Background

Consider a DC motor with armature resistance , torque constant , and back emf constant being driven by a voltage source .

J represents the combined moment of inertia of the motor armature and the load. Then we have

. From the coupling equations we have

and

. Combining these along with Ohm's law, we get:

where

and

In terms of the angle of rotation of the drum

, the equation describing the system is

where

: is the summing amplifier gain
: is the equivalent moment of inertia seen at the shaft, including the effects of the load mass.
: is the gain of the potentiometer ( ).
: is the equivalent torque caused by the weight of the load.
: is the desired angle of the shaft.

Making the substitutions

and

we get

Using superposition we define

with

and

with

. Then

. Evaluating

and

we have

and

Some observations

In the steady state, , i.e. there is no steady state gain error regardless of A. If there will be a constant offset, but .
falls off as , i.e. disturbance rejection increases with increasing gain.
System dynamics depend on A . The characteristic equation is . This is our familiar second order system with and . So as A increases, the system eventually becomes underdamped and will oscillate.