Inflated inner tubes, aluminum tape, putty, dryer ducts, Ikea bowls, I have seen them all used to make DIY toroids for Tesla coils. All yield, in the best of cases, fairly poor results. Functional, but not good looking.
I have, personally, never seen a solid set of instruction on how to build one of those cool "cage" style toroids that some people, ArcAttack, for example, use this kind in their SSTCs.
I offer you my - untested - toroid design.
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Step 1: Modeling the Spiral
"helical toroid" ... "toroidal helix", yeah I think its "toroidal helix".
I did my modeling in Grasshopper, but the concept would work in any CAD program that allows scripting.
Maybe, scripting the helix in Processing and exporting it would work too.
The equations above is fairly straightforward. The outputs are the X, Y, Z coordinates, from top to bottom.
When describing Tesla coil toroids it's done in terms of diameters, A being major, C being minor.
The equation does things in terms of radii.
The last thing of note is the "sample range", that is the resolution. My example uses around 700 points, the more the better. I then interpolate a curve through the points to make my geometry.
The number of turns was not important to me. I kept them as dense as possible, with a separation between turns of 0.5", that is, so the teeth of the combs that hold the whole thing together are .25"x.25".
Step 2: The Results You Are Seeking
The way you should make the profiles is relative to the inside measurements of the coil... I'll let the image explain.
Here are my profiles.
They are really similar, but the slight slant and offset in the teeth really pays off when putting the coil together. The files attached are a 4.5" minor diam 18" major diam toroid, with a 1.5" separation.
"top" and "bottom" are kind of meaningless, so long as they work together with the right tooth offset it makes no difference.
Step 3: Laser Cut Them!
Allways laser cut in cardboard first in case there is something wrong!
You don't want to waste precious acrylic.
Step 4: Make the Aluminum Coil
Get a pipe that proximates your desired coil diameter.
I got an abs pipe with a 4.5" exterior diam.
Turns out that yields a larger coil, I got a 5" factoring in the width of the actual aluminum tubing... but it worked just the same.
I made some holes and twisted some wire through some holes to hold the aluminum to the abs.
Tip: keep the pipe steady and work the spool of tubing around the pipe. You want to keep the curvature that the tubing already has, the more you work it the harder it gets. Not to mention kinks won't ever go away.
Step 5: Putting It All Together.
How do people manage to pause and take pictures?... once I'm on a roll I always forget.
You start by... finding that the 3d modeled spiral and the spiral you wound..aren't in the same direction...but do not panic...its an easy fix. Just flip and turn your combs, and it will work out.
Make sure, when you bolt them together the comb tooth offset it the right amount.
Once all the orienting is right, and you have spacers and bolts in your combs you can start assembling.
All those holes around the edge are for zip ties... I thought the coil would resist a lot more, and that I would need a zip tie per turn, but once you stretch and ease the coil in, it almost holds itself in place.
In my case, I had to do 2 coils and joined the ends with shaft couplers to get all the way around. I thought it was going to be a problem but they came together easily.
Use a tool that cuts really cleanly, and doesn't leave burrs or deform the tube, otherwise, the ends won't fit into the shaft coupling.
Step 6: You Just Saved Yourself ~200 Dollars!
Here it is, posing with my secondary coil.
If you got the money, a smooth toroid is still the nicest.
But this is a really nice solution for much less money.
For coils up to 750kV, 18" x 4.5" with ¼" Center Hole (no seam) ... $312.50
The 18" toroid (with a visible seam) ... $169.95
Those prices are from https://www.amazing1.com/toroids.html