All the things getting out of a motor (Torque, speed and stuff)

This post is the follow-up of the Motors-in-robotics’ serie. Although what I will present you here is still basic notions and can be addressed to anyone keen on knowing more about it, don’t hesitate to have a look at the previous posts (types of motors in robotics, DC motor direction and speed control, how to read a datasheet), they will help you expand your knowledge about electric motors and start on good basis.

Your DC brushed motor is really full of surprises: you think that if feeding it with some electric power, it kindly provides a turning shaft. Sometimes it spins very fast, sometimes slower, and with more or less force — actually torque —. That’s something amazing and that’s good enough.

But that’s actually not quite everything.

Remember this image I put in one of the first post about motors?

Starring, from left to right: a transmission tower as the input electric power; a nice black box as a DC brushed motor; and a spinning top as the output mechanic power.

It’s highly simplified, though, and reality is slightly different.
What I will detail here is still simplified, but however a little more precise.

A motor generates more than a rotation, more than only torque and speed. There are many other things getting out of it, and they go from things that you don’t really care to things you wish never exist. Let’s review everything what really comes out from your motors.

Torque and speed

We begin with the basics. Both these outputs are the most important, because they are the ones we want and we use. A motor is supposed to rotate, it rotates, cool, now let’s have a coffee.

What are they?
Torque is the force applied at the distance to an object’s pivot —hence it’s a Force times a Distance —, allowing the object to rotate. In International System of Units measure, it’s represented in Newton-meter (N.m) (1). Applied to your motor, torque is also the rotational force provided by the rotor.

Speed, is actually the rotational speed of the rotor. Nothing much to say about it, because you already know from a previous post that you can control that speed with easy electronics and PWM. You can express it in radians per second (rad/s), or most of the time in revolutions per minute (rpm) (1).

Let me remind you that in a DC motor, high speed means low torque and inversely. On the characteristic curve, it looks like this:

This blue curve shows all the theoretical functioning points (defined by a torque and a speed) of a motor at a given voltage.

This curve is represented by the following function:

N is speed (Y-axis) and T is torque (X-axis). N_0 is no-load speed, T_stall is stall torque.

We already discussed about this in a previous post, but it’s interesting to precise that this linear curve is theoretical. In your real world, it may look like that:

Moment of inertia

Because while a motor is working, its rotor is a spinning mass. And any rotating mass has a moment of inertia. You can see it as a factor to calculate how much torque is needed to give a movement to an object. Its unit is kg.m² (SI), or lb.ft².

Moment of inertia — also called rotational inertia — of an object depends on three factors: the mass of the object, the position of its axis of rotation, and its mass distribution around this axis. For example, more torque is needed to rotate a free horizontal bicycle wheel than to rotate a small rotor. Both masses are evenly distributed, but the bicycle wheel is heavier and some of its mass is far away from the axis of rotation; for the rotor, it’s really light and the mass is very close to its axis of rotation.

Another example: which banana is simpler to rotate, i.e. needs less torque? The answer is the banana on the right, only because of the position of rotation axis, implying a different distribution of the mass in each case.

In order to ease the painful life of scientists, the moment of inertia calculation can be simplified, or assimilated to a known geometry reference solid (like a sphere, a cube, a tore, etc.). A banana is not a geometry reference.

Angular acceleration

DC motors need to provide acceleration in order to change their speed. For example, from idle speed to whatever speed a rotor wants to get, the transition is not spontaneous (because an infinite acceleration, as well as a null moment of inertia, are strongly unlikely), it needs an acceleration to reach a given speed. In the case of reducing the speed, a deceleration is only a negative acceleration.

Acceleration — actually angular acceleration —, torque and moment of inertia are closely linked together by this equation:

See how life is simple?

Torque (T) (in N.m) is equal to moment of inertia (I) (in kg.m²) times angular acceleration (alpha) (in m/s², all the units here are form SI). This equation appears in Newton’s law of motion.

Back electromotive force

Also called counter electromotive force, back-EMF or CEMF.
Back-EMF is a voltage (in V), a difference of potential. In the coils of a motor, it appears when electrons are moving through the wires while under a motion of magnetic field (Lenz’ law may explain it better than me).

This one is tricky because electromotive force and back electromotive force are the same phenomenon. Only one of them (back-EMF) exerts a mechanical force opposing the motion of the motor, while the other is giving a rotation to the motor (you already know the process, by the way).

A back-EMF can be measured by a voltage generated at the output of the motor which, in certain cases, can damage the electronic circuits.

However, back-EMF is proportional to the rotational speed of the rotor, making it in sometimes very useful to measure.


Heat is a recurrent encounter when dealing with electrical power. Why is that? It’s because electrical power is always put face-to-face with electric components, and many of them have more or less resistance to electricity. This resistance means electrical power transformed into heat. A famous equation called Joule’s first law is explaining it:

In International System of units, P_th is in Joule, R in Ohm, and i in Ampere.

