FACTA UNIVERSITATIS (NIS)

SER.: ELEC. ENERG. vol. 23, no. 3, Decmber 2010, 259-272

Magnetic Field Determination for Different BlockPermanent Magnet Systems

Ana Mladenovic Vuckovic and Slavoljub Aleksic

Abstract: The paper presents magnetic field calculation of three characteristic per-manent magnet systems, which component parts are block magnets homogeneouslymagnetized in arbitrary direction. Method used in this publication is based on a sys-tem of equivalent magnetic dipoles. The results obtained using this analytical methodare compared with results obtained using COMSOL Multiphysics software. Magneticfield and magnetic flux density distributions of permanent magnet systems are alsoshown in the paper.

Keywords: Magnetic field, permanent magnet, magnetic dipole.

1 Introduction

PERMANENT magnetism is one of the oldest continuously studied branches ofthe science. There are many properties of permanent magnetsthat are takeninto consideration when designing a magnet for a certain device. Most often thedemagnetization curve is the one that has the greatest impact on its usability. Curveshape contains information on how the magnet will behave under static and dy-namic operating conditions, and in this sense the material characteristic will con-strain what can be achieved in the devices design. TheB (magnetic flux density)versusH (magnetic field) loop of any permanent magnet has some portions whichare almost linear, and others that are highly non-linear [1].

Magnetic materials are vital components of most electromechanical machines.An understanding of magnetism and magnetic materials is therefore essential forthe design of modern devices. The magnetic components are usually concealed

Manuscript received on June 22, 2010.Authors are with Faculty of Electronic Engineering, University of Nis, Aleksandra

Medvedeva 14, 18000 Nis, Serbia (e-mail:[ana.vuckovic, slavoljub.aleksic]@elfak.ni.ac.rs).

259

260 A. Mladenovic Vuckovic and S. Aleksic:

in subassemblies and are not directly apparent to the end user. Permanent magnetshave been used in electrical machinery for over one hundred years. Scientific break-throughs in materials study and manufacturing methods fromthe early 1940s to thepresent have improved the properties of permanent magnets and made the use ofmagnetic devices common. This work is motivated by the need for different shapedpermanent magnets in great number of electromagnetic devices. Knowledge of themagnetic scalar potential, magnetic flux density or magnetic field is required tocontrol devices reliably [2].

Determination of the magnetic field components in vicinity of permanent mag-nets, starts with presumption that magnetization,MMM, of permanent magnet is known.The following methods can be used in practical calculation:

(a) Method based on determining distribution of microscopic amperes current;

(b) Method based on Poisson and Laplace equations, determining magnetic scalarpotential; and

(c) Method based on a system of equivalent magnetic dipoles [3].

The third method that is mentioned in the paper for magnetic field calculationis based on superposition of elementary results obtained for elementary magneticdipoles.

Elementary magnetic dipole (Fig. 1) has magnetic moment

dmmm= MMMdV (1)

0

x

z

y

rr

R

d = dVm M

P

Fig. 1. Elementary magnetic dipole.

This magnetic moment produces, at field pointP, elementary magnetic scalarpotential

dm =1

4RRRdmmmR3

=1

4RMRMRMR3

dV, (2)

Magnetic Field Determination for Different Block Permanent Magnet ... 261

whereRRR = |rrr rrr | is distance from the point where the magnetic field is beingcalculated to elementary source, andRRR= rrr rrr .

After integration magnetic scalar potential is obtained as

m =1

4

V

RMRMRMR3

dV. (3)

Magnetic field vector can be expressed as

HHH = gradm. (4)

In publications [4]- [5] magnetic field components are determined for differentshaped permanent magnets using methods presented above. This paper describesthree typical examples of block permanent magnet systems.

2 Problem definition

The aim of this paper is magnetic field components determination of block perma-nent magnet systems. To determine the magnetic field of a system the permanentmagnet presented in the Fig. 2 will be considered first, [6]. The method describedabove is used for determining the magnetic field components of the block perma-nent magnet magnetized in arbitrary direction (Fig. 2).

x

z

M

y

y

x x

y

1

1

2

2

Fig. 2. Block permanent magnet magnetized in arbitrary direction.

