步进电机

Oscillation, Instability and Control of Stepper MotorsLIYU CAO and HOWARD M. SCHWARTZDepartment of Systems and Computer Engineering, Carleton University, 1125 Colonel By Drive,Ottawa, ON K1S 5B6, Canada(Received: 18 February 1998; accepted: 1 December 1998) Abstract. A novel approach to analyzing instability in permanent-magnet stepper motors is presented. It is shown that there are two kinds of unstable phenomena in this kind ofmotor: mid-frequency oscillation and high-frequency instability. Nonlinear bifurcation theory is used to illustrate the relationship between local instability and midfrequency oscillatory motion. A novel analysis is presented to analyze the loss of synchronism phenomenon, which is identified as high-frequency instability. The concepts of separatrices and attractors in phase-space are used to derive a quantity to evaluate the high-frequency instability. By using this quantity one can easily estimate the stability for high supply frequencies. Furthermore, a stabilization method is presented. A generalized approach to analyze the stabilization problem based on feedback theory is given. It is shown that the mid-frequency stability and the high-frequency stability can be improved by state feedback. Keywords: Stepper motors, instability, nonlinearity, state feedback.1. IntroductionStepper motors are electromagnetic incremental-motion devices which convert digital pulse inputs to analog angle outputs. Their inherent stepping ability allows for accurate position control without feedback. That is, they can track any step position in open-loop mode, consequently no feedback is needed to implement position control. Stepper motorsdeliver higher peak torque per unit weight than DC motors; in addition, they are brushless machines and therefore require less maintenance. All of these properties have made stepper motors a very attractive selection in many position and speed control systems, such as in computer hard disk drivers and printers, XY-tables, robot manipulators, etc.Although stepper motors have many salient properties, they suffer from an oscillation or unstable phenomenon. This phenomenon severely restricts their open-loop dynamic performance and applicable area where high speed operation is needed. The oscillation usually occurs at stepping rates lower than 1000 pulse/s, and has been recognized as a mid-frequency instability or local instability [1], or a dynamic instability [2]. In addition, there is another kind of unstable phenomenon in stepper motors, that is, the motors usually lose synchronism at higher stepping rates, even though load torque is less than their pull-out torque. This phenomenon is identified as high-frequency instability in this paper, because it appears at much higher frequencies than the frequencies at which the mid-frequency oscillation occurs. The high-frequency instability has not been recognized as widely as mid-frequency instability, and there is not yet a method to evaluate it.Mid-frequency oscillation has been recognized widely for a very long time, however, a complete understanding of it has not been well established. This can be attributed to the nonlinearity that dominates the oscillation phenomenon and is quite difficult to deal with.384 L. Cao and H. M. SchwartzMost researchers have analyzed it based on a linearized model [1]. Although in many cases, this kind of treatments is valid or useful, a treatment based on nonlinear theory is needed in order to give a better description on this complex phenomenon. For example, based on a linearized model one can only see that the motors turn to be locally unstable at somesupplyfrequencies, which does not give much insight into the observed oscillatory phenomenon. In fact, the oscillation cannot be assessed unless one uses nonlinear theory.Therefore, it is significant to use developed mathematical theory on nonlinear dynamics to handle the oscillation or instability. It is worth noting that Taft and Gauthier [3], and Taft and Harned [4] used mathematical concepts such as limit cycles and separatrices in the analysis of oscillatory and unstable phenomena, and obtained some very instructive insights into the socalled loss of synchronous phenomenon. Nevertheless, there is still a lack of a comprehensive mathematical analysis in this kind of studies. In this paper a novel mathematical analysis is developed to analyze the oscillations and instability in stepper motors.The first part of this paper discusses the stability analysis of stepper motors. It is shown that the mid-frequency oscillation can be characterized as a bifurcation phenomenon (Hopf bifurcation) of nonlinear systems. One of contributions of this paper is to relate the midfrequency oscillation to Hopf bifurcation, thereby, the existence of the oscillation is provedtheoretically by Hopf theory. High-frequency instability is also discussed in detail, and a novel quantity is introduced to evaluate high-frequency stability. This quantity is very easyto calculate, and can be used as a criteria to predict the onset of the high-frequency instability. Experimental results on a real motor show the efficiency of this analytical tool.The second part of this paper discusses stabilizing control of stepper motors through feedback. Several authors have shown that by modulating the supply frequency [5], the midfrequencyinstability can be improved. In particular, Pickup and Russell [6, 7] have presented a detailed analysis on the frequency modulation method. In their analysis, Jacobi series was used to solve a ordinary differential equation, and a set of nonlinear algebraic equations had to be solved numerically. In addition, their analysis is undertaken for a two-phase motor, and therefore, their conclusions cannot applied directly to our situation, where a three-phase motor will be considered. Here, we give a more elegant analysis for stabilizing stepper motors, where no complex mathematical manipulation is needed. In this analysis, a d–q model of stepper motors is used. Because two-phase motors and three-phase motors have the same q–d model and therefore, the analysis is valid for both two-phase and three-phase 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 improve mid-frequency stability, but also effective to improve high-frequency stability.2. Dynamic Model of Stepper MotorsThe stepper motor considered in this paper consists of a salient stator with two-phase or threephase windings, and a permanent-magnet rotor. A simplified schematic of a three-phase motor with one pole-pair is shown in Figure 1. The stepper motor is usually fed by a voltage-source inverter, which is controlled by a sequence of pulses and produces square-wave voltages. Thismotor operates essentially on the same principle as that of synchronous motors. One of major operating 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 and instability problems usually arise.Figure 1. Schematic model of a three-phase stepper motor.A mathematical model for a three-phase stepper motor is established using q–d framereference transformation. The voltage equations forthree-phase windings are given byv a = Ri a + L*di a /dt − M*di b/dt − M*di c/dt + dλpma/dt ,v b = Ri b + L*di b/dt − M*di a/dt − M*di c/dt + dλpmb/dt ,v c = Ri c + L*di c/dt − M*di a/dt − M*di b/dt + dλpmc/dt ,where R and L are the resistance and inductance of the phase windings, and M is the mutual inductance between the phase windings. _pm a, _pm b and _pm c are the flux-linkages of thephases due to the permanent magnet, and can be assumed to be sinusoid functions of rotor position _ as followλpma = λ1 sin(Nθ),λpmb = λ1 sin(Nθ− 2π错误！未找到引用源。