【正文】 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。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。 (20)where D1NarccosZ2IqCR 1!1VmZ: (21)Because Bm2Sm,sothatDes m。 (19)where, d is the distance operator in fourdimensional space, Smdenotes separatrix curve andindex mindicates that multiple separatrices exist. Desreflects the stable margin of the equilibrium. This is because if the motion at the equilibrium is disturbed and 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。 /。2, ....In other words,Oscillation, Instability and Control of Stepper Motors 393a0a1b1separatrix?δ?ωseparatrixfailure trajectoryxFigure 7. The definition of separatrix. 1 0706050403020100?δ (rad)?ω (rad/s)Figure 8. A failure trajectory when !1D2826 rad/s.thereissomeunstable region in the phase space, where a stepper motor cannot keep stablerotation and tends to stall. Figure 8 shows a failure trajectory in 1 。!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。 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。2. . //=d 6D0.Then there is a birth of limit cycles at .y0。 /, y2Rn, 2R has an equilibrium.y0。p2p3。2,....ThetermZ is the transferred impedance given byZDpR2C.!1L1/2: (15)388 L. Cao and H. M. SchwartzTable 1. The parameters of athreephase stepper motor.N 50R Omega1L1 mH 11:77 10?3VsBf1:9 10?3Nms/radJ 400 gcm2Vm Vand ’ is its phase angle defined by’Darctan!1L1R: (16)Equations (12) and (13) indicate that multiple equilibria exist, which means that these equilibria can never be globally stable. One can see that there are two groups of equilibria asshown in Equations (12) and (13). The first group represented by Equation (12) correspondsto the real operating conditions of the motor. The second group represented by Equation (13)is always unstable and does not relate to the real operating conditions. In the following, wewill concentrate on the equilibria represented by Equation (12).The stability of these equilibria can be examined based on the linearized version of Equation (10) about the equilibria, which is given by1PXDAl1XCB1u。 (14)where, mD0。 (11)N 0D?’CarccosZ2IqCRN 1!0VmZ 2m (12)D?’?arccosZ2IqCRN 1