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?

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:

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