Two theoretical optimal antenna questions

Discussion in 'Antennas, Feedlines, Towers & Rotors' started by M0AGP, Jul 28, 2020.

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  1. M0AGP

    M0AGP Ham Member QRZ Page

    Very interesting! I hope Ma has published some papers with his method - at $145 that book looks a bit pricey.

    I wonder if one could add a constraint or modify Ma's approach to get past the ill-conditioned nature of his optimal solution?

    Maybe optimize the average of a statistical ensemble of antennas where the currents across the ensemble for a given element are caused to fluctuate by1% on average.

    The result might be lower performance but higher error tolerance...
  2. N3OX

    N3OX Ham Member QRZ Page

    Paperback's better :) But if you click through to the Google Colab notebook, I've got some equations. The basic thing is to compute the matrix B and the vector e and then the optimal vector of currents is i_{opt} = B^-1 e

    I had some bad equation renders in there but I think I fixed them.

    Yeah I think adding some noise would be a great help.
  3. M0AGP

    M0AGP Ham Member QRZ Page

    OK - given that there is a chance this super simple calculation I have done may be interesting to look at (not a substitute for EZNEC type calcs of course), I thought to post a couple plots from it.

    For all the idealizations and approximations involved, please see my earlier post in this thread.

    So the array I am looking at is the 8-circle array which has 8 radiators spaced a quarter wave from its nearest neighbor, arranged in a circle. Please excuse the ugly Excel graphics (shown for 20 meter band and distance is in meters):
    Using equal currents in the eight radiators but optimizing phases for maximum directionality I find this azimuthal pattern:
    Adding in the third dimension I find the radiation pattern in the vertical plane looks like this:
    So the pattern looks a bit like a beached whale :)

    The units on these charts is supposed to be proportional |E|^2 and putting all the current into a single radiator gives you a circular pattern with magnitude 1.00. The radiators are isotropic.

    I guess therefore that "8" should correspond to a bit over 9dBi.

    The half-power beam width is about 64 degrees (directly at the horizon).

    The optimal phasing I find is (for the eight radiators, going counter-clockwise from the one directly in the "beam path":
    0, 83, 118, 83, 0, -83, -118, -83.

    From ON4UN's book "Low Band DXing", section 4.14.2 "Optimizing phasing, 8-Circle array" the maximum gain is stated as 9.2 dBi.
    The "3-dB forward angle:" (same as half power beam?) width is listed at 46 degrees.

    However the array in the optimal example in ON4UN's book is different: only 4 elements at a time are used.

    Then again in my model the radiators I think are equivalent to "point dipoles" (imagine a very short radiator with a superconducting loading coil!) so I guess a dipole (or theoretical perfect 1/4 wave vertical) will have a gain over an isotropic radiator with size much less than a quarter wavelength.

    If I allow the optimizer to play with currents, it seems to always want equal currents (they are constrained so that the sum of power from each antenna is a constant - conservation of energy), then it seems to always wander back to equal currents as a solution.

    If the optimizer is allowed to play with the "nearest neighbor distance" it always wanders back to the original quarter wave spacing for optimality.

    But I expect there are very many local maxima in this optimization - it would be interesting to start with a random element placement and see what the optimizer comes up with.

    If I randomly increase or decrease currents by 2.5% there is no observable difference in the radiation pattern, so this solution seems relatively robust.

    Actually if my radiators are isotropic, that isn't really the same as a point dipole is it? Even a point dipole should not have an isotropic radiation pattern... I guess my radiators would be like a point particle whose charge increases and decreases in magnitude at 14 MHz... hmmm....
  4. N3OX

    N3OX Ham Member QRZ Page

    This is the pattern I get (on ARRL-style log) for the Ma solution for a 1/4-wave spaced 8-circle on 20m.

    I also find that this one is nice and stable with respect to current perturbations. I think the ratio of minimum to maximum current amplitude is a good indicator of whether or not this will be the case. The optimal currents here are

    Source 1 amp, phase: 0.762845, -24.790210
    Source 2 amp, phase: 0.865042, 156.701616
    Source 3 amp, phase: 1.000000, 0.000000
    Source 4 amp, phase: 0.865042, -156.701616
    Source 5 amp, phase: 0.762845, 24.790210
    Source 6 amp, phase: 0.865042, -156.701616
    Source 7 amp, phase: 1.000000, -0.000000
    Source 8 amp, phase: 0.865042, 156.701616

    Lots of the fragile optimal arrays have a 10:1 or more ratio of min to max current.

    The eight-circle with 1/4-wave spacing has a gain of just over 12dBi and a 31.6 degree beamwidth if implemented as lossless halfwave vertical dipoles in EZNEC:


    This is what I'm getting with uniform currents and your phases:


    For reference, here's the Ma directivity-optimal pattern vs. what I'm getting for yours on a linear radial scale in E^2 units (normalized to 8):


    Those computations are pattern superposition with infinitesimal dipoles.