The intensity of magnetic field depends on magnetization pattern and the mag-net shape. In this case it is assumed that the magnitude of themagnetizationMhas the same value everywhere, it is oriented inxy plane and its direction can varygiving

MMM = M(cos()x+sin()y). (5)

Outside the permanent magnet, magnetic scalar potential, at the field pointP(x,y,z), could be presented using the expression (3).

262 A. Mladenovic Vuckovic and S. Aleksic:

As magnetization has two components,x andy, dot productRRRMMM is formed as

RRRMMM = [(xx)x+(yy)y+(zz)z]M(cos()]x+sin()y), (6)

therefore,RRRMMM = M((xx)cos()+ (yy)sin()). (7)

Distance from the point where the magnetic field is being calculated to elemen-tary source is

R=

(xx)2 +(yy)2 +(zz)2. (8)

Finally, substituting the expressions Eq.7 and Eq. 8 in Eq. 3, magnetic scalarpotential produced by a block magnet is formed as

m =M4

x2

x1

y2

y1

z2

z1

(xx)cos()+ (yy)sin()[(xx)2 +(yy)2 +(zz)2]

32

dxdydz. (9)

The solution of this integral is

m(x,y,z) =M4

((V[xx2,yy1,yy2,zz1,zz2]

V[xx1,yy1,yy2,zz1,zz2])cos+(V[yy2,xx1,xx2,zz1,zz2]

V[yy1,xx1,xx2,zz1,zz2]))sin ,

(10)

where functionV has the following form:

V(a,x1,x2,z1,z2) =x2 lnC2C3

+x1 lnC1C4

+z1 lnC5C8

+z2 lnC6C7

2|a|arctanC5C8 +a2 +z12+z1(C5 +C8)

|a|(C8C5)

+2|a|arctanC7C6 +a2 +z22+z2(C6 +C7)

|a|(C6C7),

(11)

andC1 = z1 +

a2 +x12 +z12;

C3 = z1 +

a2 +x22 +z12;

C5 = x1 +

a2 +x12 +z12;

C7 = x1 +

a2 +x12 +z22.

C2 = z2 +

a2 +x22 +z22;

C4 = z2 +

a2 +x12 +z22;

C6 = x2 +

a2 +x22 +z22;

C8 = x2 +

a2 +x22 +z12.

Magnetic Field Determination for Different Block Permanent Magnet ... 263

3 Magnetic field determination

3.1 Example I

The first system which is considered is presented in the Fig. 3. The magnetic scalarpotential solution obtained for the permanent magnet presented above can be usedfor each block of the system if it is taken that = 0 or = [7]. Therefore, eachof these block permanent magnets is homogeneously magnetized in longitudinaldirection,

MMM = M(x). (12)

The certain modification of this system might find its application in modernhard drives.

x

2a

a

a

M

yP x y z( , , )

z

Fig. 3. First block permanent magnet system.

Magnetic scalar potentials of the block magnets may be presented using thefunctionV, Eq. 11, as

m1 =M4

(V[x+4a,y+a2,y

a2,z+

a2,z

a2]

V[x+2a,y+a2,y

a2,z+

a2,z

a2]),

(13)

m2 =M4

(V[x,y+a2,y

a2,z+

a2,z

a2]

V[x+2a,y+a2,y

a2,z+

a2,z

a2]),

(14)

m3 =M4

(V[x2a,y+a2,y

a2,z+

a2,z

a2]

V[x,y+a2,y

a2,z+

a2,z

a2]),

(15)

m4 =M4

(V[x2a,y+a2,y

a2,z+

a2,z

a2]

V[x4a,y+a2,y

a2,z+

a2,z

a2]).

(16)

264 A. Mladenovic Vuckovic and S. Aleksic:

Magnetic scalar potential of the whole system is the sum of magnetic scalarpotentials obtained for each magnetized block.

