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Discussionabouttheanalyticalcalculationofthemagneticfieldcreatedbypermanentmagnets

ARTICLEinPROGRESSINELECTROMAGNETICSRESEARCHB·JANUARY2009

DOI:10.2528/PIERB08112102·Source:OAI

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GuyLemarquand

UniversitéduMaine

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V.Lemarquand

InstituteofElectricalandElectronicsEngineers

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ClaudeDepollier

UniversitéduMaine

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Retrievedon:13November2015

Progress In Electromagnetics Research B, Vol. 11, 281–297, 2009

DISCUSSION ABOUT THE ANALYTICALCALCULATION OF THE MAGNETIC FIELD CREATEDBY PERMANENT MAGNETS

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

Laboratoire d’Acoustique de l’Universite du Maine, UMR CNRS 6613Avenue Olivier Messiaen, 72085 Le Mans, France

Abstract—This paper presents an improvement of the calculation ofthe magnetic field components created by ring permanent magnets.The three-dimensional approach taken is based on the CoulombianModel. Moreover, the magnetic field components are calculatedwithout using the vector potential or the scalar potential. It is notedthat all the expressions given in this paper take into account themagnetic pole volume density for ring permanent magnets radiallymagnetized. We show that this volume density must be taken intoaccount for calculating precisely the magnetic field components in thenear-field or the far-field. Then, this paper presents the componentswitch theorem that can be used between infinite parallelepipedmagnets whose cross-section is a square. This theorem implies that themagnetic field components created by an infinite parallelepiped magnetcan be deducted from the ones created by the same parallelepipedmagnet with a perpendicular magnetization. Then, we discuss thevalidity of this theorem for axisymmetric problems (ring permanentmagnets). Indeed, axisymmetric problems dealing with ring permanentmagnets are often treated with a 2D approach. The results presentedin this paper clearly show that the two-dimensional studies dealingwith the optimization of ring permanent magnet dimensions cannot betreated with the same precisions as 3D studies.

1. INTRODUCTION

This paper continues the papers written by Babic and Akyel [1]and Ravaud et al. [2]. Ring permanent magnets axially or radiallymagnetized are commonly used for creating magnetic fields in magneticbearings [3, 4], in flux confining devices [5–9], in sensors [10–12],

282 Ravaud et al.

in electrical machines [13–16] and in loudspeakers [17–19]. Thecalculation of the magnetic field created by such structures canbe done by using numerical methods or analytical methods. Thenumerical methods are often based on a finite element method butthe evaluation of the magnetic field components with such methodshas a very high computational cost. Analytical methods are eitherbased on the Coulombian model [20–23], or the Amperian model [24–33], in a 2D or 3D approach. Authors have often used 2D-analyticalmethods for optimizing ring permanent magnet dimensions becausetheir expressions are fully analytical. However, these expressions arenot valid in two cases: for small ring permanent magnets and whenthe magnetic field is calculated far from the magnets. Consequently,3D-analytical methods are required.

As emphasized at the beginning of the introduction, such methodshave already been presented, but this paper improves once more theway of calculating the magnetic field created by a ring permanentmagnet radially magnetized. Indeed, the magnetic pole volumedensity is taken into account and we show that this magnetic chargecontribution is necessary to calculate the magnetic field componentsin the near-field and the far-field. Then, we discuss the validity of thecomponent switch theorem for ring permanent magnets and we showthat this theorem cannot be used in three-dimensions whereas it canbe used in two dimensions between two infinite parallelepiped magnets.The main reason lies in the fact that the magnetization is uniform for aring permanent magnet axially magnetized whereas it is not for a ringpermanent magnet radially magnetized. Such results are importantfor modeling the magnetic field created by cylindrical structures. Itis noted that the 2D approximation consists in representing a ringpermanent magnet by an infinite parallelepiped as shown in Fig. 1.With this approximation, the two magnetic components Hr(r, z) andHz(r, z) created by an infinite parallelepiped are assumed to bethe same as the ones created by a ring permanent magnet radiallymagnetized (see Fig. 2). In fact, this approximation is false for tworeasons. The first reason has been studied in a previous paper [2]: themagnitude of the magnetic field created by a ring permanent magnetis under-estimated with the 2D approximation. In addition, there isanother physical reason which can be demonstrated mathematically.With the 2D approximation, the magnetic field components Hr(r, z)and Hz(r, z) verify the component switch theorem. This theoremimplies that the magnetic field components created by an infiniteparallelepiped magnet can be deducted from the ones created bythe same parallelepiped magnet with a perpendicular magnetization.However, the theorem cannot be used for modelling the magnetic field

