Copper loss minimizing torque control of IPMSM based on flux variables

The voltage and torque equation are written in terms of flux variables (λ_d, λ_q), instead of currents. Also, the voltage and current limits are depicted in the plane of (λ_d, λ_q). Then the voltage limits appear as circles centered at the origin, whereas the current limit appear as an ellipse. In the field weakening region, the voltage is utilized to the maximum. Hence, the only thing allowed to change in the field weakening region is the angle of the voltage so as to accommodate changes in speed and torque. In the flux setting, the voltage angle can be determined by an intersection point between the voltage circle and a torque line. That solution will be the copper loss minimizing point. However, it requires to solve a four order polynomial equation. In this work, a Taylor series approximation method is utilized to circumvent the difficulty of solving the fourth order equation. Desired performances were demonstrated by some experimental results.