Lossless transmission line at 400 kHz with L = 0.5 mH/km and C = 0.08 μF/km: compute the phase constant β (radians per km).

Introduction / Context:For a lossless transmission line (R ≈ 0, G ≈ 0), the propagation constant is purely imaginary (γ = jβ). The phase constant β determines wavelength and phase shift per unit length, and depends on frequency and the line's distributed inductance and capacitance.

Given Data / Assumptions:

• Frequency f = 400 kHz ⇒ ω = 2πf.
• L = 0.5 mH/km = 0.0005 H/km.
• C = 0.08 μF/km = 0.08 × 10^-6 F/km.
• Lossless line: β = ω √(LC).

Concept / Approach:

In a lossless line: γ = jβ with β = ω √(LC). Units must be consistent per kilometer to get β in radians/km. Accurate unit conversion is crucial.

Step-by-Step Solution:

Verification / Alternative check:

Wavelength λ = 2π/β ≈ 2π/15.9 ≈ 0.395 km; phase velocity v_p = fλ ≈ 400 kHz × 395 m ≈ 1.58 × 10^8 m/s, plausible for a cable (below c), validating the computation.

Why Other Options Are Wrong:

• 31.8, 63.6, 105.4 radians/km: correspond to scaling β by 2×, 4×, and ~6.6×; these result from arithmetic or unit mistakes.
• 7.95 radians/km: half of the correct β, often from forgetting a factor of 2 in ω = 2πf.

Common Pitfalls:

• Misreading μF as mF; always check prefixes.
• Using β = √(LC) without multiplying by ω.

Final Answer:

15.9 radians/km