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Estimation of non-linear effective permeability of magnetic materials with fine structure
ARTICLE IN PRESS
Physica B 372 (2006) 383–387 www.elsevier.com/locate/physb
Estimation of non-linear effective permeability of magnetic materials with fine structure H. Waki, H. Igarashi, T. Honma Division of System Science and Informatics, Graduate School of Information Science and Technology, Hokkaido University, Kita 14, Nishi 9, Kita-ku, Sapporo 060-0814, Japan
Abstract This paper describes a homogenization method for magnetic materials with fine structure. In this method, the structures of the magnetic materials are assumed to be periodic, and the unit cell is defined. The effective permeability is determined on the basis of magnetic energy balance in the unit cell. This method can be applied not only for linear problems but also for non-linear ones. In this paper, estimation of the effective permeability of non-linear magnetic materials by using the homogenization method is described in detail, and then the validity for the non-liner problems is tested for two-dimensional problems. It is shown that this homogenization method gives accurate non-linear effective permeability. r 2005 Elsevier B.V. All rights reserved. PACS: 41.20.Gz Keywords: Homogenization; Effective permeability; Non-linear analysis; Magnetic shielding
1. Introduction Magnetic composite materials such as concrete mixed with iron ore grain, synthetic resins including ferrite particles and so on, have been used for electromagnetic shielding in recent years. These materials will work as shields against magnetostatic field, for e.g., terrestrial magnetism and magnetic fields generated by feeding currents of railways. The magnetic shielding effects can be numerically analyzed by using the finite element method (FEM), etc. However, the analyses are time-consuming, because the composite materials have fine structure. Simplification of the structure by homogenization makes it possible to analyze them efficiently [1]. A lot of homogenizing rules have been introduced in order to estimate the effective material parameters of the composite materials and mixtures so far [2]. The homogenization is also studied in fields of remote sensing [3,4]. The most well-known rules might be the Maxwell–Garnett formula [3,5] and the Bruggeman formula [3,6]. In recent Corresponding author. Tel.: +81 11 706 7692; fax: +81 11 709 6280.
E-mail address: [email protected] (H. Waki). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.091
years, Sareni et al. [7] have introduced a homogenization method in which the effective permittivity is obtained on the basis of potential. The authors have introduced a homogenization method to estimate the effective permeability of linear magnetic materials with fine structure [8]. This method has possibility to apply for any periodic structure [8,9], and the effective permeability is more accurate than conventional methods. The homogenization method can be applied not only for linear problems but also for nonlinear problems [8]. However, our previous reports suggested just applicability of the homogenization method for non-linear materials. In this paper, a non-linear analysis based on the homogenization method is discussed in detail, and then the validity for non-linear problems is tested for a twodimensional (2D) numerical example. 2. Non-linear effective permeability Fig. 1 shows a 2D model of the composite material, which includes cylindrical steel rods with isotropic magnetism. In the homogenization method, the structure of the
ARTICLE IN PRESS H. Waki et al. / Physica B 372 (2006) 383–387
384
1
Magnetic substance
Γorig
1
Γhomo
Steel rod 4
Γhomo 4
∝m µ
Γorig µ0
Sorig
Air Original
Periodical structure
∝effeff µ 2 Γorig
3
Γorig
Unit cell
Sorig
2
Shomo Γhomo = Shomo
Original cell
3
Γhomo
Homogenized cell
Fig. 1. Definition of unit cell. Fig. 2. Homogenization of cell.
