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Generators and transformers

Electricity and magnetism · level 7, lesson 5 of 6

A question to start

Almost all the electricity in the world's power grids is made by spinning something: a turbine turned by steam, water or wind. What does the spinning do?

The idea

A generator is a motor run backwards. A coil spins in a magnetic field, so the flux through it keeps changing, and an alternating voltage is induced. For \(N\) turns of area \(A\) spinning at \(\omega\) radians per second in a field \(B\), the peak voltage is

\[\varepsilon_0 = NBA\omega.\]

A transformer changes the size of an alternating voltage. An alternating current in the primary coil makes a changing flux in an iron core, which induces a voltage in the secondary coil:

\[\frac{V_s}{V_p} = \frac{N_s}{N_p}.\]

An ideal transformer wastes no energy, so \(V_p I_p = V_s I_s\): step the voltage up and the current goes down.

Power lines run at very high voltage for that reason. A smaller current wastes less energy as heat, since the loss is \(I^2R\).

primary:few turnssecondary:many turnsiron core

Worked examples

1. A generator. 500 turns, \(B = 0.2\) T, \(A = 0.01\) m², spinning 60 times a second (\(\omega = 377\) rad/s): \(\varepsilon_0 = 500 \times 0.2 \times 0.01 \times 377 = 377\) V.

2. A transformer. 120 V to 12 V needs a turns ratio of 10:1. With 1,000 primary turns, the secondary has 100.

3. Why high voltage. Sending 1 MW at 10,000 V needs 100 A. At 100,000 V, only 10 A: a tenth of the current and a hundredth of the \(I^2R\) loss.

Try it

  1. Find the peak voltage of a 200-turn coil of area 0.05 m² spinning at 100 rad/s in a 0.1 T field.
  2. A transformer has 400 primary turns and 20 secondary turns. What does it give from 240 V?
  3. That transformer delivers 2 A from its secondary. What current does the primary draw?
  4. Why does a transformer not work on a steady current from a battery?
Answers
  1. \(200 \times 0.1 \times 0.05 \times 100 = 100\) V.
  2. \(240 \times 20/400 = 12\) V.
  3. Power in = power out: \(12 \times 2 = 24\) W, so \(24/240 = 0.1\) A.
  4. A steady current makes a steady flux, which induces nothing in the secondary.

Common mistake

"A step-up transformer makes extra energy." It raises the voltage but lowers the current by the same factor. The power out is never more than the power in.

Where it leads