PaperlayerView exact revision on arXiv

Source-linked AI summary

Noninvasive Anisotropic Identification of Magnetic Properties in Toroidal Shaped Magnetic Steel

Xusen Qin, Reza Zeinali, Mitrofan Curti

arXiv:2609.07476v2eess.SYcond-mat.mtrl-scics.SYphysics.app-ph

TL;DR

Manufacturing can make a component’s effective magnetic properties differ from nominal data, motivating non-invasive identification from measurements on the component. The paper independently reconstructs internal B and H as harmonic fields from boundary Cauchy data, then uses them to recover principal-direction B–H characteristics. Numerical results report high accuracy on unperturbed data and an overall mean error below 5% at 10% noise.

  • Problem

    Manufacturing effects can make a component’s effective magnetic characteristics deviate from nominal material data, motivating identification from measurements on the actual component.

    Source: §I · p. 1

  • Method

    The method independently reconstructs internal B and H as harmonic fields from boundary Cauchy data, then samples the fields along selected lines to recover principal-direction B–H characteristics.

    Source: §I · p. 1, A. Harmonic Approximation and Field… · p. 3

  • Results

    The overall mean reconstruction error remains below 5% at 10% noise, while unperturbed-data reconstruction has a maximum mean relative error of 0.12%.

    Source: §VI · p. 5, A. Reconstruction from Unperturbed Data · p. 5

  • Takeaways & Limitations

    The harmonic reconstruction provides a simple, computationally efficient framework for identifying nonlinear anisotropic magnetic characteristics from boundary data.

    Source: §VI · p. 5, §VII · p. 6

    The study considers only single-valued B–H curves; demagnetization behavior and hysteresis are outside its scope.

    Source: §VI · p. 5

Abstract

from arXiv · show

This research proposes a model-free inverse approach for identifying the nonlinear anisotropic single-valued magnetic constitutive characteristics of soft magnetic materials from boundary measurements. Using the harmonic approximation, the internal magnetic flux density B and magnetic field strength H are independently reconstructed from the measured boundary data, thus avoiding the limitation of using a predefined parametric B--H mapping function. Subsequently, the reconstructed fields are spatially averaged along selected collection lines to recover the B--H characteristics of the principal directions. Numerical results demonstrate high reconstruction accuracy for unperturbed data and good robustness against Gaussian perturbations of the boundary measurements, with the overall mean error remaining below 5% for a 10% noise level. These results indicate that under the harmonic-field assumptions, the field-based reconstruction provides a simple and computationally efficient framework for non-invasive identification of nonlinear anisotropic magnetic constitutive characteristics.

I. INTRODUCTION

Manufacturing can make a component’s effective magnetic characteristics differ from nominal material data, motivating identification directly from measurements on the component. This work proposes a model-free approach to recover nonlinear anisotropic characteristics without a predefined constitutive form.

I. INTRODUCTION

  • Manufacturing effects can make a component’s effective B–H characteristics deviate from nominal material data, motivating non-invasive identification on the actual component.The cited examples include mechanical stress, cutting, and assembly.Source: §I · p. 1
  • Earlier inverse and coupled measurement approaches may yield homogenized global properties and have flexibility constrained by predefined analytical or parametric models.The cited approaches include methods using global or local magnetic measurements on assembled devices.Source: §I · p. 1
  • The proposed method uses boundary Cauchy data from a toroidal soft magnetic core to identify its unknown single-valued characteristics without a predefined model.The Cauchy data comprise normal flux density Bn and tangential field strength Ht, used to reconstruct fields inside the core.Source: §I · p. 1

A. Identification of the Anisotropic Single-Valued Magnetic Constitutive Law

  • The response is assumed separable along the principal material axes, with cross-component coupling neglected.For laminated electrical steel, the rolling and transverse directions are the principal material directions.Source: A. Identification of the Anisotropic… · p. 1
  • The model-free mapping constructs the anisotropic constitutive response from measurement data, and the principal-direction curves generally differ because of material anisotropy.The identified mapping is used to evaluate flux density for an arbitrary magnetic-field vector in the material plane.Source: A. Identification of the Anisotropic… · p. 2

B. Toroidal Geometry and Measurement Setup

The study models the core as an in-plane, two-dimensional toroid with principal material axes aligned to the transverse and rolling directions. Its forward problem represents nonlinear anisotropic behavior in the core and uses vacuum permeability in non-magnetic regions.

