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91 lines
3.5 KiB
91 lines
3.5 KiB
id: opt-ex-thevenin-complete
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type: multi-part
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difficulty: hard
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points: 40
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related_lesson: opt-03
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question: |
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COMPLETE THÉVENIN ANALYSIS
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You measured the following Thévenin parameters for your DRSSTC at 188 kHz:
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- Z_th = 115 - j2300 Ω (drive OFF, 1V test source)
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- V_th = 340 kV (drive ON, no load)
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The spark has:
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- C_mut = 8 pF, C_sh = 5 pF (from FEMM)
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- R = 65 kΩ (assumed operating resistance)
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Tasks:
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(a) Calculate the spark admittance Y_spark
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(b) Convert to Z_spark
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(c) Calculate total circuit impedance Z_total = Z_th + Z_spark
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(d) Calculate current through the spark
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(e) Calculate voltage across the spark
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(f) Calculate real power dissipated in the spark
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(g) Compare R to R_opt_power for these capacitances
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hints:
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- "Use admittance formulas for parallel (R || C_mut) then series with C_sh"
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- "Add impedances in series: Z_total = Z_th + Z_spark"
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- "Current divider applies: I = V_th / Z_total"
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- "Voltage across spark: V_spark = I × Z_spark"
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- "Power: P = 0.5 × |I|² × Re{Z_spark}"
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solution:
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steps:
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- "Part (a): Calculate Y_spark"
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- "ω = 2π × 188×10³ = 1.181×10⁶ rad/s"
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- "G = 1/65000 = 15.38 μS"
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- "B₁ = 1.181×10⁶ × 8×10⁻¹² = 9.45 μS"
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- "B₂ = 1.181×10⁶ × 5×10⁻¹² = 5.91 μS"
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- "Denom: G² + (B₁+B₂)² = 236.5 + 236.2 = 472.7 μS²"
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- "Re{Y} = 15.38 × 34.93 / 472.7 = 1.14 μS"
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- "Im{Y} = 5.91 × [236.5 + 145.2] / 472.7 = 4.77 μS"
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- "Y_spark = 1.14 + j4.77 μS"
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- "Part (b): Convert to Z_spark"
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- "|Y| = √(1.14² + 4.77²) = 4.90 μS"
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- "|Z_spark| = 1/4.90×10⁻⁶ = 204 kΩ"
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- "φ_Y = atan(4.77/1.14) = 76.6°"
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- "φ_Z = -76.6°"
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- "R_eq = 204 × cos(-76.6°) = 47.6 kΩ"
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- "X_eq = 204 × sin(-76.6°) = -198 kΩ"
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- "Z_spark = 47.6 - j198 kΩ"
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- "Part (c): Calculate Z_total"
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- "Z_total = Z_th + Z_spark"
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- "= (115 - j2300) + (47600 - j198000)"
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- "= (47715 - j200300) Ω"
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- "= 47.7 - j200.3 kΩ"
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- "|Z_total| = √(47.7² + 200.3²) = 205.9 kΩ"
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- "Part (d): Calculate current"
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- "I = V_th / Z_total = 340×10³ / 205.9×10³ = 1.65 A peak"
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- "Part (e): Calculate voltage across spark"
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- "V_spark = I × Z_spark = 1.65 × 204×10³ = 337 kV"
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- "Part (f): Calculate power"
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- "P = 0.5 × I² × R_eq = 0.5 × 1.65² × 47.6×10³"
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- "= 0.5 × 2.72 × 47.6×10³ = 64.8 kW"
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- "Part (g): Compare to R_opt_power"
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- "R_opt = 1/(ω × (C_mut + C_sh))"
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- "= 1/(1.181×10⁶ × 13×10⁻¹²) = 65.1 kΩ"
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- "Actual R = 65 kΩ ≈ R_opt_power ✓"
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- "Operating at optimal resistance for maximum power transfer!"
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answer_a: "1.14 + j4.77 μS"
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answer_b: "204 kΩ ∠-76.6° or 47.6 - j198 kΩ"
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answer_c: "205.9 kΩ"
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answer_d: "1.65"
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unit_d: "A peak"
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answer_e: "337"
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unit_e: "kV"
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answer_f: "64.8"
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unit_f: "kW"
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answer_g: "R_opt = 65.1 kΩ, actual = 65 kΩ, at optimum!"
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tolerance: 3.0
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explanation: |
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This complete Thévenin analysis demonstrates the power of the equivalent circuit
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method. Key insights: (1) Most voltage appears across the spark (337 kV out of
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340 kV) because |Z_spark| >> |Z_th|, (2) The actual R ≈ R_opt_power suggests
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the plasma self-optimized to maximize power extraction, (3) Power dissipated
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(64.8 kW) is substantial, (4) Strongly capacitive spark (φ_Z = -76.6°) is typical.
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This analysis predicts performance without full coupled simulation.
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related_concepts: ["thevenin-method", "complete-analysis", "power-prediction", "self-optimization"]
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