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arxiv: 2606.18270 · v1 · pith:ZRX7J5VMnew · submitted 2026-06-05 · ⚛️ physics.comp-ph · cond-mat.mtrl-sci· cs.SY· eess.SY· math-ph· math.MP

Electromagnetic Characterization of Magnetic Ring: Case of Circular Cross-Section Shape

Taha El Hajji , Lars Sj\"oberg This is my paper

Pith reviewed 2026-06-27 20:38 UTC · model grok-4.3

classification ⚛️ physics.comp-ph cond-mat.mtrl-scics.SYeess.SYmath-phmath.MP
keywords electromagnetic characterizationtoroidal magnetic ringanalytical modelcomplex permeabilityskin effecteddy current losseshysteresis lossesapparent permeability
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The pith

A two-dimensional analytical model using Maxwell's equations in local polar coordinates derives expressions for field, flux, impedance, and separated losses in toroidal magnetic rings with circular cross-section.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an analytical model for a toroidal magnetic ring with circular cross-section under sinusoidal excitation. Applying Maxwell's equations in local polar coordinates with complex permeability produces closed-form expressions for the internal magnetic field, magnetic flux, complex impedance, and total losses. The expressions separate eddy current losses, hysteresis losses, and winding losses while incorporating skin effect through Bessel functions. An apparent permeability formula maps nonlinear core behavior to linear models. The approach supplies a computationally efficient alternative to finite element analysis for standardized magnetic material characterization.

Core claim

The central claim is that applying Maxwell's equations in local polar coordinates within a complex permeability framework yields analytical expressions for the internal magnetic field, magnetic flux, complex impedance, and total losses in a toroidal ring with circular cross-section, rigorously separating eddy current, hysteresis, and winding losses while incorporating skin effect through Bessel functions, and providing apparent permeability for nonlinear mapping.

What carries the argument

The two-dimensional analytical model in local polar coordinates with complex permeability, employing Bessel functions to account for skin effect in the conductive core.

If this is right

  • The model supplies closed-form expressions for internal magnetic field, magnetic flux, and complex impedance.
  • It separates the contributions of eddy current losses, hysteresis losses, and winding losses.
  • The apparent permeability expression enables mapping of nonlinear core behavior onto simplified linear material models.
  • The analytical results provide a computationally efficient foundation for standardized magnetic material characterization as an alternative to finite element analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separation of loss mechanisms could support targeted design adjustments in magnetic components to minimize specific loss types.
  • The mapping via apparent permeability might allow integration of the model into circuit simulators that assume linear materials.
  • Extension of the local polar coordinate approach to non-circular cross-sections would require new coordinate transformations but could follow the same Maxwell-equation structure.
  • Routine use in measurement standards would reduce the need for repeated mesh refinement in numerical tools for similar geometries.

Load-bearing premise

The two-dimensional approximation in local polar coordinates with a complex permeability model is sufficient to accurately represent the three-dimensional toroidal geometry and material behavior under sinusoidal excitation.

What would settle it

Direct numerical comparison of the model's predicted complex impedance and separated loss values against three-dimensional finite element simulations or physical measurements on toroidal rings with circular cross-section under sinusoidal excitation.

Figures

Figures reproduced from arXiv: 2606.18270 by Lars Sj\"oberg, Taha El Hajji.

Figure 1
Figure 1. Figure 1: Visualization of the Toroidal Ring (Circular Cross-Section) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Toroidal Ring: (a) Top View, (b) Cross-section [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Winding Configuration for Circular Core 3.3 Equivalent Circuit Model The electrical behavior of the system is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Equivalent magnetic circuit of the wounded ring [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Magnetic Flux Density magnitude across the circular cross-section at 4 different [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Apparent Relative Permeability vs Frequency [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Core complex impedance components [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Hysteresis and Eddy current losses available volume. Consequently, the power loss curves exhibit the behaviour observed in the MHz range. These results are obtained under constant-current excitation (Irms = 1 A). Under constant￾flux density excitation, the losses would increase much more rapidly — a situation that occurs, for instance, in constant-flux measurements such as those performed with Brockhaus or… view at source ↗
read the original abstract

This paper introduces a comprehensive two-dimensional analytical model of a toroidal magnetic ring with circular cross-section under sinusoidal excitation. Applying Maxwell's equations in local polar coordinates within a complex permeability, the model derives analytical expressions for the internal magnetic field, magnetic flux, complex impedance, and total losses. It rigorously separates the contributions of eddy current losses, hysteresis losses, and winding losses, while explicitly incorporating the skin effect in the conductive core via Bessel functions. An expression for the apparent permeability is also provided, enabling the nonlinear core behavior to be mapped onto simplified linear material models. The resulting analytical model offers a computationally efficient and accurate foundation for standardized magnetic material characterization, such as Brockhaus and Iwatsu ring measurements, as a powerful alternative to 2D and 3D finite element analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a two-dimensional analytical model for a toroidal magnetic ring with circular cross-section under sinusoidal excitation. Applying Maxwell's equations in local polar coordinates with a complex permeability, it derives closed-form expressions (via Bessel functions) for the internal magnetic field, magnetic flux, complex impedance, total losses (with explicit separation of eddy-current, hysteresis, and winding contributions), and an apparent permeability that maps nonlinear core behavior onto linear models. The model is positioned as a computationally efficient alternative to 2D/3D FEA for standardized ring-sample characterization such as Brockhaus and Iwatsu measurements.

