Minimization of eddy currents in permanent magnets of an electric machine with shape derivatives

TL;DR

This paper tackles minimizing eddy-current losses in the magnets of an interior permanent magnet synchronous machine (IPMSM) by incorporating time-dependent eddy-current effects through a magneto-quasi-static PDE and a time-stepping formulation. It advances a gradient-based, shape-derivative optimization framework that couples forward time-dependent state equations with backward-in-time adjoint equations, enabling the computation of domain sensitivities for designing rotor air-pocket shapes. Through a scalarized objective that combines average eddy-current dissipation and average torque, the method achieves a ~17% reduction in eddy losses and a torque increase, demonstrating the practical viability of time-dependent, shape-optimized IPMSMs. Limitations related to topology changes and manufacturing feasibility motivate future work on mechanical-stiffness constraints and time-periodic problem formulations, as well as potential remeshing strategies to sustain optimization progress.

Abstract

In this work we deal with the shape optimization of an electric machine considering time-dependent effects such as eddy currents. The considered electric machine is an interior permanent magnet synchronous machine and we minimize the average dissipated power due to the eddy currents in the magnets over a period of time corresponding to a rotation, while at the same time maximizing the average torque. Our approach is based on the computation of the shape derivative which -- beside the computation of a time discretization of time-dependent state problem -- also involves solving a a time discretization of a time-dependent adjoint problem. The challenge of this problem is related to the dependency of each one of the N time steps of the adjoint problem on two different time steps, due to the use of finite difference in the calculation of eddy current losses.

Figures (5)

• Figure 1: Design of the interior permanent magnet synchronous machine (IPMSM) under study. Ferromagnetic material is depicted in red, air is depicted in blue, permanent magnets are depicted in light blue and the coils are depicted in yellow.
• Figure 2: Eddy currents $\tilde{J}_e$ in the permanent magnets at a fixed time step (a). The current is assumed to be purely directed along the z-axis and the loop current to be closed at infinity. Power density related to eddy currents $\tilde{J}_e$ in the permanent magnets at a fixed time step (b).
• Figure 3: Solution of the problem \ref{['eq.pbu_sequence_weak']} at two different step with the locked step method. Antiperiodicity condition is imposed at the left and right sides, as well as at the non connected interface between rotor and stator.
• Figure 4: Plot of the convergence history of the optimization process. The average dissipated power due to the eddy currents in the magnets over an electric period is reduced from $P_{it0}=0.367 \hbox{W}$ to $P_{it53}=0.314 \hbox{W}$, while the average torque increases from $T_{it0}=574.68 \hbox{Nm}$ to $T_{it53}=587.30 \hbox{Nm}$.
• Figure 5: Superposition of the initial design of the machine under study and the optimal design obtained with the proposed shape optimization algorithm.