Introduction to Line Characteristics

Understanding how signals behave as they travel through a conductor is crucial for designing any wire-based telecommunication system. Transmission lines, such as copper cables, are characterized by a set of continuous electrical properties distributed along their length. These properties, often called primary parameters (resistance, inductance, capacitance, and conductance), define the effect of an infinitely small segment of a line on a signal.

These parameters determine a line's ability to conduct current, store energy in electric and magnetic fields, and dissipate energy.

Resistance (R) - Opposing Current Flow

Resistance (R), measured in $\Omega/\text{km}$, describes the electrical resistance that a conductor opposes to the flow of current. This resistance converts electrical energy into heat (Joule's effect), causing signal loss.

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 ($A$) (diameter) of the conductor.
  DC resistance formula: $R = \frac{\rho L}{A}$.
  For a loop: $R_{loop,km} = 2 \cdot \frac{\rho \cdot 1000}{A}$ (total resistance of 1 km of both wires).
• 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 ideal symmetric pairs, both wires should have identical resistance. However, small manufacturing differences create asymmetry, which is the ratio of the difference to the sum of the resistances (e.g., $\frac{R_1 - R_2}{R_1 + R_2} \cdot 100\%$). Low asymmetry is crucial for rejecting common-mode noise, as resistance asymmetry can transform common-mode noise into unwanted differential signals, degrading performance.

Inductance (L) - Storing Magnetic Energy

Inductance (L), measured in $\text{H}/\text{km}$, represents a conductor's ability to store energy in a magnetic field. When current flows, it generates a magnetic field, and a changing magnetic field induces a voltage.

Components and Factors Affecting:

• 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 typically the dominant component and depends mainly on the line geometry.
  For a two-conductor pair, $L_e$ is dependent on the distance between the wires $(D)$ and their diameter $(d)$, increasing with greater separation.
• 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 $\mu_r \approx 1$.

Inductance generally decreases slightly with frequency due to skin effect, becoming relatively constant at very high frequencies because internal inductance becomes negligible. Cables with widely spaced wires (e.g., overhead lines) have higher inductance, while tightly packed wires (e.g., twisted pairs) have lower inductance.

Capacitance (C) - Storing Electric Energy

Capacitance (C), measured in $\text{F}/\text{km}$ (or often $\text{nF}/\text{km}$), describes a transmission line's ability to store energy in an electric field between its conductors. The wires act like plates of a capacitor, separated by the insulating dielectric material.

Components and Factors Affecting:

• 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 wire and a common ground reference or shielding. In multi-pair cables, there's also capacitance between different pairs.
  The effective capacitance of a symmetric pair is often calculated using specialized methods like the Gauss method, accounting for all these components.
• Dielectric Material: The insulating material between conductors significantly impacts capacitance. Its relative permittivity ($\epsilon_r$) is a key factor.
• Geometry: The distance between wires ($D$) and their diameter ($d$), along with the type of twisting (e.g., star quad, pair twist), influence capacitance.
  Approximate formula for a coaxial line: $C \approx \frac{2 \pi \epsilon_0 \epsilon_r}{\ln(D/d)}$ (per unit length).

Capacitance Asymmetry

Capacitance asymmetry refers to the difference in capacitance-to-ground between the two wires of a pair. High asymmetry makes the cable more susceptible to external electromagnetic interference, as common-mode interference can be converted into unwanted differential signals, degrading performance. It is a critical parameter often with strict requirements to minimize interference susceptibility.

Conductance (G) - Dielectric Leakage and Losses

Conductance (G), measured in $\text{S}/\text{km}$ (Siemens per kilometer), represents the electrical leakage through the dielectric material that insulates the conductors. This is effectively energy loss within the dielectric itself.

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 becomes dominant at higher frequencies. It results from the polarization and reorientation of dielectric molecules in a rapidly changing electric field, dissipating 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 ($2\pi f$).

While often negligible at lower frequencies, conductance becomes an important factor for overall signal loss and channel performance in high-frequency applications where dielectric losses can be substantial.