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The scope of this question may sound weird.

Let's say there are two very giant weird-shaped antennas (imagine they are both 50 meters long): one for transmission and the other for receiving. On a flat surface, one antenna is placed towards the other one while the other is set point to the sky. In this case, how can I determine the distance R in Friis Equation? (Or how to find the distance of arbitrary antenna pairs in 3d space?)

The example picture is shown as below. Don't mind my drawing, as I just want to show an uncommon case where the tiny distance difference is not negligible.

Transmission example

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    \$\begingroup\$ I believe the Friis equation applies in the far-field of the antenna, where the wavefront is curved (if relatively close to the antenna), or essentially flat (if very far away). The picture you've shown would be a near-field interaction between the two antennas. \$\endgroup\$ – SteveSh Jun 11 '20 at 20:42
  • \$\begingroup\$ Also, when the two antennas are separated by distances much greater then the individual legs/arms of the arbitrarily shaped antennas you've drawn, the individual arms "disappear" and you're left with something close to a point source who's phase and amplitude (antenna pattern) is a complex sum of the different arms/legs of the antenna. \$\endgroup\$ – SteveSh Jun 11 '20 at 20:45
  • \$\begingroup\$ The graph is a simple drawing. The scale may not be applied so you can assume the distance is long enough to be far-field but short enough so that the details on the antenna is not negligible. Is this how the existing antenna types determine the point of counting distance? And I could simplify the antennas into "black-box" points when I know how this works. \$\endgroup\$ – ONLYA Jun 11 '20 at 21:15
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A Friis reflective path null occurs when the incident and reflected signal are both, but one is delayed by any multiple of 1/2 wavelength such that they cancel out.

Your hypothetical problem shows one line of sight path of TX antenna to the fringe side of Rx antenna. This ground fringes are more likely to equal and cancel line of sight path fringe loss than skyward reflections.

With an irregular shape antenna that is likely to have many irregular null lobes, unless you know that your Line of sight path is on a good side lobe, you can expect poor reception as well as motion Rayleigh loss and static Friis loss.

An accurate analysis requires and accurate antenna gain phase dispersion plot.

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  • \$\begingroup\$ There may be too many irregular LoS paths in these cases that some of the paths will remain after all cancelations. However, as @SteveSh says, the total path could be the vector sum of all the paths. But the directivity of the antenna should be considered as well. For the arbitrary cases, I prefer the Electromagnetic analysis for the general cases while the antenna analysis is the easiest for specified pairs. \$\endgroup\$ – ONLYA Jun 11 '20 at 21:28

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