Magnetic properties of matter
The area of the hysteresis loop is energy lost on every cycle. Multiplied by the frequency, it gives the watts that heat the core, and from that number comes the choice of material for every transformer.
01Three responses to the field
Every material responds to a magnetic field, even though in almost all of them the response is imperceptible. The quantity that measures it is the relative permeability, and according to its value materials split into three very unequal groups.
| Type | \( \mu_r \) | Why | Examples |
|---|---|---|---|
| Diamagnetic | Barely less than 1 | The field induces opposing moments: it repels | Copper, water, graphite, bismuth |
| Paramagnetic | Barely greater than 1 | Permanent moments that partly align | Aluminum, platinum, liquid oxygen |
| Ferromagnetic | From 100 to 100,000 | Coupled moments that align as a block | Iron, nickel, cobalt, ferrites |
The difference between the first two rows and the third is five orders of magnitude, and that is why in practice there are only two categories: magnetic materials and all the others. An aluminum shield is useless against a static field for exactly that reason: its permeability is practically that of vacuum.
In a ferromagnet the moments are not disordered but grouped into domains: regions a few microns across where all of them point the same way. A demagnetized piece has its domains oriented at random and the net field is zero. When \( H \) is applied, the favorably oriented domains grow at the expense of the others by moving their walls, and that motion is not reversible: the walls get caught on impurities and grain boundaries. Everything in the next section comes from that irreversibility.
02The hysteresis loop
Since the motion of the domain walls costs energy and does not undo itself, \( B \) does not depend only on \( H \): it also depends on how you got there. Going through a complete cycle of excitation, the curve encloses a loop.
| Parameter | What it is | What it determines |
|---|---|---|
| \( B_{sat} \) | Saturation flux density | The limit of the core: beyond it, it contributes nothing |
| \( B_r \) | Remanence: what is left at \( H = 0 \) | Whether the material works as a permanent magnet |
| \( H_c \) | Coercivity: the \( H \) that demagnetizes it | Whether it is soft or hard |
| Loop area | Energy per unit volume per cycle | The hysteresis losses |
The last row is the key for electronics: the area of the loop is energy lost in every cycle, and multiplied by the frequency it gives dissipated power. That is why transformer cores are made of narrow-loop materials —magnetically soft— and permanent magnets of wide-loop ones.
Six real materials with their saturation and their coercivity. You vary the excitation and the loop is traced in full; below appear the loss per cycle and the power dissipated at the chosen frequency.
03Soft and hard
| Magnetically soft | Magnetically hard | |
|---|---|---|
| \( H_c \) | Less than 1000 A/m | More than 10,000 A/m |
| Loop | Narrow, low losses | Wide: that is what is wanted |
| What for | Cores of transformers and inductors | Permanent magnets, motors, loudspeakers |
| Examples | Silicon steel, ferrite, mu-metal | Alnico, hard ferrite, neodymium |
The Curie temperature marks the end of all this: above it, thermal agitation overcomes the coupling between moments, the domains disappear and the material becomes a plain paramagnet. It is 770 °C for iron, 358 for nickel and barely 310 for a common neodymium magnet, which is why a permanent-magnet motor has a strict thermal limit: past a certain point the magnets degrade and the motor is never the same again.
At 50 Hz the hysteresis losses of silicon steel are tolerable and its saturation of 2 T makes it compact. But those losses grow with frequency, and at 100 kHz they would be two thousand times greater. Ferrite has a much narrower loop and, above all, is a ceramic that barely conducts, so it has no eddy currents either. It pays for this with a saturation of only 0.4 T —five times lower—, but at that frequency the core needs much less flux and it does not matter.
04In the lab
With a transformer whose core is accessible, send the drop across a series resistor to the X axis —proportional to \( H \)— and the output of an RC integrator on the secondary to the Y axis —proportional to \( B \)—. View the loop in XY mode and measure \( B_r \) and \( H_c \).
Increase the voltage applied to the primary and observe the current with the oscilloscope. Identify the moment when it stops being sinusoidal and spikes appear: that is saturation. Explain why the current shoots up.
Heat a nail held by a magnet with a torch until it drops off. Estimate the temperature from the color of the metal and compare it with the 770 °C of iron. Verify that when it cools it is attracted again.
05Common mistakes
- Confusing \( B \) with \( H \). One is the resulting flux density; the other, the excitation that you apply.
- Believing that \( \mu_r \) is a constant. It depends on \( H \), on history and on temperature.
- Shielding a static field with aluminum or copper. They are diamagnetic: \( \mu_r \approx 1 \) and they do not divert anything.
- Ignoring saturation when designing an inductor. Beyond \( B_{sat} \) the inductance collapses and the current shoots up.
- Using silicon steel at high frequency. The losses grow with frequency; ferrite goes there.
- Forgetting the Curie temperature of a magnet. Neodymium degrades well before iron does.
06Self-assessment
What does the area of the hysteresis loop represent?
The energy lost per unit volume in each cycle. Multiplied by the frequency it gives the dissipated power.
What distinguishes a magnetically soft material?
Its low coercivity and narrow loop: it demagnetizes easily and loses little energy per cycle.
Why is aluminum no use as a magnetic shield?
Because it is paramagnetic with \( \mu_r \) almost 1: it does not divert the field lines.
What happens above the Curie temperature?
Thermal agitation overcomes the coupling, the domains disappear and the material becomes paramagnetic.
Why do switching power supplies use ferrite and not sheet steel?
Because of the losses: ferrite has a narrow loop and barely conducts, so it has no eddy currents either.
What happens to an inductor that saturates?
Its inductance collapses and the current shoots up, because the core can no longer contribute more flux.
07Further 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. Magnetism in matter, domains, hysteresis and the Curie temperature.
- William D. Callister and David G. Rethwisch. Materials Science and Engineering: An Introduction. 9th ed., Wiley, 2014 (Spanish edition, “Introducción a la ciencia e ingeniería de los materiales”, Reverté, 2019). Soft and hard magnetic materials, with their values and their industrial applications.
- Colonel Wm. T. McLyman. Transformer and Inductor Design Handbook. 4th ed., CRC Press, 2011. The real design of a core: choice of material, saturation, losses and air gap.