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  "title": "Antennas — Dipole + Linear Array — EMFT",
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  "description": "The Antennas workbench covers the half-wave dipole (the canonical antenna) and N-element linear arrays. Drag the dipole length from 0.1λ to 1.4λ and watch the donut radiation pattern split into lobes. Add elements (N = 2 to 16), set the spacing d (in wavelengths), and the progressive phase β to steer the beam. Grating-lobe warning when d > λ/2.",
  "keywords": [
    "emft",
    "antenna",
    "dipole",
    "half wave",
    "radiation pattern",
    "directivity",
    "hpbw",
    "linear array",
    "array factor",
    "broadside",
    "end fire",
    "pattern multiplication",
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    "formulas": [
      "Half-wave dipole: L = λ/2, Z_in ≈ 73 Ω, directivity ≈ 2.15 dBi, HPBW ≈ 78°",
      "Array factor: AF(θ) = Σ_{n=0}^{N-1} e^{j·n·(k·d·cos θ − β)}",
      "Main lobe: cos θ₀ = β/(k·d); β = 0 → broadside, β = kd → end-fire",
      "HPBW (broadside, uniform): ≈ 0.886·λ / (N·d) (radians)",
      "Directivity (uniform broadside): ≈ 2·N·d/λ",
      "Grating lobe condition: d > λ/2 (Nyquist for arrays)"
    ],
    "citations": [
      "Balanis, C. A., \"Antenna Theory: Analysis and Design,\" 4th ed., Wiley, 2016.",
      "Stutzman, W. L., Thiele, G. A., \"Antenna Theory and Design,\" 3rd ed., Wiley, 2012."
    ],
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        "q": "Why does the dipole pattern split into lobes for L > 0.5λ?",
        "a": "Current distribution is no longer a single half-sine; it has multiple half-cycles, each radiating. Pattern multiplication: element pattern × current distribution."
      },
      {
        "q": "How do I steer the beam electronically?",
        "a": "Change the progressive phase β. A phase shifter per element is the standard approach. Beam squint is a concern for broadband signals (use time-delay steering, not phase steering)."
      }
    ]
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