Laminated Steel

Electrical Conductivity and Magnetic Permeability

The electrical conductivity and magnetic permeability of an electrical sheet of steel are generally orthotropic:

Figure 1. EQUATION_DISPLAY

\bar{\bar{σ_{m}}} = {σ_{m x}, σ_{m y}, σ_{m z}}

(4324)

Figure 2. EQUATION_DISPLAY

\bar{\bar{μ_{m}}} = {μ_{m x}, μ_{m y}, μ_{m z}}

(4325)

To define the electrical conductivity, the lamination normal orientation, $\hat{n}$ , is specified as a unit vector orthogonal to the lamination layers. The positive orientation can be arbitrary, however, in a local coordinate system the positive orientation direction can be classed as vertical, or the z axis.

All three diagonal tensor quantities for the electrical conductivity and magnetic permeability in the Normal ( $\hat{z} = \hat{n}$ ), Rolling ( $\hat{x}$ ), and Transverse ( $\hat{y}$ ) direction are somewhat different, especially in grain orientated electrical steel.

Magnetic permeability in most cases is highly nonlinear. Due to this, manufactures may provide a single average electrical conductivity and a single B-H scalar table for the material. For convenience, the effective electrical conductivity of the lamination stack is assumed to be zero in the direction normal to the lamination stack:

Figure 3. EQUATION_DISPLAY

\bar{\bar{σ}} = {σ_{m}, σ_{m}, 0}

(4326)

Lamination Stack

The thickness of the metal sheet from which each layer is made is defined as the layer metal thickness ( $h_{m}$ ). It is assumed that the layers all have the same thickness.

The total layer thickness,

h

, can be considered the sum of the metal layer thickness and the insulation layer thickness,

h_{i}

.

Figure 4. EQUATION_DISPLAY

h = h_{m} + h_{i}

(4327)

The stacking factor, otherwise known as metal fill factor, can be defined as a fraction of metal thickness in total thickness:

Figure 5. EQUATION_DISPLAY

s = \frac{h_{m}}{h}

(4328)

If the insulation thickness is negligible or unknown,

h_{i} \approx 0

:

Figure 6. EQUATION_DISPLAY

s \approx 1.0

(4329)

Magnetic Field

The differential mode current is replaced with the laminated steel bulk model magnetic augmentation term. Theoretically this is defined using a time derivative anisotropic relationship:

Figure 7. EQUATION_DISPLAY

H_{L S} = \frac{s^{3} \bar{\bar{σ}} h^{2}}{12} \frac{\partial B}{\partial t}

(4330)

From this the total magnetic field in the laminations becomes:

Figure 8. EQUATION_DISPLAY

H = B \bar{\bar{μ^{- 1}}} + H_{L S}

(4331)

where $B$ is the magnetic flux density and the steel inverse permeability $\bar{\bar{μ^{- 1}}}$ is resolved using B-H tables.