How do I work out an $X$-$Y$ position for a given length around a spiral?

Question

I'm attempting to simulate an antenna, based on this paper.

The antenna is a spiral with a track width of $1$mm and a gap of $0.5$, with a shorting pin inserted at a given length around the spiral, e.g. $58$mm.

I have extracted the following details regarding the spiral:

• inner diameter: $1.5$mm
• Number of turns: ~$4.3$
• Width of the track: $1$mm
• The gap between tracks: $0.5$mm
• Length of the full Spiral: $104$mm

As the shorting pin wants to be central to the track, I believe we can assume a line spiral with the distance between each turn being $1.5$mm.

I wish to be able to alter the tuning of the antenna to do this I'll need to be able to move the shorting pin around the spiral. The simulation software that I am using is parametric, so if I know the formula, and constants (for this case) I would be able to specify the length and be provided with $(X,Y)$ coordinates. Unfortunately, I'm not that familiar with spiral mathematics so I would appreciate some help to calculate what is needed.

EDIT: I've looked further and I'm aware that I can easily convert from polar to cartesian, and from polar I can use

$L=\frac{\alpha}{2}*(\theta*sqrt(1+\theta^2)+ln(\theta+sqrt(1+\theta^2)))$

where for the spiral that I'm looking at ${\alpha} = \frac{0.75}{\pi}$

I'm just not sure how to rearrange this to give me $\theta$ as the desired value.

Thanks!

Answer 1

Since the gap between tracks is a constant, I assume this spiral is created by semicircles of increasing diameter being attached so that the curve is not discontinuous.

Length on one turn = $\pi r_n$

where $r_n = r_{n-1} + \frac{d}{2}$

$d$ is the distance between tracks

Hence, length of spiral for $N$ turns is

$$L_t = \sum_{i=0}^N \pi r_i$$

You can find this using sum of an arithmetic progression

Now, given a track length, you need to find the turns to reach it. Once you do, you can calculate the new centre of the current turn, and use the length difference to find the coordinates of the point w.r.t the centre, and then shift to the origin using translation