Radar range refers to the maximum distance at which a radar system can detect and accurately locate a target. It is a crucial parameter in radar technology that determines the system's effectiveness in various applications such as weather monitoring, air traffic control, and military operations. The radar range depends on several factors including the transmitter power, antenna characteristics, target properties, and environmental conditions.

The radar range equation is given by:

\[ R = \left(\frac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 S_{min}}\right)^{1/4} \]

Where:

- \( R \) is the maximum radar range (in meters, m)
- \( P_t \) is the transmitter power (in watts, W)
- \( G \) is the antenna gain (dimensionless)
- \( \lambda \) is the wavelength of the radar signal (in meters, m)
- \( \sigma \) is the radar cross section of the target (in square meters, m²)
- \( S_{min} \) is the minimum detectable signal power (in watts, W)

Let's calculate the maximum radar range for a typical radar system:

- Given:
- Transmitter power (\( P_t \)) = 1000 W
- Antenna gain (\( G \)) = 30
- Wavelength (\( \lambda \)) = 0.1 m
- Radar cross section (\( \sigma \)) = 1 m²
- Minimum detectable signal (\( S_{min} \)) = 1 × 10⁻¹⁰ W

- Apply the radar range equation: \[ R = \left(\frac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 S_{min}}\right)^{1/4} \]
- Substitute the known values: \[ R = \left(\frac{1000 \times 30^2 \times 0.1^2 \times 1}{(4\pi)^3 \times 1 \times 10^{-10}}\right)^{1/4} \]
- Simplify: \[ R = \left(\frac{9 \times 10^5}{1.98 \times 10^{-9}}\right)^{1/4} \]
- Calculate the final result: \[ R \approx 135,720 \text{ m} \approx 135.7 \text{ km} \]

Let's visualize the radar range concept:

This diagram illustrates:

- The radar station (green dot at the center)
- The maximum radar range (\( R \)) represented by the red arrow
- The circular detection area (blue dashed line)
- A target at the maximum detection range (blue dot)

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