Field calculator + engineering guide
Estimate peak air-gap flux density. Compare geometry and single or dual rotors, then check the model limits before selecting magnets.
0.5092 T
At mean radius 70.0 mm; z = 2 mm from the magnet face.
Ideal open-circuit screening estimate; peak normal component, not RMS. Real assembly accuracy is unknown. Default estimate; JavaScript enables editing.
Next: validation checklistChange geometry to compare scenarios. In dual mode both identical arrays reinforce Bz at the gap midpoint; the entered distance is half the total gap.
Isolated-array B0
0.6400 T
Decay length 1/k
8.75 mm
Thickness factor
49.6%
Segmentation factor
97.45%
Br at temperature
1.3233 T
Magnets per rotor
64
At one decay length the single-source field retains 36.8%. Dual mode adds the two sources at their common midpoint.
Scroll the table on narrow screens. At physical edges these are local planar references; radial fringing is excluded.
| Radius (mm) | Pitch (mm) | Peak Bz (T) |
|---|---|---|
| Inner: 50.0 | 19.63 | 0.5779 |
| Mean: 70.0 | 27.49 | 0.5092 |
| Outer: 90.0 | 35.34 | 0.4462 |
Rear leakage, harmonic amplitudes/THD, force, torque and demagnetization margin: not predicted. Use a geometry-specific field solution and measured Hall map.
First recorded: · Reviewed: · Sources checked: October 27, 2026 · Next review: April 27, 2027
Prepared and maintained by Halbach Array, a commercial magnetic assembly supplier. We may quote assemblies discussed here. The first recorded date is the repository record; public publication has not been independently verified. Calculated examples are software evidence, not customer results or hardware qualification.
| Conclusion | Evidence / scope | Decision |
|---|---|---|
| Distance and pole pitch act together | Planar fundamental model | Compare kz = pz/r, with clearance fixed by the real assembly. |
| Thickness gains saturate | Calculated from 1-exp(-kh); see Section 3 | Compare incremental field with added volume and retention load. |
| Dual arrays add at a common plane | Linear superposition under the stated free-space assumptions | Define the total gap and phase before comparing results. |
| Finite radial edges need 3D treatment | Axial-machine 3D analysis | Treat the radial table as a trend, then verify a finite rotor. |
| A catalog temperature ceiling is not a safety margin | Supplier grade data | Check actual hot B-H curves and reverse-field load cases. |
The tool applies a local planar fundamental model to an annulus. The planar amplitude and decay follow Post and Ryutov; replacing the linear wavenumber with p/r is the approximation made on this page. It assumes uniform equal-angle segments, full fill, no iron, no current and magnet relative recoil permeability of one.
r_mean = (R_in + R_out) / 2; k = p / r_mean; pole pitch = pi / k
M = 2 * segments_per_pole; S = sin(pi/M) / (pi/M)
Br(T) = Br20 * [1 + alpha * (T - 20) / 100]
B0 = Br(T) * S * [1 - exp(-k*h)]
Single: Bz(z) = B0 * exp(-k*z)
Dual: Bz(s) = B0 * [exp(-k*s) + exp(-k*(g-s))]
Dual midpoint: s = z = g/2; Bz = 2 * B0 * exp(-k*z)
Use mm for every length so k is in inverse mm; the exponents remain dimensionless. The result is the angular peak of the normal fundamental at the selected radius and plane. B0 always refers to an isolated array; it is not the total surface field of the dual assembly.
With the defaults: Br20 = 1.39 T, alpha = -0.12%/C, T = 60 C, radii 50/90 mm, p = 8, M = 8, h = 6 mm and z = 2 mm, the model gives Br(T) = 1.3233 T, B0 = 0.6400 T and Bz = 0.5092 T.
