Modern Control Engineering

aa

Section 5–3 / Second-Order Systems 165

+–

r

K

1

s(Js+B)

ecT

J

B

(a)

+

• 
  

R(s)

R(s)

C(s)

C(s)

T(s)

(b)

K

K

+– s(Js+B)

(c)

Figure 5–5
(a) Servo system;
(b) block diagram;
(c) simplified block
diagram.

Step Response of Second-Order System. The closed-loop transfer function of

the system shown in Figure 5–5(c) is

(5–9)

which can be rewritten as

The closed-loop poles are complex conjugates if B^2 -4JK<0and they are real if

B^2 -4JK0. In the transient-response analysis, it is convenient to write

wheresis called the attenuation;vn, the undamped natural frequency; and z, the damp-

ing ratioof the system. The damping ratio zis the ratio of the actual damping Bto the

critical damping or

z=

B

Bc

=

B

21 JK

Bc= 21 JK

K

J

=v^2 n ,

B

J

= 2 zvn= 2 s

C(s)

R(s)

=

K

J

cs+

B

2J

+

B

a

B

2J

b

2

-

K

J

dcs+

B

2J

-

B

a

B

2J

b

2

-

K

J

d

C(s)

R(s)

=

K

Js^2 +Bs+K