DC Motor Equations

Content

1DC Motors 2

1.1Gearbox 2

1.2Reversibility 2

1.3Efficiency 2

2Datasheet 3

2.1Performance Chart and Operating Regions 4

2.2Performance parameters 6

2.3RC Car DC Motors Datasheet (Bad) 7

2.4Units of Measure 8

3DC Motor Equations 9

3.1DC Motor Model Parameters 10

3.2DC Motor Model 11

3.3DC Motor Equations 12

3.4DC Motor Power and Efficiency 13

3.5Torque for Peak Efficiency 14

3.6Peak Efficiency 17

3.7Max Power 19

3.8Current draw at Max Power and Peak Efficiency 20

3.9Motor Speed at Max Power and Peak Efficiency 21

3.10Power at Max Power and Peak Efficiency 22

3.10.1Electrical Power in Max Power region 22

3.10.2Torsional Power output in Max Power region 23

3.10.3 Heat dissipation in Max Power region 24

3.10.4Electrical Power in Peak Efficiency region 25

3.10.5Torsional Power output in Peak Efficiency region 26

3.10.6Heat dissipation in Peak Efficiency operating region 27

4DC Motor Equations Recap 28

4.1Glossary 28

4.2DC Motor Model equations 29

4.3Max Power Equations Recap 30

4.4Peak Efficiency Equations Recap 31

5Conclusions 32

1DC Motors

A DC Motor is a transformer that converts electrical power into torsional power.
In the electric domain, the winding voltage [V] is the effort, the winding current [A] is the flux. Their product is the electrical power [W] that must be fed to the motor.
In the torsional domain, the axle torque [Nm] is the effort, the axle rotational speed [π/s] [Radians per second] is the flux. Their product is mechanical torsional power [W] at the axle.

1.1Gearbox

usually a DC motor is to fast and too weak to be used directly. A gearbox lowers the speed while increasing the torque. In principle, a gearbox N:1 will divide the speed by a factor of N and multiply the torque by a factor of N.
Due to losses inside the gearbox, the actual output torque is lower than the theoretical value while the speed is exactly the theoretical value. The power loss is dissipated as heat inside the gearbox.

1.2Reversibility

A DC motor is fully reversible, so injecting torsional power at the axle will result in electrical power leaving from the windings, a fact that can be exploited for electrodynamic breaking, regenerative breaking and power generation.

1.3Efficiency

A DC motor is not perfectly efficient. Some energy is lost in the winding, the bearings and the gears if the motor includes a reducer meaning that some input power is converted into heat. For an electric motor, conversion efficiencies of about 40% to 90% are typical.
Cheaper motors are less efficient due to cheaper bearings with more drag, thinner wires and weaker stator magnets.
Motors with gearbox are less efficient. Cheaper gearboxes are much less efficient due to cheaper bearings with more drag.

2Datasheet

The starting point is to understand the motor data manufacturers provide in their datasheet.

2.1Performance Chart and Operating Regions

A good motor datasheet also provides a performance chart. In this simple model a fixed voltage is fed to the windings of the motor and an increasing torque load is applied to the axle.
This has two effects:

Speed decreases linearly until it reaches zero at the Stall Torque
Current increases linearly from the No Load current to the Stall current at the Stall Torque

A DC Motor has four basic regions of operation:

No Load: The axle is free to turn unimpeded at its maximum speed. The motor only draw power because of internal losses. The speed is high, but the torque is zero so the actual output torsional power is 0 [W].
Stall: Condition opposite to No Load. The axle is loaded to the point the motor does not have enough power to turn it, and the speed is zero. In this region, current and torque are maximum. Torque is high, but speed is zero so the actual output torsional power is 0 [W]. In this condition, the motor is dissipating all of the input power as heat.
Power: Torque is half of the Stall torque and speed is half of No Load speed. In this condition the motor is outputting its maximum power to the axle.
Efficiency: At this load and speed condition, the motor is achieving the best ratio of output power over input power. It's the point where efficiency is highest.

2.2Performance parameters

A good motor datasheet will list the key parameters of the motors. The current, torque and speed in each of the motor operating region. This is an example of a good motor datasheet with all parameters of interest.

