Three sentences first. The wall of every pull-down (attraction) system is P ∝ h2 — double the gap, quadruple the drive power. Breakthrough level one: let a permanent-magnet ring carry the static load, and power falls with (1−η)2 — moving the bias ratio η from 80% to 95% is a 16× reduction. Level two: split the correction work across a multi-matrix coil array, and total drive power falls as 1/N — double the channels, halve the power.
The wall: the square law of the gap
The full derivation chain, four steps:
Maxwell stress (magnetic pressure): B = flux density at the pole face · A = pole area · μ₀ = permeability of free space
Balance condition F = mg solved for B. Note what is not in this formula: the gap. Required B is set by payload and pole area alone
Air-gap ampere-turns: N = turns, I = current, h = gap length. Two gaps in series → double it
Copper loss: ρ = resistivity, ℓ = mean turn length, kfill = winding fill factor, Aw = winding window. Net result: Pcu ∝ (NI)2 ∝ h2
The key insight: the B you need has nothing to do with the gap — the entire price is paid in pushing that B across the air.
| Gap h (mm) | 15 | 20 | 25 | 30 | 40 | 50 |
|---|---|---|---|---|---|---|
| Relative power (P ∝ h2) | 1.0 | 1.8 | 2.8 | 4.0 | 7.1 | 11.1 |
| Engineering meaning | Natural cooling | Acceptable | Cooling must be considered | Heatsink + airflow | Thermal design becomes a main cost | Impractical at consumer grade |
Level one: the magnet ring carries the static load (bias ratio)
Bpm = field contributed by the permanent magnet ring · Bcoil = field the coil must add · η = bias ratio · P₀ = power of a pure electromagnet doing the same job
| Bias ratio η | 0 (pure EM) | 50% | 80% | 90% | 95% | 99% |
|---|---|---|---|---|---|---|
| Coil must supply | 100% | 50% | 20% | 10% | 5% | 1% |
| Relative power | 100% | 25% | 4% | 1% | 0.25% | 0.01% |
P/P₀ = (1−η)2 on a log scale: the leverage is in the last few points of bias.
The leverage point: moving η from 80% to 95% shifts only 15 more points of load onto the magnet — and cuts coil power sixteen-fold.
Bonus: linearity comes along
Three terms: static lift (free) · controllable force proportional to coil current, with gain amplified by Bpm · a small quadratic remainder
While ΔB ≪ Bpm, the controllable force is linear in current — and its gain is amplified by Bpm. A pure electromagnet is a square-law, single-polarity, asymmetric actuator; add bias and it becomes a linear, bidirectional one.
Level two: multi-matrix coils
Fix the total correction force Fc and split it across N channels: current per channel scales as 1/N, copper loss per channel as (1/N)2, and N channels together:
P₁ = power of a single channel doing all the correction work — double the channels, halve the total power
| Channels N | 1 | 2 | 4 (common today) | 8 | 12 |
|---|---|---|---|---|---|
| Current per channel | 1.00 | 0.50 | 0.25 | 0.125 | 0.083 |
| Total power | 1.000 | 0.500 | 0.250 | 0.125 | 0.083 |
Two extra dividends:
- Angular uniformity. The circumferential force distribution carries ripple whose harmonic order equals N; higher orders decay faster — a denser array makes force ripple finer and smaller.
- Redundancy. One dead channel still holds the float. For unattended displays this is worth more than the power saving.
The costs, honestly:
- Power devices, PCB area and assembly steps grow with N.
- Channel-to-channel consistency (gain, phase, magnetic symmetry) gets tighter.
- Array density is capped by the bore — never trade effective pole area for channels.
How float distance goes up
After two stages of power saving, height stops being a power problem and becomes a geometry problem again.
A = pole area · Ro = outer radius · Ri = inner (bore) radius — the pivot of the height-versus-load trade-off
A ring magnet’s field lines leave the outer edge, arch over the top, and return to the inner edge — how high they can arch sets the usable height. A larger bore arches higher, but shrinks the pole area.
| Orientation | Bore | Pole area A | Usable height | Load capacity |
|---|---|---|---|---|
| Compact | Small | Large | Lower | High |
| Balanced | Medium | Medium | Medium | Medium |
| High-float | Large | Small | High | Must be bought back with outer diameter |
Public material reference: hard ferrite remanence ≈ 0.4 T; sintered NdFeB N42 ≈ 1.3 T — roughly three times. At equal pole area there is real headroom in both height and load; the price is material cost and thermal management.
Conclusion: height can only be bought with area and magnet grade — high float is naturally a premium tier.
The cost ledger
| Item | Change | Note |
|---|---|---|
| Magnet ring | ↑ | Larger volume, higher grade |
| Coils & drivers | ↑ | Channel count N grows |
| Power adapter | ↓↓ | Power falls with (1−η)2·(1/N) — the adapter spec drops sharply |
| Thermal structure | ↓↓ | Less heat → simpler heatsinks, airflow paths and vents |
| Long-term reliability | ↑ | Lower temperature rise → longer life and steadier performance |
Conclusion: BOM goes up, but the system-level total cost can tie — or even win. For certified, always-on display products it is especially worthwhile.
Formula cheat sheet
| Relation | Expression | Use |
|---|---|---|
| Magnetic pressure | F = B2A/(2μ₀) | Load capacity |
| Required flux density | Bneed = √(2μ₀mg/A) | Design starting point — independent of the gap |
| Air-gap ampere-turns | NI = B·h/μ₀ | Turns and current |
| Copper loss | Pcu ∝ (NI)2 | → yields P ∝ h2 |
| Bias benefit | P/P₀ = (1−η)2 | Direct conversion of the magnet’s share |
| Array benefit | P ∝ 1/N | Double the channels, halve the power |
| Pole area | A = π(Ro2−Ri2) | Pivot of the height-load trade-off |
The honesty clause
That physics honesty is also the natural hand-off to our sister article: Next-Generation Push-Up Levitation: Magnet Matrix + Coil Arrays.
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: a real pull-down project — the 42cm Giant Levitating Globe for a school STEM lab — and our levitation module lineup.