Transmission line propagation constant – Formula in terms of primary constants Given the per-unit-length parameters R (series resistance), L (series inductance), G (shunt conductance), and C (shunt capacitance), what is the correct expression for the complex propagation constant γ (in nepers/m + j radians/m)?
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Aγ = sqrt((R + jωL) * (G + jωC))
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Bγ = (R + jωL) / (G + jωC)
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Cγ = (G + jωC) / (R + jωL)
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Dγ = sqrt(RG) + j * sqrt(LC)
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Eγ = j * sqrt(LC)
Answer
Correct Answer: γ = sqrt((R + jωL) * (G + jωC))
Explanation
Introduction:The propagation constant γ characterizes how voltage and current waves attenuate and change phase as they travel along a transmission line. It depends on both the series and shunt per-unit-length parameters of the line and varies with frequency in a way captured by the telegrapher's equations. Engineers use γ to predict signal integrity, dispersion, and attenuation.
Given Data / Assumptions:
- Uniform transmission line with parameters R, L, G, C (per meter).
- Sinusoidal steady state with angular frequency ω = 2πf.
- Linear, time-invariant behavior; standard reference impedance.
Concept / Approach:
From the telegrapher's equations, the dispersion relation is γ^2 = (R + jωL) * (G + jωC). Taking the principal square root gives γ = α + jβ, where α is the attenuation constant and β is the phase constant. Special cases include the low-loss line approximation and the lossless line where R = G = 0. These limits provide useful checks on the general expression.
Step-by-Step Solution:
1) Write per-unit-length impedance Z′ = R + jωL and admittance Y′ = G + jωC.2) Use the relation γ = √(Z′Y′) → γ = √((R + jωL)(G + jωC)).3) Decompose into α and β if needed for design: γ = α + jβ.Verification / Alternative check:
In the lossless limit (R = G = 0): γ = j * ω * √(LC). In low-loss lines, α ≈ (R/2) * √(C/L) + (G/2) * √(L/C) and β ≈ ω * √(LC), consistent with the exact formula, providing a sanity check for approximations used in hand calculations.
Why Other Options Are Wrong:
Ratios (options B, C) are dimensionally inconsistent for γ. Option D incorrectly separates α and β as independent square roots. Option E applies only to the lossless special case, not the general expression that includes R and G.
Common Pitfalls:
Forgetting frequency dependence; misusing the lossless formula when R and G are nonzero; confusing α and β with impedance/admittance magnitudes rather than with the complex square root of their product.
Final Answer:
γ = sqrt((R + jωL) * (G + jωC)).