3.2 Example II

The second example that is considered is calculation of the external magnetic fieldgenerated by domains of periodic parallel structure with domains widtha, whosemagnetizationMMM is the same in magnitude but reverses orientation from domain todomain (Fig. 4) [8],

MMM = M(y). (17)

ab

l

a a a

a a a0 0 0

x

M

y

Fig. 4. Second block permanent magnet system.

It is considered the case of magnetization vectorMMM orientation perpendicularto a sample surface as shown in the figure. It is also assumed that the thickness ofthe block isb lengthl and the distance between two neighboring blocks isa0 [7].

The magnetic scalar potential generated by the block permanent magnet is mag-netized in positive direction ofy axis is obtained from the expression (10) when = /2 , and magnetic scalar potential of the block magnetized in opposite di-rection in the case when = /2. Using the derivation presented above themagnetic scalar potential obtained for blocks are given by following expressions:

m1 =M4

(V[yl2,x+2a+

3a02

,x+a+3a02

,z+b2,z

b2]

V[y+l2,x+2a+

3a02

,x+a+3a02

,z+b2,z

b2]),

(18)

m2 =M4

(V[y+l2,x+a+

a02

,x+a02

,z+b2,z

b2]

V[yl2,x+a+

a02

,x+a02

,z+b2,z

b2]),

(19)

Magnetic Field Determination for Different Block Permanent Magnet ... 265

m3 =M4

(V[yl2,x

a02

,xaa02

,z+b2,z

b2]

V[y+l2,x

a02

,xaa02

,z+b2,z

b2]),

(20)

m4 =M4

(V[y+l2,xa

3a02

,x2a3a02

,z+b2,z

b2]

V[yl2,xa

3a02

,x2a3a02

,z+b2,z

b2]).

(21)

In cases where system hasN block permanent magnets, the magnetic scalarpotential of the whole system is equal to sum of magnetic scalar potentials formedby all of its magnetized parts:

m =N

i=1

mi . (22)

3.3 Example III

The method that is described above is also used for determining the magnetic fieldcomponents of the block permanent magnet system presented in the Fig.5 [9].

x

y

Fig. 5. Third block permanent magnet system.

With multiple translations and rotations of coordinate system, the block shownin the Fig. 2 may be positioned like its shown on Fig. 5 to obtain suitable system.

If the original point isP(x,y,z) it can be translated to pointP(x,y,z). Newcoordinates are

x = xx1, y = yy1, z

= z. (23)

266 A. Mladenovic Vuckovic and S. Aleksic:

If the coordinate system is rotated clockwise around centerpoint placed in thecenter of coordinate system new coordinates will be

x = xcos()+ysin(), y = xsin()+ycos(), z = z. (24)

Therefore, using expression (10) and relations (23) and (24) magnetic scalarpotential of each block of the treated system can be calculated [10]. As in previoustwo examples, the total magnetic scalar potential is equal to the sum of magneticscalar potentials generated by all permanent magnet blocks(22). After determin-ing magnetic scalar potential magnetic field component generated by permanentmagnet systems may be calculated using the expression (4).

4 Numerical results

For the first considered permanent magnet system distribution of magnetic field, inx0y plane, outside the system is illustrated in the Fig. 6. It is obtained using theanalytical method for magnetic field determination. Magnetic field lines for thesame system, obtained using the COMSOL Multiphysics software are presented inthe Fig. 7. Distribution of magnetic flux density is shown in the same figure witharrows and its intensity is presented with gradient of gray.Magnetization of eachblock in the system is 750kA/m. Comparing these two figures itcan be concludedthat results of the analytical method are confirmed in satisfactory manner usingCOMSOL Multiphysics software.

Fig. 6. Distribution of magnetic field obtained using the analytical method.

Fig. 8 presents magnetic flux density inside the system alongthe directiony= 0.It is obvious that magnetic flux density has the highest values inside of blocks andthe lowest values in the border area between neighboring blocks.

The pictures from Fig. 9 to Fig. 12 present results obtained for the second ex-ample of permanent magnet system. The system consists of four blocks which di-mensions area,b = a andl = 3a and the distance between two neighboring blocksis a0 = a/5. Magnetization of each block in the system is 150kA/m. Fig.9 showsmagnetic field distribution of the system as the result of theanalytical method.

