【正文】 est, 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。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!。 0/, and near .y0。vdDVm /。 pmbD 1 ?2 =3/。 pmcD 1 C2 =3/。 (9)where Vmis the maximum of the sine wave. With the above equation, we have changed theinput voltages from a function of time to a function of state, and in this way we can representthe dynamics of the motor by a autonomous system, as shown below. This will simplify themathematical analysis.Oscillation, Instability and Control of Stepper Motors 387From Equations (5), (7), and (8), the statespace model of the motor can be written in amatrix form as followsP。 0/there is locally only one limitcycle for each . The initial period (of the zeroamplitude oscillation) is T0D2 = .It is easy to check that for the system under consideration, condition 1 of the Hopf theoremis satisfied. To check condition 2, one first observes that the eigenvalues of Alare continuousfunctions of !1. Hence, if condition 2 is not satisfied, then 3/ would have an extremumor an inflection at !11and !12. However, Figure 3 indicates that this is not true.One can use the Hopf theorem to determine that when !12T!11。1 plane (Figure 7) is2 . Sometimes it is convenient to use instead of 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。 (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 。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。vcDRicCLdicdt?Mdiadt?MdibdtCd pmcdt。 (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 state