Compare dimensionless ratios before changing a design. At fixed Br and segmentation, distance multiplies the field by exp(-kz), while thickness multiplies it by 1-exp(-kh). These factors are calculated from the equations above; they are not measured machine efficiencies.
| Ratio | Distance retained at z/pitch | Thickness factor at h/pitch |
|---|---|---|
| 0.1 | 73.0% | 27.0% |
| 0.25 | 45.6% | 54.4% |
| 0.5 | 20.8% | 79.2% |
| 1 | 4.3% | 95.7% |
Going from h = 0.5 pitch to h = 1.0 pitch raises the thickness factor from 79.2% to 95.7%: 16.5 percentage points, or about 20.8% relative field gain. Magnet volume doubles at unchanged planform. A cost increase needs a supplier quote; it cannot be inferred from volume alone.
The equal-step fundamental factor approaches one as M rises. The planar report supports this screening factor; a full harmonic solution needs the actual magnetization and observation region. The tool does not infer a total harmonic distortion from two selected sidebands.
| Segments / pole | M / pole pair | Fundamental factor | Magnets / rotor |
|---|---|---|---|
| 2 | 4 | 90.03% | 32 |
| 3 | 6 | 95.49% | 48 |
| 4 | 8 | 97.45% | 64 |
| 6 | 12 | 98.86% | 96 |
| 8 | 16 | 99.36% | 128 |
Choose segmentation against the waveform requirement, available magnetization directions, bond lines, inspection and assembly cost. A higher factor alone does not establish low eddy-current loss or low machine torque ripple. Rear leakage remains unknown without finite geometry and a measurement plane.
| Question | Single array | Matched dual arrays |
|---|---|---|
| Field at distance z | B0 * exp(-kz) | 2 * B0 * exp(-kz) at midpoint; g = 2z |
| Magnet inventory | One rotor | Two identical rotors; twice the magnet count |
| Mechanical review | Retention and any loads from surrounding material | Rotor-to-rotor attraction, disc deflection and assembly fixturing |
| When the simple comparison fails | Iron, current, edge fringing or partial fill | All single-array limits plus phase mismatch or unequal arrays |
Symmetry does not remove rotor attraction. Evaluate force using the full field and a defined integration surface, then check discs, bearings, retention and fault loads. The force calculation guide explains the separate force task.
These presets are representative supplier catalog inputs, checked October 27, 2026. Br refers to 20 C; coefficients are in percent per C. A listed maximum is a screening ceiling, not an assembly qualification. Supplier ranges, product revisions and actual B-H curves take precedence.
| Preset | Br20 (T) | Coefficient (%/C) | Ceiling (C) | Source |
|---|---|---|---|---|
| NdFeB N52 | 1.45 | -0.12 | 80 | Alliance |
| NdFeB N48SH | 1.39 | -0.12 | 150 | Arnold Neo |
| NdFeB N42SH | 1.30 | -0.11 | 150 | Alliance |
| NdFeB N38UH | 1.26 | -0.12 | 180 | Arnold Neo |
| Sm2Co17 RECOMA 30S | 1.12 | -0.035 | 350 | Arnold RECOMA |
The reversible correction uses the signed temperature difference, including below 20 C. It is only a linear approximation over a supplier-supported range. Coercivity and irreversible losses are not calculated. Enter a custom material when its Br, coefficient or documented ceiling differs.
For demagnetization review, obtain temperature-dependent curves and check local operating points against worst-case reverse fields. IEC 60404-5 describes material measurement scope; IEC 60404-8-1:2023 covers material specifications. Neither proves that this assembly is safe at the catalog ceiling.
This calculator adapts a planar fundamental. It does not solve the finite annulus. Xia, Zhu and Howe show the importance of material and boundary conditions in cylindrical machines; Okita and Harada provide an axial-specific 3D treatment. Those papers do not give this tool a universal error percentage.
| Quantity or risk | Status here | Minimum next step |
|---|---|---|
| Peak fundamental and geometry trends | Calculated under the displayed assumptions | Match peak/RMS convention, radius, angular location and probe plane. |
| Radial edge field and finite-geometry error | Unknown; no calibrated correction | Model finite sectors and scan across the radial span. |
| Rear leakage and spatial THD | Unknown; fundamental only | Specify rear probe distance; solve and measure the full waveform. |
| Hot demagnetization / fault current | Not calculated | Use hot supplier curves, local working points and current load cases. |
| Torque, force and mechanical safety | Not calculated | Use full-field stress integration and mechanical load analysis. |
| Manufacturing cost and assembly yield | Unknown without a quote | Request segmented-part, tooling, fixture and inspection costs. |
| Slotted or iron-cored machine fit | Outside the free-space assumptions | Include nonlinear materials, slots, windings and armature reaction. |
A narrow annulus, few pole pairs or large radial span can weaken the planar approximation. A 3D magnetic model with mesh convergence and measurement comparison is the practical alternative. More decimal places in the calculator do not establish greater physical accuracy.
