Home: https://ky8d.net/spiral
Series:
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[2]
[3]
[4]
[5]
Iteration Set #1: Firstly I'll iterate over a range of spiral expansion factors (symbol '𝜏', governed by the CLI arg '--tau').* Then after seeing the plots, I'll know the cost/benefit ratio for wire length in terms of gain, radiation pattern and take-off angle.
* If confused, it's likely because you arrived here other than via the home page for this series. Click on this page's top-most link.
Color Key for Paragraph Text: White (greater contrast) presents information crucial to interpretation. Red presents a delving into various fiddly details about the surounding main text in white. Stuff which I feel some readers may find self-evident, uninteresting, or even beside-the-point. Tidbits I couldn't convince myself to simply delete.
Update! Spiral-expansion ratios (for which I here employ the Greek letter '𝜏' tau) are quantifiable plural ways. Two such would be: A) ratio of radius-lengths between any two equal divisions of a circle; or B) the ratio of two (overlapping) radii such as define a full circle. Formerly I had employed A. Since then, however, I've switched to B. Not that the antennas themselves care either way. This change is apparent only from the perspective of someone using my program to design spiral antennas, outputting those as *.nec and *.ant files. Think of the change being as if I once gave out city maps marked in miles, then later reissued the same maps with kilometers showing in addition. But still with the names of those maps titled as 'miles'. So it is that plots displayed here now show both ways to quantify tau while keeing their original file names. Hoping that helps.
Thus you will find that no longer do graphics simply say merely 'tau', alone by itself, but specify accordingly how the ratio is measured. Whether according to system A, as measured between adjacent radii, thus: tau(45°). Or instead according to system B, as measured on a unit circle, thus: tau(360°). Only just here, though. Not anywhere else. All my future usage of the term tau in connection with spiral antennas shall employ the full-circle quantifier.
Why the change? My antenna-generating program must necessarily solve spiral geometries from one radius intersection to the next. Hence method A. Later, however, I discovered that basing 𝜏 on a unit circle isntead makes more sence conceptually. Doing so segregates spiral tightness from any dependency upon the number of radii. It wasn't until conducting a different experiment (this time iterating on --div) that I made the discovery. My penance for this lack of forsight was three hours consumed in redacting 284 graphics. Happily all were SVG format, and so easily done. File names, however, remain as before.
Note: If two SWR curves show, each for a different Zin, 1st represents lowest average, 2nd the lowest absoute.
Info: These SWR plots are produced by a Perl script I wrote which reads a table of NEC data generated by Nec2Go. It's a somewhat smart Perl script in that it makes its own choice of which Zin to plot SWR cruves against. The script has a wide range of choices, as follows: 25Ω , 50Ω , 75Ω , 100Ω , 150Ω , 200Ω , 300Ω , 450Ω , 600Ω , 800Ω , 900Ω , 1200Ω , 1600Ω . I afforded it just only those for the obvious reason. They represent values of Zin for which a balun can be constructed to match either 50Ω or 75Ω feedlines.
My Perl script therefor wholly ignores that Zin value supplied by the user to Nec2Go when making the sweep and instead tests against all the above. Then in accordance with its ownfindings, makes a choice of one or two Zin values to plot against. The script will plot for both the lowest average value of SWR and also the lowest absolute SWR. Like that because I thought both might be useful to know.
Thus do I continue on with yet further plots, even though already we've crossed above 2:1 on part of one sweep. I am curious as to whether I might discover a pleasant surprise. It does seem rather unlikely, but still I must put the nagging question to rest. Will the slowly loosening spiral unwind its way into a lower, more useful range of Zin? Let's find out.
A Conflict of Parameters: Not really a conflict, per se. Just something I didn't foresee (although I should have). Not until graphing the data did it hit me over the head.
My Perl script's spiral-generating algorithm enforces a rule that wires must terminate at a polygon radius. This for convenience of construction. In cases above, an increase of tau45° by 0.005 per iteration allowed for steady wire terminations always at the next radius anti-clockwise. For values of tau45° > 1.1, this is sometimes not the case. False economies can therefor result: small increases of both OD and wire length. Like so for one or more iterations until the spiral is able to meet or cross its max-frequency lambda/2 diameter at a point clockwise of the next polygon radius.
Two approaches alone would have prevented this: (A) increasing the magnitude of iteration steps to greater than 0.005; or (B) better approximating a circle by employing a higher-order polygon. Neither means of redress to the problem have been implemented here. And in any case, both are just kicking the can a little ways further down the same road. And likely why my Internet searches failed to locate a single other instance of log-spiral antennas used for HF. Being both curious and sturbborn, though, I'll persist further yet.
Similar occurances manifest plurally as the progression continues. I will group and lable these.
1st group of 2 terminating at a common 1-of-8 radius boundary.
2nd group of 2 terminating at a common 1-of-8 radius boundary.
3rd group of 2 terminating at a common 1-of-8 radius boundary.
4th and final group of just only 2 terminating at a common 1-of-8 radius boundary.
1st group of 4 terminating at a common 1-of-8 radius boundary.
