Closed-Form Demagnetization-Safety Criteria for Dy-Reduced Nd–Fe–B Permanent Magnets Under Nonuniform Reverse Fields
This article presents a compact analytical framework for evaluating demagnetization safety in Dy-reduced Nd–Fe–B permanent magnets under a representative nonuniform reverse-field loading condition. The formulation combines temperature-dependent intrinsic-property degradation, a microstructure-sensitive coercivity model, and a normalized reverse-field map to derive a closed-form safety criterion for zero hazard. The hazard-area ratio is shown to be the upper-tail measure of the normalized reverse-field distribution, which leads directly to an explicit safe/unsafe boundary in the parameter space of Dy reduction and grain-boundary-diffusion (GBD) reinforcement. The closed-form expressions are further obtained for the critical reinforcement level required to eliminate hazardous regions and for the critical reverse-field suppression factor that preserves full safety. The 2-D finite-element simulations on a current-free magnetostatic prototype confirm that the proposed closed-form boundary accurately separates safe and unsafe regimes and captures the hazard-area evolution with respect to Dy reduction and reinforcement strength. The proposed framework provides a concise theoretical basis for demagnetization-aware design of low-Dy permanent magnets under high-field operation.