自動化專業(yè)外文翻譯---步進電機的振蕩、不穩(wěn)定以及控制

1、<p><b>　　電子與信息工程學院</b></p><p>　　本科畢業(yè)論文(設計)</p><p>　　外 文 文 獻 翻 譯</p><p>　　譯文題目：Oscillation, Instability and Control of Stepper Motors

2、 </p><p>　　學生姓名： 宋 海 軍 </p><p>　　專 業(yè)： 電氣工程及其自動化 </p><p>　　指導教師： 李 東 京

3、 </p><p><b>　　2010年2月 </b></p><p>　　Oscillation, Instability and Control of Stepper Motors</p><p>　　LIYU CAO and HOWARD M. SCHWARTZ</p><p&

4、gt;　　Department of Systems and Computer Engineering, Carleton University, 1125 Colonel By Drive,</p><p>　　Ottawa, ON K1S 5B6, Canada</p><p>　　(Received: 18 February 1998; accepted: 1 December 19

5、98)</p><p>　　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 of motor: mid-frequen

6、cy oscillation and high-frequency instability. Nonlinear bifurcation theory is used to illustrate the relationship between local instability and mid-frequency oscillatory motion. A novel analysis is presented to analyze

7、the loss of synchronism phenomenon, which is identified as high-frequency instability.</p><p>　　Keywords: Stepper motors; instability; nonlinearity; state feedback</p><p>　　1. Introduction</p

8、><p>　　Stepper 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 feed

9、back. 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 unit weight than DC motors; in addition,

10、they are brushless machines and therefore require less maintenance. All of th</p><p>　　Although stepper motors have many salient properties, they suffer from an oscillation or unstable phenomenon. This pheno

11、menon 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

12、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,</p><p>　　Mid-frequency oscillation has been re

13、cognized 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

14、deal with.</p><p>　　384 L. Cao and H. M. Schwartz</p><p>　　Most researchers have analyzed it based on a linearized model [1]. Although in many cases, this kind of treatments is valid or useful,

15、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 s

16、ome supply</p><p>　　frequencies, which does not give much insight into the observed oscillatory phenomenon. In fact, the oscillation cannot be assessed unless one uses nonlinear theory.</p><p>　

17、　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

18、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 s

19、till a lack of a comprehensive mathematical analysis in this </p><p>　　The first part of this paper discusses the stability analysis of stepper motors. It is shown that the mid-frequency oscillation can be c

20、haracterized as a bifurcation phenomenon (Hopf bifurcation) of nonlinear systems. One of contributions of this paper is to relate the mid-frequency oscillation to Hopf bifurcation, thereby; the existence of the oscillati

21、on is proved</p><p>　　Theoretically 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 easy

22、</p><p>　　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 of this analytical tool.</p><

23、p>　　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 mid-frequency</p><p>　　Insta

24、bility 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,

25、 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

26、a three-phase motor will be considered. Here, we give a</p><p>　　2. Dynamic Model of Stepper Motors</p><p>　　The stepper motor considered in this paper consists of a salient stator with two-phas

27、e or three phase 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 cont

28、rolled by a sequence of pulses and produces square-wave voltages. This motor operates essentially on the same principle as that of synchronous motors. One of major operating manner for stepper mo</p><p>　　Fi

29、gure 1. Schematic model of a three-phase stepper motor.</p><p>　　A mathematical model for a three-phase stepper motor is established using q–d frame reference transformation. The voltage equations for three-

30、phase windings are given by</p><p>　　va = Ria + L*dia /dt ? M*dib/dt ? M*dic/dt + dλpma/dt ,</p><p>　　vb = Rib + L*dib/dt ? M*dia/dt ? M*dic/dt + dλpmb/dt ,</p><p>　　vc = Ric + L*di

31、c/dt ? M*dia/dt ? M*dib/dt + dλpmc/dt ,</p><p>　　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

