Magnetic induction
Interrupting one ampere in one microsecond through a 10 mH coil generates ten thousand volts. The coil produces whatever voltage it takes so that its current does not change abruptly, and if nobody gives it a path, it opens one itself.
01Faraday’s law
A magnetic field that changes in time creates an electric field. That is the discovery that joins electricity and magnetism into a single subject instead of two separate ones, and it is what makes it possible to generate electric power at scale.
The flux can change in three ways, and all three are used: by varying the field —the transformer—, by varying the area —a circuit that deforms— or by varying the angle between the two, which is the generator of every power plant.
The induced current flows in the direction that opposes the change that produced it. It is not a matter of sign convention: it is the conservation of energy. If it were the other way around, the change would reinforce itself and we would have free energy. That is why it takes effort to move a magnet inside a short-circuited coil, and that mechanical work is exactly the electrical energy that appears.
02Self-inductance
A coil produces flux with its own current, so when that current changes, an EMF is induced in the coil itself. The constant of proportionality is the inductance:
The first expression has a brutal consequence. If the current is cut off suddenly, \( di/dt \) becomes enormous and so does the induced voltage. A coil does not allow its current to change instantaneously: if you insist, the coil generates whatever voltage it takes —thousands of volts— until something breaks or an arc jumps.
An RL circuit whose current is cut off. You vary the inductance and the turn-off time, and the voltage spike that the coil generates appears. Then try the flyback diode and see what changes.
Any inductive load switched by a transistor —a relay, a motor, a valve, a solenoid— needs a path for the current to keep flowing when the transistor opens. That path is the flyback diode in antiparallel with the coil: the current closes through it and decays smoothly, and the voltage across the transistor is limited to the supply voltage plus the 0.7 V of the diode. Without it, the spike destroys the transistor on the first cycle. It is the most expensive beginner’s mistake there is, and the lab above lets you see the order of magnitude of what is at stake.
03Mutual inductance and the transformer
When the flux of one coil passes through another, the current in the first induces an EMF in the second. If the two share an iron core, practically all the flux is shared and the voltages end up in the ratio of the turns:
And since the transformer needs the flux to vary, it does not work on direct current. Connecting it to DC is putting a short circuit across the source through the resistance of the winding, which is tiny. That is why power distribution is AC: only that way can the voltage be stepped up for transmission and stepped down for consumption.
04Eddy currents
In a solid conductor subjected to a varying flux, the induced currents circulate in swirls within the material itself. They dissipate energy and, by Lenz’s law, brake whatever produced them.
| Where | Effect | What is done |
|---|---|---|
| Transformer core | Losses and heating | Laminate it: sheets insulated from one another that cut the swirls |
| Core at high frequency | The losses grow with \( f^2 \) | Ferrite: a magnetic ceramic that barely conducts |
| Bus brake | Brakes without contact or wear | It is put to use |
| Induction cooktop | Heats the pot and not the burner | It is put to use |
The first two rows explain why 50 Hz cores are made of stacked sheets and those of a 100 kHz switching power supply are ferrite: at that frequency, a solid iron core would become a heater. It is the same phenomenon solved in two ways depending on the frequency.
05In the lab
Drop a neodymium magnet down a copper tube and down a plastic one of the same length, and time it. Explain the difference with Lenz’s law and estimate the energy dissipated in the tube.
Switch a relay coil with a transistor and use the oscilloscope to observe the collector voltage when it opens, first without a flyback diode and then with it. Measure the spike in both cases and compare it with the maximum voltage of the transistor.
With a transformer that has two secondaries, measure all the no-load voltages and verify the turns ratio. Then load one secondary and measure the drop, obtaining the equivalent impedance of the winding.
06Common mistakes
- Believing that the field induces an EMF. What induces it is its change: a stationary loop in a constant field generates nothing.
- Taking the minus sign as a convention. It is Lenz’s law, and it is the conservation of energy.
- Switching an inductive load without a flyback diode. The spike destroys the transistor on the first cycle.
- Connecting a transformer to direct current. Without a change in flux, it is just the resistance of the winding.
- Using a solid iron core at high frequency. Eddy currents heat it up; use ferrite.
- Forgetting the \( N^2 \) in the inductance. Doubling the turns quadruples it.
07Self-assessment
What induces an electromotive force?
The change of the magnetic flux in time, not the flux itself.
What does the minus sign in Faraday’s law mean?
That the induced current opposes the change that produced it. It is the conservation of energy.
Why does a coil generate a spike when its current is cut off?
Because \( v = L\,di/dt \): if \( di/dt \) becomes enormous, so does the voltage.
What is the flyback diode for?
To give the coil current a path along which to keep flowing, and thus limit the voltage across the transistor.
How is an impedance reflected through a transformer?
With the square of the turns ratio.
Why are 50 Hz cores laminated and 100 kHz cores ferrite?
Because of eddy currents, whose losses grow with the square of the frequency. Lamination is enough at 50 Hz; at 100 kHz a material that does not conduct is needed.
08Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics. 10th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage, 2019. Faraday’s law, Lenz’s law, inductance, magnetic field energy and RL circuits.
- Edward M. Purcell and David J. Morin. Electricity and Magnetism. 3rd ed. (Spanish edition, “Electricidad y magnetismo”), Cambridge University Press, 2016. Induction treated with care, including the cases where the flux rule is misleading.
- Marty Brown. Power Supply Cookbook. 2nd ed., Newnes, 2001. Inductors, high-frequency transformers and the handling of the energy stored in a switching power supply.