A new approximate method is developed for calculating three-dimensional magnetic fields in magnetic circuits composed of connected axisymmetric and rectangular regions. Using this new method, fairly accurate solutions can be obtained when the leakage flux from the magnetic circuit is small. In this paper, the new method is explained and then the usefulness of the technique is clarified by comparing calculated and measured flux densities.</p

Funakoshi, H.

Ito, S.

Kawase, Y.

Nakata, Takayoshi

Takahashi, N.

Okayama University Scientific Achievement Repository

PhysicsElectricity & Magnetism fieldsOkayama University Year 1985Finite element analysis of magneticcircuits composed of axisymmetric andrectangular regionsTakayoshi Nakata∗ N. Takahashi† Y. Kawase‡H. Funakoshi∗∗ S. Ito††∗Okayama University†Okayama University‡Okayama University∗∗Okayama University††Fukiage Factory, Fuji Electric Company LimitedThis paper is posted at eScholarship@OUDIR : Okayama University Digital InformationRepository.http://escholarship.lib.okayama-u.ac.jp/electricity and magnetism/67IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-21, NO. 6, NOVEMBER 1985 2199 FINITE  ELEMENT  ANALYSIS OF MAGNETIC  CIRCUITS  COMPOSED  OF AXISYMMETRIC  AND  RECTANGULAR  REGIONS T.Nakata,  N.Takahashi,  Y.Kawase,  H.Funakoshi and S.Ito* ABSTRACT A new approximate  method is developed  for calculating  three-dimensional  magnetic  fields  inmagnetic circuits composed of connected axisymmetric and  rectangular regions.  Using  this  new  method,  fairly accurate solutions can be obtained when the leakage flux  from  the  magnetic  circuit s small. In  this paper, the  new method is explained  and then the usefulness  of  the  technique is clarified by comparing  calculated  and  measured  flux  densities. 1. INTRODUCTION -._._I__ I__.-I_ Magnetic  circuits are often  composed  of  regions  of axisymmetric shape and others of rectangular shape. Though  three-dimensional  analysis is required  for  such magnetic  circuits,  the  magnetic  fields  have  often  been analyzed by the axisymmetric or the two-dimensional. method  to  reduce  computer  storage  and  computing  time. A  more  accurate  approximate  method  for  analyzing such  magnetic  circuits  has  been  developed by modifying and  combining  the  axisymmetric  and  the  two-dimensional finite  element methods. 2. METHOD OF ANALYSIS -. ..--_I _____ In this  Section,  first  the  new  method  is  outlined. Next, a new vector potential is introduced in order to satisfy the continuity of flux on the boundary between the axisymmetric  and the rectangular regions. Finally,  the  condition  for  the  continuity  of  flux is derived,  and  the  Rayleigh-Ritz  equations are given. 2.1 Outline of  the Method -----l-l_ I___..-- The  new method  can be illustrated  by  an  example shown  in Fig.1. Figure l(a) shows the  magnetic circuit,  composed of a  limb  with  circular  section a d a yoke  with  rectangular section. A  coil is wound  around the limb.  Figure  l(b)  shows  the  cross-section  of  the region to be  analyzed. Our  method is derived by combining  and  modifying the usual axisymmetric  and  the  two-dimensional  finite element  methods,  inorder to calculate three-dimensional magnetic fields within almost the same computing  time  and  program  size as the two-dimensional or the axisymmetric finite element method.  An axisymmetric finite element method is applied to the region a-b-c-d-a, where the flux distribution is assumed to be axisymmetric, and a two-dimensional  finite  element  method is applied  to  the region d-c-e-f-d, where the flux distribution is assumed to be two-dimensional. If both methods are simply  combined, the flux  is  not  always  continuous  on the  boundary  between  two  regions.  