B. Toroidal Geometry and Measurement Setup

  • The core is assumed to be an in-plane 2D toroid, with transverse and rolling directions along the x- and y-axes, respectively, and a symmetrically placed excitation coil.The geometry can be approximated by a sufficiently long toroid to make end effects negligible.Source: B. Toroidal Geometry and Measurement… · p. 2
  • The computational domain is decomposed into boundary, core, and outer regions, with the core denoting the ferromagnetic material.The inner and outer core interfaces are distinguished, and the entire domain has an external boundary Γ.Source: B. Toroidal Geometry and Measurement… · p. 2

A. Forward Problem: Magnetostatic Model

  • The forward model represents the core’s magnetic response with a nonlinear anisotropic constitutive mapping and assumes vacuum permeability in non-magnetic regions.The magnetostatic problem is solved with the piecewise constitutive relation.Source: A. Forward Problem: Magnetostatic Model · p. 2
  • Under the two-dimensional assumption, the magnetic vector potential is expressed along the z-direction, and a homogeneous Dirichlet condition is imposed on the external boundary.The potential formulation is used with the magnetostatic problem.Source: A. Forward Problem: Magnetostatic Model · p. 2

B. Boundary Measurements and Cauchy Data

Boundary Cauchy data comprise normal magnetic flux density and tangential magnetic field strength on the core interfaces. The study describes a Hall-probe measurement route and also uses synthetic boundary traces for its numerical reconstruction study.

B. Boundary Measurements and Cauchy Data

  • With no surface current at the interfaces, the normal component of B and tangential component of H are continuous across each interface.These continuous components form the boundary quantities required by the inverse problem.Source: B. Boundary Measurements and Cauchy Data · p. 2
  • The Cauchy data on the two interfaces consist of Bn and Ht, with n and t denoting the interface’s unit normal and tangential directions.The data are collected on the inner and outer core interfaces.Source: B. Boundary Measurements and Cauchy Data · p. 2
  • A calibrated three-axis Hall probe could scan near both interfaces from the air side, where B = µ0H and tangential-field continuity allows Ht to be inferred without an internal sensor.For the numerical study, synthetic Cauchy traces are extracted from the forward solution; the inverse reconstruction uses those traces, not the predefined constitutive laws.Source: B. Boundary Measurements and Cauchy Data · pp. 2–3

IV. INVERSE PROBLEM FORMULATIONS

The inverse formulation reconstructs magnetic field strength and flux density independently as harmonic fields in the current-free core, using boundary Cauchy data. The resulting fields are then used for field collection and recovery of the nonlinear constitutive characteristics.

A. Harmonic Approximation and Field Reconstruction

  • The method approximates H and B independently as divergence-free and curl-free harmonic fields in the core, driven by Cauchy data from both interfaces.This avoids explicitly specifying the underlying nonlinear constitutive law during field reconstruction.Source: A. Harmonic Approximation and Field… · p. 3
  • The H reconstruction uses a vector potential under the 2D plane assumption; the absence of source current in the core leads to a Laplace equation for the potential.The potential representation automatically satisfies the stated field constraint.Source: A. Harmonic Approximation and Field… · p. 3
  • Tangential H supplies a Neumann boundary condition for the H-potential reconstruction, with an additional condition fixing its additive constant.The boundary relation follows from the rotational relation between the reconstructed field and the potential gradient.Source: A. Harmonic Approximation and Field… · p. 3
  • The B reconstruction uses a harmonic vector potential that automatically satisfies the divergence-free condition and is additionally assumed to satisfy the local harmonic constraint.The potential is represented under the two-dimensional plane assumption.Source: A. Harmonic Approximation and Field… · pp. 3–4
  • Integrating measured normal flux density along each interface determines the B-potential boundary trace up to an interface-dependent additive constant.The trace is expressed using the oriented arc-length coordinate and a reference point on each interface.Source: A. Harmonic Approximation and Field… · p. 4
  • The B-potential is reconstructed by solving Laplace’s equation in the core with the derived traces as Dirichlet boundary values on the interfaces.These are the boundary conditions for the harmonic-field approximation of B.Source: A. Harmonic Approximation and Field… · p. 4
  • The reconstructed harmonic fields are subsequently used for field collection and reconstruction of the nonlinear magnetic constitutive characteristics.The method applies the corresponding local harmonic systems with Cauchy data obtained from the boundary measurements.Source: A. Harmonic Approximation and Field… · p. 4

B. Data Collection and B–H Curve Reconstruction

The method samples reconstructed fields along three core collection lines, using their principal orientations and spatial averages to build operating points for the nonlinear B–H characteristics.

