## Different polyhedra require different strength magnets

Discuss the technical details of an "open source" community-driven design of a polywell reactor.

Moderators: tonybarry, MSimon

alexjrgreen
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### Different polyhedra require different strength magnets

The distance from the centre of a face to the centre of a polygon varies, and this affects the strength of the magnets you need to build a polywell.

For polyhedra of side 1m, that "inradius" is

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``````tetrahedron	0.204124m
cube          0.500000m
octahedron    0.408248m
dodecahedron  1.113516m
icosahedron   0.755761m``````
However, if we want to compare equal sized coils, a coil of radius 1m fits inside a triangle of of side 3.464102m, a square of side 2m or a pentagon of side 1.453085m.

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``````tetrahedron   0.707106m
cube          1.000000m
octahedron    1.414213m
dodecahedron  1.618033m
icosahedron   2.618033m``````
This makes a big difference to the strength of field applied at the centre of the polygon.

For a Helmholtz coil of radius R, made of n turns of wire carrying a current I, the axial field B at distance x is

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``````     mu_zero*n*I*R^2
B = -----------------
2*(R^2+x^2)^(3/2)
``````
If R = 1m as above, then the relative field strength b = B/(mu_zero*n*I) is

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``````tetrahedron   0.272166
cube          0.176777
octahedron    0.096225
dodecahedron  0.072654
icosahedron   0.022716``````
and the relative field applied by all the coils at the centre of the polygon is

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``````tetrahedron   1.088663
cube          1.060660
octahedron    0.769801
dodecahedron  0.871852
icosahedron   0.454314``````
From which it appears that a tetrahedral configuration is slightly preferable to a cube, and probably somewhat cheaper.
Ars artis est celare artem.

rcain
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... and fewer cusps to worry about.

( i suggested tetrahedrons were nice (without the numbers of course, and for different reasons), some while ago - but people said they were a rubbish idea because it was the same as a truncated cube. well, there you go, it isn't, 'quite' )

D Tibbets
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Actually a cube , or if you prefer a truncated cube has the fewest cusps. Machines with more faces have more cusps, but the argument is that the losses per cusp decreases faster than the number increases. Perhaps by a factor of 3-5 X based on R. Bussard's predictions (other facters may contribute to his prediction).

The numbers may be changed a lot based on the shape of the coils, more square, thicker, etc.

Also, keep in mind that line cusps leak a lot more than point cusps (in the center of the magnets). This is modified by the chariteristics of the Polywell, but the corner/ funny cusps will never be quite as good as point cusps. Add to that the interconnects/ nubs that hold the magnes together. These may be a major factor in calculating the realative losses.

Dan Tibbets
To error is human... and I'm very human.

alexjrgreen
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I'm intrigued. A cube has six centre cusps and eight corner cusps.

Surely a tetrahedron has four centre cusps and four corner cusps?
Ars artis est celare artem.

KitemanSA
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rcain wrote:... and fewer cusps to worry about.

( i suggested tetrahedrons were nice (without the numbers of course, and for different reasons), some while ago - but people said they were a rubbish idea because it was the same as a truncated cube. well, there you go, it isn't, 'quite' )
If someone (heck, could have been me on one of my sloppier days) said that a rectified tetrahedron was the same as a rectified cube, they would be wrong. A cube truncates to an OCTOhedron, not a tetrahedron. What truncs to a Tet is ... a Tet!

rcain
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... that may have been my mistake, i cant recall 'exactly' what they said, it was some while ago. But, the idea of using tetrahedra was rubished as 'passe' at the time. I was in favour of exploring further, since they seemed simpler to analyse (geometry, charge, fields, cusps, etc - at least for the pupose of a simplified model); and here is Alex suggesting they do indeed possess an advantage (also) on Bfield (inradius). which is nice.

re. corner cusps leaking more than face cusps - has always made some sense to me. (imagine attempting to make a sealed polywell sphere by origami!)

the question as to whether more cusps (higher order polyhedra), end up leaking less in total than a configuration with fewer cusps - that was a question of interest to me/possible counterintuition. I note though, that Bussard seems to have factored such an analysis in to his calculations/discounted it in favour of more perfect sphericity - hence, i read, more efficient WB formation/cusp constriction.

ps. i thought we had all more or less agreed, there are no real 'line' cusps' in a (cubical) polywell. only point cusps and 'funy'/corner cusps.

KitemanSA
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I wrote: If someone (heck, could have been me on one of my sloppier days) said that a rectified tetrahedron was the same as a rectified cube, they would be wrong. A cube truncates to an OCTAhedron, not a tetrahedron. What truncs to a Tet is ... a Tet!
