4

I'm trying to use the parametric equations for the sinky curve to construct a meshable model for 3D magneto-static FEA of a transformer coil. I have the modellling and meshing covered but, I need to intertwine two conductors (twisted pair) and whilst I have gotton close by trial and error, I've yet to get the two conductors to intertwine closely without intersecting. (The red areas in the image below show the slight intersection between them.) enter image description here

So my question is, can a mathematician or two, help me avoid that intersection?

The equations used are:

x = ([r]+[a]Cos([w][t]))*Cos([t])

y = ([r]+[a]Cos([w][t]))*Sin([t])

z = [h]*[t]+[a]Sin([w][t])

Where:

a = diameter of the spiral = 0.25

h = the slope angle of the helix = 0.08 (1 translates to 2pi in the z per revolution)

r = the radius of the helix - 3.95

w = the number of turns of the spiral per revolution of the helix = 4

t runs from 0 to 14 pi.

These parameters give the curve below, and second copy rotated 45° gives the intertwine:enter image description here

The software that generates the model and detects the interferance is using math to do so; but it is closed source.

Can a mathematician show me how to adjust the parameters I am using to avoid the interferance?

I realise that this is couched in 'engineering terms', but I am an engineer. I've provided as much of the math as I understand; can you nudge me the rest of the way?

Buk

Update: Thanks to Jyrki's assistance below I've now produced my model of the transformer coil:

enter image description here

Buk
  • 91
  • 1
    I think I can do something easily using the parametrization of a tubular surface around the helix. The one I described here. – Jyrki Lahtonen Nov 01 '21 at 16:26
  • 1
    Is the following mental image about right: imagine that tube around a helix, the have the pair of wires run along its surface, in such a way that they are always on opposite sides of the tube? – Jyrki Lahtonen Nov 01 '21 at 16:28
  • 1
    Of course, making the tube thinner, and the helix rise much less per revolution than in the linked images. – Jyrki Lahtonen Nov 01 '21 at 16:31
  • 1
    A possible cause of your problem is the 45 degree rotation. Exactly how you do that? It won't necessarily achieve what I think is necessary? – Jyrki Lahtonen Nov 01 '21 at 17:13
  • 1
    If I diagnosed the problem correctly, the following might work: Use your chosen value for $a$ in your formula to get one of the wires. Then, instead of the 45 degree rotation, use the same formula but with $-a$ instead of $a$ for the other wire. Posting images and adding an explanation ASAP. – Jyrki Lahtonen Nov 01 '21 at 17:18
  • It would never have crossed my engineer's mind to use a negatve diameter! The result of just that single change has intersections, but a single width-of-wire move up or down of one wire corrects that. Still not perfect as the results is more interweaved than intertwined, but a great start. I'd post images, but these silly comments don't let me. – Buk Nov 01 '21 at 19:46
  • 1
    You can either edit the question or post an answer, if you want to add more images. – Jyrki Lahtonen Nov 01 '21 at 20:43
  • 1
    Changin $a$ to $-a$ does not really give you a negative diameter. It's effect is more like shifting the phase of that additional $(\cos\omega t,\sin\omega t)$ wave to an opposite one. There is no need for that to be in synch with the underlying helical coil $(r\cos t,r\sin t,ht)$. – Jyrki Lahtonen Nov 01 '21 at 20:48
  • 1
    By the way, what's the difference between interweaving and intertwining? The continuous contact? That can be arranged, I think. – Jyrki Lahtonen Nov 01 '21 at 20:50
  • @Jyrki Lahtonen Re: "negative diameter"; I get that -- just draw the curve that becomes the path for my wire, 180° around the encompassing tube circumference -- having read your explanation below; but I would not have thought of it that way to try it on my own. – Buk Nov 01 '21 at 23:41
  • 1
    @Jyrki Lahtonen Re: interweaving and intertwining? Intertwinging: simple twisted pair. Interweaving: One wire crossing/wrapping more that one other. Something like plaiting. Something like type 7 litz wire: https://i.stack.imgur.com/eIUN5.png The reason for the twisting is similar to the use of Litz wire; avoidance of capacitive and proximity effects. – Buk Nov 01 '21 at 23:47
  • 1
    Ok. That's beyond me, but you (and undoubtedly others before you) have made the observation that twisting is beneficial. Multiple layers, too! Glad to hear you found something that works. – Jyrki Lahtonen Nov 02 '21 at 04:21
  • Let us continue this discussion in chat. – Buk Nov 02 '21 at 15:34

1 Answers1

2

I think the problem can be solved by using the tube around the helix from an old answer and then placing the two wires on diametrically opposite sides of that tube to prevent contact.

Here's what it looks like. First from the side:

enter image description here

Then a view from above:

enter image description here

On with the explanation. There is a helix curve in between the red and blue wires in my image. It has the usual parametrization $$ \vec{r}=(R\cos t, R \sin t, ht), $$ where $R$ is the radius of the tube the helix sits on, and the curve will rise $2\pi h$ per revolution (using a negative $h$ reverses the handedness of the screw).

Differentiation w.r.t. $t$ gives the tangent vector along the central helix $$ \vec{t}=(-R\sin t, R\cos t, h). $$ We record that this has constant length $\sqrt{R^2+h^2}$. We see that the derivative of the tangent, after normalizing to unit length, gives $$ \vec{n}=(-\cos t,-\sin t,0) $$ as the normal (perpendicular to $\vec{t}$). The binormal (again normalized) is the cross product $$ \vec{b}=\frac1{||\vec{t}||}\vec{t}\times\vec{n}=\frac1{\sqrt{R^2+h^2}}(h \sin t,-h\cos t,R). $$ We see that $\vec{n}$ and $\vec{b}$ are perpendicular to each other and to $\vec{t}$. Therefore the points given by vectors like $$\vec{r}+ a\cos u\vec{n}+a\sin u\vec{b}\qquad(*)$$ $0\le u\le 2\pi$ form a circle centerd at $\vec{r}$ and perpendicular to the helix. By also varying $u$ they thus parametrize that tubular surface. We also see that by replacing $u$ with $u+\pi$ we get two points on the opposite sides of the helix. We see that the same effect is gotten by replacing $a$ with $-a$.

To get the wires intertwined around the central helix you want $u=k t$ for some parameter $k$. This is thus my suggested answer:

wire number 1: $$ \vec{r}+a\cos(kt)\vec{n}+a\sin(kt)\vec{b}, $$ wire number 2: $$ \vec{r}-a\cos(kt)\vec{n}-a\sin(kt)\vec{b}. $$ This way they are separated from each other by a constant distance $2a$ (assuming that $2a$ is smaller than the rise per revolution $2\pi h$).

When $R$ is much bigger than $h$ you may get away with using $\vec{b}'=(0,0,1)$ instead of the 'technically correct' binormal $\vec{b}$ I used. The distortion won't probably be much. It seems to me that the formulas you showed do just that.

In my images, $R=10$, $h=1/\pi$, $a=0.6$, $k=4$.

Below there is a small section of the above image with a partially transparent (Opacity[0.6] in Mathematica) tube between the red and blue wires.

enter image description here

Jyrki Lahtonen
  • 133,153
  • Thank you Jarki. With your information I've succeeded in producing my transformer coil model. See the OP for an image in and out of it pot core. I still have some work to do, to connected the layers and break out some taps on the primary. I'd love to explain further, but these comments are not conducive to discussion. – Buk Nov 01 '21 at 22:35