DC Motor Modeling

V is voltage supply for motor, i is current, R is armature resistance, L is armature inductance, vb is voltage due to back EMF (electric magneticforce), and:
Where kb is back EMF constant, is angular velocity of armature. So, we have:
Because motor inductance is very small and not convenient to measure, we ignore it. Thus, we have:
It is known that relationship between motor torque and armature current is:
Where kt is torque constant. Assume there are no electromagnetic losses, which means mecanical power is equal to electrical power dissipated by the back EMF in the armature:
Let's denote
From eq(1), eq(2) and eq(3), we get:
Therefore, we need to measure two parameters: resistance R and torque constant K.
Experiment 1--find R
Take a multimeter and measure at the terminals. Give the armature a random turn, measure it; give it another random turn, measure it, etc. Then take the average. I've measure it 50 times, and here is what I got: (unit: Ohm)
7.4 | 9.5 | 8.5 | 9.8 | 9.1 | 18.7 | 6.4 | 8.4 | 11.2 | 6.9 |
8.3 | 9.4 | 15.4 | 7.2 | 15.3 | 13.1 | 11.6 | 12.1 | 9.2 | 7.7 |
10.8 | 9.1 | 18.6 | 11.0 | 16.1 | 12.1 | 10.1 | 8.1 | 9.6 | 10.0 |
9.9 | 16.8 | 16.5 | 10.3 | 17.5 | 13.8 | 9.9 | 14.0 | 8.5 | 15.3 |
14.2 | 16.0 | 29.5 | 13.8 | 14.3 | 11.9 | 31.3 | 18.3 | 10.9 | 10.9 |
The average is 12.486. So,
Experiment 2--find K
Denote moment of inertia of motor armature J, thus
Take Laplace transform,
From eq(1), we get
From eq(4) and eq(5), we get
So, give the motor a step input, and measure the response (motor angluar velocity):
The step input is 7.23V, I know this value by measuring the voltage inputs to motor using a multimeter. A 7.23V step input written in Laplace form is:
Inserted it into eq(6), we have:
I wrote an Arduino program to grasp angular velocity data. Here is a video explains how to do this experiment:
I did this experiment 6 times. These are the angular velocity data plots:

The steady state velocities are a little jerky, so, I calculate the average value. This is result I get: (K values for the 6 experiments)
0.2670 | 0.2663 | 0.2677 | 0.2659 | 0.2669 | 0.2653 |
The average of these 6 values is 0.2665, thus:
Programs of this experiment is available in DOWNLOAD section.