专业英语电气工程P2U3教学课件.ppt
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1、自动化专业英语教程,教学课件,July 28, 2007,P2U3A The Root Locus 第二部分第三单元课文A 根轨迹,A 根轨迹 1.课文内容简介:主要介绍自动控制原理中根轨迹的定义、幅角与幅值判据、绘制根轨迹的规则、根轨迹法用于系统设计和补偿等内容。 2.温习自动控制原理中有关根轨迹的内容。 3. 生词与短语 factored adj. 可分解的 depict v. 描述 conjugate adj. 共轭的 vector n. 矢量 argument n. 辐角,相位 counterclockwise adj. 逆时针的 odd multiple 奇数倍 even multi
2、ple 偶数倍,P2U3A The Root Locus 第二部分第三单元课文A 根轨迹,plot v. 绘图 n. 曲线图 sketch v., n. (绘)草图,素描 facilitate v. 使容易,促进 coincide v. 一致 asymptote n. 渐进线 integer n. 整数 intersect v. 相交 real axis 实轴 symmetrical adj. 对称的 breakaway point 分离点 arrival point 汇合点 departure angle 出射角 arrival angle 入射角 thereof adv. 将它(们) im
3、aginary axis 虚轴 passive adj. 被动的,无源的,P2U3A The Root Locus 第二部分第三单元课文A 根轨迹,active adj. 主动的,有源的 network n. 网络,电路 phase-lead n. 相位超前 phase-lag n. 相位滞后 4. 难句翻译 1 as any single parameter, such as a gain or time constant, is varied from zero to infinity. 当任意单一参数,如增益或时间常数,从零变到无穷时。 2 These effects increase
4、in strength with decreasing distance. 随着到原点距离的减小,它们的作用强度会增加。 此处distance指零(极)点到原点的距离。 3 Ignoring for the weaker effect of the added pole, which is often placed at 10 times the distance to the origin, the zero 忽略常被置于10倍于零点到原点距离处的附加极点的微弱作用,零点,The Root Locus,Introduction,The three basic performance crit
5、eria for a control system are stability, acceptable steady-state accuracy, and an acceptable transient response. With the system transfer function known, the Routh-Hurwitz criterion will tell us whether or not a system is stable. If it is stable, the steady-state accuracy can be determined for vario
6、us types of inputs. To determine the nature of the transient response, we need to know the location in the s plane of the roots of the characteristic equation. Unfortunately, the characteristic equation is normally unfactored and of high order.,The root locus technique is a graphical method of deter
7、mining the location of the roots of the characteristic equation as any single parameter, such as a gain or time constant, is varied from zero to infinity.1 The root locus, therefore, provides information not only as to the absolute stability of a system but also as to its degree of stability, which
8、is another way of describing the nature of the transient response. If the system is unstable or has an unacceptable transient response, the root locus indicates possible ways to improve the response and is a convenient method of depicting qualitatively the effects of any such changes.,The Angle and
9、Magnitude Criteria,Without transport lag the transfer function of a system can be reduced to a ratio of polynomials such that,The root locus technique is developed by expressing the characteristic function D(s) as the sum of the integer unity and a new ratio of polynomials in s. The characteristic e
10、quation will be written as,where K is the parameter of interest, -z1, -z2 . are the (open-loop) zeros and p1, -p2, are the (open-loop) poles. K is independent of s and must not appear in the polynomials Z(s) and P(s). The form of KZ (s) and P(s) is important; these poles and zeros may be real or com
11、plex conjugates. Note in Eq. (2-3A-2) that the coefficient of s is always set equal to unity for root locus operations.,A zero is a value of s that makes Z(s) equal to zero and is given the symbol o. Do not automatically assume that this zero is also a closed-loop zero that makes N(s) equal to zero
12、in the system (closed-loop) transfer function; it may be, but is not necessarily so. A pole is a value of s that makes P(s) equal to zero and is given the symbol x. The sn term represents n poles, all equal to zero and located at the origin of the s plane. A root of the characteristic equation has p
13、reviously been defined as a value of s that makes D(s) equal to zero.,Since s is a complex variable and the poles and zeros may be complex, KZ(s)/P(s) is a complex function and may, therefore, be handled as vector having a magnitude and an associated angle or argument. Each of the factors on the rig
14、ht side of Eq.(2-3A-2)can also be treated as vector with an individual magnitude and associated angle, as shown in Fig.2-3A-1. Notice that the angle is measured from the horizontal and is positive in the counterclockwise direction. If we express each factor in polar form, then,If we now collect the
15、magnitudes together and multiply the exponentials together, we can write Returning to the characteristic equation of Eq. (2-3A-3) and solving for KZ(s)/P(s) yields,k=0,1,2,Since -1 can be reprsented by a vector of unity magnitude and an angle that is an odd multiple of 180o. According to Eq.(2-3A-3)
16、 and Eq.(2-3A-4),we can find two criteria that make the characteristic function D(s) equal to zero , i.e, there are two criteria which can find system (close-loop) poles as K is increased from 0 to .,Magnitude critertion:,Rules for Root Locus Plotting,Applying angle and magnitude criterion, the root
17、 loci can obviously be plotted by a computer, however, well introduce rapid sketching techniques. The following guides are provided to facilitate the plotting of root loci: 1. For K=0 the close-loop poles coincide with the open-loop poles, 2. For K close-loop poles approach the open-loop zero. 3. Th
18、ere are as many as locus branches as there are open-loop poles. A branch starts, for K=0, at each open-loop pole. As K is increased, the closed-loop pole positions trace out loci, which end, for K, at the open-loop zeros.,4. If there are fewer open-loop zeros than poles (ji), those branches for whic
19、h there are no open-loop zeros left to go to tend to infinity along asymptotes. The number of asymptotes is (i-j). 5. The directions of the asymptotes are found from the angle condition. The vectors from all m open-loop zeros and n open-loop poles to s have the same angle noted . Hence the asymptote
20、 angles must satisfy,(K=any integer). The angles are uniformly distributed.,6. All asymptotes intersect the real axis at a single point, at a distance o to the origin:,7. Loci are symmetrical about the real axis since complex open-loop poles and zeros occur in conjugate pairs. 8. Sections of the rea
21、l axis to the left of an odd total number of open-loop pole and zeros on this axis form part of the loci, because any trial point on such sections satisfies the angle condition. 9. If the part of the real axis between two o.l. poles (o.l. zeros) belongs to the loci, there must be a point of breakawa
22、y from, or arrival at, the real axis. If no other poles and zeros are close by, the breakaway point will be halfway. In Fig. 2-3A-2d, adding the pole p3 pushes the breakaway point away; a zero at the position of p3 would similarly attract the breakaway point.,10. The angle of departure of loci from
23、complex o.l. poles (or of arrival at complex o.l. zeros) is a final significant feature. Apply the angle condition to a trial point very close to p1 in Fig. 2-3A-3. Then the vector angles from other poles and zeros are the same as those to p1. The angle from p1 to this point must satisfy the follows
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