Well, I went back and thought about this a bit more and something funny occured. When truncating a cube one eventually reaches an octahedron. Half way there (i.e., at rectification) one achieves a cuboctahedron which is a polywell shape (i.e., having an even number of faces at each vertex). The same when truncating an octahedron, but in reverse. When truncating a TETRAhedron, one eventually reaches another tetrahedron. But the rectified state is... wait for it... an octahedron, the only Platonic solid that is a Polywell shape. Neat, no?

tombo
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Hmmmmm...
Good thoughts but.....

Code:
tetrahedron 0.707106m
cube 1.000000m
octahedron 1.414213m
dodecahedron 1.618033m
icosahedron 2.618033m
and the relative field applied by all the coils at the centre of the polygon is

Code:
tetrahedron 1.088663
cube 1.060660
octahedron 0.769801
dodecahedron 0.871852
icosahedron 0.454314

From which it appears that a tetrahedral configuration is slightly preferable to a cube, and probably somewhat cheaper.
This says that for the dodec the field is half that of a tetrahedron but that the reactor is almost 4 times the radius or 4^3=64 times the volume. That says to me that the dodec is better. (not counting cusps and assuming the fields add up in a simple manner at the center and assuming the the field at the center point of the overall reactor is the important field location to compare etc.)

IMO:
The calculation should be normalized to the overall reactor radius.
The important field location to compare is near the magrid where it shields the magrid from ions and where it strengthens the cusps.
That probably requires some FEA.

I see it as a trade-off between higher fields reducing the losses (for a given maximum current) due to smaller coils vs more cusps increasing losses. (I'm pretty sure Art C. had some numerical arguments along those lines.)
-Tom Boydston-
"If we knew what we were doing, it wouldn’t be called research, would it?" ~Albert Einstein

alexjrgreen
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If you shrink the dodecahedron to the same size as the cube, each coil is going to have a smaller cross section and so carry a smaller maximum current. That will make the field at the centre smaller still.
Ars artis est celare artem.

charliem
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alexjrgreen wrote:If you shrink the dodecahedron to the same size as the cube, each coil is going to have a smaller cross section and so carry a smaller maximum current. That will make the field at the centre smaller still.
Smaller cross section, yes, but smaller diameter also, compensating, at least partially, the magnetic strength loss.

MSimon
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The field at the center of the device is zero.

===

A tetrahedron is not so hot due to no opposing faces.
Engineering is the art of making what you want from what you can get at a profit.

KitemanSA
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MSimon wrote:The field at the center of the device is zero.
===
A tetrahedron is not so hot due to no opposing faces.
He keeps saying tet, but what he describes is a quasi-rectified tet. A rectified tet is an octahedron, which is the polywell that Dr.B. patented. Opposing faces are not required, only even numbers of faces around each vertex. The octahedron (rectified tet) satisfies that.

However, I still think more is better in that sphericity improves.

TallDave
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http://en.wikipedia.org/wiki/Truncated_cube

http://en.wikipedia.org/wiki/Rectified_cube

http://en.wikipedia.org/wiki/Dodecahedron

http://en.wikipedia.org/wiki/File:Recti ... hedron.png

The WBs appear to be truncubes or cuboctahedrons. Well, technically the coils are round so they aren't anything.
Opposing faces are not required
What happen to the field between unlike faces? Wouldn't it gradually fall to zero at the middle where the two sides cancel? That would mean no cusp and no confinement.

KitemanSA
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TallDave wrote: The WBs appear to be truncubes or cuboctahedrons. Well, technically the coils are round so they aren't anything.
A fully truncated (aka rectified) cube is a cuboctahedron. So is a rectified octahedron for that matter! And Dr.B. didn't really like the round magnets. He wanted a rounded corner square plan-form magnet for the cuboctahedron. But round is so much easier to make, in most peoples minds.
TallDave wrote:
I wrote:Opposing faces are not required
What happen to the field between unlike faces? Wouldn't it gradually fall to zero at the middle where the two sides cancel? That would mean no cusp and no confinement.
I am not quite sure what you mean by "the middle" here. Opposing faces are not required. Most folks think that if there is a "north in" magnet on one face, there needs to be an "north in" on the opposing side of the poyhedron, thru the center of the volume (reactor). This is not so. All that needs to happen is that there is a vector sum to zero at the center of the reactor. That can be done in many ways without directly opposing faces.

But the main feature of a Polywell is that there are an even number of faces (pairs of opposite polarity) around each vertex. The faces may be either all real, or half real and half virtual. But always two or more pairs of fields.

charliem
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KitemanSA wrote:... the main feature of a Polywell is that there are an even number of faces (pairs of opposite polarity) around each vertex. The faces may be either all real, or half real and half virtual. But always two or more pairs of fields.
Have to admit that I still fail to see the meaning of that.

In a cube, cuboctahedron, or dodecahedron I can only picture 3 faces touching each corner (or vertex), neither 2 nor 4, so... what am I missing?

And what exactly means "of opposite polarity"?

May he have meant edge instead of vertex?