The Dynamic Stability of Step Motors

Mid-frequency instability (MFI) generally limits the application of step motors with constant-voltage drive to a speed range well below the theoretical limit imposed by inductance, back EMF, and friction torque. Analytical studies of MFI in specific step motor types and configurations have been reported in the literature, each focused on a particular method of analysis. The goal of this paper is to develop a single model structure and method of dynamic stability analysis, applicable to all step motors. To this end, a generalized $n$ -phase step motor dynamical system model applicable to a broad range of permanent-magnet hybrid (PMH) and variable-reluctance (VR) step motors is developed. Beginning with a physical model and working from first principles, the ( $n+2$ )th-order nonlinear time-varying (NLTV) state equation is developed. Next, a Park transform for the generalized $n$ -phase step motor is applied, resulting in a fourth-order NLTV state equation and a set of $n$ -2 first-order LTI state equations. The fourth-order NLTV system is then solved at constant speed and linearized about the solution, resulting in a fourth-order linear time-invariant (LTI) state equation. Finally, the dynamic performance and stability are determined by examining the eigenvalues of the LTI system. The scope of parameterization includes: motor type (PMH or VR); number and arrangement of poles, stator teeth, and rotor teeth; number of electrical phases ( $n =2$ or odd); magnetic circuit type (uni-phase or multi-phase); winding configuration; and electrical state variable (current or flux linkage). Examples are given to illustrate the method and computer simulations are used to verify the results.