How Many Watts Actually Arrive
Module 7 gave you gain and effective aperture and stopped one step short. Here is the missing sentence: so many watts in, two antennas, d metres of nothing — how much comes out?
Power Spreads Over a Sphere
All of Pt is still there at distance d — smeared over 4πd² square metres. Nothing was absorbed. The power is thinner, not smaller.
Times the Aperture
M7-L1 gave you Ae = λ²Gr/4π. Multiply density by area and Friis falls out in one line. Note where λ entered: the antenna, not the space.
Path Loss in Decibels
FSPL = (4πd/λ)², so FSPL(dB) = 20log d + 20log f + a constant that depends only on which units you typed. Metres and hertz, then +60 dB per rescale.
2.4 GHz Across 100 m
- 20log(0.1 km) = −20.00 dB
- 20log(2.4) = +7.60 dB
- 92.45 − 20.00 + 7.60 = 80.05 dB
Check it from the definition: λ = 12.5 cm, 4πd/λ = 10,053, and 20log(10,053) = 80.05 dB. One part in a hundred million.
Does the Link Close?
1 W transmitter, receiver sensitivity −100 dBm. Sweep distance and frequency; the dashed line is where the budget runs out.
36,000 km at 12 GHz
- 92.45 + 91.13 + 21.58 = 205.2 dB of path loss
- 20 + 35 − 205.2 − 1.5 + 40 − 0.5 = −112.2 dBW
- That is −82.2 dBm, about 6 pW
- Noise floor kTB at 150 K, 27 MHz: −132.5 dBW
A carrier-to-noise ratio of 20.3 dB — it closes, and the 13 dB spare is what rain will take.
Six Decibels a Doubling
Twice the distance is four times the area, and four times in power is 6.02 dB. Ten times the distance is 20 dB. So loss climbs at a steady 20 dB per decade — a straight line on a log axis.
High Frequencies Do Not Lose More
Nothing in free space absorbs. The +20log f came from Ae = λ²G/4π — a fixed-gain antenna shrinks. Fix the dish size instead and λ moves to the denominator.
What you learned
- Friis = spreading (4πd²) then collecting (Ae)
- FSPL in dB: −147.55, +32.45 or +92.45
- 20 dB per decade; 80.05 dB at 100 m and 2.4 GHz
- A 12 GHz geostationary hop costs 205.2 dB
- Fixed gain loses 14 dB; fixed dish gains 14 dB