These are calculated examples using the displayed equations, not customer case studies, simulations or measured performance. Each starts from the default N48SH inputs: 50/90 mm radii, eight pole pairs, M = 8, 60 C, 6 mm thickness and 2 mm distance.
Single array; change z from 2 to 4 mm.
0.4051 T
Baseline 0.5092 T; change -20.4%.
Keep structural clearance requirements, then compare the loss against the field target. Do not reduce a clearance solely to recover field.
Two identical reinforcing arrays; total gap g = 4 mm, midpoint z = 2 mm.
1.0184 T
Baseline 0.5092 T; change 100.0%.
The ideal midpoint field doubles relative to a single source at the same distance. Review rotor attraction, alignment and assembly fixtures before selecting this topology.
Single array; change thickness from 6 to 12 mm at fixed z = 2 mm.
0.7657 T
Baseline 0.5092 T; change 50.4%.
Compare this incremental field gain with magnet volume, rotor mass, retention and supplier pricing. The tool does not predict machine efficiency or torque.
If the calculator rejects the geometry, document those same inputs and request a geometry-specific solution. This provides a usable next step without extrapolating the tool outside its ranges.
Request a field review with these inputsThe answers use the assumptions, equations and linked sources above. They do not replace a machine-specific field and load analysis.
The peak fundamental normal flux density Bz at the mean radius and specified plane. It is a local planar, open-circuit estimate with full magnet fill and relative recoil permeability set to one. It is not RMS, pole-average field, torque or a certified field map.
For one array, B0 = Br(T) * sin(pi/M)/(pi/M) * (1 - exp(-k*h)), and Bz(z) = B0 * exp(-k*z). The local wavenumber is k = p/r, with pole pairs p, radius r and M segments per pole pair. Use consistent length units.
Two identical arrays phased to reinforce the normal field add at the same plane: Bz(s) = B0 * [exp(-k*s) + exp(-k*(g-s))]. At s = g/2 the result is 2*B0*exp(-k*g/2). Enter z = g/2 in dual mode. Iron and load-current effects are excluded.
For a single array, z is magnet-face-to-evaluation-plane distance. For dual arrays, z is half the total face-to-face separation g = 2z. Include winding, encapsulation and clearance when locating the plane; do not enter just rotor-to-stator clearance.
Flux density B is measured in tesla; magnetic field strength H is measured in A/m. This estimate describes the amplitude of the normal fundamental component. Comparing it with probe readings requires matching the radius, plane and angular peak.
No. The radial table changes the local pole pitch with radius, but omits finite-edge fringing. Inner and outer rows are ideal trend references. Use 3D analysis and a radial Hall scan for physical edge values.
In the source-free planar model each spatial Fourier component decays with distance. The fundamental retains exp(-k*z); at z = 1/k it retains 36.8% of its isolated-array surface amplitude. Radial edge effects can change real measurements.
No. The thickness factor is 1-exp(-k*h). Doubling h from half a pole pitch to one pole pitch increases this factor from 79.2% to 95.7%, a relative field increase of about 20.8% at fixed geometry and Br. This is a calculated trend, not a cost quote.
More equal magnetization steps raise the ideal fundamental factor toward one. M=4 gives 90.03%, M=8 gives 97.45%, and M=16 gives 99.36%. Additional magnets also add placement, bonding and inspection work; decide using a quoted assembly and measured field target.
No. A fundamental-only model does not determine a complete harmonic spectrum or finite-rotor rear leakage. Those require specified magnetization, clearances, geometry and a field solution or measurement. No universal shielding ratio is assumed.
