Magnetic Levitation
Magnetic Levitation: Theoretical Foundations
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."
Computational Methods for Magnetic Levitation
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.