4.3.5 Magnetic Losses and Frequency Behavior

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.
	In contrast to dielectric materials, we will start with looking at magnetic losses first.

Hystereses Losses

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 currents. This is different from dielectrics, which we always took to be insulators: ferromagnetic materials are usually conductors.

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 f in both cases, because every time you change the field you incur the same losses per cycle.

2. The maximum magnetic flux B_max in both cases.

3. The conductivity s = 1/r for the eddy currents, and

4. The magnetic field strength H for the magnetic losses.

More involved calculations (see the advanced module) give the following relation for the total ferromagnetic loss P_Fe per unit volume of the material

P_Fe	»	P_eddy + P_hyst »	p · d² 6r	· (f · B_max)² + 2f · H_C · B_max

With d = thickness of the material perpendicular to the field direction, H_C = coercivity.

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 d 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.

We will not consider eddy current losses further, but now look at the remaining hystereses losses P_hyst

The term H_C · B_max 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.

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?

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.

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. More to that in an advanced module.

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. Some information about the complex magentic permeability is contained in an advanced module.

Questionaire
Multiple Choice questions to 4.3.5