Magnetic field and force evaluation in open boundary axisymmetric structures
Magnetic field and force evaluation in open boundary axisymmetric structures
Magnetic field and force evaluation in open boundary axisymmetric structures
Magnetic field and force evaluation in open boundary axisymmetric structures

Magnetic field and force evaluation in open boundary axisymmetric structures

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  • 8/7/2019 Magnetic field and force evaluation in open boundary axisymmetric structures


    IEEE MELECON 2004, May 12-15, 2004, Dubrovnik, Croatia

    Magnetic Field and Force Evaluation in OpenBoundary Axisymmetric Structures

    A. L opes R ibe i roInstituto Superior TCcnicolDep. of Electrical and Computer Eng ineering, Lisbon, Portugale-mail: arturlrO,ist.utl.Dt

    Abstract-This paper presents a metho d to obtain themagnetic field resulting from currents in coils andpermanent magnets in structures with axial symmetry andopen boundaries. The finite element method (FEM) is usedinside a sphere containing the structures and a Legendrepolynomial expansion is used to represent the field in theoutside region. The polynomial expansion is truncated to anorder equal to the num ber of grid no des along the sphericalboundary resulting from the FEM discretization. Thepolynomial coefficients are determined in order toguarantee the field continuity in those nodes. T he method isthen utilized to preview the interaction force between aSamarium-Cobalt (Sm-CO) magnet and a solenoid.

    I . INTRODUCTIONDifferent numerical methods to obtain the magneticforces can be found in the literature. The finite differencemethod (FDM)[1,2] and the finite element method [3,4]were used to obtain the field quantities necessary to derivethose forces. In [ I ] the FDM was used to obtain the fieldin terms of the scalar potential. The torque/displacementrelation in saturated variable reluctance step motors wasobtained using the Maxwell field stress method. In [2] theproblem is addressed, being the magnetic forcesoriginated by currents, changes of material permeabilityand the presence of permanent magnets. In [3] the

    necessity of computing the field twice, inherent to thevirtual displacement method is avoided by determining thelocal jacobian derivative. In [4] the magnetic forcedistribution is obtained via the flux densities in each oneof the finite elements and the magnetizing currents alongthe border lines of each element.The comparison of the main three methods (energy,Maxw ell's stress and equivalent currents ) is made in [5,7]and the physical and mathematical meaning of theequivalent magnetizing currents for the force calculationis discussed in [6]. A presentation of the equivalentsources methods is made in [SI for the case ofaxisymmetric problems and in [9] the magnetostriction istaken into account to obtain the deformations inferromagnetic materials.This paper presents a method to obtain the magneticfield in open boundary structures with axisymmetricgeometry [lo] as shown in figure 1. The open space isdivided by a spherical surface, enclosing all the fieldsources. These sources may consist of electrical currentsor magnetized media. In the sphere interior the field isobtained by using the FEM [3,4,11], and in the outsidevolume the field is developed using a series of orthogonalLegendre polynomials. In the two zones the field isobtained in terms of the azimuthal vector magnetic

    potential or in terms of an equivalent flux function. Thecorrect field distribution is finally obtained by imposingthe appropriate boundary conditions along the sphericalboundary surface.

    Figure 1. Relative position of th e solenoid and th e Sm-Co magnet.Magnet data: B ~ 1 . 0 2 , Hc=720 W m .The method was used to determine the spatialdistribution of the magnetic field due to a cylindncalSamarium-Cobalt magnet and a coaxial solenoid, and topreview the interaction force between them [121.The demagnetizing characteristic of the Sm-CO magnetis a straight line between the points (H=O, B=B,) and

    ( H = - H c , B = O ) . The magnet material may beconsidered as a constant magnetization in a linear mediumwith magnetic permeability very close to po. Therefore itwas possible to take advantage from linearity separatingthe field in two parts, one originated by the solenoid andthe other by the magnet.11. FIE L D VAL UAT IONN T H E INTERNALPHE RE

    To evaluate the axisymm etric magnetostatic field in theinternal zone the finite element method was used. Thefield may be described in terms of the azimuthalcomponent A+ of the vector mag netic potential or in termsof the scalar flux hnction IU being A = A,& andy~ = 2npA+, where p is the cylindncal radius and \v is themagnetic flux across a circle of radius p (see Fig.2).In terms of the flux y ~ , he variational principleequivalent to the given boundary-value problem may beformulated as

    being rl the external boundary segment coincident withthe radial symmetry axis where the magnetic potential(and the \v) must vanish. The functional F ( y ) s given by0-7803-827 1-4/04/$20.00 02 00 4 IEEE 71

  • 8/7/2019 Magnetic field and force evaluation in open boundary axisymmetric structures