magnetic composite materials is assumed to be periodic, and the unit cell, which is the minimum volume to represent the overall statistics, is defined. Here, it is assumed that one cylinder is located in the center of the cell, and the cell has also structural isotropy. The unit cell is regarded as a homogeneous magnetic substance with the effective permeability as shown in Fig. 2. The effective permeability is defined on the basis of magnetic energy balance in the unit cell: it is assumed that the original cell and the homogenized cell include equivalent magnetic energy when both unit cells are immersed in equivalent magnetic field. In an actual estimation, the solution of the Laplace equation is computed by FEM. In this analysis, vector potentials are unknowns, and assuming the applied field is unidirectional at least in the cell, say, in the x-direction, the boundary conditions are set as A ¼ A1
on
G1orig ; G1homo ,
(1)
A ¼ A3
on
G3orig ; G3homo ,
(2)
qA ¼ 0 on G2orig ; G4orig ; G2homo and G4homo , (3) qn where A represents the vector potential, n the normal vector to Gtorig or Gthomo . The boundary conditions are equally set to both the cells, and the applied field Bap is determined, which is equal to the magnetic flux density in the homogenized cell Bhomo. These vectors have only xcomponents. Thus, for simplicity, the index x will be omitted hereafter. When the homogenization method is applied to non-linear magnetic materials, the magnetic field in the unit cell is evaluated by the Newton–Raphson technique. The magnetic energy in the original cell is represented as follows: Z Z U orig ðBap Þ ¼ Horig dBorig dSorig , (4) S orig
Borig
where Borig represents the magnetic flux density, Horig the magnetic field, Sorig the area of the original cell. The magnetic energy in the homogenized cell is represented as follows: Z U homo ðBap Þ ¼ S homo H homo dBhomo , (5) Bhomo
where Hhomo represents the magnetic field, Shomo the area of the homogenized cell. In the homogenization method, it is assumed that the magnetic energy in both cells is equivalent. Therefore a relation of the energy balance U orig ðBap Þ ¼ U homo ðBap Þ
(6)
is organized. For the linear problems, from Eqs. (4), (5) and (6), the effective permeability is given as follows: R ðBhomo Þ2 dS homo S meff ¼ R homo , (7) S orig Borig Horig dS orig which holds for any applied field [8]. For the non-linear problem, the applied field is increased in the following form: ( 0 ðj ¼ 0Þ j Bap ¼ . (8) j1 Bap þ DBap ðj ¼ 1; 2; . . .Þ When the jth magnetic field Bjap , which is equal to Bjhomo , is applied to the cells, the magnetic energy in the homogenized cell is equal to U jhomo as shown in Fig. 3, which is equal to summation of the magnetic energy DU jhomo . Thus, the jth magnetic energy in the homogenized cell is represented as follows: U jhomo ¼
X H j1 þ H j homo homo DBjhomo S homo . 2 j
(9)
The magnetic energy in the original cell is also obtained by this approach as follows: U jorig ¼
X X He;origj1 þ He;origj e
j
2
DBe;origj ¼ Be;origj Be;origj1 ,
DBe;origj S eorig ,
(10)
(11)
where e represents the element in the original cell. From Eqs. (6), (8–11), the magnetic field in the homogenized cell is given by P e;j1 e;j e;j e e ðHorig þ Horig Þ DBorig S orig j H j1 (12) H homo ¼ homo . DBjhomo S homo The B–H property of the homogenized substance can be obtained from Eq. (12), and then the effective permeability
ARTICLE IN PRESS H. Waki et al. / Physica B 372 (2006) 383–387
cell is set as shown in Fig. 5. The homogenized substance is also immersed in the magnetic field as well as the original sample material. In these examples, the magnetic fields are analyzed by using FEM, and then the errors of those magnetic fields in
j
j-1 Bhomo
j
∆Bhomo jj-11 U Uhomo homo
Steel rod (Magnetic substance m)
0.018 [m]
Air 0
0.02 [m]
Magnetic flux densityB
∆Uhomo j Bhomo
385
0
j-1 Hhomo
j Hhomo
Sample material
Magnetic field H
0.02 [m] Unit cell
Fig. 5. Unit cell of sample material.
