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I am currently trying to design, in theory, a large wireless power spot beaming antenna for a satellite to transfer 200 MW at a wavelength of 116 mm. I was wondering how I could design it so that a ground station to receive the broadcast from 36,000 km away would be at max a few kilometers squared. I am a bit confused about the equation below from Wikipedia's article covering Friis Transmission. If I wanted to collect all of my power transmitted, the equations seems a bit off.

How do you tell the Solid Angle generated by an antenna array?

$$ P_r = \frac{A_tA_r}{r^2\lambda^2}P_t $$

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You should place a link to any information you are given, can you give a link to the wikipedia article that gave you that equation? –  Kortuk Jan 17 '13 at 18:37
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That formula isn't useful to you in its current form because your question (about directivity -- via "solid angle") is consumed in the aperture parameters. That equation basically answers this question:

"If I know the directivity, efficiency, and relative position of both my transmitting and receiving antennae, how much power do I need to ensure I receive at least this much when transmitting through free space?"

You need to refine your design problem since you have too many unknowns at present (design of the TX system, design of the RX system, frequency, etc...). All you've specified is the separation, received power, and physical receive aperture (note: that isn't the same as the Ar in your equation).

There are an infinite set of solutions.

In a theoretical antenna design, would you be able to recommend how I could get started in refining the problem. Is there a specific gain I need on the transmitting system in order to obtain ground rx area?

Ceteris paribus, ground RX area = Aphys in the article you mentioned. To get to Aeff you need to determine the efficiency. That requires either a computational or analytical solution to your antenna design (determine the current moments and from that the flux density).

But I think the question you wanted to ask is, "What order should I perform the design in?" You've got more contraints on your receiver than transmitter, so I would start there, but to do that, you must determine your frequency.

In free space, this is easy. Since you want small apertures, you need ridiculously high frequencies. If you want to go through Earth's atmosphere you need to go through "the window", which places a practical upper bound.

enter image description here

  1. Frequency
  2. RX size/design --> efficiency --> Ar
  3. With Ar, r, and lambda known, design transmitter antenna (perhaps it is also size constrained?) --> At
  4. Pr is known/desired, solve for Pt
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In a theoretical antenna design, would you be able to recommend how I could get started in refining the problem. Is there a specific gain I need on the transmitting system in order to obtain ground rx area? –  d.mc2 Jan 17 '13 at 19:17
    
@d.mc2 -- I expanded my answer to cover your comment. Cheers. –  DrFriedParts Jan 17 '13 at 19:29
    
Thank you. I was just wondering, if 200 MW is desired, would there be more power needed to be transmitted. Is there a way to narrow down the beam to have less dispersion or just have the receiver collect the entire transmission? –  d.mc2 Jan 17 '13 at 19:36
    
@d.mc2 -- In a theoretical (read "ideal") sense, "yes." It is possible to collect all of the transmitted energy. Practically, you won't get anywhere near it. Most certainly not over the distances you specify. You can focus the beam through a number of different techniques (if you've seen a "satellite dish", you've seen an example of this), but they either require substantial size (relative to wavelength) or introduce losses themselves. Aircraft-mounted RADAR systems are probably a good example system of what you're trying to accomplish if you are looking for ideas. –  DrFriedParts Jan 17 '13 at 19:51
    
I think the first thing to design is at the transmitter end : if the antenna guidance ever mistracks, you need a completely reliable way of disconnecting the power. Otherwise it's a case of - if you can forgive this - DrFriedPeople. –  Brian Drummond Jan 17 '13 at 22:33
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