id: model-ex-lumped-complete
type: multi-part
difficulty: hard
points: 50
related_lesson: model-02
question: |
  LUMPED MODEL COMPLETE WORKFLOW

  You extracted the following Maxwell capacitance matrix from FEMM for a 1.2 m
  (4-foot) spark with a toroid topload:

  Maxwell Matrix (pF):
           Topload  Spark
  Topload  [  32.5   -9.2 ]
  Spark    [  -9.2   15.6 ]

  Operating frequency: f = 185 kHz

  Tasks:
  (a) Validate the matrix (check symmetry, signs, physical reasonableness)
  (b) Extract C_mut and C_sh for the lumped circuit model
  (c) Compare C_sh to the empirical 2 pF/foot rule
  (d) Calculate R_opt_power and R_opt_phase
  (e) Build the lumped model with R = R_opt_power and calculate Z_spark

hints:
  - "Maxwell matrix has negative off-diagonals"
  - "C_mut = |C₁₂|, C_sh = C₂₂ - |C₁₂|"
  - "Check if C_sh ≈ 2 pF/foot × 4 feet = 8 pF"
  - "Use admittance formulas for part (e)"

solution:
  steps:
    - "Part (a): Matrix validation"
    - "Symmetry: C₁₂ = C₂₁ = -9.2 pF ✓"
    - "Diagonal positive: C₁₁ = 32.5 > 0, C₂₂ = 15.6 > 0 ✓"
    - "Off-diagonal negative: C₁₂ = -9.2 < 0 ✓"
    - "Row sums: R₁ = 32.5 - 9.2 = 23.3, R₂ = -9.2 + 15.6 = 6.4 (ground contribution) ✓"
    - "Matrix is valid"
    - "Part (b): Extract lumped parameters"
    - "C_mut = |C₁₂| = |-9.2| = 9.2 pF"
    - "C_sh = C₂₂ - |C₁₂| = 15.6 - 9.2 = 6.4 pF"
    - "Part (c): Compare to empirical rule"
    - "Empirical: C_sh ≈ 2 pF/foot × 4 feet = 8 pF"
    - "FEMM: C_sh = 6.4 pF"
    - "Ratio: 6.4/8 = 0.8 (within factor 2, acceptable) ✓"
    - "Part (d): Calculate optimal resistances"
    - "ω = 2π × 185×10³ = 1.162×10⁶ rad/s"
    - "C_total = 9.2 + 6.4 = 15.6 pF"
    - "R_opt_power = 1/(ω × C_total) = 1/(1.162×10⁶ × 15.6×10⁻¹²) = 55.2 kΩ"
    - "Product: C_mut(C_mut + C_sh) = 9.2 × 15.6 = 143.5 pF²"
    - "R_opt_phase = 1/(ω × √143.5×10⁻¹²) = 1/(1.162×10⁶ × 11.98×10⁻¹²) = 71.9 kΩ"
    - "Part (e): Calculate Z_spark at R_opt_power"
    - "Use R = 55.2 kΩ, so G = 18.12 μS"
    - "B₁ = ω × C_mut = 1.162×10⁶ × 9.2×10⁻¹² = 10.69 μS"
    - "B₂ = ω × C_sh = 1.162×10⁶ × 6.4×10⁻¹² = 7.44 μS"
    - "Denominator: G² + (B₁+B₂)² = 328.3 + 328.1 = 656.4 μS²"
    - "Re{Y} = 18.12 × 55.35 / 656.4 = 1.53 μS"
    - "Im{Y} = 7.44 × [328.3 + 193.7] / 656.4 = 5.92 μS"
    - "Y = 1.53 + j5.92 μS"
    - "|Y| = 6.11 μS, |Z| = 163.6 kΩ"
    - "φ_Y = atan(5.92/1.53) = 75.5°, φ_Z = -75.5°"
    - "Z_spark = 163.6 kΩ ∠-75.5°"

  answer_b: "C_mut = 9.2 pF, C_sh = 6.4 pF"
  answer_c: "6.4 pF vs 8 pF empirical, ratio 0.8, acceptable"
  answer_d: "R_opt_power = 55.2 kΩ, R_opt_phase = 71.9 kΩ"
  answer_e: "163.6 kΩ ∠-75.5°"
  tolerance: 3.0

explanation: |
  This complete workflow demonstrates lumped model extraction from FEMM. Key points:
  (1) Matrix validation catches errors early, (2) Sign conversion is critical
  (C_mut = |C₁₂|, not C₁₂), (3) FEMM values within factor 2 of empirical rules is
  normal, (4) Both critical resistances are calculated for optimization, (5) Final
  impedance is strongly capacitive (φ_Z = -75.5°) as expected. The 4-foot spark
  shows typical behavior with r = C_mut/C_sh = 1.44, giving φ_Z,min ≈ -75°.

related_concepts: ["FEMM-extraction", "lumped-model", "matrix-validation", "complete-workflow"]