**General Remarks**

So far we have avoided to consider the frequency behavior of the magnetization, i.e. we did not discuss what happens if the external field oscillates! | ||

The experience with electrical polarization can be carried over to some magnetic behaviour,
of course. In particular, the frequency response of paramagnetic material will be quite
similar to that of electric dipole orientation, and diamagnetic materials show close parallels to the electronic polarization
frequency behaviour. | ||

Unfortunately, this is of (almost) no interest whatsoever. The "almost" refers to
magnetic imaging employing magnetic resonance
imaging (MRI)
or nuclear spin resonance imaging - i.e. some kind of "computer
tomography". However, this applies to the paramagnetic behavior of the magnetic moments of the nuclei, something we haven't even discussed so far. | ||

What is of interest, however, is what happens in a ferromagnetic
material if you have expose it to an changing, i.e. oscillating magnetic field. H
= H_{o} · exp(iwt) | ||

Nothing we discussed for dielectrics corresponds to this questions. Of course, the frequency behavior of ferroelectric materials would be comparable, but we have not discussed this topic. | ||

Being wise from the case of dielectric materials, we suspect that the frequency behavior and
some magnetic energy losses go in parallel, as indeed they do. |
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In contrast to dielectric materials, we will start with looking at magnetic losses
first. | ||

If we consider a ferromagnetic material with a given hysteresis curve exposed
to an oscillating magnetic field at low frequencies - so we can be sure that the internal magnetization can instantaneously
follow the external field - we may consider two completely independent mechanisms causing
losses. | ||||||||

1. The changing magnetic field induces currents wandering around in the material -
so called eddy
. This is different from dielectrics, which we
always took to be insulators: ferromagnetic materials are usually conductors.currents | ||||||||

2. The movement of domain walls needs (and disperses) some energy, these are the intrinsic magnetic losses or hystereses losses. | ||||||||

Both effects add up; the energy lost is converted into heat. Without going into details, it is clear that the losses encountered increase with | ||||||||

1. The frequency in fboth cases,
because every time you change the field you incur the same losses per cycle. |
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2. The maximum magnetic flux in both cases.B_{max} |
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3. The conductivity s = 1/r
for the eddy currents, and | ||||||||

4. The magnetic field strength for the magnetic losses.H |
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More involved calculations (see the advanced
module) give the following relation for the total ferromagnetic loss per unit volume of
the materialP_{Fe} | ||||||||

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With thickness of the material perpendicular to the field direction, d = coercivity.H_{C}
= | ||||||||

It is clear what you have to do to minimize the eddy current losses: | ||||||||

Pick a ferromagnetic material with a high resistivity - if
you can find one. That is the point where ferrimagnetic materials come in. What you
loose in terms of maximum magnetization, you may gain in reduced eddy losses, because many ferrimagnets are ceramics with
a high resistivity. | ||||||||

Make small by stacking insulated thin sheets of the (conducting) ferromagnetic
material. This is, of course, what you will find in any run-of-the-mill transformer.d | ||||||||

We will not consider eddy current losses further, but now look at the remaining
hystereses losses P_{hyst} |
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The term is pretty much the area
inside the hystereses curve. Multiply it with two times the frequency, and you have the hystereses losses in a good approximation.H_{C} · B_{max} |
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In other words: There is nothing you can do - for a given material with its given hystereses curve. | ||||||||

Your only choice is to select a material with a hystereses curve that is just right. That leads to several questions: | ||||||||

1. What kind of hystereses curve do I need for the application I have
in mind? | ||||||||

2. What is available in terms of hystereses curves? | ||||||||

3. Can I change the hystereses curve of a given material in a defined way? |
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The answer to these questions will occupy us in the next subchapter; here we will just finish with an extremely cursory look at the frequency behavior of ferromagnets. | ||||||||

**Frequency Response of Ferromagnets**

As already mentioned, we only have to consider ferromagnetic materials - and that means the back-and-forth movement of domain walls in response to the changing magnetic field. | ||||||||||

We do not have a direct feeling for how fast this process can happen; and we do not have any
simplified equations, as in the case of dielectrics, for the forces acting on domain walls. Note that the atoms do not move if a domain wall moves - only the direction of the magnetic moment that they carry. |
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We know, however, from the bare fact that permanent magnets exist, or - in other words - that coercivities can be large, that it can take rather large forces to move domain walls - they might not shift easily. | ||||||||||

This gives us at least a feeling: It will not be easy to move domain walls fast
in materials with a large coercivity; and even for materials with low coercivity we must not expect that they can take large
frequencies, e.g. in the optical region | ||||||||||

There are materials, however, that still work in the GHz region. | ||||||||||

And that is where we stop. There simply is no general way to express the frequency dependence of domain wall movements. | ||||||||||

That, however, does not mean that we cannot define a complex
magnetic permeability µ = µ' + iµ'' for a particular magnetic material. | ||||||||||

It can be done and it has been done. There simply is no general
formula for it and that limits its general value. | ||||||||||

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© H. Föll (Electronic Materials - Script)