Reactive Behavior of a Short, Open-Circuited Line A transmission line segment of length less than λ/4 that is open-circuited at the far end behaves equivalently as which lumped element at the input?

Introduction / Context:
Short transmission-line stubs are widely used to synthesize reactive elements for matching. The sign of the reactance seen at the input depends on whether the stub is short- or open-circuited and on its electrical length relative to λ/4.

Given Data / Assumptions:

• Lossless line with characteristic impedance Z0.
• Open-circuited at the far end.
• Physical length ℓ is small (ℓ < λ/4) at the operating frequency.

Concept / Approach:

The input reactance of an open-circuited lossless line is X_in = − Z0 * cot(βℓ), where β = 2π/λ. For small electrical lengths (βℓ ≪ π/2), cot(βℓ) is large and positive, so X_in is negative (capacitive). Thus, a short open stub emulates a capacitor; by contrast, a short short-circuited stub (Z_in = j Z0 tan βℓ) is inductive for small βℓ.

Step-by-Step Solution:

Verification / Alternative check:

Smith chart movement from an open at the right edge toward the generator by a small fraction of λ rotates downward along the susceptance circle, confirming capacitive susceptance.

Why Other Options Are Wrong:

Inductance (A) corresponds to a short-circuited short stub; series/parallel L–C models (C, D) do not represent the pure reactance of a very short, lossless open stub; a pure resistor (E) is incorrect except at special lengths with loss.

Common Pitfalls:

Confusing λ/4 transformations (where opens convert to shorts) with sub-quarter-wave behavior; forgetting sign conventions for reactive input impedance.

Final Answer:

capacitance C