Line Characteristics

Primary and Secondary Line Parameters

A metallic transmission line is described by four primary parameters per unit length: resistance $R$, inductance $L$, capacitance $C$, and conductance $G$. Their values depend on conductor geometry, insulation, temperature, and frequency.

The primary parameters determine the secondary parameters, characteristic impedance and propagation constant. For a uniform line, $Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}}$ and $\gamma = \sqrt{(R + j\omega L)(G + j\omega C)}$. The units below refer to one kilometre of cable unless stated otherwise. ETSI TS 102 080 gives capacitance values of 25 to 120 $\text{nF}/\text{km}$ at 800 Hz for representative local-loop cable sections.

Resistance (R) and Conductor Loss

Resistance (R) is measured in $\Omega/\text{km}$ and represents the opposition to current flow. It converts electrical energy into heat, which contributes to attenuation.

Factors Affecting Resistance

• Material: Conductors, such as copper, have lower resistivity ($\rho$) than, for example, aluminum.
• Geometry: Resistance is directly proportional to the length $(L)$ and resistivity, and inversely proportional to the cross-sectional area ($A$) of the conductor. For a round wire, the area grows with the square of its diameter.
  For conductor length $L$ in metres, the DC resistance is $R = \frac{\rho L}{A}$.
  For a 1 km loop formed by two equal conductors, $R_{\text{loop},\text{km}} = 2 \cdot \frac{\rho \cdot 1000}{A}$. The factor 2 includes both conductors.
• Temperature: Resistance in metallic conductors increases with temperature.
• Frequency (AC Effects):
  • Skin Effect: At higher frequencies, current tends to flow closer to the surface of the conductor, effectively reducing the cross-sectional area and increasing resistance.
  • Proximity Effect: In multi-wire cables, the magnetic fields of adjacent wires further disrupt the current distribution, increasing resistance.

Resistance Asymmetry

In an ideal symmetric pair, both conductors have the same resistance. A useful magnitude measure is $\frac{|R_1 - R_2|}{R_1 + R_2} \cdot 100\%$. Lower asymmetry improves common-mode rejection because unequal series resistance converts part of a common-mode voltage into a differential voltage.

Inductance (L) and Magnetic Energy

Inductance (L) is measured in $\text{H}/\text{km}$, commonly in $\text{mH}/\text{km}$ or $\mu\text{H}/\text{km}$. It describes energy storage in the magnetic field and the voltage induced by a changing current.

Components and Factors Affecting Inductance

• Internal Inductance $(L_i)$: It arises from the magnetic field inside the conductor material. Skin Effect causes $L_i$ to decrease with frequency.
• External Inductance $(L_e)$: It arises from the magnetic field around the conductors. It is usually the dominant component and depends mainly on line geometry.
  For a two-conductor pair, $L_e$ depends on conductor spacing $(D)$ and diameter $(d)$. Greater spacing generally increases the external inductance.
• Magnetic Permeability: The ability of a material to support the creation of a magnetic field. For non-magnetic materials, such as copper, the relative permeability $\mu_r \approx 1$.

Internal inductance decreases as frequency increases because of skin effect. At high frequency the external component dominates. Cable spacing and dielectric geometry then determine most of the pair inductance.

Capacitance (C) and Electric Energy

Capacitance (C) is measured in $\text{F}/\text{km}$, commonly in $\text{nF}/\text{km}$. It describes electric-field energy storage between conductors and between each conductor and its surroundings.

Components and Factors Affecting Capacitance

• Direct Capacitance $(C_{ab})$: The capacitance directly between the two active wires of a pair.
• Capacitance to Ground $(C_{ag}, C_{bg})$: The capacitance between each conductor and ground or a shield. Multi-pair cables also have capacitance between separate pairs.
  The effective pair capacitance depends on all these electric-field paths and on the cable geometry.
• Dielectric Material: The insulating material between conductors affects capacitance. Its relative permittivity ($\epsilon_r$) is a key factor.
• Geometry: Conductor spacing, diameter, twist, and dielectric permittivity determine capacitance.
  For a coaxial line, $C' \approx \frac{2 \pi \epsilon_0 \epsilon_r}{\ln(D/d)}$ per unit length, where $D$ is the inner diameter of the outer conductor and $d$ is the inner conductor diameter.

Capacitance Asymmetry

Capacitance asymmetry is the difference between the capacitance to ground of the two conductors. It converts part of a common-mode disturbance into a differential voltage and therefore reduces immunity to external interference.

Conductance (G) and Dielectric Loss

Conductance (G) is measured in $\text{S}/\text{km}$, commonly in $\text{mS}/\text{km}$. It models leakage through the dielectric and the frequency-dependent dielectric loss.

Components and Frequency Dependence

• DC Leakage $(G_{DC})$: This component is due to the finite (non-ideal) resistance of the insulating material. For high-quality insulators, $G_{DC}$ is typically very small and often negligible.
• AC Dielectric Losses $(G_{AC})$: This component grows with frequency because polarization in the dielectric dissipates energy as heat. $G_{AC}$ is proportional to frequency, capacitance, and the dielectric loss tangent ($\tan \delta$):
  $G_{AC} = \omega C \tan \delta$
  where $\omega$ is the angular frequency ($2\pi f$).

The total conductance combines DC leakage with dielectric loss. Its contribution to attenuation grows with frequency and depends on the insulation material.