【正文】 his paper, because it appears at muchhigher frequencies than the frequencies at which the midfrequency oscillation occurs. Thehighfrequency instability has not been recognized as widely as midfrequency instability,and there is not yet a method to evaluate it.Midfrequency oscillation has been recognized widely for a very long time, however, aplete understanding of it has not been well established. This can be attributed to thenonlinearity that dominates the oscillation phenomenon and is quite difficult to deal with.384 L. Cao and H. M. SchwartzMost researchers have analyzed it based on a linearized model [1]. Although in many cases,this kind of treatments is valid or useful, a treatment based on nonlinear theory is neededin order to give a better description on this plex phenomenon. For example, based on alinearized model one can only see that the motors turn to be locally unstable at some supplyfrequencies, which does not give much insight into the observed oscillatory phenomenon. Infact, the oscillation cannot be assessed unless one uses nonlinear theory.Therefore, it is significant to use developed mathematical theory on nonlinear dynamics tohandle the oscillation or instability. It is worth noting that Taft and Gauthier [3], and Taft andHarned [4] used mathematical concepts such as limit cycles and separatrices in the analysis ofoscillatory and unstable phenomena, and obtained some very instructive insights into the socalled loss of synchronous phenomenon. Nevertheless, there is still a lack of a prehensivemathematical analysis in this kind of studies. In this paper a novel mathematical analysis isdeveloped to analyze the oscillations and instability in stepper motors.The first part of this paper discusses the stability analysis of stepper motors. It is shownthat the midfrequency oscillation can be characterized as a bifurcation phenomenon (Hopfbifurcation) 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 anovel 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 throughfeedback. 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 presenteda detailed analysis on the frequency modulation method. In their analysis, Jacobi series wasused to solve a ordinary differential equation, and a set of nonlinear algebraic equations hadto 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 threephasemotor 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 ofstepper motors is used. Because twophase motors and threephase motors have the sameq–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 improvemidfrequency 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 permanentmag rotor. A simplified schematic of a threephase motorwith one polepair is shown in Figure 1. The stepper motor is usually fed by a voltagesourceinverter, 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 majoroperating 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 andinstability problems usually arise.Oscillation, Instability and Control of Stepper Motors 385pulsemandacbInverterSupplySNFigure 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 byvaDRiaCLdiadt?Mdibdt?MdicdtCd pmadt。vcDRicCLdicdt?Mdiadt?MdibdtCd pmcdt。 pmbD 1 ?2 =3/。 (2)whereN is number of rotor teeth. The nonlinearity emphasized in this paper is represented bythe above equations, that is, the fluxlinkages are nonlinear functions of the rotor position.By using the q。b。d frame is given by [8]TrD23 / ?2 3/ C2 3/ / ?2 3/ C2 3/: (3)For example, voltages in the q。b。 therefore, theabove transformation from three variables to two variables is allowable. Applying the abovetransformation to the voltage equations (1), the transferred voltage equation in the q。vdD RidCL1diddt?NL1iq!。 (7)whereBfis the coefficient of viscous friction, andTlrepresents load torque, which is assumedto be a constant in this paper.In order to constitute the plete state equation of the motor, we need another statevariable that represents the position of the rotor. For this purpose the so called load angle [8] is usually used, which satisfies the follow