How electric motors work: from magnetic force to shaft torque

The Lorentz force on a conductor

Every electric motor rests on one physical effect. When a current I flows through a conductor of length L that sits in a magnetic field of flux density B, and the conductor is perpendicular to the field, it is pushed sideways by a force

F = B × I × L

with F in newtons, B in tesla, I in amperes and L in metres. The direction follows the left-hand rule for motors: point the first finger along the field (north to south), the second finger along the current, and the thumb shows the direction of the force. Reverse either the current or the field and the force reverses; reverse both and it stays the same.

A small example: a 0.1 m conductor carrying 10 A in a 0.8 T field feels F = 0.8 × 10 × 0.1 = 0.8 N. That is less than 1 N, which is why real motors use many turns of wire and strong fields.

From force to torque

A single straight conductor can only push. To get rotation, the conductors are placed on a cylinder (the rotor or armature) that is free to turn about a shaft. A force F acting at radius r produces a torque

T = F × r

If the winding has N conductors under the poles, each carrying current I in a field B, the total torque is roughly

T = N × B × I × L × r

This shows the three levers every motor designer has: more magnetic field, more current, or more conductor length and radius. It also explains why motor torque is almost proportional to current, and why motors with strong permanent magnets or large frames give more torque. Mechanical power then follows from torque and angular speed:

P = T × ω

where ω = 2π × N / 60 is the speed in radians per second and N is in rpm. A motor that produces 2 N·m at 3000 rpm delivers P = 2 × 314.2 ≈ 628 W.

Stator, rotor and the air gap

Nearly every rotary motor has the same two main parts.

• The stator is the fixed part. It carries either windings or permanent magnets and creates one of the two interacting magnetic fields.
• The rotor is the part on the shaft. It carries the other field, produced by permanent magnets, a winding with its own current, or currents induced from the stator.
• The air gap is the thin space between them, typically a fraction of a millimetre to a few millimetres. The magnetic field crosses it, and torque is produced there.

Torque is, in the end, the tendency of two magnetic fields to line up. The stator field and the rotor field try to align like two bar magnets, and as long as the stator field keeps moving ahead of the rotor field, the rotor keeps chasing it. Different motor types are simply different ways of keeping the stator field one step ahead.

Commutation: keeping the torque going the same way

If the current in a rotating loop never changed direction, the torque would flip sign every half turn and the rotor would just oscillate. Commutation is the switching of current that prevents this.

In a brushed DC motor, commutation is mechanical. Carbon brushes press on a segmented copper commutator fixed to the rotor. As the rotor turns, the segments move under the brushes and reverse the current in each coil exactly as it passes the neutral position, so the torque keeps the same sign. It is simple and cheap, but brushes wear, spark and limit speed.

In a brushless motor, commutation is electronic. The magnets are on the rotor and the windings are on the stator. Hall sensors, or a position estimate, tell an inverter where the rotor is, and the inverter energizes the stator phases in the right sequence. There are no brushes to wear out, so efficiency and life improve, at the cost of drive electronics.

AC induction and synchronous motors need no commutator at all: a three-phase supply creates a rotating field directly, which is covered in the rotating magnetic field guide.

A motor is also a generator

The same machine can be run backwards. If the shaft is turned by an engine, turbine or the motor's own load, the conductors cut the magnetic field and a voltage is induced in them, E = B × L × v. This is Faraday's law of induction, and it is the voltage a generator produces.

This happens in every running motor too. As the rotor spins, the windings generate a voltage that opposes the supply, called the back-EMF. It is the reason a motor's current falls as it speeds up, and the reason a motor can regenerate energy when it is driven faster than its no-load speed, for example an electric vehicle braking downhill. Motoring and generating are the same energy-conversion process with the power flowing in opposite directions.

For a motor, the electrical power drawn from the supply is converted to mechanical power plus losses. For a generator the order is reversed: mechanical power in, electrical power out.

Families of electric motors

Motors differ in how they create and keep the stator field ahead of the rotor.

• DC motors: brushed DC and, with electronic commutation, brushless DC. Speed follows voltage and torque follows current.
• Induction (asynchronous) motors: the rotor has no supply; its current is induced by the stator field. Robust, cheap and the workhorse of industry. They run slightly slower than the field.
• Synchronous motors: the rotor locks to the rotating field and turns at exactly the synchronous speed. This family includes wound-field motors, permanent-magnet synchronous motors (PMSM) and reluctance motors.
• Stepper and servo motors: built for precise position control, either by stepping through discrete positions or by closed-loop feedback.
• Universal motors: series-wound commutator motors that run on AC or DC, common in drills and vacuum cleaners.
• Linear motors: the same principles unrolled flat to produce straight-line force instead of torque.

The choosing a motor type guide compares them in detail.

Try it in the Curvo Studio simulators

Open the brushed DC motor simulator and watch the brushes and commutator turn in the 3D scene. Raise the Supply voltage and the Mechanical load, and read the Torque, Armature current and Back EMF values. Use the Polarity control to switch between Positive and Reversed and see the rotation direction change.

Open the brushless DC motor simulator to see electronic commutation. Change the PWM duty cycle and the Commutation sequence, and watch the phases U, V and W being energized in turn as the rotor follows.

Finally, open the DC generator simulator and compare its Generated EMF and Terminal voltage readouts with the motor's Back EMF: the same machine, with energy flowing the other way.

Try it in the simulator