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1、Oscillation, Instability and Control of Stepper MotorsAbstract. 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. Nonlin

2、ear 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

3、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 feedb

4、ack 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

5、 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 motors deliver higher peak torque per u

6、nit 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 ma

7、nipulators, 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 ra

8、tes 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

9、 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

10、 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 phenome

11、non 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 comp

12、lex phenomenon. For example, based on a linearized model one can only see that the motors turn to be locally unstable at some supplyfrequencies, which does not give much insight into the observed oscillatory phenomenon. In fact, the oscillation cannot be assessed unless one uses nonlinear theory.The

13、refore, 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 unstab

14、le 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 i

15、nstability 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 m

16、idfrequency 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 c

17、an 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 modulat

18、ing 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

19、 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, whe

20、re no complex mathematical manipulation is needed. In this analysis, a dq model of stepper motors is used. Because two-phase motors and three-phase motors have the same qd model and therefore, the analysis is valid for both two-phase and three-phase motors. Up to date, it is only recognized that the

21、 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

22、 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 pro

23、duces 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, oscillat

24、ion 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 qd framereference transformation. The voltage equations for three-phase windings are given byva = Ria + L*dia /dt M*dib/dt M*d

25、ic/dt + dpma/dt ,vb = Rib + L*dib/dt M*dia/dt M*dic/dt + dpmb/dt ,vc = Ric + L*dic/dt M*dia/dt M*dib/dt + dpmc/dt ,where R and L are the resistance and inductance of the phase windings, and M is the mutual inductance between the phase windings. _pma, _pmb and _pmc are the flux-linkages of thephases

26、due to the permanent magnet, and can be assumed to be sinusoid functions of rotor position _ as followpma = 1 sin(N),pmb = 1 sin(N 2/3),pmc = 1 sin(N - 2/3),where N is number of rotor teeth. The nonlinearity emphasized in this paper is represented by the above equations, that is, the flux-linkages a

27、re nonlinear functions of the rotor position.By using the q; d transformation, the frame of reference is changed from the fixed phase axes to the axes moving with the rotor (refer to Figure 2). Transformation matrix from the a; b; c frame to the q; d frame is given by 8For example, voltages in the q

28、; d reference are given byIn the a; b; c reference, only two variables are independent (ia C ib C ic D 0); therefore, the above transformation from three variables to two variables is allowable. Applying the abovetransformation to the voltage equations (1), the transferred voltage equation in the q;

29、 d frame can be obtained asvq = Riq + L1*diq/dt + NL1id + N1,vd=Rid + L1*did/dt NL1iq, (5)Figure 2. a, b, c and d, q reference frame.where L1 D L CM, and ! is the speed of the rotor.It can be shown that the motors torque has the following form 2T = 3/2N1iqThe equation of motion of the rotor is writt

30、en asJ*d/dt = 3/2*N1iq Bf Tl ,where Bf is the coefficient of viscous friction, and Tl represents load torque, which is assumed to be a constant in this paper.In order to constitute the complete state equation of the motor, we need another state variable that represents the position of the rotor. For

31、 this purpose the so called load angle _ 8 is usually used, which satisfies the following equationD/dt = 0 ,where !0 is steady-state speed of the motor. Equations (5), (7), and (8) constitute the statespace model of the motor, for which the input variables are the voltages vq and vd. As mentioned be

32、fore, stepper motors are fed by an inverter, whose output voltages are not sinusoidal but instead are square waves. However, because the non-sinusoidal voltages do not change the oscillation feature and instability very much if compared to the sinusoidal case (as will be shown in Section 3, the osci

33、llation is due to the nonlinearity of the motor), for the purposes of this paper we can assume the supply voltages are sinusoidal. Under this assumption, we can get vq and vd as followsvq = Vmcos(N) ,vd = Vmsin(N) ,where Vm is the maximum of the sine wave. With the above equation, we have changed th

