Electromagnetic Characterization of Magnetic Bar: Case of Square Cross-Section Shape

Bruno Ricardo Marques; Lars Sj\"oberg; Taha El Hajji

arxiv: 2606.22117 · v1 · pith:T233EPTBnew · submitted 2026-06-20 · 🧮 math-ph · cond-mat.mtrl-sci· cs.SY· eess.SY· math.MP

Electromagnetic Characterization of Magnetic Bar: Case of Square Cross-Section Shape

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

Pith reviewed 2026-06-26 11:13 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.mtrl-scics.SYeess.SYmath.MP

keywords electromagnetic characterizationmagnetic barssquare cross-sectionapparent permeabilitycomplex permeabilityeddy currentshysteresis lossesskin effect

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The pith

A 2D model of square magnetic bars produces an apparent permeability that enables fast characterization without finite element analysis.

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

The paper develops a complete two-dimensional theoretical model for square-section solid magnetic bars under sinusoidal loading. It applies Maxwell's equations in Cartesian coordinates together with complex permeability to derive exact expressions for mutual impedance, internal magnetic fields, flux, and core losses. Hyperbolic functions separate variables to represent edge flux accumulation and the 2D skin effect while decoupling eddy current and hysteresis losses. This yields a new apparent permeability parameter that supplies a rapid method for magnetic steel characterization.

Core claim

Through a two-dimensional Cartesian treatment of Maxwell's equations with integrated complex permeability, exact mathematical expressions are obtained for the electromagnetic quantities in square magnetic bars, leading to an apparent permeability parameter that enables efficient material characterization.

What carries the argument

The apparent permeability parameter, obtained by integrating complex permeability into the 2D hyperbolic-function solutions, which quantifies the effective response for fast characterization.

Load-bearing premise

The model assumes purely sinusoidal loading together with a strictly two-dimensional Cartesian treatment that ignores three-dimensional end effects and nonlinear material responses.

What would settle it

Direct comparison of the model's predicted apparent permeability values against laboratory measurements on actual square-section magnetic bars or against full three-dimensional finite element simulations would test whether the parameter matches observed behavior.

Figures

Figures reproduced from arXiv: 2606.22117 by Bruno Ricardo Marques, Lars Sj\"oberg, Taha El Hajji.

**Figure 1.** Figure 1: 3D view of the studied bar. Assuming the infinite solenoid approximation holds true, which requires l1 ≫ a alongside measurement locations positioned far from the physical ends, the magnetic field impressed upon the surface of the core is assumed to be both spatially uniform and purely longitudinal [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: Front view of the wound bar. 4 Modeling and Mathematical Proof 4.1 Assumptions and Operational Constraints The analytical framework for the square magnetic bar operates under the following key assumptions, which are direct extensions of the standard 1D cylindrical bar model [1]: • Infinite Solenoid Approximation: To neglect longitudinal edge effects and fringing fields at the ends of the bar, it is assume… view at source ↗

**Figure 3.** Figure 3: Apparent relative permeability versus frequency. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Real and imaginary parts of the equivalent primary core impedance versus frequency. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

read the original abstract

This paper presents a complete two-dimensional theoretical model for the electromagnetic behavior of square-section solid magnetic bars under sinusoidal loading. Through the application of Maxwell's equations within a Cartesian coordinate system and the integration of complex permeability, exact mathematical expressions are derived for mutual impedance, internal magnetic fields, flux, and core losses. Hyperbolic functions are utilized to separate the variables, enabling the accurate representation of edge flux accumulation and the 2D skin effect. In addition to mathematically decoupling eddy current and hysteresis losses, this formulation yields a new apparent permeability parameter. This parameter establishes a fast, reliable method for magnetic steel characterization that bypasses the extensive processing times associated with Finite Element Analysis (FEA). Numerical results over 1 Hz-1 MHz show the apparent relative permeability decreasing from 500 to 300 and a characteristic resistance peak near 700 kHz, marking the transition from volumetric to surface-dominated loss regimes.

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.

Desk Editor's Note private letter to a colleague

They derive closed-form 2D expressions for fields, impedance, and losses in square-section bars via complex permeability and hyperbolic functions, then define an apparent permeability from the results.

read the letter

The paper's core output is a set of exact 2D Cartesian solutions for a square bar under sinusoidal excitation. They solve Maxwell's equations with complex permeability, use separation of variables with hyperbolic functions to handle the geometry, and obtain expressions for internal fields, flux, mutual impedance, and separated eddy plus hysteresis losses. From the integrated quantities they introduce an apparent permeability that they say lets you characterize the material without running FEA.

That analytical route is the useful part. For someone who already works with 2D skin-effect models on round or rectangular sections, the square-case extension is a straightforward next step and the closed forms could save time in preliminary design loops. The numerical plots (permeability dropping from 500 to 300, resistance peaking near 700 kHz) are at least internally consistent with the model.

