步进电机的振荡不稳定以及控制

1、榆仲辗亥漾癫拣雪旅葜褂岌橥歉顿弗瘪阿压岐沉殉毕业设计(论文)外文资料翻译硭挺蜾夫驯骞蹯僬芤铢宿吊缎崧奇耦酵怦谇文菜昭系 （院）：昆铣谈钡嫣笃锚截碚叽馁菘淆伟娘吆鳢萜笛舣桥唛专 业：隽鸨雌帧泰芯墁逊酊洞妻电气工程及其自动化硝裎绌桉霆示啬雎睛厢杪姓 名：赣澳堀厢黥诤康府雾桴藓纷送丹偶液溱旖滢诲缙蓍学 号：洁蹙弗谋洳暖错肥首贴鳍氤俳媲塄尢奈摔岜甲谏撕外文出处：儇多拒夤堂农胯么檀烛懂Nonlinear Dynamics18:383-404,1999帮怠晌田忪疸谣拙邺鄯拳(用外文写)径昱杜猜熊蓣柰五飑掮苤话陲咝徒蔓授累煨浅邕孥附 件：迮哕空砾媳撤扔衢鹱缗碹1.外文资料翻译译文；2.外文原文。畈居酴衬夷兑

2、焚锌科梦瞎遘尕冻伛奄勘瓶蔺丛佑坠煅挟狡徜媵泺幌显苹握蛉吓谧苫酴宕诃鹊腧洚桓帆指导教师评语：陛藉过鲛骺选誊谇筷翦赆 外文资料翻译内容基本接近课题，具有一定的外文检索能力，中文译文语句基本通顺，但有个别语句不符合中文习惯表达。妊骠毫摆濂喁皙妤珙訇几2009饼嘲耽峋豆渝聋嵩浒胖呻年芈菥累瞻躏嫫审桂阋峭咳3腺均吏务酃臼巡旨交霭汰月既辶由鲱吐魃琳酪均晖灯15后驰畛溢仍啜莫鄯搋颠吸日伤擂浣桥蛱铜茎动菲氘濯签名： （手写签名）狺瓣妓钙跫刭玎错孵岑攉 祉悛倜绡穗淋经杀毕抄锬注：请将该封面与附件装订成册。附件1：外文资料翻译译文歉尝为蘑瓠衿拜庐锬楣卦噍蝾簇笏陋直训龇惋幸蚵步进电机的振荡、不稳定以及控制焊却雕澜盯

3、褐鸯琴檠动驮害墟崂楷塍帏痪以指叹泡摘要：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。谐阈雾呖獯杯纥元彀垡缡关键词：步进电机，不稳定，非线性，状态反馈。涌谗寻歪呦伯筱爰饶辽祆1.

4、介绍萦诈帼嵌甚框踩憧毯胺戡步进电机是将数字脉冲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.枣哒仿本洁昔皿皆盐鹋堕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

5、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，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。哧镅注勃恣罹窕闪嘧细阡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

6、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。窗体顶端雳敲惺霍己叹乘囡啄泷嫒窗体底端优嗤抠曩貌蟀乔坐沏儆州因此，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

7、被开发了用于分析步进电机的振动和不稳定性。哇毂栾荽颖繁艽搀稆史淆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（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。醚安沂谦冻丨枚媸伯蟊民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Pic

8、kup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。羿犴础虞辆韩汝缴粹

9、弘颁2. 动态模型的步进电机茸芨龆舅遇峥拈惘崩取蘸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。召猴龅砗癫糸瘴隗戕锨鲕六洪苁隧镭椤醑谘虮轱恸图1.三相电动机的图解模型 迕陵辖荻炼迈烦脾搭嵛尉用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晨喽祉溘伤赖墨獍鸸

10、迈铆va = Ria + L*dia /dt M*dib/dt M*dic/dt + dpma/dt ,翅裂圾贾荽折痱脾缩怏哪vb = Rib + L*dib/dt M*dia/dt M*dic/dt + dpmb/dt ,螽税显瞑莅发懊螭戟犴敖vc = Ric + L*dic/dt M*dia/dt M*dib/dt + dpmc/dt , (1) 抟膣胺刺羲盾糍睚堇摺泥其中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。算缘闯箕临伽狮踯磙钪蕻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谄娓疝挪每暇躲赭砑舌董pma = 1

11、 sin(N),牧惧锑瘴庄筇师瓶蜞钝珐pmb = 1 sin(N 2/3),耸舂砺凳牵喋炙偎狼我锰pmc = 1 sin(N - 2/3), (2)郸歌睬弛柚晖奂唑妆大亏其中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。秫砍喉凰忒死贤蘑攮缏乏使用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顺蚬报览膜鹘磲下傣脬谲 (3)搁绒地膏回幼蹶氍畔蒲肫例如，给出了q,d参考里的电压杯涑苹掳痊十嘣恍睬沮阼 (4)占象穑健岩程谦焯活用纳在a,b,c参考中，只有两个变量是独立的（ia + ib + ic

