3.1 Transmission Lines: Coaxial, Twin-Lead, and Characteristic Impedance

3.1 Transmission Lines: Coaxial, Twin-Lead, and Characteristic Impedance

ACMA Exam Focus: Transmission lines (feedlines) are critical links in every amateur radio station. Understanding how physical geometry and dielectric materials dictate characteristic impedance ($Z_0$), velocity factor ($VF$), physical-to-electrical length conversions, and attenuation at higher frequencies is essential for the ACMA Standard Theory examination.

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1. Purpose of Transmission Lines

A transmission line (or feedline) is a specialised conductor system designed to transport radio frequency (RF) electromagnetic energy from a transmitter to an antenna, or from an antenna to a receiver, with maximum efficiency and minimal power loss.

An ideal transmission line accomplishes two fundamental goals:

1. Zero Radiation Loss: It guides RF energy without radiating electromagnetic fields into space along its length.
2. Zero Signal Pickup: It prevents ambient electromagnetic noise (man-made interference, impulse noise, neighbouring transmissions) from coupling into the receiver.

In practical amateur radio installations, transmission lines fall into two primary structural categories: unbalanced coaxial cables and balanced parallel-conductor lines (twin-lead, ladder line, and open-wire line).

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2. Coaxial Cable Construction

Coaxial cable (coax) is the most widely used feedline in modern radio stations. Its name derives from the coaxial geometry: both inner and outer conductors share the exact same central longitudinal axis.

  +-------------------------------------------------------------+
  | Outer Protective Jacket (PVC / PE)                         |
  |  +-------------------------------------------------------+  |
  |  | Outer Metallic Braid / Shield (Grounded Return)      |  |
  |  |  +-------------------------------------------------+  |  |
  |  |  | Solid or Foam Dielectric Insulator (er)         |  |  |
  |  |  |  +-------------------------------------------+  |  |  |
  |  |  |  | Centre Conductor (Solid / Stranded Copper) |  |  |  |
  |  |  |  +-------------------------------------------+  |  |  |
  |  |  +-------------------------------------------------+  |  |
  |  +-------------------------------------------------------+  |
  +-------------------------------------------------------------+

A standard coaxial line consists of four concentric layers:

1. Inner (Centre) Conductor: Solid or stranded high-conductivity copper (or copper-clad steel). It carries the primary RF signal voltage and current.
2. Dielectric Insulator: A flexible insulating sheath surrounding the inner conductor. Common materials include Solid Polyethylene ($\varepsilon_r \approx 2.25$), Foam Polyethylene ($\varepsilon_r \approx 1.3 - 1.5$), and PTFE / Air-Spaced Polyethylene ($\varepsilon_r \approx 1.1 - 1.5$). The dielectric maintains exact concentric spacing between conductors.
3. Outer Shield (Braid / Foil): Woven tinned-copper braid, aluminium foil, or solid corrugated copper (hardline). The outer shield acts as the return path for RF current and forms an electromagnetic shield. Because the outer shield is held at ground potential, coax is an unbalanced line.
4. Outer Protective Jacket: Tough, weather-resistant Polyvinyl Chloride (PVC) or Polyethylene (PE) jacket protecting the cable against moisture ingress, ultraviolet (UV) degradation, and mechanical abrasion.

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3. Parallel-Conductor Lines: Twin-Lead and Open-Wire Line

Parallel-conductor lines consist of two identical parallel wires separated by a uniform insulating gap along their entire length.

• Twin-Lead Ribbon Line: Two stranded copper wires embedded within a continuous flat ribbon of solid polyethylene insulation.
• Open-Wire / Ladder Line: Two solid or stranded wires held at a fixed spacing by plastic or ceramic insulating spacers (spreaders) positioned at regular intervals. Air acts as the primary dielectric between conductors.

Balanced vs. Unbalanced Operation

Parallel-conductor lines are balanced transmission lines. Equal and opposite RF currents flow in the two conductors relative to ground. The electromagnetic fields produced by the equal and opposite currents cancel each other completely in space, preventing line radiation provided the conductor spacing is small relative to the operating wavelength ($S \ll \lambda$).

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4. Characteristic Impedance ($Z_0$)

Definition

The characteristic impedance ($Z_0$) of a uniform transmission line is the ratio of RF voltage to RF current of a single wave propagating along an infinitely long loss-free line:

$Z_0 = \frac{V_{\text{forward}}}{I_{\text{forward}}} = \sqrt{\frac{L}{C}}$

where $L$ is inductance per unit length (henries/metre) and $C$ is capacitance per unit length (farads/metre).

CRITICAL ACMA CONCEPT: Characteristic impedance is determined exclusively by the physical dimensions (conductor diameter and spacing) and the relative permittivity ($\varepsilon_r$) of the dielectric material. It is completely independent of physical cable length. A 1-metre length of RG-58 coaxial cable has an identical characteristic impedance ($50,\Omega$) to a 100-metre roll of RG-58.

