Page 222 - Servo Motors and Industrial Control Theory -

220 Appendix C

Resistance = 0.07 Ω
Inductance = 0.00125

Determine the linear position accuracy that can be achieved with the mentioned
motor.
In practice, a series of pulses is fed to the power units where each pulse repre-
sents one step rotation of the motor. Another pulse must be fed to the power unit
which determines the direction of motion. The power unit is made of electronics
with high power transistors to produce output voltage with high currents. The
electronics determine which phases of the motor must be energized and which
phases must be de-energized.
For mathematical model, it can be assumed that the demand position is in ra-
dians and output position is also in radians. Assume that the current of 2 amps
produces an output torque 5 N m.
Determine the response characteristics of the motor response for the application
of a single step. In principle the torque is in sinusoid form and you should linear-
ize this nonlinear function and find the coefficients of the linearized model and
then solve the equation of motion.
Assume that the mechanical part of the system is rigid and with first approxima-
tion assume that the inductance is negligible. Define variables as required for
derivation of the mathematical model and define those parameters that are not
defined above and set engineering values for them. Derive the open loop transfer
function which relates the output velocity to the demand velocity. Determine the
damping ratio of the open loop velocity control system. For position control, it
must be assumed that there is an internal position feedback so that the position
can be controlled. With this in mind design a proportional controller with gain K
and find the new transfer function for the system which relates the output posi-
tion to the demand position and determine the maximum value of the gain that
gives a damping ratio at least 0.7. Determine the response time for a unit step of
demand position.
To increase the speed of response and to increase the damping ratio of the system
a velocity feedback a gain of K might be used. Repeat the analysis and deter-
d
mine the two gains so as to achieve fast response with sufficient damping in the
system.
Repeat the above analysis when the inductance of the motor cannot be ignored.
In this case you will find a third order transfer function indicating that the closed
loop system might become unstable. Select the values of the proportional gain
and derivative feedback gain so that the system remains stable with sufficient
damping in the fundamental roots of the characteristic equation.
2. A stepping motor is selected to control the position of a rotary device through a
gear box as shown below. The stepping motor selected has a power of 2 kW, a
stepping angle of 1.8° and the input output velocity ratio of the gearbox is 20.
2
The moment of inertia of the rotary device is 4 kg·m and the total viscous fric-
tion referred to the motor shaft is 0.001 Nm/rev. Calculate the moment of inertia