3.3 Coaxial transmission lines

Coaxial cable structure

Coaxial transmission lines carry high-frequency signals with minimal interference by confining electromagnetic fields between two concentric conductors. Their controlled impedance environment makes them a workhorse in RF, microwave, and broadband systems. Understanding their structure, field behavior, and performance parameters is central to transmission-line analysis in electromagnetism.

Inner and outer conductors

The inner conductor (solid or stranded wire) carries the signal. The outer conductor surrounds it coaxially, serving as both the return current path and an electromagnetic shield. Both are typically copper or aluminum, chosen for high conductivity.

The ratio of the outer conductor's inner radius to the inner conductor's outer radius directly sets the cable's characteristic impedance and frequency response. Standard designs target 50 Ω or 75 Ω depending on the application.

Dielectric insulation layer

The dielectric separates the two conductors, providing electrical insulation and mechanical support. Common materials include polyethylene (PE), polytetrafluoroethylene (PTFE), and foamed PE.

Two dielectric properties matter most:

• Dielectric constant ($\epsilon_r$): controls the velocity of propagation and characteristic impedance
• Loss tangent ($\tan \delta$): determines how much energy the dielectric absorbs, contributing to attenuation

The dielectric thickness is engineered to maintain the target impedance for a given conductor geometry.

Braided shield and jacket

Many coaxial cables add a braided shield (woven conductive mesh) around the outer conductor for extra EMI rejection and mechanical durability. The outermost layer is a protective jacket, usually PVC, that guards against physical damage, moisture, and UV exposure. Jacket color often identifies the cable type (e.g., black for RG-58, white for RG-6).

Electromagnetic fields in coaxial lines

The coaxial geometry confines all electromagnetic energy between the conductors, which is why coaxial lines radiate so little compared to open wire pairs.

TEM mode propagation

Coaxial lines support the transverse electromagnetic (TEM) mode, where both $\vec{E}$ and $\vec{H}$ are entirely perpendicular to the propagation direction. The TEM mode has no cutoff frequency, so coaxial cables operate from DC up to frequencies where higher-order modes (TE/TM) begin to propagate. That upper limit depends on the cable dimensions; for typical cables it can reach tens of GHz.

Because TEM is non-dispersive (phase velocity is independent of frequency), coaxial lines maintain consistent signal integrity across a broad bandwidth.

Radial electric field

The electric field points radially from the inner conductor to the outer conductor. Its magnitude falls off as:

$E(r) = \frac{V}{r \ln(b/a)}$

where $r$ is the radial distance, $a$ is the inner conductor radius, and $b$ is the outer conductor radius. This $1/r$ dependence means the field is strongest near the inner conductor. The radial electric field establishes the voltage between conductors and determines the cable's capacitance per unit length.

Circumferential magnetic field

The magnetic field wraps circumferentially around the inner conductor, also falling off as $1/r$:

$H(r) = \frac{I}{2\pi r}$

This field is generated by the current flowing through the conductors and determines the cable's inductance per unit length. The orthogonality of $\vec{E}$ (radial) and $\vec{H}$ (circumferential) is the hallmark of TEM propagation and ensures efficient energy transfer with minimal crosstalk.

Characteristic impedance

The characteristic impedance $Z_0$ is the ratio of voltage to current for a wave propagating along the line. When the load impedance equals $Z_0$, there are no reflections, and maximum power transfers to the load.

Impedance formula derivation

Starting from the per-unit-length capacitance and inductance of the coaxial geometry, the characteristic impedance works out to:

$Z_0 = \frac{1}{2\pi} \sqrt{\frac{\mu}{\epsilon}} \ln\left(\frac{b}{a}\right)$

where:

• $a$ = outer radius of the inner conductor
• $b$ = inner radius of the outer conductor
• $\mu$ = permeability of the dielectric (usually $\mu_0$)
• $\epsilon$ = permittivity of the dielectric ($\epsilon_0 \epsilon_r$)

For a non-magnetic dielectric ($\mu = \mu_0$), this simplifies to:

$Z_0 = \frac{60}{\sqrt{\epsilon_r}} \ln\left(\frac{b}{a}\right) \quad [\Omega]$

Dependence on conductor dimensions

$Z_0$ depends on the logarithm of the ratio $b/a$, not on the absolute sizes. Increasing $b/a$ raises the impedance; decreasing it lowers the impedance. A higher dielectric constant $\epsilon_r$ also lowers $Z_0$, since $\epsilon$ appears in the denominator under the square root.

