id: fund-ex-04b
type: multi-part
difficulty: medium
points: 15
related_lesson: fund-04
question: |
  A spark has φ_Z = -60°. The impedance magnitude is |Z| = 150 kΩ.

  (a) Find R and X (rectangular components)
  (b) Calculate the power factor

hints:
  - "Use R = |Z| × cos(φ_Z) and X = |Z| × sin(φ_Z)"
  - "Power factor = cos(φ_Z)"
  - "Negative angle means capacitive reactance (X < 0)"

solution:
  steps:
    - "Part (a): Calculate resistance"
    - "R = |Z| × cos(φ_Z) = 150 × cos(-60°) = 150 × 0.5 = 75 kΩ"
    - "Calculate reactance"
    - "X = |Z| × sin(φ_Z) = 150 × sin(-60°) = 150 × (-0.866) = -130 kΩ"
    - "Rectangular form: Z = 75 - j130 kΩ"
    - "Part (b): Calculate power factor"
    - "Power factor = cos(φ_Z) = cos(-60°) = 0.5"

  answer_R: "75"
  answer_X: "-130"
  unit: "kΩ"
  power_factor: "0.5"
  tolerance: 2.0

explanation: |
  With a -60° phase angle, this spark is significantly capacitive. The resistance
  (75 kΩ) equals half the impedance magnitude, while the capacitive reactance
  (-130 kΩ) is 1.73× the resistance. The power factor of 0.5 means only 50% of
  the apparent power (V×I) is real power dissipated in the plasma. The other 50%
  is reactive power - energy oscillating in the capacitances. This is typical for
  Tesla coil sparks, which operate with power factors in the 0.25-0.70 range.

related_concepts: ["power-factor", "rectangular-components", "capacitive-impedance"]