aa166 Chapter 5 / Transient and Steady-State Response AnalysesR(s) E(s) vn C(s)
s(s+ 2 zvn)2
+Figure 5–6
Second-order system.In terms of zandvn, the system shown in Figure 5–5(c) can be modified to that shown
in Figure 5–6, and the closed-loop transfer function C(s)/R(s)given by Equation (5–9)
can be written
(5–10)
This form is called the standard formof the second-order system.
The dynamic behavior of the second-order system can then be described in terms of
two parameters zandvn. If 0<z<1, the closed-loop poles are complex conjugates
and lie in the left-half splane. The system is then called underdamped, and the tran-
sient response is oscillatory. If z=0, the transient response does not die out. If z=1,
the system is called critically damped. Overdamped systems correspond to z>1.
We shall now solve for the response of the system shown in Figure 5–6 to a unit-step
input. We shall consider three different cases: the underdamped (0<z<1), critically
damped(z=1), and overdamped (z>1)cases.
(1)Underdamped case(0<z<1): In this case,C(s)/R(s)can be written
where The frequency vdis called the damped natural frequency.For
a unit-step input,C(s)can be written
(5–11)
The inverse Laplace transform of Equation (5–11) can be obtained easily if C(s)is writ-
ten in the following form:
Referring to the Laplace transform table in Appendix A, it can be shown that
l-^1 c
vd
As+zvnB^2 +v^2 d
d =e-zvn^ tsinvd t
l-^1 c
s+zvn
As+zvnB^2 +v^2 d
d =e-zvn^ tcosvd t
=
1
s
-
s+zvn
As+zvnB^2 +v^2 d
-
zvn
As+zvnB^2 +v^2 d
C(s)=
1
s
-
s+ 2 zvn
s^2 + 2 zvn s+v^2 n
C(s)=
v^2 n
As^2 + 2 zvn s+v^2 nBs
vd=vn 21 - z^2.
C(s)
R(s)
=
v^2 n
As+zvn+jvdBAs+zvn-jvdB
C(s)
R(s)
=
v^2 n
s^2 + 2 zvn s+v^2 n
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