Post
by **Chris Bradley** » Sat Jun 08, 2013 11:09 am

Rich,

You'll find that I've answered and/or already given leads on how to determine those questions, but I'll go over them again here, and see if I can describe how I look at this problem, step by step.

However, just a mention on your electromagnet design first - depending on what you want the field to do, you can also use focussing poles. So if the windings wrap around a pole piece where it has a diameter D, but the piece then necks down conically to a diameter D/2, then you will increase the field level by x4.

But focussing a field to increase the strength with conical poles is not used for cyclotrons (excepting to shape the fields at the edge) because if you crunch the maths, you'll find that it makes no difference to the energy of particles you can accelerate - for a higher field strength it is only in proportion to an increase in the required centripetal force for a given particle velocity/energy. So instead using a focussed field only makes the applied frequencies higher and the engineering for the cyclotron itself smaller, which may be advantages or disadvantages depending on your objectives.

I am not convinced that there is an optimised way of balancing total conductor mass versus power like you said. Whenever I have done the calculation, whatever you do to increase the conductor thickness you reduce turns in direct proportion. All that does is give you control on the voltage/current combination, but not the power.

Your questions -

1. If you want actual measured values on my inverted yoke, then do the search and I'm sure you'll find them. The results tallied with the predictions that Maxwell SV gave me, close enough (which is a freeware student version of Maxwell which I recommend to you to do your analyses).

2. I will talk in 'practical' expressions here fit for an amateur builder, rather than explicitly scientific: A PM magnet has a certain amount of 'magnetic energy' locked into it. Magnetic fields can do no work, so that energy is never 'taken out' of the magnet. All that happens is that the magnetic energy in the magnet is re-distributed around it, according to the relative permeabilities of material in that space.

A given magnetic field in a given space [including the space of the magnet] *is* a given energy. The energy density is defined as B^2/2u (where B is the magnetic flux density [Teslas], u is the permeability). Now imagine you build a yoke like the ones you show here, with a gap of two inches and around one inch wide. Say you attach a magnet of one inch thickness and width to one of the poles so that there is a one inch gap left. What is the field strength?

OK, so imagine how the magnetic flux circuit flows around the yoke. There will be the same flux all around, it's like current the same all around. But now imagine the different parts are like resistors, and the air gap is the bigger resistor where the 'current' does the 'work'. So the magnet has this fixed energy that it will 'share out' to wherever its flux flows. In the metal yoke, the u, permeability, is very high, maybe 10,000's [relative permeability], so for a given B (which we are aiming to work out what it is) the magnetic energy in the yoke is very low. Whereas in the air gap, the u is 1, so the energy of the field in the gap is very high. Remember the B is the same all around the circuit. It's a magnetic flux density. It's like a superconducting current flux (it does no work). Let's say the flux path in the yoke is 10 inches and the u is 10,000. So that means the air gap energy is 99.9% of the total magnetic energy in the loop. We'll ignore the yoke from now on (of course, you can only do this if the yoke is made from high u material).

3. So now we examine the magnet's performance. Answering Q3, you can be sure that neodymium magnets you buy are rated for around the 1.1 to 1.3T range, and that they are like that, because it is simply down to the way that they are made and the nature of the materials. Let's say it is 1.1T, so it's energy density is “1.1T per magnet volume”. Now, IF that 1 inch gap was completely filled with the magnetic flux from the magnet as it flowed around that magnetic circuit, and there was no stray fields bulging outwards, then that'd mean the field level would be approx 0.55T, because that is simply the same volume as the magnet in the same space, and most of the energy of the circuit is focussed in the air gap and the magnet.

It's 0.55T because the permeability of the magnet itself is around 1, pretty much the same as the air gap. It's a bit like a current source with a big internal resistance. It has its own 'u'. The 1.1T 'current' that you see in magnet specs is equivalent to the 'short circuit/maximum current', so to speak. So whatever you do to make the magnetic flux flow around as smoothly and as unobstructed as you can, the magnet itself will limit how much energy density can flow into any air gap because it has only so much energy to 'give out', some of which it 'needs' to support the fields within it, itself.

So if you made the gap 2 inches for a one inch high magnet, the field would be ~0.36 T (ignoring field fringing effects). But if you put two one inch magnets on top of each other and a one inch gap, then there would be '2 units' worth of 1.1T/per inch of flux path now 'flowing' through 3 inches (viz. the two magnets and the one inch air gap). So now the flux would be ~0.7T.

This is saying that the flux through the magnet itself changes, according to the materials around it. A '1.1T magnet' only actually has 1.1T flux in it when it is clamped into a high permeability yoke with no air gaps. If you take a direct measurement of its surface, you might measure the fields as they 'short circuit' directly back into the magnet, and they don't do this evenly, depending on the magnet geometry. So if you had a 'perfect' magnetometer which did not interfere with the field itself, you'd find 'dead spots' on the surface of the magnet where the flux line is heading straight out of the magnet on some wild, long distance path through free space.

Let's go back to the one magnet scenario with one inch air gap. The bulging fields might double the volume if the magnet is one inch wide across a one inch gap. This would result in a, possibly disappointing, ~0.28T for those expecting to 'see a 1.1T field'. If you close that gap up, the relative percentage of 'bulge' to actual air space filled with a field would come down progressively until it can be almost neglected. Say you have a ¼ inch air gap. So now the 'one inch's worth of magnetic energy is being spread around 1.25 inch of permeability = 1 (we are still ignoring the energy in the rest of the yoke, because it will be relatively so low providing it is a high permeability). So now we'll get 1.1T x 4/5 = ~0.9T.

To hang some real numbers off this, sintered neo magnets are around the 300kJ/m^3 range (~N38). If we want to estimate the flux in a yoked air gap of 10 cm^3 total volume, and the total magnetic material in the yoke is 20 cm^3 of 300kJ/m type, then the total magnetic energy in the circuit = 300kJ x 0.00002 = 6J.

We use the formula E = 6J = B^2/2u, where u = 4x10^-7 pi. So B = sqrt{(6 x 2 x 4x10^-7 pi)] /0.00003} = ~0.7T.

Note on units: The energy content of magnetic materials is often quoted as Mega Gauss Oersted MGO. I believe the conversion is 1 MGOe = ~7.95 kJ/m^3. 'N38' *means* 38 MGO, so by definition, for example, N45 would be 357 kJ/m^3. If the energy content of a neo is 357 kJ/m^3, that means* it is *N45, and if it isn't then it isn't an N45!

(Just for completeness, here are the temperature ratings too: H = 120C, SH = 150C, UH = 180C & EH = 200C. Now you know what 'N45 SH' actually means!)

I've not seen discussions on magnets in such simple terms like this. I hope it helps serve to 'demystify' the workings of permanent magnets!