12、= 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撵觌例嶂匝痘诘溱煸丕昂vq = Riq + L1*diq/dt + NL1id + N1,妆碛谎搋仪阅棠嵩坯豹螃vd = Rid + L1*did/dt NL1iq, (5) 欷渭缔昀暖谚羡转筅词砻妊关囊锿喝仫阄挽泶烟吕图2，a,b,c和d,q参考框架慈奁呙埃备妫订痞揪瘁町其中L1 = L + M，且是电动机的速度。缙雉缸夙钩菔闫裁匣恃悝有证据表明,电动机的扭矩有以下公式魁蛾神骒逵捃呦芒好水嶙T = 3/2N1iq . (6)苇笱屡钣卢扫恚宝玺枵靶转子电动机的方程为苑

13、湿八计桦锑娘颧俯繁癜J*d/dt = 3/2*N1iq Bf Tl , (7) 桥胲嗷鄢汐厂涿阡荡裉缴如果Bf是粘性摩擦系数，和Tl代表负荷扭矩（在本文中假定为恒定）。凉夥楫疋艚阎璨莱憷鼙蓑为了构成完整的电动机的状态方程，我们需要另一种代表转子位置的状态变量。为此，通常使用满足下列方程的所谓的负荷角8皋獒蟛溉疫捣爵壕暑私胃D/dt = 0 , (8) 丁焦察膏熙迩治耽怔葜皲其中0是电动机的稳态转速。方程（5），（7），和（8）构成电动机的状态空间模型，其输入变量是电压vq和vd.如前所述，步进电机由逆变器供给，其输出电压不是正弦电波而是方波。然而，由于相比正弦情况下非正弦电压不能很大程度地改变

14、振荡特性和不稳定性（如将在第3部分显示的，振荡是由于电动机的非线性），为了本文的目的我们可以假设供给电压是正弦波。根据这一假设，我们可以得到如下的vq和vd撰镘犹磔港街殳铺僬仿谩vq = Vmcos(N) ,搋瘙盲慷悄昝截纱崴罪堂vd = Vmsin(N) , (9) 挽鞫世戆昙囤毓憝唾弓龈其中Vm是正弦波的最大值。上述方程，我们已经将输入电压由时间函数转变为状态函数，并且以这种方式我们可以用自控系统描绘出电动机的动态，如下所示。这将有助于简化数学分析。无柞童茫酏单黄斤眢瞠郓根据方程（5），（7），和（8），电动机的状态空间模型可以如下写成矩阵式咂拙灿源鹾谀骡沁圃硐衽 = F(X,u) = A

15、X + Fn(X) + Bu , (10) 佟欠翥善罚盛盔弓貉斑籁其中X = iq id T, u = 1 Tl T 定义为输入，且1 = N0 是供应频率。输入矩阵B被定义为厂槿瘵褡估佐壅男蛮泻戡蚰腾挠睿碉饧衡寺轷岱僻矩阵A是F(.)的线性部分，如下炫撇柁乏樘颓绡菟秤儋案涉挨惯祭曰髹际浇堵髻迩Fn(X)代表了F(.)的线性部分，如下鲑筷呤烫虏情苛琅钅府郭岫孽婧睹舫适荇谷縻墼拟输入端u独立于时间，因此，方程（10）是独立的。愦锈泼哥准荜垴芩始仁喃在F(X,u)中有三个参数，它们是供应频率1，电源电压幅度Vm和负荷扭矩Tl。这些参数影响步进电机的运行情况。在实践中，通常用这样一种方式来驱动步进电

16、机，即用因指令脉冲而变化的供应频率1来控制电动机的速度，而电源电压保持不变。因此，我们应研究参数1的影响。镇菅脸痰痛惑蜓屿副泐袭3.分叉和中频振荡，遢荀狈榫泱博述艟鬼渴肢设=0，得出方程（10）的平衡佛荻铛疤逻罅傩鸩薹洼枥套崇闯蝼循业账蒈匹薷顽缏滋嚎雌脊湿袜榭聪堑且是它的相角，搛莰嗽斩屯羊骑铧秸愁饿 = arctan(1L1/R) . (16) 骤恒绉防闩筛蛸忘赫亢揉方程（12）和（13）显示存在着多重均衡，这意味着这些平衡永远不能全局稳定。人们可以看到，如方程（12）和（13）所示有两组平衡。第一组由方程（12）对应电动机的实际运行情况来代表。第二组由方程（13）总是不稳定且不涉及到实际运作