Anew  vector potential is introduced to satisfy the condition of continuity of flux. If the leakage flux from the magnetic  circuit is small  due  to  the high  permeability of  the  steel,  fairly  accurate  solutions  can be obtained by  the  new  method. The  authors are with  the  Department  of  Electrical The author* is With the Fukiage Factory, Fuji Engineering,  Okayama  University,  Okayama 700, Japan. Electric  Company  Limited,  Saitama 369-01; Japan. X (a) shape of model axisymmetric  two-d mensional (bl analyzed  region Fig.1 Analyzed  model. If  the  vector  potential  on the boundary  b-a-f-e  in Fig.l(b) is assumed to be  zero,  the  vector  potential AR defined by the  following  equation  corresponds to  the flux  per  radian  in  the  limb t11. A R  = r A e ,  (1) where A g . i s  the  circumferential  component  of  the  vector potential in the axisymmetric field and r is the radius. The total  flux t~~~ within  the  radius  R in the limb  can  be  written,  using  the  vector  potential AR, by the  following  equation [ l l ,  * r t =  2 EAR . ( 2 )  The vector potential Az in the two-dimensional field corresponds to the flux in the yoke per unit length  in the z-direction. The total  flux Q x y  in  the yoke  can be written,  using  the  vector  potential Az, by * M Y =  t o A z ,  (3) where t o  is  the  thickness of the yoke. The following relationship  between Q r z  and px,, can  be  obtained  from the  continuity  of  flux, c P r a / 2 = 6 x ~ .  ( 4 )  By substituting ( 2 )  and (3) into ( 4 ) ,  the following equation  can  be  obtained, n A R  =toAz, ( 5 )  When AR and Az are used  for the analysis of the respective  regions, AR should be  equal to  Az  on the boundary.  However,  Eq.(5) indicates  hat  thecontinuity  of  flux  cannot  be  satisfied,  when Am is  set to be  equal to Az. Therefore,a new  vector  potential  A which corresponds to the flux in a thickness t in the z-direction, is introduced  in  therectangular regionf21.  A is defined by the equation, 0018-9464/85/1100-2199$01.0001985 IEEE 2200 where Bx and By  are  the x- and  y-components of the flux density in the rectangular region, respectively. The total  flux $.xy in the yoke  can  be  written,  using  the new vector  potential  A,  by $ x y = A  t o / t  . ( 7 )  From Eqs.(2), (4) and (7), the  following  relationship is  obtained: n’AR = A t o / t  . (8) As AR and  A  denote  the  fluxes in adjoining  magnetic regions, AR and  A  should be equal on the  boundary 9-d between the two regions. Therefore, the. thickness t which  satisfies the continuity of flux  can  be  derived from (8) as follows: t = t o / n  . ( 9 )  From Eqs.(7) and (9 ) ,  Q x y  is finally  written as $ x y =  P A .  Continuity  of  flux c-g in the air. AR and c-g . _.___..____I. 2.3 Formulations Poisson’s equation field can be  written as (1.0) is also assumed on the  boundary A are equal on the boundary for the axisymmetric magnetic follows [ l l :  where v and J o e  denote the reluctivity and the circumferential  component  of  the  current  density, respectively. Using the new vector potential A, Poisson’s equation  for  the  two-dimensional  magnetic  field  can be written as follows: where, Joz is the z-component of the  current  density. 2.4 Rayleigh-Ritz  Equations __...____._._.______.-I_- The Rayleigh-Ritz equations can be obtained by minimizing  the  total  energy X as follows: where Ai  is  the  vector  potential at node i, and  n is the number of nodes at which  the  vector  potentials  are unknown. Ai corresponds to AR in the axisymmetric region’and to A in the  rectangular  region. x is given by X =  X,,+ X x y  (14) X r z  is  the  energy  corresponding to Eq.(ll) and is given by where Srz denotes  the  axisymmetric  region. x, is the energy corresponding to Eq. (121, in which  the  thickness  t  is  replaced by Eq.(9), and  is given by where Sxy denotes the rectangular region. The flux densities Brz and  Bxy in the  respective  regions  can  be represented  by 2 n a A  2 Bxy= ( - - ) + (- dA )* t o  a y  t o  a x  . _____.