Demagnetization Curve
Theory and Physics
What is a Demagnetization Curve?
Professor, a demagnetization curve is part of the B-H curve, right?
The second quadrant of the B-H hysteresis loop is called the demagnetization curve (demag curve). It is the most critical characteristic for determining the operating point of a permanent magnet.
Key parameters:
- $B_r$ (Remanent Flux Density): Magnetic flux density at H=0. For NdFeB magnets, 1.0 to 1.4 T
- $H_{cB}$ (Coercivity): Magnetic field strength where B=0
- $H_{cJ}$ (Intrinsic Coercivity): Magnetic field strength where magnetization J=0. An indicator of heat resistance
- $(BH)_{max}$: Maximum energy product. A comprehensive indicator of magnet performance
A magnet with high $B_r$ and high $H_{cJ}$ is ideal, isn't it?
Yes. However, both decrease as temperature rises. For NdFeB, the temperature coefficient is approximately $-0.12$%/°C for $B_r$ and about $-0.6$%/°C for $H_{cJ}$. It is essential to use the demagnetization curve at the operating temperature.
Summary
- B-H curve in the second quadrant — Operating characteristics of permanent magnets
- $B_r$, $H_{cJ}$, $(BH)_{max}$ — The three major parameters for magnet selection
- Temperature dependence — Risk of irreversible demagnetization at high temperatures
Why Neodymium Magnets are the "Strongest" – The Exquisite Balance of Br and Coercivity
The shape of the demagnetization curve is determined by two factors: remanent flux density Br and coercivity Hc. Ferrite magnets have relatively high Hc but low Br. Alnico magnets have high Br but very low Hc, making them prone to demagnetization even with a small reverse field. Neodymium magnets (Nd₂Fe₁₄B), discovered in 1984 by Dr. Masato Sagawa and others at Sumitomo Special Metals, achieved the ideal combination of both high Br and high Hc. Their BHmax (maximum energy product) exceeds 400 kJ/m³, allowing them to store about 10 times the energy of ferrite magnets of the same volume—this is the theoretical basis for calling them the "strongest magnets."
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 time-varying 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 $\partial D/\partial t$ becomes non-negligible, 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 on hair makes hair stand up due to static electricity—electric field lines radiate from the charged sheet (charge), exerting force on the light hair. Capacitor design uses this law to calculate the electric field distribution between electrodes. ESD (electrostatic discharge) countermeasures are also based on electric field analysis derived from Gauss's law.
- Magnetic flux conservation $\nabla \cdot \mathbf{B} = 0$: States that magnetic monopoles do not exist. 【Everyday Example】Cutting a bar magnet in half does not create a magnet with only an N pole or only an S pole—N and S poles 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): Valid when current direction is uniform and end effects can be ignored
- Isotropy assumption: Anisotropic materials (e.g., silicon steel rolling direction) require direction-specific property definitions
- Non-applicable cases: Additional constitutive relations are needed 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 unit 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
Handling Demagnetization Curves in FEM
How do you incorporate a demagnetization curve into FEM?
Permanent magnets are handled using an equivalent current model. Constitutive relation for magnets:
$\mathbf{M}_0$: Remanent magnetization vector. The slope of the demagnetization curve is the recoil permeability $\mu_r$ (approx. 1.05 for NdFeB).
How do you determine irreversible demagnetization?
Irreversible demagnetization occurs when the operating point falls below the knee point of the demagnetization curve. In JMAG or Maxwell, the operating point of each element can be plotted on the demagnetization curve to visualize regions below the knee point. For demagnetization analysis including temperature distribution, a temperature-dependent demagnetization curve is assigned to each element.
Summary
- Equivalent current model — $\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M}_0)$
- Knee point determination — Evaluation criterion for irreversible demagnetization
- Temperature coupling — Use temperature-corresponding demagnetization curves for each element
How to Obtain Demagnetization Curve Data – Don't Blindly Trust Manufacturer Catalogs
Demagnetization curve data input into FEA is often taken from manufacturer datasheets, but actual magnets can deviate from catalog values due to lot variations and manufacturing temperature history. Particularly for high-grade products with BHmax (maximum energy product) above 50 MGOe, cases have been reported where in-house measurements of small samples are 5-10% lower than catalog values. For critical designs, it is more reliable to perform in-house measurements using a VSM (Vibrating Sample Magnetometer) and feed the data, including temperature dependence, back into the FEA. Designs based solely on catalog data can lead to unexpected demagnetization issues during mass production.
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 in 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)
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
Reduced to a steady-state problem by assuming time-harmonic conditions. Requires complex number operations, but wideband characteristics are obtained via time-domain analysis.
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