    I do think this is about practical feeding and good-enough performance.

    The power from each element? Or just constraining the whole antenna?

    The in-situ impedances and drive powers of the optimal antenna are pretty weird. Most of the elements have very low drive impedances in the optimal-directivity antenna and the forward element Element 5 is sinking 73 watts! I think ON4UN's book talks about negative-power elements a bit, I think it's just a thing that happens. It's not lost power, it's recirculating power, but it shows some of the issues with feeding these things.


    Attached Files:

    M0AGP likes this.
  5. M0AGP

    M0AGP Ham Member QRZ Page

    Wow this is so interesting! Thanks for all the work you did to produce the calculation and plots. I think if I had toys like this to play with earlier I might have veered into EE rather than physics.

    I suspect I am making some fundamental dumb mistake in the way I set up the optimization here. Putting the optimal currents and phases you found, I get (on my linear scale) the following pattern that looks pretty clean! But weirdly the strength of the main lobe is relatively weak, and no doubt incorrect....
    I suspect my antenna numbering scheme is different from yours, hence the different direction for the beam heading when I put your numbers in.

    My logic was that the radiated E field strength is proportional to the current, so power is proportional to current^2. So if the sum of "power radiated by each antenna" is proportional to total energy (as there are no Joule losses by assumption), then to enforce conservation of energy I need to ensure that the sum of squares of currents is a constant, which I set to 1.

    I think I have goofed up here... with the wrong normalization, the optimizer will be working under the wrong constraints.

    Maybe I should have integrated the radiated power over all 4 pi steradians and kept that constant as a constraint? (Maybe 2 pi given these are ground mounted verticals)

    Thoughts? (And I hope others find this interesting too!)
    Last edited: Aug 7, 2020
    N3OX likes this.
  6. N3OX

    N3OX Ham Member QRZ Page

    Yes, I do think that you need to normalize by total power.

    The calculation I'm doing is supposed to maximize the intensity in the desired direction divided by the total power:

    I'm using this expression for E:
    So I think in general, E^2 should have lots of cross terms which won't be taken into account if you sum I_i^2.
    M0AGP likes this.
  7. N3OX

    N3OX Ham Member QRZ Page

    Missing image above for directivity:
  8. M0AGP

    M0AGP Ham Member QRZ Page

    Ah - literally a schoolboy error! This boy has been out of school too long... Thanks!
  9. N3OX

    N3OX Ham Member QRZ Page

    I made a significantly easier-to-use version of my Google Colab calculator [here].
    M0AGP likes this.
  10. M0AGP

    M0AGP Ham Member QRZ Page

    Thanks for this - very interesting - you have done a lot of work here! It looks like there is a python version of NEC in there and you have 248 segments - this could be useful to many hams! I was unaware it existed, though never thought to search for it. I can see that the function f(phi, theta) must be the absolute magnitude of the spherical harmonic Y_11(theta, phi) or 1/4pi * sin^2(theta).

    I will have a play!

    This would allow us to provide an answer to one of the two theoretical questions I asked at the beginning, at least in the restricted case of the 20m by 20m back yard which is allowed to have as many verticals as desired.

    I suppose we could further assume that we will use many 1/8 wavelength radials, which then leaves us space within the back yard to have a 4 by 4 array of quarter wave verticals spaced 1/4 wavelength apart.

    I will try and use your code to create that theoretical array (too complex to actually build I expect in terms of phasing lines needed for steering) and see how much gain we can get out of it.

    I think with that result we can consider that part of the question answered, to a reasonable degree.

    This would not be a practical solution, but maybe knowing what it looks like, one can approximate it with fewer elements.

    We need to not let the perfect be the enemy of the good, as the old Italian saying goes that Voltaire quoted...

    I expect the answer could easily be that on a practical basis, an elevated radial version of the 4-square array ends up being optimal.

    My educational spreadsheet
    As an aside, I converted my spreadsheet to integrate over all directions as a normalization. The result makes much more sense, and playing with it, it becomes quickly clear that there are many local maxima in the 16 dimensional space of currents and phases for the directionality function!

    If I start at your optimal solution and let it optimize, it walks slightly away from it and gets a solution that looks very similar. Here is how your optimal solution (left) and my solution look on a fully logarithmic plot:
    upload_2020-8-8_11-28-37.png upload_2020-8-8_11-29-21.png
    Mine is probably different due to inaccuracy in integration, using isotropic radiators (though I could now add the Ylm form factor in so as to use point dipoles...)

    My method is in fact only of educational use to me and maybe anyone else interested!

    One other question: what is the exact mathematical definition of the ARRL method for charting the dB polar plots?
    N3OX likes this.

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