__ 3. FACTORS AFFECTING THE ACCURACY The effects of the position of the boundary between  the  axisymmetric  and  the  rectangular  regions, and  the  shape of the  magnetic  circuit on the  accuracy of the  calculated  results  are  investigated. 3.1 Position of the  boundary __ - __ - ___ -___I -The flux distribution in the magnetic circuit shown in Fig.1 has been  analyzed. The radius of the limb and radius of the coil are 19 and 39(mm), respectively. The thickness of the yoke is 80(mm). The  boundary  positions  investigated  are  denoted  by  the broken  lines  in Fig.2. Figure 3 shows the  effects of the  position R of the  boundary on the average  flux  densities  Bbh  and  Bie of the  limb  and  the yoke derlotcd in Fig.2. The  ampere turn  Ixn  of  the coil is 0.25~850 and lx850(AT). The solid lines denote the calculated results and the broken  lines denote the measured  ones.  The  shape  of the  yoke  insidelthe  boundary is different  from  the  true one.  As  the  percentage of such  a  part  of  the  yoke is small compared with the total magnetic circuit, the accuracies  of  Bbh  and Bie are not affected  much by the position R. position R=O voke 1-4 Fig.2 Position of the  boundary. position R(mm) position R (mm) (a) limb (b) yoke Fig.3 Effects of the  position  of  the  boundary on  the  flux  densities. Figure 4 shows the x-component Bx of the flux density at the  point  which is x(mm)  from  the  center line as denoted ,in Fig.l(b). Bx is increased  with x during x<l6(mm). When x,l6(mm), the increase of the cross-sectional  area  becomes  larger  than  that of the amount of the flux. Therefore, Bx has the  peak  value near x=16(mm). In this model, the cross-sectional areas of the  axisymmetric  and  rectangular  regions on the boundary are equal to each other when R is 25.46(mm). The assumed sections of the limb at the positions  R=19, 25.46,  60(mm), for  examples,  are  shown in Fig.5. Although  the  continuity  of  the  flux at the boundary is considered as denoted in 2.2, the flux density  Bx is not continuous  at  the  boundary  when R is not equal  to 25.46(mm) as  shown  in Fig.4. Therefore, the boundary should be set on the position where respective  cross-sectional areas of the  boundaries of both  regions  are  the  same. 2201 different  from  the  experimental  ones.  This is because the new method  can  take  account of the cross-sectional areas of the  limb  and the yoke; 0- 19 39 60 101 X (mm) (b) R 2 3 9  Fig.4 Effects of the  position  R  on the  flux  distribution. axisymmetric region (a) R = 1 9  (b)  R=25.46  (c) R=60 Fig.5 Assumed sections of theaxisymmetric and rectangular  regions at each  position of the  boundary. 3.2 Shape of the  magnetic  circuit -. ._ - . .- . . . . - .- - The dimension of the magnetic circuit shown in Fig. 1 is determined  by  the  radius  r  of  the  limb,  and the  length L, the  width D, the  thickness t o  and the height H of the yoke. When,  the  conventional two-dimensional  method is applied to the  whole  region, the  rror  due  to the  radius of the  limb is considerable,  because  the  whole  region  is  assumed to be rectangular. If the magnetic field of this magnetic circuit is analyzed using the axisymetric or the two-dimensional method, there  are  some  errors in the calculated  flux densities because  such  a  method assumes that  the  whole  region  is  axisymmetric  or  rectangular. The error  of  the  cross-sectional  area  of  the  yoke is mainly due to  the  radius  r  of  the  limb,  and  the  length L and the thickness t o  of the yoke. Therefore, the effects  of r, L and t o  on the  flux  densities  Bbh  and Bie  are  analyzed.  The  current in the  coil  is 1(A), and the boundary is at R=25.46(m). Figures 6, 7 and 8 show  the  effects  of  the radius r of the  limb  and  the  length L and  the  thickness t o  of the  yoke on Bbh  and  Bie.  Results  using  the nc?w n i c C h o d  are  denoted  by "A". "0" and " x "  denote  flux  densities calculated  using the axisymmetric  and  thetwo-dimensional  method,  respectively.  