DC Motor

Basic Modeling of a DC motor

A DC brushed motor may be modeled as a mechanical second-order system and an electrical first-order system. The equation for the rotational acceleration of the motor with the motor torque, a viscous friction, and a static friction, may be defined as:

${\ddot {\theta }}={\frac {1}{J}}(\tau _{m}-B_{v}{\dot {\theta }}-F_{s}sign({\dot {\theta }}))$ Where the motor torque is defined by the current and motor constant as:

$\tau _{m}=K_{a}i_{a}$ The derivative of the current in the motor is a function of the input voltage, the motor back-electromotive force, and the resistance of the motor coils as:

${\dot {i}}_{a}={\frac {1}{L}}(u_{a}-e_{a}-R_{a}i_{a})$ Where the back-emf is defined by the angular velocity and the motor constant as:

$e_{a}=K_{a}{\dot {\theta }}$ Identification of DC motor Parameters

As Motors do not always come with datasheets with exact parameters, the specific details of a motor often need to be measured.

Motor Coil Resistance

The motor coil resistance can most easily be measured through use of a multimeter. Alternatively, if testing the resistance with a microcontroller, a known resistor can be used to create a voltage divider, the voltage drop over which may be used to determine the coil resistance. This measurement, if the current is not kept low through a high resistance, or a low input voltage, will require the motor to be manually stalled in order for the back-emf to not skew the measurement.

$R_{a}={\frac {R_{1}V_{in}}{V1}}-R_{1}$ 