The soft spot is the leap from the 2D linear model to a practical characterization method. The formulation assumes infinite length and purely sinusoidal linear response; real bars have end effects that change the field distribution, and magnetic steels are nonlinear and can show non-sinusoidal behavior even in the 1 Hz–1 MHz range they plot. The apparent permeability is defined inside the same 2D framework, so it is not an independent check on the material—it is a derived quantity that inherits those assumptions. Without 3D comparisons or measurements in the text, it is not yet clear how much error those omissions introduce.

This is the kind of paper that belongs in a specialized electromagnetics or magnetic-component journal. A reader who needs quick analytical estimates for square bars will get value from the expressions; someone looking for validated material models or 3D accuracy will not. The derivations look worth a referee's time to verify the algebra and boundary conditions, even if the applicability claims need tempering.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a two-dimensional Cartesian model for square-cross-section magnetic bars under sinusoidal excitation, solving Maxwell's equations with complex permeability to derive exact expressions (via hyperbolic functions) for mutual impedance, internal fields, flux, and core losses. It introduces a new apparent permeability parameter claimed to enable fast magnetic steel characterization without FEA, with numerical results over 1 Hz–1 MHz showing apparent relative permeability dropping from 500 to 300 and a resistance peak near 700 kHz.

Significance. If the 2D derivations are rigorous and the apparent permeability is independently useful, the analytical expressions could aid design of linear, infinite-length bars. However, the claimed practical significance for bypassing FEA in real steel characterization does not hold without addressing the model's omissions, limiting impact to theoretical cases.

major comments (2)

[Abstract] Abstract: the central claim that the apparent permeability 'establishes a fast, reliable method for magnetic steel characterization that bypasses the extensive processing times associated with Finite Element Analysis (FEA)' is unsupported, as no equations demonstrate independent derivation (versus reduction to the input complex permeability), and no validation against 3D FEA or experiments is provided.
[Abstract] Abstract: the strictly 2D Cartesian formulation (with separation via hyperbolic functions) assumes infinite length and linear sinusoidal response, but the claim for practical characterization requires that neglected z-direction end effects and nonlinearity do not alter the internal field distribution or effective impedance; this assumption is load-bearing and untested.

minor comments (1)

[Abstract] Abstract: the assertion of 'mathematically decoupling eddy current and hysteresis losses' via complex permeability requires clarification, as the complex permeability approach typically models their combined effect rather than separating them.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on the abstract. We address each major comment below and revise the manuscript to ensure claims accurately reflect the 2D analytical scope.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the apparent permeability 'establishes a fast, reliable method for magnetic steel characterization that bypasses the extensive processing times associated with Finite Element Analysis (FEA)' is unsupported, as no equations demonstrate independent derivation (versus reduction to the input complex permeability), and no validation against 3D FEA or experiments is provided.

Authors: The apparent permeability is derived by equating the 2D analytically computed flux (from the hyperbolic-function solutions for the square geometry) to an equivalent uniform-field expression, yielding a geometry-dependent effective parameter that incorporates edge effects and 2D skin effect. It is not a direct reduction of the input complex permeability. We agree the manuscript provides no experimental validation or 3D FEA comparisons, as the focus is the exact 2D derivation. We will revise the abstract to limit the claim to 'within the 2D infinite-length model, offering a rapid analytical alternative to FEA for such cases.' revision: partial
Referee: [Abstract] Abstract: the strictly 2D Cartesian formulation (with separation via hyperbolic functions) assumes infinite length and linear sinusoidal response, but the claim for practical characterization requires that neglected z-direction end effects and nonlinearity do not alter the internal field distribution or effective impedance; this assumption is load-bearing and untested.

Authors: The model is formulated under the explicit assumptions of infinite z-length and linear response with complex permeability, enabling the exact variable separation. These assumptions are stated in the manuscript. We acknowledge that end effects and nonlinearity in real finite bars remain untested here and could affect impedance. We will revise the abstract and add a clarifying sentence in the introduction noting the idealized scope. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation is self-contained

full rationale

The provided abstract and description show a standard analytical derivation: Maxwell's equations are solved in 2D Cartesian coordinates for a square bar using complex permeability as an input material property, hyperbolic functions for separation of variables, and resulting expressions for impedance, fields, flux, and losses. The apparent permeability is described as an output parameter yielded by this formulation, not defined in terms of itself or obtained by fitting a subset of data and renaming the fit. No self-citation chains, uniqueness theorems, or ansatzes smuggled via prior work are referenced. The model assumptions (sinusoidal loading, 2D treatment) are stated explicitly as limitations rather than hidden equivalences. This is the common case of an independent analytical model whose central claim does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, background axioms, or new postulated entities; the only modeling choices visible are the adoption of complex permeability and hyperbolic separation of variables.

pith-pipeline@v0.9.1-grok · 5702 in / 928 out tokens · 34250 ms · 2026-06-26T11:13:45.615512+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 6 canonical work pages

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IEEE Access , volume=

Per-Unit Hysteresis and Eddy Loss Method Based on 3D Finite Elements for Non-Symmetric Toroidal Magnetic , author=. IEEE Access , volume=. 2020 , publisher=. doi:10.1109/ACCESS.2020.2975063 , url=

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