A complex load Z_L = 75 − j50 Ω is connected to a lossless 75 Ω coaxial line at 10 GHz. What is the most practical single-stub matching method at microwave frequencies to achieve a perfect match?
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AAdd a series inductance right at the load
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BConnect a short-circuited shunt stub exactly at the load flange
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CUse a short-circuited shunt stub placed at a specific distance from the load
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DUse a series capacitance at some distance from the load
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EUse an open-circuited series stub at the load
Answer
Correct Answer: Use a short-circuited shunt stub placed at a specific distance from the load
Explanation
Introduction / Context:At microwave frequencies, distributed matching techniques are favored because lumped components become lossy, parasitic-dominated, and hard to realize. A single-stub tuner on a transmission line is a standard and flexible solution.
Given Data / Assumptions:
- Line is lossless with Z0 = 75 Ω.
- Load is Z_L = 75 − j50 Ω.
- Operating frequency is 10 GHz (wavelength is short; distributed elements are convenient).
Concept / Approach:
The single-stub shunt tuner works by first moving along the line from the load to a point where the normalized admittance has a real part of 1 (or a convenient value). A short-circuited shunt stub at that point cancels the residual susceptance to achieve a match.
Step-by-Step Solution:
1) Normalize the load: z_L = Z_L/Z0 = (75 − j50)/75 = 1 − j(50/75) = 1 − j0.667.2) Convert to admittance at the load: y_L = 1/z_L (complex inversion).3) Move a distance d toward the generator until Re{y(d)} = 1 (conductance equal to 1/Z0).4) At that point, add a short-circuited shunt stub with susceptance b_stub = −Im{y(d)} to cancel the reactive part.5) With the susceptance canceled, the input admittance equals 1/Z0, giving a perfect match.Verification / Alternative check:
This procedure is exactly what the Smith chart single-stub design accomplishes: rotate from the load toward the generator to a unity-conductance circle, then pick the stub length to supply equal and opposite susceptance.
Why Other Options Are Wrong:
- Add a series inductance at the load: Practical but less flexible at 10 GHz; hard to realize accurately and does not exploit the line transformation.
- Short-circuited shunt stub exactly at the load: May not provide the precise susceptance needed because the transformed admittance at the load is not at a convenient point.
- Series capacitance at some distance: Series elements are less convenient for coax implementations and often require tight tolerances.
- Open-circuited series stub: Series stubs are harder to implement; open stubs risk radiation and dispersion.
Common Pitfalls:
Forgetting to move to a unity-conductance point before adding the stub; choosing an open stub in exposed environments; ignoring line losses and connector repeatability.
Final Answer:
Use a short-circuited shunt stub placed at a specific distance from the load