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Quarter-Wave Transformer Design: Match a 200 Ω load to a 50 Ω line—what should be the transformer’s characteristic impedance?

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:

  • Load impedance ZL = 200 Ω.
  • Line impedance Zi = 50 Ω.
  • Quarter-wave (λ/4) transformer section, lossless.


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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