Although **Magnum **and **PerMag **are magnetostatic codes, they can often be used to find AC magnetic field distributions (*e.g.*, transformers and motors). A static field calculation provides a good approximation when the electromagnetic wavelength is much larger than the system scale length. The issue is complicated by the presence of iron and steel, because the speed of light is reduced by a factor 1/?r, where ?r is the relative magnetic permeability. The quantity is defined as the ratio of the total magnetic flux density inside the iron compared to the applied value:

?r = *B*/*B0*. [1]

The permeability characterizes the contribution of material currents to the total magnetic field.

An AC magnetic field at frequency *f0* (Hz) can penetrate a block of iron or steel to a distance given by the skin depth:

?= ?[?/(?*?r*?r**f0*)]. [2]

where ? is the volume resistivity in ?-m. The steels used in transformers have high ?r (several thousand) and low resistivity. For example, nickel steel has ?r = 8000 and ? = 45 × 10[-7] ?-m. At *f0* = 1.0 KHz, the skin depth is only ? = 0.12 mm (4.7 mil). Therefore, steel structures used for AC magnets are *laminated*, fabricated from thin sheets separated by insulators. The orientation of the laminations follows the direction of the magnetic field in the magnetic circuit, as in the figure below. If the lamination thickness is smaller than ?, the field penetrates each lamination so that the magnetic properties are almost the same as for static fields.

In a practical finite-element calculation of a macroscopic device, we approximate cores with many thin lamina as a homogeneous object. In principle, a laminated core has an anisotropic permeability, with ?r » 1 along the lamina and ?r ~ 1 perpendicular to them. In practice, the field lines and lamina are generally in the same direction in the yokes and poles of magnetic circuits. In consequence, we can neglect the normal field component and use a single value of ?r for the component parallel to the lamina.

The relative magnetic permeability of the insulating layers of a laminated core is ?r ? 1.0. The fill fraction *f* is defined as the fraction of the core cross-section occupied by steel. In magnet calculations, we must account for the missing core material. To begin, suppose the system operates with magnetic flux density *B* in the lamina well below the saturation value. In this case, we can use a fixed value of ?r throughout the core material. The effective magnetic permeability of the laminated core is defined as average magnetic flux density inside the core compared to the applied value.

<?r> = <*B*>/*B0* = [?r**B0***f* + *B0**(1-*f*)]/*B0* = ?r**f* + (1-*f*). [3]

The second term on right-hand side is usually small compared to first. To include the effect of the fill fraction, we use a reduced value of relative permeability for the core. For example, if ?r = 2000 and *f* = 0.80, then <?r> = 1600.

Next, consider a core represented by a magnetization curve. The curve has the form ?r(*B0*) in **Magnum **and ?r(<*B*>) in **PerMag**. For **Magnum **calculations, the code determines a local value of ?r for a known value of *B0*. To represent a laminated core, we prepare a table where no changes are made to the independent values (*B0*) and ?r values are adjusted according to Eq. [3]. For a **PerMag **calculation, must make two changes to the table. First, if the calculation were expressed as ?r(*B*) (where *B* is the magnetic flux density *inside *the lamination), the dependent values of ?r would be reduced according to Eq. [3]. On the other hand, the code uses the value of <*B*> for the interpolation, where

<*B*> = *B***f* + (*B*/?r)*(1-*f*) ?r*B***f. *[4]

Therefore, we should multiply values along the abscissa by *f*. Forexample, if a material saturates at a field of 2.0 tesla inside the lamination, saturation would occur at an average magnetic flux density <*B*> = 1.5 tesla for f = 0.75.

In summary, here is how to modify magnetization tables when there is a fill fraction *f*:

- For
**Magnum**tables of ?r(*B0*), multiply ?r values by*f*. - For
**PerMag**tables of ?r(*B*), multiply both ?r and*B*values by*f*.

For more information on **Magnum** and **PerMag**, please see http://www.fieldp.com/magnum.html and http://www.fieldp.com/permag.html.