Link Budget Antenna: use of effective aperture and the gain in a problem

[Master1022]

Homework Statement

If we have a system where we receive power $x$ W at the receiver. Find the power at the input if the receiver and transmitting antennae are parabolic and the same size / transmission efficiency.

Relevant Equations

Power density = Power / Area

Hi,

I was just attempting this problem and was confused about the calculation process involved.

Context:
In earlier parts of the question, we calculate the gain $G$ and effective aperture $A$ for the parabolic antennae.

My Attempt:
We are given the power at the output $x$ in Watts
1) The receiving antenna has a gain $G$ and therefore we must divide by that to get the power at the entry to the receiving antenna $\frac{x}{G}$
2) Convert the power to power density $$ P_{receiver} = P_r = \frac{x}{G \cdot A} $$
3) We know that:
$$ P_r = \frac{P_t G}{4 \pi R^2} $$ and therefore, we can find
$$ P_t = \frac{4 P_r \pi R^2}{G} $$

However, the answer only includes the gain $G$ once, that is it uses the formula:
$$ P_t = \frac{4 x \pi R^2}{G A} $$

I cannot understand why this is the case. I know that the effective aperture and gain related to one another, but I thought the effective aperture was about the effectiveness of the physical antenna and the gain was about the amplification of the signal.

Should I not be counting the gain twice?

Any help is greatly appreciated

 

[Master1022]

I have answered this question and I should not include the gain twice. The antenna does not (according to my notes) act like an op-amp which physically amplifies the signal. The gain is a measure of directivity of the beam, which can also be represented by its physical characteristics (eg. its effective area). Therefore, there is no need to include the gain twice.

 

[berkeman]

It seems like you might want to apply the gain twice if you are accounting for both the TX and RX antennas? But I can't tell it that's what you are wanting to do.

 

[Master1022]

It seems like you might want to apply the gain twice if you are accounting for both the TX and RX antennas? But I can't tell it that's what you are wanting to do.

Apologies, my post may have not been too clear. I also haven't seen the terms TX and RX, but I assume they stand for transmitting and receiving. So yes, we need to account for gain twice, but we shouldn't be including both gain and the effective aperture for the same antenna (as they are related to one another and describe the same phenomenon). So in my formula I should have included $G_t$ and only one of $G_r$ or $A_{eff}$ (with the respective conversion formula).

Otherwise, I could have used the effective aperture for antennae as they have the same dimensions.