How does an electric motor turn? Inside a brushed DC motor

Put the batteries in a toy car and the wheels start turning straight away. The same thing happens when you lower a car window with a button, switch on a hair dryer or pull the trigger of a drill: electricity is turned into the rotation of a shaft. The machine that does this is called an electric motor. In this post we open up the simplest kind, the brushed direct current (DC) motor, and look inside. What does each part do, why does the motor keep turning, and why does it get hot when it is jammed?

If you want to remind yourself what a current is before we start, the How does electricity flow? journey builds up the idea of a closed circuit step by step. Here we add a magnet and a wire that can turn to that same circuit.

A magnet, a wire and a force

In 1820 the Danish physicist Hans Christian Ørsted saw that a compass needle next to a wire carrying a current was deflected. An electric current creates a magnetic field around it. The reverse is also true: a wire carrying a current inside a magnet’s field feels a force.

Three properties of this force are the whole secret of the motor:

• Size: When the field is perpendicular to the wire, the force is $F = B \cdot I \cdot L$. Here $B$ is the strength of the magnetic field, $I$ the current and $L$ the length of wire inside the field. Double the current and the force doubles too.
• Direction: The force is perpendicular to both the current and the field. Schools teach Fleming’s left-hand rule to find it: if your first finger points along the field (from N to S) and your second finger along the current, your thumb points along the force.
• Reversal: Reverse the current and the force reverses too.

In 1821 Michael Faraday used this force to make a wire that kept going round and round a magnet. It was one of the first devices to turn electricity into continuous motion.

How does a loop turn?

Now picture, instead of a straight wire, a rectangular loop of wire, a coil, mounted on a shaft. Put the coil between two magnets. On one side of the coil the current flows towards us, and on the other side it flows away from us. Because the currents are opposite, the forces on the two sides are opposite too: one side is pushed one way and the other side the other way. Together the two forces try to turn the coil around the shaft. This turning effect is called torque.

The size of the force does not change as the coil turns, but its turning effect does. When the plane of the coil is parallel to the field, the forces act on the sides with a lever arm at right angles and the torque is at its largest. When the coil has turned so that its plane is perpendicular to the field, the two forces pull against each other along the same line and the torque drops to zero. It is like pushing a door right at its hinge: however hard you push, it will not turn.

The real problem starts after this point. When the coil coasts past this “dead point”, its sides have swapped places. If the current still flows the same way, the forces now try to turn the coil back. The coil rocks to and fro and finally stops at the dead point. For a motor that keeps turning, something has to reverse the current every half turn.

The commutator: a reversal every half turn

The part that does this is the commutator. It is a copper ring fixed to the shaft and split in two, with a thin insulating gap between the halves. The two ends of the coil are connected to the two half rings. Two brushes, small blocks usually made of graphite or carbon, are pressed against the ring by springs and carry the current from the outside supply to the turning ring.

The brushes stay put and the ring turns with the coil. As the coil passes the dead point, the gaps in the ring pass under the brushes and each brush starts touching the other half. So the current in the coil reverses at exactly that moment. The result: whichever side is on top always carries current in the same direction, the force always pushes the same way, and the coil keeps turning in the same direction.

In the drawing above, turn the voltage up and press “Slow motion”. The purple arrows show the force on the sides of the coil and the white dots show the direction of the current. The gold half of the commutator is the segment touching the positive brush. As the coil becomes horizontal and passes the dead point, you can see the gold segment change hands and the current in the sides reverse.

Real motors do not have just one coil. They have many coils wound into slots in an iron core, connected to many commutator segments. Because some coil is always producing torque, there is no dead point and the motor runs more smoothly. The iron core also strengthens the magnetic field.

Voltage, speed and torque

Three numbers describe a motor: the applied voltage, the speed of the shaft (revolutions per minute, rpm) and the torque the shaft can deliver. In a simple DC motor they are linked by two short equations:

The first says that torque is proportional to current; $k$ is a constant for each motor (it depends on the number of turns, the magnet and the size of the coil). In the second, $R$ is the resistance of the winding and $\omega$ is the angular speed of the shaft (radians per second). The last term of this equation, $k \cdot \omega$, deserves a section of its own.

Back EMF: a motor is also a generator

A coil turning in a magnet’s field produces a voltage inside itself, just like a dynamo. This is electromagnetic induction, discovered by Faraday in 1831. In a motor this voltage opposes the supply voltage, so it is called the back electromotive force (back EMF). Its size is proportional to the speed: the faster the motor turns, the larger the back EMF.

