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 and differ only in how the in-sheet eddy-current correction is wired into the formulation — as a separate weak form in the time domain, as an imaginary part of the complex reluctivity in the frequency domain.

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 \(B_\perp\)):

The permeability tensor used is therefore

Electric conductivity.

Electric current cannot flow across the insulating interlayers, so \(\sigma_\perp = 0\). In-plane the conductivity is reduced by the volume fraction of conducting material:

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 by solving 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 is given as an augmentation of 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 and is therefore 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, \(\delta_{\alpha\beta} \approx 1.0\,\mathrm{mm}\) and the ratio is \(\sim 0.5\) — 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, and the boundary layer at the stack ends should be resolved 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])

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¶