Fig. 3. Estimation of magnetic energy in homogenized cell.
y Applied field Btest [T] Sample material (-5,5)
(5,5)
x
0 .4 O
1
Serror (5,-5)
(-5,-5) The error in this area is evaluated. Unit: [m] Fig. 4. Test problem.
is given as follows: meff ðBjap Þ ¼
DBjhomo DH jhomo
.
(13)
3. Two-dimensional test problem The homogenization method is applied to 2D test problems as shown in Fig. 4 in order to evaluate the validity. In the test problems, a sample material composed of steel rods is immersed in the magnetostatic field Btest. The sample material is homogenized by Eq. (12). The unit
Fig. 6. B–H property. (a) Steel rod, (b) homogenized substance.
ARTICLE IN PRESS H. Waki et al. / Physica B 372 (2006) 383–387
386
the area Serror shown in Fig. 3 are evaluated. The errors are here defined by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , u X 0 uX 0 0 e e0 2 e0 t (14) E¼ jB B j S jBe j2 S e , homo
e0 0
orig
error
orig
error
Eq. (7). The above-mentioned non-linear approach is compared with the linear approximation. 4. Result and discussion
e0 0
where S eerror is the area of the element e0 , Beorig represents the 0 magnetic field in case of the original, Behomo is the magnetic field in case of the homogenized substance. The area Serror has an inner circle. If the radius of this circle is reduced towards the sample material, the error E will increase because of ripples in the magnetic field near the surface of the steel rods [8]. However this paper concerns shielding problems where far fields are important. This is the reason why the region near the steel rods is excluded in the error estimation. In the test problem, the effective permeability based on linear approximated magnetic property is also estimated by
Fig. 7. B–H property of homogenized substance. (a) f ¼ 0:478, (b) f ¼ 0:636.
Fig. 6(a) shows the B–H property of the steel rod. In the linear approach, the permeability is set to 1000m0. The B–H property of the homogenized substance is obtained as shown in Fig. 6(b). Fig. 7 shows those m–H properties. The volume fraction f is equal to the volume ratio of the steel rod to the cell. The differences between the non-linear approach and the linear approximation increase with the applied field. This tendency is intensified for large value of f. Fig. 8 shows the error E versus the applied field Btest. The error obtained by the non-linear approach is held below 2 104. This method gives accurate result being independent of Btest and f. The error obtained by the linear approximation increases with Btest, and this tendency becomes remarkable for large value of f.
Fig. 8. Error E vs. applied field B. (a) f ¼ 0:478 [m]. (b) f ¼ 0:636 [m].
ARTICLE IN PRESS H. Waki et al. / Physica B 372 (2006) 383–387
5. Conclusion The present method has been applied for the non-linear magnetic materials with fine structure. This method gives accurate non-linear effective permeability being independent of the applied field. Moreover, the linear approximation yields remarkable errors for large value of the volume fraction f while the present non-linear approach gives accurate results under such conditions. References [1] H. Waki, H. Igarashi, T. Honma, Comput. Eng. I, JASCOME (2004) 117.
387
[2] E. Tuncer, Y.V. Serdyuk, S.M. Gubanski, IEEE Trans. Dielectr. Electr. Insul. 9 (5) (2002) 809. [3] A. Sihvola, IEE Electromagn. Waves Ser. 47 (1999) 63. [4] A.H. Sihvola, J.A. Kong, IEEE Trans. Geosci. Remote Sensing 26 (4) (1988) 420. [5] J.C.M. Garnett, Trans. Roy. Soc. 53 (1904) 385. [6] D.A.G. Bruggeman, Ann. Phys. 5. Folge, Band 24 (1935) 636. [7] B. Sareni, L. Krahenbuhl, A. Beroual, A. Nicolas, C. Brosseau, IEEE Trans. Magn. 33 (1997) 1580. [8] H. Waki, H. Igarashi, T. Honma, IEEE Trans. Magn. 41 (5) (2005) 1520. [9] H. Waki, H. Igarashi, T. Honma, COMPEL 24 (2) (2005) 566.
