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The Radar Range Equation
Short Course on Radar and Electronic Warfare
Kyle Davidson
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Introduction
• The goal is understanding and applying the radar range equation
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What is the radar range equation?
𝑅4 =𝑃𝑡𝜏𝐺𝑡𝐺𝑅𝜎𝜆
2
4𝜋 3𝑘𝑇𝐹(𝑆/𝑁)
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What does it mean?
• It expresses the relationship between the radar detection range and the radar and the target’s characteristics
• There are many forms, this is one of the most common
• Next building it…
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Factors in the Range Equation
𝜎
𝜏
𝑃𝑡, 𝐹, 𝑆/𝑁
𝐺𝑡 = 𝐺𝑟 𝜆
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Round Trip Time
𝑅 =𝑐Δ𝑡
2• R = range
• c = speed of light in a vacuum (3 x 108 m/s)
• Δ𝑡 = round trip time
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Power Density
𝑝 =𝑃𝑡𝐺𝑡4𝜋𝑅2
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Power Radiated from the Target
𝑃𝑟 = 𝑝𝜎 =𝑃𝑡𝐺𝑡𝜎
4𝜋𝑅2
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Power Density at Radar Receiver
𝑝𝑟 =𝑃𝑡𝐺𝑡4𝜋𝑅2
𝜎1
4𝜋𝑅2
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Received Signal Power
𝑆 =𝑃𝑡𝐺𝑡4𝜋𝑅2
𝜎1
4𝜋𝑅2𝐴𝑒𝑓𝑓
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Simplifying the Received Power…
𝐴𝑒𝑓𝑓 =𝐺𝑟𝜆
2
4𝜋
𝑆 =𝑃𝑡𝐺𝑡4𝜋𝑅2
𝜎1
4𝜋𝑅2𝐺𝑟𝜆
2
4𝜋
𝑆 =𝑃𝑡𝐺𝑡𝐺𝑅𝜎𝜆
2
4𝜋 3𝑅4
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What about the Noise?
• The received noise power is:𝑁 = 𝑘𝑇𝐵𝐹
• 𝑘 = 1.38 × 10−23 𝑊/(𝐻𝑧𝐾)
• T = 290 K
• B = receiver bandwidth
• F = noise figure
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Effects of Noise (SNR = 10 dB)
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Power vs. Energy
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Signal to Noise Ratio
• Ratio of signal to noise power
• Strong indicator of ability to detect a radar return
𝑆/𝑁 =𝑆
𝑘𝑇𝐵𝐹=𝑆𝑇𝑜𝑏𝑠𝑘𝑇𝐹
=𝐸
𝑘𝑇𝐹=
𝐸
𝑁0
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Received Signal Energy
• Taking into account the duty cycle to determine the average transmitted power
• The received energy is then:
𝐸 =𝑃𝑡𝜏𝐺𝑡𝐺𝑅𝜎𝜆
2
4𝜋 3𝑅4
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The Complete RRE
• Assuming a matched filter is used:
𝑆/𝑁 =𝑃𝑡𝜏𝐺𝑡𝐺𝑅𝜎𝜆
2
4𝜋 3𝑘𝑇𝐹𝑅4
• Then solving for the range:
𝑅 =4 𝑃𝑡𝜏𝐺𝑡𝐺𝑅𝜎𝜆
2
4𝜋 3𝑘𝑇𝐹 𝑆/𝑁
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Incorporating the Antenna Scan
• Time on target
𝑇𝑜𝑡 =𝜃𝐵𝜔𝑎
• 𝜃𝐵 = antenna beam width
• 𝜔𝑎 = antenna rotation rate
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Integrating the Pulses
• Coherent receivers𝑁𝑖 = 𝑓𝑟𝑇𝑜𝑡
• 𝑓𝑟 = PRF
• Non-coherent receivers use a 𝑁𝑖 factor as
an estimate of the effects of non-coherent integration
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Integrating the Pulses and Losses
𝑅 = 4 𝑁𝑖𝑃𝑡𝜏𝐺𝑡𝐺𝑅𝜎𝜆2
4𝜋 3𝑘𝑇𝐹𝑆𝑁 𝑃𝑑
𝐿𝑖
• The loss factor incorporates losses from all aspects of the radar system:– Transmitter
– Receiver
– Radar channel
– And more!
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But, the Radar Horizon…
• The earth is curved!
• Range is limited by this curve, and the height of the radar and target above it
𝑅𝐻 ≈ 4123 × 𝐻𝑟