Progress In Electromagnetics Research B, Vol. 11, 2009 283

0

0

u z

u

u r r1

r

2

2

z

z

1

u r

uz

Figure 1. Approximation of a ring permanent magnet with an infiniteparallelepiped (the 2D approximation).

components created by ring permanent magnets radially magnetized.The first section presents the component switch theorem for

the case of two infinite parallelepiped magnets. Then, the secondsection presents the expressions of the magnetic field componentscreated by ring permanent magnets radially magnetized. Then, thissection discusses the validity of the component switch theorem for ringpermanent magnets axially and radially magnetized.

2. COMPONENT SWITCH THEOREM IN TWODIMENSIONS

This section presents the component switch theorem that can be usedfor calculating the magnetic components created by infinite magnetizedparallelepipeds.

2.1. The 2D Approximation

Let us consider the two geometries shown in Fig. 2. The uppergeometry (Fig. 2(a)) corresponds to an infinite parallelepiped whose

284 Ravaud et al.

x

uz

uz

x1x 2

0u

z1

z2

x 2x1

0u

z1

z 2

x

(a)

(b)

Figure 2. Representation of two infinite parallelepiped magnets whosemagnetization undergoes a rotation of 90 degrees between the upperand lower configurations.

polarization is colinear with the z direction. The lower geometry(Fig. 2(b)) corresponds to an infinite parallelepiped whose polarizationis colinear with the x direction. As the two configurations studiedare infinite along one direction (y direction), the magnetic field theycreate does not depend on y. Moreover, only two magnetic fieldcomponents Hx(x, z) and Hz(x, z) exist. These components can bedetermined analytically by using the coulombian model. By denotingH

(1)x (x, z) and H

(1)z (x, z), the magnetic components created by the

configuration shown in Fig. 2(a) and H(2)x (x, z) and H

(2)z (x, z), the

magnetic components created by the configuration shown in Fig. 2(b),

Progress In Electromagnetics Research B, Vol. 11, 2009 285

we have:

H(1)x (x, z) = J̃

(∫ ∫S1+

PMi,i,2

|PMi,i,2|3dS1+ −

∫ ∫S1−

PMi,i,1

|PMi,i,1|3dS1−

)· �ux (1)

H(1)z (x, z) = J̃

(∫ ∫S1+

PMi,i,2

|PMi,i,2|3dS1+ −

∫ ∫S1−

PMi,i,1

|PMi,i,1|3dS1−

)· �uz (2)

H(2)x (x, z) = J̃

(∫ ∫S2+

PM1,i,i

|PM1,i,i|3dS2+ −

∫ ∫S2−

PM2,i,i

|PM2,i,i|3dS2−

)· �ux (3)

H(2)z (x, z) = J̃

(∫ ∫S2+

PM1,i,i

|PM1,i,i|3dS2+ −

∫ ∫S2−

PM2,i,i

|PM2,i,i|3dS2−

)· �uz (4)

where J̃ = J4πµ0

, PMα,β,γ = (x− xα)�ux + (y − yβ)�uy + (z − zγ)�uz anddS1+ = dS1− = dxidyi and dS2+ = dS2− = dyidzi.

