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i=l n - m
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(15.15)
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These asymptotic lines make angles of 7r[(2k + l)l(n - m)] with the real axis and are, therefore, equally spaced at angles 2d(n - m) to each other (k = 0,1,2..., n - m - 1).
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RULE 5. BREAKAWAY POINT. The point at which two root loci, emerging
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from adjacent poles (or moving toward adjacent zeros) on the real axis, intersect and then leave (or enter) the real axis is determined by the solution of the equation (15.16) These loci leave (or enter) the real axis at angles of -+42. Equation (15.16) is solved by trial by checking it for various test points, s = s c, on the real axis between the poles (or zeros) of interest. For real poles or zeros, the terms in the denominator of Eq. (15.16) are obtained by simply measuring distances along the real axis between the test point and the poles and zeros. If a pair of complex poles, pi = ui jbi., am present, add to the right side of Eq. (15.16) the term 2(S - Ui) (S - ai)* + bf
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*A pole pa of order q is present in the open-loop transfer function if the denominator of G contains (S - p,)q. A zero za of order q is present if the numerator of G contains (S - zn)q.
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(This term accounts for both poles of the complex pair.) This term is merely the result of simplifying the sum 1 1 + s - ai - jbi S - Qj + jbi For a pair of complex zeros, add a similar term to the left side of Eq. (15.16).
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RULE 6. ANGLE OF DEPARTURE OR APPROACH. There are q loci emerging from each qth-order open-loop pole at angles determined by
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+[ C2k
+ l)n + T&@a i=l
Zi) - 2 &@a - pj)]
j=l it=a
(15.17)
k =0,1,2
,..., q - l
where pa is a particular pole of order q. Each of the m loci that do not approach the asymptotes will terminate at one of the m zeros. They Will approach their particular zeros at angles
(2k+1)~+~4(zb-pj)-~4(zb-zi)
j=l i=l i f b
(15.18)
k = 0, 1,2, . . . , v - 1
where Zb is a particular zero of order v. For simple poles (or zeros) on the real axis, the angle of departure (or approach) will be 0 or 7r. An analog from potential theory is useful in plotting a root-locus diagram. It may be shown that the loci correspond to the paths taken by a positively charged particle in an electrostatic field which is established by poles (positive charges) and zeros (negative charges). In general, we may expect a locus to be repelled by a pole and attracted toward a zero. Another general aid to plotting the loci is to be aware of the fact that for n - m 2 2,thesumoftheroots(rt+r2+ * 1. + r,) is constant, real, and independent of K. This requires that motion of branches to the right be counterbalanced by the motion of other branches to the left. Most of the open-loop transfer functions encountered in single-loop chemical process control systems will have all their poles on &he real axis. In exceptional cases where the feedback path includes second-order measuring elements, such as a pressure transmitter, the open-loop transfer function will contain complex poles, but very often they will be located so far from the remaining dominant poles that they can be ignored. These rules and guides will now be explained by applying them to specific examples.
ROOTLOCUS
- (a+l)(r+2)(~+3)
-2-1 -2
(cl : FIGURE 15-5 Root-locus constmction for Example 15.1.
Example 15.1. Plot the root-locus diagram for the open-loop transfer function:* K G = (3 + l)(S + 2)(s + 3)
In general, our stepwise procedure will follow the same order in which the rules
were presented. 1. Plot the open-loop poles as shown in Fig. 15.5a. The poles are indicated by X . There are no open-loop zeros for this example. 2 . (Rule 1) Since we have three poles, there are three branches. 3 . (Rule 3) A portion,of the locus is on the real axis between -1 and -2 and another portion is to thk left of -3.
*To grasp more easily the graphical procedure for plotting the mot locus, the reader should actually plot these examples according to the steps given in the solution. Also note that this is the same example that was treated by algebraic methods at the beginning of this chapter.
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