This paper presents explicit algebraic methods for approximating optimal dq-axis current references in permanent magnet synchronous motors (PMSMs) under given torque commands. The proposed approach addresses three key optimal control strategies: maximum torque per ampere (MTPA), maximum torque per voltage (MTPV), and loss-minimization control. For MTPA operation, a closed-form explicit formula is derived to approximate the d-axis current that minimizes copper losses. For MTPV operation, an analytical expression is developed to approximate the optimal current vector, effectively addressing iron losses in the high-speed region. Furthermore, a simplified formulation for loss-minimization control is proposed to enhance overall efficiency by balancing both copper and iron losses. These formulas are computationally efficient and eliminate the need for iterative numerical procedures while maintaining high accuracy. Supplementary expressions are also provided to facilitate practical implementation under current and voltage constraints. The mathematical fidelity and computational efficiency of the proposed formulas are rigorously validated through numerical simulations using representative PMSM models. The results demonstrate that the proposed explicit approximations closely match the true numerical optimal trajectories, offering a practical alternative to complex iterative methods without the need for extensive experimental characterization.
Bae et al. (Mon,) studied this question.