The analytical integrations of (1), (2), (3) and (4) give thefollowing expressions:

H(1)x (x, z) = log

[(x − x2)

2 + (z − z2)2

(x − x1)2 + (z − z2)2

]+ log

[(x − x1)

2 + (z − z1)2

(x − x2)2 + (z − z1)2

](5)

H(1)z (x, z) = arctan

[x − x1

z − z2

]− arctan

[x − x2

z − z2

]− arctan

[x − x1

−z + z2

]

+arctan

[x − x2

−z + z2

]− arctan

[x − x1

z − z1

]+ arctan

[x − x2

z − z1

]

− arctan

[x − x1

−z + z1

]+ arctan

[x − x2

−z + z1

](6)

H(2)x (x, z) = arctan

[x − x1

z − z2

]− arctan

[x − x2

z − z2

]− arctan

[x − x1

−z + z2

]

+arctan

[x − x2

−z + z2

]− arctan

[x − x1

z − z1

]+ arctan

[x − x2

z − z1

]

− arctan

[x − x1

−z + z1

]+ arctan

[x − x2

−z + z1

](7)

H(2)z (x, z) = − log

[(x − x2)

2 + (z − z2)2

(x − x1)2 + (z − z2)2

]− log

[(x − x1)

2 + (z − z1)2

(x − x2)2 + (z − z1)2

](8)

Thus, we deduct the following expressions that constitute thecomponent switch theorem.

H(1)z (x, z) = H(2)

x (x, z) (9)

and

H(1)x (x, z) = −H(2)

z (x, z) (10)

286 Ravaud et al.

The relations (9) and (10) are valid only if the cross-section of thebar-shaped magnet is a square. The calculations of such expressionsare well-known in the literature. As many authors have modelled ringpermanent magnets with parallelepiped magnets (2D approximation),it can be interesting to know if the component switch theorem can beused for ring permanent magnets radially and axially magnetized. Forthis purpose, we propose in the next section to use the coulombianmodel for calculating the magnetic field expressions created by aring permanent magnet radially magnetized. It is noted that thiscalculation has been improved because the magnetic pole volumedensity is taken into account in this paper and the expressions havebeen simplified.

3. THREE-DIMENSIONAL EXPRESSIONS OF THEMAGNETIC FIELD COMPONENTS CREATED BYRING PERMANENT MAGNETS

3.1. Notation and Geometry

The geometry considered and the related parameters appear in Fig. 3.The ring inner radius is r1 and the ring outer one is r2. Its height ish = z2−z1. In addition, the axis z is an axis of symmetry. Calculationsare obtained by using the Coulombian model. Consequently, the ringpermanent magnet is represented by two curved planes that correspond

v

0

uz

u

u r r1

r

2

2

z

z

1

*s

s*

*

Figure 3. Representation of the configuration studied; the innerradius is r1, the outer radius is r2, the height is z2−z1, σ∗

s = �J ·�n = 1 T,σ∗

v = −∇ �J = Jr .

Progress In Electromagnetics Research B, Vol. 11, 2009 287

to the inner and outer faces of the ring and a charged ring with amagnetic pole volume density +σ∗

v . The inner face is charged with asurface magnetic pole density +σ∗

s and the outer one is charged withthe opposite surface magnetic density −σ∗

s . By denoting r0, a pointthat belongs to the ring permanent magnet, the magnetic field �H(r, z)created by the ring permanent magnet at any point of the space isexpressed as follows:

�H(r, z) =∫

(V )

σ∗v(�r0)(�r − �r0)

4πµ0|�r − �r0|3d3 �r0 +

∫(Sin)

σ∗s(�r0)(�r − �r0)

4πµ0|�r − �r0|3d2 �r0

−∫

(Sout)

σ∗s(�r0)(�r − �r0)

4πµ0|�r − �r0|3d2 �r0 (11)

with

(�r − �r0)|�r − �r0|3

=(r − ri cos(θ)) �ur − ri sin(θ)�uθ + (z − zs)�uz

(r2 + r2i − 2rri cos(θ) + (z − zs)2)

32

(12)

where i = 1 for d�r02 = dSin = r1dθsdzs, i = 2 for d�r0

2 = dSout =r2dθsdzs, i = s for d�r3

0 = rsdrsdθsdzs.

