【正文】 a novel quantity is introduced to evaluate highfrequency stability. This quantity is very easy to calculate, and can be used as a criteria to predict the onset of the highfrequency instability. Experimental results on a real motor show the efficiency of this analytical tool. The second part of this paper discusses stabilizing control of stepper motors through feedback. Several authors have shown that by modulating the supply frequency [5], the midfrequency Instability can be improved. In particular, Pickup and Russell [6, 7] have presented a detailed analysis on the frequency modulation method. In their analysis, Jacobi series was used to solve a ordinary differential equation, and a set of nonlinear algebraic equations had to be solved numerically. In addition, their analysis is undertaken for a twophase motor, and therefore, their conclusions cannot applied directly to our situation, where a threephase motor will be considered. Here, we give a more elegant analysis for stabilizing stepper motors, where no plex mathematical manipulation is needed. In this analysis, a d–q model of stepper motors is used. Because twophase motors and threephase motors have the same q–d model and therefore, the analysis is valid for both twophase and threephase motors. Up to date, it is only recognized that the modulation method is needed to suppress the midfrequency oscillation. In this paper, it is shown that this method is not only valid to improve midfrequency stability, but also effective to improve highfrequency stability. 咸寧學院本科畢業(yè)論文（設計）：外文翻譯 2. Dynamic Model of Stepper Motors The stepper motor considered in this paper consists of a salient stator with twophase or three phase windings, and a permanentmag rotor. A simplified schematic of a threephase motor with one polepair is shown in Figure 1. The stepper motor is usually fed by a voltagesource inverter, which is controlled by a sequence of pulses and produces squarewave voltages. This motor operates essentially on the same principle as that of synchronous motors. One of major operating manner for stepper motors is that supplying voltage is kept constant and frequency of pulses is changed at a very wide range. Under this operating condition, oscillation and instability problems usually arise. Figure 1. Schematic model of a threephase stepper motor. A mathematical model for a threephase stepper motor is established using q–d frame reference transformation. The voltage equations for threephase windings are given by va = Ria + L*dia /dt ? M*dib/dt ? M*dic/dt + dλpma/dt , vb = Rib + L*dib/dt ? M*dia/dt ? M*dic/dt + dλpmb/dt , vc = Ric + L*dic/dt ? M*dia/dt ? M*dib/dt + dλpmc/dt , where R and L are the resistance and inductance of the phase windings, and M is the mutual inductance between the phase windings. _pma, _pmb and _pmc are the fluxlinkages of the phases due to the permanent mag, and can be assumed to be sinusoid functions of rotor position _ as follow λpma = λ1 sin(Nθ), λpmb = λ1 sin(Nθ ? 2 /3), λpmc = λ1 sin(Nθ 2 /3), where N is number of rotor teeth. The nonlinearity emphasized in this paper is represented by the above equations, that is, the fluxlinkages are nonlinear functions 咸寧學院本科畢業(yè)論文（設計）：外文翻譯 of the rotor position. By using the q。 state feedback 1. Introduction Stepper motors are electromagic incrementalmotion devices which convert digital pulse inputs to analog angle outputs. Their inherent stepping ability allows for accurate position c