Aleksić, Slavoljub R.

Mladenović, Ana N.

Digitale Bibliothek Thüringen

  PROCEEDINGS 11-15 September 2006      FACULTY OF ELECTRICAL ENGINEERING AND INFORMATION SCIENCE    INFORMATION TECHNOLOGY AND ELECTRICAL ENGINEERING - DEVICES AND SYSTEMS, MATERIALS AND TECHNOLOGIES FOR THE FUTURE      Startseite / Index: http://www.db-thueringen.de/servlets/DocumentServlet?id=12391 51. IWK Internationales Wissenschaftliches Kolloquium International Scientific Colloquium Impressum  Herausgeber: Der Rektor der Technischen Universität llmenau  Univ.-Prof. Dr. rer. nat. habil. Peter Scharff  Redaktion: Referat Marketing und Studentische Angelegenheiten  Andrea Schneider   Fakultät für Elektrotechnik und Informationstechnik  Susanne Jakob  Dipl.-Ing. Helge Drumm  Redaktionsschluss: 07. Juli 2006  Technische Realisierung (CD-Rom-Ausgabe):  Institut für Medientechnik an der TU Ilmenau  Dipl.-Ing. Christian Weigel  Dipl.-Ing. Marco Albrecht  Dipl.-Ing. Helge Drumm  Technische Realisierung (Online-Ausgabe):  Universitätsbibliothek Ilmenau    Postfach 10 05 65  98684 Ilmenau  Verlag:   Verlag ISLE, Betriebsstätte des ISLE e.V.  Werner-von-Siemens-Str. 16  98693 llrnenau   © Technische Universität llmenau (Thür.) 2006  Diese Publikationen und alle in ihr enthaltenen Beiträge und Abbildungen sind urheberrechtlich geschützt. Mit Ausnahme der gesetzlich zugelassenen Fälle ist eine Verwertung ohne Einwilligung der Redaktion strafbar.   ISBN (Druckausgabe): 3-938843-15-2 ISBN (CD-Rom-Ausgabe): 3-938843-16-0  Startseite / Index: http://www.db-thueringen.de/servlets/DocumentServlet?id=12391  51st Internationales Wissenschaftliches Kolloquium Technische Universität Ilmenau  September 11 – 15, 2006  A. N. Mladenović, S. R. Aleksić   Magnetic field calculation of permanent magnet    INTRODUCTION   The paper presents calculation the magnetic field of permanent magnet, homoge-neously magnetized in known direction.  Method used in the paper is based on a system of equivalent magnetic dipoles. The results that are obtained using this analytical method are compared with results obtained using program packet FEMLAB. Magnetic field and magnetic flux density distributions of permanent magnet are also shown in the paper.  To determine the magnetic field components in vicinity of permanent magnets, it starts from supposition that magnetization of permanent,M , magnet is known. The following methods are useful in practical calculation: a) Method based on determining distribution of microscopic Ampere's current; b) Method based on Poisson's and Laplace's equations, determining magnetic scalar potential; and c) Method based on a system of equivalent magnetic dipoles.  OUTLINE OF THE METHODS    Magnetic field inside and outside the permanent magnet, if magnetization of permanent magnet is known, can be calculated using equivalent system of volume and surface microscopic Ampere's currents, which are determined as  ( ) ( )'rot'a rMrJ = , and (1)  ( ) ( ) nrMJ ˆ''sa ×=r , (2) where nˆ  is unit vector of outgoing normal (Fig.1). These currents produce magnetic vector potential:  ∫∫ ′′πµ+′′πµ=SVRSRV d)(4d)(4)( sa0a0 rJrJrA , (3) where rr ′−=R . Fig.1 - Permanent magnet  Fig.2 - Elementary magnetic dipole Magnetic flux density is   )(rot)( rArB = . (4) Inside a permanent magnet, magnetic field can be determined using relation  MBH −µ= 0 . (5) Outside a permanent magnet, magnetic field can be determined using relation  0µ= BH . (6)  The second method is based on determining magnetic scalar potential mϕ . Inside a permanent magnet magnetic scalar potential satisfies Poisson's equation  Mdivm =ϕ∆ . (7) Magnetic field vector can be presented as  mgradϕ−=H . (8) Outside a permanent magnet 0=M , because of that magnetic scalar potential, 0mϕ , satisfies Laplace's equation,  00m =ϕ∆ , (9) where   0m0 gradϕ−=H . (10)  The third method that is mentioned in the paper for magnetic field calculation is based on superposition of elementary results obtained for elementary magnetic dipoles. Elementary magnetic dipole (Fig.2) has magnetic moment  V ′= dd Mm .  (11) This magnetic moment produces, at field point P , elementary magnetic scalar potential  VRR 3′π=π=ϕ d41d41d 3mRMmR ,  (12) where rr ′−=R  is distance from the point where the magnetic field is being cal-culated to elementary source, and rrR ′−= . After integration magnetic scalar potential is obtained as  VRR VV′π=π=ϕ ∫∫ d41d41 33m RMmR .  (13)  PROBLEM DEFINITION   Permanent magnet that is observed in the paper is homogeneously magnetized in known direction. It consists of three parts that are made of ferromagnetic material. Two ends are magnetized in different direction and the third part is magnetized in angular direction. Dimensions of the permanent magnet are presented in the Fig.3.  Fig.3 – Permanent magnet  Outside the permanent magnet magnetic scalar potential, at field point ),,P( zyx , can be determined using superposition of results obtained for each magnetized part,  3m2m1mm ϕ+ϕ+ϕ=ϕ ,  (14)  where 1mϕ and 2mϕ  are magnetic scalar potentials of two ends and 3mϕ  is magnetic scalar potentials that originates from the part which is magnetized in angular direction.  These magnetic scalar potentials can be determined using the expression (13), where  ( ) ( ) ( )222 ''' zzyyxxR −+−+−=′−= rr ,and  (15)  ( ) ( ) ( )zyxrrR ˆ'ˆ'ˆ' zzyyxx −+−+−=′−= . (16) The first part of the permanent magnet is homogeneously magnetized in negative direction of x-axes,  xM)−=M . (17) Scalar product is  )( xxM ′−−=RM . (18) Substituting expressions (15) and (18) in (13), magnetic scalar potential 1mϕ  is obtained  ( ) ( ) ( )( ) zyxzzyyxx xxMhhRRl′′′−+−+−′−π−=ϕ ∫ ∫ ∫−ddd'''42221 0232221m. (19) The second part of the permanent magnet is homogeneously magnetized in positive direction of x-axes,  xM)=M . (20) Scalar product is  )( xxM ′−=RM . (21) Substituting expressions (15) and (21) in (13), magnetic scalar potential 2mϕ  is obtained  ( ) ( ) ( )( ) zyxzzyyxx xxMhhRRl′′′−+−+−′−π=ϕ ∫ ∫ ∫−−−ddd'''42212 0232222m. (22) The third part of the permanent magnet is homogeneously magnetized in angular direction  θˆM=M .  (23) Relations between coordinates x , y , z  and cylindrical coordinates θ,r  and z are  θcosrx = , θsinry =  and zz = .  (24) Using these relations, distance from the point where the magnetic field is being calculated to elementary source, can be presented as  22 )'()θ'cos(θ'2' zzrrrrR 2 −+−−+= .  (25) As magnetization has only angular component θ , scalar product RM  is obtained as  ( ) ( ) ( )[ ] θˆˆ'ˆ'ˆ' Mzzyyxx zyxRM −+−+−= .  (26) Relation between unite vectors xˆ , yˆ , zˆ  and θˆ  is  ∂∂+∂∂+∂∂= zyx ˆθˆθˆθ1θˆ zyxr,  (27) and the following expressions are also satisfied  θ1ˆθˆ ∂∂= xrx , θ1ˆθˆ ∂∂= yry , θ1ˆθˆ ∂∂= zrz .  (28) Using relations (17) and (21), the following relations are obtained  θsinˆθˆ −=x , θcosˆθˆ =y , 0ˆθˆ =z ,  (29) and scalar product can be presented as  )'sin(' θ−θ= MrRM .  (30) Substituting expressions (25) and (30) in (13), magnetic scalar potential is obtained as   ( ) ( )[ ]∫ ∫ ∫π−+−−+θ−π=ϕ21220232223m'θ'θcos'2''d'd'd)θ'sin(θ'4RRhh-zzrrrrzrrM . (31) The solutions of the integrals presented in the expressions (19), (22) and (31) are very complex. Because of that the expression for magnetic scalar potential (14) is very large and it won’t be shown in the paper, but it is used for determining the components of magnetic field.  