Motod Dc

Example: DC Motor Speed Modeling

Physical setup and system equations
Design requirements
Matlab representation and open-loop response

Physical setup and system equations

Photocourtesy: Pope Electric Motors Pty Limited

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 transitionalmotion. The electric 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 physicalparameters. These values were derived by experiment from an actual motor in Carnegie Mellon's undergraduate controls lab.

* moment of inertia of the rotor (J) = 0.01 kg.m^2/s^2
* damping ratio of themechanical system (b) = 0.1 Nms
* electromotive force constant (K=Ke=Kt) = 0.01 Nm/Amp
* electric resistance (R) = 1 ohm
* electric inductance (L) = 0.5 H
* input (V): Source Voltage
* output (theta):position of shaft
* The rotor and shaft are assumed to be rigid

The motor torque, T, is related to the armature current, i, by a constant factor Kt. The back emf, e, is related to the rotationalvelocity by the following equations:

In SI units (which we will use), Kt (armature constant) is equal to Ke (motor constant).

From the figure above we can write the following equations based onNewton's law combined with Kirchhoff's law:

1. Transfer Function
Using Laplace Transforms, the above modeling equations can be expressed in terms of s.

By eliminating I(s) we can get thefollowing open-loop transfer function, where the rotational speed is the output and the voltage is the input.

2. State-Space
In the state-space form, the equations above can be expressed by choosing therotational speed and electric current as the state variables and the voltage as an input. The output is chosen to be the rotational speed.

Design requirements
First, our uncompensated motor can...