17、情况来代表。在下面，我们将集中精力在由方程（12）代表的平衡上。 鲋罹斜振罢瓮赦拭威璃砝附件2：外文原文佥渐鼬碜蹇撺椿洱众锘绌 Oscillation, Instability and Control of Stepper Motors偾濡邵蛾惰衄枋鹗皖忸籍LIYU CAO and HOWARD M. SCHWARTZ转禚搌嗷艇板筅僬岷攴嘈Department of Systems and Computer Engineering, Carleton University, 1125 Colonel By Drive,雏舟暧疯仙噎脸磷肄从瘛Ottawa, ON K1S 5B6, Canada谣

18、炽妊獠哓璃赶赵垴句岐(Received: 18 February 1998; accepted: 1 December 1998)缛卜斓扇事舞陈婢迎斛钌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-frequen

19、cy instability. Nonlinear bifurcation theory is used to illustrate the relationship between local instability and midfrequency缨方媵娃洌予擒桉赐谁阚oscillatory motion. A novel analysis is presented to analyze the loss of synchronism phenomenon, which is identified as high-frequency instability. The concepts of

20、 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 stab

21、ilization problem based on feedback theory is given. It is shown that the mid-frequency stability头晡分菌坎遥出鳐遴俳艏and the high-frequency stability can be improved by state feedback. Keywords: Stepper motors, instability, nonlinearity, state feedback.痒瞽栖戤史规徽辑虔胙邂1. Introduction圻秭影止撇葚嫔躺扳憝刊Stepper motors are

22、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 p

23、osition control. Stepper motors deliver 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

24、 in computer hard disk drivers and printers, XY-tables, robot manipulators, etc.罾咏碛廑蹿此腐銎姿枰多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 hi

25、gh 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

26、 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

27、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.蔼糸昆鳔黻薤捅侄俭金酌Mid-frequency oscillation has been recognized widely for a very long time, however, a complete understanding of it has not been well establis

28、hed. This can be attributed to the nonlinearity that dominates the oscillation phenomenon and is quite difficult to deal with.惭稻腹按蒋槲扪潘徽朝替384 L. Cao and H. M. Schwartz轩嬉狯颈攒尝挡刑别掀尖Most researchers have analyzed it based on a linearized model 1. Although in many cases, this kind of treatments is valid o

29、r 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 some supply攉足佳讲拉歙桑隋阁舜柜frequencies, which does not give much insight int

30、o the observed oscillatory phenomenon. In fact, the oscillation cannot be assessed unless one uses nonlinear theory.皖诊执黎疙形硬架呢憋宾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

31、, 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 mathema

32、tical analysis in this kind of studies. In this paper a novel mathematical analysis is developed to analyze the oscillations and instability in stepper motors.昶篪歇镌靴阉荤汞隆阴授The first part of this paper discusses the stability analysis of stepper motors. It is shown that the mid-frequency oscillation ca

33、n 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 proved被嗜俘嚷籽懿昆箱操部济theoretically by Hopf theory. High-frequency instabi

34、lity is also discussed in detail, and a novel quantity is introduced to evaluate high-frequency stability. This quantity is very easy颦哟升袱捧盾踣歆枧雳丝to 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

35、 of this analytical tool.骧将搜汨才棣烷尝瓢聘醉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 midfrequency苦锻盖颇圹堋圜碥踱撅酚instability can be improved. In particular, Pickup and Russell 6, 7 hav

36、e 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 there

37、fore, 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 dq model of stepper motors is used. Because

38、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 modulation method is needed to suppress the midfrequency oscillation. In this paper, it is shown that this method

39、is not only valid to improve mid-frequency stability, but also effective to improve high-frequency stability.祆蟀捻姹蘸呐嗓畴刺篮母2. Dynamic Model of Stepper Motors承蚣肢支睥萜魅拶嗜瘴伯The stepper motor considered in this paper consists of a salient stator with two-phase or threephase windings, and a permanent-magnet r

40、otor. 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. This泞琵妻蚓竭疯觯寮晒撖痪motor operates essentially on the same principle a

41、s that of synchronous motors. One of major operating manner for stepper motors is that supplying voltage is kept constant and frequency拮醺妹趣镔揎夏拣舸灞完of pulses is changed at a very wide range. Under this operating condition, oscillation and instability problems usually arise.粉飓康老蛋碡翟皋岗臂襻匠笛瞬慢憬窦瞥包飒纭疬Figure

42、 1. Schematic model of a three-phase stepper motor.恼琪主吖丐凰舔胶淳青唆A mathematical model for a three-phase stepper motor is established using qd framereference transformation. The voltage equations for three-phase windings are given by生椹扩橥贞探袭捐场梧缬va = Ria + L*dia /dt M*dib/dt M*dic/dt + dpma/dt ,撞友珐畛漯珲莲家奴具

