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  "slug": "emft-smith",
  "title": "Smith Chart — EMFT",
  "category": "electromagnetics",
  "category_label": "Electromagnetics",
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  "description": "The Smith Chart is a 2D plot of the complex reflection coefficient Γ, drawn on a circle of radius 1. Every passive impedance (R ≥ 0, X ∈ ℝ) is inside the unit circle. The chart lets you visualize matching networks, compute input impedance transformations, and design single-stub tuners without algebra. Use it as a calculator for transmission-line problems.",
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    "formulas": [
      "Γ = (Z_L − Z₀) / (Z_L + Z₀) (reflection coefficient)",
      "z = Z / Z₀ (normalized impedance)",
      "Γ = (z − 1) / (z + 1) (normalized)",
      "VSWR = (1 + |Γ|) / (1 − |Γ|)",
      "Return loss: RL = −20·log₁₀|Γ| (dB)",
      "Quarter-wave transformer: Z_T = √(Z₀·Z_L)",
      "Single-stub matching: move toward generator (clockwise rotation) until G = 1, then add susceptance to cancel B"
    ],
    "citations": [
      "Pozar, D. M., \"Microwave Engineering,\" 4th ed., 2011, Ch. 3 (Smith chart section).",
      "Smith, P. H., \"Electronic Applications of the Smith Chart,\" Noble Publishing, 1995."
    ],
    "faq": [
      {
        "q": "Where on the Smith chart is Z = Z₀ (matched)?",
        "a": "At the center (Γ = 0). The workbench highlights the center as the matching target."
      },
      {
        "q": "Why does the Smith chart rotate clockwise as I move toward the generator?",
        "a": "Because Γ(z) = Γ_L·e^{−j·2·β·z}. Moving toward the generator (negative z) subtracts phase, which is a clockwise rotation on the chart."
      }
    ]
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  "last_updated": "2026-07-05"
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