32、flux-linkages of the</p><p>　　phases due to the permanent magnet, and can be assumed to be sinusoid functions of rotor position _ as follow</p><p>　　λpma = λ1 sin(Nθ),</p><p>　　λpmb

33、 = λ1 sin(Nθ ? 2 /3),</p><p>　　λpmc = λ1 sin(Nθ - 2 /3),</p><p>　　where N is number of rotor teeth. The nonlinearity emphasized in this paper is represented by the above equations, that is, the

34、flux-linkages are nonlinear functions of the rotor position.</p><p>　　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

35、 Figure 2). Transformation matrix from the a; b; c frame to the q; d frame is given by [8]</p><p>　　For example, voltages in the q; d reference are given by</p><p>　　In the a; b; c reference, on

36、ly 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 above transformation to the voltage equations (1), the transferred

37、 voltage equation in the q; d frame can be obtained as</p><p>　　vq = Riq + L1*diq/dt + NL1idω + Nλ1ω,</p><p>　　vd=Rid + L1*did/dt ? NL1iqω, (5)</p>&

38、lt;p>　　Figure 2. a, b, c and d, q reference frame.</p><p>　　where L1 D L CM, and ! is the speed of the rotor.It can be shown that the motor’s torque has the following form [2]</p><p>　　T = 3/

39、2Nλ1iq</p><p>　　The equation of motion of the rotor is written as</p><p>　　J*dω/dt = 3/2*Nλ1iq ? Bfω – Tl ,</p><p>　　where Bf is the coefficient of viscous friction, and Tl represen

40、ts load torque, which is assumed to be a constant in this paper.</p><p>　　In order to constitute the complete state equation of the motor, we need another state variable that represents the position of the r

41、otor. For this purpose the so called load angle _ [8] is usually used, which satisfies the following equation</p><p>　　Dδ/dt = ω?ω0 ,</p><p>　　where !0 is steady-state speed of the motor. Equat

42、ions (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 inverter, whose output voltages are not sin

43、usoidal 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 os

44、cillation </p><p>　　vq = Vmcos(Nδ) ,</p><p>　　vd = Vmsin(Nδ) ,</p><p>　　where Vm is the maximum of the sine wave. With the above equation, we have changed the input voltages from a

45、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.</p><p>　　From Equatio

46、ns (5), (7), and (8), the state-space model of the motor can be written in a matrix form as follows</p><p>　　? = F(X,u) = AX + Fn(X) + Bu , (10)</p><p>　　where

47、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</p><p>　　The matrix A is the linear part of F._/, and is given by</p>&l

48、t;p>　　Fn.X/ represents the nonlinear part of F._/, and is given by</p><p>　　The input term u is independent of time, and therefore Equation (10) is autonomous.</p><p>　　There are three parame

49、ters 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 usually driven in suc

50、h a way that the supply frequency !1 is changed by the command pulse to control the motor’s speed, while the supply voltage is kept constant. Therefore, we shall investigate the effect of parameter !1.</p><p&g

51、t;　　3. Bifurcation and Mid-Frequency Oscillation</p><p>　　By setting ! D !0, the equilibria of Equation (10) are given as</p><p>　　and ' is its phase angle defined by</p><p>　　φ

52、 = arctan(ω1L1/R) . (16) </p><p>　　Equations (12) and (13) indicate that multiple equilibria exist, which means that these equilibria can never be globa

53、lly stable. One can see that there are two groups of equilibria as shown in Equations (12) and (13). The first group represented by Equation (12) corresponds to the real operating conditions of the motor. The second grou

54、p 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 rep</p><p>　　步進電機的振蕩、不穩(wěn)定以及控制</p><p&g