Magnetic field vector can be expressed as  mgradϕ−=H ,  (32) therefore its components are  xH ∂ϕ∂−= mx , yH ∂ϕ∂−= my , zH ∂ϕ∂−= mz . (33)  NUMERICAL RESULTS   Distribution of magnetic flux density outside the permanent magnet, obtained using the analytical method is presented in the Fig.4, for the following dimension of permanent magnet 6,4 21 == hRhR  and 6=hl .   Fig.4 – Distribution of magnetic flux density  Fig.5 – Distribution of magnetic flux density (FEMLAB) Magnetic field lines have the same form and the same direction as magnetic flux density lines, outside the magnet. The Fig.5 presents distribution of magnetic flux density (arrow) and magnetic potential (contour) obtained using program packet FEMLAB. Comparing these figures the excellent agreement between analytical method results and FEMLAB results is evident. Table I Normalized magnetic field values along the direction 0,0 == zy  x l/ H M/0.10.20.30.40.50.60.70.80.91.01.11.21.31.41.50.4361330.2069130.1248550.0810540.0536260.0350400.0218940.0124330.0057110.0011190.0017670.0032190.0037750.0037530.003425  Table   II Normalized magnetic field values along the direction 0,67 == zlx  y l/ H M/0.10.20.30.40.50.60.70.80.91.01.11.21.31.41.50.0038390.0065140.0109710.0183810.0311670.0520690.0745360.0843510.0806230.0637390.0406320.0239960.0149830.0104170.008101           In the Table I and Table II, magnetic field values along characteristic direction, for mentioned dimensions of permanent magnet, are presented.                                             CONCLUSION   Permanent magnet, homogeneously magnetized in known direction is observed in the paper. 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. Magnetic flux density distribution of permanent magnet is also presented in the paper. Magnetic field lines have the same form and the same direction as magnetic flux density lines, outside the magnet. Results obtained by analytical method are satisfactory confirmed using program packet FEMLAB.       References:    [1] Textbook of Electromagnetics, Edited by Hermann Uhlmann (Veličković D. M., Chapter 3.2: Magnetostatic field), Technische Universitaet Ilmenau. [2] A. N. Mladenović, S. R. Aleksić: “Rod shaped permanent magnet”, 48. Conference ETRAN 2004, Čačak, 6th-11th Jun, Proceedings of full papers, pp. 233-236. (In Serbian) [3] A. N. Mladenović, Slavoljub R. Aleksić: Toroidal shaped permanent magnet with air gap, International Conference on Applied Electromagnetics PES2005, Niš 23rd-25th May 2005, Proceedings of Extended Abstracts, pp. 23-24. [4] Ana N. Mladenović, Slavoljub R. Aleksić:” Methods for magnetic field calculation”, 11th International Conference on Electrical Machines, Drives and Power Systems ELMA 2005, Sofia, Bulgaria, 15-16 September 2005, Vol.2, pp. 350-354. [5] Ana N. Mladenović: “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.   Authors:    Ana N. Mladenović   Slavoljub R. Aleksić  Faculty of Electronic Engineering, Aleksandra Medvedeva 14, P.O.B. 73 18000, Niš, Serbia&Montenegro Phone: +381 18 529 423 Fax: +381 18 588 399 E-mail: anam@elfak.ni.ac.yu, as@elfak.ni.ac.yu 

Magnetic field calculation of permanent magnet

https://www.db-thueringen.de/servlets/MCRFileNodeServlet/dbt_derivate_00017776/IWK_2006_3_3_11.pdf

Magnetic field calculation of permanent magnet

Abstract

Similar works

Full text

Available Versions

Digitale Bibliothek Thüringen