Which expression represents the Friis transmission equation for received power in free space?

• A. Pr = Pt Gt Gr (4πR/λ)^2
• B. Pr = Pt Gt Gr (λ/(4πR))
• C. Pr = Pt Gt Gr (λ/(4πR))^2
• D. Pr = Pt Gt Gr (λ^2/(16π^2 R^2))

Answer: (C)

The main idea is that in free space the received power is determined by the transmitted power, the antenna gains, the wavelength, and the distance between antennas, with the power falling off as the square of the distance. In free space, the power density at distance R is Pt Gt divided by 4πR^2. The receiving antenna then captures power proportional to its effective aperture, which relates to Gr and to wavelength by Ae = Gr λ^2/(4π). Multiplying these together gives Pr = Pt Gt Gr λ^2 /(16 π^2 R^2), which is the same as Pr = Pt Gt Gr (λ/(4πR))^2. This form cleanly shows how Pr scales with wavelength and distance: longer λ or larger gains increase Pr, while increasing distance decreases Pr with the square of R. The equivalent form λ^2/(16π^2 R^2) is just another way to write the same relationship. The other expressions either reverse the dependence on R, omit the square, or compress it into a form that’s less directly tied to the 1/R^2 spreading and the receiving aperture.