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).
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A15.9 radians/km
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B31.8 radians/km
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C63.6 radians/km
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D105.4 radians/km
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E7.95 radians/km
Answer
Correct Answer: 15.9 radians/km
Explanation
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:
Compute ω: ω = 2π × 400 × 10^3 ≈ 2.513 × 10^6 rad/s.Compute LC: L × C = 0.0005 × 0.08 × 10^-6 = 4 × 10^-11.Compute √(LC): √(4 × 10^-11) = 2 × 10^-5.5 ≈ 6.324 × 10^-6.Compute β: β = ω √(LC) ≈ 2.513 × 10^6 × 6.324 × 10^-6 ≈ 15.9 radians/km.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