id: fund-ex-04a
type: calculation
difficulty: easy
points: 8
related_lesson: fund-04
question: |
  An impedance is measured as Z = 60 + j40 kΩ.

  Calculate the phase angle φ_Z. Is this inductive or capacitive?

hints:
  - "Use φ_Z = atan(X/R)"
  - "Positive X means inductive, negative X means capacitive"
  - "The sign of φ_Z tells you about the reactive component"

solution:
  steps:
    - "Identify components: R = 60 kΩ, X = +40 kΩ"
    - "Calculate phase: φ_Z = atan(X/R) = atan(40/60) = atan(0.667) = 33.7°"
    - "Since X > 0, this is inductive"
    - "Positive phase angle confirms inductive behavior"

  answer: "33.7"
  unit: "degrees"
  type_answer: "inductive"
  tolerance: 1.0

explanation: |
  The phase angle is calculated from the ratio of reactance to resistance. The
  positive value of both X and φ_Z indicates inductive impedance - the current
  lags the voltage. This would be unusual for a Tesla coil spark circuit, which
  are typically capacitive (negative φ_Z). An inductive impedance might appear
  in the primary circuit or at very low frequencies where inductance dominates.

related_concepts: ["phase-angle", "inductive-vs-capacitive", "impedance-components"]