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

This paper presents a complete two-dimensional theoretical model for the electromagnetic behavior of square-section magnetic bars under sinusoidal loading, deriving exact mathematical expressions for key parameters and introducing a new apparent permeability parameter that enables rapid magnetic steel characterization while bypassing the computational costs of Finite Element Analysis.

Original authors: Taha El Hajji, Bruno Ricardo Marques, Lars Sjöberg

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Taha El Hajji, Bruno Ricardo Marques, Lars Sjöberg

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a solid, square-shaped metal bar, like a thick chocolate bar made of magnetic steel. Now, imagine wrapping a wire around the middle of it and running an electrical current through that wire. This creates a magnetic field, kind of like turning the bar into a temporary magnet.

This paper is about figuring out exactly what happens inside that square bar when you do this, especially when you change how fast the electricity is switching on and off (the frequency).

Here is the breakdown of their discovery, using simple analogies:

The Problem: Squares are Tricky

Scientists have known for a long time how to calculate what happens inside a round metal rod. It's like rolling a ball; the math is smooth and predictable. But when the metal bar is square (like a chocolate bar), the corners create a mess.

When the electricity switches on and off quickly, it creates tiny swirling currents inside the metal called "eddy currents." Think of these like whirlpools in a river. In a round bar, these whirlpools are neat. But in a square bar, the magnetic field gets "stuck" or piles up at the sharp corners, creating a very uneven mess inside. Traditional math tools (which work for round bars) break down here.

The Solution: A New Mathematical Recipe

The authors created a new, precise mathematical recipe (a "theoretical model") to solve this square-bar puzzle without needing to run slow, heavy computer simulations.

They used a clever math trick called "separation of variables." You can think of this like solving a complex jigsaw puzzle by separating the edge pieces from the center pieces. By doing this, they were able to write down exact formulas that describe:

How the magnetic field moves inside the bar.
How much energy is lost as heat (due to the swirling currents and the material's internal friction).
How the bar resists the electricity flowing through the coil.

The Key Discovery: The "Skin" Effect

The most interesting part of their findings is what happens as the electricity switches faster and faster (from 1 Hz to 1 million Hz).

The Slow Speed (Low Frequency): When the current changes slowly, the magnetic field penetrates the whole bar evenly, like water soaking into a sponge. The bar acts like a solid block of magnetic material.
The Fast Speed (High Frequency): As the speed increases, the "whirlpools" (eddy currents) get stronger. They push the magnetic field away from the center and force it to hug the outside edges of the bar. This is called the Skin Effect.
- The Analogy: Imagine a crowded dance floor. When the music is slow, people can dance anywhere. When the music gets frantic and fast, everyone crowds against the walls to avoid bumping into each other, leaving the center of the room empty.
- The Result: Because the magnetic field is pushed to the edges, the "effective" size of the bar shrinks. The paper calculates that the bar's ability to store magnetic energy (permeability) drops significantly as it gets faster.

The "Apparent" Permeability

The authors introduced a new concept called Apparent Permeability.

Think of the bar's "real" magnetic strength as a fixed number (like 500).
However, because of the "crowding" at the edges at high speeds, the bar acts like it has a lower strength (dropping down to about 300 in their tests).
Their formula allows engineers to calculate this "fake" or "apparent" strength instantly, rather than waiting hours for a computer to simulate it.

The Energy Loss Peak

They also found a specific point where the energy loss (heat) behaves strangely.

At low speeds, the bar loses a little heat.
As speed increases, the heat loss shoots up.
The Peak: Around 700,000 cycles per second (700 kHz), the resistance hits a maximum peak.
Why? This is the exact moment where the "crowd" (magnetic field) has completely moved from the center of the bar to the edges. After this point, the way the bar loses heat changes its behavior again.

Why This Matters (According to the Paper)

The main goal of this paper isn't to invent a new machine, but to provide a fast, accurate calculator.

Currently, if an engineer wants to know how a square magnetic bar behaves, they often have to use Finite Element Analysis (FEA), which is like using a supercomputer to simulate every single atom. It takes a long time.
This paper gives them a direct math formula (like a shortcut) that gives the same accurate answer in a fraction of a second.

