Gradient Networks for Universal Magnetic Modeling of Synchronous Machines

This paper presents a physics-constrained neural network framework for dynamic modeling of saturable synchronous machines, including spatial harmonics. The proposed architecture embeds gradient networks directly into the fundamental machine equations to model nonlinear, coupled electromagnetic behavior. By learning the gradient of magnetic field energy, the model satisfies reciprocity and energy-balance constraints by construction. The approach can universally approximate any physically feasible magnetic characteristics while offering key advantages over lookup tables and conventional black-box networks: monotonicity, smooth outputs, and improved generalization from limited data. These properties also support robust model inversion and trajectory optimization for control. The method is validated using measured and finite-element-method (FEM) data from a 5.6-kW permanent-magnet (PM) synchronous reluctance machine and is further demonstrated experimentally in real-time closed-loop operation on an embedded control platform. The results show accurate and physically consistent modeling performance, even with limited training data.