# Mathematical model of a DC motors

DC (Direct Current) or Unidirectional motors belong to rotary electric machines (electric motors and electric generators) in which the electric power is converted into the mechanical energy of rotational motion. They belong to double-coils systems because they have two coils, one on the stator and one on the rotor. See Image 1. When the coils are connected to the voltage source the current flow will be established through them. Image 1. Schematic of armature and field coil of a DC motor

The basic model of a DC machine with one coil on the stator and one coil on the rotor is described by the following equations: Legend:
uS – field winding voltage
uR – armature winding voltage
iS – field winding current
iR – armature winding current
RS – field winding resistance
RR – armature winding resistance
LS – field coil inductance
LR – armature coil inductance
MSR – mutual inductance
ωm – angular speed of the engine rotation
mC, mm – moment of conversion and load
Jm – total engine inertia torque
Bm – coefficient of viscous friction

# Types of a DC Motors regarding the field coil connection

Depending on the way of connecting the field winding, there are several types of DC motors:

• Independent wound DC motors,
• Series wound DC motors (the field and armature windings are connected in series)
• Shunt wound DC motors (the field and armature windings are connected in parallel) and
• Compound wound DC motors.

All these solutions have a serious disadvantage from the point of view of the application in robotics. The field that generates the stimulating flux requires its own power supply, which increases engine losses. Because of this flaw, instead of a field winding, a permanent magnet is used to obtain a constant flux source that does not require its own power supply. This type of motor is most commonly applied in robotics.

# Equations of a DC motor with permanent magnet

The mathematical equations of a DC motor with permanent magnet instead of the field winding follow directly from the model equations (1 – 4). In the static state, all variables are constant, so that the parts representing the differential part of the equations are equal to zero. kE – constant of velocities,
kM – constant of torques

Since the flux constant is dependent on the engine geometry. The kE and kM constants are not independent of each other. The following formula describes the relationship between these two constants: If the equation (6) is replaced by (5), the angular velocity can be expressed depending on the voltage on the armature.
From the equation (10) it can be seen that the rotational speed can be controlled by changing the voltage on the armature in case of constant load. Of course, this is true if the friction is negligible, and in many cases this is approximately satisfied. If friction can’t be ignored, a slight deviation from the linear dependence of the rotational speed on the armature voltage will be indicated in the final account. It is always preferable to have a fast return speed, because in many cases the engine requires the exact number of rotations.

# MAXON RE35 motor and GP32C reducer

Image 2 presents the inner and outer of a RE 35 DC motor of 90W designed by MAXON. It is 35 mm diameter DC motor with graphite brushes. More about RE35 motor you can find on MAXON Motor page. See image below for the details. Image 2. The inner and outer of a RE 35 DC motor

Usually, MAXON RE35 motor is manufactured with the GP 32 C reducer.  The GP 32 C of the MAXON reducer is shown in the image 3. Image 3. MAXON GP 32 C reducer

More about the relationship between electrical quantities of DC motors, you can read in our article “DC Motors – voltage, current, speed, power, losses and torque relationships“.
Often DC motor is manufactured with incremental encoders. More details about basic operation principle of encoders , you can find in our tutorial “Optical digital incremental encoder“.