Magnetic Field Determination for Different Block Permanent Magnet ... 267

Fig. 7. Distribution of magnetic field for the first permanentmagnetsystem (COMSOL Multiphysics software).

Fig. 8. Magnetic flux density inside the blocks of the system along thedirectiony = 0.

This system is also analyzed using COMSOL Multiphysics software and theresult is shown in the Fig. 10. The same picture presents the magnetic flux density,marked with arrows, while its intensity is presented with different colors. Thelight yellow color presents the highest magnetic flux density while the black colorexpresses the lowest magnetic flux density.

Fig. 11 and Fig. 12 represent magnetic flux density between two neighboringblocks, positioned in the center of the system, forx= 0, z= 0 andl/2 y l/2.

268 A. Mladenovic Vuckovic and S. Aleksic:

Fig. 9. Distribution of magnetic field for second permanent magnet system (ana-lytical method).

Fig. 10. Distribution of magnetic field for second permanentmagnet system(COMSOL Multiphysics software) .

The first one is result of COMSOL Multiphysics software while, the second oneis obtained using analytical method. Comparing these two diagrams it may beconcluded that alignment exists between analytical and numerical methods of cal-culation. Certain deviation is present on block ends.

Table in the Figure 13 presents normalized magnetic field values for the second

Magnetic Field Determination for Different Block Permanent Magnet ... 269

Fig. 11. Magnetic flux density between two neighboring blocks (COMSOL Multiphysicssoftware).

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

Mag

netic

flux

den

sity

[T]

Length

Fig. 12. Magnetic flux density between two neighboring blocks (analytical method).

block permanent magnet system with dimensions given above.It is obtained usingthe analytical method. The values along the directionx = 0, z = 0, in the gapbetween two neighboring blocks, are shown in the table.

Distribution of magnetic field generated by the third system, in x0y plane, out-side the permanent magnet is illustrated in the Fig. 14. It isobtained using theanalytical method for magnetic field determination. Magnetic field distribution forthe same system, obtained using the COMSOL Multiphysics software is presentedin the Fig. 15. Distribution of magnetic flux density is shownin the same figurewith arrows and its intensity is presented with gradient of gray. Magnetization ofeach block in the system is 750kA/m. Comparing these two figures it is obvious

270 A. Mladenovic Vuckovic and S. Aleksic:

y/a H/M

0.1 0.0025690.2 0.0054400.3 0.0089460.4 0.0134810.5 0.0195450.6 0.0277960.7 0.0391350.8 0.0548220.9 0.0766721.0 0.1073731.1 0.1510411.2 0.2142331.3 0.3074861.4 0.4420881.5 0.548363

Fig. 13. Normalized values ofmagnetic field along the directionx = 0, z= 0.

Fig. 14. Distribution of magnetic field for third perma-nent magnet system (analytical method).

Fig. 15. Distribution of magnetic field for third permanent magnet system (COM-SOL Multiphysics software).

that results of the analytical method are confirmed in satisfactory manner usingCOMSOL Multiphysics software.

This system may be applicable for approximation of the permanent magnetssystem found in rotation motors.

Magnetic Field Determination for Different Block Permanent Magnet ... 271

5 Conclusion

Three different permanent magnet systems which consist of block permanent mag-nets, homogeneously magnetized in known direction, are observed in the paper.Method that is used for magnetic field determination is basedon superposition ofresults that are obtained for elementary magnetic dipoles.Magnetic field and mag-netic flux density distributions of permanent magnet systems are also presentedin the paper. Magnetic field lines have the same form and the same direction asmagnetic flux density lines, outside the system. Results obtained by the analyticalmethod are satisfactory confirmed using COMSOL Multiphysics software. Thiswork is motivated by the need for different shaped permanentmagnets in greatnumber of electromagnetic devices [11].

Acknowledgments

The author would like to acknowledge the support of the Ministry of Science andTechnological Development, Serbia.

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