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Lossless line, L Z0 β Γ

Short answer

L = 1/(v²C) = 0.533 μH/m. Z₀ = √(L/C) = 133.3 Ω. β = ω/v = 2.51 rad/m at 100 MHz. Γ = (50 − 133.3)/(50 + 133.3) = −0.455.

The velocity of propagation in a lossless transmission line is 2.5×10⁸ m/s. If the capacitance of the line is 30 pF/m, find: (a) inductance of the line (b) characteristic impedance (c) phase constant at 100 MHz (d) reflection coefficient if the line is terminated with a resistive load of 50 Ω.

Given: v = 2.5×10⁸ m/s, C = 30 pF/m, lossless line.

For a lossless line v = 1/√(LC) and Z₀ = √(L/C).

(a) Inductance

L = 1/(v² C) = 1 / [(2.5×10⁸)² × 30×10⁻¹²]
  = 1 / (6.25×10¹⁶ × 3×10⁻¹¹)
  = 1 / (1.875×10⁶)
  = 0.5333×10⁻⁶ H/m

L = 0.533 μH/m

(b) Characteristic impedance

Z₀ = √(L/C) = √(0.5333×10⁻⁶ / 30×10⁻¹²) = √17778 = 133.3 Ω

(Also Z₀ = 1/(vC) = 1/(2.5×10⁸ × 30×10⁻¹²) = 133.3 Ω ✓.)

Z₀ = 133.3 Ω

(c) Phase constant at 100 MHz

β = ω/v = 2π × 100×10⁶ / 2.5×10⁸ = 2.513 rad/m

β = 2.51 rad/m (λ = 2.5 m)

(d) Reflection coefficient for R_L = 50 Ω

Γ = (R_L − Z₀)/(R_L + Z₀) = (50 − 133.3)/(50 + 133.3) = −83.3/183.3
  = −0.4545

Γ = −0.455 (magnitude 0.455, angle 180°), and S = (1 + 0.4545)/(1 − 0.4545) = 2.67.

✓ Answer checked by Claude Opus 5.5 (high effort)

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