Laminated Material¶

The Lamination module ships two parallel material classes — one per magnetic model — that share the same parameter set and the same underlying physics. See the overview for the laminate geometry, the two eddy-current channels and the relevant references.

The two classes implement the same constitutive law. They differ only in how the in-sheet eddy-current correction is wired into the formulation. The time-domain class uses a separate weak form; the frequency-domain class uses an imaginary part of the complex reluctivity.

Effective constitutive tensors¶

In the local frame both tensors are diagonal with two equal in-plane components and a distinct out-of-plane component.

Magnetic permeability.

The in-plane direction is an arithmetic mean (continuity of \(H_\parallel\) across iron/air interfaces); the out-of-plane direction is a harmonic mean (continuity of \(B_\perp\)):

So the model uses this permeability tensor:

Electric conductivity.

Electric current cannot flow across the insulating interlayers, so \(\sigma_\perp = 0\). The volume fraction of conducting material reduces the in-plane conductivity:

This \(\bar{\boldsymbol{\sigma}}\) accounts only for the homogenised \(\vec{j}_{\alpha\beta}\) — the macroscopic eddy-current loops in the lamination plane.

In-sheet eddy-current correction¶

The component \(\vec{j}_{\beta\gamma}\) — the in-sheet loop driven by the parallel flux variation \(\partial_t \vec{B}_\alpha\) — has a known analytical structure¹. It comes from the 1-D Faraday equation across the sheet thickness. In the thin-sheet limit the in-sheet current density varies linearly with the through-thickness coordinate. The correction augments the magnetic-field strength²: $$ \vec{H} = \bar{\boldsymbol{\mu}}^{-1}\,\vec{B} + \underbrace{\frac{\sigma\,d^2}{12}\,\partial_t \vec{B}}{\displaystyle \vec{H}\text{LS}}, $$

where \(\vec{H}_\text{LS}\) is the lamination-stack augmentation that accounts for the in-sheet eddy currents.

Validity range¶

The first-order augmentation \(\vec{H}_\text{LS}\) neglects the in-sheet skin effect. So it is valid when the in-sheet skin depth \(\delta_{\alpha\beta} = \sqrt{2 / (\omega\,\mu_\perp\,\sigma_\parallel)}\) exceeds about half the sheet thickness:

For 0.5 mm silicon-steel (\(\sigma = 5 \times 10^6\,\mathrm{S/m}\), \(\mu_r = 1000\)) at 50 Hz, the in-sheet skin depth is \(\delta_{\alpha\beta} \approx 1.0\,\mathrm{mm}\). The ratio is then \(\sim 0.5\) — well inside the validity range. For thinner sheets or lower frequencies the approximation is better; for thicker sheets or frequencies above a few kHz at these material parameters the linear \(\gamma\)-expansion breaks down.

A second length scale, the penetration depth into the stack \(\delta_\gamma = \sqrt{2 / (\omega\,\mu_\parallel\,\sigma_\parallel)}\), governs the mesh size near the stack ends. Because \(\mu_\parallel \gg \mu_\perp\) for typical iron laminates, \(\delta_\gamma \ll \delta_{\alpha\beta}\). The perpendicular flux decays much faster from the stack ends than the in-plane skin depth would suggest. You must resolve the boundary layer at the stack ends accordingly.

Parameters¶

All scalar / vector parameters accept the standard mufem coefficient inputs (CffConstantScalar, CffConstantVector, field-dependent coefficients, …).

Time-Domain Magnetic¶

import mufem
from mufem import Vol
from mufem.electromagnetics.module.lamination import (
    TimeDomainMagneticLaminationMaterial,
)

iron_material = TimeDomainMagneticLaminationMaterial(
    name="Core",
    marker="Core" @ Vol,
    sheet_permeability=mufem.CffConstantScalar(1000.0),
    sheet_conductivity=mufem.CffConstantScalar(5.0e6),
    stacking_direction=mufem.CffConstantVector(0.0, 0.0, 1.0),
    stacking_factor=mufem.CffConstantScalar(0.95),
    lamination_thickness=mufem.CffConstantScalar(0.5e-3),
)

time_domain_magnetic_model.add_materials([iron_material])

The Magnetic Energy Density coefficient gives the reversible stored energy density of the homogenised stack:

• \(w\) – magnetic energy density [J/m³]
• \(\vec{B}\) – magnetic flux density [T]
• \(\bar{\boldsymbol{\mu}}^{-1}\) – homogenised reluctivity tensor, the inverse of the effective permeability tensor above [m/H]

The value does not include the in-sheet eddy-current correction, because that term dissipates energy and does not store it. In versions before 0.6.0, this coefficient was zero in laminated regions.

Time-Harmonic Magnetic¶

import mufem
from mufem import Vol
from mufem.electromagnetics.module.lamination import (
    TimeHarmonicMagneticLaminationMaterial,
)

iron_material = TimeHarmonicMagneticLaminationMaterial(
    name="Core",
    marker="Core" @ Vol,
    sheet_permeability=mufem.CffConstantScalar(1000.0),
    sheet_conductivity=mufem.CffConstantScalar(5.0e6),
    stacking_direction=mufem.CffConstantVector(0.0, 0.0, 1.0),
    stacking_factor=mufem.CffConstantScalar(0.95),
    lamination_thickness=mufem.CffConstantScalar(0.5e-3),
)

time_harmonic_magnetic_model.add_materials([iron_material])

References¶