Optimal current waveforms for torque control of permanent magnet synchronous machines with any number of phases in open circuit

Polyphase permanent magnet synchronous motors are well suited for electromechanical actuation systems demanding a high level of reliability. They are indeed able to run on a reduced number of phases and therefore to make the actuation system fault tolerant. The paper introduces a Lagrangian formulation for determining the optimal waveforms of the phase currents allowing the motor to develop the needed torque with one or more than one phase lost in open circuit. The method is then adapted to find sub-optimal sinusoidal currents in fault tolerant mode for machines with sinusoidal EMFs. Eventually the paper compares for this type of machine the oversizing imposed for maintaining the performance unchanged in fault tolerant operation mode, with or without imposing to the currents to remain sinusoidal.