43、靶vb = Rib + L*dib/dt M*dia/dt M*dic/dt + dpmb/dt ,机榨芽稂驭铒淦盆蕖箫霸vc = Ric + L*dic/dt M*dia/dt M*dib/dt + dpmc/dt ,琼芈给嫁蜡蟠悔劲孪弹邝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 the鸵

44、毽丝袅萋逆厶欧米赂耦phases due to the permanent magnet, and can be assumed to be sinusoid functions of rotor position _ as follow楂笙瞧饺叫面偬象篷戡很pma = 1 sin(N),屺民胍謇瘩哗浒莪绥吊薪pmb = 1 sin(N 2/3),碚苇堤硬绻茶篌狄跆鞍垧pmc = 1 sin(N - 2/3),禀悔押敌噬萜播俯舫尿已where N is number of rotor teeth. The nonlinearity emphasized in this paper is rep

45、resented by the above equations, that is, the flux-linkages are nonlinear functions of the rotor position.洲卧汲礁瞟健赀贝佳水麾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;

46、b; c frame to the q; d frame is given by 8禽俺速韦继凌砭缠醑昵氆桁鄂霁趋妆席彷旒荒剑鼙For example, voltages in the q; d reference are given by税勰鼙蘼鏖坪琚蕃搞拚蔬舶樽钷丌桦椁姿冂梳且虾In 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

47、allowable. Applying the above闻险磕苏窒壹东散殊世裟transformation to the voltage equations (1), the transferred voltage equation in the q; d frame can be obtained as罗鲆唛恝距蝰光玷冤矛么vq = Riq + L1*diq/dt + NL1id + N1,赁哇陛淳八博奖毹樱埴旱vd=Rid + L1*did/dt NL1iq, (5)玛迹喔朊星沧谬怵沓豳售谲蛋柔丑鞍哮窕幅糕馆俐Figure 2. a, b, c and d, q reference fr

48、ame.坍然铰埠鱿垢庄段蘅港沾where L1 D L CM, and ! is the speed of the rotor.It can be shown that the motors torque has the following form 2舌喂宛四抗矫漉虿洄碍粞T = 3/2N1iq嘤蓁毗州崦懊堙榆魇黻咩The equation of motion of the rotor is written as戆疋锥鹨渐畲拄刘双奢J*d/dt = 3/2*N1iq Bf Tl ,绉善洞钗铯桔篮略惧锿拐where Bf is the coefficient of viscous fricti

49、on, and Tl represents load torque, which is assumed to be a constant in this paper.璨咴餍疒寝瘀猸肀瘢喂君In order to constitute the complete state equation of the motor, we need another state variable that represents the position of the rotor. For this purpose the so called load angle _ 8 is usually used, whic

50、h satisfies the following equation鼓痿豫眸勘徜审臣蔸庳划D/dt = 0 ,逼盔霉胜晒窭架摭酝庠逦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 before, stepper motors are fed by an inverte

51、r, 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 oscillation is due to the nonlinearity of the

52、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 follows是蛙妊尼妗炽妇眸礓蛟嗔vq = Vmcos(N) ,腰贶兰哮术钧猊虎徘曜罄vd = Vmsin(N) ,韭霎湎荑胰嵯痱掳枫嘞弹where Vm is the maximum of the sine wave. With the above equation, we have changed the input v

53、oltages 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.垒迨窆感鏊坠嬴苤屎桌佳From Equations (5), (7), and (8), the state-space model of the motor can be written in a

54、matrix form as follows芹掩剧妇喝昌枇落磲镩慊 = F(X,u) = AX + Fn(X) + Bu , (10)狸叽做回碑赋糊垮跤屎集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 by鸺锁髭苋啷哏倨啐笋濠珊培鞔囊闹姻跸屙妮幻叨跪The matrix A is the linear part of F._/, and is given by并遒硒叟杷蕞歃厶波削谢

55、阢笮酞唳撩芜掖阑锭衡剡Fn.X/ represents the nonlinear part of F._/, and is given by堵貌奄旱附关逵析镥灸焐膪谯凝绌硬迪误嘲崇烫先The input term u is independent of time, and therefore Equation (10) is autonomous.巢卑雪舢竺碲钿陈呲锈左There are three parameters in F.X;u/, they are the supply frequency !1, the supply voltage magnitude Vm and the l

56、oad torque Tl . These parameters govern the behaviour of the stepper motor. In practice, stepper motors are 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.砌蚝活愆瞿魑肢咯老祓锿3. Bifurcation and Mid-Frequency Oscillation蠹呱侉阡疫芳己筠夭怕偈By setting ! D !