0.024m,

zd zc = 0.003m, zb za = 0.003m, za = 0.001m, r = 0.024m, J1 = J2 = 1T

where dT(1) , T

(2) , dT

(3) and dT

(4) are given in the appendix. Strictly speaking, this expression is three-71

dimensional and we take into account all the contributions between the two tiles. However, it is noted that72

all the contributions have not the same weight and the interaction between the surface densities located73

on the inner and outer faces of each tile permanent magnet is in fact the most important. Furthermore, it74

is noted that this expression could probably still simplified and led to incomplete elliptical integrals of the75

first, second and third kind. In addition, we use the Cauchy principal value for determining the singular76

cases that appear for example when one tile permanent magnet is exactly in front of the other one.77

III. COMPARISON OF THE TORQUE TRANSMITTED BETWEEN TWO TILES RADIALLY MAGNETIZED78

AND TWO TILES UNIFORMLY MAGNETIZED79

Tiles uniformly magnetized are generally less efficient than tiles radially magnetized for magnetic80

couplings. However, this loss of torque depends greatly on the tile angular width. Therefore, it is interesting81

to determine this loss of torque for different tile angular widths. For this purpose, we represent in the same82

figure the torque transmitted between two tiles radially magnetized and the torque transmitted between83

two tiles uniformly magnetized. On Figure 3, the angular width of each tile is the same ( pi12 ). Then,84

we represent in Figs 4, 5 and 6 the torque transmitted between tiles whose angular widths are pi6 ,pi485

and pi3 . Figs 3, 4, 5 and 6 show clearly that the more the tile angular width is important, the less86

the magnetic torque between tiles uniformly magnetized is important. Consequently, we deduct that a87

manufacturer should use tiles uniformly magnetized only if their widths are small. Furthermore, Figs 3,88

70 0.25 0.5 0.75 1 1.25 1.5Angle @radD

0

0.0025

0.005

0.0075

0.01

0.0125

0.015

Torq

ue@N

.mD

Fig. 4. Representation of the torque transmitted between two tiles ; thick line: the magnetization is uniform, dashed lines:the magnetization is radial; 4 3=2 1=

pi

6, rin1 = 0.025m, rout1 = 0.028m, rin2 = 0.021m, rout2 = 0.024m,

zd zc = 0.003m, zb za = 0.003m, za = 0.001m, r = 0.024m, J1 = J2 = 1T

0 0.25 0.5 0.75 1 1.25 1.5Angle @radD

0

0.0025

0.005

0.0075

0.01

0.0125

0.015

Torq

ue@N

.mD

Fig. 5. Representation of the torque transmitted between two tiles ; thick line: the magnetization is uniform, dashed lines:the magnetization is radial; 4 3=2 1=

pi

4, rin1 = 0.025m, rout1 = 0.028m, rin2 = 0.021m, rout2 = 0.024m,

zd zc = 0.003m, zb za = 0.003m, za = 0.001m, r = 0.024m, J1 = J2 = 1T

0 0.25 0.5 0.75 1 1.25 1.5Angle @radD

-0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015

Torq

ue@N

.mD

Fig. 6. Representation of the torque transmitted between two tiles ; thick line: the magnetization is uniform, dashed lines:the magnetization is radial; 4 3=2 1=

pi

3, rin1 = 0.025m, rout1 = 0.028m, rin2 = 0.021m, rout2 = 0.024m,

zd zc = 0.003m, zb za = 0.003m, za = 0.001m, r = 0.024m, J1 = J2 = 1T

8

12

6

4

3

Angular Width @radD102030405060

Loss of Torque @%D

p = 6

p = 8

p = 12

p = 24

Fig. 7. Representation of the mean loss of efficiency of the torque transmitted between two tiles uniformly magnetized versus theangular width of the tiles. This loss of torque is calculated in comparison with the torque transmitted between two tiles radiallymagnetized. The points correspond to the figures 3, 4, 5 and 6. We have : rin1 = 0.025m, rout1 = 0.028m, rin2 = 0.021m,rout2 = 0.024m, zb za = zd zc = 0.003m, J1 = J2 = 1T.

