【正文】 2 rad=s !12and the system is unstable at the first group of equilibriumOscillation, Instability and Control of Stepper Motors 391? ?2 ? ?1 ? 0 1 2 x 10?3?2??1?012load angle error(rad)speed error(rad/s)Figure 5. A trajectory approaching the origin in the 1 , 1! plane.points. The trajectory starts with a nonzero initial condition and approaches a limit cycle(periodic oscillation). These results coincide with what the Hopf theorem says.One should be aware of the disadvantages of the Hopf theorem, that is: although it canpredict unspecified regions of a parameter (in our case, it is !1) where limit cycles exist, itgives little help in determining their size. On the other hand, it is worth noting that Mees[12] has presented several versions of the Hopf theorem with an attempt to overe thedisadvantages of the original version. In particular, his harmonic balance method seems tohave potential for providing more information (for example, the frequency and the amplitude)for the oscillatory phenomenon under consideration.4. Separatrix and HighFrequency StabilityHighfrequency stability relates closely to the socalled loss of synchronism phenomena. Theloss of synchronism phenomena happens occasionally in synchronous motors and steppermotors, where a rotor must catch up with the rotation of the mag field produced by thestator’s current. Otherwise, the rotor will stall. The phenomenon that a motor fails, becauseof some reason (for example, a disturbance in load torque), to keep the same rotation as thestator’s field is called loss of synchronism or failure. When the motor fails, while its rotor isbasically at rest, a very small oscillation can be observed in its speed and position, becausethe stator’s field is still in rotation.392 L. Cao and H. M. Schwartz? ? ? ? 0 ?15?10?5051015load angle error(rad)speed error(rad/s)Figure 6. A trajectory approaching a limit cycle in the 1 , 1! plane.In practice, it is often found that a stepper motor loses synchronism at higher frequenciesthan that predicted by the mid frequency unstable region. Obviously, this phenomenon cannotbe explained as midfrequency instability, and little attention has been paid to it. Here, wecall this phenomenon the highfrequency ‘instability’, and analyze it using the concept of aseparatrix in phase space. We also introduce a novel quantity, which is easily calculated andappears to be very useful in evaluating the highfrequency instability.As stated previously, dependent on the values of the load angle , the equilibria can bedivided into two groups, as represented by Equations (12) and (13). We will call them Amand Bm, respectively, the index m corresponds to the m in Equations (12) and (13). We havediscussed the Amgroup in the previous section. By puting the eigenvalues of the matrixAlvalued at Bm, we can find that Bmis unstable for all operating conditions. Therefore, Bmdoes not relate to steady state operation. When Amis stable, some trajectories in the 1!,1 plane are shown in Figure 7, where a0and a1are the projections of A0and A1in the1!, 1 plane respectively, b0is defined in the same way and appears as a saddle point. Thetwo trajectories that tend asymptotically to b1as t !C1are examples of separatrices. Ingeneral, a separatrix divides phase space into basins of attraction of different attractors [9].Therefore, the stability boundary of an attractor is formed by the separatrices. In Figure 7, theseparatrices distinguish the basin of the equilibrium a0from that of the equilibrium a1.The failure phenomenon in a stepper motor shows that there exists a region in the phasespace which does not belong to the basin of attraction of Am, mD 1。1!plane when the supplyfrequency is !1D 2826 rad/s, where the initial condition of the trajectory is nonzero. Theboundary between the stable region (basin of attraction) and unstable region of an attractor isalso determined by the separatrices around it. In the following, it is shown that as the supplyfrequency !1increases, the distance between a stable equilibrium and its stability boundarywill tend to zero. This highlights the fact that when supply frequency is high a stepper motorgets many more chances to lose synchronism.394 L. Cao and H. M. SchwartzTaft and Gauthier [3] and Taft and Harned [4] have given a phase portrait including separatrices for stepper motors in the planar case, and have shown that if a trajectory crosses aseparatrix the motor will lose synchronism. Their analysis is very insightful in the sense ofconnecting separatrices with the loss of synchronism phenomenon. However, because in theiranalysis the dynamic behaviours of the motor’s currents were not considered, it cannot beapplied directly to our situation. Here, we have a fourdimensional phase space and the phaseportrait cannot be visualized as in the case of twodimensional space. In our case, it is verydifficult to inspect the separatrices relying on a geometrical grasp of the phase portrait. For thisreason, we develop a very simple analytic method which is relevant and useful in evaluatingthe failure behaviour.Consider the distance from a equilibrium point to the nearest separatrix, which can bewritten asDesm。 2Smg。Bm/D2s 2C VmZ2 sin2.N /。Bm/. Therefore, m。Bm/ as well as Des.Thatis,as!1increases one gets m。Bm/can be used as an indicatorfor highfrequency stability margin. However, it should be noted that this stability is distinctfrom the normal definition of stability. It represents the ability to tolerate disturbances.From the above analysis, one can see that plays an important role in determining the highfrequency stability margin. The distance between a1and b1in the 1!。B