# Lesson 3 Flashcards

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<summary><strong>1. Write the radar range equation for received power.</strong></summary>
<div class="card-answer"><p><span class="math notranslate nohighlight">\(P_r = \dfrac{P_t G_t G_r \lambda^2 \sigma}{(4\pi)^3 R^4}\)</span>.</p></div>
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<summary><strong>2. Where does the <span class="math notranslate nohighlight">\(1/R^4\)</span> come from?</strong></summary>
<div class="card-answer"><p>Two-way spreading: the signal spreads as <span class="math notranslate nohighlight">\(1/R^2\)</span> on the way out and again <span class="math notranslate nohighlight">\(1/R^2\)</span> on the way back. The product is <span class="math notranslate nohighlight">\(1/R^4\)</span>.</p></div>
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<summary><strong>3. What is radar cross section (RCS)?</strong></summary>
<div class="card-answer"><p><span class="math notranslate nohighlight">\(\sigma\)</span> (m²): the equivalent area that, capturing the incident density and re-radiating it isotropically, would produce the observed echo. It encodes how strongly a target reflects.</p></div>
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<summary><strong>4. Write the maximum detection range.</strong></summary>
<div class="card-answer"><p><span class="math notranslate nohighlight">\(R_\text{max} = \left[\dfrac{P_t G_t G_r \lambda^2 \sigma}{(4\pi)^3 S_\text{min}}\right]^{1/4}\)</span>, equivalently <span class="math notranslate nohighlight">\(R_\text{max} = K\,\sigma^{1/4}\)</span>.</p></div>
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<summary><strong>5. By what factor must transmit power rise to double detection range?</strong></summary>
<div class="card-answer"><p>×16 (+12 dB), because every term sits under a fourth root. The same factor applies to <span class="math notranslate nohighlight">\(G_t\)</span>, <span class="math notranslate nohighlight">\(G_r\)</span>, or <span class="math notranslate nohighlight">\(1/S_\text{min}\)</span>.</p></div>
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<summary><strong>6. What is <span class="math notranslate nohighlight">\(S_\text{min}\)</span>?</strong></summary>
<div class="card-answer"><p>The minimum detectable signal — the smallest received power at which the radar declares a detection. It is bounded by the receiver noise floor; coherent integration lowers it somewhat.</p></div>
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<summary><strong>7. Why is RCS the target's best lever despite the fourth root?</strong></summary>
<div class="card-answer"><p>The radar's power/gain levers run into hard thermal and physical limits, while shaping and materials keep reducing <span class="math notranslate nohighlight">\(\sigma\)</span>. Both sides are fourth-rooted, but the target's lever has room left.</p></div>
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<summary><strong>8. In the worked example, what is <span class="math notranslate nohighlight">\(R_\text{max}\)</span> for a 1 m² target?</strong></summary>
<div class="card-answer"><p>About 472 km (S-band, <span class="math notranslate nohighlight">\(P_t=1\)</span> MW, <span class="math notranslate nohighlight">\(G_t=G_r=30\)</span> dBi, <span class="math notranslate nohighlight">\(\lambda=0.1\)</span> m).</p></div>
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<summary><strong>9. Dropping that target to −30 dBsm changes <span class="math notranslate nohighlight">\(R_\text{max}\)</span> to roughly what?</strong></summary>
<div class="card-answer"><p>About 84 km. A 1000× RCS cut scales range by <span class="math notranslate nohighlight">\((10^{-3})^{1/4} \approx 0.178\)</span>, i.e. ~5.6× shorter.</p></div>
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<summary><strong>10. What is the effective aperture of a receive antenna?</strong></summary>
<div class="card-answer"><p><span class="math notranslate nohighlight">\(A_e = G_r \lambda^2 / 4\pi\)</span>. Multiplying the echo power density by <span class="math notranslate nohighlight">\(A_e\)</span> gives the received power <span class="math notranslate nohighlight">\(P_r\)</span>.</p></div>
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<summary><strong>11. Why is the Detect link the most leverage-rich part of the kill chain?</strong></summary>
<div class="card-answer"><p>The fourth-power law makes detection range extremely sensitive in operational terms: small engineering changes to <span class="math notranslate nohighlight">\(\sigma\)</span> reshape the entire defended volume, and detection gates everything downstream.</p></div>
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