2nd and final group of 4 terminating at a common 1-of-8 radius boundary.
All spirals remaining terminate at a common 1-of-8 radius boundary.
A value for tau45° of 1.20 marks the upper boundary for this set of iterations.
So I guess that's it, with no fortuitous circumstances of SWR having arisen. Just possibly it might have done had I divided the circle into more divisions than just only eight. False economies (in terms of wire length) would also have proven fewer, as groupings into two-or-more all sharing a common radius boundary would not have begun to occur until further along in the progresson.
Reminder: My Perl script's spiral-generating algorithm enforces a rule that wires must terminate at a polygon radius. In cases above, an increase of tau45° by 0.005 per iteration allowed for steady wire terminations always at the next radius anti-clockwise. For values of tau45° > 1.1, this is no longer the case. False economies sometimes result: small increases of both OD and wire length. Like so for one or more iterations until the spiral is able to meet or cross its max-frequency lambda/2 diameter at a point clockwise of the next polygon radius.
Lone Wolf Approach: Here yet again, I have gone my own separate way. But, as elsewhere, I arrive at much the same place come the end. I chose to employ an expansion factor which (for lack of knowing a better term) I'm calling tau. Employing that identifier so as to maintain commonality with log-periodic arrays.
Paramter Change: As explained near the top of this web page, during this experiment only do I specify functionally-identical but numerically-different values for tau. An old way, now abandoned, is to specify tau as the expantion ratio between spiral arm intersections of adjacent radii. Which here worked out to ratio increases of 0.005 every 45° (the --div param having been called out as 8). The new way is to specify tau equivallently as the ratio once per full turn (every 360°). The change is made so as to allow for iterations on the --div param to produce spirals all wound with similar tightness based only upon tau by itself.
The relationship between these two is as follows. If --div 8 should be in force, the number of radii will be 8, spaced 45° apart. The value for tau(360°) is obtained by raising tau(45°) to the power of 8. My program handles it in reverse. It will accept user's inputs of --tau 4.3 and --div 8, knowing to re-generate tau for its own use as 8th root of 4.3. Or were those inputs instead --tau 4.3 and --div 6, then the 6th root of 4.3. And in either case, spiral-tightness will come out the same, just segmented into either 8 or 6 straight lines. Clear as mud?
This experiment having already been performed, to allay (possibly augment?) confusion, I have redacted 284 graphics. Any as formerly presented just simply tau now display that value as tau(45°) And wheresoever space did allow, alongside it appears the newer, more universal, functional equivalent labled as tau(360°). Never elsewhere will you see that. Only just here. Forever more the new method will display as just simply tau.
Construction Minutea: Central wires all measure lamda/2 at 50MHz. This so as to be having elbow room at the high end. Half of that is the inner radius, that measure which my design program first multiplies by it's on interpretation of tau ... in this case the 8th root of --tau since my choice for --div I'd chosen 8. Thus is chosen how far out along the next anti-clockwise radius the segmented spiral shall extend towards. And so on, and so forth.
Connections continue spiraling outward until a distqance equal to lambda/4 of the low-frequency limit (in this case study 13.5MHz). But as this anti-clockwise position would not likely fall on 45° exactly ... and we don't want the wire left dangling ... my program extends the wire so as to interesect the next anti-clockwise radius. Always this results in a false economy in terms of wire length.
The files themselves were batch-generated by a custom Perl script of my own devising. Said script can also output conical spirals. It likewise outputs text-macro files compatible with the 3D CAD platform Rhinoceros 6. Output from Rhino as SVG, I thereby obtained spiral geometries for visual display (after tweaking in Inkscape). Thus all are exactly to scale, each with the others. Other Perl scripts call Nec2Go via command-line, generate SVG plots from Nec2Go data, etc.
Note: Only even-numbered divisions of a circle ever prove practical. Any employment of odd divisions always results in a geometry greatly inconvenient. The number of required radii for odd values is always double. Below I diagram the such a result. In this example tau45° = 1.100 while div = 9.
Firstly: All taken with all, I was hereby inclined to repeat the experiment for iterations of tau45° starting at 1.09 (tau360° 1.993) for the sake of wire economy, this time dividing the circle into ten radii. And hence did I discover the non-universiality of system A, revising my program to system B. Live and learn.
Secondly: Much more economically, I could instead consider setting aside all concerns regarding Zin and SWR, construct a log-sprial antenna according to tau45° = 1.180 (tau360° 3.759) without including a balun, supplying it instead with balanced parallel feedline (one wire either side of the spiral, wires of the spiral passing between always at perfect right angles), and tune for an impedance match from inside the shack.
I could save a great deal of wire with miniscule sacrifice of radiation pattern, take-off angle, and gain. So, I think, I'll stick with that. And for simplicty's sake, henceforth employ a CLI arg of '--tau 3.7'. The expansion factor now having been decided, next next I'll try varying curve resolution. That is to say, with a range of less and more flat sides per full turn.
This now decided, next next I'll try out spirals of varying resolution.: Iterating on Div.