If I want to reduce the diameter of a satellite dish, how much power should I add to the transmitter? how is this formula? As I have already been informed a little in this other question I asked: X-Band range and data I would need a 3 meter diameter parabolic antenna and 20W of power roughly, how can I add power, for example 200W and reduce the diameter of the parabolic antenna to 0.5 meters?
We assume a transmitter with 20 W on one side and a receiving antenna with 3 m diameter on the other side.
When we increase the power from 20 W to 200 W, we have 10 times more power. We may reduce the area of the antenna to 1/10 and its diameter to 1/sqrt(10). Instead of 3 m we may use now 0.95 m.
We need 2 KW instead of 20 W to reduce the diameter by 1/sqrt(100) = 1/10, from 3 m to only 0.3 m.
The distance between transmitter and receiver is the same for 20 W, 200 W and 2 kW. The diameter of the transmitting antenna is not changed.
But what if both the transmitting and the receiving antennas diameter is decreased? The beam width of the transmitter would increase and the received signal would be weaker. For small beam widths a reduction of the antenna diameter by a factor of 0.5 would double the beam width.
If we use 1.5 m diameter instead of 3 m on both sides, we need to increase the power by a factor of 4 (the square of 2) for both sides. So need 16 times the power to compensate, 320 W instead of 20 W.
For a reduction by the factor of 0.1, from 3 m down to 0.3 m, we need 10000 times more power, 200 kW instead of 20 W.
So exchange diameter for power is impossible for satellites.