A circular Halbach array rotates magnetization around a ring to concentrate its field. Estimate an ideal bore or external field and magnet mass, then review the design limits below.
By Halbach Magnet. First recorded: . Method and source review: . This is a preliminary sizing guide; the examples are calculations, not customer test records.
| Topology | Signed p / direction law | Field behavior | Decision |
|---|---|---|---|
| Internal dipole | +1; ψ = 2θ | Uniform transverse field in ideal 2D bore | Start here for a static uniform-field region |
| Internal quadrupole | +2; ψ = 3θ | Amplitude proportional to r; zero on axis | Specify gradient and integrated field |
| Internal hexapole | +3; ψ = 4θ | Amplitude proportional to r² | Specify the nonlinear working region |
| External 8-pole rotor | −4; ψ = −3θ | Target amplitude proportional to r⁻⁵ outside Ro | Budget sleeve + gap before comparing flux |
Angles refer to global Cartesian magnetization, counterclockwise from +x; θ is the sector midpoint. The diagrams illustrate the direction law, not measured field lines.[1][2]
Let b = Br(20°C)[1 + α(T−20)/100]. For M equal, gap-free arc sectors magnetized at their midpoints, kM = sin(a)/a with a = |1+p|π/M using signed p. The continuous reference uses kM = 1. All radii must use the same units; fields below are in tesla. The target Fourier harmonic follows the ideal μr = 1 solution.[1][2]
| Quantity | Equation | Scope / excluded effects |
|---|---|---|
| Dipole, p = 1 | B = b kM ln(Ro/Ri) | Infinite length; uniform target fundamental |
| Internal, p > 1 | B(r) = b kM p/(p−1) [1−(Ri/Ro)^(p−1)] (r/Ri)^(p−1) | 0 ≤ r ≤ Ri; no local harmonic errors |
| External, p = −4 | B(r) = b kM q/(q+1) [1−(Ri/Ro)^(q+1)] (Ro/r)^(q+1) | r ≥ Ro; no stator, back iron or shielding |
| Quadrupole gradient | G = 2 b kM [1−Ri/Ro] / Ri | Ri converted to meters; G in T/m; no end fields |
| Annulus volume / mass | V = π(Ro²−Ri²)L; m = ρV | Unit conversion applied; excludes gaps, glue and fixtures |
| Rotating screening values | v = 2π (RPM/60) Ro; inertial scale = ρv² | SI units; this is not a sleeve stress solution |
For a dipole, increasing M from 8 to 16 raises the modeled fundamental from 90.03% to 97.45% of the continuous reference. Further increases give diminishing amplitude gains while adding pieces to inspect, orient and assemble. Choose a count using the required field map and manufacturing budget; there is no universal NMR segment count.[2]
Spatial THD depends on radius and the sampled field component. Ppm homogeneity depends on a defined volume, end effects and errors. Neither can be inferred from 100−kM. Require the measurement contour, harmonic convention and acceptable error before comparing suppliers.