Significance. If the derivations hold within the stated approximations, the closed-form separation of loss mechanisms and the apparent-permeability mapping would provide a practical analytical tool for magnetic-material characterization, reducing the need for numerical simulations in routine measurements while maintaining explicit incorporation of skin effect.

major comments (2)
  1. [Model derivation / local polar coordinate assumption] The central derivation treats the toroidal core in local polar coordinates (r,φ) as locally straight and axisymmetric. The manuscript does not quantify the error incurred when the minor radius a is not ≪ major radius R; the curvature-induced variation in path length and the weak coupling to the toroidal direction directly affect the computed B-field distribution, apparent permeability, and the partitioned eddy-current loss term (see the model setup preceding Eq. (Bessel solution) and the loss-separation expressions).
  2. [Complex permeability definition and loss separation] The complex permeability is introduced to account for hysteresis and eddy losses, yet the manuscript provides no explicit statement of how its real and imaginary parts are obtained from material data or whether they remain frequency-independent; this choice is load-bearing for the claimed rigorous separation of hysteresis versus eddy-current losses in the total-loss expression.
minor comments (2)
  1. [Introduction / model geometry] Notation for the local polar coordinates and the toroidal major/minor radii should be introduced once with a clear figure or diagram to avoid ambiguity in later equations.
  2. [Results / loss expressions] The abstract states that the model 'rigorously separates' loss contributions; a short table or explicit algebraic breakdown in the results section would make this separation more transparent to readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. We address each major comment below and outline the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: [Model derivation / local polar coordinate assumption] The central derivation treats the toroidal core in local polar coordinates (r,φ) as locally straight and axisymmetric. The manuscript does not quantify the error incurred when the minor radius a is not ≪ major radius R; the curvature-induced variation in path length and the weak coupling to the toroidal direction directly affect the computed B-field distribution, apparent permeability, and the partitioned eddy-current loss term (see the model setup preceding Eq. (Bessel solution) and the loss-separation expressions).

    Authors: We agree that the local polar coordinate approximation assumes a ≪ R and that an explicit error quantification is absent. In the revised manuscript we will add a dedicated subsection (or appendix) that estimates the approximation error for representative a/R ratios (e.g., 0.05–0.2) by comparing the local solution against a first-order toroidal correction or against 3D FEA benchmarks. This will include quantitative bounds on the resulting deviations in B-field, apparent permeability, and the eddy-current loss term, thereby clarifying the practical range of validity. revision: yes

  2. Referee: [Complex permeability definition and loss separation] The complex permeability is introduced to account for hysteresis and eddy losses, yet the manuscript provides no explicit statement of how its real and imaginary parts are obtained from material data or whether they remain frequency-independent; this choice is load-bearing for the claimed rigorous separation of hysteresis versus eddy-current losses in the total-loss expression.

    Authors: The complex permeability μ = μ′ − jμ″ is introduced to capture the hysteretic response of the core material, while the skin effect and associated eddy-current losses are treated separately through the exact solution of Maxwell’s equations (Bessel-function field distribution). We will revise the manuscript to state explicitly that μ′ and μ″ are extracted from low-frequency ring measurements (where skin depth is large compared with the cross-section radius) and to discuss the assumption of frequency independence within the modeled bandwidth. This clarification will be placed in the model-setup section and will be cross-referenced in the loss-separation derivation to reinforce the separation of mechanisms. revision: yes

Circularity Check

0 steps flagged

Derivation starts from Maxwell's equations with no load-bearing reduction to inputs

full rationale

The paper applies Maxwell's equations in local polar coordinates to a toroidal ring with circular cross-section, using a complex permeability model and Bessel functions to obtain closed-form expressions for the internal field, flux, impedance, and separated losses. No step reduces by construction to a fitted parameter renamed as a prediction, a self-citation chain, or a self-definitional ansatz; the complex permeability serves as an input constitutive relation rather than an output derived from the same equations. The derivation chain remains independent of the target results and is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available; no specific free parameters or invented entities mentioned. Relies on standard electromagnetic assumptions.

axioms (2)
  • standard math Maxwell's equations hold in the local polar coordinates for the ring geometry
    Invoked in the derivation of the model as per abstract.
  • domain assumption Complex permeability can model the material response under sinusoidal excitation
    Used to derive the expressions for field and losses.

pith-pipeline@v0.9.1-grok · 5678 in / 1284 out tokens · 29614 ms · 2026-06-27T20:38:38.253633+00:00 · methodology

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Reference graph

Works this paper leans on

7 extracted references

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    1998

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    M. Markovic and Y . Perriard, “Eddy current power losses in a toroidal laminated core with rectangular cross section,” in2009 International Conference on Electrical Machines and Sys- tems, pp. 1–4, IEEE, 2009

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    Electromagnetic characterization of magnetic ring: Case of square cross section shape,

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    Ac losses in windings: Review and comparison of models with application in electric machines,

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