Symbolic Modeling, Linearization, and Small-Signal Analysis of LSPMSM Dynamics

This paper presents a comprehensive symbolic and numerical analysis of the dynamic behavior of Line-Start Permanent Magnet Synchronous Motors (LSPMSMs). Unlike conventional modeling approaches that rely heavily on numerical simulation, we develop a symbolic model that captures the electrical and mechanical dynamics of the motor with enhanced analytical clarity. Using the dq0 transformation, the motor differential equations are formulated in the synchronous reference frame. These equations are then linearized around a steady-state operating point to derive a small-signal model, facilitating deeper insights into local stability and transient response characteristics. A significant contribution of this work lies in the derivation of the Jacobian matrix and its eigenvalue analysis, which allows for direct observation of system stability under different operating conditions. By symbolically computing the Jacobian, we avoid numerical approximation errors and preserve parameter dependencies, making the model highly adaptable for control design and optimization studies. We further demonstrate how the linearized model can predict small perturbations in system behavior, offering a practical tool for early-stage control system development. Simulation results validate the accuracy of the symbolic and linearized models by comparing them against the nonlinear system response under typical startup and load conditions. The results show that the linearized system closely approximates the behavior of the full nonlinear model within a defined operating region, confirming the reliability of the approach.