Typical impedance values

Attenuation in coaxial lines

Attenuation quantifies signal power loss per unit length, expressed in dB/m (or dB/ft). Two mechanisms dominate, and their relative importance shifts with frequency.

Conductor losses

Current flowing through the finite-conductivity inner and outer conductors dissipates power as heat. Because of the skin effect (see below), the effective conductor resistance rises with frequency. Conductor loss is the dominant attenuation mechanism at lower frequencies (roughly below 1 GHz for typical cables).

Dielectric losses

The dielectric material absorbs a fraction of the electromagnetic energy each cycle. This fraction is characterized by the loss tangent $\tan \delta$. Dielectric loss scales linearly with frequency, so it becomes the dominant loss mechanism at higher frequencies (above ~1 GHz). Low-loss materials like PTFE and foamed PE are chosen specifically to push this crossover higher.

Skin effect and proximity effect

At high frequencies, current crowds toward the conductor surfaces. The skin depth is:

$\delta_s = \sqrt{\frac{2}{\omega \mu \sigma}}$

where $\omega$ is the angular frequency and $\sigma$ is the conductor's conductivity. As frequency rises, $\delta_s$ shrinks, the effective cross-section carrying current decreases, and resistance increases.

Proximity effect further distorts the current distribution when conductors are close together, adding to high-frequency losses. Silver-plated or tinned conductors and stranded designs help mitigate both effects by improving surface conductivity and current distribution.

Velocity of propagation

Signals in a coaxial cable travel slower than light in vacuum because the dielectric slows the wave.

Velocity factor

The velocity factor relates the cable's propagation speed to $c$:

$VF = \frac{v_p}{c} = \frac{1}{\sqrt{\epsilon_r}}$

Typical values:

• Solid PE ($\epsilon_r \approx 2.25$): $VF \approx 0.66$
• Foam PE ($\epsilon_r \approx 1.3$): $VF \approx 0.88$
• PTFE ($\epsilon_r \approx 2.1$): $VF \approx 0.69$

Relation to dielectric constant

Since $v_p = c / \sqrt{\epsilon_r}$, a higher dielectric constant means slower propagation. For a given physical cable length, a slower velocity factor means a longer electrical length (more wavelengths fit in the same cable). This matters when you need cables cut to specific electrical lengths for phasing or delay matching.

Comparison to free-space velocity

In vacuum, $v_p = c \approx 3 \times 10^8$ m/s. Inside any dielectric-filled coaxial cable, $v_p < c$. This velocity reduction is not a design flaw; it's an inherent consequence of energy storage in the dielectric. Accurate knowledge of $v_p$ is essential when designing cable assemblies with precise electrical lengths or phase characteristics.

Reflections and standing waves

Whenever a propagating wave encounters a change in impedance, part of its energy reflects back toward the source. The interference between forward and reflected waves creates standing wave patterns along the line.

Impedance mismatches

A mismatch occurs when the load impedance $Z_L$ differs from the cable's characteristic impedance $Z_0$. Common causes include improper terminations, damaged connectors, or transitions between cables of different impedance. Even small mismatches degrade signal quality and waste power.

Reflection coefficient

The reflection coefficient $\Gamma$ quantifies how much of the incident wave reflects:

$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$

• $\Gamma = 0$: perfect match (no reflection)
• $\Gamma = +1$: open circuit (total reflection, in phase)
• $\Gamma = -1$: short circuit (total reflection, inverted)

The fraction of reflected power is $|\Gamma|^2$, so even a modest $|\Gamma| = 0.3$ means about 9% of the power bounces back.

Voltage standing wave ratio (VSWR)

VSWR converts the reflection coefficient into the ratio of maximum to minimum voltage along the line:

$VSWR = \frac{1 + |\Gamma|}{1 - |\Gamma|}$

• VSWR = 1:1 is a perfect match.
• VSWR = 2:1 corresponds to $|\Gamma| = 1/3$, meaning about 11% reflected power.
• High VSWR increases cable losses, distorts signals, and can damage transmitters or amplifiers that aren't designed to handle reflected power.

In practice, most RF systems aim for VSWR below 1.5:1 at the operating frequency.

Power handling capacity

Every coaxial cable has limits on how much power it can carry before the dielectric breaks down or the conductors overheat.

Average and peak power limits

• Average power limit: the maximum continuous power the cable can handle without overheating. Set by thermal dissipation: the cable must radiate or conduct away the heat generated by attenuation losses.
• Peak power limit: the maximum instantaneous power before dielectric breakdown or arcing occurs. This is relevant for pulsed signals (radar, pulsed RF) where peak power can far exceed the average.

