Low-loss line identity — geometric-mean relation For a (low-loss) transmission line section, the characteristic impedance satisfies Z0 = sqrt(Zoc * Zsc). State whether this relation is true or false.

Introduction / Context:When measuring an unknown transmission line, one classic method uses its input impedance under open-circuit and short-circuit terminations. For a uniform low-loss section, there is a convenient identity relating these two measurements to the characteristic impedance Z0. This question asks whether that identity is valid.

Given Data / Assumptions:

• Uniform, low-loss line section at a given frequency.
• Measured input impedances: Z_oc (with load open) and Z_sc (with load short).

Concept / Approach:The transmission-line input impedance formula yields, after algebraic manipulation for open and short cases, a product relation. Under the low-loss assumption, magnitudes (or complex values near the real axis) satisfy:

This serves as a practical way to extract Z0 from bench measurements without directly accessing internal L and C values.

Step-by-Step Solution:

Verification / Alternative check:Laboratory practice frequently uses the geometric-mean method as a quick Z0 check for low-loss cables and waveguide sections.

Why Other Options Are Wrong:

• False: would reject a well-established identity used in RF measurements; deviations occur only when losses are non-negligible or measurements are off-frequency for the assumed electrical length.

Common Pitfalls:Applying the identity to highly lossy lines without correction; confusing magnitudes with complex values when significant attenuation is present.

Final Answer:True