 Open Access
 Total Downloads : 471
 Authors : Mr. Saurabh Patel, Mr. C. P.Dewan, Prof. D. A. Patel
 Paper ID : IJERTV2IS4083
 Volume & Issue : Volume 02, Issue 04 (April 2013)
 Published (First Online): 18042013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Analytical And Finite Element Comparison Of Magnetic Flux Density And Magnetic Field Intensity Of Permanent Magnet
Mr. Saurabh Patel 
Mr. C. P.Dewan 
Prof. D. A. Patel 
Research Scholar 
Head, OPMG/MESA 
Associate Professor 
SPCE, Visnagar 
ISRO, Ahmedabad Head, 
Mech Dept, SPCE, Visnagar 
Abstract
The paper presents calculation the magnetic field and magnetic field intensity of permanent magnet. Consider permanent magnet homogeneously magnetized in Z axis. In charge method the magnet is reduced to distribution of equivalent charge. Analytical calculation based upon charge method is compared with results obtained in COMSOL MULTYPHYSICS. Magnetic flux density distributions of permanent magnet are also shown in thepaper.
To determine the magnetic field components in vicinity of permanent magnets, itstarts from supposition that
Substitute this into Eq. (2), taking into account the constitutive relation B 0 (H M ) .These yields
2 A (.A) (J M )… (3) Next, impose the coulomb gauge condition .A 0 and obtain
0
0
2 A (J M ) … (4)
If there is no free current (J=0), and if we assume an infinite homogeneous material (no boundaries), then the solution to Eq. (4) can be written in integral form using the freespace Greens function. For operator 2
the green function can be written as,
magnetization of permanent, M, magnet is known. Thefollowing methods are useful in practical
G(x, x ' ) 1
4
1
x x '
calculation:

Method based on determining distribution of
Thus, magnetic vector potential can be written as
microscopic Ampere's current or current method;

Method based on determining magnetic scalar
A(x) 0
J (x ' )
dv ' .. (5)
m
m
potential or charge method.
Keywords: Permanent Magnet, Charge Method, Finite Element Method
4 x x '
Now B A and obtain
J (x ' ) 1

Analytical Method
And,
m J x x ' m
(x ' )
x x '
Derivation of Magnetic field density from charge method and current method is shown below.
1
x x '
(x x ' )
3
3
x x '

CURRENT METHOD:
Now, consider a stationary, homogeneous and isotropic material with a linear constitutive relation
Where, x is the observation point and x' is the source point, indicates differentiation with respect to the unprimed variables.
m
m
So,
B 0 (H M ) . Introduce the vector potential A in
0 '
(x x ' )
………….. (6)
equation (1).
B(x) J (x )
4
dv'
3
3
x x '
.B 0 B A (2)
If the magnetization M is confined to a volume V (of permeability 0 ), and falls abruptly to zero outside of V, then Eqs. (5) and (6) reduce to
J (x' )
j (x' )
m .M
A(x) 0 m dv' 0 m
ds' .. (7)
.. (14)
And
4 v
x x'
4 s
x x'
m M . n
0
0
' (x x' )
' (x x' )
..(8)
From Eq. (13) and (14) magnetic flux density can be written as
m4
m4
B(x) 0 J (x )
v
x x'
3 dv' j (x )
m4
m4
s
x x'
3 ds'
B(x) 0
m (x')(x x') dv' 0
m (x')(x x')ds'.. (15)
v
v
s
s
4
x x' 3 4
x x' 3
In these expressions S is the surface of the magnet, and Jm andjmare equivalent volume and surface current densities. These are defined in the following:
Jm M . (9)
Now, the magnetic field above a rectangular magnet is find out using charge method is depicted below. The magnet is polarized along its axis with uniform magnetization as shown in figure 1. Assume that magnetization is
jm M n
M Ms z (16)
First determine charge densities. From Eq. (14)

