Hysteresis Modeling
Hysteresis Modeling: Theoretical Foundations
Magnetic Hysteresis
Professor, hysteresis is the phenomenon where magnetization doesn't follow the magnetic field, right?
When an alternating magnetic field is applied to a ferromagnetic material, the B-H curve forms a loop. The area of this loop corresponds to the hysteresis loss.
Major models:
- Jiles-Atherton (J-A) Model — 5 parameters. A differential equation model based on domain wall pinning.
- Preisach Model — Superposition of hysterons. Can reproduce minor loops of arbitrary shape.
- Play/Stop Hysteron Model — Analogy to mechanical hysteresis.
Is the J-A model the most commonly used?
The J-A model is widely used due to its ease of integration into FEM. However, the Preisach model offers higher accuracy. JMAG adopts the Play model.
Summary
- Hysteresis Loss — Area of the B-H loop.
- J-A Model — 5-parameter differential equation model.
- Preisach Model — High-fidelity reproduction of minor loops.
The Physics of Hysteresis—Domain Wall Pinning and "Magnetic Memory"
The hysteresis loop of a magnetic material is a phenomenon where "domain wall motion and magnetization rotation" respond with a delay to the external magnetic field. When domain walls are "pinned" by grain boundaries, inclusions, or defects, the change in magnetization becomes irreversible, generating coercivity (Hc). Soft magnetic materials (electrical steel sheets) have small domain wall pinning and thus small hysteresis loop area (core loss), while hard magnetic materials (neodymium magnets) have large pinning and strong coercivity. Preisach (1935) mathematically modeled hysteresis as the "superposition of an infinite number of switching elements," laying the foundation for modern hysteresis CAE models.
Computational Methods for Hysteresis Modeling
Integration into FEM
How is a hysteresis model implemented in FEM?
The B-H history is tracked at each Gauss point (integration point). For each time step:
1. Calculate a provisional B.
2. Determine the corresponding H using the hysteresis model.
3. Evaluate the residual and perform Newton-Raphson iteration.
That seems computationally expensive.
Compared to a normal B-H curve (single-valued), computation time increases by 3 to 5 times. The Preisach model requires storing hysteron history data at each integration point, increasing memory usage as well. JMAG implements this efficiently using the Play model.
Summary
- History tracking at each integration point — Update B-H relation for each time step.
- Newton-Raphson iteration — Nonlinear convergence.
- Computational cost — 3 to 5 times that of a normal B-H curve.
Numerical Implementation of Hysteresis Models—Preisach Model and Jiles-Atherton
The two main approaches for hysteresis modeling in CAE are the "Preisach model" and the "Jiles-Atherton model." The Preisach model inversely analyzes a density function from measured initial magnetization curves, accurately reproducing arbitrary magnetization processes, but at high computational cost. The Jiles-Atherton model describes magnetization with a 5-parameter differential equation, is easy to integrate into commercial FEM solvers, and is computationally fast. While the Preisach model excels in high-precision prediction of magnetic hysteresis loss, the Jiles-Atherton model is widely adopted for practical analysis in motor design.
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