Magnetic Levitation
Theory and Physics
Principles of Magnetic Levitation
Professor, how does the levitation in linear motor cars work?
There are mainly three methods.
1. EDS (Electrodynamic Suspension) — When superconducting magnets pass over ground coils, repulsive force is generated by eddy currents. Used in JR's linear system.
2. EMS (Electromagnetic Suspension) — Attracted to the rail by electromagnetic attraction. Used in Transrapid.
3. Eddy Current Repulsion Method — Permanent magnets/electromagnets move over a conductor plate to achieve levitation.
Eddy current repulsive force:
The levitation force increases with speed $v$. However, drag force (braking force) is also generated simultaneously.
So it doesn't levitate unless it's above a certain speed.
For EDS, the levitation start speed is about 150 km/h. It runs on wheels at low speeds.
Summary
- EDS — Eddy current repulsion. Superconducting magnets + ground coils.
- EMS — Electromagnetic attraction. Feedback control is essential.
- Levitation force $\propto v B^2$ — Speed dependent.
The Mechanics of Magnetic Levitation—Earnshaw's Theorem and Why "Static Magnetic Fields Cannot Levitate"
"Earnshaw's theorem," proven by Samuel Earnshaw in 1842, demonstrates that charged or magnetic bodies cannot be stably levitated using static electromagnetic forces alone. This means magnetic levitation is fundamentally a dynamic (control-dependent) technology. Levitation using superconductors via perfect diamagnetism (Meissner effect) or diamagnetic materials stands as an exception to this theorem. Superconducting magnetic levitation in linear motor cars can be understood in this context, providing the theoretical basis for the advantage of superconductors: "stable levitation without control."
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 where 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 heating.
- Magnetic Field Term $\nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t$: Ampère–Maxwell 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 significant, describing electromagnetic wave radiation.
- Gauss's Law $\nabla \cdot \mathbf{D} = \rho_v$: States that electric charge is the source of divergence of electric flux. 【Everyday Example】 Rubbing a plastic sheet against hair causes static electricity, making hair stand up—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 $\nabla \cdot \mathbf{B} = 0$: Indicates the absence of magnetic monopoles. 【Everyday Example】 Cutting a bar magnet in half does not create a magnet with only a N pole or only a 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.
- Isotropic assumption: Direction-specific property definitions are needed for anisotropic materials (e.g., rolling direction of silicon steel sheets).
- 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 Oersted (Oe): 1 Oe = 79.577 A/m |
| Current Density $J$ | A/m² | Calculated from conductor cross-sectional area 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
Levitation Force Analysis with FEM
How do you perform FEM analysis for magnetic levitation?
Solve the interaction between moving magnets/coils and the conductor. Methods:
- Sliding Mesh — Separate meshes for moving and fixed bodies, coupled at the interface.
- Convection Term — Fix the conductor and represent motion via $\sigma(\mathbf{v}\times\mathbf{B})$.
- Time-Domain Transient Analysis — Dynamic analysis including speed changes.
Levitation force is calculated using Maxwell's stress tensor or the virtual work method.
Can EMS control simulation also be done with FEM?
For EMS, electromagnet current is controlled via PID feedback. Couple FEM with control circuit software (e.g., MATLAB/Simulink). JMAG has Simulink co-simulation capabilities.
Summary
- Sliding Mesh — FEM handling of moving bodies.
- Maxwell Stress — Calculation of levitation force.
- FEM-Control Coupling — Feedback control for EMS.
FEM for Magnetic Levitation—Generating Lookup Tables for Nonlinear Magnetic Forces
In FEM analysis of magnetic levitation systems, it is common to generate a lookup table (LUT) of electromagnetic forces for combinations of electromagnet current values and levitation gaps. The resolution of the LUT (grid count for current and gap) and interpolation accuracy determine the precision of the control simulation. To accurately incorporate nonlinear magnetic saturation, Newton's method must converge at each point, requiring large-scale parallel computation of hundreds to thousands of FEM calculations.
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).
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