Tutorial 6 - DC Motor Control

This tutorial is designed to teach you the basics of controlling the
velocity of a DC motor. An electro-mechanical model of a DC motor is
shown below

**Electro-mechanical mode of a DC motor**

V = input voltage (V)

R = nominal resistance (W)

L = nominal inductance (H)

J = Inertial load (kg*m^2/s^2)

Vemf = back emf voltage (V)

b = damping constant (Nms)

t = motor output torque (Nm)

q = motor shaft angle (rad)

The motor torque, t, is linearly
proportional the current through the armature. This is expressed as

As the armature rotates, a voltage difference, proportional to the
rotational velocity of the armature shaft, is created. This voltage
is called the *back* *emf *voltage, where emf stands for electromotive
force.

By using Newton's and Kirchoff's Laws, the following equations can
be derived

By making use of the Laplace transform, the above two equations can
be rewritten as

The last step is to combine these two equations, cancelling the I(s)
term. This yields the transfer function for the DC motor.

Observe that the denominator of the transfer function is a second-order
polynomial, which means there are two poles in the s-plane. Note that
the numerator does not contain any s terms, therefore both zeros occur
at s = infinity.

We can use MATLAB to plot the step response of the motor transfer function.
First, we will choose approximate constant values for a hypothetical
motor. In the MATLAB command window type (note, >>
denotes the command prompt)

>>
J = 0.01;

>> b = 0.1;

>> K = 0.01;

>> R = 1;

>> L = 0.5;

Now that the constants are defined, enter the
numerator and denominator of the transfer function.

>>
num = K;

>> den = [J*L J*R+L*b K^2+b*R];

>> G = tf(num,den);

The step response of the motor can now be simulated
by the command

>>
step(G);

MATLAB will produce the following plot

Notice from the plot that the system
is overdamped, this can be verified by finding the roots of the denominator
with the command

>>
roots(den)

ans =

-9.9975

-2.0025

The roots are both unique and negative, which translates into a overdamped,
stable transfer function.