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41 lines
1.4 KiB
41 lines
1.4 KiB
id: fund-ex-04b
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type: multi-part
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difficulty: medium
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points: 15
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related_lesson: fund-04
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question: |
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A spark has φ_Z = -60°. The impedance magnitude is |Z| = 150 kΩ.
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(a) Find R and X (rectangular components)
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(b) Calculate the power factor
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hints:
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- "Use R = |Z| × cos(φ_Z) and X = |Z| × sin(φ_Z)"
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- "Power factor = cos(φ_Z)"
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- "Negative angle means capacitive reactance (X < 0)"
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solution:
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steps:
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- "Part (a): Calculate resistance"
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- "R = |Z| × cos(φ_Z) = 150 × cos(-60°) = 150 × 0.5 = 75 kΩ"
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- "Calculate reactance"
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- "X = |Z| × sin(φ_Z) = 150 × sin(-60°) = 150 × (-0.866) = -130 kΩ"
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- "Rectangular form: Z = 75 - j130 kΩ"
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- "Part (b): Calculate power factor"
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- "Power factor = cos(φ_Z) = cos(-60°) = 0.5"
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answer_R: "75"
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answer_X: "-130"
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unit: "kΩ"
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power_factor: "0.5"
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tolerance: 2.0
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explanation: |
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With a -60° phase angle, this spark is significantly capacitive. The resistance
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(75 kΩ) equals half the impedance magnitude, while the capacitive reactance
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(-130 kΩ) is 1.73× the resistance. The power factor of 0.5 means only 50% of
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the apparent power (V×I) is real power dissipated in the plasma. The other 50%
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is reactive power - energy oscillating in the capacitances. This is typical for
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Tesla coil sparks, which operate with power factors in the 0.25-0.70 range.
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related_concepts: ["power-factor", "rectangular-components", "capacitive-impedance"]
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