Motor Modeling 1:
Current-controlled vs Voltage-controlled
Problem 1 on ps #2 distinguishes between "voltage-controlled" and "current-controlled" motors.
What does this mean?
For ps #1, we assumed a "current-controlled" model for the motor was valid.
That is, we could specific a current, i, directly, and the torque was given as:
T = K * i
In ps #2, we will consider the more complicated (but also, alas, more typical) "voltage-controlled" motor.
Here, we command a specific voltage, V, directly, but how this then maps into a current (to produce torque via that same relationship, T=K*i) is more complicated. We must now consider the "back EMF".
Voltage-controlled Motor Model
As the armature of the motor turns, it produces this back electro-motive force (EMF) which feeds back to act against the whatever voltage we command. When we commanded a current, we could immediately use that (known and commanded!) value of current to find the motor torque. (once again, because of the linear relationship T=K*i)
Here is a mechanical analogy, if it sounds a little confusing:
Imagine a system where you can command either the velocity or the force of some element in the system. Let's say the velocity is ultimately the output of interest to us, just as the current is of interest (so we can use T=K*i). Since we control the velocity directly, we know the velocity is whatever we commanded. (duh.)
But now, let's say we control the force acting on some element, but we still wish to find the velocity. With friction acting proportional to velocity (F_friction = B * (dx/dt)) and AGAINST our commanded force, we have a natural feedback occurring in the system. (This is same type of situation you modeled in ps #1, where a damping force acts against the torque output of the motor...)
Controlling the voltage rather the directly controlling the current input to the motor requires modeling of the same type feedback, only now it the back EMF (rather than a friction force) which feed back against the commanded voltage (rather than against the motor torque).
Note in the figure above that the electric circuit model for the motor is given (using Laplace notation) as:
