DC Motor Position: System Modeling
Key MATLAB commands used in this tutorial are: tf , ss
Contents
Physical setup
System equations
Design requirements
MATLAB representation
Physical setup
A common actuator in con
...
DC Motor Position: System Modeling
Key MATLAB commands used in this tutorial are: tf , ss
Contents
Physical setup
System equations
Design requirements
MATLAB representation
Physical setup
A common actuator in control systems is the DC motor. It directly provides rotary motion and, coupled with wheels or drums
and cables, can provide translational motion. The electric equivalent circuit of the armature and the free-body diagram of the
rotor are shown in the following figure.
For this example, we will assume the following values for the physical parameters. These values were derived by
experiment from an actual motor in Carnegie Mellon's undergraduate controls lab.
(J) moment of inertia of the rotor 3.2284E-6 kg.m^2
(b) motor viscous friction constant 3.5077E-6 N.m.s
(Kb) electromotive force constant 0.0274 V/rad/sec
(Kt) motor torque constant 0.0274 N.m/Amp
(R) electric resistance 4 Ohm
(L) electric inductance 2.75E-6H
In this example, we assume that the input of the system is the voltage source (V) applied to the motor's armature, while the
output is the position of the shaft (theta). The rotor and shaft are assumed to be rigid. We further assume a viscous friction
model, that is, the friction torque is proportional to shaft angular velocity.
System equations12/21/2015 Control Tutorials for MATLAB and Simulink- Motor Position: System Modeling
http://ctms.engin.umich.edu/CTMS/index.php?example=MotorPosition§ion=SystemModeling 2/4
(4)
(7)
(8)
(9)
(10)
(1)
(2)
(3)
(5)
(6)
In general, the torque generated by a DC motor is proportional to the armature current and the strength of the magnetic field.
In this example we will assume that the magnetic field is constant and, therefore, that the motor torque is proportional to only
the armature current i by a constant factor Kt as shown in the equation below. This is referred to as an armature-controlled
motor.
The back emf, e, is proportional to the angular velocity of the shaft by a constant factor Kb.
In SI units, the motor torque and back emf constants are equal, that is, Kt = Ke; therefore, we will use K to represent both the
motor torque constant and the back emf constant.
From the figure above, we can derive the following governing equations based on Newton's 2nd law and Kirchhoff's voltage
law.
1. Transfer Function
Applying the Laplace transform, the above modeling equations can be expressed in terms of the Laplace variable s.
We arrive at the following open-loop transfer function by eliminating I(s) between the two above equations, where the
rotational speed is considered the output and the armature voltage is considered the input.
However, during this example we will be looking at the position as the output. We can obtain the position by integrating the
speed, therefore, we just need to divide the above transfer function by s.
2. State-Space
The differential equations from above can also be expressed in state-space form by choosing the motor position, motor
speed and armature current as the state variables. Again the armature voltage is treated as the input and the rotational
position is chosen as the output.
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