In summary: The paper says, "We figured out the exact math for how magnetic fields behave inside square metal bars. We found that at high speeds, the magnetic field hides in the corners, making the bar act weaker and hotter than we thought, and we now have a fast formula to predict exactly when and how this happens."

Technical Summary: Electromagnetic Characterization of Magnetic Bar with Square Cross-Section

Problem Statement
The accurate evaluation of soft magnetic components is critical for the performance of modern electromagnetic machinery. While established one-dimensional mathematical models utilizing Bessel functions effectively describe cylindrical geometries and ring cores with circular cross-sections, these simplifications fail for square cross-sections. In square bars, particularly as operating frequencies increase, the skin effect induces eddy currents that repel magnetic flux toward the geometric extremities. This results in severe two-dimensional flux accumulation at the edges and a drastically non-uniform internal field. Furthermore, standard models often struggle to decouple hysteresis and eddy current losses or to account for the phase delay introduced by complex permeability in these non-cylindrical topologies. Consequently, there is a need for an advanced multi-dimensional modeling approach that avoids the computational intensity of Finite Element Analysis (FEA).

Methodology
The authors present a complete two-dimensional theoretical model for solid magnetic bars with square cross-sections under sinusoidal loading. The approach relies on solving Maxwell's equations within a Cartesian coordinate system under the following constraints:

Geometry: A long, straight bar with side length $a$ and length $L$ , modeled with an infinite solenoid approximation ( $l_1 \gg a$ ) to neglect longitudinal edge effects.
Material Properties: The core is assumed to have homogeneous, isotropic conductivity ( $\sigma$ ) and a linearized, frequency-dependent complex permeability ( $\mu_c = \mu' - j\mu''$ ) to capture hysteresis.
Mathematical Derivation: The authors apply the method of separation of variables to the magnetic diffusion equation derived from Ampère's and Faraday's laws. By utilizing hyperbolic functions, the variables are separated to accurately represent the 2D nature of the flux distribution.
Loss Separation: The model mathematically decouples eddy current and hysteresis losses by evaluating the limit of the impedance as conductivity approaches zero.

Key Contributions

Closed-Form Analytical Expressions: The paper derives exact mathematical expressions for the internal magnetic field ( $H_z$ ), flux density ( $B_z$ ), total magnetic flux ( $\Phi$ ), and mutual complex impedance ( $Z_m$ ) specifically for square cross-sections.
Loss Decoupling: A formulation is provided to separate core losses into distinct hysteresis ( $P_h$ ) and eddy current ( $P_e$ ) components based on the real part of the core admittance.
Apparent Permeability Parameter: The study introduces a new parameter, apparent permeability ( $\mu_{app}$ ), derived from the reactive part of the impedance. This parameter quantifies the reduction in energy-storage capability due to eddy currents without requiring iterative numerical processing.
Computational Efficiency: The model offers a rigorous alternative to FEA, promising significantly faster processing times for magnetic steel characterization.

Results and Analysis
Numerical simulations were conducted over a frequency range of 1 Hz to 1 MHz using a square bar with a 10 mm side length and specific material properties ( $\mu'_r = 500$ , $\mu''_r = 50$ , $\sigma = 90$ S/m). Key findings include:

Apparent Permeability: At low frequencies (up to ~10 kHz), the apparent relative permeability remains constant at the intrinsic value of 500. As frequency increases, the skin effect forces flux to the surface, causing $\mu_{app}$ to drop, reaching approximately 300 at 1 MHz.
Impedance Behavior: The equivalent core resistance ( $Re(Z_{core})$ ), representing total losses, rises sharply after 10 kHz as eddy current losses (scaling with $f^2$ ) dominate hysteresis losses. The resistance peaks near 700 kHz.
Transition Regime: The peak in resistance marks the transition from a volumetric loss dependency to a surface-area dependency ( $\propto \sqrt{f}$ ), occurring when the skin depth becomes comparable to the geometric dimensions of the bar.

Significance
The paper claims that this mathematical formulation provides a fast, reliable, and highly rigorous method for characterizing magnetic steel in square cross-sections. By bypassing the extensive processing times associated with FEA, the model enables the rapid determination of mutual impedance, internal field distributions, and separated loss components. The introduction of the apparent permeability parameter establishes a direct link between measurable impedance and the physical phenomenon of flux shielding, facilitating more efficient material testing and design optimization for electromagnetic machinery.