ELEC 242 Lab

Background

The System

The figure below is a schematic of the position control system. As a simplification we have omitted the gearbox and absorbed its effect into the values of $K_e$ and $K_t$ . We have also absorbed the reflected moment of intrtia of the gears and motor armature into $J'$ .

\includegraphics[scale=1.000000]{scan8.1.ps}

This translates into the following block diagram

\includegraphics[scale=0.500000]{ckt8.7.ps}
Ignore for now the effects of the mass $m$ (i.e. assume there are no weights on the hook and that the weight of the cord and the hook are negligable). Then the motor, gearbox, and drum can be modeled by the first order transfer function $\displaystyle \frac{K_m}{1+s\tau}$ . Integrating the angular velocity $\omega$ to give the shaft angle $\theta$ corresponds to the transfer function $\frac{1}{s}$ . Finally the mapping of the shaft angle into the potentiometer output voltage $v_{act}$ is given by $K_s$ . Putting it all together, we have the following block diagram for the closed loop control system:
\includegraphics[scale=0.650000]{ckt8.6.ps}
One thing we have not modeled is the solid friction between the various mechanical components of the system. Unlike viscous friction which is proportional to velocity, solid friction is a threshold force which must be exerted before any motion can be initiated. This produces a dead zone in the relation between applied force and resulting acceleration similar to that in the complimentary emitter follower. We have seen this effect before in the speed vs. voltage curve of the motor. It will be particularly pronounced in this system because of the additional friction introduced by the gear train and the potentiometer and will make it difficult to control the position manually. As in the case of the motor amplifier where we reduced the crossover distortion by placing the emitter follower inside the feedback loop, feedback will reduce this effect in the closed loop system.

Sensing Angle

Previously we have used another motor as a tachogenerator to measure speed. We will be using a type of variable resistor known as a potentiometer to measure the angle of the gearbox output. A potentiometer (or pot for short) is a fixed value resistor with a third, movable contact or slider which may be positioned anywhere along the resistive element. If we represent the position of the slider by $\alpha$ , where $\alpha$ varies between 0 (fully counterclockwise) and 1 (fully clockwise), then the resistance between the lower end of the resistor and the slider will be $\alpha R$ and between the slider and the upper end will be $(1-\alpha) R$ , where $R$ is the total resistance of the potentiometer.

If we connect the two fixed contacts to a voltage source and measure the output between the movable contact and one fixed contact, we get a variable voltage divider:

\includegraphics[scale=0.650000]{pot1.ps}

Then the output is

$\displaystyle v_{out} = \frac{R_2}{R_1 + R_2} v_{in} = \frac{\alpha R}{(1-\alpha)R + \alpha R} v_{in}= \alpha v_{in}$

The potentiometer we are using is a 10-turn pot, which means that ten turns ($3600^\circ$ ) of the shaft are required to go from $\alpha = 0$ to $\alpha = 1$ . We will be using -5 V for $v_s$ , so

$\displaystyle v_{out} = -5\frac{\theta}{3600^\circ}$