Module 4
Second – Order System
(1 hours)
Poles and zeros
Second order systems,damping,
natural frequency
Step response of second order system
Second order systems –
systems described by a second
order differential equation
Second order systems contain two
kinds of energy storage elements:
Lump mass and spring
Capacitor and inductor
Others
4.1 Second-Order System Transfer Function
22
2
2 2)()(1
)(
)(
)(
nn
n
ssKsTs
K
sHsG
sG
sR
sC



)1()( Tss
KsG
Where,
K – system total gain (actual parameter)
T – time constant (actual parameter)
– the damping ratio (characteristic parameter)
– the natural frequency (characteristic
parameter)
n?
It can be drawn into a feedback system block
diagram:
KsTs
K
2
R C
)1(?Tss
K
R C
R C
)2(
2
n
n
ss
R C
22
2
2 nn
n
ss

TK
n
n

2
1;
2
The closed-loop characteristic equation:
22
2
2 2)()(1
)(
)(
)(
nn
n
ssKsTs
K
sHsG
sG
sR
sC



0)()(1 sHsG 02 22 nn ss
dnnn jjss
2
21 1,
21
nd
– the damped natural frequency
4.2
tsts eAeA 21
211
1,221 nnss
2
2
1
11)(
ss
A
ss
A
s
sC

nss21,
Case 4,The system is undamped.
0
ttC n?co s1)(
2
0
2
1
njss21,
Fig,4.16 shows the step response for various
values of damping ratio and for the
particular case =1 rad/s,At this stage,it is
important to recognize that when <1,the
figure shows that
The system oscillates at frequency,
The response decays due to the exponent,
The overshoot depends on the value of,
Conclusions (P65)
n?
d?
n
Instructional objectives:
At the end of this lecture students should be able to
identify the dominant poles of a system
determine the damping ratio and natural
frequency of a second – order system and
classify it as,underdamped,overdamped,
critically damped,or undamped.
apply the formulas for the %Overshoot