Solve Link Budget: EIRP 100000 W, -75 dBm Receiver Sensitivity

[ENGRstudent]

I think I've done this right, but I'm looking for some reassurance because I often make stupid mistakes when converting from dB to watts and back again.

Thank you in advance! I'm about to graduate in 2 weeks, and this forum has been AWESOME throughout my college career. Kudos to everyone who takes the time to help out.

The problem statement reads:
"A satellite in synchronous orbit outputs a 4 GHz signal with an EIRP of 100000 W. If the ground based receiver has a sensitivity of -75 dBm, what is the gain required of the receiving antenna?"

My equations:
(P_receiver) = (EIRP) x (G_free space) x (G_receiving antenna)

(G_free space) = [λ/(4*pi*d)]^2

P_receiver = receiving antenna power
G_free space = free space gain
G_receiving antenna = receiving antenna gain
λ = c/f
d = 35,800,000 meters (height of orbit for geosynchronous satellite courtesy of Wiki)

Converting receiver sensitivity to Watts:

-75 dBm = 30 + 10 * log[(P_receiver)] ==> P_receiver = 3.16 x 10^-11 W

Solving for G_free space:

(G_free space) = [(3.0 x 10^8)/(4 x 10^9)]/[(4*pi*35,800,000)]^2
= 2.78 x 10^-20

Solving for G_receiving antenna:

(G_receiving antenna) = (P_receiver)/[(EIRP)*(G_free space)]
(G_receiving antenna) = (3.16 x 10^-11)/[(100000)(2.78 x 10^-20)]
= 11367
= 10 * log(11367) = 40.56 dB

So my final answer is 40.56 dB. Yes? No? Maybe?

 

[light_bulb]

i get -128.5dBm without an antenna on the ground, thing is I'm stumped on how to calculate needed gain for the antenna.

so i just added the arrl antenna book to my wish list, hopefully this should be covered.

edit: shouldn't it be -75-(-128.5)=53.5dB of gain to meet the receiver minimum?

 

[ENGRstudent]

Thanks for taking the time to look over the problem. Where did you get -128.5 from?

I just worked it again and found an error in my last step, and the new value I got for the gain is 40.56 dB.

I have attached my work as a PDF to make it more clear. Please review it to see where I may have made a mistake?

Thanks again!

EDIT: I made a mistake in my correction, 40.56 dB is still my final answer.

 

[light_bulb]

the equations are a little different in my book, they want a ratio for both antenna gains which i left at 1.

Pr=Pt*Gt*Gr*λ^2/16*pi^2*d^2

 

[ve7lyd]

Don't know if its too late to answer this but here it goes. 1st convert everything to dB, adding and subtracting is way easier then multiplying and dividing. So 100000 Watts EIRP = 80 dBm. Free space loss of 4 GHz at 35,800 Km is 196.2 db. So 80 - 196.2 = -116.2 dbm. Your receiver needs -75 so you need a -75 - -116.2 = 41.2 dBi antenna. (dBi because transmit power was in EIRP)

Equations:
Free space loss in dB: P(dBm) = 32.43 + 20LogD(km) + 20LogF(MHz)
watts to dbm: dBm = (10 log (power watts)) + 30