Five data taken in the No Load and Stall regions are enough to compute the characteristic of the motor in between:

Nominal Voltage [V]: Rated voltage at which motor parameters are computed
No Load Speed [RPM]: Speed of the motor at rated voltage when there is no load applied
No Load Current [A]: Current drawn at rated voltage and no load
Stall Torque [mNm][gcm]: Torque at rated voltage when the motor is stalled
Stall Current [A]: Current drawn at rated voltage in stall condition.

Some datasheet may specify other parameters and/or omit the stall/no load parameters

2.3RC Car DC Motors Datasheet (Bad)

Motors meant for RC Hobbyist usually have rubbish/useless datasheet.

This datasheet only specify a speed and a current, it has ambiguous voltage ratings and has no torque ratings, making it impossible to extract the performance and efficiency of the motor.
RC Car motors specify a T ratings which is the number of turns of the windings of the rotor. Higher T rating (80T) means lower speed but more torque at a given voltage.
This still gives no information about the actual torque performance of the motor and without it its impossible to extract the motor performance. One can make an educated guesses about efficiency, but that's it. A Guess.
There just isn't enough data provided in the datasheet.

2.4Units of Measure

Typical unit of measures used for motor parameters

3DC Motor Equations

From the No Load and Stall parameters it's possible to extract more fundamental parameters of the motor. All calculation will use SI units, and conversions must be done before hand.

3.1DC Motor Model Parameters

Extract DC motor parameters from No Load and Stall datasheet parameters.

3.2DC Motor Model

The model used for the scope of this document is simple and does not take into account many phenomenons. The scope of this document is just to extrapolate the performance of the motor from five datasheet values and nothing more, so this model is enough.

Rloss is computed from No Load parameters and models the internal losses of the motor. It causes the Inoload [A]. The motor losses are caused by the bearings, and increase linearly with speed. By modelling the losses with a resistor, the losses increase linearly with the input motor voltage, which is fair enough. A more accurate model should take into account electrical losses and bearing losses separately, but it would be a lot more complex and outside the scope of this document.
Rwinding is computed from the No Load and Stall current. It models the actual current that contributes to the torque.
The transformer has three equations:

The No Load speed is obtained by multiplying the motor voltage by a motor parameter. This models the No Load speed of the motor
The actual speed decreases linearly, from the No Load speed at zero torque to zero at the Stall torque
The current drawn by the motor is the torque multiplied by a motor parameter

Together those three equations model the basic behaviour of the motor and allow to compute the efficiency and the performance chart.
This model also allows to extend the behaviour of the motor to a voltage lower than the rated voltage (undervolting) or above the rated voltage (overvolting).

3.3DC Motor Equations

Equations that govern the behaviour of a DC motor

3.4DC Motor Power and Efficiency

Compute electrical and torsional power of the motor and efficiency of the motor.

3.5Torque for Peak Efficiency

Compute the torque at which the derivative of the efficiency is zero
First Compute general form of derivative of function

3.6Peak Efficiency

Compute the peak efficiency of the DC Motor

This is the peak efficiency of the motor in function of fundamental parameters.
KL tells how big the loss resistance is compared to the winding resistance. The higher the better.
KT is the torque per ampere, KW is the speed per volt. Their product is dimensionless. It's now clear their product should be less than 1 if the motor comply with the laws of thermodynamics.

3.7Max Power

Since Speed decrease linearly with torque, the product torque speed (torsional power) is maximized when the torque equals half the stall torque.

3.8Current draw at Max Power and Peak Efficiency

Current drawn by the motor in Max Power and Peak Efficiency operating region

3.9Motor Speed at Max Power and Peak Efficiency

Motor speed in Max Power and Peak Efficiency operating region

3.10Power at Max Power and Peak Efficiency

Extract all the power equations for electrical torsional and loss in the Max Power and Peak Efficiency operating regions.

3.10.1Electrical Power in Max Power region

Power drawn by the motor in the Max Power operating region

3.10.2Torsional Power output in Max Power region

Power output at the axle in the Max Power operating region

3.10.3 Heat dissipation in Max Power region

Power dissipated as heat when working in the Max Power region

3.10.4Electrical Power in Peak Efficiency region

Power drawn by the motor in the peak Efficiency operating region

3.10.5Torsional Power output in Peak Efficiency region

Power output at the axle in the Peak Efficiency operating region

3.10.6Heat dissipation in Peak Efficiency operating region