34、e input voltages from a function of time to a function of state, and in this way we can represent the dynamics of the motor by a autonomous system, as shown below. This will simplify the mathematical analysis.From Equations (5), (7), and (8), the state-space model of the motor can be written in a ma

35、trix form as follows = F(X,u) = AX + Fn(X) + Bu , (10)where X D Tiq id ! _UT , u D T!1 TlUT is defined as the input, and !1 D N!0 is the supply frequency. The input matrix B is defined byThe matrix A is the linear part of F._/, and is given byrepresents the nonlinear part of F._/, and is given byThe

36、 input term u is independent of time, and therefore Equation (10) is autonomous.There are three parameters in F.X;u/, they are the supply frequency !1, the supply voltage magnitude Vm and the load torque Tl . These parameters govern the behaviour of the stepper motor. In practice, stepper motors are

37、 usually driven in such a way that the supply frequency !1 is changed by the command pulse to control the motors speed, while the supply voltage is kept constant. Therefore, we shall investigate the effect of parameter !1.3. Bifurcation and Mid-Frequency OscillationBy setting ! D !0, the equilibria

38、of Equation (10) are given asand is its phase angle defined by = arctan(1L1/R) . (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 as shown in Equations (12) and (1

39、3). The first group represented by Equation (12) corresponds to 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, we will concentrate on the equilibria represented by Eq

40、uation (12).步进电机的振荡、不稳定以及控制摘要：本文介绍了一种分析永磁步进电机不稳定性的新颖方法。结果表明，该种电机有两种类型的不稳定现象：中频振荡和高频不稳定性。非线性分叉理论是用来说明局部不稳定和中频振荡运动之间的关系。一种新型的分析介绍了被确定为高频不稳定性的同步损耗现象。在相间分界线和吸引子的概念被用于导出数量来评估高频不稳定性。通过使用这个数量就可以很容易地估计高频供应的稳定性。此外，还介绍了稳定性理论。广义的方法给出了基于反馈理论的稳定问题的分析。结果表明，中频稳定度和高频稳定度可以提高状态反馈。关键词：步进电机，不稳定，非线性，状态反馈。1. 介绍步进电机是将数字脉冲

41、输入转换为模拟角度输出的电磁增量运动装置。其内在的步进能力允许没有反馈的精确位置控制。 也就是说，他们可以在开环模式下跟踪任何步阶位置，因此执行位置控制是不需要任何反馈的。步进电机提供比直流电机每单位更高的峰值扭矩;此外，它们是无电刷电机，因此需要较少的维护。所有这些特性使得步进电机在许多位置和速度控制系统的选择中非常具有吸引力，例如如在计算机硬盘驱动器和打印机，代理表，机器人中的应用等.尽管步进电机有许多突出的特性，他们仍遭受振荡或不稳定现象。这种现象严重地限制其开环的动态性能和需要高速运作的适用领域。 这种振荡通常在步进率低于1000脉冲/秒的时候发生，并已被确认为中频不稳定或局部不稳定1

42、，或者动态不稳定2。此外，步进电机还有另一种不稳定现象，也就是在步进率较高时，即使负荷扭矩小于其牵出扭矩，电动机也常常不同步。该文中将这种现象确定为高频不稳定性，因为它以比在中频振荡现象中发生的频率更高的频率出现。高频不稳定性不像中频不稳定性那样被广泛接受，而且还没有一个方法来评估它。中频振荡已经被广泛地认识了很长一段时间，但是，一个完整的了解还没有牢固确立。这可以归因于支配振荡现象的非线性是相当困难处理的。大多数研究人员在线性模型基础上分析它1。尽管在许多情况下，这种处理方法是有效的或有益的，但为了更好地描述这一复杂的现象，在非线性理论基础上的处理方法也是需要的。例如，基于线性模型只能看到电