Use it only as a free-space screening reference. Iron permeability, saturation, slot geometry and armature reaction alter the boundary conditions. Use a model that includes those materials and currents before making an acceptance decision.
No. Symmetry can reduce net stator force in an ideal arrangement, while the rotors still attract each other. Size retention, discs and bearings with a geometry-specific stress calculation, assembly load cases and mechanical analysis.
Br(T) = Br20 * [1 + alpha*(T-20)/100], where alpha is entered in percent per C. The signed correction also applies below 20 C, with an extrapolation warning. It represents reversible change only and requires supplier confirmation over the actual temperature range.
No. Reverse field, operating point, temperature-dependent B-H curves and transient current all matter. The ceiling is only a screening input. Check local magnet operating points for the worst thermal and electrical load cases.
Supply Br at 20 C, its reversible temperature coefficient and a documented temperature ceiling. Those fields stay visible alongside operating temperature. Replace the prefilled screening assumptions with the actual supplier data.
Compare specific documented grades when reversible temperature stability or hot operation matters. Use the table as a starting point; coercivity, operating point, corrosion protection, mechanical strength and procurement conditions still need separate review.
Define a calibrated Hall-probe scan at agreed radii, angular positions and axial planes, recording temperature, probe orientation, standoff and uncertainty. Compare the fundamental and full waveform with 3D predictions; request supplier material curves separately.
Send radii, pole pairs, segments and magnetization directions, thickness, total separation and probe plane, material curves, target peak or RMS convention, operating temperatures, surrounding materials, current load cases, retention details and quantity.
Publication records and public catalog scopes checked October 27, 2026. The page uses the cited scope shown below; numerical comparisons are analytical calculations, not claimed measurements. Unavailable or unverified hardware outcomes remain unknown.
R. F. Post and D. D. Ryutov, Lawrence Livermore National Laboratory · 2000
Primary report describing the planar Halbach fundamental, exponential distance decay and segmentation factor. Applying k = p/r to an annulus here is a local planar approximation, not a 3D solution.
J. C. Mallinson, IEEE Transactions on Magnetics 9(4), 678–682 · 1973
Foundation for one-sided flux distributions under ideal magnetization conditions. It does not establish a rear leakage allowance for a finite manufactured rotor.
K. Halbach, Nuclear Instruments and Methods 169(1), 1–10 · 1980
Foundational oriented permanent multipoles; background for magnetization design, not validation of this calculator or an axial machine performance guarantee.
Z. P. Xia, Z. Q. Zhu and D. Howe, IEEE Transactions on Magnetics 40(4), 1864–1872 · 2004
Polar-coordinate models include recoil permeability and compare with FEA and measurements. Its internal/external rotor analysis should not be read as an axial-annular error bound.
T. Okita and H. Harada, IEEE Access 11, 2078–2091 · 2023
Axial-machine-specific 3D formulation illustrates why finite geometry needs its own field model. No universal radial correction percentage is inferred from it here.
Arnold Magnetic Technologies · Live manufacturer catalog; checked October 27, 2026
N48SH and N38UH preset Br, reversible coefficient and listed temperature ceiling. Supplier data, not a finished assembly qualification.
Alliance Magnets · Live supplier catalog; checked October 27, 2026
N52 and N42SH ranges, coefficients and listed temperature ceilings. Preset Br is representative within the published range.
Arnold Magnetic Technologies · Live manufacturer catalog; checked October 27, 2026
RECOMA 30S sheet, PDF page 14: nominal Br 1.12 T, coefficient -0.035%/C measured between 20 and 150 C, listed ceiling 350 C. The ceiling does not validate linear extrapolation to 350 C.
International Electrotechnical Commission, TC 68 · 2015
Official public scope covers flux density, polarization, field strength, demagnetization curves and recoil lines. This page does not claim to reproduce or certify the paid standard.
International Electrotechnical Commission, TC 68 · 2023
Current cataloged edition replaces the 2015 material specification. Its scope covers principal properties and dimensional tolerances, not guaranteed operating limits of an assembled rotor.