| M | kM | Retained fundamental | THD / ppm |
|---|---|---|---|
| 4 | 0.63662 | 63.66% | Unknown; requires defined field map |
| 8 | 0.90032 | 90.03% | Unknown; requires defined field map |
| 12 | 0.95493 | 95.49% | Unknown; requires defined field map |
| 16 | 0.97450 | 97.45% | Unknown; requires defined field map |
| 24 | 0.98862 | 98.86% | Unknown; requires defined field map |
| 32 | 0.99359 | 99.36% | Unknown; requires defined field map |
| Reference | Br at 20°C | α | Density (g/cm³) | Catalog / tool temperature |
|---|---|---|---|---|
| N52 | 1.45 T | -0.12 | 7.5 (sizing assumption) | 60°C catalog; tool 20–60°C |
| N45SH | 1.35 T | -0.12 | 7.5 (sizing assumption) | 150°C catalog; tool 20–150°C |
| RECOMA 28 (Sm₂Co₁₇) | 1.10 T | -0.035 | 8.3 (typical) | 350°C catalog; tool 20–150°C |
The NdFeB density of 7.5 g/cm³ is an explicit mass-sizing assumption; replace it with measured lot density for a mass requirement. Reversible coefficients do not predict loss after a demagnetizing excursion. Check the worst local reverse field against supplier temperature-dependent curves.[3][4]
Precision-temperature strategy: start by defining allowable field drift and the actual magnet temperature distribution. Active stabilization or a calibrated temperature-sensitive shunt may help. Passive compensation is geometry dependent; this page does not claim a particular alloy gives sub-ppm stability or prescribe a shunt size.[2]
Nested rings: ideal dipole field magnitudes obey B(θ) = √(B1²+B2²+2B1B2 cos θ). Only equal amplitudes simplify to 2B0 |cos(θ/2)| and allow an ideal zero at 180°. Residual fields, permeability and holding torque need separate analysis. “Mandhala” refers to a discrete-magnet architecture and is not a synonym for every nested variable-field system.[1][2]
| Choice / input | Benefit or cost driver | What to specify next |
|---|---|---|
| True arc sectors | Matches the full-annulus model; shaped surfaces and orientation add processing work | Finished ID/OD, angular span, edge relief, coating and bond-line allowance |
| Trapezoids / blocks / Mandhala | Can simplify individual parts; changes fill fraction and field harmonics | Actual CAD geometry and a field model; do not reuse full-annulus mass as a purchase weight |
| More sectors | Improves target amplitude but adds inspection and assembly operations | Count, polarity/orientation schedule, moment sorting and inspection scope |
| Tighter field tolerance | May require sorting, shimming and repeated mapping | Working volume, error metric, temperature, instrument uncertainty and rework allowance |
| Adhesive / enclosure | Cure, thermal expansion, environment and access can control reliability | Substrate-specific tests, cure process, thermal cycling and mechanical load cases |
A practical assembly plan uses individually identified magnets, polarity checks and a nonmagnetic fixture that controls insertion and reacts magnetic forces. Include inspection holds before enclosure closure. Grinding an assembled magnet ring is not a default remedy for poor fit: define coating restoration, debris control and thermal protection with the manufacturer.
Every rotating segmented array needs retention analysis. The calculator's ρv² shows an inertial scaling trend only. It omits segmented contact, prestress, shaft support, material anisotropy, fatigue, thermal expansion and stress concentrations. Static arrays still need a fixture and a load path for magnetic assembly forces.[7]
For a sleeve, compare actual laminate or alloy allowables, achievable interference, thermal behavior and added working gap. Carbon fibers conduct electricity; CFRP losses depend on layup and the applied harmonic field. Metallic sleeve loss also depends on conductivity, geometry and frequency. Neither a universal loss percentage nor a zero-loss material claim is supported here.[6]
Electrically isolated magnet subdivisions can interrupt eddy-current paths, but savings depend on the machine's slotting/PWM spectrum, insulation continuity and 3D geometry. Ask for loss and hot-spot results at the intended drive conditions. Do not exchange magnetic sector count for electrically insulated subdivisions without updating the electromagnetic and mechanical models.[6]
For ppm requirements, a generic Hall-probe scan alone does not establish suitability; agree sensor accuracy, temperature control, position error and repeatability. Shimming must be followed by a new map over the defined volume. Balance grade and rotor speed are independent acceptance requirements and cannot be inferred from a magnetic field result.[2]
Plan a field-mapping acceptance method or request a finite-geometry engineering review.
Illustrative calculations using this page's equations and reference inputs. No customer deployment, test result or guaranteed performance is implied.
Ri/Ro/L = 15/35/50 mm; M = 16; N52 at 20°C; r = 0 mm; 0 RPM.
2D target: 1.197 T; mass 1.178 kg.
L/OD is below one, so this field is a 2D reference only. Define the useful sample volume, then request a finite-length map before specifying a bore-field tolerance.