43、动机在某些供应频率下转向局部不稳定，并不能使被观测的振荡现象更多深入。事实上，除非有人利用非线性理论，否则振荡不能评估。窗体顶端窗体底端因此，在非线性动力学上利用被发展的数学理论处理振荡或不稳定是很重要的。值得指出的是，Taft和Gauthier3，还有Taft和Harned4使用的诸如在振荡和不稳定现象的分析中的极限环和分界线之类的数学概念，并取得了关于所谓非同步现象的一些非常有启发性的见解。尽管如此，在这项研究中仍然缺乏一个全面的数学分析。本文一种新的数学分被开发了用于分析步进电机的振动和不稳定性。本文的第一部分讨论了步进电机的稳定性分析。结果表明，中频振荡可定性为一种非线性系统的分叉现象

44、（霍普夫分叉）。本文的贡献之一是将中频振荡与霍普夫分叉联系起来，从而霍普夫理论从理论上证明了振荡的存在性。高频不稳定性也被详细讨论了，并介绍了一种新型的量来评估高频稳定。这个量是很容易计算的，而且可以作为一种标准来预测高频不稳定性的发生。在一个真实电动机上的实验结果显示了该分析工具的有效性。本文的第二部分通过反馈讨论了步进电机的稳定性控制。一些设计者已表明，通过调节供应频率 5 ，中频不稳定性可以得到改善。特别是Pickup和Russell 6，7都在频率调制的方法上提出了详细的分析。在他们的分析中，雅可比级数用于解决常微分方程和一组数值有待解决的非线性代数方程组。此外，他们的分析负责的是双相

45、电动机，因此，他们的结论不能直接适用于我们需要考虑三相电动机的情况。在这里，我们提供一个没有必要处理任何复杂数学的更简洁的稳定步进电机的分析。在这种分析中，使用的是d-q模型的步进电机。由于双相电动机和三相电动机具有相同的d-q模型，因此，这种分析对双相电动机和三相电动机都有效。迄今为止，人们仅仅认识到用调制方法来抑制中频振荡。本文结果表明，该方法不仅对改善中频稳定性有效，而且对改善高频稳定性也有效。2. 动态模型的步进电机本文件中所考虑的步进电机由一个双相或三相绕组的跳动定子和永磁转子组成。一个极对三相电动机的简化原理如图1所示。步进电机通常是由被脉冲序列控制产生矩形波电压的电压源型逆变器供

46、给的。这种电动机用本质上和同步电动机相同的原则进行作业。步进电机主要作业方式之一是保持提供电压的恒定以及脉冲频率在非常广泛的范围上变化。在这样的操作条件下，振动和不稳定的问题通常会出现。图1.三相电动机的图解模型 用qd框架参考转换建立了一个三相步进电机的数学模型 。下面给出了三相绕组电压方程va = Ria + L*dia /dt M*dib/dt M*dic/dt + dpma/dt ,vb = Rib + L*dib/dt M*dia/dt M*dic/dt + dpmb/dt ,vc = Ric + L*dic/dt M*dia/dt M*dib/dt + dpmc/dt , (1)

47、其中R和L分别是相绕组的电阻和感应线圈，并且M是相绕组之间的互感线圈。pma, pmb and pmc 是应归于永磁体 的相的磁通，且可以假定为转子位置的正弦函数如下pma = 1 sin(N),pmb = 1 sin(N 2/3),pmc = 1 sin(N - 2/3), (2)其中N是转子齿数。本文中强调的非线性由上述方程所代表，即磁通是转子位置的非线性函数。使用Q ,d转换，将参考框架由固定相轴变换成随转子移动的轴（参见图2）。矩阵从a,b,c框架转换成q,d框架变换被给出了8 (3)例如，给出了q,d参考里的电压 (4)在a,b,c参考中，只有两个变量是独立的（ia + ib + ic = 0），因此，上面提到的由三个变量转化为两个变量是允许的。在电压方程（1）中应用上述转换，在q,d框架中获得转换后的电压方程为vq = Riq + L1*diq/dt + NL1id + N1,vd = Rid + L1*did/dt NL1iq, (5) 图2，a,b,c和d,q参考框架其中L1 = L +