oscillation，instabilityandcontrolofsteppermotors-外文文献(编辑修改稿)

oscillation，instabilityandcontrolofsteppermotors-外文文献(编辑修改稿)内容摘要：

0) is autonomous.There are three parameters。 u/, they are the supply frequency !1, the supply voltagemagnitude Vmand the load torque Tl. These parameters govern the behaviour of the steppermotor. In practice, stepper motors are usually driven in such a way that the supply frequency!1is changed by the mand pulse to control the motor’s speed, while the supply voltage iskept constant. Therefore, we shall investigate the effect of parameter !1.3. Bifurcation and MidFrequency OscillationBy setting !D!0, the equilibria of Equation (10) are given asIqDf!0CTl/3N 1。 (11)N 0D−’CarccosZ2IqCRN 1!0VmZ 2m (12)D−’−arccosZ2IqCRN 1!0VmZ 2m。 (13)IdD .Vm 0/CNL1!0Iq/=R。 (14)where, mD0。 1。 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。 (17)where 1XDT1iq1id1! 1 UT,andAlis defined byAlDAC@Fn@XD26666664−RL1−!1−N. 1L1CId/ −NVmL1 0/!1−RL1NIqNVmL1 0/32N 1J0 −BfJ000 1 037777775: (18)Assume that all of the eigenvalues of Alhave no zero parts, then the system defined inEquation (10) is stable if and only if all these eigenvalues have negative real parts [9, 10].For a threephase stepper motor whose parameters are shown in Table 1, the calculatedeigenvalues of Alfor some !1based on the first group of equilibra are given in Table 2. Theeigenvalues can also be divided into two groups: one group given by p1and p2, correspondsto the electrical subsystems of the motor, and the other group given by p3and p4correspondsto the mechanical subsystems. When all the real parts of these eigenvalues are negative, theequilibrium is stable and is called an attractor. This indicates that motor is at steadystate operation with the speed!D!1=N. As shown in Table 2,p1andp2are close to−R=L1 j!1tosome extent, and are always stable for all !1. However, the real parts of p3and p4are positivefor some range of!1. The phenomenon that the real parts ofp3andp4bee positive relatesthe observable midfrequency oscillation, which has been analyzed by many authors. At thisOscillation, Instability and Control of Stepper Motors 389Table 2. Eigenvalues of Alevaluated at some !1.!1p1。 p2p3。 p4750 −272 j845 −23:4 j7071000 −305 j1037 9:5 j6591250 −296 j1265 9:0 j6041500 −289 j1507 −6:7 j5514 2 0 2 4 6 8 10560580600620640660680700ω1=806rad/sω1=1371rad/sReal PartImaginary Partω11ω12Figure 3. The locus of p3for !1D806 1371 rad/s.point, we have not presented any new contribution. The above stability analysis based on alinearized model is just what had been done by many authors. From now on, we will presentour new observations for this problem based on bifurcation theory.Bifurcation is an important phenomenon in nonlinear systems. It is defined as changes thatoccur in the qualitative structure of the trajectories when some control parameters vary in adynamic system. As shown in Figure 3, at some value of !1, the real part of the eigenvaluep3(p4) changes from negative to positive. At larger values of !1, the real part of the eigenvalue p3.p4/changes from positive to negative. These changes in the eigenvalues imply someimportant changes in the behaviour of the motor. These are the Hopf bifurcation phenomenon.Physically, they relate to the midfrequency oscillation.From Figure 3, we see that at some !1, the real part ofp3.p4/is just zero. At these !1,thatis, !1D!11and !1D!12, we cannot draw conclusions about the stability of system (10)by examining the eigenvalues of Al. This problem was ignored in the literature. However, theoperating points !11and !12are important theoretically because they represent a bifurcationpoint of equilibrium. The bifurcation of this kind of equilibrium is called Hopf bifurcations[9], which represents a kind of ‘bifurcation’ from point equilibrium to periodic motion. Hopfbifurcation for the stepper motor is illustrated in Figure 4.As will be shown below, !11and !12determine the range where the midfrequency oscillation occurs. Although the linearized model tells us that system (10) is locally unstable for!11!1!12, however, what this means physically is a question that cannot be answered390 L. Cao and H. M. 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