  • 8/7/2019 Magnetic field and force evaluation in open boundary axisymmetric structures


    term x,H is taken into account by considering a relativepermeability p, = 1.127 in the volume occupied by themagnet, as shown in figure 3 .

    v. CONTINUITY IN THE SPHERICAL BOUNDARYTo solve the stationary condition (3) for a given

    elementary discretization of the internal zone, it isnecessary to relate the nodal values of the zenithalmagnetic field HB to the nodal values of the magnetic fluxv, along the circular boundary segm ent r2.Considering the polynomial decomposition (7) of A+along l-1 ( p r o ) , nd truncating it to an order NT 2 equal tothe number of nodes in the boundary segment T2 we have

    Taking into account the relation between He and4 , he tangential field is expressed as

    and by substitution of (9) in (10) we obtain a similarpolynomial expansion for He :

    n= l

    Considering that A4 is linearly interpolated between theboundary nodes along r2, he coefficients C, may berelated to the actual values of the potential in thediscretized internal zone. Considering the orthogonalproperty of the polynomials of different order and havingpreviously normalised them, we obtain

    being Ai a piecewise linear approximation for the truepotential A+ .The numerical evaluation of (12) produces asquare matrix [MI of order (Nr2xNr2) elating the NT2nodal values of Ab with the NT2coefficients C, :

    (C)=[MI(A$ (13)According to the equation (11) the coefficients R,appearing in the tangential field He along the circularboundary r2 re closely related to the coefficients C, ofthe polynomial decomposition of A + . Therefore therelation between them m ay be expressed as

    being [D] a diagonal matrix with D,, = ,uro/n Whendetermining the relation between the polynomialcoefficients R, and the actual nodal values of thetangential field H g , the procedure is exactly the sam e asused before in (12), and the sam e matrix [M I is obtained,being

    Relating (15) with (13 ) we obtain

    Remembering that \v is related to the potential A , byty = 2npiA, , p i =yo sinei (17)

    being pi and Bi the axial distance and the zen ithal angle inthe circular boundary node (i), the flux nodal values arerelated to the potential values by the m atrix equation

    being [F] diagonal with Fi,=2n ro sin Oi . The final relationbetween the flux nodal values cy) and the zenithalmagnetic field (He) is given by

    VI. NUMERICALESULTSThe m agnetic flux lines were obtained and the resultswere presented in figure 4 for the attractive and repulsivecases respectively. Howev er, to obtain the attraction or therepulsive forces it is only necessary to obtain the magnetic

    field to the coil current and compute the resulting forceover the magnet equivalent superficial current.The coil field was obtained separately for a structurewith dimension given in figure 5. With this dimensionsthe repulsion force could be compared to the results in

    [ I l l .

    Figure 4: Magnetic flux density lines in the attra ctive (left) andrepulsive (right) configurations. Ea ch flux tube encloses AV = 60 pw b .


  • 8/7/2019 Magnetic field and force evaluation in open boundary axisymmetric structures


    VII. CONCLUSIONSIn this paper the repulsion force between a magnet anda solenoid was estimated. The problem presentsaxisymmetric geometry with open boundaries. TO obtainthe magnetic field the structure was enclosed by aspherical surface. The field was determined using the

    FEM in the sphere interior and a Legendre polynomialexpansion in the external zone.The determined results agree with those obtained usingan experimental apparatus.The present work show s that the coil resistance must bereduced to obtain optimum electromagnetic damping.Therefore we intend to study a new structure with a ring toobtain the necessary mechanical damping by eddycurrents.

    I I

    &....-..__-.-.AR L ,

    Figure 5: Relative position of the coil and the Sm-CO magnet. Magnetdata: H , = 720 kA / m , R, = 1S O mm , H , = 1.60 mm .Coil data: 280

    turns, H s = 1.52 mm , R , , , = 1.52 mm , R, = 1.98 mm , ntensity1 = 1 0 0 d.

    The Sm-Co magnet field was also obtained separatelyand the magnetic flux across the coil was computed.These results allow the calculation of the repulsion forcebetween the magnet and the coil as a function of thedistance d between the two components. For equal forces,a systematic difference, of the order of one tenth ofmillimeter, was observed betw een o ur results and those inFigure 6 show s the repulsion force between the Sm-Comagnet and the coil, and the m agnet flux across the 280-turn coil.

    ~ 3 1 .

    60 1

    'Od.0 ' 0:2 ' Oj4 ' 016 ' 0:s ' 1:O ' I100distance d [mm]

    Figure 6: Axial force (lower curve) and the magnetic flux (upper curve)across the coil turns due to the magnet field.

    For a repulsion force of 35m N , corresponding to aoverall mass of 3.56 grams, the derivatives of th e twocurves were calculated to obtain the equivalent springconstant and the corresponding minimum dampingco