Stub behavior: A lossless transmission line that is slightly shorter than λ/4 and short circuited at its far end behaves as which lumped element at the input?

Introduction / Context:
Quarter-wave stubs are building blocks for impedance matching in RF and microwave circuits. By choosing the termination and slightly adjusting the electrical length relative to λ/4, a designer can realize an effective lumped inductance or capacitance from a distributed transmission line segment. This saves components at high frequency and leverages the low loss of transmission lines.

Given Data / Assumptions:

• Lossless transmission line with characteristic impedance Z0 and propagation constant β.
• Short circuit termination at the far end of the stub.
• Electrical length slightly less than λ/4 so that βl is a little smaller than π/2.

Concept / Approach:

The input impedance of a short circuited, lossless line is Z_in = j Z0 tan(βl). For lengths from 0 to just under λ/4, tan(βl) is positive, producing a positive imaginary input impedance, which is inductive. Exactly at λ/4 the stub transforms a short to an open, and beyond λ/4 the sign flips to capacitive. Thus the specified length and termination produce an inductive effect.

Step-by-Step Solution:

Verification / Alternative check:

On a Smith chart normalized to Z0, a short circuited stub traces the inductive arc as its length increases from 0 to λ/4, crossing to the capacitive side only after passing λ/4. Lab measurements of standing waves confirm this behavior.

Why Other Options Are Wrong:

• Capacitance applies to an open circuited stub below λ/4 or to a short circuited stub above λ/4.
• Series or parallel L–C require two reactive elements. A single short stub near λ/4 is purely reactive of one sign.
• Pure resistance equal to Z0 would require a special lossy condition or a particular transformer arrangement, not present here.

Common Pitfalls:

Mixing up the sign for open and short stubs, and forgetting that exactly at λ/4 the impedance inverts. Also, some learners assume losses introduce resistance in the ideal formula; while losses exist in practice, the theoretical input impedance here is purely imaginary.

Final Answer:

Inductance L (inductive reactance)