Generators and AC induction motors

The AC generator

A generator is a motor run backwards. Instead of supplying a current to make a coil turn, we turn the coil and an emf is induced in it. A coil of N turns and area A rotating at angular speed ω in a uniform field B has a flux through it of $\Phi = BA\cos\theta$, where θ = ωt is the angle between the field and the normal to the coil. The flux changes continuously, so by Faraday's law an emf is induced.

Using calculus (not required for the HSC, but it shows where the formula comes from), $\varepsilon = -N\dfrac{d\Phi}{dt} = NBA\omega\sin\omega t$. The emf varies sinusoidally, with a peak value

$$\varepsilon_{0} = NBA\omega = 2\pi NBAf$$

• The emf is zero when the plane of the coil is perpendicular to the field. Here the flux is a maximum, but for an instant it isn't changing, because the sides of the coil are moving parallel to the field lines.
• The emf is a maximum when the plane of the coil is parallel to the field. The flux is zero but changing fastest, because the sides are cutting straight across the field lines.
• The emf reverses every half turn, because the side that was moving up is now moving down. The output is alternating current with the same frequency as the rotation.

Slip rings connect the rotating coil to the external circuit. Each end of the coil is permanently joined to its own complete ring, and a brush rests on each ring. Because each brush always connects to the same end of the coil, the reversal of the emf in the coil appears in the external circuit as AC.

The peak emf can be increased by using more turns, a stronger field (electromagnets and an iron core), a larger coil, or by turning it faster. Turning it faster also increases the frequency, which is why power station generators are driven at a fixed speed: in Australia, 3000 rpm for a two-pole generator gives 50 Hz.

The DC generator

Replace the slip rings with a split-ring commutator and the generator gives DC. Just as the emf in the coil reverses, the commutator swaps the connections to the brushes. The output never changes direction: it is a series of half-sine "humps" that rise from zero to the peak and back, twice per revolution. It is DC in the sense that it flows one way, but it is not steady.

Practical DC generators smooth this ripple by using many coils at different angles, each with its own commutator segments, so the output stays close to the peak. Today, most DC is made by generating AC and converting it with diode rectifiers, because brushes and commutators spark and wear at high currents. A car alternator works this way.

Generators and conservation of energy

With nothing connected, a generator coil spins freely; only friction opposes it. Connect a load and a current flows in the coil. Now the coil is a current-carrying conductor in a magnetic field, so it experiences the motor effect. By Lenz's law, the induced current flows in the direction that opposes the rotation that produces it, so the forces on the coil produce a torque against the rotation.

To keep the generator turning at the same speed, whatever drives it (steam turbine, wind, falling water) must supply a matching torque and do work. The more current drawn, the larger the opposing torque. This is conservation of energy: the electrical energy delivered to the load comes from the mechanical work done turning the coil. If the induced current aided the rotation instead, the generator would speed itself up while delivering energy, getting something for nothing.

The same thing happens in a power station. When demand rises across the grid, the generators slow slightly and the frequency drops below 50 Hz. Control systems respond by sending more steam or water to the turbines. The grid operator keeps the frequency very close to 50 Hz by matching generation to demand second by second.

A generator and a motor are the same device. A spinning motor generates a back emf, and a loaded generator experiences a back torque. Electric cars and trains use this in regenerative braking: when the driver brakes, the motor is switched to run as a generator, charging the battery, and the opposing torque slows the vehicle.

The AC induction motor

The AC induction motor is the workhorse of industry: pumps, fans, compressors, conveyor belts, washing machines and many electric cars. It is cheap, tough and needs almost no maintenance, because its rotor has no electrical connection to the supply. No brushes, no commutator, no slip rings.

The stator and the rotating field

The stator has sets of coils around the outside, connected to a three-phase AC supply. The three currents peak one after the other, a third of a cycle apart, so the combined magnetic field they produce has a constant strength but a direction that rotates. With a 50 Hz supply and one pair of poles, the field turns 50 times a second, or 3000 rpm. With two pairs of poles (a four-pole motor), the field turns half as fast. The speed of the rotating field is the synchronous speed:

$$n_{sync} = \frac{120f}{p}$$

where n is in rpm, f is the supply frequency in hertz and p is the number of poles. A two-pole motor on 50 Hz has a synchronous speed of 3000 rpm; a four-pole motor, 1500 rpm.

The squirrel-cage rotor

The rotor is a cylinder of laminated iron with conducting bars (aluminium or copper) running along it, joined at each end by a short-circuiting ring. Without the iron it looks like an exercise wheel for a pet, hence "squirrel cage". The iron concentrates the field through the rotor; the laminations reduce eddy currents in the iron itself.

How it turns

1. The stator's field rotates past the bars of the stationary rotor, so the flux through each loop of the cage changes.
2. By Faraday's law an emf is induced in the bars, and because the cage is a closed circuit with very low resistance, large currents flow.
3. These currents are in a magnetic field, so they experience a force (the motor effect).
4. By Lenz's law the forces act to reduce the relative motion between the field and the rotor, so they drag the rotor round in the same direction as the field.

This is the same physics as the magnet dragging a metal plate along in magnetic braking, turned into a motor.

Slip

The rotor can never quite catch up with the field. If it turned at the synchronous speed, there would be no relative motion between the bars and the field, no change in flux, no induced current and no torque. Friction would slow it down. So the rotor always turns a little slower than the field. The difference, as a fraction of the synchronous speed, is the slip:

$$s = \frac{n_{sync} - n_{rotor}}{n_{sync}}$$

A typical motor at full load has a slip of a few percent. When the load increases, the rotor slows, the slip increases, the field cuts the bars faster, and the induced current and torque increase to match the load. Like the DC motor and its back emf, the induction motor automatically draws more power when it has more work to do.