Coaxial transmission lines are essential for transmitting high-frequency signals with minimal interference. These cables consist of concentric conductors separated by a dielectric, creating a controlled impedance environment for signal propagation.

The unique structure of coaxial cables supports TEM wave propagation, confining electromagnetic fields within the cable. This design minimizes radiation losses and interference, making coaxial lines ideal for various RF and microwave applications.

Coaxial cable structure

• Coaxial cables are widely used in RF and microwave applications for transmitting signals with minimal interference and loss
• They consist of concentric conductors separated by a dielectric insulation layer, providing a controlled impedance environment for signal propagation

Inner and outer conductors

• The inner conductor is a solid or stranded wire that carries the signal
• Surrounded by the outer conductor, which serves as a shield and return path for the signal current
• Conductor materials are typically copper or aluminum, chosen for their high electrical conductivity and mechanical strength
• Dimensions of the conductors determine the cable's characteristic impedance and frequency response (50 Ω, 75 Ω)

Dielectric insulation layer

• Separates the inner and outer conductors, providing electrical insulation and mechanical support
• Common dielectric materials include polyethylene (PE), polytetrafluoroethylene (PTFE), and foam PE
• Dielectric constant and loss tangent of the insulation material affect the cable's velocity of propagation and attenuation
• Dielectric thickness is designed to maintain the desired characteristic impedance (1.5 mm for 50 Ω cable)

Braided shield and jacket

• The braided shield is a woven mesh of conductive wires that surrounds the outer conductor
• Provides additional shielding against external electromagnetic interference (EMI) and improves mechanical strength
• The outer jacket is a protective layer made of PVC or other durable materials
• Protects the cable from physical damage, moisture, and UV exposure
• Color-coded jackets are used to identify different cable types and applications (black for RG-58, white for RG-6)

Electromagnetic fields in coaxial lines

• Coaxial cables support transverse electromagnetic (TEM) wave propagation, with electric and magnetic fields perpendicular to each other and the direction of propagation
• The unique geometry of coaxial lines confines the electromagnetic fields within the cable, minimizing radiation losses and interference

TEM mode propagation

• In TEM mode, the electric and magnetic fields are entirely transverse to the direction of propagation
• This mode has no cutoff frequency, allowing coaxial cables to operate from DC to high frequencies (up to 100 GHz)
• TEM mode is the dominant mode of propagation in coaxial lines, ensuring low dispersion and consistent signal integrity

Radial electric field

• The electric field in a coaxial cable is radially oriented between the inner and outer conductors
• Field intensity is inversely proportional to the radial distance from the center conductor ($E \propto 1/r$)
• The radial electric field is responsible for the voltage difference between the conductors and the cable's capacitance per unit length

Circumferential magnetic field

• The magnetic field in a coaxial cable is circumferentially oriented around the center conductor
• Field intensity is inversely proportional to the radial distance from the center conductor ($H \propto 1/r$)
• The circumferential magnetic field is associated with the current flowing through the conductors and the cable's inductance per unit length
• The orthogonal orientation of electric and magnetic fields in TEM mode ensures efficient energy transfer and low crosstalk between adjacent cables

Characteristic impedance

• The characteristic impedance ($Z_0$) is a fundamental property of a coaxial cable that determines its voltage-to-current ratio during signal propagation
• It is a function of the cable's geometry and dielectric properties, and is designed to match the impedance of connected devices to minimize reflections

Impedance formula derivation

• The characteristic impedance of a coaxial cable is given by: $Z_0 = \frac{1}{2\pi} \sqrt{\frac{\mu}{\epsilon}} \ln\left(\frac{b}{a}\right)$
• Where $\mu$ is the permeability of the dielectric, $\epsilon$ is the permittivity of the dielectric, $b$ is the inner radius of the outer conductor, and $a$ is the outer radius of the inner conductor
• This formula is derived from the cable's capacitance and inductance per unit length, which are determined by the conductor dimensions and dielectric properties

Dependence on conductor dimensions

• The characteristic impedance is primarily determined by the ratio of the outer conductor radius to the inner conductor radius ($b/a$)
• Increasing the $b/a$ ratio results in a higher characteristic impedance, while decreasing it leads to a lower impedance
• The dielectric constant of the insulation material also affects the impedance, with higher dielectric constants resulting in lower impedance values

Typical impedance values

• Common characteristic impedance values for coaxial cables are 50 Ω, 75 Ω, and 93 Ω
• 50 Ω cables (RG-58, RG-174) are widely used in RF and microwave applications, test equipment, and antenna feedlines
• 75 Ω cables (RG-6, RG-11) are used in video and cable television (CATV) systems for signal distribution
• 93 Ω cables (RG-62) are used in specialized applications, such as high-speed digital data transmission and timing reference distribution

Attenuation in coaxial lines

• Attenuation is the loss of signal power as it propagates through a coaxial cable, expressed in decibels per unit length (dB/m or dB/ft)
• It is caused by conductor and dielectric losses, as well as skin effect and proximity effect at high frequencies

