This part is about understanding the derivation of the motor model. (Actually this is obtained from Control System Engineering Book by Nise, so credit to that fellow). Someone new to control theory might wonder what the heck is a model. Let's put that a model is used to calculate an output given an input. Like a simple function like y=f(x). The input is x and output is y. But in control model, it is more to dynamic model or model which involves ordinary differential equation. Let's proceed.
Modeling Electrical Characteristics

Electrical Schematic 
Starting off with the electrical characteristics. From the motor schematics, we can draw the Kirchoff Voltage Law (KVL) line.

Equation 1 
According to KVL, the total voltage in the line is zero.

Equation 2 
Therefore, we can come up with Equation 2.

Equation 3 
Taking the form of La Place in Equation 3 with all initial values zeroed.

Equation 4 
The equation then can be arrange in term of current (Equation 4),

Equation 5 
Or in the form of transfer function (Equation 5).
Modeling Mechanical Characteristics
After the electrical part, let's move on to the mechanical part:

Mechanical Schematic 
The schematic above shows the free body diagram.

Equation 6 
Start off with Newton's law, where torque equals moment of inertia, angular acceleration

Equation 7 
In Equation 7, there are three acting torque, (1) the torque from the motor, (2) the viscous friction from where the higher the speed of rotation the higher the friction force is, and (3) the Coulomb friction where the friction counters the movement. For simplicity, the Coulomb friction can be assumed constant or zero.

Equation 8 
Rearranging in terms of motor speed, omega.

Equation 9 
Applying La Place transform with zero initial values.

Equation 10 
Rearranging in terms of omega,

Equation 11 
Or in terms of transfer function.

Equation 12 
Equation 12 is an auxiliary equation, just for completeness, where speed is the derivation of motor position, theta

Equation 13 
Applying La Place transform with zero initial value,

Equation 14 
And arranging it in transfer function form.

Equation 15 
Equation 15 and 16 are additional equations where they are need to "connect" the electrical and mechanical parts. Equation 15 connects the motor current to the torque produced.

Equation 16 
Equation 16 is the back electromotive force produces with a certain motor speed.
The Complete Model
Arranging Equation 5, 11, 14, 15 and 16, we obtain the motor model like in the figure below.
From this figure, it can be seen that a motor have six unknown parameters, (1) motor resistance, (2) motor inductance, (3) load moment of inertia, (4) load viscous friction, (5) torque constant, Kt, and (6) speed constant, Ke. This is when the external torque, T1 is assumed zero.
In the next part, the simulation and interpretation will be shown, using some values for all these parameters. It is also interesting to know that there are topics in parameter estimation or system identification to identify the parameter values. Most probably this will not be covered here because it is quite extensive.
Check out the full series
Part I
Part II
Part III
Part II, III  Interlude
Part IV
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