DC motor model ETEC6419

Motors of Models There are many different models of DC motors that use differential equations. During this set of slides we will only focus on first and second order differential equations that are linear and ordinary differentials.

Motors are common place Electromechanical systems such as electric motors and electric pumps are used in most industrial and commercial applications.

The figure below shows a simple d.c. motor circuit. The torque produced by the motor is proportional to the applied current and is given by (1) where T is the torque produced, k t is the torque constant and i is the motor current Torque derived from current component in DC motor model

Torque derived from angular speed and moment of inertia component in the DC motor model

Back EMF component in DC motor model As the motor armature coil is rotating in a magnetic field there will be a back e.m.f. induced in the coil in such a way as to oppose the change producing it. This e.m.f. is proportional to the angular speed of the motor and is given by: (4) where v b is the back e.m.f., k e is the back e.m.f. constant, and ω is the angular speed of the motor.

Summing voltages in a voltage loop for a DC motor Using Kirchhoff’s voltage law, we can write the following equation for the motor circuit: (5) where V a is the applied voltage, v b is the back e.m.f., and L and R are the inductance and the resistance of the armature circuit, respectively.

Using the previous component models Using equation (3) and rearranging it to make i the subject (6) Combining equations 4,5,6 we get (7) Equation (7) is the second order differential equation model for a simple d.c. motor, describing the change of the angular velocity with the applied voltage.

The first order model of the motor In many applications the motor inductance is small and can be neglected. The model then becomes (8) which is a first order differential equation model describing the change in angular velocity with the voltage applied.

Converting the second order model into the laplace domain

Re-arranging into transfer function form

Matlab implementation Assuming the motors values are known for inductance L, motor resistance R, motor back e.m.f constant k e, this model for the motor can be used in a simulation. It can be entered directly into matlab’s (depending on the version) and used to simulate the response of the motor to controller configurations.

First order model of the motor in laplace domain

Re-arranging into transfer function form