740 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 6, NOVEMBER 1998 Brief Papers Torque Maximization for Permanent Magnet Synchronous Motors Alexander Verl and Marc Bodson Abstract—This paper discusses the problem of maximizing the torque of permanent magnet synchronous motors in the presence of voltage and current constraints. Formulas are given that are suitable to operation with voltage and current source inverters and to real-time computation. Generally, the available torque is limited by current constraints at low speeds and by voltage con- straints at high speeds, and there is often an intermediate range of speeds where both the voltage and the current constraints are active. The paper specifies how optimal operation can be achieved in all three ranges, and at what speeds the optimal operation transitions from one mode to another. An example is discussed to illustrate the contributions of the paper and experimental results demonstrate the usefulness of the formulas in a high-performance robotic application. Index Terms—Brushless motors, motor control, robotics, syn- chronous machines, torque control. I. INTRODUCTION T HE MAXIMIZATION of the torque produced by electric motors is an important practical consideration, since optimization may yield the use of a smaller motor for a given application or a faster operation for a given motor. In this paper, we consider the problem for permanent magnet synchronous motors, which include brushless DC motors and stepper motors, and are commonly used in robotic applications. At high speeds, the available torque of electric motors decreases because of the back-emf voltage and the limits on the source voltage. Maximization of the torque requires the use of field-weakening which, for synchronous motors, can be achieved optimally using relatively simple strategies. These strategies are derived assuming that the speed is constant, an assumption that is often not satisfied in practice. However, the speed usually varies sufficiently slowly that the formulas are useful [1]. At low speeds, the current limits restrict the torque available, and torque maximization can be achieved by aligning the current vector with the direction perpendicular to the permanent magnet. Interestingly, however, one finds that torque optimization throughout the speed range may involve an intermediate region where both the voltage and the current limits are active and where the low-speed and Manuscript received October 7, 1996; revised October 12, 1997. Recom- mended by Associate Editor, G. J. Rogers. This work was supported in part by the Air Force Office of Scientific Research under Grant F49620-95-1-0341. A. Verl was with the Institute for Robotics and System Dynamics, German Aerospace Research Establishment (DLR), D-82234, Wessling, Germany. He is now with AMATEC GmbH, D-82234 Wessling, Germany. M. Bodson is with the Department of Electrical Engineering, University of Utah, Salt Lake City, UT 84112 USA. Publisher Item Identifier S 1063-6536(98)09033-2. high-speed solutions are not applicable. The objective of this paper is to solve the optimization problem in all three speed ranges, and to calculate the location of the transition points between the regions. The paper by [2] also partly addresses the same problem. However, since the derivations are based on a geometric approach, the stator resistance had to be neglected. In addition to the nonzero resistances of the stator windings, this paper provides explicit, analytic formulas for operation in voltage and current controlled modes which are suitable for real-time implementation. II. STATE-SPACE MODEL AND PROBLEM STATEMENT For a two-phase motor, the DQ transformation is given by (1) where and are the currents in phases and , respectively, is the rotor angular position, and is the number of pole pairs (or the number of rotor teeths for a stepper motor). The currents and are the transformed currents in the and (for direct and quadrature) reference frame. In the same way, the voltages and applied to phases and can be transformed into the voltages and . With the rotor speed, the inductance of a stator winding, the resistance of a stator winding, the moment of inertia of the rotor, the load torque, the state-space model of a permanent magnet synchronous motor is given by (2) (3) (4) (5) For simplicity, the model assumes that the rotor is smooth and that the magnetics are linear. For constant speed , (2)–(4) yield (6) (7) (8) In (8), is the electrical torque developed by the motor. The voltage and current constraints are incorporated in the problem statement by using the fact that, at constant speed, the 1063–6536/98$10.00  1998 IEEE