A general magnetic-energy-based torque estimator: validation via a permanent-magnet motor drive

This paper describes the use of the current–flux-linkage ($i{-}psi$ ) diagram to validate the performance of a general magnetic-energy-based torque estimator. An early step in the torque estimation is the use of controller duty cycles to reconstruct the average phase-voltage waveform during each pulsewidth-modulation (PWM) switching period. Samples over the fundamental period are recorded for the estimation of the average torque. The fundamental period may not be an exact multiple of the sample time. For low speed, the reconstructed voltage requires additional compensation for inverter-device losses. Experimental validation of this reconstructed waveform with the actual PWM phase-voltage waveform is impossible due to the fact that one is PWM in nature and the other is the average value during the PWM period. A solution to this is to determine the phase flux-linkage using each waveform and then plot the resultant $i{-}psi$ loops. The torque estimation is based on instantaneous measurements and can therefore be applied to any electrical machine. This paper includes test results for a three-phase interior permanent-magnet brushless ac motor operating with both sinusoidal and nonsinusoidal current waveforms