Generators and Motors: AC Generator, DC Motor and Back EMF

Turning Motion into Electricity, and Back Again

Almost all the electricity we use, whether from the national grid, a petrol generator or a hydroelectric dam, is produced by spinning a coil in a magnetic field, or a magnet inside a coil. And most of the machines that use electricity, from ceiling fans and pumps to blenders and electric cars, contain motors that do the reverse. Generators and motors are two sides of the same physics: electromagnetic induction and the motor effect. This tutorial explains how a.c. and d.c. generators and d.c. motors work, the role of slip rings and the commutator, the formula for the e.m.f. of a rotating coil, back e.m.f. in motors, and why a motor draws a large current when it starts.

Generators and Motors: The Two Principles

Electromagnetic induction (generators): when the magnetic flux through a coil changes, an e.m.f. is induced, equal to the rate of change of flux linkage (Faraday's law), in a direction that opposes the change (Lenz's law).
The motor effect (motors): a current-carrying conductor in a magnetic field experiences a force F = BIL, in a direction given by Fleming's left-hand rule.

A generator turns mechanical energy into electrical energy; a motor turns electrical energy into mechanical energy. The same machine can often work as either.

The Simple a.c. Generator

A rectangular coil of wire is rotated at a steady rate between the poles of a magnet. Its ends are connected to two slip rings, which turn with the coil, and carbon brushes press against the rings to carry the current to the external circuit.

As the coil turns, the flux through it changes continuously:

when the plane of the coil is perpendicular to the field, the flux through it is greatest, but it is momentarily not changing, so the e.m.f. is zero;
when the plane of the coil is parallel to the field, the flux is zero but changing fastest, because the sides of the coil are cutting the field lines at right angles, so the e.m.f. is greatest.

Each half turn the sides of the coil swap places, so the e.m.f. reverses. The output is a sinusoidal alternating current, completing one cycle per revolution.

E.m.f. of a Rotating Coil

For a coil of N turns and area A rotating with angular velocity ω in a uniform field B, the flux linkage is NBA cos ωt. Differentiating gives the induced e.m.f.:

E = NBAω sin ωt, with peak value E₀ = NBAω.

The peak e.m.f. can be increased by using more turns, a stronger magnet, a larger coil or faster rotation. Faster rotation also raises the frequency.

Example 1. A generator coil has 100 turns, each of area 0.020 m², and rotates at 50 revolutions per second in a field of 0.50 T. Find the peak and r.m.s. e.m.f.

ω = 2π × 50 = 314 rad/s. E₀ = 100 × 0.50 × 0.020 × 314 = 314 V. The r.m.s. value is E₀/√2 ≈ 222 V, close to mains voltage, at a frequency of 50 Hz.

Example 2. If the coil is turned at half the speed, what happens?

ω halves, so the peak e.m.f. halves to 157 V and the frequency halves to 25 Hz. This is why a petrol generator must run at a governed, steady speed: in a simple two-pole machine, 50 Hz output needs 3,000 revolutions per minute.

Real Power Station Generators (Alternators)

Large generators, called alternators, usually keep the output coils still (the stator) and rotate an electromagnet inside them (the rotor). Only a small current is needed to energise the rotor through slip rings, while the large output current is taken directly from the fixed stator coils, avoiding big sliding contacts. Steam turbines (in gas-fired plants) or water turbines (at Kainji, Jebba and Shiroro dams) provide the rotation. Generators with more magnetic poles can run slower: a four-pole machine gives 50 Hz at 1,500 rpm, which suits slow water turbines.

The d.c. Generator and the Commutator

To get current that flows in one direction only, replace the slip rings with a split-ring commutator: a single ring cut into two halves, each connected to one end of the coil. Every half turn, just as the e.m.f. in the coil reverses, the coil's connections to the brushes swap over. The output is therefore always in the same direction, although it still rises and falls (a "bumpy" d.c.). Using many coils at different angles, each with its own commutator segments, smooths it out.

The d.c. Motor

A d.c. motor has the same construction as a d.c. generator: a coil on an axle between magnet poles, a split-ring commutator and brushes. When current flows in the coil, the two sides carry current in opposite directions, so the motor effect pushes one side up and the other down. This pair of forces forms a couple that turns the coil.

Without the commutator, the coil would turn only until its plane was perpendicular to the field and then rock back. The commutator reverses the current every half turn, at exactly that point, so the turning effect always acts in the same direction and the coil keeps spinning.

The turning effect (torque) is increased by more turns, a larger current, a stronger field and a larger coil area. Practical motors use many coils wound on a soft iron core (the armature), and curved pole pieces so that the field is radial and the torque more even.

Back e.m.f. in Motors

A spinning motor coil is also a coil turning in a magnetic field, so it acts as a generator too. It induces an e.m.f. which, by Lenz's law, opposes the supply voltage. This is the back e.m.f., Eb.

If the armature has resistance R and the supply voltage is V, the current is

I = (V − Eb) / R.

The back e.m.f. is proportional to the speed of rotation:

At switch-on, the motor is not turning, so Eb = 0 and the current I = V/R is very large.
As the motor speeds up, Eb rises and the current falls.
When a load is added and the motor slows, Eb falls and the current rises to supply more power.

Example 3. A 12 V d.c. motor has an armature resistance of 2.0 Ω. At full speed it draws 1.0 A. Find (a) the starting current, (b) the back e.m.f. at full speed, (c) the mechanical power output, and (d) the efficiency.

(a) At start, Eb = 0: I = 12/2.0 = 6.0 A.
(b) At full speed: 12 − Eb = 1.0 × 2.0, so Eb = 10 V.
(c) The power converted to mechanical work is Eb × I = 10 × 1.0 = 10 W. (The input power is 12 W, and I²R = 2 W is wasted as heat in the armature.)
(d) Efficiency = 10/12 ≈ 83%.

This explains several everyday observations:

Lights in a house may dim briefly when a large motor (a pumping machine or an air conditioner compressor) starts, because of the large starting current.
If a motor is jammed so it cannot turn, there is no back e.m.f., the current stays at its large starting value, and the windings overheat and can burn out. Fans and pumps that are stuck should be switched off quickly.
Large motors use starter circuits that add resistance at first, or "soft starters" and variable-speed drives, to limit the starting surge. Inverter sizing for solar systems must allow for this surge.

Generator Effect in Electric Vehicles

Because generators and motors are the same machine, electric cars and e-bikes use regenerative braking: when the driver brakes, the motor is switched to act as a generator, turning the car's kinetic energy back into electrical energy to recharge the battery, while the opposing force (from Lenz's law) slows the car.

Summary

Generators and motors are essentially the same machine: one turns motion into electricity, the other turns electricity into motion.
Generators use electromagnetic induction; motors use the force on a current in a magnetic field.
An a.c. generator uses slip rings; its e.m.f. is E = NBAω sin ωt, with peak NBAω.
A split-ring commutator gives one-way output in a d.c. generator and keeps a d.c. motor turning in one direction.
A running motor produces a back e.m.f.: I = (V − Eb)/R, so the starting current is much larger than the running current.
Mechanical power output of a motor = Eb × I.