Physica B 372 (2006) 383–387 www.elsevier.com/locate/physb
Estimation of non-linear effective permeability of magnetic materials with fine structure H. Waki, H. Igarashi, T. Honma Division of System Science and Informatics, Graduate School of Information Science and Technology, Hokkaido University, Kita 14, Nishi 9, Kita-ku, Sapporo 060-0814, Japan
Abstract This paper describes a homogenization method for magnetic materials with fine structure. In this method, the structures of the magnetic materials are assumed to be periodic, and the unit cell is defined. The effective permeability is determined on the basis of magnetic energy balance in the unit cell. This method can be applied not only for linear problems but also for non-linear ones. In this paper, estimation of the effective permeability of non-linear magnetic materials by using the homogenization method is described in detail, and then the validity for the non-liner problems is tested for two-dimensional problems. It is shown that this homogenization method gives accurate non-linear effective permeability. r 2005 Elsevier B.V. All rights reserved. PACS: 41.20.Gz Keywords: Homogenization; Effective permeability; Non-linear analysis; Magnetic shielding
1. Introduction Magnetic composite materials such as concrete mixed with iron ore grain, synthetic resins including ferrite particles and so on, have been used for electromagnetic shielding in recent years. These materials will work as shields against magnetostatic field, for e.g., terrestrial magnetism and magnetic fields generated by feeding currents of railways. The magnetic shielding effects can be numerically analyzed by using the finite element method (FEM), etc. However, the analyses are time-consuming, because the composite materials have fine structure. Simplification of the structure by homogenization makes it possible to analyze them efficiently [1]. A lot of homogenizing rules have been introduced in order to estimate the effective material parameters of the composite materials and mixtures so far [2]. The homogenization is also studied in fields of remote sensing [3,4]. The most well-known rules might be the Maxwell–Garnett formula [3,5] and the Bruggeman formula [3,6]. In recent Corresponding author. Tel.: +81 11 706 7692; fax: +81 11 709 6280.
E-mail address: [email protected] (H. Waki). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.091
years, Sareni et al. [7] have introduced a homogenization method in which the effective permittivity is obtained on the basis of potential. The authors have introduced a homogenization method to estimate the effective permeability of linear magnetic materials with fine structure [8]. This method has possibility to apply for any periodic structure [8,9], and the effective permeability is more accurate than conventional methods. The homogenization method can be applied not only for linear problems but also for nonlinear problems [8]. However, our previous reports suggested just applicability of the homogenization method for non-linear materials. In this paper, a non-linear analysis based on the homogenization method is discussed in detail, and then the validity for non-linear problems is tested for a twodimensional (2D) numerical example. 2. Non-linear effective permeability Fig. 1 shows a 2D model of the composite material, which includes cylindrical steel rods with isotropic magnetism. In the homogenization method, the structure of the
ARTICLE IN PRESS H. Waki et al. / Physica B 372 (2006) 383–387
384
1
Magnetic substance
Γorig
1
Γhomo
Steel rod 4
Γhomo 4
∝m µ
Γorig µ0
Sorig
Air Original
Periodical structure
∝effeff µ 2 Γorig
3
Γorig
Unit cell
Sorig
2
Shomo Γhomo = Shomo
Original cell
3
Γhomo
Homogenized cell
Fig. 1. Definition of unit cell. Fig. 2. Homogenization of cell.
