Three sentences before the details. First: push-up (repulsion) levitation gets one axis for free — the vertical direction is passively stable, so static power draw approaches zero in theory, because the quantity being controlled is supposed to be zero at steady state. Second: float height is decided by the magnets — pole area, inner-bore span, energy product. It is a clean geometry bill that no coil or control trick can pay. Third: stability is decided by the coils, and the difficulty rises with a high power of the float height. That is why our next-generation push-up platform is built on two ideas that are optimized separately: a high-energy magnet matrix and denser coil arrays.
One axis is free
Earnshaw’s theorem (1842) states that no arrangement of static magnets can be stable in every direction at once — at least one degree of freedom must be handed to active control or to a non-magnetic force. The real engineering question is which degree of freedom you give away.
| Architecture | Vertical axis Z | Horizontal X/Y | Static power |
|---|---|---|---|
| Pull-down (attraction) | Active control | Passively stable | Control loop must stay powered |
| Push-up (repulsion) | Passively stable | Needs correction | Approaches zero in theory |
Push-up architecture spends its free pass on the direction that carries weight. The vertical restoring force comes from permanent-magnet repulsion itself — rise and the repulsion weakens, sink and it strengthens; the physics is a spring. The electronics only have to mind the horizontal plane, where the force required at the ideal centered position is exactly zero.
What the field above a ring magnet looks like
Field lines leave one side of the ring, arch up over the center, and come back down on the other side — a family of closed arches. Two direct consequences:
- The arch height is the usable float height. The arch span is set by the inner bore — a wider bore arches higher.
- Field strength decays fast with height. Hugging the surface, the field concentrates above the ring band as two peaks; higher up the peaks merge and the central region becomes the usable working zone.
Ring magnet in cross-section: field lines arch over the bore; the floater works at height h in the merged-field zone.
A question every customer asks: “The floater is visibly smaller than the ring bore — isn’t that wasting the field?” No. The floater is positioned where the field gradient is strongest. Levitation runs on the rate of change of field strength with position, not on raw strength itself. A larger floater magnet mostly adds self-weight without adding proportional usable force. The criterion is not “fill the bore” — it is whether force ÷ self-weight is optimal.
Area, bore and height — a trade you can compute
A = pole area · Ro = outer radius · Ri = inner (bore) radius — the geometric base of load capacity
Far-field dipole approximation: m₁, m₂ = magnetic moments, h = separation. A real ring deviates in the near field — use it for orders of magnitude: double the height, force falls to about 1/16
| Adjustment | Pole area A | Usable height h | Load capacity | Cost |
|---|---|---|---|---|
| Larger outer diameter | ↑ | ↑ (wider arch span) | ↑ | Volume, cost |
| Larger inner bore | ↓ | ↑↑ | ↓ | Load drops; outer diameter must buy it back |
| Thicker magnet | ≈ | ↑ (diminishing) | ↑ (diminishing) | Weight, cost |
| Higher energy-product grade | ≈ | ↑ | ↑ | Material cost, thermal management |
Public material reference: hard ferrite remanence ≈ 0.4 T; sintered NdFeB N42 ≈ 1.3 T — roughly three times. One accuracy note: material does not change the shape of the field distribution (geometry owns the shape); it scales the whole working point upward.
Stability is a separate budget
Correction-force budget: kneg = horizontal negative stiffness of the magnet field · xmax = worst-case horizontal excursion · Fdisturb = disturbance allowance — the starting point for coil sizing
Coil output also decays fast with height — the quantitative way of saying “higher float is harder to hold”
Engineering rule of thumb: size peak actuator force at 1.5–2.5× the floater’s weight. It looks generous until you remember the floater can swing and spin; real peak demand routinely exceeds intuition.
The electrical time constant sets control bandwidth; too little bandwidth and the correction force arrives late
Damping ratio: ζ too small means slow settling. Vertical damping is inherently weak — the root of micro-bounce — and needs extra means such as eddy-current damping in a conductor
Corollary: when float height goes up, control difficulty does not rise linearly — it rises with a high power of h. To keep the same stability margin, the correction system must grow in step.
The magnet matrix: array the permanent side too
Today’s platform: one large ring magnet plus four coil sets. The next generation turns the permanent side into a matrix of high-energy magnet units arranged around the circle, with coil groups matched to them.
| Dimension | One solid ring | Magnet matrix |
|---|---|---|
| Circumferential uniformity | Depends on single-piece manufacturing consistency | Array averaging dilutes local errors |
| Height / load decoupling | Strongly coupled — change one, move the other | Zone-configurable, more design freedom |
| Coil pairing | Layout constrained by ring geometry | Units map one-to-one to coils; shorter correction path |
| Redundancy | One failed piece kills the whole | Unit-level redundancy, higher reliability |
Why arraying directly helps stability — the core insight of this article. The circumferential field of any real ring carries periodic ripple, and its harmonic order equals the number of array units N. A general rule: higher harmonic orders decay faster with distance. Densify the array and the force ripple becomes finer and smaller.
The costs, honestly:
- More magnet units → tighter assembly and magnetization consistency requirements.
- Drive channels grow with coil groups → more circuit and tuning complexity.
- Array density is capped by the bore — never sacrifice the central working area for density.
Formula cheat sheet
| Relation | Expression | Use |
|---|---|---|
| Pole area | A = π(Ro2−Ri2) | Geometric base of load capacity |
| Far-field decay | F ∝ m₁m₂/h4 | Double the height → force ≈1/16 (order of magnitude) |
| Magnetic pressure ceiling | F = B2A/(2μ₀) | Force ceiling on a given pole area |
| Correction budget | Fcoil ≥ kneg·xmax + Fdisturb | Coil selection starting point |
| Control bandwidth | fc ≈ R/(2πL) | Whether the response keeps up |
| Damping ratio | ζ = c/(2√(km)) | How fast jitter dies out |
| Output decay | Fcoil ∝ 1/hn, n≈3–4 | Why “higher float is harder to hold” |
| Engineering margin | Fpeak ≈ (1.5–2.5)·mg | Peak actuator force rule of thumb |
Closing
Height to the magnets — bought with pole area, bore span and energy product: a clean geometry bill. Stability to the coils — held with array density, control bandwidth and a correction-force budget, while demand grows with a high power of h.
Once the two are separated, “float higher” and “float steadier” stop dragging each other down: the magnets lift the operating point, and the coil matrix keeps order at that higher position.
This article is a technology outlook based on published physics and public material data. It describes design directions — not product specifications or delivery commitments.
Related reading: our current push-up modules in production — the PU300G 300g platform — plus Why Does It Spin? Rotation in PU Levitation Systems and the Global Adapter Guide Levitation Module Selection Guide. Sister article: Next-Generation Pull-Down Levitation: Magnetic Bias + Multi-Matrix Coils.