You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
79 lines
3.3 KiB
79 lines
3.3 KiB
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"]
|