Analytical And Finite Element Comparison Of Magnetic Flux Density And Magnetic Field Intensity Of Permanent Magnet

Analytical And Finite Element Comparison Of Magnetic Flux Density And Magnetic Field Intensity Of Permanent Magnet

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 free-space 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

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 '

   1. 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)

   2. 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

   current-free 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 free-space 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(°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(°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

   Table-1: 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 z-axis.

2. 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.

   1. 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 rare-earth permanent magnet materials are shown in Table-1.

      Table 2: Properties of NdFeB Magnet

   2. Magnet dimensions:

      HEIGHT	L	0.012m
      LENGTH	b	0.025m
      WIDTH	a	0.0125m

      Figure 2: Rectangular bar magnet, (a) physical magnet dimensions

3. 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

4. References

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

2. 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, 15-16 September 2005, Vo.2, pp. 350-354.

3. Ana N. Mladenovi: Toroidal shaped permanent magnet with two air gaps, International PhD-Seminar Numerical Field Computation and Optimization in Electrical Engineering, Proceedings of Full Papers, Ohrid, Macedonia, 20-25 September 2005, pp.153-158.