【正文】 menon. In fact, the oscillation cannot be assessed unless one uses nonlinear theory.Therefore, it is significant to use developed mathematical theory on nonlinear dynamics to handle the oscillation or instability. It is worth noting that Taft and Gauthier [3], and Taft and Harned [4] used mathematical concepts such as limit cycles and separatrices in the analysis of oscillatory and unstable phenomena, and obtained some very instructive insights into the socalled loss of synchronous phenomenon. Nevertheless, there is still a lack of a prehensive mathematical analysis in this kind of studies. In this paper a novel mathematical analysis is developed to analyze the oscillations and instability in stepper motors.The first part of this paper discusses the stability analysis of stepper motors. It is shown that the midfrequency oscillation can be characterized as a bifurcation phenomenon (Hopf bifurcation) of nonlinear systems. One of contributions of this paper is to relate the midfrequency oscillation to Hopf bifurcation, thereby, the existence of the oscillation is provedtheoretically by Hopf theory. Highfrequency instability is also discussed in detail, and a novel quantity is introduced to evaluate highfrequency stability. This quantity is very easyto 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 midfrequencyinstability 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.2. Dynamic Model of Stepper MotorsThe stepper motor considered in this paper consists of a salient stator with twophase or threephase windings, and a permanentmagnet 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. Thismotor 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 frequencyof 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 framereference transformation. The voltage equations for threephase windings are given byva = 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 thephases due to the permanent magnet, and can be assumed to be sinusoid functions of rotor position _ as followλpma = λ1 sin(Nθ),λpmb = λ1 sin(Nθ ? 2 π/3),λpmc = λ1 sin(Nθ 2 π/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 of the rotor position.By using the q。 b。 d frame is given by [8]For example, voltages in the q。 b。 therefore, the above transformation from three variables to two variables is allowable. Applying the abovetransformation to the voltage equations (1), the transferred voltage equation in the q。u/, they are the supply frequency !1, the supply voltage magnitude Vm and the load torque Tl . These parameters govern the behaviour of the stepper motor. In practice, stepper motors are usually driven in such a way that the supply frequency !1 is changed by the mand pulse to control the motor’s speed, while the supply voltage is kept constant. Therefore, we shall investigate the effect of parameter !1.3. Bifurcation and MidFrequency OscillationBy setting ! D !0, the equilibria of Equation (10) are given asand 39