Transmission Line Parameters

A transmission line is not simply a wire — it is a distributed electromagnetic structure whose electrical behaviour depends on its physical geometry and the surrounding medium. The four distributed parameters R, L, G, and C per unit length form the foundation of all transmission line analysis. Without understanding these parameters physically, the telegrapher's equations and their solutions remain abstract symbols disconnected from real engineering.

Core Concept Explanation

A transmission line supports a guided wave whose fields are largely confined between two conductors — a signal conductor and a return conductor. Unlike lumped circuits where voltage and current are the same everywhere in a wire, on a transmission line these quantities vary continuously along the length. This is because the line dimensions are comparable to or larger than the signal wavelength. The distributed model captures this by representing every infinitesimal segment Δz of the line with four small lumped elements: RΔz, LΔz, GΔz, and CΔz.

The series resistance R (Ω/m) accounts for the ohmic losses in the conductors due to finite conductivity. At high frequencies, the current crowds into a thin skin depth near the conductor surface, increasing the effective resistance. This is the skin effect, and it causes R to increase with frequency as sqrt(f). For ideal lossless lines, R = 0.

The series inductance L (H/m) represents the magnetic flux linkage per unit length between the two conductors. It arises from the magnetic field surrounding the conductors when current flows. L depends on the geometry: the spacing between conductors and the conductor radius for a two-wire line, or the inner and outer radii for a coaxial line. For a lossless line, L stores and releases energy without dissipation.

The shunt conductance G (S/m) represents the leakage of current through the dielectric separating the conductors. A perfect dielectric has G = 0. In practice, all dielectric materials have a small loss tangent, and G is proportional to the imaginary part of the complex permittivity. G also increases with frequency for most real dielectrics.

The shunt capacitance C (F/m) represents the electric charge stored per unit voltage between the two conductors. It arises from the electric field between the conductors. C depends on geometry and permittivity of the dielectric. For a lossless line, C also stores and releases energy without loss.

Mathematical Expressions

For standard geometries, the RLGC parameters can be calculated from material and geometric properties. For a coaxial cable with inner radius a, outer radius b, conductivity of conductors σc, and dielectric permittivity ε = εᵣε₀, loss tangent tanδ:

L = (μ/2π) ln(b/a) H/m. C = 2πε/ln(b/a) F/m. G = 2πωε tanδ/ln(b/a) S/m. R = (1/2π)(1/σc·δs)(1/a + 1/b) Ω/m, where δs = sqrt(2/ωμσc) is the skin depth.

Notice that LC = με and G/C = ωtanδ for any TEM line regardless of geometry. The characteristic impedance Z₀ = sqrt((R + jωL)/(G + jωC)), which for a lossless line reduces to Z₀ = sqrt(L/C). The propagation constant γ = sqrt((R + jωL)(G + jωC)) = α + jβ, where α is the attenuation constant and β is the phase constant.

Practical Understanding

In practice, transmission line parameters are specified per unit length in datasheets. A typical 50 Ω coaxial cable operating at 1 GHz might have L ≈ 250 nH/m, C ≈ 100 pF/m, R ≈ 0.8 Ω/m, and G ≈ 0.1 mS/m. From these, Z₀ = sqrt(250n/100p) = 50 Ω, confirming the geometric design.

The LC product determines wave velocity on the line: v_p = 1/sqrt(LC) = 1/sqrt(με). For a coaxial line filled with PTFE (εᵣ ≈ 2.1), v_p = c/sqrt(2.1) ≈ 0.69c. This is the velocity factor quoted in cable specifications.

Minimising conductor losses (R) requires large conductor cross-sections and high-conductivity materials (silver, copper). Minimising dielectric losses (G) requires low-loss dielectrics such as PTFE or air. High-power RF cables often use air-spaced or foam dielectric to keep both R and G low.

Given:
Coaxial cable:
  Inner conductor radius: a = 0.5 mm
  Outer conductor radius: b = 3.5 mm
  Dielectric: PTFE, εᵣ = 2.1, tanδ = 0.0002
  Frequency: f = 1 GHz → ω = 2π × 10⁹ rad/s

Why this formula applies:
Coaxial geometry → use standard coaxial RLGC formulas

Formula:
L = (μ₀/2π) ln(b/a)
C = 2πε₀εᵣ / ln(b/a)
G = ωC·tanδ
Z₀ = sqrt(L/C)  [lossless approximation]

Substitution:
ln(b/a) = ln(3.5/0.5) = ln(7) = 1.9459
L = (4π×10⁻⁷ / 2π) × 1.9459 = 2×10⁻⁷ × 1.9459
C = (2π × 8.854×10⁻¹² × 2.1) / 1.9459

Calculation:
L = 3.892 × 10⁻⁷ H/m ≈ 389 nH/m
C = (2π × 18.593×10⁻¹²) / 1.9459 = 116.8×10⁻¹²/1.9459 ≈ 60.0 pF/m
G = 2π×10⁹ × 60×10⁻¹² × 0.0002 = 0.0754 mS/m
Z₀ = sqrt(389×10⁻⁹ / 60×10⁻¹²) = sqrt(6483) ≈ 80.5 Ω

Final Answer:
L ≈ 389 nH/m, C ≈ 60 pF/m, G ≈ 0.075 mS/m
Characteristic impedance Z₀ ≈ 80.5 Ω

Exam Tip: GATE frequently uses the result LC = με for a TEM line — this is geometry-independent. Also remember that for a lossless line, Z₀ = sqrt(L/C) and v_p = 1/sqrt(LC). A very common GATE trap is to confuse Z₀ = sqrt(L/C) with Z₀ = sqrt(R/G) — the latter applies only under the distortionless line condition (R/L = G/C), not in general.

Mechanism — Parameter Physical Origins

• R arises from conductor ohmic dissipation and increases with frequency due to the skin effect — at high frequency, current flows only in a thin surface layer of depth δs = sqrt(2/ωμσ).
• L is determined by magnetic flux linkage between conductors. For coax: L = (μ₀/2π)ln(b/a). It is largely frequency-independent for good conductors.
• C is determined by the electric field energy stored in the dielectric between conductors. For coax: C = 2πε/ln(b/a).
• G represents dielectric loss. For a dielectric with loss tangent tanδ, G = ωC·tanδ, meaning G increases linearly with frequency.
• The product LC = με is a universal result for TEM lines, independent of geometry. This links line parameters to the medium's material properties.

Quick Revision

• Four parameters: R (Ω/m) — series conductor loss; L (H/m) — series inductance; G (S/m) — shunt dielectric loss; C (F/m) — shunt capacitance.
• Lossless line: R = 0, G = 0. Distortionless line: R/L = G/C (phase distortion eliminated).
• Z₀ = sqrt(L/C) [lossless]; γ = sqrt((R+jωL)(G+jωC)); v_p = 1/sqrt(LC) = 1/sqrt(με).
• LC = με always holds for TEM transmission lines, regardless of geometry.
• R increases with sqrt(f) due to skin effect; G increases linearly with f.
• Coax formulas: L = (μ/2π)ln(b/a), C = 2πε/ln(b/a), G = ωC·tanδ.
• Common GATE trap: Z₀ = sqrt(R/G) is the distortionless condition, NOT the general formula for Z₀.