Mathematical Formulas for $Z_0$

1. Coaxial Cable

For a coaxial line with inner conductor diameter $d$, outer shield inner diameter $D$, and dielectric relative permittivity $\varepsilon_r$:

$Z_0 = \frac{138}{\sqrt{\varepsilon_r}} \log_{10}\left(\frac{D}{d}\right)$

2. Parallel-Conductor Line (Twin-Lead / Open-Wire)

For two parallel conductors of diameter $d$ separated by centre-to-centre distance $S$ in a dielectric of relative permittivity $\varepsilon_r$:

$Z_0 = \frac{276}{\sqrt{\varepsilon_r}} \log_{10}\left(\frac{2S}{d}\right)$

From these formulas, we observe:

• Increasing conductor spacing ($S$ or $D$) increases $Z_0$.
• Increasing conductor diameter ($d$) decreases $Z_0$.
• Increasing dielectric constant ($\varepsilon_r$) decreases $Z_0$.

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5. Standard Transmission Line Impedances

In amateur and commercial radio engineering, specific standard impedances are used:

• $50,\Omega$ Coax (RG-58, RG-213, LMR-400): Standard for amateur HF/VHF/UHF transceivers. $50,\Omega$ is an optimum engineering compromise between maximum power-handling capability (which occurs at $\sim 30,\Omega$ for air-dielectric coax) and minimum attenuation loss (which occurs at $\sim 77,\Omega$).
• $75,\Omega$ Coax (RG-59, RG-6): Standard for television distribution and matching half-wave dipole antennas in free space (which have a natural resonant feedpoint impedance of $\approx 73,\Omega$).
• $300,\Omega$ Twin-Lead: Standard ribbon line used for folded dipoles and FM receiver antennas.
• $450,\Omega$ Ladder Line / $600,\Omega$ Open-Wire: Extremely low-loss balanced feedlines used with antenna tuning units (ATUs) for multi-band doublet antennas.

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6. Velocity Factor ($VF$) and Physical vs. Electrical Length

Velocity Factor Definition

Radio waves travel in a vacuum at the speed of light ($c \approx 3 \times 10^8\text{ m/s}$). However, when an electromagnetic wave travels through a dielectric material inside a transmission line, it slows down.

The Velocity Factor ($VF$) is the ratio of the speed of an electromagnetic wave in the transmission line ($v$) to the speed of light in a vacuum ($c$):

$VF = \frac{v}{c} = \frac{1}{\sqrt{\varepsilon_r}}$

Dielectric Velocity Factors

Physical vs. Electrical Length Calculation

Because the wave travels slower in the cable, the wavelength in the transmission line ($\lambda_{\text{line}}$) is shorter than the wavelength in free space ($\lambda_{\text{free}}$):

$\lambda_{\text{line}} = \lambda_{\text{free}} \times VF$

To calculate the physical length ($L_{\text{phys}}$) required for a specific electrical fraction of a wavelength (such as a quarter-wave matching stub, $\frac{1}{4}\lambda$):

$L_{\text{free}} = \frac{300}{f_{\text{MHz}}} \text{ metres}$

$L_{\text{phys}} = L_{\text{elec}} \times VF = \left( \frac{300}{f_{\text{MHz}}} \times \text{fraction} \right) \times VF \text{ metres}$

Worked Example

Problem: Calculate the physical length in metres of a quarter-wavelength ($\frac{1}{4}\lambda$) matching stub cut for $14.1\text{ MHz}$ using solid polyethylene coaxial cable ($VF = 0.66$).

Solution:

1. Calculate free-space wavelength: $\lambda_{\text{free}} = \frac{300}{14.1} = 21.277\text{ metres}$.
2. Calculate electrical quarter-wavelength: $L_{\text{elec}} = \frac{21.277}{4} = 5.319\text{ metres}$.
3. Multiply by Velocity Factor: $L_{\text{phys}} = 5.319 \times 0.66 = \mathbf{3.51\text{ metres}}$.

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7. Dielectric Loss and Frequency Attenuation

Transmission lines suffer signal loss (attenuation), expressed in decibels per 100 metres ($\text{dB}/100\text{ m}$). Feedline attenuation increases significantly as frequency increases due to two primary physical mechanisms:

1. Skin Effect (Conductor Loss): At direct current (DC), electric current flows uniformly across the entire cross-sectional area of a wire. As frequency increases, electromagnetic induction forces alternating RF current to flow exclusively in a paper-thin layer on the outer surface of the inner conductor and inner surface of the shield. This drastically reduces the effective conductive cross-section, causing AC resistance to rise proportional to \sqrt{f}.
2. Dielectric Hysteresis Loss: Alternating electric fields cause rapid polarisation cycles of molecules within the dielectric insulator, generating heat. Dielectric loss increases linearly with operating frequency ($f$).

Attenuation Comparison Across Bands ($\text{dB}/100\text{ m}$)

Practical Takeaway: Thin RG-58 is acceptable for short HF patch leads, but unusable for long VHF/UHF feedline runs due to excessive attenuation.