Electric Motors and Generators
An electric motor converts electrical energy into rotational kinetic energy using the magnetic force on a current-carrying coil in a magnetic field, while a generator does the reverse, converting rotational energy into electrical energy by rotating a coil in a field to induce an emf. In brief: the torque on a motor coil is τ = NBIA sin θ. A generator coil rotating at angular speed ω in a field B produces an alternating emf ε = NBAω sin ωt, with a peak value NBAω. In a DC motor, a split-ring commutator reverses the current every half turn to keep the torque in one direction; in an AC generator, slip rings give an alternating output.
Almost all electricity is produced by generators, and electric motors run industrial machines, appliances and electric vehicles. This guide explains how motors and generators work, the roles of the commutator and slip rings, compares the two machines in a table, works through numerical examples for torque and emf and corrects common mistakes.
How does a DC motor work?
A simple DC motor has a rectangular coil between the poles of a magnet. When a current flows, the force on the two long sides of the coil is equal and opposite (by Fleming’s left-hand rule, because the currents are in opposite directions), producing a torque that makes the coil rotate. The torque for a coil of N turns, area A, carrying current I, at an angle θ between the plane normal and the field: τ = NBIA sin θ. The torque is maximum when the coil is parallel to the field and zero when it is perpendicular. At the zero-torque position, inertia carries the coil over, and the split-ring commutator, with brushes that touch the two halves, reverses the current in the coil, so that the torque keeps the same sense and the rotation continues. The speed can be increased with the current or voltage, the strength of the field and the number of turns. Practical motors use many coils, curved pole pieces and iron cores to obtain smooth torque. As the coil rotates in the field, an emf is induced in it that opposes the supply voltage (the back emf), which limits the current as the motor speeds up (a stalled motor draws a very large current).
How does a generator work?
In a generator (dynamo or alternator), the coil is rotated by an external source of mechanical energy (a turbine driven by steam, water, wind, or an engine). The flux through the coil is Φ = BA cos ωt, and the induced emf, by Faraday’s law, is ε = −N dΦ/dt = NBAω sin ωt. The output is alternating, with a peak emf ε0 = NBAω, and frequency f = ω ÷ 2π. The root-mean-square value is εrms = ε0 ÷ √2 = 0.707ε0. In an AC generator, the ends of the coil are connected to two slip rings, which keep contact with the external circuit throughout the rotation. A DC generator uses a split-ring commutator to give an output of one polarity (pulsating). Large power station generators use rotating electromagnets (the rotor) inside fixed coils (the stator), with three-phase outputs, producing 50 Hz or 60 Hz AC. The output can be increased by increasing the speed, the field, the area or the number of turns.
Motors and generators compared
| Feature | Motor | Generator |
|---|---|---|
| Energy conversion | Electrical to mechanical | Mechanical to electrical |
| Principle | Force on a current in a field | Induction by a changing flux |
| Input | Current from a supply | Rotation of the coil |
| Output | Rotation (torque) | Emf and current |
| Key device | Split-ring commutator (DC) | Slip rings (AC) |
Worked example: torque and peak emf
- Motor torque. A coil with N = 50 turns and area 0.0040 m2 carries 3.0 A in a field of 0.20 T. The maximum torque (coil parallel to the field, sin θ = 1) is τ = NBIA = 50 × 0.20 × 3.0 × 0.0040 = 0.12 N·m.
- Power. If the motor runs at 1,500 rpm (ω = 1,500 × 2π ÷ 60 = 157 rad/s) with an average torque of 0.08 N·m, the mechanical power is P = τω = 0.08 × 157 = 12.6 W.
- Generator emf. A coil has N = 100 turns, area 0.020 m2, and rotates in a field of 0.50 T at 50 revolutions per second. ω = 2π × 50 = 314 rad/s. Peak emf = NBAω = 100 × 0.50 × 0.020 × 314 = 314 V. The rms value = 314 ÷ 1.414 = 222 V.
- Double the speed. At 100 revolutions per second, the peak emf is 628 V and the frequency doubles to 100 Hz.
- Back emf. A motor on a 12 V supply with coil resistance 2.0 Ω draws 6.0 A when stalled (I = V ÷ R). At full speed, the back emf is 10 V, so the current is (12 − 10) ÷ 2.0 = 1.0 A.
How were they invented?
Michael Faraday demonstrated the first electric motor in 1821 (a wire rotating around a magnet) and the first generator in 1831. Hippolyte Pixii built a hand-cranked dynamo with a commutator in 1832. Werner von Siemens introduced the self-excited dynamo in 1866, and Nikola Tesla developed AC induction motors and polyphase systems in the 1880s, with George Westinghouse. Today electric motors use about half of the world’s electricity, and are essential in electric vehicles.
Which mistakes are common?
Mistake 1: Confusing motors and generators
A motor uses current to produce motion, and a generator uses motion to produce current. The device is the same in principle.
Mistake 2: Saying that the commutator makes the current AC
In a DC motor, it reverses the current in the coil so that the torque stays in the same direction.
Mistake 3: Forgetting the back emf
A running motor draws less current than a stalled one.
Mistake 4: Mixing peak and rms values
The mains voltage quoted (such as 230 V) is the rms value. The peak value is 1.41 times larger (325 V).