id: opt-ex-thevenin-complete
type: multi-part
difficulty: hard
points: 40
related_lesson: opt-03
question: |
  COMPLETE THÉVENIN ANALYSIS

  You measured the following Thévenin parameters for your DRSSTC at 188 kHz:
  - Z_th = 115 - j2300 Ω (drive OFF, 1V test source)
  - V_th = 340 kV (drive ON, no load)

  The spark has:
  - C_mut = 8 pF, C_sh = 5 pF (from FEMM)
  - R = 65 kΩ (assumed operating resistance)

  Tasks:
  (a) Calculate the spark admittance Y_spark
  (b) Convert to Z_spark
  (c) Calculate total circuit impedance Z_total = Z_th + Z_spark
  (d) Calculate current through the spark
  (e) Calculate voltage across the spark
  (f) Calculate real power dissipated in the spark
  (g) Compare R to R_opt_power for these capacitances

hints:
  - "Use admittance formulas for parallel (R || C_mut) then series with C_sh"
  - "Add impedances in series: Z_total = Z_th + Z_spark"
  - "Current divider applies: I = V_th / Z_total"
  - "Voltage across spark: V_spark = I × Z_spark"
  - "Power: P = 0.5 × |I|² × Re{Z_spark}"

solution:
  steps:
    - "Part (a): Calculate Y_spark"
    - "ω = 2π × 188×10³ = 1.181×10⁶ rad/s"
    - "G = 1/65000 = 15.38 μS"
    - "B₁ = 1.181×10⁶ × 8×10⁻¹² = 9.45 μS"
    - "B₂ = 1.181×10⁶ × 5×10⁻¹² = 5.91 μS"
    - "Denom: G² + (B₁+B₂)² = 236.5 + 236.2 = 472.7 μS²"
    - "Re{Y} = 15.38 × 34.93 / 472.7 = 1.14 μS"
    - "Im{Y} = 5.91 × [236.5 + 145.2] / 472.7 = 4.77 μS"
    - "Y_spark = 1.14 + j4.77 μS"
    - "Part (b): Convert to Z_spark"
    - "|Y| = √(1.14² + 4.77²) = 4.90 μS"
    - "|Z_spark| = 1/4.90×10⁻⁶ = 204 kΩ"
    - "φ_Y = atan(4.77/1.14) = 76.6°"
    - "φ_Z = -76.6°"
    - "R_eq = 204 × cos(-76.6°) = 47.6 kΩ"
    - "X_eq = 204 × sin(-76.6°) = -198 kΩ"
    - "Z_spark = 47.6 - j198 kΩ"
    - "Part (c): Calculate Z_total"
    - "Z_total = Z_th + Z_spark"
    - "= (115 - j2300) + (47600 - j198000)"
    - "= (47715 - j200300) Ω"
    - "= 47.7 - j200.3 kΩ"
    - "|Z_total| = √(47.7² + 200.3²) = 205.9 kΩ"
    - "Part (d): Calculate current"
    - "I = V_th / Z_total = 340×10³ / 205.9×10³ = 1.65 A peak"
    - "Part (e): Calculate voltage across spark"
    - "V_spark = I × Z_spark = 1.65 × 204×10³ = 337 kV"
    - "Part (f): Calculate power"
    - "P = 0.5 × I² × R_eq = 0.5 × 1.65² × 47.6×10³"
    - "= 0.5 × 2.72 × 47.6×10³ = 64.8 kW"
    - "Part (g): Compare to R_opt_power"
    - "R_opt = 1/(ω × (C_mut + C_sh))"
    - "= 1/(1.181×10⁶ × 13×10⁻¹²) = 65.1 kΩ"
    - "Actual R = 65 kΩ ≈ R_opt_power ✓"
    - "Operating at optimal resistance for maximum power transfer!"

  answer_a: "1.14 + j4.77 μS"
  answer_b: "204 kΩ ∠-76.6° or 47.6 - j198 kΩ"
  answer_c: "205.9 kΩ"
  answer_d: "1.65"
  unit_d: "A peak"
  answer_e: "337"
  unit_e: "kV"
  answer_f: "64.8"
  unit_f: "kW"
  answer_g: "R_opt = 65.1 kΩ, actual = 65 kΩ, at optimum!"
  tolerance: 3.0

explanation: |
  This complete Thévenin analysis demonstrates the power of the equivalent circuit
  method. Key insights: (1) Most voltage appears across the spark (337 kV out of
  340 kV) because |Z_spark| >> |Z_th|, (2) The actual R ≈ R_opt_power suggests
  the plasma self-optimized to maximize power extraction, (3) Power dissipated
  (64.8 kW) is substantial, (4) Strongly capacitive spark (φ_Z = -76.6°) is typical.
  This analysis predicts performance without full coupled simulation.

related_concepts: ["thevenin-method", "complete-analysis", "power-prediction", "self-optimization"]