4, 5 and 6 show that the torque transmitted between two tile permanent magnets radially magnetized89

does not depend very much on their angular widths whereas it decreases greatly with the increase in the90

angular width when the tiles are uniformly magnetized.91

We can also represent this decrease by calculating its rate of loss versus the angular width of tiles (Fig92

7).93

Figure 7 is consistent with Figs 3, 4, 5 and 6. Indeed, when the angular width of the tile permanent94

magnets tends to zero, we can use either tiles radially magnetized or tiles uniformly magnetized. This is95

the only case in which the choice has not a great importance. Consequently, as tiles uniformly magnetized96

are cheaper to fabricate, their use is more judicious. However, when the angular width of the tiles used97

becomes greater, the choice is certainly more difficult and other considerations must be taken into account.98

For example, if the first property required in a coupling is really the value of the torque transmitted, the99

magnetization of the tiles should be radial and not uniform.100

IV. APPLICATION : ALTERNATE MAGNET STRUCTURES USING TILES RADIALLY OR UNIFORMLY101

MAGNETIZED102

We can illustrate the expression established in the previous section by studying the torque transmitted103

in an alternate magnet structure with 8 tile permanent magnets on each rotor. For this purpose, we use104

the principle of superposition with (7) for the calculation of the torque transmitted between the leading105

9-3 -2 -1 0 1 2 3Angle @radD

-0.4

-0.2

0

0.2

0.4

Torq

ue@N

.mD

Fig. 8. Representation of the total torque transmitted between an eight tile rotor and an eight tile stator versus the angle .The tiles are either radially magnetized (dashed line) or uniformly magnetized (thick line) rin1 = 0.025m, rout1 = 0.028m,rin2 = 0.021m, rout2 = 0.024m, zb za = zd zc = 0.003m, J1 = J2 = 1T

rotor and the lead rotor. We represent this torque versus the shifting angle in Fig 8.106

Figure 8 is consistent with the previous representations of the torque transmitted between two tiles107

radially or uniformly magnetized. First, we note that the torque transmitted in the alternate magnet structure108

is sixteen times greater than the one transmitted between only two tile permanent magnets. Then, we note109

that the torque transmitted between tiles uniformly magnetized is smaller than the one transmitted between110

tiles radially magnetized, which is still consistent with Fig 4.111

V. CONCLUSION112

This paper has presented a three-dimensional expression of the torque transmitted between two tile113

permanent magnets uniformly magnetized. Then, we have compared the torque transmitted between two114

tiles uniformly magnetized with two tiles radially magnetized. An alternate magnet structure has been115

studied to illustrate the three-dimensional expression of the torque transmitted between tiles radially or116

uniformly magnetized. Some theoretical results have been analytically determined. First, tiles uniformly117

magnetized are less interesting than tiles radially magnetized for realizing couplings. Indeed, the more the118

tile angular widths are important, the less the torque transmitted between two tiles uniformly magnetized119

is important in comparison with the torque transmitted between two tiles radially magnetized. However,120

the cost of the magnets must be taken into account. Tiles uniformly magnetized are easier to fabricate121

10

than tiles radially magnetized (and thus cheaper...).122

APPENDIX123

We give the expressions of the parameters used for calculating the torque transmitted between two124

tile permanent magnets uniformly magnetized. The torque can be expressed in terms of semi-analytical125

expressions using one or two numerical integrations or not.126

T =J1J2

40

( 21

43

dT(1) + T

(2) +

zbza

dT(3) +

21

dT(4)

)(9)

where127

dT(1) = + cos(

1 + 22

) cos( 3 + 4

2) sin(

)A[rin1, rin2]dd

cos( 1 + 22

) cos( 3 + 4

2) sin(

)A[rout1, rin2]dd

cos( 1 + 22

) cos( 3 + 4

2) sin(

)A[rin1, rout2]dd

+cos( 1 + 22

) cos( 3 + 4

2) sin(

)A[rin2, rout1]dd

(10)

128

T(2) = +sin(

2 12

) sin(4 3

2)B[2, 4]

+ sin(2 1

2) sin(

4 32

)B[1, 4]

+ sin(2 1

2) sin(

4 32

)B[2, 3]

+ sin(2 1

2) sin(

4 32

)B[1, 3]

(11)

129

dT(3) = sin(

2 12

) cos(2 1

2 4 3

2)C[2, rin2]dz

+ sin(2 1

2) cos(

2 12

3 + 42

)C[1, rin2]dz

sin(2 12

) cos(2 1

2 3 + 4

2)C[2, rout2]dz

sin(2 12

) cos(2 1

2 3 + 4

2)C[1, rout2]dz

11

(12)

