【正文】 its trajectory crosses theseparatrix near to it, then it is very possible, especially at high frequencies of the supply, thatthe trajectory enters the unstable region and loss of synchronism happens. However, becausethe separatrices are curves in fourdimensional space and an analytic expression for themare generally not available, the distance from a equilibrium point to a separatrix cannot beobtained analytically. Therefore, we have to seek an alternative to the distance Des,whichshould preserve the qualitative property of Des.Consider the distance from a stable equilibrium point Amto its adjacent unstable equilibrium point Bm. This distance can be obtained easily from Equations (12), (13), and (14) asfollowsm。Bm/ is a supremum onDes. Let us examine . Observe that Equation (21) has a solution if and only ifZ2IqCR 1!1 VmZ: (22)This represents possible operation condition for stepper motors. Furthermore, it is assumedthat only the supply frequency !1changes, and the supply voltage Vmand the load torqueTlkeep constant. Under these conditions, from Equations (11) and (15) one can see that as!1increases the value of .Z2IqCR 1!1/=Vm=Z increases monotonically and will tend to 1.Therefore, as !1increases one gets !0. For the motor shown in Table 1, the graph as afunction of !1was calculated for TlD0 and the result is shown in Figure 9. One can see thatwhen !1is near to 2500 rad/s, goes down very quickly and tends to 0.From Equation (20) and the above analysis one can predict the asymptotic property ofm。1 plane (Figure 7) is2 . Sometimes it is convenient to use instead of m。Bm/! 0andDes! 0.Oscillation, Instability and Control of Stepper Motors 395φ0 500 1000 1500 2020 2500 30000ω1Figure 9. versus !1.These mean that the equilibrium Amwill go to the separatrix Smas!1increases. This explainswhy at high frequencies a stepper motor easily loses its synchronism. One can conclude thatthe higher!1is, the less the stability margin. Therefore,m。 (20)where D1NarccosZ2IqCR 1!1VmZ: (21)Because Bm2Sm,sothatDes m。 /。!12U,where is a positive number, a periodic oscillation exists, and the period canbe predicted approximately by the eigenvalues listed in Table 2. Up to now, although themidfrequency oscillation has been known for a very long time, a theoretical proof of theexistence of it has not been available. While using the linearization method only, we can getnothing but the instability of the equilibria [1]. We cannot explain the connections between theinstability and the oscillation. Therefore, the Hopf theorem plements our understandingand prediction of the onset of instability.Some simulation results using Equation (10) for the threephase motor are shown in Figures 5 and 6. Only projections of the calculated phase space trajectories in load angle error1 .D ? 0/versus speed error1!.D!?!0/plane are given. In Figure 5,!1D733 rad=s !11and the first group of equilibrium points is stable, hence the trajectory starts with a nonzero initial condition and approaches the equilibrium point .0。2. . //=d 6D0.Then there is a birth of limit cycles at .y0。p2p3。 (14)where, mD0。 (10)where XDTiqid! UT, uDT!1TlUTis defined as the input, and !1DN!0is the supplyfrequency. The input matrix B is defined byBD2666400000 ?1=J?1=N 037775:The matrix A is the linear part of F. /, and is given byAD26664?R=L10 ?N 1=L100 ?R=L10032N 1=J 0 ?Bf=J 000 1037775:Fn.X/ represents the nonlinear part of F. /, and is given byFn.X/D26664?Nid!CVm=L1 /Niq!CVm=L1 /0037775:The input term u is independent of time, and therefore Equation (10) is autonomous.There are three parameters 。 (8)where !0is steadystate speed of the motor. Equations (5), (7), and (8) constitute the statespace model of the motor, for which the input variables are the voltages vqand vd.Asmentioned before, stepper motors are fed by an inverter, whose output voltages are not sinusoidal but instead are square waves. However, because the nonsinusoidal voltages do notchange the oscillation feature and instability very much if pared to the sinusoidal case(as will be shown in Section 3, the oscillation is due to the nonlinearity of the motor), forthe purposes of this paper we can assume the supply voltages are sinusoidal. Under thisassumption, we can get vqand vdas followsvqDVm /。d framecan be obtained asvqD RiqCL1diqdtCNL1id!CN 1!。d reference are given by vqvd DTr24vavbvc35: (4)In the a。d transformation, the frame of reference is changed from the fixed phaseaxes to the axes moving with the rotor (refer to Figure 2). Transformation matrix from thea。 (1)where R and L are the resistance and inductance of the phase windings, and M is the mutualinductance between the phase windings. pma, pmband pmcare the fluxlinkages of thephases due to the permanent mag, and can be assumed to be sinusoid functions of rotorposition as follow pmaD 1 /。 accepted: 1 December 1998)Abstract. A novel approach to analyzing instability in permanentmag stepper motors is presented. It is shownthat there are two kinds of unstable phenomena in this kind of motor: midfrequency oscillation and highfrequencyinstability. Nonlinear bifurcation theory is used to illustrate the relationship between local instability and midfrequency oscillatory motion. A novel analysis is presented to analyze the loss of synchronism phen