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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
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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)
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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).
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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.
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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)
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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)
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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)
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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.
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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.
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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
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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
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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.
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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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