A uniform transmission line has a characteristic impedance of 400 Ω for a 10 km length. If the line length is increased to 100 km (same construction), what is the new characteristic impedance?

Introduction / Context:
Characteristic impedance Z0 is a property of a transmission line per unit length and depends on its geometry and materials. It does not depend on the overall physical length of the line segment used, a fact often misunderstood by beginners.

Given Data / Assumptions:

• Uniform line construction (same conductor size, spacing, dielectric).
• Same operating frequency band (so per-unit-length parameters remain the same).
• Initial Z0 = 400 Ω for the given line type.

Concept / Approach:

The formula Z0 = sqrt((R + jωL) / (G + jωC)) is based on per-unit-length parameters R, L, G, and C. If the physical construction and materials are unchanged, these parameters per unit length are unchanged, and thus Z0 remains the same, independent of total length.

Step-by-Step Solution:

Verification / Alternative check:

Measurement with a network analyzer shows that at a fixed frequency, the input impedance of a very long matched line equals Z0 irrespective of length. Length affects phase delay and loss, not Z0 itself.

Why Other Options Are Wrong:

• 4000 Ω, 40 Ω, 4 Ω: Incorrectly assume Z0 scales with length.
• 'Depends on frequency only': While Z0 depends on frequency through R, G, L, C, here frequency is unchanged and length does not alter Z0.

Common Pitfalls:

Confusing characteristic impedance with input impedance of a finite length (which is length-dependent unless terminated in Z0); mixing series/parallel lumped logic with distributed line behavior.

Final Answer:

400 Ω