Boundary Condensation of a Two-layer Dirichlet-to-Neumann Air-Gap Operator onto a Reluctance Network: Trace Conformity and Validation on a Surface-Magnet Synchronous Machine

Abstract

A reluctance network discretises the iron of an electrical machine but imports its air-gap coupling from a closure whose parameters the geometry does not fix. This paper closes the gap with the exact Dirichlet-to-Neumann operator of the two-layer annulus formed by the air gap and the surface magnets, the magnetisation entering as a source of the same surface map, and condenses it onto the bore nodes of a nonlinear reluctance network solved by Newton on the differential reluctance. The operator is of order one, so its condensation is defined only on a trial space contained in ; one potential per node imposes a piecewise-constant bore datum, which lies outside that space. It is proved in closed form that the condensed self-energy of a column grows as  with the harmonic truncation , the divergent part being  times the sum of the squared jumps of the bore potential; that locking the truncation to the tiling makes the operator converge to a definite limit which is not the continuous one; and that a continuous piecewise-linear basis, still with one unknown per node, restores an absolutely convergent series with a  tail and a proved error estimate. Direct summation reproduces the closed forms to between  and  at  with no fitted constant. On a 750 W, 15-slot/14-pole machine the condensed model, with no fitted air-gap constant, deviates from two-dimensional finite elements by 0.10 % on synchronous inductance, 0.24 % on flux linkage and 0.31 % on the fundamental air-gap flux density, each below the dispersion the tiling alone produces and hence reported as a bound, while the slot sidebands and the cogging torque, governed by the tooth-tip corner singularity, are bracketed. Rotor-position invariance makes a 721-position sweep cost one factorisation, and a five-test procedure audits the finite-element reference without re-solving it.