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Primary constants from Z0, α, β

Short answer

Using Z = γZ₀ and Y = γ/Z₀ gives R ≈ 13.7 Ω/m, L ≈ 3.49 mH/m, C ≈ 8.16 nF/m, and G ≈ 0.

Determine the primary constants (R, L, C and G) on the transmission line when the measurement on the line at 1 kHz gave the following results: Z0 = 710∠−16°, α = 0.01 neper/m and β = 0.035 rad/m.

Given at f = 1 kHz: Z₀ = 710∠−16° Ω, α = 0.01 Np/m, β = 0.035 rad/m.

ω = 2π × 1000 = 6283.2 rad/s
γ = α + jβ = 0.01 + j0.035 = 0.03640 ∠74.05° m⁻¹
Z₀ = 710∠−16° = 682.50 − j195.70 Ω

Relations: Z₀ = √(Z/Y), γ = √(ZY), so

Z = R + jωL = γ Z₀
Y = G + jωC = γ / Z₀

Series impedance

Z = 0.03640∠74.05° × 710∠−16°
  = 25.84 ∠58.05°
  = 13.67 + j21.93 Ω/m

So R = 13.67 Ω/m and ωL = 21.93 Ω/m:

L = 21.93 / 6283.2 = 3.49×10⁻³ H/m

Shunt admittance

Y = 0.03640∠74.05° / 710∠−16°
  = 5.127×10⁻⁵ ∠90.05°
  = −4.9×10⁻⁸ + j5.127×10⁻⁵ S/m

The real part is essentially zero (G ≈ 0, a numerical rounding result), and

C = 5.127×10⁻⁵ / 6283.2 = 8.16×10⁻⁹ F/m

Answer: R ≈ 13.7 Ω/m, L ≈ 3.49 mH/m, C ≈ 8.16 nF/m, G ≈ 0.

✓ Answer checked by Claude Opus 5.5 (high effort)

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