In transmission-line theory, when is the input impedance of a line exactly equal to the line’s characteristic impedance Z0? (Consider steady-state sinusoidal operation and negligible losses.)

Introduction:
Matching a transmission line to its load is a foundational idea in RF and microwave engineering. The question tests the concept of characteristic impedance Z0 and how proper termination eliminates reflections so that the input impedance seen at the source equals Z0.

Given Data / Assumptions:

• Uniform transmission line with characteristic impedance Z0.
• Steady-state sinusoidal operation.
• Losses are negligible for conceptual clarity.

Concept / Approach:

When a line is terminated in a load ZL equal to Z0, the reflection coefficient at the load, Gamma_L, is zero: Gamma_L = (ZL − Z0) / (ZL + Z0) = 0. With no reflected wave, the voltage/current ratio is constant along the line and equals Z0 everywhere, including at the input.

Step-by-Step Solution:

Verification / Alternative check:

Using the general input-impedance formula for a lossless line: Zin = Z0 * (ZL + j Z0 tan βl) / (Z0 + j ZL tan βl). Substituting ZL = Z0 gives Zin = Z0 for any electrical length βl.

Why Other Options Are Wrong:

• Open-circuit: |Gamma| = 1, Zin varies with length and can be reactive or infinite.
• Short-circuit: |Gamma| = 1, Zin varies with length and can be reactive or zero.
• Pure inductance / capacitance: reactive mismatch ⇒ reflections, Zin ≠ Z0 in general.

Common Pitfalls:

Confusing “no reflection” with “no loss.” Matching does not mean zero loss; it means the absence of reflections. Also, assuming a certain length (like λ/4) can fix mismatch; only ZL = Z0 guarantees Zin = Z0 for any length.

Final Answer:

line terminated in Z0 (matched load)