Difficulty: Easy
Correct Answer: 100 Ω
Explanation:
Introduction / Context:Quarter-wave transformers provide narrowband impedance matching between a load and a transmission line. The transformer's characteristic impedance is chosen to minimize reflections at the operating frequency.
Given Data / Assumptions:
Concept / Approach:For a λ/4 transformer, the characteristic impedance Zt is the geometric mean of the load and line impedances: Zt = sqrt(ZL * Zi). This ensures Zin seen by the line equals Zi at the design frequency.
Step-by-Step Solution:
Zt = sqrt(ZL * Zi) = sqrt(200 * 50).Compute product: 200 * 50 = 10000.Take square root: sqrt(10000) = 100 Ω.Verification / Alternative check:
If Zt = 100 Ω, the transformer input at resonance becomes Zi, eliminating reflections (Γ ≈ 0).Why Other Options Are Wrong:
40 Ω or 4 Ω: Too low and will not match 200 to 50 Ω.10000 Ω: Nonsensical magnitude for RF lines.150 Ω: Not the geometric mean; would leave a mismatch.Common Pitfalls:
Using arithmetic mean instead of geometric mean; forgetting quarter-wave constraint (only exact at the design frequency).Final Answer:
100 Ω
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