Calculate one-sided planar array fields, explore the exponential decay characteristics, and understand manufacturing limits.
The magnetic field of a planar array drops exponentially with distance from the surface. The decay rate is strictly governed by the spatial wavelength (λ). Larger gaps necessitate larger wavelengths to project field effectively.
Increasing magnet thickness (d) beyond λ/2 yields almost no additional magnetic field, following the 1-e^(-kd) relationship. Optimizing thickness prevents wasting expensive NdFeB material.
Analytical equations assume infinite arrays. Real, finite arrays exhibit flux leakage at the boundaries and non-uniform field profiles near the ends. FEA is also needed to check any nearby ferromagnetic part for local saturation before extra magnet volume is treated as useful field.
The calculator is a first-pass sizing tool. Use it to compare design direction, then validate finite geometry, tolerances, and production constraints before release. Source review date: 2026-07-19.
| Conclusion | Calculator Assumption | Evidence Basis | Required Verification |
|---|---|---|---|
| Working-gap field decays with distance. | The calculator uses k = 2π/λ and B_y = B_0 e^(-ky) for the first harmonic of a long planar Halbach array. | Classical one-sided Halbach field theory and later 1D Halbach array studies support the sinusoidal working-side approximation. | Run 3D FEA or a gauss-map scan when the array has few poles, narrow width, curved geometry, or nearby steel. |
| More magnet thickness eventually stops helping. | The calculator applies B_0 = Br(1 - e^(-kd)) times a discrete-segment coefficient. | The relationship follows the first-harmonic analytical model; it is a sizing estimate, not a material guarantee. | Check temperature, grade demagnetization margin, and mechanical packaging before buying more magnet volume. |
| Discrete segment count changes field quality. | The tool uses sin(π/M)/(π/M) to approximate loss from blocky magnetization directions. | Discrete Halbach magnet research reports that angular distribution, material imperfections, and finite geometry affect homogeneity. | Specify magnetization angle tolerance, segment gap, inspection method, and acceptance field map in the RFQ. |
| Design Situation | Use the Tool For | Next Engineering Step |
|---|---|---|
| Early sizing for linear motor or maglev track | Use the calculator to compare λ, gap, thickness, Br, and M=4 versus M=8. | Shortlist two or three geometries and request FEA for thrust ripple, leakage, and steel saturation. |
| Axial-flux disk, rotor, or ring array | Treat the planar result as a local estimate at one radius, not the final disk field. | Model curvature, radial pole pitch change, retainers, adhesive gaps, and end effects in 3D. |
| Uniform-field MRI, NMR, or sensor volume | Use the calculator only to understand wavelength and thickness sensitivity. | Optimize angular magnetization distribution and verify homogeneity over the target volume. |
| Shielding or one-sided leakage constraint | Use the one-sided model to estimate the intended working-face advantage. | Measure backside leakage; Halbach layouts reduce reverse flux but do not make it exactly zero. |
Axial (planar) Halbach arrays are fundamental to high-performance flat applications like linear motors, axial-flux generators, and maglev systems.
By arranging trapezoidal or rectangular blocks in a disk, an axial Halbach array provides a strong sinusoidal field in the axial direction. This is highly effective for coreless axial-flux motors (AFPM) because it eliminates the need for heavy rotor back-iron, drastically reducing inertia and weight.
In linear arrays, the working side interfaces with the moving stator coils. A 4-segment-per-wavelength (M=4) array is standard, balancing field strength and assembly cost. The strong one-sided flux interacts with the coils to produce high thrust without saturating adjacent structural steel.
| Segments (M) | Harmonic Coeff. | Assembly Complexity | Typical Use Case |
|---|---|---|---|
| M = 2 (Alternating) | 0.636 | Low (N-S sequence) | Low-cost sensors, conventional speakers |
| M = 4 (90° steps) | 0.900 | Medium | Standard linear motors, most Halbach rotors |
| M = 8 (45° steps) | 0.974 | High | High-precision undulators, premium AFPM |
| Ideal (Continuous) | 1.000 | Theoretical only | Mathematical limit |
*Harmonic Coefficient equals sin(π/M) / (π/M). Higher values extract more usable fundamental flux from the given magnet remanence.
Transitioning from theoretical models to physical implementation introduces significant mechanical, thermal, and cost challenges that must be evaluated during the design phase.
Unlike conventional N-S alternating arrays, Halbach segments heavily oppose each other during assembly. This requires robust mechanical retention systems (e.g., carbon fiber banding, titanium sleeves, or custom interlocking geometries) and massive assembly fixtures.
