Arc & Segment Rotor Calculator
Go from rotor diameter and pole count to a segment drawing. This tool returns the included angle, arc length, chord and mass for every segment, sizes the back iron from the flux it has to carry, and checks whether the adhesive or the retaining sleeve actually holds the magnets on at your maximum speed.
interactive — segment geometry, retention & yoke
Centrifugal load scales with the square of speed and linearly with radius. A rotor that is perfectly safe on adhesive at 3,000 rpm may need an Inconel or carbon sleeve at 12,000 rpm on the same diameter. Check the retention verdict before you commit to a bonding process — retrofitting a sleeve changes the air gap, and therefore the whole magnetic design.
segment geometry and nesting
An arc segment is defined by four numbers on the drawing: outer radius, inner radius, included angle and axial length. Everything else — chord width, arc length, volume — falls out of those. What is not obvious is how much the pole arc ratio costs you in material and how much it buys you in waveform quality.
| pole arc ratio | effect |
|---|---|
| 1.0 — full arc | Maximum flux and maximum material. Segments touch, which makes assembly fiddly and leaves no room for tolerance. Rarely used. |
| 0.85–0.95 | The usual production band. Small inter-pole gap eases assembly and trims the harmonic content slightly. |
| 0.70–0.85 | Noticeably lower cogging and a cleaner back-EMF waveform. Costs fundamental amplitude, saves material. |
| below 0.70 | Waveform shaping territory. Flux drops fast — usually only justified where torque ripple is the binding constraint. |
Bread-loaf and inset variants change the picture. A bread-loaf segment — curved outer face, flat inner face — is easier to grind and seats on a machined flat rather than a curved surface, at the cost of non-uniform thickness. Inset magnets recessed into the yoke get mechanical retention from the iron between poles and gain reluctance torque, but the inter-pole iron also provides a leakage path.
retention at speed
A surface-mounted magnet is held on by whatever resists its own centrifugal load. Adhesive alone works to a point; above it you need a sleeve, and the sleeve has to carry both the magnet load and its own.
| option | where it fits | trade |
|---|---|---|
| adhesive only | low surface speed, moderate radius | cheapest; bond line is a single point of failure and degrades with thermal cycling |
| stainless sleeve | modest speeds | conductive — eddy losses in the sleeve, same physics as a coupling containment shell |
| inconel sleeve | high speed, high temperature | strong and higher resistivity than stainless, but still conductive; expensive |
| titanium sleeve | high speed, weight-sensitive | good strength-to-density; conductive |
| carbon fibre wrap | very high speed | strongest and non-conductive — no eddy loss; needs interference fit and careful thermal matching |
| inset pockets | any speed, if the design tolerates it | mechanical retention with no sleeve; adds inter-pole leakage |
Every millimetre of sleeve is a millimetre of magnetic gap. Add a 0.6 mm sleeve to a 1 mm gap design and you have a 1.6 mm gap — the flux density falls accordingly and the machine loses torque. Size the sleeve at the same time as the magnet, not after. The gap penalty is already included in the flux figure above.
sizing the back iron
The rotor yoke carries half the flux of one pole, splitting left and right around the circumference. Undersize it and the iron saturates, shunting flux away from the air gap and quietly costing torque that no amount of extra magnet will recover. The recommended thickness above comes straight from that flux balance.
Two practical notes. A solid steel hub is normal on a rotor because the field is largely static in the rotor frame, so lamination is usually unnecessary. And the hub also has to carry the shaft torque and the interference fit, so the structural requirement frequently exceeds the magnetic one — take the larger of the two.
reference — the formulas used
| segment geometry | θ = α_p · 360° / N R_o = OD/2, R_i = R_o − t_m arc length = θ_rad · (R_o + R_i) / 2 chord = 2 · R_o · sin(θ/2) volume = (θ_rad / 2) · (R_o² − R_i²) · L |
| air-gap flux density | B_g = (4/π) · sin(α_p·π/2) · Br(T) · t_m / (t_m + µ_rec · g_eff)g_eff includes the sleeve wall |
| yoke thickness | t_yoke = B_g · τ_p / (2 · B_sat), τ_p = π · OD / Nhalf the pole flux travels each way around the yoke |
| centrifugal pressure on the sleeve | p = ρ_m · t_m · ω² · R_mean, ω = 2π·n/60magnet mass per unit area × centripetal acceleration |
| sleeve hoop stress | σ = p · R / t_s + ρ_s · v²magnet load plus the sleeve's own centrifugal self-load, v = surface speed |
| bond stress, no sleeve | σ_bond = pthe adhesive carries the full radial load in tension |
The retention model treats the sleeve as a thin ring in pure hoop tension and ignores interference-fit pre-stress, thermal growth differentials and any stress concentration at segment edges. Interference fit in particular changes the picture significantly — it pre-loads the sleeve, which raises working stress at rest but keeps the magnets seated at speed. Treat the numbers as a screening calculation and have a structural analysis done before committing a high-speed rotor to tooling.
going deeper — segment tolerancing and coating choices are covered in arc & segment magnets, bonding and retention practice in bonding & mounting magnets. If the pole count is repeating and the volume is real, compare against a one-piece sintered ring in the multipole ring visualizer — it frequently wins on installed cost. Check the magnet operating point at temperature with the demagnetization calculator before finalising the grade.