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57 lines
2.0 KiB
57 lines
2.0 KiB
id: fund-ex-05a
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type: multi-part
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difficulty: hard
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points: 20
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related_lesson: fund-05
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question: |
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Calculate the topological phase constraint for a spark circuit with:
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- Frequency: f = 150 kHz
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- Mutual capacitance: C_mut = 12 pF
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- Shunt capacitance: C_sh = 8 pF
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(a) Calculate the capacitance ratio r
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(b) Calculate the minimum achievable phase angle φ_Z,min
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(c) Calculate R_opt_phase that achieves this minimum angle
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hints:
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- "Ratio r = C_mut / C_sh"
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- "Minimum phase: φ_Z,min = -atan(2√[r(1+r)])"
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- "Optimal resistance: R_opt_phase = 1/[ω√(C_mut(C_mut+C_sh))]"
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solution:
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steps:
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- "Part (a): Calculate ratio"
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- "r = C_mut / C_sh = 12 pF / 8 pF = 1.5"
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- "Part (b): Calculate minimum phase"
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- "φ_Z,min = -atan(2√[r(1+r)])"
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- "= -atan(2√[1.5 × 2.5])"
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- "= -atan(2√3.75)"
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- "= -atan(2 × 1.936)"
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- "= -atan(3.873)"
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- "= -75.5°"
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- "Part (c): Calculate R_opt_phase"
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- "ω = 2πf = 2π × 150×10³ = 9.425×10⁵ rad/s"
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- "R_opt_phase = 1/[ω√(C_mut(C_mut+C_sh))]"
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- "= 1/[9.425×10⁵ × √(12×10⁻¹² × 20×10⁻¹²)]"
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- "= 1/[9.425×10⁵ × √(240×10⁻²⁴)]"
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- "= 1/[9.425×10⁵ × 15.49×10⁻¹²]"
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- "= 1/(14.60×10⁻⁶)"
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- "= 68.5 kΩ"
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answer_r: "1.5"
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answer_phi_min: "-75.5"
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answer_R_opt: "68.5"
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unit_R: "kΩ"
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unit_phi: "degrees"
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tolerance: 3.0
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explanation: |
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With r = 1.5, this circuit cannot achieve -45° (which requires r < 0.207). The
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minimum achievable phase is -75.5°, which is quite capacitive. This occurs when
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R = R_opt_phase = 68.5 kΩ. Any other resistance value will result in a phase
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angle with magnitude greater than 75.5°. This topological constraint is fundamental
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to the circuit structure - it's impossible to overcome by changing component
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values. The ratio r = 1.5 is typical for medium Tesla coils with moderate-length
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sparks.
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related_concepts: ["topological-constraint", "phase-optimization", "R_opt_phase", "capacitance-ratio"]
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