Hysteresis Modeling
Theory and Physics
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.
Physical Meaning of Each Term
- Electric Field Term $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$: Faraday's law of electromagnetic induction. A time-varying magnetic flux density generates an electromotive force. 【Everyday Example】A bicycle dynamo (generator) produces voltage in a nearby coil by rotating a magnet—a direct application of this law stating that a temporally changing magnetic field induces an electric field. Induction cooking (IH) heaters also use the same principle, where high-frequency magnetic field changes induce eddy currents in the pot bottom, heating it via Joule heat.
- Magnetic Field Term $\nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t$: Ampère-Maxwell's law. Electric current and displacement current generate a magnetic field. 【Everyday Example】When current flows through a wire, a magnetic field is created around it—this is Ampère's law. Electromagnets operate on this principle, passing current through a coil to create a strong magnetic field. Smartphone speakers also apply this law: current → magnetic field → force on the diaphragm. At high frequencies (e.g., GHz-band antennas), the displacement current $\partial D/\partial t$ cannot be ignored, describing electromagnetic wave radiation.
- Gauss's Law $\nabla \cdot \mathbf{D} = \rho_v$: Indicates that electric charge is the divergence source of electric flux. 【Everyday Example】Rubbing a plastic sheet against hair causes static electricity, making hair stand up—charged sheet (electric charge) radiates electric field lines outward, exerting force on light hair. Capacitor design calculates the electric field distribution between electrodes using this law. ESD (electrostatic discharge) countermeasures are also based on electric field analysis following Gauss's law.
- Magnetic Flux Conservation $\nabla \cdot \mathbf{B} = 0$: Expresses the absence of magnetic monopoles. 【Everyday Example】Even if you cut a bar magnet in half, you cannot create a magnet with only an N pole or only an S pole—they always exist as a pair. This means magnetic field lines form "closed loops with no start or end points." In numerical analysis, the formulation using vector potential $\mathbf{B} = \nabla \times \mathbf{A}$ is used to satisfy this condition, automatically guaranteeing magnetic flux conservation.
Assumptions and Applicability Limits
- Linear material assumption: Permeability and permittivity are independent of magnetic/electric field strength (nonlinear B-H curve needed in saturation region).
- Quasi-static approximation (low frequency): Displacement current term can be ignored ($\omega \varepsilon \ll \sigma$). Common in eddy current analysis.
- 2D assumption (cross-section analysis): Effective when current direction is uniform and end effects can be ignored.
- Isotropic assumption: For anisotropic materials (e.g., silicon steel rolling direction), direction-specific property definitions are needed.
- Non-applicable cases: Additional constitutive laws are required for plasma (ionized gas), superconductors, nonlinear optical materials.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Magnetic Flux Density $B$ | T (Tesla) | 1T = 1 Wb/m². Permanent magnets: 0.2 to 1.4T |
| Magnetic Field Strength $H$ | A/m | Horizontal axis of B-H curve. Conversion from CGS Oe (Oersted): 1 Oe = 79.577 A/m |
| Current Density $J$ | A/m² | Calculated from conductor cross-section and total current. Note non-uniform distribution due to skin effect. |
| Permeability $\mu$ | H/m | $\mu = \mu_0 \mu_r$. In vacuum $\mu_0 = 4\pi \times 10^{-7}$ H/m |
| Electrical Conductivity $\sigma$ | S/m | Copper: approx. 5.96×10⁷ S/m. Decreases with temperature rise. |
Numerical Methods and Implementation
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.
Edge Elements (Nedelec Elements)
Elements specialized for electromagnetic field analysis. Automatically guarantee continuity of tangential components and eliminate spurious modes. Standard for 3D high-frequency analysis.
Nodal Elements
Used for scalar potential formulations. Effective for scalar potential methods in magnetostatics and electrostatic field analysis.
FEM vs BEM (Boundary Element Method)
FEM: Handles nonlinear materials and inhomogeneous media. BEM: Naturally handles infinite domains (open boundary problems). Hybrid FEM-BEM is also effective.
Nonlinear Convergence (Magnetic Saturation)
The nonlinearity of the B-H curve is handled by the Newton-Raphson method. Residual criterion: $||R||/||R_0|| < 10^{-4}$ is typical.
Frequency Domain Analysis
Reduces to a steady-state problem using time-harmonic assumption. Requires complex number operations, but broadband characteristics are obtained via time-domain analysis.
Time Domain Time Step
A time step smaller than 1/20 of the highest frequency component is required. Implicit time integration allows larger steps but requires attention to accuracy.
Choosing Between Frequency Domain and Time Domain
Frequency domain analysis is like "tuning a radio to a specific frequency"—it can efficiently compute the response at a single frequency. Time domain analysis is like "recording all channels simultaneously"—it can reproduce transient phenomena containing all frequency components, but at a high computational cost.
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