Page 481 - Mechanical Engineers' Handbook (Volume 2)

472 Closed-Loop Control System Analysis

2
2
s 2 s 0 (68)
n
n
giving the poles
s 1.2 j
1 j d (69)
2
n
n
2
where and
1 , 1.
n d n
The two complex-conjugate poles given by Eq. (69) are located in the complex s-plane,
as shown in Fig. 23.
It can be easily seen that
2
2
2
s
n (70)
n
d
1.2
and
1
cos 0 –
(71)
2
When the system has no damping, 0, the impulse response is a pure sinusoidal oscil-
lation. In this case the undamped natural frequency is equal to the damped natural fre-
n
quency .
d
5.5 Step Input Response of a Standard Second-Order System
When the transfer function has complex-conjugate poles, the step input response rather than
the impulse response is used to characterize its transients. Moreover, these transients are
almost always used as time-domain design specifications. The unit step input response of
the second-order system given by Eq. (67) can be easily shown to be

Figure 23 Compelx-conjugate poles in the s-plane.