id: fund-ex-05a
type: multi-part
difficulty: hard
points: 20
related_lesson: fund-05
question: |
  Calculate the topological phase constraint for a spark circuit with:
  - Frequency: f = 150 kHz
  - Mutual capacitance: C_mut = 12 pF
  - Shunt capacitance: C_sh = 8 pF

  (a) Calculate the capacitance ratio r
  (b) Calculate the minimum achievable phase angle φ_Z,min
  (c) Calculate R_opt_phase that achieves this minimum angle

hints:
  - "Ratio r = C_mut / C_sh"
  - "Minimum phase: φ_Z,min = -atan(2√[r(1+r)])"
  - "Optimal resistance: R_opt_phase = 1/[ω√(C_mut(C_mut+C_sh))]"

solution:
  steps:
    - "Part (a): Calculate ratio"
    - "r = C_mut / C_sh = 12 pF / 8 pF = 1.5"
    - "Part (b): Calculate minimum phase"
    - "φ_Z,min = -atan(2√[r(1+r)])"
    - "= -atan(2√[1.5 × 2.5])"
    - "= -atan(2√3.75)"
    - "= -atan(2 × 1.936)"
    - "= -atan(3.873)"
    - "= -75.5°"
    - "Part (c): Calculate R_opt_phase"
    - "ω = 2πf = 2π × 150×10³ = 9.425×10⁵ rad/s"
    - "R_opt_phase = 1/[ω√(C_mut(C_mut+C_sh))]"
    - "= 1/[9.425×10⁵ × √(12×10⁻¹² × 20×10⁻¹²)]"
    - "= 1/[9.425×10⁵ × √(240×10⁻²⁴)]"
    - "= 1/[9.425×10⁵ × 15.49×10⁻¹²]"
    - "= 1/(14.60×10⁻⁶)"
    - "= 68.5 kΩ"

  answer_r: "1.5"
  answer_phi_min: "-75.5"
  answer_R_opt: "68.5"
  unit_R: "kΩ"
  unit_phi: "degrees"
  tolerance: 3.0

explanation: |
  With r = 1.5, this circuit cannot achieve -45° (which requires r < 0.207). The
  minimum achievable phase is -75.5°, which is quite capacitive. This occurs when
  R = R_opt_phase = 68.5 kΩ. Any other resistance value will result in a phase
  angle with magnitude greater than 75.5°. This topological constraint is fundamental
  to the circuit structure - it's impossible to overcome by changing component
  values. The ratio r = 1.5 is typical for medium Tesla coils with moderate-length
  sparks.

related_concepts: ["topological-constraint", "phase-optimization", "R_opt_phase", "capacitance-ratio"]