Impedance Inversion Techniques on Transmission Lines Which of the following realizations provides impedance inversion (Z_in ≈ Z0^2 / ZL) at a design frequency?

Introduction / Context:
Impedance transformation is central to RF matching. A specific and powerful case is impedance inversion, where a load impedance is transformed roughly to Z0^2 / ZL using a transmission-line section.

Given Data / Assumptions:

• Lossless line sections at a specific frequency.
• Characteristic impedance Z0 chosen for desired transformation.
• Single-frequency (narrowband) behavior is acceptable for design.

Concept / Approach:

A quarter-wave (λ/4) line produces the inversion: at electrical length 90 degrees, Z_in = (Z0^2) / ZL. This is the classic impedance inverter used in filters and matching networks. Half-wave lines repeat the load (Z_in ≈ ZL), while single stubs (open or short) mainly provide reactive compensation rather than full inversion.

Step-by-Step Solution:

Verification / Alternative check:

Filter theory and matching texts describe quarter-wave inverters as fundamental elements in band-pass/band-stop networks.

Why Other Options Are Wrong:

Short/open stubs: supply reactive tuning, not inversion. Half-wave line: repeats the load. L-pad: resistive network causing loss, not ideal inversion.

Common Pitfalls:

Assuming any 90-degree stub acts as an inverter; only the through line of 90 degrees with Z0 set appropriately provides Z0^2 / ZL transformation.

Final Answer:

a quarter wave line