Transfer Function of a DC Motor.
Consider the model presented in Figure 10:
Figure 10. Model for a DC Machine [4]
Let’s determine the Transfer Function of the DC Motor from Figure 10. Since the current-carrying armature is rotating in a magnetic field, its voltage is proportional to its speed. That is the back electromotive force as it was established in equation 7:
Equations 12
Taking the Laplace Transform we get:
Equations 13
The torque developed by the motor is proportional to the armature current, as it was said in Equations 9:
Equations 14
Transforming every impedances of Figure 10 into their Laplace Transform equivalent , we find the voltage equation for the loop around the armature circuit:
Equations 15
Now, we substitute Equations 13 y 14 en 15:
Equations 16
We need Tm in terms of 
in order to find 
. That can be get using the equivalent model for mechanical loading on a motor as shown in Figure 11:
Figure 11. Typical equivalent mechanical loading for a DC Machine [4]
Where Jm and Dm are mechanical constant which can be derived from a typical configuration such as:
Figure 12. A DC Motor driving a rotational mechanical load [4]
Considering Figure 12, Jm and Dm are:
Equations 17
Now, from Figure 11 we can find the relationship between Tm and :
Equations 18
Substituting Equations 18 in 16 we get:
Equations 19
In the most cases La is too small compared with Ra, so Equations 19 can be simplified and rearrange as:
Equations 20
Now from Equations 20 we obtain the Transfer Function for a DC Motor as follow:
Equations 21
The electrical constants of the motor Kt y Kb can be found with the following relations:
Equations 22
Where Tstall, Ea y Wno-load, use to be derive from a Graphic Speed Vs Torque such as:
Figure 13. Torque-speed curves with an armature voltage Ea as a parameter [4]
As an example, consider the case of Figure 14:
Figure 14. Torque-speed curves and system example [4]
Hence:
And using the gear ratio N1/N2=1/10:
[4] Control Systems Engineering, Norman Nise
Written by: Larry Francis Obando – TSU
Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, Caracas.
Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.
Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.
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