Peak limits are typically orders of magnitude higher than average limits.

Factors affecting power handling

• Frequency: higher frequencies mean higher attenuation, so more heat is generated per unit length, reducing the average power limit.
• Ambient temperature: a hotter environment leaves less thermal headroom for heat dissipation.
• Altitude: lower air pressure at altitude reduces the dielectric strength of any air gaps, lowering peak power limits.
• Cable length: longer runs accumulate more total loss and heat; the hottest point is at the input end where power is highest.

High-power coaxial cable designs

For high-power applications (broadcast transmitters, particle accelerators, high-power radar), cables use:

• Larger conductor cross-sections to reduce resistive loss
• Corrugated outer conductors for flexibility and increased surface area for heat dissipation
• High-thermal-conductivity dielectrics (e.g., ceramic-loaded materials)
• Copper-clad aluminum (CCA) conductors to reduce weight while maintaining conductivity
• Specialized connectors rated for the power and thermal environment

Coaxial cable types and applications

Rigid and flexible coaxial cables

Rigid coaxial cables have solid tubular outer conductors. They offer the lowest loss and highest power handling but cannot be bent after manufacture. You'll find them in broadcast transmitter installations and fixed antenna feeds.

Flexible coaxial cables use braided or corrugated outer conductors. They're easier to route and install, making them the standard choice for most lab, field, and commercial applications.

Common coaxial cable standards

High-frequency and broadband applications

Coaxial cables serve a wide range of systems that demand consistent impedance and low distortion:

• Microwave communications and radar links
• Satellite ground station feeds
• Cable television and broadband internet distribution
• RF test and measurement setups

For demanding high-frequency work, specialized cables with phase stability over temperature and ultra-low loss (e.g., semi-rigid cables, conformable cables) are available.

Connectors and terminations

The connector is often the weakest link in a coaxial system. A poorly chosen or installed connector introduces reflections, loss, and potential failure points.

Coaxial connector types

Impedance matching terminations

Terminations provide a known load at the end of a cable:

• Resistive terminations (50 Ω or 75 Ω loads): absorb all incident power, preventing reflections. Used at unused ports and during testing.
• Short-circuit and open-circuit terminations: produce total reflection with known phase. Used as calibration standards for VNA measurements.
• Reactive terminations: provide frequency-dependent matching for broadband systems or filter networks.

Connector installation and maintenance

Proper installation is critical. The basic steps:

1. Cable preparation: strip the jacket, dielectric, and braid to the dimensions specified for the connector type.
2. Connector assembly: insert the prepared cable and crimp, solder, or clamp the contacts per the manufacturer's procedure.
3. Weatherproofing (for outdoor use): apply heat-shrink tubing or self-amalgamating tape to seal against moisture.
4. Verification: check continuity, measure return loss or VSWR to confirm a good connection.

Use the correct tools (calibrated crimp tools, torque wrenches for precision connectors) and inspect connector interfaces regularly for contamination or mechanical damage.

Measurement techniques

Two primary instruments characterize coaxial cable performance: the time-domain reflectometer and the vector network analyzer.

Time-domain reflectometry (TDR)

TDR sends a fast-rise-time pulse (or step) down the cable and records reflections as a function of time. Since you know the cable's velocity factor, you can convert time to distance.

What TDR reveals:

• Location of impedance discontinuities (connectors, splices, damage)
• Type of discontinuity: a capacitive fault shows a downward dip then recovery; an inductive fault shows an upward bump
• Cable length: the round-trip time to the open or shorted end gives the total length
• Impedance profile: the reflected amplitude at each point maps out $Z$ along the line

TDR is the go-to tool for field troubleshooting because it pinpoints where a problem is, not just that a problem exists.

Vector network analyzer (VNA) measurements

A VNA measures the complex (magnitude and phase) scattering parameters (S-parameters) of a cable or component over a swept frequency range.

Key measurements include:

• Return loss ($S_{11}$): quantifies impedance match at a port. Higher return loss (in dB) means less reflection.
• Insertion loss ($S_{21}$): measures total attenuation through the cable as a function of frequency.
• Phase response: reveals group delay and phase linearity, critical for phase-matched cable assemblies.

VNA measurements require careful calibration (typically using open, short, and load standards) to remove the effects of test cables and adapters from the data. Together, TDR and VNA give a complete picture of a coaxial line's performance in both time and frequency domains.