In this equation, R is the resistance of any conductive component, and i is the current which flows through this component. P_th is the thermal power, i.e. the power (2) which is transformed in pure heat, emitted from the component.

In a DC brushed motor, electricity is flowing into the winding; having a resistance, this winding tends to heat according to the amount of current.

To summarize:

A part of electrical power is always transformed in heat if passing through electric components.

By the way, the more a component (like a motor) tends to heat, the more its efficiency — which is the ratio of output power by input power— is said to be low. High-efficiency component will emit very little heat.

Also, a DC brushed motor, while running, has a constant inside-friction between brushes and commutators, and between shaft and housing through plain- or ball-bearings. This friction is a heat generator as well, i.e. an efficiency-killer.

This was the third thing that comes out of a motor, and wait here, there is more.

Magnetic field

Of course, you remember this one from one of my previous post about a terrified about-to-be-lifted banana, most of the electric motors run with magnetic field — not black magic.

The electricity running through the winding generates magnetic field, and provides motion to electrically-induced or permanent magnets. One sentence to summarize how a motor works! Sweet.

Let’s emphasize that only the permanent magnets produce a “static” magnetic field. The rest and particularly the magnetic field from the winding is more likely to be called “electromagnetic”, we will see that in a short while.

Anyway, all this magnetic mess is not disappearing away, it forms kind of imaginary lines from one pole to another (negative to positive), that we call a magnetic field, all around the motor. Its strength is measured in Amperes per meter (A/m). The representation of lines, though imaginary, is a nice model to help to understand how a magnetic field would look if you had super-powers.

The Earth happens to have something similar:

Beware: Earth’s magnetic field is not due to the high number of DC motors on it surface. (Source)

Any magnetic material (e.g. an iron-made object) put near a motor will interact with its magnetic field. Although no material can stop and cancel a magnetic field, it’s possible to create shields that are able to redirect the magnetic lines.


So now who even is this Emi and what does she have to tell us about motors?

EMI stands for Electromagnetic Interference. It’s a blend between electronic and magnetic fields, both being perpendicular from each other and to their direction of propagation.

E is the direction of electric field, B is the direction of magnetic field, and C is the direction of their propagation. Here λ is the period and I won’t talk about it.

To simplify, we can say it’s a radiation of energy through the air around the motor. It’s caused by the electricity running through winding, and also through commutators, brushes, terminals and wires.

This kind of interference is not really welcomed, if not totally unwanted. It can highly affect electronic circuits so that it would go as far as stop working. Fortunately, electromagnetic shields can be applied to reduce their propagation.

Note: Communication buses used to send signal—eg. I2C — can be really sensitive to interference. However, some of them provide a differential pair, which is basically the same signal transiting on two very close wires of the same length, one signal being opposed from the other one. The electromagnetic interference being the same on both wires, the difference of signal at reception allows to get a clean signal. This is why the bus associated called Robus (from Luos Robotics) uses a differential pair.

Bonus: Noise, vibrations, light, and other stuff


There is not much to say about the noise: only that it’s generated mostly from the friction of brushes on commutators, and of shaft on bearings. Also, some vibrations can appear on the shaft at high speed and increase the noise.

One last thing: the frequency of electrical signal through the winding may sometimes generate a noise, particularly between 100 Hz and 8000 Hz (more or less the audible range of human hear). Controlling motors at 20 kHz is a way to avoid high-pitched noises.


As motorization implies motion, and despite the fact that the axis of rotation of the rotor is fairly situated at its center of mass, the inertia is never perfectly balanced. Every motor generates more or less vibrations.


Technically, heat generation implies photon generation, i.e. light. But this would be quibbling to say that the winding could heat at the point of generating visible light, and that the light would be visible through the interstices of the motor’s housing, not to mention this would imply the motor to be melting and… burning. So not working anymore. But, technically, yes, “light”. Also, most of the light is infrared anyway, so you can stop awkwardly staring at your rotating motors.


Talking about quibbling, ever try to smell a running motor? Actually, the kind of plastic shield around the coils’ wire could emits some smelly particles while heating.

And I think it’s time to stop here, please don’t even try to taste a motor.

That is all for today.

For the lazy ones like me, or for the ones in a hurry, here is a TL;DR:

An electric motor provides (what is wanted):

  • Torque
  • …and speed, both of them are highly expected from a healthy running motor.
  • Moment of inertia: because of the mass of the rotor.
  • Acceleration, in order to transit from any speed to another.

But also (what is sometimes not wanted):

  • “Back” EMF.
  • Heat, because of Joule effect and frictions.
  • Magnetic field, generated by winding and magnets.
  • EMI, from electricity flowing everywhere it can flow.
  • Noise, vibrations, “ light ”, and smell…

An electric motor does not provide:

  • Black magic, sorry.

Vibrations and light (and smell) were voluntarily omitted on this figure.

Thank you for reading.

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Note (1): For a quick overview of the different measurement systems and units, go see the section Units are hell at the end of this post.

Note (2): A power is the energy per unit of time, so the thermal power is the power converted from electrical energy to thermal energy.