Page 540 - Mechanical Engineers' Handbook (Volume 2)

5 Root Locus 531

From the root locus it is clear that at a certain gain value the system goes unstable.
From Fig. 24c we know that when the gain is doubled, the system has two closed-loop
eigenvalues at A and A . From the impulse response trace the frequency of oscillation is
2

/10
or
20 rad/s
Thus we know that when the gain is doubled, there are two closed-loop poles at j20.
The corresponding CLCE therefore is
P(s) (s a)(s j20)(s j20) 0
or
3
2
s as 400s 400a 0
We also know that
2K
P(s) 1 0
s(s )(s 40)
or
2
3
s (40 )s 40 s 2K 0
By matching coefficients
40

40 400
2K 400a
Therefore 10, a 50, and K 10,000. Hence the open-loop transfer function is
10,000
G(s)
s(s 10)(s 40)

5.3 Time-Domain Response versus s-Domain Pole Locations
Given a transfer function G(s) the pole locations can be found. These pole locations essen-
tially describe the type of time response to be expected. The basic response can be effectively
characterized by the impulse response g(t) given by

1
g(t) L [G(s)]
If
K m (s z )
G(s) i 1 i n m
n (s p )
j 1 j
then