magnetic composite materials is assumed to be periodic, and the unit cell, which is the minimum volume to represent the overall statistics, is defined. Here, it is assumed that one cylinder is located in the center of the cell, and the cell has also structural isotropy. The unit cell is regarded as a homogeneous magnetic substance with the effective permeability as shown in Fig. 2. The effective permeability is defined on the basis of magnetic energy balance in the unit cell: it is assumed that the original cell and the homogenized cell include equivalent magnetic energy when both unit cells are immersed in equivalent magnetic field. In an actual estimation, the solution of the Laplace equation is computed by FEM. In this analysis, vector potentials are unknowns, and assuming the applied field is unidirectional at least in the cell, say, in the x-direction, the boundary conditions are set as A ¼ A1
on
G1orig ; G1homo ,
(1)
A ¼ A3
on
G3orig ; G3homo ,
(2)
qA ¼ 0 on G2orig ; G4orig ; G2homo and G4homo , (3) qn where A represents the vector potential, n the normal vector to Gtorig or Gthomo . The boundary conditions are equally set to both the cells, and the applied field Bap is determined, which is equal to the magnetic flux density in the homogenized cell Bhomo. These vectors have only xcomponents. Thus, for simplicity, the index x will be omitted hereafter. When the homogenization method is applied to non-linear magnetic materials, the magnetic field in the unit cell is evaluated by the Newton–Raphson technique. The magnetic energy in the original cell is represented as follows: Z Z U orig ðBap Þ ¼ Horig dBorig dSorig , (4) S orig
Borig
where Borig represents the magnetic flux density, Horig the magnetic field, Sorig the area of the original cell. The magnetic energy in the homogenized cell is represented as follows: Z U homo ðBap Þ ¼ S homo H homo dBhomo , (5) Bhomo
where Hhomo represents the magnetic field, Shomo the area of the homogenized cell. In the homogenization method, it is assumed that the magnetic energy in both cells is equivalent. Therefore a relation of the energy balance U orig ðBap Þ ¼ U homo ðBap Þ
(6)
is organized. For the linear problems, from Eqs. (4), (5) and (6), the effective permeability is given as follows: R ðBhomo Þ2 dS homo S meff ¼ R homo , (7) S orig Borig Horig dS orig which holds for any applied field [8]. For the non-linear problem, the applied field is increased in the following form: ( 0 ðj ¼ 0Þ j Bap ¼ . (8) j1 Bap þ DBap ðj ¼ 1; 2; . . .Þ When the jth magnetic field Bjap , which is equal to Bjhomo , is applied to the cells, the magnetic energy in the homogenized cell is equal to U jhomo as shown in Fig. 3, which is equal to summation of the magnetic energy DU jhomo . Thus, the jth magnetic energy in the homogenized cell is represented as follows: U jhomo ¼
X H j1 þ H j homo homo DBjhomo S homo . 2 j
(9)
The magnetic energy in the original cell is also obtained by this approach as follows: U jorig ¼
X X He;origj1 þ He;origj e
j
2
DBe;origj ¼ Be;origj Be;origj1 ,
DBe;origj S eorig ,
(10)
(11)
where e represents the element in the original cell. From Eqs. (6), (8–11), the magnetic field in the homogenized cell is given by P e;j1 e;j e;j e e ðHorig þ Horig Þ DBorig S orig j H j1 (12) H homo ¼ homo . DBjhomo S homo The B–H property of the homogenized substance can be obtained from Eq. (12), and then the effective permeability
ARTICLE IN PRESS H. Waki et al. / Physica B 372 (2006) 383–387
cell is set as shown in Fig. 5. The homogenized substance is also immersed in the magnetic field as well as the original sample material. In these examples, the magnetic fields are analyzed by using FEM, and then the errors of those magnetic fields in
j
j-1 Bhomo
j
∆Bhomo jj-11 U Uhomo homo
Steel rod (Magnetic substance m)
0.018 [m]
Air 0
0.02 [m]
Magnetic flux densityB
∆Uhomo j Bhomo
385
0
j-1 Hhomo
j Hhomo
Sample material
Magnetic field H
0.02 [m] Unit cell
Fig. 5. Unit cell of sample material.