B. Data Collection and B–H Curve Reconstruction

C. Quantitative Comparison and Error Metrics

The evaluation uses reference B–H curves for the principal directions, including a controlled threefold anisotropic scaling, and defines reconstruction error against reference flux density across excitations.

C. Quantitative Comparison and Error Metrics

  • For each excitation, reconstructed field strength is mapped through the prescribed constitutive law to obtain reference flux density, and mean relative error is defined over K excitation currents.Source: C. Quantitative Comparison and Error… · p. 4
  • The NO27 electrical-steel reference curve represents the easier-magnetization rolling direction, while the transverse direction requires three times higher field strength for the same flux density.The reference curve is assigned to the y-direction (rolling direction) and the transverse direction corresponds to x.Source: C. Quantitative Comparison and Error… · p. 5
  • The two principal-direction reference characteristics form a controlled anisotropic test case for evaluating the identification method.Source: C. Quantitative Comparison and Error… · p. 5

A. Reconstruction from Unperturbed Data

With unperturbed Cauchy data, reconstructed curves follow the prescribed B–H characteristics across the collection lines, with mean relative errors ranging from 0.08% to a maximum of 0.12%.

A. Reconstruction from Unperturbed Data

B. Reconstruction from Perturbed Data and Robustness Analysis

With 10% Gaussian noise on boundary measurements, the overall mean reconstruction error remains below 5%. The discussion presents model-free harmonic reconstruction as a simplified inverse formulation, while noting limits tied to its assumptions and measurement scope.

B. Reconstruction from Perturbed Data and Robustness Analysis

  • Independent Gaussian noise is added to the Cauchy data at 3%, 5%, and 10%; systematic effects such as probe misalignment and finite stand-off are excluded.The noise represents random uncertainty from the Hall probe and data-acquisition system.Source: B. Reconstruction from Perturbed Data… · p. 5
  • 4.63% overall mean error at 10% noise; the error rises from 1.39% at 3% to 2.32% at 5%, while the nonlinear trend remains identifiable on all three lines.The 90° collection line, corresponding to the harder transverse direction, consistently gives the smallest error.Source: Table II · p. 5, B. Reconstruction from Perturbed Data… · p. 5, Figure 4 · p. 6

VI. DISCUSSION

  • The inverse procedure is model-free: it reconstructs internal magnetic characteristics directly from measured Cauchy data without a predefined nonlinear constitutive model.Source: §VI · p. 5
  • The inverse problem is reduced to linear harmonic reconstruction problems for B and H, avoiding iterative solution of a nonlinear constitutive problem.Source: §VI · p. 5
  • The harmonic approximation adds assumptions, and error may increase for highly non-uniform materials, anisotropic materials, or geometrically complex settings.Source: §VI · p. 5
  • The study addresses only single-valued B–H curves; demagnetization behavior and hysteresis effects are outside its scope.Source: §VI · p. 5
  • The method requires normal B and tangential H measurements on both core interfaces; practical measurement errors may exceed the independent Gaussian perturbations considered.Potential effects include finite probe size and stand-off, positioning and orientation errors, limited access near the winding, calibration, and temperature drift.Source: §VI · p. 6

VII. CONCLUSION

The work presents a model-free approach to identify nonlinear anisotropic magnetic constitutive characteristics from boundary Cauchy data. Its accuracy depends on the harmonic and locally unidirectional field approximations.

VII. CONCLUSION

  • The model-free inverse approach identifies nonlinear anisotropic magnetic constitutive characteristics from boundary Cauchy data.It avoids directly using the unknown constitutive law.Source: §VII · p. 6
  • Independent harmonic reconstructions of magnetic flux density and field strength are sampled along collection lines to recover principal-direction B–H characteristics.Source: §VII · p. 6
  • Numerical results show high accuracy on unperturbed data and good robustness under measurement noise.Source: §VII · p. 6
  • Accuracy depends on the validity of the harmonic and locally unidirectional field approximations.Source: §VII · p. 6
Loading 2609.07476v2…