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E&M theory: resonator question

Has anyone ever seen analysis, formulas, data, etc. concerning (if it's even a word) helicotoroidal resonators?

Banal example: any toroidal inductor, single layer winding. Example: chokes with single layer windings, most CTs.

The simplest case ought to be the thin toroid (the physicist's old standby): if the properties of a thin (or infinite) solenoid (helix) are known, it should be easy enough to apply periodic boundary conditions, making it into a loop (a thin toroid). So instead of infinite propagating modes, standing waves occur.

The frequencies of those standing waves will depend on the dispersion of the helix, which I understand is not the same as an ideal transmission line, so they won't be a harmonic series. I would SWAG the resonances occur at Bessel function zeroes, or something like that. But that doesn't help much. More importantly, they will depend on geometry and stuff.

I would of course be most interested in what an actual toroidal winding (of finite size and thickness, wire and turns, all around a permeable core of known properties) does, but if I can hand-wave some ideas it would be great.

Tim

--
Seven Transistor Labs 
Electrical Engineering Consultation 
Website: http://seventransistorlabs.com
Reply to
Tim Williams
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Hi Tim, I'm a bit confused. Do you want the cavity modes for a toroid? (Something like a donut covered with copper.) Or the self resonant frequency of a 'real' torodial inductor. Or something else?

George H.

Reply to
George Herold

Are you essentially asking for the solutions of the 2D wave equation on an annulus?

Reply to
bitrex

Subject to the boundary conditions implied by Maxwell's equations, of course.

Reply to
bitrex

Yes, but one wound with a helix of conductive material.

If it were simply the various modes inside a toroidal cavity, or a pipe bent 'round, even accounting for permeability of the core, resonances would be through the roof -- the fact that it's wound, potentially with hundreds or thousands of turns, can push those modes down into the 100s of kHz -- which you can imagine isn't good news for chokes or transformers operating in that range.

The characteristics of a helicotoroidal resonator proper (probably one inside a shielded box, with no permeable core), optimized for Q and size, might be interesting for RF purposes, but I would guess because only full standing waves are permitted, such a design will be larger than a regular old helical resonator (which permits 1/4 wave modes).

Apparently there's such a thing as a two layer counter-wound toroidal antenna (Corum and others).

Tim

--
Seven Transistor Labs 
Electrical Engineering Consultation 
Website: http://seventransistorlabs.com
Reply to
Tim Williams

Yes.

Tim

--
Seven Transistor Labs 
Electrical Engineering Consultation 
Website: http://seventransistorlabs.com
Reply to
Tim Williams

I think the case of a 2 dimensional annulus wound into a helix might be able to be solved analytically. Take the 3 dimensional wave equation in polar coordinates - the boundary conditions would be periodicity in theta with boundaries at the top and bottom of the helix, boundaries in R at the edges of the annulus, and periodicity in Z, with the Z axis terminating after however many turns of the helix. In addition the tangential components of E and the normal components of H must vanish at all the boundaries of R (edges of the helix) and theta (top and bottom of the helix).

Reply to
bitrex

I have a reference that may be helpful:

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Reply to
bitrex

Oooh, good ol' BSTJ. Thanks!

Yikes, pages and pages of equations... and it looks like they just dive right in. I'd have to read a lot of E&M and microwave stuff to get there.. hmm..

Tim

--
Seven Transistor Labs 
Electrical Engineering Consultation 
Website: http://seventransistorlabs.com
Reply to
Tim Williams

Here's another more recent paper:

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Still lots of equations, but there's a section with numerical results at the end and some pictures...

Reply to
bitrex

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