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E&M theory: resonator question

Started by Tim Williams October 9, 2013
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 


On Wednesday, October 9, 2013 9:47:58 PM UTC-4, Tim Williams wrote:
> 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. >
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.
> > Tim > > > > -- > > Seven Transistor Labs > > Electrical Engineering Consultation > > Website: http://seventransistorlabs.com
On 10/9/2013 9:47 PM, Tim Williams wrote:
> 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 >
Are you essentially asking for the solutions of the 2D wave equation on an annulus?
On 10/10/2013 8:57 AM, bitrex wrote:
> On 10/9/2013 9:47 PM, Tim Williams wrote: >> 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 >> > > Are you essentially asking for the solutions of the 2D wave equation on > an annulus?
Subject to the boundary conditions implied by Maxwell's equations, of course.
"bitrex" <bitrex@de.lete.earthlink.net> wrote in message 
news:eaydnUH2kPj9OcvPnZ2dnUVZ_qqdnZ2d@earthlink.com...
> Are you essentially asking for the solutions of the 2D wave equation on > an annulus?
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
"George Herold" <gherold@teachspin.com> wrote in message 
news:bc108024-d3dc-4f03-a4ea-1211ce43bde6@googlegroups.com...
> Or the self resonant frequency of a 'real' torodial inductor.
Yes. Tim -- Seven Transistor Labs Electrical Engineering Consultation Website: http://seventransistorlabs.com
On 10/10/2013 9:47 AM, Tim Williams wrote:
> "bitrex" <bitrex@de.lete.earthlink.net> wrote in message > news:eaydnUH2kPj9OcvPnZ2dnUVZ_qqdnZ2d@earthlink.com... >> Are you essentially asking for the solutions of the 2D wave equation on >> an annulus? > > 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 >
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).
On 10/9/2013 9:47 PM, Tim Williams wrote:
> 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 >
I have a reference that may be helpful: http://www3.alcatel-lucent.com/bstj/vol37-1958/articles/bstj37-6-1599.pdf
"bitrex" <bitrex@de.lete.earthlink.net> wrote in message 
news:Qv2dnT8H2sPsr8XPnZ2dnUVZ_tCdnZ2d@earthlink.com...
> I have a reference that may be helpful: > > http://www3.alcatel-lucent.com/bstj/vol37-1958/articles/bstj37-6-1599.pdf
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
On 10/11/2013 4:25 PM, Tim Williams wrote:
> "bitrex" <bitrex@de.lete.earthlink.net> wrote in message > news:Qv2dnT8H2sPsr8XPnZ2dnUVZ_tCdnZ2d@earthlink.com... >> I have a reference that may be helpful: >> >> http://www3.alcatel-lucent.com/bstj/vol37-1958/articles/bstj37-6-1599.pdf > > 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 >
Here's another more recent paper: http://cdn.intechopen.com/pdfs/42731/InTech-Analyzing_wave_propagation_in_helical_waveguides_using_laplace_fourier_and_their_inverse_transforms_and_applications.pdf Still lots of equations, but there's a section with numerical results at the end and some pictures...