Conductor losses

• Conductor losses are due to the resistance of the inner and outer conductors, which dissipate power as heat
• The resistance is a function of the conductor material, cross-sectional area, and frequency (due to skin effect)
• Conductor losses increase with frequency and cable length, and are the dominant loss mechanism at lower frequencies (below 1 GHz)

Dielectric losses

• Dielectric losses are caused by the dissipation of energy in the insulation material between the conductors
• The loss is determined by the dielectric constant and loss tangent of the insulation material
• Dielectric losses increase with frequency and are the primary loss mechanism at higher frequencies (above 1 GHz)
• Low-loss dielectric materials (PTFE, foam PE) are used to minimize dielectric losses in high-frequency applications

Skin effect and proximity effect

• Skin effect is the tendency of high-frequency currents to flow near the surface of a conductor, reducing the effective cross-sectional area and increasing resistance
• The skin depth decreases with increasing frequency, leading to higher conductor losses at high frequencies
• Proximity effect is the influence of nearby conductors on the current distribution, further increasing high-frequency losses in tightly packed cables
• The use of stranded conductors and specialized cable designs (tinned copper, silver-plated conductors) can help mitigate skin effect and proximity effect losses

Velocity of propagation

• The velocity of propagation ($v_p$) is the speed at which signals travel through a coaxial cable, typically expressed as a fraction of the speed of light in vacuum ($c$)
• It is determined by the dielectric constant of the insulation material and affects the cable's electrical length and phase response

Velocity factor

• The velocity factor ($VF$) is the ratio of the signal velocity in the cable to the speed of light in vacuum: $VF = \frac{v_p}{c} = \frac{1}{\sqrt{\epsilon_r}}$
• Where $\epsilon_r$ is the relative dielectric constant of the insulation material
• Typical velocity factors range from 0.66 (for solid PE) to 0.88 (for foam PE), with lower values indicating slower signal propagation

Relation to dielectric constant

• The velocity of propagation is inversely proportional to the square root of the dielectric constant
• Higher dielectric constants result in slower signal propagation and shorter electrical lengths for a given physical length
• The dielectric constant of the insulation material is a key factor in determining the cable's phase response and delay characteristics

Comparison to free space velocity

• In vacuum or air, electromagnetic waves propagate at the speed of light ($c \approx 3 \times 10^8$ m/s)
• In coaxial cables, the velocity of propagation is always lower than the speed of light due to the presence of the dielectric material
• The velocity reduction is necessary to maintain the cable's characteristic impedance and minimize reflections at the connections
• Understanding the velocity of propagation is essential for designing cable assemblies with specific electrical lengths and phase characteristics

Reflections and standing waves

• Reflections occur when a signal encounters an impedance mismatch along the coaxial cable, causing a portion of the signal to be reflected back towards the source
• Standing waves are the result of the interaction between the forward and reflected waves, leading to voltage and current variations along the cable

Impedance mismatches

• Impedance mismatches happen when the characteristic impedance of the cable does not match the impedance of the connected devices (source, load, or other cables)
• Mismatches can occur due to improper terminations, connector issues, or cable damage
• Impedance mismatches cause reflections, which can lead to signal distortion, power loss, and increased VSWR

Reflection coefficient

• The reflection coefficient ($\Gamma$) quantifies the fraction of the incident signal that is reflected at an impedance mismatch: $\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$
• Where $Z_L$ is the load impedance and $Z_0$ is the characteristic impedance of the cable
• The reflection coefficient ranges from -1 (short circuit) to +1 (open circuit), with 0 indicating a perfect match
• The magnitude of the reflection coefficient determines the severity of the mismatch and the amount of reflected power

Voltage standing wave ratio (VSWR)

• VSWR is a measure of the impedance mismatch and the resulting standing wave pattern on a coaxial cable: $VSWR = \frac{1 + |\Gamma|}{1 - |\Gamma|}$
• It is the ratio of the maximum to minimum voltage (or current) along the cable
• A VSWR of 1:1 indicates a perfect match, while higher values indicate more severe mismatches and greater reflected power
• High VSWR can lead to increased cable losses, signal distortion, and potential damage to connected devices
• Minimizing VSWR is essential for optimal signal transmission and system performance in RF and microwave applications

Power handling capacity

• The power handling capacity of a coaxial cable determines the maximum amount of power that can be safely transmitted without causing damage or excessive heating
• It is affected by factors such as cable size, dielectric material, frequency, and environmental conditions

Average and peak power limits

• The average power limit is the maximum continuous power that a cable can handle without overheating or degradation
• It is determined by the cable's cross-sectional area, dielectric properties, and thermal dissipation characteristics
• The peak power limit is the maximum instantaneous power that a cable can withstand without dielectric breakdown or arcing
• Peak power limits are typically much higher than average power limits and are important for pulsed or modulated signals

Factors affecting power handling

• Frequency: Higher frequencies result in greater losses and reduced power handling capacity due to skin effect and dielectric losses
• Ambient temperature: Higher ambient temperatures decrease the cable's ability to dissipate heat, reducing its power handling capacity
• Altitude: Lower air pressure at high altitudes reduces the dielectric strength of air, increasing the risk of arcing and lowering power handling limits
• Cable length: Longer cables have greater total losses and may require lower input power to avoid overheating or signal degradation

