Digital Signal Processing Reference
In-Depth Information
x
(
t
)
y
(
t
)
H
(
s
)
Figure 3.21
LTI system.
The form of the transfer function in (3.70), which is a ratio of polynomials, is
called a
rational function
. The transfer function of a continuous-time LTI system
described by a linear differential equation with constant coefficients, as in (3.50),
will
always
be a rational function.
We now consider two examples to illustrate the preceding developments.
Transfer function of a servomotor
EXAMPLE 3.16
In this example, we illustrate the transfer function by using a physical device. The device is a
servomotor, which is a dc motor used in position control systems. An example of a physical
position-control system is the system that controls the position of the read/write heads on a
computer hard disk. In addition, the audio compact-disk (CD) player has three position-
control systems. (See Section 1.3.)
The input signal to a servomotor is the armature voltage
e(t),
and the output signal is
the motor-shaft angle
u(t).
The commonly used transfer function of a servomotor is second
order and is given by [3]
K
H(s) =
+ as
,
s
2
where
K
and
a
are motor parameters and are determined by the design of the motor. This
motor can be represented by the block diagram of Figure 3.22, and the motor differential
equation is
d
2
u(t)
dt
2
+ a
du(t)
dt
= Ke(t).
This common model of a servomotor is second order and is of adequate accuracy in most
applications. However, if a more accurate model is required, the model order is usually
increased to three [3]. The second-order model ignores the inductance in the armature circuit,
while the third-order model includes this inductance.
■
Servomotor
e
(
t
)
(
t
)
K
s
2
as
Armature
voltage
Shaft
angle
Figure 3.22
■
System for Example 3.16.
Sinusoidal response of an LTI system
EXAMPLE 3.17
In this example, we calculate the system response of an LTI system with a sinusoidal excita-
tion. Consider a system described by the second-order differential equation
d
2
y(t)
dt
2
+ 3
dy(t)
dt
+ 2y(t) = 10x(t).
