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46 lines
1.8 KiB
46 lines
1.8 KiB
id: fund-ex-03b
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type: calculation
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difficulty: medium
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points: 12
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related_lesson: fund-03
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question: |
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An admittance is measured as Y = 2.0 + j4.5 μS.
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Convert this to impedance Z in both rectangular and polar forms.
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hints:
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- "Use |Z| = 1/|Y| for the magnitude"
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- "Use φ_Z = -φ_Y for the phase angle"
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- "Calculate |Y| = √(Re{Y}² + Im{Y}²)"
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- "For rectangular: Z = R + jX where R = |Z|cos(φ_Z), X = |Z|sin(φ_Z)"
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solution:
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steps:
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- "Calculate magnitude of Y: |Y| = √(2.0² + 4.5²) = √(4 + 20.25) = √24.25 = 4.92 μS"
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- "Calculate magnitude of Z: |Z| = 1/|Y| = 1/(4.92×10⁻⁶) = 203 kΩ"
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- "Calculate admittance phase: φ_Y = atan(4.5/2.0) = atan(2.25) = 66.0°"
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- "Calculate impedance phase: φ_Z = -φ_Y = -66.0°"
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- "Polar form: Z = 203 kΩ ∠-66.0°"
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- "Calculate rectangular components:"
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- "R = |Z| × cos(φ_Z) = 203 × cos(-66°) = 203 × 0.407 = 82.6 kΩ"
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- "X = |Z| × sin(φ_Z) = 203 × sin(-66°) = 203 × (-0.914) = -185.5 kΩ"
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- "Rectangular form: Z = 82.6 - j185.5 kΩ"
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answer_polar: "203 kΩ ∠-66.0°"
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answer_rectangular: "82.6 - j185.5 kΩ"
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magnitude: "203"
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phase: "-66.0"
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resistance: "82.6"
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reactance: "-185.5"
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unit: "kΩ"
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tolerance: 2.0
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explanation: |
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This conversion demonstrates the fundamental relationship between admittance and
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impedance: they are reciprocals in the complex plane. The key relationships are
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|Z| = 1/|Y| and φ_Z = -φ_Y. Note the opposite sign of the phase angle - this is
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critical! A positive admittance phase (capacitive susceptance) corresponds to a
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negative impedance phase (capacitive reactance). The negative reactance confirms
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this is a capacitive impedance, as expected for spark circuits.
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related_concepts: ["admittance-to-impedance", "complex-reciprocal", "phase-relationship", "polar-rectangular"]
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