Radar Received Power Calculation

This solution details the calculation of received power in a radar system using the standard radar range equation, based on the provided parameters.

Problem Parameters Analysis

Identify the key parameters given in the radar problem:

• Radar Frequency ($f$): 5 GHz = $5 \times 10^9$ Hz
• Antenna Gain ($G$): 150 (interpreted as linear power gain)
• Range ($R$): 1 km = $1000$ m
• Radar Cross-Section ($\sigma$): 3 m²
• Transmitted Power ($P_t$): 100 kW = $10^5$ W

Calculating Radar Wavelength

First, determine the wavelength ($\lambda$) of the radar signal. The speed of light ($c$) is approximately $3 \times 10^8$ m/s.

$ \lambda = \frac{c}{f} = \frac{3 \times 10^8 \text{ m/s}}{5 \times 10^9 \text{ Hz}} = 0.06 \text{ m} $

Applying the Radar Range Equation

The received power ($P_r$) is calculated using the radar range equation:

$ P_r = \frac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 R^4} $

Calculating Received Power ($P_r$)

Substitute the given parameters and the calculated wavelength into the radar range equation:

$ P_r = \frac{(10^5 \text{ W}) \times (150)^2 \times (0.06 \text{ m})^2 \times (3 \text{ m}^2)}{(4\pi)^3 \times (1000 \text{ m})^4} $

Calculate the terms:

• $G^2 = 150^2 = 22500$
• $\lambda^2 = (0.06)^2 = 0.0036$
• $(4\pi)^3 \approx 1983.6$
• $R^4 = (1000)^4 = 10^{12}$

Now, compute $P_r$:

$ P_r = \frac{(10^5) \times (22500) \times (0.0036) \times 3}{(1983.6) \times (10^{12})} $

$ P_r = \frac{2.43 \times 10^7}{1.9836 \times 10^{15}} \text{ W} $

$ P_r \approx 1.225 \times 10^{-8} \text{ W} $

Converting Power to MicroWatts

The question asks for the received power in microWatts (µW). Use the conversion factor $1 \text{ W} = 10^6 \text{ µW}$.

$ P_r (\text{µW}) = P_r (\text{W}) \times 10^6 $

$ P_r \approx (1.225 \times 10^{-8}) \times 10^6 \text{ µW} $

$ P_r \approx 0.01225 \text{ µW} $