# DC Motors – current, voltage, speed, power, losses and torque relationships

This article presents basic physical sizes of DC motor with permanent magnet on the stator. This type of motor is very suitable for driving autonomous robots. A main power of the robot is a battery (DC voltage), as well as the power of these engines. In this article, a RE 35 motor with a GP 32 C reducer with a transmission ratio of 1: 14 and a 1: 33 designed and manufactured by MAXON was used as an example.

# How is torque related to current? Image 1. Relationship between torque and armature current for the MAXON RE35

Image 1. shows dependence of the armature current from the engine torque for the MAXON RE35 when the armature winding voltage is 12V. The increase in the torque on the motor shaft results in the linear increase in the armature current.  It is also shown in the equation (8) from the previous tutorial. The current function I, depending on the torque M, shows that more current flowing through the motor will produce a higher torque.  Part of the diagram that is painted in yellow is the area in which the engine is not allowed to operate for a long time (short term operation).

# How is torque related to speed? Image 2. Relationship between speed and torque for a MAXON RE35

For each DC motor, a function of speed n can be plotted, depending on the torque M (mechanical engine characteristics). Image 2 shows the dependence of the speed n on the torque M at a constant voltage of 12V. It can be noted that the speed decreases linearly when the torque increases.
To draw the curve, two endpoints are used. The first point is when the torque is zero. The second point is when the speed is zero. The image shows that the armature speed is 405 rpm (rotation per minute) when the torque is zero. The torque is 7050 mNm when the speed is zero. This is not presented in the Image 2. If the armature motor voltage is changed, speed and torque also proportionally change. The relation between the speed without load (n0) and armature voltage (U) is given in the following equation: The mechanical power at the output is derived from the input electrical power and power losses (Joule losses) in the motor according to the equation (12). If we use the equations (8) (9) from our previous tutorial and (11) we can calculate output mechanical power with equation (12). Legend:
Pel – input electrical power
Pj – Power losses in motor
Pmeh – output mechanical power
n – speed
R – armature resistance
I – armature current

Using the equation (8) and integrating the values for input electrical power and power losses into equation (12), we obtain velocity expressed via the torque: The mechanical output power is calculated over the speed n and torque M according to the following equation (14): # How is mechanical output power related to the torque?

Figure 3 shows how the mechanical output power depends on the torque for the MAXON RE35 DC motor. The curve is plotted when applied voltage is 12V and ambient temperature is 25 Celsius degrees. Figure 3. Relation between the mechanical output power and the torque for the MAXON RE35

# How is coefficient of efficiency related to the torque?

The coefficient of efficiency describes the relationship between the mechanical power obtained at the output and the electric power applied to the motor input connections. Dependence of the coefficient of efficiency on the torque for the MAXON RE35 is given in Figure 4. Figure 4. Relation between the coefficient of efficiency and the torque for the MAXON RE35

The expression for the coefficient of efficiency is given in the following formula (15): # How is armature resistance related to torque? Figure 5. Relation between the armature resistance and the torque for the MAXON RE35

Dependence of the armature resistance on the torque for the DC motor MAXON RE35 is given in Figure 5.

# How is winding temperature related to torque? Figure 6. Relation between the coil temperature and the torque for the MAXON RE35

Dependence of the armature winding temperature on the torque for the MAXON RE35 is given in Figure 6.

Usually, encoder is used for measuring speed and position of motor shaft. More about basic operation principle of an encoder, you can find in our tutorial “Optical digital incremental encoder“.