Magnetic Coupling Torque Calculator
Size a synchronous permanent-magnet coupling — coaxial or disc — from rotor geometry, pole count, magnet grade and gap. The calculator returns peak air-gap flux density, air-gap shear stress, pull-out torque, the full torque-versus-load-angle curve, and eddy-current loss in a metallic containment shell.
interactive — pull-out torque estimate
The analytic 2-D result is an upper bound. Production couplings typically measure 40–60 % of the ideal value once three-dimensional end leakage, inter-pole leakage, yoke saturation and bond-line gaps are included — which is why the realism factor defaults to 0.50. Treat the output as sizing guidance, then confirm with a prototype. Need the rings? custom arcs and radially-oriented rings are our core business.
Coaxial coupling, developed flat. Each rotor carries an alternating-polarity magnet array on a steel yoke; torque is transmitted across the gap and through the non-magnetic containment shell.
method — how the estimate works
The model is the tangential Maxwell stress integral, the same relation used in published coaxial-coupling and magnetic-gear analysis. Four steps, no empirical curve fitting until the final realism factor:
| step | what is computed |
|---|---|
| 1 — series magnetic circuit | Both magnet arrays sit in series across the total mechanical gap. Ampère's law around the loop gives the aligned square-wave gap flux density from magnet thickness, recoil permeability and total clearance. |
| 2 — fundamental & leakage | The square-wave distribution is reduced to its fundamental (a 4/π factor scaled by pole arc ratio), then penalised for inter-pole leakage, which grows as the gap approaches the pole pitch. Short pole pitch — a high pole count on a small diameter — leaks badly. |
| 3 — Maxwell stress | For a pure multipole field in the annulus the radial and tangential components have equal amplitude, so the tangential stress integrates to a closed form. Torque varies as sin δ in electrical angle, so the mechanical breakaway angle is δ/p. |
| 4 — shell losses | A conductive containment shell sees a field pattern sweeping past at rotor surface speed. The induced current density gives a volumetric loss that scales with speed squared and wall thickness. |
Temperature enters through the remanence coefficient of the selected material. Because torque scales with the square of flux density, a 5 % loss of Br costs roughly 10 % of pull-out torque — the single most common reason a coupling that passed on the bench slips in service. Check the magnet operating point separately with the demagnetization & permeance calculator.
reality check — what the analytic value misses
| effect | typical impact on measured pull-out torque |
|---|---|
| axial end leakage | flux escapes the ends of a short rotor — 10–25 % loss when axial length is comparable to pole pitch |
| inter-pole leakage | partly modelled, but rises sharply once total gap exceeds roughly a quarter of pole pitch |
| yoke saturation | a thin steel back-iron saturates and shunts flux — 5–20 % loss; size yoke thickness to at least half the pole pitch |
| bond line and encapsulation | adhesive, potting and a can liner each add effective gap — often 0.3–0.8 mm total |
| magnet tolerance and arc fit | arc segments that do not seat cleanly on the yoke add a parasitic gap |
| elevated temperature | force and torque scale with B², so Br derating roughly doubles in effect |
| eddy-current drag | a metallic shell removes real shaft power and heats the magnets — see below |
| pole count | this model favours fewer poles monotonically, because it only penalises leakage. Real couplings have an optimum, usually 6–12 poles: too few and the yoke must be thick to carry the flux and the breakaway angle becomes coarse; too many and leakage wins. Treat a two-pole-pair result as an upper bound you will not reach in practice. |
Size pull-out torque at a minimum of 1.5–2× the worst-case load torque, evaluated at the highest expected magnet temperature and the largest gap the shell can reach under pressure. A synchronous coupling that slips a pole does not re-synchronise on its own — the drive must be stopped and restarted, and the slip event dumps heat into the shell and magnets.
containment shells — the eddy-current penalty
If the barrier between the rotors is metallic and conductive, the rotating field induces circulating currents in the wall. That loss appears as drag torque on the driver, heat in the shell, and — because the shell sits millimetres from the magnets — a temperature rise that erodes the very flux producing the torque.
| shell material | resistivity Ω·m | notes |
|---|---|---|
| peek / pps | insulating | zero eddy loss; pressure and temperature limited |
| ceramic / cfrp | insulating | zero eddy loss; brittle or anisotropic, higher cost |
| hastelloy c-276 | 1.25 × 10−6 | highest resistivity of the common alloys — lowest metallic loss |
| 316 stainless | 7.4 × 10−7 | default choice; moderate loss, excellent availability |
| titanium grade 2 | 5.6 × 10−7 | higher loss than Hastelloy, lighter and strong |
The calculated loss is a classical upper bound: it neglects the armature reaction of the induced currents, which partially shields the shell and reduces loss at high speed. Expect real losses of roughly 50–80 % of the figure shown for thin walls at moderate speed, approaching the calculated value as wall thickness and frequency fall.
reference — the formulas used
| series gap flux density | B_sq = Br(T) · (h₁ + h₂) / (h₁ + h₂ + µ_rec · g_mech)h = magnet thickness per rotor, g_mech = total mechanical clearance plus shell wall, µ_rec ≈ 1.05 |
| per-rotor fundamental | B̂ = ½ · (4/π) · sin(α_p · π/2) · B_sq · k_leak k_leak = 1 / (1 + 2·g_mech / τ_p), τ_p = π·D_g / (2p) |
| pull-out torque — coaxial | T(δ) = (2π · L · R_g² · B̂² / µ₀) · sin(δ) T_max = 2π · L · R_g² · B̂² / µ₀δ in electrical radians; mechanical breakaway angle = 90° / p |
| pull-out torque — disc | T_max = (2π · B̂² / (3µ₀)) · (R_o³ − R_i³)same stress integrated over an annular face instead of a cylinder |
| air-gap shear stress | σ = B̂² / µ₀useful sanity check — well-built NdFeB couplings land in the 20–100 kPa band |
| shell eddy loss | P = π · R · L · t · v² · B̂² / ρ, v = 2π·n·R / 60classical upper bound, neglects reaction field and skin effect |
| remanence vs. temperature | Br(T) = Br₂₀ · [1 + α · (T − 20) / 100] |
going deeper — magnetic-circuit theory, load lines and permeance coefficients are covered in magnets 201 — advanced magnetics engineering. For the application context — sealless pumps, agitators and HVAC drives — see pumps, HVAC & magnetic couplings, and for the arc and ring geometry itself, arc & segment magnets.