3.2. Components Along the Three Directions �ur, �uθ, �uz

The calculation of (11) leads to the magnetic field components alongthe three defined axes: Hr(r, z), Hθ(r, z) and Hz(r, z). It is noted thatthe azimuthal component Hθ(r, z) equals 0 because of the cylindricalsymmetry.

3.3. Radial Component Hr(r, z)

The radial component Hr(r, z) is given by

Hr(r, z) =2∑

i=1

2∑j=1

(−1)(i+j)(Sr

i,j + V ri,j + Ni,j

)(13)

where

Sri,j = α

(0)i,j

(α

(1)i F∗

[α

(2)i,j , α

(3)i

]+ α

(4)i Π∗

[α

(2)i,j , α

(5)i , α

(6)i

])(14)

V ri,j = f(z − zj , r

2 + r2i + (z − zj)2, 2rri,−1)

−f(z − zj , r2 + r2

i + (z − zj)2, 2rri, 1) (15)

288 Ravaud et al.

with

f(a, b, c, u) = −η(β(1) + β(2)

)(16)

η = β(3)[(b − c)E∗

[β(4), β(5)

]+ cF∗

[β(4), β(5)

]]+β(6)

[(b − a2)F∗

[β(7), β(8)

]+ (b − a2 + c)Π∗

[β(9), β(7), β(8)

]]−β(10) − β(11) (17)

Table 1. Parameters used for calculating the surface contribution ofthe radial component Hr(r, θ, z).

Parameters

α(0)i,j

J√

24πµ0

ri(−z+zj)

(2rir)3/2α(1)i

α(1)i r2

i + r2 + 2rir

α(2)i,j

√4rir

α(1)i +(z−zj)2

α(3)i

√α1

i +(z−zj)2

4rir

α(4)i 2rir

2 − ri(r2i + r2)

α(5)i

α(1)i +(z−zj)

2

α(1)i

α(6)i

√2(α

(1)i +(z−zj)2)

4rir

Ni,j =∫ −1

1(1 − u2) arctan

[(ri − ru)(z − zj)√

r2(u2 − 1)ξ

]du (18)

ξ =√

r2 + r2i − 2rriu + (z − zj)2 (19)

3.4. Axial Component Hz(r, z)

Hz(r, z) =2∑

i=1

2∑j=1

(−1)(i+j)(Sz

i,j + V zi,j

)(20)

Progress In Electromagnetics Research B, Vol. 11, 2009 289

Table 2. Parameters used for calculating the volume contribution ofthe radial component Hr(r, θ, z).

Parameters

β(1) a√

1 − u2√

b−cub+c + a

√c(1+u)

c√

1−u2

β(2) a(a2+b) arcsin[u]

c√

b+c

√b − cu

β(3) (1 + u)√

c(u−1)b−c

β(4) arcsin[√

b−cub+c

]β(5) b+c

b−c

β(6)√

1 − u2

√c(1+u)

b+c

β(7) arcsin[√

1+u2

]β(8) 2c

b+c

β(9) 2cb+c−a2

β(10) −2√

1 − u2 log[a +√

b − cu]

β(11) −√

xc log

[4c2(c+a2u−bu+

√x√

1−u2)x

32 (a2−b+cu)

]x −a4 + 2a2b − b2 + c

with

Szi,j =

2ri

(r − ri)2 + (z − zj)2K∗

[− 4rri

(r − ri)2 + (z − zj)2

](21)

V zi,j =

∫ 2π

0tanh−1

√r2 + r2

i + (z − zj)2 − 2rri cos(θs)

ri − r cos(θs)

(22)

The special functions used are defined as follows:

K∗ [m] = F∗[π

2,m

](23)

F∗ [φ, m] =∫ φ

0

1√1 − m sin(θ)2

dθ (24)

Π∗ [n, φ, m] =∫ φ

0

1(1 − n sin(θ)2)

√1 − m sin(θ)2

dθ (25)

290 Ravaud et al.

3.5. Comparison of the Magnetic Field Created by RingPermanent Magnets Axially and Radially Magnetized

This section discusses the validity of the component switch theoremfor ring permanent magnets axially and radially magnetized. For thispurpose, we represent in Figs. 4 and 5 the magnetic field modulusH(r, z) created by either a ring radially magnetized or a ring axiallymagnetized.