Tight segment pitch, bondline, and magnetization angle controls are required to minimize air gaps between blocks. The correct tolerance depends on wavelength, air-gap target, ripple budget, and inspection method; do not apply a universal micrometer number without a field-map acceptance plan.
| Risk | Misuse Signal | Mitigation |
|---|---|---|
| Assembly force exceeds fixture capacity | Segments rotate, jump, or open adhesive gaps during bonding. | Plan mechanical capture features, staged assembly, post-assembly magnetization, or supplier-built tooling. |
| Tolerance stack distorts the harmonic field | Measured ripple or sideband harmonics exceed the motor/sensor budget. | Control segment pitch, magnetization angle, bondline thickness, and final field-map acceptance. |
| Analytical model overpredicts finite array performance | Required field is near the calculated value with no margin for edges, heat, or nearby steel. | Use FEA before release and reserve magnet grade or geometry margin for production variation. |
Use these adjacent pages when the estimate turns into a product choice, field-map requirement, or manufacturability review.
An axial Halbach array (also known as a planar array) consists of permanent magnets arranged on a flat plane or a ring so their magnetization directions rotate incrementally. This augments the magnetic field on one working side while reducing, but not perfectly eliminating, reverse-side leakage.
The magnetic field at the surface increases with array thickness (d) following the factor (1 - e^(-kd)). Once thickness exceeds half the spatial wavelength (λ/2), adding more magnet material provides severely diminishing returns.
For a planar array, the field solves Laplace’s equation, yielding a pure exponential decay term e^(-ky) where y is the distance from the magnet surface and k is the wavenumber (2π/λ). Large air gaps require correspondingly large spatial wavelengths.
M=4 (90° rotations) provides about 90% of the ideal continuous Halbach field and is cheaper to manufacture. M=8 (45° rotations) provides about 97% of the ideal field but requires twice as many individual blocks and significantly more complex assembly fixtures.
For a disk or ring, λ is the local pole-pair spatial period at the radius being evaluated. Because the circumference changes with radius, a planar calculator should be treated as a local estimate before a full 3D model is built.
Start with magnet grade remanence Br, target air gap, available magnet thickness, and spatial wavelength or pole pitch. If these are unknown, use the defaults to understand sensitivity, then refine them from the mechanical package.
It is most useful for long, wide planar arrays with many repeated poles and no nearby ferromagnetic parts. Confidence drops for short arrays, curved rings, small pole counts, tight homogeneity targets, or steel structures near the working gap.
Br is a material remanence value. The useful air-gap field is lower because spatial wavelength, magnet thickness, segment discretization, finite edges, temperature, and nearby materials all change how much flux reaches the measurement plane.
Choose the smallest mechanically stable gap that still allows clearance, thermal expansion, coating, vibration, and assembly tolerance. Once gap becomes a large fraction of λ, the exponential decay term dominates the design.
Assembly forces. Bringing large magnetized blocks together requires massive fixtures to resist the extreme repulsive forces during bonding. For very large arrays, we magnetize post-assembly or use segmented mechanical retainers.
The calculator gives the fundamental harmonic amplitude for an infinitely wide array. Real arrays are finite, so edge effects will reduce field uniformity and magnitude near the perimeter. Always request 3D FEA before finalizing a design.
Ask for the modeled field map, assumed magnet grade and temperature, magnetization angle tolerance, mechanical datum scheme, bondline or gap limits, inspection plane, and measured gauss-map acceptance criteria.
No. M=8 usually improves the first-harmonic approximation, but it doubles part count, increases angular orientation risk, and raises assembly cost. M=4 is often the better sourcing choice when the application tolerates more ripple.
Sometimes, especially in weight-sensitive coreless designs. The decision depends on allowed leakage, structural stiffness, saturation limits, and cost. Backside flux is reduced rather than guaranteed to be zero.
Provide target field, measurement plane, allowable ripple or homogeneity, envelope, operating temperature, coating, retained geometry, production quantity, and whether the array can be magnetized after assembly.
Sources last reviewed on 2026-07-19. These references support the qualitative model and validation cautions; production release should still rely on application-specific FEA and measured field data.
Our engineering team builds custom FEA models to validate analytical results, taking into account precise geometry, edge fringing, and thermal conditions.
If you already know the target field and want a quote-ready checklist, use the buy Halbach array sourcing configurator to capture material, retention, QA, and lead-time assumptions.
