Consider the Friis’ transmission equation 𝑃𝑅=(𝑃𝑇𝐺𝑇𝐺𝑅𝜆^2) (4𝜋𝐷)^2 ⁄ , where P𝑅 and 𝑃𝑇 are the received and the transmitted powers ...

 

Q. Consider the Friis’ transmission equation 𝑃𝑅=(𝑃𝑇𝐺𝑇𝐺𝑅𝜆2) (4𝜋𝐷)2 ⁄ , where P𝑅 and 𝑃𝑇 are the received and the transmitted powers, respectively. G𝑇 and 𝐺𝑅 are the gain of transmitting and receiving antennas, respectively, 𝐷 is the distance between the transmitting and receiving antennas, and 𝜆 is the wavelength in free space. Given: 𝐺𝑇=𝐺𝑅=1.0,𝜆=0.30 m and PT = +10 dBm. Choose the distance (D), in km, from the following options at which the received power, PR = −90 dBm?

Solution

Given,

• $P_T=+10\text{dBm}$

• $P_R=-90\text{dBm}$

• $G_T=G_R=1$ (linear gain)

• $\lambda =0.30\text{m}$

$P_R(in\; dBm)\; -P_T$ (in dBm): Power difference in dB

dB = 10^(dB/10) (in ratio)

Pr = Pt.Gt.Gr.λ^2 / (4.π.D)^2
Or, Pr/Pt = Gt.Gr.λ^2 / (4.π.D)^2
Or, 10^(-90-10)/10 = Gt.Gr.λ^2 / (4.π.D)^2
Or, 10^-10 = 1.1.(0.3)^2 / (4.π.D)^2 
Or, D = 15/2π
Answer: Option B