Transmission lines: Given open-circuit and short-circuit input impedances of 20 Ω and 5 Ω respectively, determine the characteristic impedance Z0 using the standard relationship for a uniform line.

Introduction / Context:
In transmission-line theory for radio-frequency and power systems, the characteristic impedance Z0 is a fundamental parameter that governs reflections, matching, and power transfer. One classical laboratory method to estimate Z0 for a uniform line with unknown distributed parameters is to measure its driving-point impedance with the far end open and then with the far end shorted. This question checks your recall of the direct formula linking those two measurements to Z0.

Given Data / Assumptions:

• The line is uniform and reciprocal, with frequency held constant.
• Measured input impedances: Z_oc = 20 Ω (far end open), Z_sc = 5 Ω (far end short).
• Losses, if any, do not affect the product relation used for Z0 estimation.

Concept / Approach:

For a uniform transmission line, the measured open-circuit and short-circuit input impedances at the same frequency satisfy the identity Z0 = sqrt(Z_oc * Z_sc). This follows from the hyperbolic expressions for input impedance and the fact that the propagation factor cancels when the two extreme terminations (open and short) are multiplied. The relationship is widely used in practical line characterization when direct measurement of Z0 is inconvenient.

Step-by-Step Solution:

Verification / Alternative check:

Check units: multiplying two impedances yields Ω^2; the square root returns Ω, which is consistent. The numerical result 10 Ω lies between the two measured values, which is typical for many real lines, providing a quick sanity check.

Why Other Options Are Wrong:

• 100 Ω and 50 Ω: these ignore the square root relationship and instead reflect the product or sum/division guesses.
• 25 Ω: would correspond to sqrt(625) and not match the given data.
• 5 √5 Ω (≈ 11.18 Ω): close but not exact; arises from incorrect arithmetic on the product.

Common Pitfalls:

• Using Z0 = (Z_oc + Z_sc)/2, which is not valid.
• Mixing up open and short results, or measuring at different frequencies so the identity no longer holds.

Final Answer:

10 Ω