High-power coaxial cable designs

• High-power coaxial cables are designed with larger conductor sizes, thicker dielectric layers, and improved heat dissipation features
• Copper-clad aluminum (CCA) conductors are used to reduce weight while maintaining high conductivity and power handling capacity
• Corrugated outer conductors improve flexibility and power handling by increasing the surface area for heat dissipation
• Dielectric materials with high thermal conductivity (boron nitride, aluminum oxide) are used to enhance heat transfer and increase power handling limits
• Specialized connectors and terminations are employed to minimize reflections and ensure reliable operation under high-power conditions

Coaxial cable types and applications

• There are various types of coaxial cables designed for different frequency ranges, power levels, and environmental conditions
• The choice of cable depends on the specific application requirements, such as signal integrity, attenuation, flexibility, and cost

Rigid and flexible coaxial cables

• Rigid coaxial cables have solid outer conductors and are used in high-power, low-loss applications (broadcast transmitters, antenna feeders)
• They offer excellent shielding and power handling but are inflexible and require specialized installation techniques
• Flexible coaxial cables have braided or foil outer conductors and are more versatile and easier to install
• They are used in a wide range of applications, from low-frequency audio to high-frequency microwave signals

Common coaxial cable standards

• RG-6: 75 Ω cable used in CATV and satellite TV installations for signal distribution
• RG-58: 50 Ω cable used in RF and low-power microwave applications, test equipment, and short antenna feedlines
• RG-174: Miniature 50 Ω cable used in portable devices, patch cables, and low-power RF applications
• RG-213: High-power 50 Ω cable used in amateur radio, military, and commercial RF applications
• LMR-400: Low-loss 50 Ω cable used in wireless communications, base station antennas, and long cable runs

High-frequency and broadband applications

• Coaxial cables are widely used in high-frequency and broadband applications due to their consistent impedance and low signal distortion
• Examples include:
  • Microwave communications systems
  • Radar and satellite links
  • Cable television networks
  • Broadband internet distribution
  • Test and measurement equipment
• Specialized coaxial cables (low-loss, phase-stable) are designed for high-frequency applications to minimize attenuation and ensure signal integrity

Connectors and terminations

• Coaxial connectors and terminations are used to connect cables to devices, adapt between different cable types, and ensure proper impedance matching
• The choice of connector depends on the frequency range, power level, and environmental requirements of the application

Coaxial connector types

• BNC (Bayonet Neill-Concelman): 50 Ω connectors used in RF and low-power microwave applications, test equipment, and video systems
• TNC (Threaded Neill-Concelman): Threaded version of BNC connectors, offering better mechanical stability and weatherproofing
• Type N: Medium-power 50 Ω or 75 Ω connectors used in microwave applications, base station antennas, and test equipment
• SMA (SubMiniature version A): Precision 50 Ω connectors used in high-frequency microwave applications, up to 18 GHz
• 7/16 DIN: High-power 50 Ω connectors used in wireless base stations, antenna systems, and high-power RF applications

Impedance matching terminations

• Terminations are used to provide a matched load at the end of a coaxial cable, preventing reflections and ensuring maximum power transfer
• Common termination types include:
  • Resistive terminations: Provide a fixed resistance (50 Ω or 75 Ω) to match the cable impedance
  • Short-circuit and open-circuit terminations: Used for calibration and reference purposes in test and measurement applications
  • Reactive terminations: Provide frequency-dependent impedance matching, such as in broadband applications or filter networks

Connector installation and maintenance

• Proper connector installation is critical for maintaining signal integrity, minimizing reflections, and ensuring reliable operation
• Key steps in connector installation include:
  • Cable preparation: Stripping the jacket, dielectric, and conductors to the correct dimensions
  • Connector assembly: Inserting the cable into the connector and crimping or soldering the contacts
  • Weatherproofing: Applying heat-shrink tubing or sealants to protect the connection from moisture and corrosion
• Regular maintenance, such as cleaning connector interfaces and checking for damage, is essential for optimal performance and longevity
• Specialized tools (crimp tools, cable strippers) and techniques (soldering, torque wrenches) are used to ensure consistent and reliable connector installation

Measurement techniques

• Various measurement techniques are used to characterize the performance of coaxial cables, detect faults, and ensure compliance with specifications
• These techniques involve specialized equipment and provide valuable insights into the cable's electrical properties and signal transmission characteristics

Time-domain reflectometry (TDR)

• TDR is a technique used to locate and characterize impedance discontinuities along a coaxial cable
• It works by sending a short electrical pulse down the cable and measuring the reflections caused by impedance mismatches
• TDR provides information on the location, type (capacitive, inductive), and severity of the discontinuities
• It is used for fault detection, cable length measurement, and impedance profile analysis

Vector network analyzer (VNA) measurements

• VNAs are used to measure the complex impedance, insertion loss, and phase response of coaxial cables and components
• They work by measuring the amplitude and phase of the transmitted and reflected signals over a range of frequencies
• VNA measurements provide