$G_T$ and $G_R$ are the gain of transmitting and receiving antennas, respectively, $D$ is the distance between the transmitting and receiving antennas, and $\lambda$ is the wavelength in free space.
Given: $G_T = G_R = 1.0, \lambda = 0.30\text{ m}$ and $P_T = +10\text{ dBm}$.
Choose the distance ($D$), in km, from the following options at which the received power, $P_R = -90\text{ dBm}$?
Friis Equation: Calculate Transmission Distance
This solution calculates the distance D using the Friis transmission equation based on given power levels, antenna gains, and wavelength.
Given Information
- Equation: $P_R = \frac{P_T G_T G_R \lambda^2}{(4\pi D)^2}$
- Parameters: $G_T=1.0$, $G_R=1.0$, $\lambda = 0.30\text{ m}$
- Power Levels: $P_T = +10\text{ dBm}$, $P_R = -90\text{ dBm}$
- Goal: Find D in km.
Power Conversion (dBm to Watts)
Convert power levels to Watts (W): $P_W = 10^{(P_{\text{dBm}}/10)} \times 10^{-3}$.
- $P_T = 10^{(10/10)} \times 10^{-3}\text{ W} = 0.01\text{ W}$
- $P_R = 10^{(-90/10)} \times 10^{-3}\text{ W} = 10^{-12}\text{ W}$
Solving for Distance D
Rearrange the Friis equation to solve for D:
$D^2 = \frac{P_T G_T G_R \lambda^2}{16\pi^2 P_R}$Substitute values and calculate D in meters:
$D^2 = \frac{(0.01\text{ W}) \times (1.0) \times (1.0) \times (0.30\text{ m})^2}{16\pi^2 \times (10^{-12}\text{ W})}$ $D^2 = \frac{0.01 \times 0.09}{16\pi^2 \times 10^{-12}} = \frac{9 \times 10^{-4}}{16\pi^2 \times 10^{-12}} = \frac{9 \times 10^8}{16\pi^2}$ $D = \sqrt{\frac{9 \times 10^8}{16\pi^2}} = \frac{3 \times 10^4}{4\pi} \text{ meters}$Convert to Kilometers
Divide the distance in meters by 1000:
$D_{\text{km}} = \frac{3 \times 10^4 / (4\pi)}{1000} = \frac{30000}{4000\pi} = \frac{30}{4\pi} = \frac{15}{2\pi} \text{ km}$The correct distance is &frac{15}{2\pi} km.
Important Questions from Antennas
Which of the following antennas is the standard reference antenna for the directiveness?
Consider the following statements:
(a) Fiber optic cable is much lighter than copper cable
(b) Fiber optic cable is not affected by power surges or electromagnetic interference
(c) Optical transmission is inherently bidirectional.
Which of the statements is (are) correct?Broadside arrays have
A. Number of dipoles of unequal size
B. Number of dipoles equally spaced
C. Collinear dipoles
D. Dipoles in phase
E. Dipoles are 90 out of phase
Choose the correct answer from the options given below:
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