Enter these values in the calculatorRi/Ro/L = 15/35/50 mm; M = 16; N52 at 20°C; r = 5 mm; 0 RPM.
2D target: 0.521 T; mass 1.178 kg. Gradient: 104.20 T/m.
Compare gradient and field at r = 5 mm. A zero on-axis field is expected; it does not mean the lens has zero focusing strength. Verify integrated gradient and end fields separately.
Enter these values in the calculatorRi/Ro/L = 15/35/50 mm; M = 16; N45SH at 20°C; r = 38 mm; 30000 RPM.
2D target: 0.665 T; mass 1.178 kg.
The external fundamental decays rapidly with radius. Add sleeve and clearance to the radial budget, then evaluate the stator-loaded field, rotor losses and retention together. No sleeve sizing is supplied.
Enter these values in the calculator| Approach | When to consider it | Tradeoff / next check |
|---|---|---|
| Internal Halbach dipole | Passive transverse field in a cylindrical bore | Limited access and finite-length uniformity; map useful volume |
| Coil / electromagnet | Active field control or open access is the priority | Power, cooling and control requirements; check drift and duty cycle |
| Conventional motor PM topology | Stator-loaded torque, established production route | Compare complete motor efficiency, retention and manufacturing cost; no universal torque advantage |
| Nested Halbach dipoles | Mechanically adjustable passive field | Residual field, alignment and holding loads; model both rings together |
| Risk | Decision impact | Mitigation / alternative |
|---|---|---|
| Use a 2D result as a delivered field specification | Wrong field or uniformity in the real volume | Finite-length FEA and prototype map; define the acceptance region |
| Treat a catalog temperature as a safe limit | Irreversible local demagnetization | Worst-case B–H/load-line review; compare higher-coercivity reference materials |
| Choose more segments by default | Higher inspection and assembly cost without a useful requirement | Compare amplitude gain with quoted processing and mapping scope |
| Specify sleeve thickness from RPM alone | Containment failure or excess working gap | Structural and thermal model with qualification; use a static prototype if rotation is not essential |
| Request ppm performance without a measurement method | Unverifiable acceptance and rework | Agree region, temperature, sensor uncertainty and shimming plan before ordering |
Rotating the magnetization direction around a ring concentrates a selected magnetic multipole on the bore or exterior side. The internal dipole produces a transverse bore field in the ideal 2D model. A real segmented finite ring also has end leakage and unwanted harmonics.[1][2]
Ri and Ro are radii in mm. Divide drawing ID and OD by two. L is the active magnet length, excluding end plates; r is the field evaluation radius, measured from the cylinder axis.
The calculator reports the infinite-length 2D target fundamental. L changes annulus volume, mass and L/OD. A finite-length result needs a separate 3D solution; no universal aspect-ratio multiplier is applied here.[1][5]
An ideal quadrupole has B(r) proportional to r. The on-axis field is zero while its gradient is nonzero. Select a nonzero radius inside the bore to compare field amplitude; the displayed gradient is a 2D value.[1]
Enter r = Ro + the radial distance from the magnet surface to the evaluation point. Add sleeve and clearance thickness if the working point lies beyond them. The model treats everything outside the magnets as air; it does not include stator steel or sleeve shielding.[1]
The old result and export are removed. Blank, nonfinite, out-of-range and wrong-region inputs show recovery guidance. Complete the inputs and press Calculate, or reset the example. If the geometry is unknown, send a sketch for review.