130

dT(4) = +cos(

1 + 22

) sin(4 3

2)D[4, rin1]d

cos( 1 + 22

) sin(4 3

2)D[4, rout1]d

+cos( 1 + 22

) sin(2 1

2)D[3, rin1]d

cos( 1 + 22

) sin(2 1

2)D[3, rout1]d

(13)

The calculation of the torque expression can be done as follows. Eqs (4) and (7) show four fundamental131

expressions, written A[r1, r2], B[i, j ], C[i, ri] and D[j , rj ] that are necessary for calculating the torque132

T. These expressions are expressed as follows:133

A[r1, r2] =

zbza

zdzc

r21r22(r21 + r

22 2r1r2 cos( ) + (z z)2)

3

2

dzdz (14)

134

B[i, j ] =

rout2rin2

zdzc

rout1rin1

zbza

rr sin(j i)(r2 + r

2 2rr cos(j i) + (z z)2) 32drdzdrdz (15)

135

C[i, ri] =

rout1rin1

zbza

zdzc

r2i r sin( i)(r2 + r2i 2rri cos( i) + (z z)2)

3

2

drdzdz (16)

136

D[j , rj ] =

rout2rin2

zdzc

zbza

r2j r sin(j i)(r

2+ r2j 2rj r cos(j ) + (z z)2)

3

2

dzdrdz (17)

The torque transmitted between two tile permanent magnets could be calculated directly by numerical137

means with (8). However, the computational cost would be too long. Therefore, we give here four reduced138

semi-analytical expressions of A[r1, r2], B[i, j ], C[i, ri] and D[j , rj ]. We obtain:139

A[r1, r2] = A(1)[za, zc]A(1)[zb, zc]A(1)[za, zd] + A(1)[zb, zd] (18)

with140

A(1)[zi, zj ] =r21r

22

r21 + r

22 + (zi zj)2 2r1r2 cos( )

r21 + r22 2r1r2 cos( )

(19)

12

141

B[i, j ] =

rout2rin2

(B(1)[za zc, rout1] + B(1)[zb zc, rout1]

)dr

+

rout2rin2

(+B(1)[za zd, rout1]B(1)[zb zd, rout1]

)dr

+

rout2rin2

(B(1)[za zc, rin1]B(1)[zb zc, rin1]

)dr

+

rout2rin2

(B(1)[za zd, rin1] + B(1)[zb zd, rin1]

)dr

(20)

with142

B[y, ri] = r

r2 2rrx + y + r2x log

[r rx +

r2 2rrx + y

]

+ri(x +1 + x2

r2 y)

21 + x2 log[A]

+ri(x +

1 + x2)

r2 y

21 + x2 log[B]

(21)

143

A =2i(rr(1 + x2) + r2(x x3 + x21 + x2))

(x +1 + x2)(r + r(x +1 + x2))(r2 y) 32

+2i1 + x2(y + i

r2 y

r2 2rrx + y)

(x +1 + x2)(r + r(x +1 + x2))(r2 y) 32(22)

144

B =2irr(1 + x2) 2ir2x(1 + x2 + x1 + x2)

(x +1 + x2)(r + r)(x +1 + x2)(r2 y) 32

+21 + x2(iy +

r2 y

r2 2rrx + y)

(x +1 + x2)(r + r(x +1 + x2))(r2 y) 32

(23)

145

x = cos(i j) (24)

13

C[i, ri] =

rout1rin1

C(1)[i, ri]dr (25)

C(1)[i, ri] = ri

r2i + r

2 + (zc z)2 2rir cos(3 i)

rir2i + r

2 + (zd z)2 2rir cos(3 i)

rir2i + r

2 + (zc z)2 2rir cos(4 i)

+ri

r2i + r

2 + (zd z)2 2rir cos(4 i)

+ri(zc z) log[zc + z +

r2i + r

2 + (zc z)2 2rir cos(3 i)]

ri(zd z) log[zd + z +

r2i + r

2 + (zd z)2 2rir cos(4 i)]

+ri(zd z) log[zd + z +

r2i + r

2 + (zd z)2 2rir cos(3 i)]

ri(zc z) log[zc + z +

r2i + r

2 + (zc z)2 2rir cos(3 i)]

(26)

D[j , rj ] =

rout2rin2

(D(2)(zb, zc)D(2)(za, zc) + D(2)(za, zd)D(2)(zb, zd)

)dr (27)

with146

D(2)(x, y) =sin(i j)r2j r

r2j +

r2

+ (x y)2 2rj r cos(j i)r2j +

r2 2rj r cos(j i)

(28)

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