Characteristic Impedance from Per-Unit-Length L and C for a Coaxial Line A coaxial line has per-unit-length parameters L = 500 nH/m and C = 50 pF/m. Compute the characteristic impedance Z0.
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A500 Ω
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B250 Ω
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C100 Ω
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D50 Ω
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E75 Ω
Answer
Correct Answer: 100 Ω
Explanation
Introduction / Context:For a lossless (or low-loss) transmission line, the characteristic impedance Z0 depends on its per-unit-length inductance L and capacitance C.
Given Data / Assumptions:
- L = 500 nH/m = 500 × 10^-9 H/m
- C = 50 pF/m = 50 × 10^-12 F/m
- Line is treated as lossless for Z0 calculation.
Concept / Approach:
For a lossless line, Z0 = sqrt(L / C). We will compute this directly using the given values.
Step-by-Step Solution:
1) Form the ratio L / C = (500 × 10^-9) / (50 × 10^-12) = (500/50) × 10^3 = 10 × 10^3 = 10^4.2) Take square root: Z0 = sqrt(10^4) = 100 Ω.3) Therefore, the characteristic impedance is 100 ohms.Verification / Alternative check:
Dimensional check: sqrt(H/F) has units of ohms. Numerical value agrees with typical coax impedances when L/C ratio is 10^4.
Why Other Options Are Wrong:
500 Ω, 250 Ω, 50 Ω, and 75 Ω do not satisfy Z0 = sqrt(L/C) for the supplied numbers.
Common Pitfalls:
Arithmetic slips with scientific notation; forgetting that Z0 depends on the ratio L/C, not on frequency for the ideal lossless case.
Final Answer:
100 Ω.