【正文】 ughfeedback. Several authors have shown that by modulating the supply frequency [5], the midfrequency instability can be improved. In particular, Pickup and Russell [6, 7] have presenteda detailed analysis on the frequency modulation method. In their analysis, Jacobi series wasused to solve a ordinary differential equation, and a set of nonlinear algebraic equations hadto be solved numerically. In addition, their analysis is undertaken for a twophase motor,and therefore, their conclusions cannot applied directly to our situation, where a threephasemotor will be considered. Here, we give a more elegant analysis for stabilizing stepper motors,where no plex mathematical manipulation is needed. In this analysis, a d–q model ofstepper motors is used. Because twophase motors and threephase motors have the sameq–d model and therefore, the analysis is valid for both twophase and threephase motors.Up to date, it is only recognized that the modulation method is needed to suppress the midfrequency oscillation. In this paper, it is shown that this method is not only valid to improvemidfrequency stability, but also effective to improve highfrequency stability.2. Dynamic Model of Stepper MotorsThe stepper motor considered in this paper consists of a salient stator with twophase or threephase windings, and a permanentmag rotor. A simplified schematic of a threephase motorwith one polepair is shown in Figure 1. The stepper motor is usually fed by a voltagesourceinverter, which is controlled by a sequence of pulses and produces squarewave voltages. Thismotor operates essentially on the same principle as that of synchronous motors. One of majoroperating manner for stepper motors is that supplying voltage is kept constant and frequencyof pulses is changed at a very wide range. Under this operating condition, oscillation andinstability problems usually arise.Oscillation, Instability and Control of Stepper Motors 385pulsemandacbInverterSupplySNFigure 1. Schematic model of a threephase stepper motor.A mathematical model for a threephase stepper motor is established using q–d framereference transformation. The voltage equations for threephase windings are given byvaDRiaCLdiadt?Mdibdt?MdicdtCd pmadt。Nonlinear Dynamics 18: 383–404, 1999.169。vbDRibCLdibdt?Mdiadt?MdicdtCd pmbdt。 pmcD 1 C2 =3/。c frame to the q。c reference, only two variables are independent (iaCibCicD 0)。 (5)386 L. Cao and H. M. SchwartzθωqdabcFigure 2. a, b, c and d, q reference frame.where L1DLCM,and! is the speed of the rotor.It can be shown that the motor’s torque has the following form [2]T D32N 1iq: (6)The equation of motion of the rotor is written asJd!dtD32N 1iq?Bf!?Tl。 (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。 (11)N 0D?’CarccosZ2IqCRN 1!0VmZ 2m (12)D?’?arccosZ2IqCRN 1!0VmZ 2m 。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。 /, y2Rn, 2R has an equilibrium.y0。 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, ....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 。 (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