55、t;　　摘要：本文介紹了一種分析永磁步進電機不穩(wěn)定性的新穎方法。結果表明，該種電機有兩種類型的不穩(wěn)定現(xiàn)象：中頻振蕩和高頻不穩(wěn)定性。非線性分叉理論是用來說明局部不穩(wěn)定和中頻振蕩運動之間的關系。一種新型的分析介紹了被確定為高頻不穩(wěn)定性的同步損耗現(xiàn)象。在相間分界線和吸引子的概念被用于導出數(shù)量來評估高頻不穩(wěn)定性。通過使用這個數(shù)量就可以很容易地估計高頻供應的穩(wěn)定性。此外，還介紹了穩(wěn)定性理論。廣義的方法給出了基于反饋理論的穩(wěn)定問題的分析。結果表明，

56、中頻穩(wěn)定度和高頻穩(wěn)定度可以提高狀態(tài)反饋。</p><p>　　關鍵詞：步進電機；不穩(wěn)定；非線性；狀態(tài)反饋</p><p><b>　　1. 介紹</b></p><p>　　步進電機是將數(shù)字脈沖輸入轉換為模擬角度輸出的電磁增量運動裝置。其內在的步進能力允許沒有反饋的精確位置控制。 也就是說，他們可以在開環(huán)模式下跟蹤任何步階位置，因此執(zhí)行位置控制

57、是不需要任何反饋的。步進電機提供比直流電機每單位更高的峰值扭矩;此外，它們是無電刷電機，因此需要較少的維護。所有這些特性使得步進電機在許多位置和速度控制系統(tǒng)的選擇中非常具有吸引力，例如如在計算機硬盤驅動器和打印機，代理表，機器人中的應用等.</p><p>　　盡管步進電機有許多突出的特性，他們仍遭受振蕩或不穩(wěn)定現(xiàn)象。這種現(xiàn)象嚴重地限制其開環(huán)的動態(tài)性能和需要高速運作的適用領域。 這種振蕩通常在步進率低于1000脈

58、沖/秒的時候發(fā)生，并已被確認為中頻不穩(wěn)定或局部不穩(wěn)定[1]，或者動態(tài)不穩(wěn)定[2]。此外，步進電機還有另一種不穩(wěn)定現(xiàn)象，也就是在步進率較高時，即使負荷扭矩小于其牽出扭矩，電動機也常常不同步。該文中將這種現(xiàn)象確定為高頻不穩(wěn)定性，因為它以比在中頻振蕩現(xiàn)象中發(fā)生的頻率更高的頻率出現(xiàn)。高頻不穩(wěn)定性不像中頻不穩(wěn)定性那樣被廣泛接受，而且還沒有一個方法來評估它。</p><p>　　中頻振蕩已經(jīng)被廣泛地認識了很長一段時間，但是，

59、一個完整的了解還沒有牢固確立。這可以歸因于支配振蕩現(xiàn)象的非線性是相當困難處理的。大多數(shù)研究人員在線性模型基礎上分析它[1]。盡管在許多情況下，這種處理方法是有效的或有益的，但為了更好地描述這一復雜的現(xiàn)象，在非線性理論基礎上的處理方法也是需要的。例如，基于線性模型只能看到電動機在某些供應頻率下轉向局部不穩(wěn)定，并不能使被觀測的振蕩現(xiàn)象更多深入。事實上，除非有人利用非線性理論，否則振蕩不能評估。窗體頂端</p><p>

60、;<b>　　窗體底端</b></p><p>　　因此，在非線性動力學上利用被發(fā)展的數(shù)學理論處理振蕩或不穩(wěn)定是很重要的。值得指出的是，Taft和Gauthier[3]，還有Taft和Harned[4]使用的諸如在振蕩和不穩(wěn)定現(xiàn)象的分析中的極限環(huán)和分界線之類的數(shù)學概念，并取得了關于所謂非同步現(xiàn)象的一些非常有啟發(fā)性的見解。盡管如此，在這項研究中仍然缺乏一個全面的數(shù)學分析。本文一種新的數(shù)學分被開