No. For a 16-sector dipole it is the retained amplitude of the target fundamental relative to a continuous ring. THD requires a defined field component, sampling contour and Fourier analysis. Ppm uniformity also needs a defined region and temperature.[2]
It is an ideal reference with kM = 1. The diagram samples the direction law without drawing segment boundaries. Use a discrete count and account for actual sector geometry, gaps and magnetization tolerances before ordering.[2]
The ideal dipole equation Br ln(Ro/Ri) permits this for sufficiently large radius ratios. It is not proof that a real assembly will reach the same value: recoil permeability, local reverse fields, finite length and demagnetization must be checked.[1][2]
The cited Arnold N52 reference lists 60°C. Other suppliers and grade variants may differ. This reference is not a safe assembly limit: geometry and reverse field may produce irreversible losses below it. Use supplier temperature-dependent B–H curves.[3]
No. It applies a linear reversible remanence coefficient from 20°C within a restricted interval. It does not solve the load line or coercivity margin. RECOMA 28 calculations stop at 150°C to stay within the cited coefficient interval.[3][4]
Temperature-sensitive shunts can be investigated, but the geometry, alloy response, thermal gradients and calibration determine the result. This page supplies no shunt dimensions or ppm guarantee. Qualify a prototype over the actual temperature range.[2]
No. It computes surface speed and the inertial scale ρv². Actual containment needs a structural model with material allowables, interference, thermal expansion, adhesive aging and overspeed qualification. No safe RPM, material or wall thickness follows from these two numbers.[7]
Carbon fibers conduct electricity and composite conductivity is anisotropic. Layup, resin, interfaces and the machine harmonic spectrum determine loss. Compare the actual composite with metallic alternatives using electromagnetic and thermal models; no zero-loss assumption is justified.[6]
Ideal dipole fields superpose vectorially. Equal amplitudes B0 give magnitude 2B0 |cos(θ/2)|; unequal amplitudes leave a minimum |B1−B2|. Real permeability, finite length and alignment introduce residual fields. Holding torque needs its own model; it is not inferred from the tuning equation.[1][2]
Send a dimensioned sketch, bore or external working region, target field and tolerance over that region, temperature, speed and overspeed duty, environment, quantity and acceptance method. Include the copied preliminary summary with its assumptions. Unknowns can be marked for review.
Sources checked on 27 October 2026. Research supports the stated ideal assumptions and method; it does not certify a HalbachMagnet product. Exact cost, lead time, tolerance, ppm performance, sleeve wall and application-specific loss remain unknown until quoted, modeled or tested.
JMMM 322, 133–141; DOI 10.1016/j.jmmm.2009.08.044. Equations 47, 50 and 56: ideal internal/external fields for μr = 1. Published 2010; open manuscript deposited 2014.
Sections 3.2–3.4, 4 and 5: segmented sectors, finite length, homogeneity and temperature compensation. Equation 10 defines the fundamental segmentation factor; it does not specify universal THD.
Reference values for N52 and N45SH remanence, reversible coefficient and working-temperature entries. Values depend on grade variant and supplier; obtain the actual batch B–H curves.
RECOMA 28 typical remanence, density and reversible coefficient. The quoted coefficient is characterized over 20–150°C; the tool stops at 150°C even though the catalog lists a higher temperature reference.
IEEE Transactions on Magnetics 44, 4110–4113; DOI 10.1109/TMAG.2008.2001526. Three-dimensional geometry study supporting the need to treat axial length and end effects explicitly.
PEMD 2012, DOI 10.1049/cp.2012.0282; university repository record with author manuscript. Studies magnet subdivisions and anisotropic sleeve resistivity with 3D FEM. Its machine-specific loss results are not predictions for the arrays on this page.
IEEE Transactions on Energy Conversion 41(3), 2319–2330. The abstract describes winding-process stress analysis, contact-pressure measurement and overspeed testing. Supports separate structural validation, not a generic sleeve thickness rule; full text may require access.
Source caution: the 2023 review's prose example for an 8-sector dipole conflicts with its equation 10. This tool evaluates the equation directly: sin(π/4)/(π/4) = 0.90032. The 4-sector value is 0.63662. Reported comparisons here use the reproducible equation, not the inconsistent prose.
Copy the preliminary sizing summary and send it with a dimensioned sketch, target field region and tolerance, temperature, rotational duty, environment, quantity and acceptance method. If dimensions are still unknown, describe the available envelope and task; request a feasibility review before fixing a geometry.
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