Earnshaw's theorem
The honest starting point for any levitation project: a set of fixed permanent magnets cannot hold another object in stable static equilibrium. Earnshaw's theorem guarantees that for any arrangement, at least one axis is unstable — nudge the floating part and it either falls or flies to a magnet.
This is why a simple stack of repelling magnets always slips sideways and why "pure magnet" perpetual levitation devices don't work. Every real levitator adds one more ingredient to defeat the instability — and that ingredient defines the system.
The four stabilization routes
Permanent magnets supply the lifting force; something else supplies the missing stability:
| Route | How it stabilizes | Where PMs fit |
|---|---|---|
| Motion / induced currents | Relative motion induces eddy currents that push back (Inductrack, EDS) | Halbach array is the moving field source |
| Diamagnetic | Diamagnets (pyrolytic graphite, superconductors) repel and self-center | Small NdFeB over graphite; flux-pinned over HTS |
| Spin | Gyroscopic stability (the Levitron top) | Ring/disc magnet spun above a base magnet |
| Active control | Sensors + electromagnets servo the gap (EMS maglev, mag bearings) | Bias flux from PMs, trim with coils |
| Mechanical constraint | One axis held by a bearing; PMs offload the rest | Passive PM bearing + one contact axis |
If a design claims stable levitation from permanent magnets alone, one of these five is hiding in it — or it doesn't work. Identify which one your system uses before sizing magnets.
Halbach / Inductrack maglev
The Inductrack approach mounts a Halbach array on the vehicle above a passive conductive track (a litz-wire or shorted-ladder circuit). The Halbach geometry concentrates flux on the track side and cancels it above, and the vehicle's motion induces currents in the track that repel the array — lifting it once speed passes a low threshold.
- Passive & fail-safe — no track power and no active control for lift; the physics does the work above threshold speed.
- Halbach is the enabling geometry — the one-sided field puts the magnetic energy where it levitates and keeps it off the passengers. See the live Halbach array visualizer and the Halbach array products.
- Array period & speed — the wavelength of the Halbach pattern and the vehicle speed set the induced-current frequency and the lift/drag ratio.
Passive magnetic bearings
Permanent-magnet bearings use repelling or attracting magnet rings to carry a shaft load without contact — cutting friction, wear, and lubrication in flywheels, turbomolecular pumps, meters, and spindles.
- They can't be stable in every axis (Earnshaw again) — a passive PM bearing always needs one axis held by a mechanical bearing, an active magnetic bearing, or a stabilized counterpart.
- Ring geometry — concentric or stacked ring magnets in repulsion (radial support) or attraction; stiffness comes from grade, gap, and ring cross-section.
- Offloading, not eliminating — the usual win is taking most of the load off a mechanical bearing to extend its life, not removing all bearings.
Field, force & gap
- Levitation force falls steeply with gap — like every magnetic force, it drops fast as the working distance grows; small gap changes are large force changes.
- Strong-side field — a Halbach array roughly doubles the useful one-sided field versus a plain bar array of the same magnets, which is why it dominates levitation designs.
- Grade & geometry — force scales with the square of working flux, so remanence and array geometry both matter; a grade step is a few percent, geometry is the rest (grades chart).
Material & environment
- NdFeB for maximum force density — the default for Halbach arrays and PM bearings.
- Temperature: induced currents heat conductive tracks and bearing structures, and that heat feeds back into the magnets; size the temperature class for real operating temperature, or step to SmCo where it runs hot.
- Retention at speed: magnets in rotating bearings or fast vehicles see centrifugal and dynamic loads — bond and mechanically capture them and qualify at max speed (mounting & retention).
- Coating: matched to the environment; outdoor transport and wet pump bearings need robust corrosion protection.
Design pitfalls
| Mistake | Consequence | Fix |
|---|---|---|
| Expecting stable static PM levitation | It slips out sideways — Earnshaw | Add motion, diamagnetism, spin, or active control |
| Ignoring the unstable axis in a PM bearing | Shaft crashes to one side | Constrain one axis mechanically or actively |
| Plain bar array instead of Halbach | Half the useful lift, field on both sides | Use a Halbach array for one-sided force |
| Ignoring eddy heating | Magnets warm, force fades | Size temperature class; cool the track/bearing |
| Adhesive-only retention at speed | Magnets migrate or fly | Bond + capture; qualify at max speed & temp |
| Designing at nominal gap only | Force collapses at max gap | Design at worst-case gap with margin |
Specifying a levitation magnet
For a levitation or bearing magnet set, alongside the RFQ checklist:
- Stabilization method: which of the four routes the system uses — it frames everything.
- Array geometry & period (Halbach) or ring dimensions (bearing), and the working gap.
- Force / stiffness target at the working gap, with margin.
- Grade & temperature class for the real thermal environment (including eddy heating).
- Retention and max speed / dynamic load for the centrifugal check.
- Coating & environment.