A Permanent Magnet Pellet Made of Ferrite or Neodymium, Iron, Boron (NIB) Composite

Permanent magnet structures

©2004 Magneticasoft

Permanent magnets generate a magnetic field without any electrical current. This is due to the intrinsic properties of the materials. The resulting magnetic field depends on the shape of the magnet. This demostration is intended to show relationships between matter properties and magnet shape.

The proposed permanent magnet structures are cylindrical pellets made with the presently most commonly used permanent magnet materials: either ferrite or a NIB composite. The pellets are 10 millimeters in diameter and 5 millimeters long.

The magnetic field calculation for permanent magnet structures can be made according to two different methods determined by the behavior of the material. The first method assumes a rigid magnetization; this is specified by using the component name AXSRigidMagneticComponent. The second method assumes a linear law of magnetization; this is specified by using the component name AXSHardMagneticComponent.

The first method is a crude approximation of the magnetic field generated by the structure, but provides a quick evaluation. The second method is much more accurate, but requires lengthy calculations.

The symbolic names proposed for these new magnetic systems are  barmagnet1, barmagnet2, and  barmagnet3.

Magnetic System Definition

barmagnet1 = 
          AXSMagneticSystem[
{AXSRigidMagneticComponent[
                Ferrit1, 0, 10, 5, 
                0]}, 
SystemLabel -> "cylindrical pellet \
FERRITE"];

barmagnet2 = 
          AXSMagneticSystem[
{AXSRigidMagneticComponent[
                NIBcomp1, 0, 10, 5, 
                0]}, 
SystemLabel -> "cylindrical pellet \
NIB compound (rigid)"];

barmagnet3 = 
          AXSMagneticSystem[
{AXSHardMagneticComponent[
                NIBcomp1, 0, 10, 5, 
                0]}, 
SystemLabel -> "cylindrical pellet \
NIB compound (linear)"];

Drawing the Magnetic Systems

DrawSystem[barmagnet1];

DrawSystem2D[barmagnet2, ShowArrows -> True];

Material Properties

The function Properties shows the physical and magnetic properties of the material, and the function MaterialMagnetizationCurves displays the magnetic properties as a function of the exciting magnetic field (called the magnetizing field to express its action on the matter).

Magnetization is a local property of the matter that arises from the tiny dipoles of the atomic substances. Volume magnetization, which represents the density of cooperative dipoles, has the same dimensional unit as the magnetic field. Therefore, it can be expressed in any relevant unit: gauss, telsa, ampere per meter, and so on.

Properties[Ferrit1]

Properties[NIBcomp1]

MaterialMagnetizationCurves[
      Ferrit1, {Mprincipal, Mtransversal}]

MaterialMagnetizationCurves[
      NIBcomp1, {Mprincipal, Mtransversal}]

These graphs show the strong anisotropic magnetic properties of permanent magnet materials. In the principal direction, the material magnetization varies slightly but always stays positive; for any other direction, the magnetization varies in a reversible way. If the magnetizing field in the principal direction is increased negatively beyond a certain intensity (called the coercive field), a catastrophic phenomenon occurs: the magnetization decreases abruptly through an irreversible process. Consequently, for any device using permanent magnet materials, it is essential to stay away from the coercitive field.

Punctual Magnetic Field Calculation

The magnetic field is calculated at the center of the pellet (that is inside the material) as well as on its surface (i.e., its north pole). The total local magnetic field inside the material (B, called induction or traditionally called flux density) arises from both the magnetization (M) and the local magnetizing field (Hm). The law of composition of magnetic field is simply B = M + H (using the appropriate magnetic field units and considering that this is a vectorial formula). Outside the material, neglecting the magnetization (very small), B = H (H is also called the field intensity).

FieldPoint[barmagnet1, {0, 0}]

FieldPoint[barmagnet2, {0, 0}]

FieldPoint[barmagnet3, {0, 0}]

In this pellet, due to the shape effect, the central flux density is about half the strength of the intrinsic remanent magnetic field. Also, in case 3, which is identical to case 2 but takes into account the linear variation of the magnetization with the local magnetizing field, there is a further decrease in the central flux density.

FieldPoint[barmagnet1, {0, 2.5}]

FieldPoint[barmagnet2, {0, 2.5}]

FieldPoint[barmagnet3, {0, 2.5}]

These values stand for the magnetic field on the north pole surface. This is the useful magnetic field produced by the pellet and the intensity is about one third of the intrinsic remanent field. Increasing the length of the pellets will improve this ratio.

The verbose function provides a textual, explanatory, and formatted output.

VerboseFieldPoint[barmagnet2, {0, 2.5}]

Field Map Graphics

2D Graphics: Field Functions along a Line

FieldMap[barmagnet3, {Bz}, Line[{0, -5}, {0, 5}]];

3D Graphics: Field Functions in a Plane

FieldMap[barmagnet3, {Bz}, 
        Plane[{-5, -5}, {5, 5}]];

ContourFieldMap[barmagnet3, {Flux}, 
        Plane[{-15, -15}, {15, 15}]];

Spherical Harmonics Field Calculation

Calculating the magnetic field using spherical harmonics is an important feature of Magnetica. Although the spherical harmonics method is an efficient way to calculate the magnetic field, this is restricted to space without a magnetic source. Consequently, two regions can be distinguished:

1. a forbidden region (due to a fundamental cause: no mathematical convergence)

2. a remote region extending from a limiting outer sphere to infinity

The next graph shows the three regions. The forbidden region (yellow) is bordered by a remote region (brown) where the precision of calculations becomes insufficient.

DrawSHLimits[barmagnet2];

The spherical harmonics method is based on a set of coefficients (the Legendre source coefficients) for Legendre polynomial development. The Legendre source coefficients measure both the intensity of the field, indicated by the zero-order coefficient, and the homogeneity of the field, shown by the coefficients of increasing order. The coefficients of the central and remote regions are two different sets of numbers (expressed by default in the current magnetic field unit).

VerboseLegendreCoefficients[barmagnet3]

Functional Magnetization

The functional magnetization is the local value of the magnetization inside the pellet. This concept applies only for structures made of materials whose magnetization varies with the applied magnetic field. Indeed, assuming a rigid magnetization (or permeability = 1, or susceptibility = 0), the magnetization is independent of the applied field. Therefore, only the case barmagnet3 is relevant to the functional magnetization concept.

The results can be expressed either in tabular form with B (flux density), M (magnetization), and H (magnetizing field) for each cell of the component or by curves as functions of the cell position.

MagnetizationTable[barmagnet3, 
      CellsList  ->  {9, 12, 13, 16}]

MagnetizationCurves[barmagnet3, {Bz}, PlotRows -> {1, 2, 3, 4}, 
        ComponentsList -> {1}];