Mathematical Modeling and Analysis of Permanent Magnet Synchronous Motor in Quaternion Space

This paper presents an innovative method for the mathematical modeling of Permanent Magnet Synchronous Motors (PMSMs) using quaternion space. Traditional modeling methods typically assume a balanced three-phase voltage source, enabling the representation of the motor's rotation in a two-dimensional dq-space vector, with the third component set to zero. However, these conventional approaches prove inadequate when confronted with an unbalanced source, as the third component becomes non-zero, necessitating a three-dimensional model for precision. To address this challenge, quaternion space is utilized to formulate a mathematical model of the PMSM in three-dimensional space. In this study, all signals such as voltage, current, and flux are expressed as pure quaternions, and rotations in three-dimensional space are executed through quaternion multiplication. As a result, a mathematical model of the Permanent Magnet Synchronous Motor (PMSM) is formulated within the realm of quaternion space. The construction of this quaternion-based model is intricately explained, providing a comprehensive representation of crucial aspects, including stator voltage, stator-rotor interactions, and critical parameters such as torque and power consumption. Simulation results demonstrate that the PMSM model based on quaternions exhibits behavior consistent with traditional models when subjected to a balanced three-phase voltage source. Notably, the developed model demonstrates adaptability in accurately representing the motor's response even in the presence of unbalanced stator voltage sources.