61、發(fā)了用于分析步進電機的振動和不穩(wěn)定性。</p><p>　　本文的第一部分討論了步進電機的穩(wěn)定性分析。結果表明，中頻振蕩可定性為一種非線性系統(tǒng)的分叉現(xiàn)象（霍普夫分叉）。本文的貢獻之一是將中頻振蕩與霍普夫分叉聯(lián)系起來，從而霍普夫理論從理論上證明了振蕩的存在性。高頻不穩(wěn)定性也被詳細討論了，并介紹了一種新型的量來評估高頻穩(wěn)定。這個量是很容易計算的，而且可以作為一種標準來預測高頻不穩(wěn)定性的發(fā)生。在一個真實電動機上的實驗結

62、果顯示了該分析工具的有效性。</p><p>　　本文的第二部分通過反饋討論了步進電機的穩(wěn)定性控制。一些設計者已表明，通過調節(jié)供應頻率[ 5 ]，中頻不穩(wěn)定性可以得到改善。特別是Pickup和Russell [ 6，7]都在頻率調制的方法上提出了詳細的分析。在他們的分析中，雅可比級數(shù)用于解決常微分方程和一組數(shù)值有待解決的非線性代數(shù)方程組。此外，他們的分析負責的是雙相電動機，因此，他們的結論不能直接適用于我們需要考

63、慮三相電動機的情況。在這里，我們提供一個沒有必要處理任何復雜數(shù)學的更簡潔的穩(wěn)定步進電機的分析。在這種分析中，使用的是d-q模型的步進電機。由于雙相電動機和三相電動機具有相同的d-q模型，因此，這種分析對雙相電動機和三相電動機都有效。迄今為止，人們僅僅認識到用調制方法來抑制中頻振蕩。本文結果表明，該方法不僅對改善中頻穩(wěn)定性有效，而且對改善高頻穩(wěn)定性也有效。</p><p>　　2. 動態(tài)模型的步進電機</p

64、><p>　　本文件中所考慮的步進電機由一個雙相或三相繞組的跳動定子和永磁轉子組成。一個極對三相電動機的簡化原理如圖1所示。步進電機通常是由被脈沖序列控制產(chǎn)生矩形波電壓的電壓源型逆變器供給的。這種電動機用本質上和同步電動機相同的原則進行作業(yè)。步進電機主要作業(yè)方式之一是保持提供電壓的恒定以及脈沖頻率在非常廣泛的范圍上變化。在這樣的操作條件下，振動和不穩(wěn)定的問題通常會出現(xiàn)。</p><p>　　圖

65、1.三相電動機的圖解模型 </p><p>　　用q–d框架參考轉換建立了一個三相步進電機的數(shù)學模型 。下面給出了三相繞組電壓方程</p><p>　　va = Ria + L*dia /dt ? M*dib/dt ? M*dic/dt + dλpma/dt ,</p><p>　　vb = Rib + L*dib/dt ? M*dia/dt ? M*dic/dt

66、+ dλpmb/dt ,</p><p>　　vc = Ric + L*dic/dt ? M*dia/dt ? M*dib/dt + dλpmc/dt , (1) </p><p>　　其中R和L分別是相繞組的電阻和感應線圈，并且M是相繞組之間的互感線圈。</p><p>　　λpma, λpmb a

67、nd λpmc 是應歸于永磁體 的相的磁通，且可以假定為轉子位置的正弦函數(shù)如下</p><p>　　λpma = λ1 sin(Nθ),</p><p>　　λpmb = λ1 sin(Nθ ? 2 /3),</p><p>　　λpmc = λ1 sin(Nθ - 2 /3), (2)<

68、/p><p>　　其中N是轉子齒數(shù)。本文中強調的非線性由上述方程所代表，即磁通是轉子位置的非線性函數(shù)。</p><p>　　使用Q ,d轉換，將參考框架由固定相軸變換成隨轉子移動的軸（參見圖2）。矩陣從a,b,c框架轉換成q,d框架變換被給出了[8]</p><p><b>　　(3)</b></p><p>　　例如，給出

