Topic: Electromagnetics
The available power required at a receiving antenna is $10^{-6}$ Watts. Transmitting and receiving antennas have gain of 40 dB each. The carrier frequency is 4 GHz and the distance between them is 30 miles. The required transmitter power is
- A. 0.64 W
- B. 0.74 W
- C. 0.78 W
- D. 0.32 W
Answer: A. 0.64 W
Explanation: Using the Friis transmission equation: $P_{r}=\frac{P_{t}G_{t}G_{r}\lambda^{2}}{(4\pi R)^{2}} \Rightarrow P_{t}=\frac{P_{r}(4\pi R)^{2}}{\lambda^{2}G_{t}G_{r}}$
Given: $P_{r}=10^{-6}~\text{W}$, $f=4~\text{GHz} \Rightarrow \lambda=\frac{c}{f}=\frac{3\times10^{8}}{4\times10^{9}}=0.075~\text{m}$.
$R=30~\text{miles} = 30\times 1.609~\text{km} = 48.27\times 10^{3}~\text{m}$.
$G_{t}=G_{r}=40~\text{dB} = 10^{4}$.
$P_{t}=\frac{10^{-6} \times (4\pi \times 48.27\times 10^{3})^{2}}{(0.075)^{2} \times 10^{4} \times 10^{4}} \approx 0.64~\text{W}$.