What sets the current in the winding is the difference between the supply voltage and the back EMF:

This one equation explains almost everything a motor does:

• The current is largest at start-up. The shaft is not turning yet, so there is no back EMF; only the small resistance of the winding limits the current. The motor draws a short surge of current the moment it starts.
• With no load the motor limits itself. As the shaft speeds up, the back EMF grows and the current and torque fall. The motor speeds up until the back EMF almost equals the supply voltage. That is why the no-load speed is roughly proportional to the voltage: $\omega \approx V / k$.
• A load makes the current rise. Brake the shaft and the motor slows down, the back EMF falls and the current rises. More current means more torque, and the motor meets the load with that torque.
• A jammed motor gets hot. If the shaft is stopped completely, the back EMF is zero and the motor keeps drawing its largest, start-up current. The energy of that current does no work; it turns into heat in the winding. That is why a jammed drill or a stuck toy motor gets hot.

Worked example

The numbers of the motor in the drawing: winding resistance $R = 2\ \Omega$, motor constant $k = 0.02$ V·s/rad, bearing and brush friction about 2 mN·m. The supply is 12 V.

At start-up. Since $\omega = 0$, $I = 12 / 2 = 6$ A. The torque is $T = 0.02 \cdot 6 = 0.12$ N·m, or 120 mN·m. This is the largest torque the motor can deliver as it starts to turn.

With no load. The motor only has to overcome friction: $I = 0.002 / 0.02 = 0.1$ A. The back EMF is $12 - 0.1 \cdot 2 = 11.8$ V, so $\omega = 11.8 / 0.02 = 590$ rad/s. In revolutions per minute that is $590 \cdot 60 / (2\pi) \approx 5630$ rpm.

Under a 60 mN·m brake. The motor has to hold both the brake and the friction: $T = 62$ mN·m, $I = 0.062 / 0.02 = 3.1$ A. The back EMF is $12 - 3.1 \cdot 2 = 5.8$ V and the speed $5.8 / 0.02 = 290$ rad/s, about 2770 rpm. The motor draws $12 \cdot 3.1 \approx 37$ W from the supply. About 19 W of that ($I^2 R = 3.1^2 \cdot 2$) turns into heat in the winding, and about 18 W reaches the shaft as mechanical power.

You can see these numbers for yourself by slowly increasing the brake in the drawing: the speed falls while the current and torque rise. When the brake reaches about 118 mN·m (120 mN·m together with friction) the shaft stops and the current stays at 6 A. In the “Your own values” boxes under the figure you can type in another voltage or load, or the winding resistance and motor constant of a different motor, and do the same calculation for it.

A simple motor at home

A small experiment that appears often in school books lets you build a single-coil motor. You need: an AA battery (1.5 V), thin varnished (enamelled) copper wire, two paper clips, a piece of tape and a small magnet.

1. Wind the wire 10–15 times around a pen and slide it off; wrap the two ends around the coil a few times so they stick straight out on opposite sides.
2. Scrape all the varnish off one end with sandpaper. On the other end, scrape the varnish off only one side, along its length.
3. Open out the paper clips and bend them into bearings, then tape them to the two ends of the battery. Put the magnet on the battery, rest the coil in the bearings and give it a gentle push.

The half-scraped end works like a simple commutator: for half a turn the coil gets current, for the other half the current is cut off and the coil coasts round. Instead of reversing the current it switches it on and off, but the result is similar.

Safety: in this circuit the battery is close to a short circuit, so the wire and the battery can get hot quickly. Keep the test short, do not hold the hot wire and take the battery out when you have finished. Strong (neodymium) magnets can pinch fingers; keep them out of reach of small children and never swallow them. If you are young, do the experiment with an adult.

Brushed and brushless motors

Because the brushes rub against the ring, they wear down over time, can give off sparks and need maintenance. That is why computer fans, drones and many newer tools use brushless (BLDC) motors. In a brushless motor an electronic circuit does the commutator’s job: it senses the position of the rotor and switches the current in the windings at the right moment. The idea is the same; transistors do the reversing instead of a copper ring. Brushed DC motors are cheap, simple and easy to drive, so they are still common in toys, in car window and wiper motors and in many small appliances.

If you are curious about how current is shared in a circuit, our article on series and parallel circuits is a good next step.

Summary

• A current-carrying wire in a magnetic field feels a force perpendicular to both the current and the field: $F = B \cdot I \cdot L$.
• Opposite forces act on the two sides of the coil, and together they make the torque that turns it.
• The commutator reverses the current in the coil every half turn, so the torque always acts in the same direction.
• Torque is proportional to current ($T = k \cdot I$); the no-load speed is roughly proportional to voltage.
• The turning coil produces a back EMF ($k \cdot \omega$). At start-up the back EMF is zero, so the current is largest; as the load grows the motor slows down and draws more current.