question about Friis equation with very large antenna gain

Hi,

in Friis equation, if the gain of transmit and receive antenna are very high, at least from this equation we may have Pr/Pt larger than 1. But since antenna is passive, it's not possible for this ratio to be larger than one. How can I understand this question? Mathematically what is limiting this ration?

Friis equation is valid for far field. Most likely you are still in the near field range if your example achieves positive power gain. Or the assumed antenna parameters are simply unrealistic.

How did you arrive at the antenna gain? Are you sure that it's valid for the transmission distance in your setup?

Exactly as FvM mentioned. If Pr/Pt > 1, then it would also mean lambda/4piD >1 ==> D<lambda/4pi
But by definition, far field extends upto infinity. Therefore the above conjecture is incorrect.