At its design frequency, a quarter-wave open-circuited stub is equivalent to which reactive network at its input?

Introduction / Context:Quarter-wave stubs are fundamental matching elements in RF/microwave design. An open-circuited λ/4 stub transforms boundary conditions and can mimic resonant L-C networks at the input, enabling compact impedance transformations.

Given Data / Assumptions:

• Lossless, uniform transmission line stub.
• Electrical length l = λ/4 at the frequency of interest.
• Open circuit at the far end.

Concept / Approach:

The input impedance of an open-circuited stub is Z_in = −j Z0 cot(β l). At l = λ/4, βl = π/2, cot(π/2) = 0, so Z_in = 0 (a short). A zero input impedance at a single frequency corresponds to a series resonance (equivalent L and C in series producing a short at resonance). Thus, the open λ/4 stub acts like a series-resonant L-C at its input.

Step-by-Step Solution:

Verification / Alternative check:

A short-circuited λ/4 stub yields Z_in → ∞ (parallel resonance). By duality, the open-ended λ/4 stub must yield series resonance at the input, consistent with design tables.

Why Other Options Are Wrong:

• Pure L or pure C: The input is a short at resonance, not purely inductive or capacitive.
• Parallel LC: That would correspond to a short-circuited λ/4 stub or open-circuited λ/2 behavior.
• Large resistor: Lossless stubs are reactive, not resistive (aside from matching transformations).

Common Pitfalls:

Confusing open and shorted stubs; forgetting that resonance type (series vs parallel) depends on termination.

Final Answer:

Inductance and capacitance in series (series resonant)