Permanent Magnet Analysis
Theory and Physics
Permanent Magnet
Professor, how are permanent magnets handled in FEM?
A permanent magnet is a material with remanent flux density $B_r$. It generates a magnetic field without an external current.
$$ \mathbf{B} = \mu_0 \mu_r \mathbf{H} + \mu_0 \mathbf{M}_r $$
Or equivalently:
$$ \mathbf{B} = \mu_0 \mu_r (\mathbf{H} + \mathbf{H}_c) $$
$\mathbf{M}_r$: Remanent magnetization, $\mathbf{H}_c$: Coercivity. In FEM, $B_r$ and the magnetization direction are specified for the magnet region.
Major Permanent Magnet Materials
Material $B_r$ [T] $H_{cJ}$ [kA/m] $(BH)_{max}$ [kJ/m³] Applications NdFeB (Sintered) 1.2 to 1.5 800 to 2500 300 to 450 Motors, Generators SmCo 0.9 to 1.1 600 to 2000 150 to 250 High-temperature applications Ferrite 0.3 to 0.4 200 to 400 25 to 40 Low-cost, Speakers Alnico 0.7 to 1.3 40 to 160 10 to 80 Instruments, Sensors
NdFeB is overwhelmingly strong.
NdFeB is essential for EV/HEV motors. However, there is a risk of price volatility for rare earth elements, and research on the resurgence of ferrite motors is also underway.
Summary
- Set $B_r$ and magnetization direction in FEM — The basics of permanent magnets
- NdFeB: $B_r = 1.2$ to 1.5 T — The strongest permanent magnet
- $(BH)_{max}$ — An indicator of the magnet's energy density
Coffee Break Yomoyama Talk
The Birth of Neodymium Magnets—How Masato Sagawa Changed the World of Electromagnetic Devices in 1982
The neodymium magnet (Nd₂Fe₁₄B), indispensable for modern high-performance motors, speakers, and MRI, was invented in 1982 by Dr. Masato Sagawa of Sumitomo Special Metals. Its maximum energy product (BHmax) was more than double that of the previous samarium-cobalt magnets, making the "miniaturization and high output" of EV motors possible at once. Its only weakness is the decrease in coercivity at high temperatures, leading to rapid demagnetization above 120-150°C. Accurately evaluating this "temperature characteristic" with CAE has become an essential task in motor design for EVs and hybrid vehicles.
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) generates voltage in a nearby coil by rotating a magnet—a direct application of this law that a 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. 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 hair with a plastic sheet creates static electricity, making hair stand on end—the charged sheet (electric charge) radiates electric field lines outward, exerting force on the 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 that magnetic monopoles do not exist. 【Everyday Example】If you cut a bar magnet in half, you cannot 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): 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 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
Professor, how are permanent magnets handled in FEM?
A permanent magnet is a material with remanent flux density $B_r$. It generates a magnetic field without an external current.
Or equivalently:
$\mathbf{M}_r$: Remanent magnetization, $\mathbf{H}_c$: Coercivity. In FEM, $B_r$ and the magnetization direction are specified for the magnet region.
| Material | $B_r$ [T] | $H_{cJ}$ [kA/m] | $(BH)_{max}$ [kJ/m³] | Applications |
|---|---|---|---|---|
| NdFeB (Sintered) | 1.2 to 1.5 | 800 to 2500 | 300 to 450 | Motors, Generators |
| SmCo | 0.9 to 1.1 | 600 to 2000 | 150 to 250 | High-temperature applications |
| Ferrite | 0.3 to 0.4 | 200 to 400 | 25 to 40 | Low-cost, Speakers |
| Alnico | 0.7 to 1.3 | 40 to 160 | 10 to 80 | Instruments, Sensors |
NdFeB is overwhelmingly strong.
NdFeB is essential for EV/HEV motors. However, there is a risk of price volatility for rare earth elements, and research on the resurgence of ferrite motors is also underway.
- Set $B_r$ and magnetization direction in FEM — The basics of permanent magnets
- NdFeB: $B_r = 1.2$ to 1.5 T — The strongest permanent magnet
- $(BH)_{max}$ — An indicator of the magnet's energy density
The Birth of Neodymium Magnets—How Masato Sagawa Changed the World of Electromagnetic Devices in 1982
The neodymium magnet (Nd₂Fe₁₄B), indispensable for modern high-performance motors, speakers, and MRI, was invented in 1982 by Dr. Masato Sagawa of Sumitomo Special Metals. Its maximum energy product (BHmax) was more than double that of the previous samarium-cobalt magnets, making the "miniaturization and high output" of EV motors possible at once. Its only weakness is the decrease in coercivity at high temperatures, leading to rapid demagnetization above 120-150°C. Accurately evaluating this "temperature characteristic" with CAE has become an essential task in motor design for EVs and hybrid vehicles.
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) generates voltage in a nearby coil by rotating a magnet—a direct application of this law that a 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. 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 hair with a plastic sheet creates static electricity, making hair stand on end—the charged sheet (electric charge) radiates electric field lines outward, exerting force on the 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 that magnetic monopoles do not exist. 【Everyday Example】If you cut a bar magnet in half, you cannot 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): 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 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
Permanent Magnets in FEM
A permanent magnet can be treated as an equivalent current source:
For uniform magnetization, the volume current is zero, and only the surface current exists. FEM solvers perform this conversion internally and automatically.
Solver Settings
Summary
- Permanent magnet = Equivalent surface current — Automatically converted internally in FEM
- Only need to specify $B_r$ and magnetization direction — Simple user setup
Demagnetization Analysis of Permanent Magnets—Checking if the Operating Point Crosses the "Knee" of the BH Curve
In demagnetization analysis of permanent magnets, it is checked whether the "operating point (B, H)" of each magnet element is above the knee point of the demagnetization curve. By calculating B and H for all magnet elements in FEM and plotting the operating points on the demagnetization curve, demagnetization risk is evaluated. The combination of high temperature, high current, and reverse magnetic field constitutes the worst-case scenario. The analysis procedure is: ① Perform FEM calculation under worst-case current and temperature, ② Extract B-H for each element, ③ Compare with temperature-specific demagnetization curves and calculate the margin. Both JMAG and ANSYS Maxwell support this Demagnetization analysis workflow as standard.
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