Page 32 - Servo Motors and Industrial Control Theory

24 2 Feedback Control Theory Continued

Fig. 2.6 Sketch of the root Imag
locus for the example consid- ..
ered in this section
Asymptote K=183.2
A 2 s=13.9
45 degree
+50
A 1
B 1 .Real
-50 +50

A 3
–50
K Increasing

are three loci. When K = 0 the loci start at the pole of the transfer equation and end to
the zero of the numerator and the remaining poles end to infinity. For this example
the pole when K = 0 are.

(s + 20 s 104 ⋅+· + ) (s 5) + K (s + 50) := 0
⋅
2
s:=− 5
1
 104

F := 20 
  (2.11)
 1 
=
s : polyroots(F)
 − 10 − 2i
s :=
2,3   − 10 + 2i 

It can be seen that even for simple second order equation the Mathcad software can
be used to determine the roots. The poles are shown on the graph paper by crosses
and zeros are shown by a circle. In this example there is only one zero and it is,
Z =−50

The poles and zeros are shown on Fig. 2.6.
Those loci which go to infinity have asymptotes determined by the following
equation,
[(2·1 1) ·180+ ]
θ=
:
1 nm−
− [(2·1 1) ·180+ ]
θ= n − m
:
2