Page 80 - Servo Motors and Industrial Control Theory -

4.5 DC Servo Motors in Closed Loop Position Control 73

R
Error := T (4.6)
KK t 1
The dynamic behavior for both input variable and external torque is determined
by the common characteristic equation, which in this case is a second order. The
properties of this kind of characteristic equation is determined by natural frequency
and damping ratio given by
KK (4.7)
:ω = t
n
JR

R K KK
:ξ= C+C t · t (4.8)
2KK m R JR
t
It can be seen as the gain K is increased, the steady state error reduces and the natu-
ral frequency increases. This is desirable but Eq. (4.8) shows that as K increases, the
damping ratio reduces. A compromise has to be made between these two conflicting
requirements. All other parameters are given by the manufacturer of DC servo mo-
tors and the design parameters of the system.
The reader is encouraged to obtain a catalogue from the manufacturer to study
the system behavior numerically. The friction of the mechanical part if the load is
attached to the motor directly may be set to zero otherwise a value must be obtained
experimentally.
If an integrator in addition to the proportional control is added, the characteristic
equation becomes third order. The system may become unstable and careful analy-
sis must be carried out. The gains of the proportional controller and the integrator
must be evaluated so the system remains stable with sufficient damping for the
complex roots. The following error and the effect of external torque in this case
becomes zero.
For position control applications, a position feedback must be used. If the motor
is attached to a rotary mechanism, a variable potentiometer or a position encoder
may be used. The position encoder is a device that generates a pulse for each unit
of angular position of the motor. This may be as small as a fraction of rotational
degrees. If the motor is connected to a load through a lead screw, a linear variable
potentiometer may be used. For both rotary and linear motion, an encoder may
be attached to the motor and calibrated according the angular position of the mo-
tor. Some electronic devices usually are used to convert the difference between the
demand position and the output position to a control signal, which determines the
required torque output of the motor.
In designing servo position control system with DC servo motors, the motor
must be matched to the load. The motors are available in different speeds and output
torque. A gearbox is usually used to reduce the speed of motor to required linear or
rotary motion. The torque applied to the motor must be calculated and the motor