id: fund-ex-03b
type: calculation
difficulty: medium
points: 12
related_lesson: fund-03
question: |
  An admittance is measured as Y = 2.0 + j4.5 μS.

  Convert this to impedance Z in both rectangular and polar forms.

hints:
  - "Use |Z| = 1/|Y| for the magnitude"
  - "Use φ_Z = -φ_Y for the phase angle"
  - "Calculate |Y| = √(Re{Y}² + Im{Y}²)"
  - "For rectangular: Z = R + jX where R = |Z|cos(φ_Z), X = |Z|sin(φ_Z)"

solution:
  steps:
    - "Calculate magnitude of Y: |Y| = √(2.0² + 4.5²) = √(4 + 20.25) = √24.25 = 4.92 μS"
    - "Calculate magnitude of Z: |Z| = 1/|Y| = 1/(4.92×10⁻⁶) = 203 kΩ"
    - "Calculate admittance phase: φ_Y = atan(4.5/2.0) = atan(2.25) = 66.0°"
    - "Calculate impedance phase: φ_Z = -φ_Y = -66.0°"
    - "Polar form: Z = 203 kΩ ∠-66.0°"
    - "Calculate rectangular components:"
    - "R = |Z| × cos(φ_Z) = 203 × cos(-66°) = 203 × 0.407 = 82.6 kΩ"
    - "X = |Z| × sin(φ_Z) = 203 × sin(-66°) = 203 × (-0.914) = -185.5 kΩ"
    - "Rectangular form: Z = 82.6 - j185.5 kΩ"

  answer_polar: "203 kΩ ∠-66.0°"
  answer_rectangular: "82.6 - j185.5 kΩ"
  magnitude: "203"
  phase: "-66.0"
  resistance: "82.6"
  reactance: "-185.5"
  unit: "kΩ"
  tolerance: 2.0

explanation: |
  This conversion demonstrates the fundamental relationship between admittance and
  impedance: they are reciprocals in the complex plane. The key relationships are
  |Z| = 1/|Y| and φ_Z = -φ_Y. Note the opposite sign of the phase angle - this is
  critical! A positive admittance phase (capacitive susceptance) corresponds to a
  negative impedance phase (capacitive reactance). The negative reactance confirms
  this is a capacitive impedance, as expected for spark circuits.

related_concepts: ["admittance-to-impedance", "complex-reciprocal", "phase-relationship", "polar-rectangular"]