Fundamentals of Soft Ferromagnetic Structures (Electromagnets)

Demonstration of Soft Ferromagnetic Structures

©2004 Magneticasoft

When excited by an external magnetic field, soft ferromagnetic materials become a secondary magnetic field source. They behave, de facto, as an amplifier of the external magnetic field. The resulting magnetic field depends on both the external magnetic field and the nature and shape of the ferromagnetic structure. This demostration considers a magnetic system with

1. a single ferromagnetic bar (central iron core)

2. the same central core but completed by iron structures around the solenoid (full iron core)

The magnetic material is a low carbon ordinary steel. The exciting solenoid is made of 1500 turns of copper, with inner and outer diameters and length, respectively, 70, 90, and 80 millimeters, and drawing a current of 0.1 ampere. The central iron core is a cylindrical bar 60 millimeters in diameter and 80 millimeters long.

The magnetic field calculation for soft ferromagnetic structures can be made assuming either a linear or a nonlinear response of the material to the exciting magnetic field. The first method is only a crude approximation of the magnetic field generated by the structure, but it provides a quick evaluation. The second method is much more accurate, but requires lengthy calculations.

The component name for any soft ferromagnetic structure is AXSSoftMagneticComponent. The specification of material behavior is done by calling the option LinearPermeability-> True or False (the default case is False).

The proposed symbolic names of these magnetic systems are electromagnet1 and electromagnet2.

Magnetic System Definition

elecmag1 = 
        AXSMagneticSystem[
{AXSSolenoid[CopperOFHC,  70, 
              90,  80,  0,  1500,  0.1, 
              FillingFactor  ->  
                0.6], 
AXSSoftMagneticComponent[
              SteelCommon1, 0, 60, 82, 0]}, 
 
          SystemLabel  ->  "Electro-Magnet with low carbon Steel central \
core"];

The full iron core consists of two flanges and an outer cylindrical shell.

\(\(elecmag2 = 
        AXSMagneticSystem[
{AXSSolenoid[CopperOFHC, \ 70, 
              90, \ 80, \ 0, \ 1500, \ 0.1, 
              FillingFactor\  -> \ 
                0.6], 
AXSSoftMagneticComponent[
              SteelCommon1, 0, 60, 82, 
              0], 
AXSSoftMagneticComponent[SteelCommon1, 
              0, 92, 10, \(-46\)], 
AXSSoftMagneticComponent[SteelCommon1, 0, 92, 10, 
              46], 
AXSSoftMagneticComponent[SteelCommon1, 
              92, 106, 102, 0]}, 
\ 
          SystemLabel\  -> \ "Electro-Magnet with low carbon Steel full core"];\)\)

Drawing the Magnetic Systems

DrawSystem[elecmag1];

DrawSystem2D[elecmag1];

DrawSystem2D[elecmag2];

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 describe 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[SteelCommon1]

MaterialMagnetizationCurves[SteelCommon1, {M, Permeability}]

This last graph shows the strong nonlinear behavior of common steel as a function of the magnetizing field. This is typical of the ferromagnetic materials. Another indication is the high value of the relative permeability.

The next graph illustrates the advantage of the Magnetica method for recording the magnetizing curve. Since it is a mathematical representation, various scales can be used such LogLog or LogLinear.

MaterialMagnetizationCurves[SteelCommon1, M, 
        MagnetizingFieldRange ->  {0.01, 10000}, 
        PlotScale -> LogLinearMap];

The comparative magnetizing curve for mu-metal is shown next. Mu-metal is typical of very high permeability materials. It is used for shielding sensitive devices (such as high-definition televisions and monitors) from the earth's magnetic field (0.5 gauss in mid latitude regions).

MaterialMagnetizationCurves[MuMetal1, {M, Permeability}]

The relative permeability reaches the 100000 mark, but this happens for a very small magnetizing field value (0.03 gauss). Further, this very high permeability value is maintained only within a very small range (about ±0.005 gauss).

Punctual Magnetic Field Calculation

Magnetic Field

The magnetic field is calculated at the center of the magnetic structure that is inside the central iron bar. The total local magnetic field inside the material (B, called induction or traditionally called flux density) arises both from the magnetization (M) and from the local magnetizing field (Hm). The law of composition of the magnetic field is simply B = M + H (using the proper magnetic field units and considering that this is a vectorial formula). Outside the material, neglecting the air magnetization (very small), B = H (H is also called field intensity).

FieldPoint[elecmag1, {0, 0}]

FieldPoint[elecmag2, {0, 0}]

The comparison of these two results shows the very important effect of closing the iron core around the exciting coil. The flux density increases by a factor 12 without changing the excitation field.

The next calculation illustrates the contribution of each component to the total flux density.

FieldPointOrigin[elecmag2, ComponentsList -> All]

Functional Magnetization

The functional magnetization is the local value of the magnetization inside the ferromagnetic structure.

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 cell position. Using the option FullDetails -> True shows the complete picture of the local magnetization expressed for the vector elements and magnitude of the various physical parameters.

MagnetizationTable[elecmag2, CellsList  ->  {65, 66, 75, 76}, 
      FullDetails  ->  True]

MagnetizationCurves[elecmag1, {Bz}, PlotRows -> {1, 2, 3}, 
      ComponentsList -> {2}]

Field Map Graphics

2D Graphics: Field Functions along a Line

FieldMap[elecmag2, {Bz}, Line[{0, -50}, {0, 50}]];

3D Graphics: Field Functions in a Plane

ContourFieldMap[elecmag1, {Flux}, 
        Plane[{-70, -70}, {70, 70}]];

VectorFieldMap[elecmag1, 
        Plane[{-70, -70}, {70, 70}]];

Inductance Calculation

Inductance is a measurement of the action of the magnetic flux on the electrical circuit generating the field. It is expressed by the ratio of flux to the current intensity. The following calculation illustrates the effect of the iron core on the resulting self-inductance.

SelfInductance[elecmag1, All, 
      InductanceUnit -> Milli Henry]

VerboseSelfInductance[elecmag1, All, 
      InductanceUnit -> Milli Henry]

VerboseSelfInductance[elecmag2, All, 
      InductanceUnit -> Milli Henry]

Using the verbose function that provides a textual, explanatory, and formatted output shows the drastic influence of the iron core on the global self-inductance. Without any iron, the self-inductance is 109 millihenry; this increases by a factor of 3 with the partial iron core and by a factor of 45 with the full iron core. The contribution of the iron core is called ferro-inductance (the method of calculation is patented).