Figure 1 below represents a schematic of a DC motor. This device converts electrical energy (when operating as a "motor") or it can convert mechanical energy into electrical energy when operating as a "generator".
 
 

In motor operation it is controlled by input to its armature (through brushes and a commutator). It also has a magnetic field, which is either supplied by a separate voltage supply or is furnished by a strong permanent magnet. When the armature is conducting current in the magnetic field, it produces a mechanical torque on the shaft proportional to the current . As it rotates in the magnetic field it also generates a voltage proportional to the speed of rotation .
 

To model the motor, both the mechanical side including any added load and the electrical circuit side must be modeled. On the circuit side, the inductor is the energy storage element so that
 
 

where the generated voltage must be subtracted from the applied armature voltage. On the mechanical side F=ma is applied in rotational form so that
 
 

These equations are then the basis of the state equations which describe the motor behavior in compact form. Before combining, we recognize that the total inertia of motor and added load is  and that the generated voltage  with the torque produced by the motor proportional to current as . The frictional torque  is generally related to motor speed but represents any type of friction tending to slow down the motor. The state equations finally become
 
 

Once appropriate parameter values have been obtained for the constants above, the equations can be used to simulate motor transient behavior with MatLab. With a linear representation for friction torque in the form , the routines lsim or step can be used. With a nonlinear friction representation (or linear as well), the ode23 or similar nonlinear solver can be used.