Fig. 3. Estimation of magnetic energy in homogenized cell.
y Applied field Btest [T] Sample material (-5,5)
(5,5)
x
0 .4 O
1
Serror (5,-5)
(-5,-5) The error in this area is evaluated. Unit: [m] Fig. 4. Test problem.
is given as follows: meff ðBjap Þ ¼
DBjhomo DH jhomo
.
(13)
3. Two-dimensional test problem The homogenization method is applied to 2D test problems as shown in Fig. 4 in order to evaluate the validity. In the test problems, a sample material composed of steel rods is immersed in the magnetostatic field Btest. The sample material is homogenized by Eq. (12). The unit
Fig. 6. B–H property. (a) Steel rod, (b) homogenized substance.
ARTICLE IN PRESS H. Waki et al. / Physica B 372 (2006) 383–387
386
the area Serror shown in Fig. 3 are evaluated. The errors are here defined by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , u X 0 uX 0 0 e e0 2 e0 t (14) E¼ jB B j S jBe j2 S e , homo
e0 0
orig
error
orig
error
Eq. (7). The above-mentioned non-linear approach is compared with the linear approximation. 4. Result and discussion
e0 0
where S eerror is the area of the element e0 , Beorig represents the 0 magnetic field in case of the original, Behomo is the magnetic field in case of the homogenized substance. The area Serror has an inner circle. If the radius of this circle is reduced towards the sample material, the error E will increase because of ripples in the magnetic field near the surface of the steel rods [8]. However this paper concerns shielding problems where far fields are important. This is the reason why the region near the steel rods is excluded in the error estimation. In the test problem, the effective permeability based on linear approximated magnetic property is also estimated by
Fig. 7. B–H property of homogenized substance. (a) f ¼ 0:478, (b) f ¼ 0:636.
Fig. 6(a) shows the B–H property of the steel rod. In the linear approach, the permeability is set to 1000m0. The B–H property of the homogenized substance is obtained as shown in Fig. 6(b). Fig. 7 shows those m–H properties. The volume fraction f is equal to the volume ratio of the steel rod to the cell. The differences between the non-linear approach and the linear approximation increase with the applied field. This tendency is intensified for large value of f. Fig. 8 shows the error E versus the applied field Btest. The error obtained by the non-linear approach is held below 2 104. This method gives accurate result being independent of Btest and f. The error obtained by the linear approximation increases with Btest, and this tendency becomes remarkable for large value of f.
Fig. 8. Error E vs. applied field B. (a) f ¼ 0:478 [m]. (b) f ¼ 0:636 [m].
ARTICLE IN PRESS H. Waki et al. / Physica B 372 (2006) 383–387
5. Conclusion The present method has been applied for the non-linear magnetic materials with fine structure. This method gives accurate non-linear effective permeability being independent of the applied field. Moreover, the linear approximation yields remarkable errors for large value of the volume fraction f while the present non-linear approach gives accurate results under such conditions. References [1] H. Waki, H. Igarashi, T. Honma, Comput. Eng. I, JASCOME (2004) 117.
387
[2] E. Tuncer, Y.V. Serdyuk, S.M. Gubanski, IEEE Trans. Dielectr. Electr. Insul. 9 (5) (2002) 809. [3] A. Sihvola, IEE Electromagn. Waves Ser. 47 (1999) 63. [4] A.H. Sihvola, J.A. Kong, IEEE Trans. Geosci. Remote Sensing 26 (4) (1988) 420. [5] J.C.M. Garnett, Trans. Roy. Soc. 53 (1904) 385. [6] D.A.G. Bruggeman, Ann. Phys. 5. Folge, Band 24 (1935) 636. [7] B. Sareni, L. Krahenbuhl, A. Beroual, A. Nicolas, C. Brosseau, IEEE Trans. Magn. 33 (1997) 1580. [8] H. Waki, H. Igarashi, T. Honma, IEEE Trans. Magn. 41 (5) (2005) 1520. [9] H. Waki, H. Igarashi, T. Honma, COMPEL 24 (2) (2005) 566.