Page 530 - Mechanical Engineers' Handbook (Volume 2)

5 Root Locus 521

Rule 9. The angle condition is made use of to determine the angle by which a branch would
depart from a pole or would arrive at a zero as K → .
A point s is considered very near the pole (zero) and the angle G(s )H(s )is
0
0
0
computed. The fact that s is very near the pole (zero) makes all but one angle fixed.
0
Thus by employing the angle condition, the unknown angle of departure (arrival) can
be computed.
An example is given next to illustrate the various rules for constructing a root locus.
Example 3 Consider CLCE
K(s 6)
1 KG(s)H(s) 1
s(s 1)(s 4)
R1: n 3 ⇒ 3 branches originating at 0, 1, 4at K 0.
R2: m 1 ⇒ 1 branch terminates at 6, at K .
R3: n m 2 branches approach along asymptotes.
R4: Hub
(0 1 4 ( 6))/2 0.5.
180 N 180 N
Asymptote angles:
n m 2
Set N 1 ⇒ 90
R1–R4: Yield the sketch of Fig. 18b.
R5: Sections on the real line are 0 to l and 4to 6.
R6: There must be a breakaway point between 0 and 1.
R7: Break points dK/ds 0.
2
3
2
⇒ (s 6)(3s 10s 4) (s 5s 4s) 0
3
2
s 11.5s 30s 12 0
(s 0.49)(s 7.89)(s 3.12) 0
Values for s of 7.89 and 3.12 are unacceptable from R5. Therefore the only breakaway
point is at 0.49. A sketch of the root loci is given in Fig. 18c.
R8: Imaginary axis crossings:
3
2
CLCE s 5s (4 K)s 6K 0
Now lets s j :
3
( j ) 5( j ) (4 K)j 6K 0
2
(6k 5 ) j [(4 K) ] 0
2
2
Therefore
6K 5 2
2
[4 K ] 0
yielding