Experimental results are denoted by " 0 " .  The results calculated using the new method show good agreement with the experimental ones for almost every r, L and t o .  The results calculated using the axisymmetric and the two-dimensional  methods,  however, are very  much E 21- radius  r (mm) radius r (mm) (a) limb  (b)  yoke A : new  method 0 : axisymmetric  method  calculated X : two-dimensional  method 0 : measured Fig.6 Effects  of  the radius r of the  limb on the  flux  densities (I=lA). length L (mm) (b)  yoke Fig.7  Effects  of  the  length L of the  yoke on the  flux  densities (I=lA). W 0 i r - - - - - - -  t 200 thickness to (mm)  thickness to (mm) (a)  limb ( b )  yoke Fig.8 Effccts of the Lhi.ckrlc?ss to of the  yokc on the  flux,  densities (I=lA). 2202 4-.- APPLICATION TO A  MAGNETIZER Figure 9 shows one quadrant of an analyzed magnetizer. The permanent  magnet to be  magnetized  is set  between the two  pole  pieces.  The  pole  piece  and the yoke are  made  of steel. The number  of  turns  of  the coil  is 840. The axisymmetric  finite  element  method  is  applied to the region b-a-f-e-d-c-b, and the two-dimensional finite  lement  method is applied  to  the  region c-d-e-f-g-h-c. The boundary e-f is chosen as the position  where  the  cross-sectional  areas of the boundaries of the axisymmetric and the rectangular regions are the same. Figure 10 shows the  flux  distributions.  The  solid line  denofes  the  flux  line in the  axisymmetric  region and the broken  line in the rectangular  region. Figure 11 shows the  calculated  and  the  measured flux  densities. R j  and  Rih denote the  flux dr?nr;jty at the  point j and  the average flux  density on the line i-h in Fig.9. As the leakage flux is distributed axisymmetrically,  the  results  calculated  using  the new method show good  agreement  with  the  experimental  ones. The new method enables u s  to analyze magnetic circuits  composed  of connected  axisymmetric  and rectangular  regions  requiring only a  small increase of computer storage and  computing  time.  The  accuracy  of the method depends on the position of the boundary between the two regions,  the  permeability, etc. A  more detailed  investigation  of the  accuracy  will  be reported  later. It  is also possible  toanalyze  the  flux distributions in magnetic circuits composed of more than two kinds of  rectangular  regions  with  different thickness [4]. The method will be improved so that magnetic  ircuits  with  more  than  two kinds of axisymmetric regions can be analyzed. pole  piece  yoke a ?+ U Boundary  conditions b-h  :Neumann  boundary b-a-g-h:uirichlet  boundary Fig.9  Model  of  a  magnetizer. O . " " " ' *  20 40 60 exciting  current I (A) (a)  flux  density Bj at  the  pole tip exciting  current  I ( A )  (b) flux  density  Bih - } calculated 2 } measured Fig.11 Excitation  characteristics. - _ _  X REFEREN-Cz T.Nakata  and  N.Takahashi:  "Finite  Element  Method in Electrical Engineering" (book, in Japanese), Morikita  Shuppan,  Tokyo, 1982. T.Nakata  et  al:  "Interdisciplinary  Finite  Element Analysis" (book), Cornel1  University,  Ithaca, 1981. T.Nakata,  Y.Kawase,  H.Funakoshi and S.Ito: "Finite Element  Analysis of a  Magnetic  Circuit  Composed of Axisymmetric  and  Cartesian  Coordinates",  Papers of  Combined  Technical  Meeting  onRotating Machines  and  Static  ApFaratus,  RM-83-39, SA-83-29, IEE, Japan, 1983. T.Nakata,  Y.Kawase,  H.Funakoshi  andS.Ito: "Improvement of the  Hybrid  Finite  Element  Method and its Application",  Papers of Combined  Technical Meeting on Rotating  Machines  and  Static Apparatus,  RM-84-22,  SA-84-6,  IEE,  Japan,  1984. Fig.10  Flux  distribution (1=20A, G=lOmm). 

Finite element analysis of magnetic circuits composed of axisymmetric and rectangular regions

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