Page 19 - Servo Motors and Industrial Control Theory

1.8 Second Order Transfer Function 11

0 0

phase angel (deg) g(ω) –90 amplitude ratio in decibels f(ω) –20

–180 –40
0.1 1 10 100 0.1 1 10 1
ω ω
frequency frequency (rad/sec)
Fig. 1.6 Phase angle and amplitude ratio versus frequency for a first order lag

frequency responses. For some mechanical systems, where the response is very slow,
the frequency response may be plotted on octave scale instead of logarithmic scale.
Therefore, by knowing the time constant τ, the frequency response and the time
response are completely known.

1.8 Second Order Transfer Function

Standard second order transfer function is written in two ways of
y = 1
x 1 s + 2ξ s + 1
2
ω 2 n ω n

or
y = ω 2 n
2
+
x s + 2ξω s ω 2
n n
where ω ξ are the natural frequency and damping ratio, respectively. It will be clear
n
later as to why the coefficients are given such names.
For step input x( t) = 1, then
1
xs() =
s

Substituting in the transfer function and taking inverse of Laplace Transform by
referring to Table 1.1, the response for ξ < 1 becomes

 1 
(
−
( ) : 1= + 
y t  ·e − ξω n .t .cos ω n . . 1 ξ +Φ ) ξ < 1
2
.
t
−
 1 ξ 
2