Magnet Eddy Current Loss and Rotor Magnet Segmentation

For each distinct rotor-frame harmonic, the radial phasor field satisfies

Here $\Omega$ is the circumferential–axial rectangle of a single insulated segment. A constant field along its boundary implies zero normal eddy current there. The field and in-plane return currents are solved together; the radial extrusion has constant properties and field.

The implementation derives a separated series for this boundary-value problem. Put $d=\min(a,b)$, $r=\max(a,b)/d$, $Q=\omega\mu\sigma d^2$, $A_n=(n\pi)^2$, $Z_n=r\sqrt{A_n+jQ}/2$, with odd $n$:

The volume-average normalized field is $1-D$. Integration by parts gives

This equals the volume integral of $|\hat J|^2/(2\sigma)$. The factor 1/2 is required for peak phasors. The infinite-slab portion is summed analytically; only the end correction is truncated. This exact rearrangement improves convergence for strong skin effect. The complex arithmetic is evaluated through equivalent real and imaginary components; decaying exponentials avoid hyperbolic overflow. A small-argument expansion avoids subtracting nearly equal numbers.

Each pair of segment counts is solved for all active harmonics. Their time-average losses add. The target is $\min(P_{\rm budget},(T_{\rm limit}-T_{\rm sink})/R_\theta)$. The search minimizes piece count among feasible designs, breaking ties by loss; it also finds the minimum loss in the bounded search. Mean rise is $\Delta T=P R_\theta$.

Ruoho et al. (2009) motivate finite current-return length; Huang et al. (2010) and Königs & Löhlein (2023), §§4–7, discuss reaction fields and nonmonotonic segmentation. This is a direct diffusion solution derived here, not their fitted circuit model. The axial segmentation paper by Ede et al. is the 2007 journal article (DOI below), rather than the 2002 control paper.

• Rectangular, identical parent magnets; equal insulated segments with negligible kerf and unchanged total magnet volume.
• Linear, isotropic operating resistivity and recoil permeability; no radial variation. The material defaults are illustrative NdFeB values, consistent with Pacific PAC's typical 150 µΩ·cm resistivity and relative recoil permeability 1.05.
• Each harmonic imposes the same peak radial field on all four segment edges. The supplied spectrum is maintained after cutting; different frequencies are orthogonal in the time average.
• The synchronous permanent magnet field is DC in the rotor frame and contributes no eddy loss.
• One steady mean temperature and one user-supplied thermal resistance for all rotor magnets. Resistivity is held at its entered operating value; temperature feedback and local hot spots are not solved.

The result is a solution of the stated 2D diffusion problem, not a universal machine loss prediction. Actual surface fields can vary with position, segment gaps and external magnetic reluctance. In particular, voltage-fed PWM and the surrounding core can change harmonic amplitudes when segmentation changes. Chen, Wang & Nair (2016), pp. 1–2 of the accepted manuscript, discuss the limitations of radial-only and reduced-dimensional assumptions. Use a coupled 3D model when radial currents, tangential harmonics, axial end fields, curvature, sleeves, proximity effects or conductive coatings matter.

Six distinct field harmonics are supported. Omitted spectral components and spatial harmonics are not synthesized from the switching frequency. Electrical bridges between segments invalidate the insulated-boundary assumption. A mean temperature below the entered limit does not establish local demagnetization margin; the limit depends on magnet grade and magnetic load line.

The default sum uses 256 odd modes per harmonic, with a doubled-term convergence comparison for the selected and uncut magnets. The integer search solves every pair, without smoothing the design map. Increase the series terms if its convergence warning appears; extreme skin depths may require more terms and longer runtime. The frequency response uses 101 solved samples at a fixed 10 mT peak field; it is a transfer comparison, not the entered spectrum. Schematic loops illustrate closed current paths rather than their local magnitudes.

Independent five-point finite-difference benchmarks verify the diffusion solution in resistance-limited and skin-limited regimes; they are numerical verification, not measured motor validation. The search assumes unchanged material volume, harmonic forcing and thermal resistance. The fewest-piece recommendation is only within the entered count and size limits; no unique finite loss-only optimum exists if arbitrarily fine, cost-free segmentation is allowed.