Bookkeeping on power from transmitter to receiver: everything in between either adds gain or subtracts loss. If what arrives meets or exceeds the receiver's sensitivity \(P_{R,\min}\), the link closes.
\(P_R = P_T G_T G_R \lambda^2/(4\pi R)^2\), with every gain a linear ratio (no units).
The surface area of an expanding sphere. Transmitted power spreads over \(4\pi R^2\) — that is the one-way spreading loss.
\(A_e = \lambda^2 G_R/4\pi\) — the effective area over which a receiving antenna collects power. Multiplying power density by \(A_e\) gives received power.
Both double \(P_R\) — these terms enter linearly.
\(P_R\) goes up by four times, since \(\lambda\) is squared. Lower frequencies travel farther for the same power.
It drops to one quarter — range enters squared.
Free-space loss goes as \(1/\lambda^2\), but dish gain goes as \(1/\lambda^2\) too. Raising frequency costs you in propagation and pays you back in antenna gain — a real design tradeoff.
\(\times 2 = +3\) dB, \(\times 4 = +6\) dB, \(\times 10 = +10\) dB, \(\times 100 = +20\) dB, \(\div 2 = -3\) dB. These are power ratios, \(10\log_{10}\); the equation sheet's \(20\log_{10}\) row is for voltage gain (Lesson 17).
Because multiplying becomes adding, and the dynamic range involved is enormous. In this course, convert a dB value to a linear ratio before it goes into an equation.
Solve Friis for \(R\) with \(P_R = P_{R,\min}\): \(R = \frac{\lambda}{4\pi}\sqrt{P_T G_T G_R/P_{R,\min}}\), in meters. It is the second Friis row on the equation sheet.
Convert first. \(R_{LOS,mi}\) takes heights in feet and gives miles; multiply by 1.61 for kilometers. Compare the two limits only in the same unit.
The difference of the two altitudes, \(R = h_\text{sat} - h_\text{aircraft}\), in meters (feet × 0.305). Straight up there is no horizon in the way, so there is no LOS limit to compute.
It needs line of sight and adequate power. The maximum range is the smaller of the two limits — and either one can bind, so compute both.