0.021 0.022 0.023 0.024r [m]

0.002

0.001

0

0.001

0.002

0.003

0.004

0.005z

[m]

−

−

Figure 4. Modulus of the magnetic field created by a ring permanentmagnet axially magnetized; we take r = 0.024 m, r1 = 0.025 m,r2 = 0.028 m, z1 = 0 m, z2 = 0.003 m, J = 1 T.

[m]

[m]

−

−0.021 0.022 0.023 0.024

r

0.002

0.001

0

0.001

0.002

0.003

0.004

0.005

z

Figure 5. Modulus of the magnetic field created by a ring permanentmagnet radially magnetized; we take r = 0.024 m, r1 = 0.025 m,r2 = 0.028 m, z1 = 0 m, z2 = 0.003 m, J = 1 T.

Figures 4 and 5 show that in the near-field, a ring permanentmagnet radially magnetized is similar to a ring permanent magnetaxially magnetized. The magnetic field becomes different when

Progress In Electromagnetics Research B, Vol. 11, 2009 291

it is calculated far from the magnets. Thereby, it implies thata ring permanent magnet radially magnetized do not create thesame magnetic field as a ring permanent magnet axially magnetized.Consequently, the choice of using a ring permanent magnet radiallyor axially magnetized depends greatly on the intended application.Moreover, the cost of the magnet must be also taken into accountsince rings radially magnetized are more expensive than rings axiallymagnetized.

−

−0.001 0 0.001 0.002 0.003 0.004

z [m]

400000

200000

0

200000

400000

Hr

[A/m

]

−

Figure 6. Radial component of a ring permanent magnet axiallymagnetized versus the axial displacement z; we take r = 0.024 m,r1 = 0.025 m, r2 = 0.028 m, z1 = 0 m, z2 = 0.003 m, J = 1 T.

−

−

[A/m

]

−0.001 0 0.001 0.002 0.003 0.004z [m]

400000

200000

0

200000

400000

Hz

Figure 7. Axial component of a ring permanent magnet radiallymagnetized versus the axial displacement z; we take r = 0.024 m,r1 = 0.025 m, r2 = 0.028 m, z1 = 0 m, z2 = 0.003 m, J = 1 T.

3.6. Magnetic Field Components Created by Rings Radiallyand Axially Magnetized

In the previous section, we have shown that a ring permanent magnetaxially magnetized does not generate the same magnetic field thana ring permanent magnet radially magnetized even though theirmagnetic field modulus seems to be nearly the same. But strictly

292 Ravaud et al.

speaking, these two configurations are different when we look at theirmagnetic components. We can see that by comparing their magneticcomponents with the help of the component switch theorem (evenif this theorem cannot be used). For this purpose, we represent inFigs. 6 and 7, the radial component of the magnetic field created bya ring axially magnetized and the axial component of the magneticfield created by a ring permanent magnet radially magnetized. Wesee that these two components seem to verify the component switchtheorem. However we will see that it is not strictly the case. Wecan also compare the axial component created by a ring permanentmagnet axially magnetized and the radial component of the magneticfield created by a ring permanent magnet radially magnetized. Thesetwo components are represented in Figs. 8 and 9.

0.001 0 0.001 0.002 0.003 0.004z [m]

250000200000150000100000

500000

50000

Hz

−

−

[A/m

]

−−−−

Figure 8. Axial component of a ring permanent magnet axiallymagnetized versus the axial displacement z; we take r = 0.024 m,r1 = 0.025 m, r2 = 0.028 m, z1 = 0 m, z2 = 0.003 m, J = 1 T.

