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)?

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:

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)).