CHARGE METHOD:
The derivation of the charge model is as follows: Start with the magnetostatic field equations for
m .M 0
To evaluate m first find surface normals
n z=0
currentfree regions H 0 and .B 0 . Now, z
introduce a scalar potential as,
z z=L
H m
m
….. (10)
x
x= b y= a
Finally, substitute Eq. (10) and constitutive relation y
B 0 (H M ) into .B 0 and obtain
m
m
2 .M . (11)
If there is no free current (J=0), and if we assume an infinite homogeneous material (no boundaries), then the solution to Eq. (11) can be written in integral form using the freespace Greens function.
m (x) G(x, x')'.M (x')dv'
1 '.M (x')dv' .. (12)
4 x x'
Where, x is the observation point and x' is the source point, ' indicates differentiation with respect to the
Figure 1: Rectangular bar magnet, (a) physical magnet; and (b) equivalent charge.
unprimed variables, and the integration is over the volume for which the magnetization exists. If the magnetization M is confined to a volume V (of permeability 0 ), and falls abruptly to zero outside of V, then Eqs. (12) becomes
1
1
1
1
From Eq. (16) and the fact that m M .n . For the top
surface m M (z=0), and m Ms for the bottom surface (z=L).
m (x)
4
' M (x')dv'
x x'
4
M (x').nds' (13)
x x'
Thus, from Eq. (15)
v s a b M z
0
0
Bz (z)
s dx' dy' (17)
3
3
Where, S is the surface that bounds V, and n is the outward unit normal to S. The form of Eq. (13) suggests the definitions of volume and surface charge densities given in Eq. (14):
4 ab x'2 y'2 z'2 2
Here, x=z z and x=x x +y y .
Material
Br(T)
Hci(kA/m)
BHmax(kJ/m3)
Tc(Â°C)
Ferrite
0.42
242
33.4
450
Alnico9
1.10
145
75.0
850
SmCo5
1.00
696
196
700
Nd2Fe14B
1.23
947
278
300
Material
Br(T)
Hci(kA/m)
BHmax(kJ/m3)
Tc(Â°C)
Ferrite
0.42
242
33.4
450
Alnico9
1.10
145
75.0
850
SmCo5 1.00
696
196
700
Nd2Fe14B
1.23
947
278
300
The integrand is even function of x and y, and therefore
a b M z
0
0
Bz (z)
s dx' dy'
3
3
4 0 0 x'2 y'2 z'2 2
M z a b
0 s
0 ( y'2
z'2 )
b 2 y'2
dy'
z 2
a
a
M
0 s tan1 by
z
b 2 y'2 z 2
0
Table1: Permanent magnet materials
z
z
MAGNETIC PROPERTY
Sign
Unit
Nom.
Min.
Saturation Magnetization
Msat
A/m
9.70e5
9.70e5
Residual Induction
Br
kG
12.3
11.9
mT
1230
1190
Coercive Force
Hci
kOe
11.9
11.3
kA/m
947
899
intrinsic Coercive Force
Hci
kOe
19
17
kA/m
1512
1353
Max. Energy Product
(BH)max
MGOe
35
33
kJ/m3
278
263
MAGNETIC PROPERTY
Sign
Unit
Nom.
Min.
Saturation Magnetization
Msat
A/m
9.70e5
9.70e5
Residual Induction
Br
kG
12.3
11.9
mT
1230
1190
Coercive Force
Hci
kOe
11.9
11.3
kA/m
947
899
intrinsic Coercive Force
Hci
kOe
19
17
kA/m
1512
1353
Max. Energy Product
(BH)max
MGOe
35
33
kJ/m3
278
263
M b 2 y 2 z 2
0 s tan 1
2
ab
In the last step tan1 (x) tan1 ( 1 x) 2 . This is the field due to the top surface. A similar analysis appliesto bottom surface with z replaced by z+L.
The total field is given by
M (z L) b2 y 2 (z L)2 z a 2 b2 c2 .
B (z) 0 s tan 1 tan 1
z
(18)
ab
ab
Here, Ms is the saturation magnetization of magnet. This is required equation to find out magnetic flux density outside the permanent magnet along zaxis.


Comparison of results of charge model with COMSOL:
The results of equation (18) are compared with COMSOL. The permanent magnet properties and dimensions are given below.

Magnet Properties:
Rare earth permanent magnet is used as source for damping. The damping coefficient is directly square proportion to the magnitude of flux density. The comparison of rareearth permanent magnet materials are shown in Table1.
Table 2: Properties of NdFeB Magnet

Magnet dimensions:
HEIGHT
L
0.012m
LENGTH
b
0.025m
WIDTH
a
0.0125m
Figure 2: Rectangular bar magnet, (a) physical magnet dimensions


Results and Conclusion:
Permanent magnet, homogeneously magnetized in known direction is applied to magnet. Method that is used for magnetic field determination is based on superposition of results that are obtained for elementary magnetic dipoles. The tables with magnetic field
values, in different points, in vicinity of permanent magnet, are shown. Results obtained by analytical method are satisfactory confirmed using program packet COMSOL MULTIPHYSICS.
DISTA NCE
MAGNET IZATION
(Msat) in (A/m)
MAGNETIC FIELD DENSITY(Bz) in TESLA
Z
Br
COMSOL
CHARGE MODEL
%ER ROR
5
970000
0.24688
0.2461638
0.29
10
970000
0.1676
0.1678099
0.13
15
970000
0.11629
0.1135211
2.38
20
970000
0.07777
0.078147
0.48
25
970000
0.05732
0.0551104
3.85
30
970000
0.03953
0.0398413
0.79
35
970000
0.02958
0.0294894
0.31
40
970000
0.02282
0.0223052
2.26
50
970000
0.01354
0.0135072
0.24
100
970000
0.00228
0.0023047
1.08
Table 3: Analytical and COMSOL result comparison of magnetic flux density
DISTAN CE
MAGNETIC FIELD INTENSITY(H)
in A/m
Z
COMSO L
CHARGE MODEL
%ERRO R
5
196460
195990.3
0.24
10
133371
133606.6
0.18
15
92543.83
90383
2.33
20
61890.9
62218.97
0.53
25
45615.69
43877.7
3.81
30
31454.79
31720.81
0.85
35
23537.94
23478.86
0.25
40
18155.6
17758.9
2.19
50
10775.14
10754.16
0.19

References

Edward P. Furlani, Permanent Magnet and Electromechanical Devices Materials, Analysis, and Applications ELSEVIER PUBLICATION, 2001.

Ana N. Mladenovi, Slavoljub R. Aleksi: Methods for magnetic field calculation, 11th International Conference on Electrical Machines, Drives and Power Systems ELMA 2005, Sofia, Bulgaria, 1516 September 2005, Vo.2, pp. 350354.

Ana N. Mladenovi: Toroidal shaped permanent magnet with two air gaps, International PhDSeminar Numerical Field Computation and Optimization in Electrical Engineering, Proceedings of Full Papers, Ohrid, Macedonia, 2025 September 2005, pp.153158.