0.001 0 0.001 0.002 0.003 0.004z [m]

250000200000150000100000

500000

50000

Hr

−

[A/m

]

−−−−

−

Figure 9. Radial component of a ring permanent magnet radiallymagnetized versus the axial displacement z; we take r = 0.024 m,r1 = 0.025 m, r2 = 0.028 m, z1 = 0 m, z2 = 0.003 m, J = 1 T.

Figures 8 and 9 show clearly that these components are nearlythe same. However, the component switch theorem cannot be usedfor studying the magnetic field created by ring permanent magnetsradially and axially magnetized.

Progress In Electromagnetics Research B, Vol. 11, 2009 293

0.016 0.018 0.02 0.022 0.024r [m]

7.5

10

12.5

15

17.5

20

22.5

25

Err

or[%

]

Figure 10. Representation of the radial component Hr(r, z) withor without the magnetic pole volume density σ∗

v versus z with thefollowing parameters: r = 0.02 m, r1 = 0.025 m, r2 = 0.028 m,z1 = 0 m, z2 = 0.003 m; J = 1 T; thick line: with σ∗

v , dashed line:without σ∗

v .

0.016 0.018 0.02 0.022 0.024r [m]

0

5

10

15

20

Err

or[%

]

Figure 11. Representation of the axial component Hz(r, z) withor without the magnetic pole volume density σ∗

v versus z with thefollowing parameters: r = 0.02 m, r1 = 0.025 m, r2 = 0.028 m,z1 = 0 m, z2 = 0.003 m; J = 1 T; thick line: with σ∗

v , dashed line:without σ∗

v .

294 Ravaud et al.

3.7. Necessity of Taking into Account the Magnetic PoleVolume Density

This section explains why the magnetic pole volume density isnecessary for calculating the magnetic field components created bya ring permanent magnet. This question is in fact crucial because thecalculation of the force or the stiffness between ring permanent magnetsis more complicated when the magnetic pole volume density is takeninto account. To do so, we plot in Fig. 10 the relative difference ofthe radial field created by a ring permanent magnet versus the radialdistance r with and without taking into account the magnetic polevolume density. In addition, we plot in Fig. 11 the relative differenceof the axial field created by a ring permanent magnet versus the radialdistance with and without the magnetic pole volume density.

We see that the farther the magnetic field is calculated from themagnets, the more the relative difference increases if the magnetic polevolume density is omitted. For a radial distance which equals 0.01 mfrom the magnets, we make an error of at least 25 per cent for the radialfield and 20 per cent for the axial field is the magnetic pole volumedensity is omitted. Consequently, we deduct that we must take intoaccount such a contribution for calculating the magnetic componentsin the far-field. For a radial distance which equals 0.001 m from themagnets, we make an error of at least 7 per cent for the radial fieldand 5 per cent for the axial field is the magnetic pole volume density isomitted. We can say that it is yet an important error for the near-field.Consequently, we must take into account the magnetic pole volumedensity for calculating the magnetic components in the near-field aswell.

4. CONCLUSION

This paper has presented an improvement of the magnetic fieldcalculation created by ring permanent magnets radially magnetized.The expressions obtained are based on real functions and the magneticpole volume density is taken into account. We discuss the importanceof taking into account such a contribution. Then, this paper discussesthe validity of the component switch theorem for ring permanentmagnets. This theorem can be used for describing the magneticfield created by infinite parallelepiped magnets (2D approach) whereasit cannot be used for modeling the magnetic field created by ringpermanent magnets (3D approach). This result implies that anoptimization of the ring dimensions with a 2D approach seems tobe difficult. Eventually, we have compared, with the exact 3Dapproach, the magnetic field created by a ring permanent magnet

Progress In Electromagnetics Research B, Vol. 11, 2009 295

radially magnetized with a ring permanent magnet axially magnetized.When their cross-section is a square, their magnetic field modulus isnearly the same in the near field whereas it is different in the far field.Such results imply that ring permanent magnets axially magnetizedcan be used in magnetic bearings in which the near-field is the mostimportant parameter to optimize. The expressions given in this paperare available online [34].

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