Page 96 - Modern Control Systems

Chapter 2 Mathematical Models of Systems

Assuming that the velocity of Mi is the output variable, we solve for \\(s) by matrix
inversion or Cramer's rule to obtain [1,3]

(M 2s + Z>! + k/s)R(s)
5
,
M(s) = ,, , , , \ w w , , , , , , TT- (2- °)
(Mis + b x + b 2)(M 2s + Z>x + k/s) - b x'
Then the transfer function of the mechanical (or electrical) system is
V x{s) (M 2s + ^ + k/s)
G(s) =
R(s) (M ts + b x + b2)(M 2s + b t + k/s) - b?
(M 2s 2 + b {s + A:)
(2.51)
(M^ + ^ + b 2)(M 2s z + b^s + k) - b?s

If the transfer function in terms of the position X\(i) is desired, then we have
Ms) V^s) G(s)
(2.52)
R(s) sR(s)
As an example, let us obtain the transfer function of an important electrical
control component, the DC motor [8]. A DC motor is used to move loads and is
called an actuator.

An actuator is a device that provides the motive power to the process.

EXAMPLE 2.5 Transfer function of the DC motor

The DC motor is a power actuator device that delivers energy to a load, as shown in
Figure 2.18(a); a sketch of a DC motor is shown in Figure 2.18(b). The DC motor
converts direct current (DC) electrical energy into rotational mechanical energy. A
major fraction of the torque generated in the rotor (armature) of the motor is
available to drive an external load. Because of features such as high torque, speed
controllability over a wide range, portability, well-behaved speed-torque charac-
teristics, and adaptability to various types of control methods, DC motors are widely
used in numerous control applications, including robotic manipulators, tape trans-
port mechanisms, disk drives, machine tools, and servovalve actuators.
The transfer function of the DC motor will be developed for a linear approxi-
mation to an actual motor, and second-order effects, such as hysteresis and the volt-
age drop across the brushes, will be neglected. The input voltage may be applied to
the field or armature terminals. The air-gap flux 4> of the motor is proportional to
the field current, provided the field is unsaturated, so that

<f> = i f. (2.53)
K f
/
The torque developed by the motor is assumed to be related linearly to <> and the
armature current as follows:
= KtfUt) = KiK fi f(tMt). (2.54)
T m