69、了q,d參考里的電壓</p><p><b>　　(4)</b></p><p>　　在a,b,c參考中，只有兩個變量是獨立的（ia + ib + ic = 0），因此，上面提到的由三個變量轉化為兩個變量是允許的。在電壓方程（1）中應用上述轉換，在q,d框架中獲得轉換后的電壓方程為</p><p>　　vq = Riq + L1*diq/dt

70、+ NL1idω + Nλ1ω,</p><p>　　vd = Rid + L1*did/dt ? NL1iqω, (5) </p><p>　　圖2，a,b,c和d,q參考框架</p><p>　　其中L1 = L + M

71、，且ω是電動機的速度。</p><p>　　有證據(jù)表明,電動機的扭矩有以下公式</p><p>　　T = 3/2Nλ1iq . (6)</p><p><b>　　轉子電動機的方程為</b></p><p>　　J*d

72、ω/dt = 3/2*Nλ1iq ? Bfω – Tl , (7) </p><p>　　如果Bf是粘性摩擦系數(shù)，和Tl代表負荷扭矩（在本文中假定為恒定）。</p><p>　　為了構成完整的電動機的狀態(tài)方程，我們需要另一種代表轉子位置的狀態(tài)變量。為此，通常使用滿足下列方程的所謂的負荷角δ[8]</p>

73、;<p>　　Dδ/dt = ω?ω0 , (8) </p><p>　　其中ω0是電動機的穩(wěn)態(tài)轉速。方程（5），（7），和（8）構成電動機的狀態(tài)空間模型，其輸入變量是電壓vq和vd.如前所述，步進電機由逆變器供給，其輸出電壓不是正弦電波而是方波。然而，由于相比正弦情況下非正弦電壓不能很大程度地改變

74、振蕩特性和不穩(wěn)定性（如將在第3部分顯示的，振蕩是由于電動機的非線性），為了本文的目的我們可以假設供給電壓是正弦波。根據(jù)這一假設，我們可以得到如下的vq和vd</p><p>　　vq = Vmcos(Nδ) ,</p><p>　　vd = Vmsin(Nδ) , (9) &l

75、t;/p><p>　　其中Vm是正弦波的最大值。上述方程，我們已經(jīng)將輸入電壓由時間函數(shù)轉變?yōu)闋顟B(tài)函數(shù)，并且以這種方式我們可以用自控系統(tǒng)描繪出電動機的動態(tài)，如下所示。這將有助于簡化數(shù)學分析。</p><p>　　根據(jù)方程（5），（7），和（8），電動機的狀態(tài)空間模型可以如下寫成矩陣式</p><p>　　? = F(X,u) = AX + Fn(X) + Bu ,

76、 (10) </p><p>　　其中X = [iq id ω δ] T, u = [ω1 Tl] T 定義為輸入，且ω1 = Nω0 是供應頻率。輸入矩陣B被定義為</p><p>　　矩陣A是F(.)的線性部分，如下</p><p>　　Fn(X)代表了F(.)的線性部分，如

77、下</p><p>　　輸入端u獨立于時間，因此，方程（10）是獨立的。</p><p>　　在F(X,u)中有三個參數(shù)，它們是供應頻率ω1，電源電壓幅度Vm和負荷扭矩Tl。這些參數(shù)影響步進電機的運行情況。在實踐中，通常用這樣一種方式來驅動步進電機，即用因指令脈沖而變化的供應頻率ω1來控制電動機的速度，而電源電壓保持不變。因此，我們應研究參數(shù)ω1的影響。</p><p&

78、gt;　　3.分叉和中頻振蕩，</p><p>　　設ω=ω0，得出方程（10）的平衡</p><p><b>　　且φ是它的相角，</b></p><p>　　φ = arctan(ω1L1/R) . (16) </p><