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48 lines
2.1 KiB
48 lines
2.1 KiB
id: fund-ex-02c
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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-02
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question: |
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A 4-foot spark is formed. The topload has C_topload = 30 pF when unloaded.
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(a) Estimate C_sh using the empirical rule
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(b) What is the total system capacitance with the spark?
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Hint: Consider how C_mut and C_sh combine in the circuit topology.
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hints:
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- "Use C_sh ≈ 2 pF/foot for part (a)"
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- "The circuit is (C_mut || R) in series with C_sh"
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- "For DC or low frequency: R || C_mut looks like just C_mut"
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- "Series capacitors: C_total = (C₁ × C₂)/(C₁ + C₂)"
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solution:
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steps:
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- "Part (a): Calculate C_sh = 2 pF/foot × 4 feet = 8 pF"
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- "Part (b): Recognize topload connects to (C_mut || R) in series with C_sh"
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- "At RF frequencies, parallel combination C_mut || R ≈ C_mut dominates"
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- "Series combination: C_series = (C_mut × C_sh)/(C_mut + C_sh)"
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- "Need to estimate C_mut. For typical geometries, C_mut ≈ 0.5 to 0.8 × C_topload"
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- "Assuming C_mut ≈ 0.6 × 30 pF = 18 pF (estimate)"
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- "C_series = (18 × 8)/(18 + 8) = 144/26 = 5.5 pF"
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- "Total system: C_total = C_topload + C_series (if in parallel to ground)"
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- "More accurately: Spark adds C_series in series, reducing total C seen from topload"
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answer_part_a: "8"
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unit_part_a: "pF"
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answer_part_b: "approximately 5.5 pF from spark circuit (but depends on C_mut estimate)"
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explanation: |
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Part (a) is straightforward using the empirical rule. Part (b) is more complex
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because the spark adds a series combination of capacitances. The exact answer
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depends on C_mut, which must be determined from FEMM or estimated based on
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geometry. The key insight is that C_mut and C_sh form a capacitive divider that
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reduces the effective capacitance seen from the topload terminal.
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In practice, the "total system capacitance" includes the topload self-capacitance
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plus the series combination of the spark circuit elements. This problem illustrates
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why FEMM extraction is necessary for accurate modeling - C_mut cannot be calculated
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from simple formulas.
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related_concepts: ["series-capacitance", "capacitive-divider", "total-capacitance", "FEMM-extraction"]
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